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36 | Through this post I am going to explain How Linear Regression works? Let us start with what is regression and how it works? Regression is widely used for prediction and forecasting in field of machine learning. Focus of regression is on the relationship between dependent and one or more independent variables. The “dependent variable” represents the output or effect, or is tested to see if it is the effect. The “independent variables” represent the inputs or causes, or are tested to see if they are the cause. Regression analysis helps to understand how the value of the dependent variable changes when any one of the independent variables is varied, while the other independent variables are kept unchanged. In the regression, dependent variable is estimated as function of independent variables which is called regression function. Regression model involves following variables.
- Independent variables X.
- Dependent variable Y
- Unknown parameter θ
In the regression model Y is function of (X,θ). There are many techniques for regression analysis, but here we will consider linear regression.
In the Linear regression, dependent variable(Y) is the linear combination of the independent variables(X). Here regression function is known as hypothesis which is defined as below.
hθ(X) = f(X,θ)
Suppose we have only one independent variable(x), then our hypothesis is defined as below.
The goal is to find some values of θ(known as coefficients), so we can minimize the difference between real and predicted values of dependent variable(y). If we take the values of all θ are zeros, then our predicted value will be zero. Cost function is used as measurement factor of linear regression model and it calculates average squared error for m observations. Cost function is denoted by J(θ) and defined as below.
As we can see from the above formula, if cost is large then, predicted value is far from the real value and if cost is small then, predicted value is nearer to real value. Therefor, we have to minimize cost to meet more accurate prediction.
Linear regression in R
R is language and environment for statistical computing. R has powerful and comprehensive features for fitting regression models. We will discuss about how linear regression works in R. In R, basic function for fitting linear model is lm(). The format is
fit <- lm(formula, data)
where formula describes model(in our case linear model) and data describes which data are used to fit model. The resulting object(fit in this case) is a list that contains information about the fitted model. The formula typically written as
Y ~ x1 + x2 + … + xk
where ~ separates the dependent variable(y) on the left from independent variables(x1, x2, ….. , xk) from right, and the independent variables are separated by + signs. let’s see simple regression example(example is from book R in action). We have the dataset women which contains height and weight for a set of 15 women ages 30 to 39. we want to predict weight from height. R code to fit this model is as below.
>fit <-lm(weight ~ height, data=women) >summary(fit)
Output of the summary function gives information about the object fit. Output is as below
Call: lm(formula = weight ~ height, data = women) Residuals: Min 1Q Median 3Q Max -1.7333 -1.1333 -0.3833 0.7417 3.1167 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -87.51667 5.93694 -14.74 1.71e-09 *** height 3.45000 0.09114 37.85 1.09e-14 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 1.525 on 13 degrees of freedom Multiple R-squared: 0.991, Adjusted R-squared: 0.9903 F-statistic: 1433 on 1 and 13 DF, p-value: 1.091e-14
Let’s understand the output. Values of coefficients(θs) are -87.51667 and 3.45000, hence prediction equation for model is as below
Weight = -87.52 + 3.45*height
In the output, residual standard error is cost which is 1.525. Now, we will look at real values of weight of 15 women first and then will look at predicted values. Actual values of weight of 15 women are as below
Output 115 117 120 123 126 129 132 135 139 142 146 150 154 159 164
Predicted values of 15 women are as below
Output 1 2 3 4 5 6 7 8 9 112.5833 116.0333 119.4833 122.9333 126.3833 129.8333 133.2833 136.7333 140.1833 10 11 12 13 14 15 143.6333 147.0833 150.5333 153.9833 157.4333 160.8833
We can see that predicted values are nearer to the actual values.Finally, we understand what is regression, how it works and regression in R.
Here, I want to beware you from the misunderstanding about correlation and causation. In the regression, dependent variable is correlated with the independent variable. This means, as the value of the independent variable changes, value of the dependent variable also changes. But, this does not mean that independent variable cause to change the value of dependent variable. Causation implies correlation , but reverse is not true. For example, smoking causes the lung cancer and smoking is correlated with alcoholism. Many discussions are there on this topic. if we go deep into than one blog is not enough to explain this.But, we will keep in mind that we will consider correlation between dependent variable and independent variable in regression.
In the next blog, I will discuss about the real world business problem and how to use regression into it.
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Would you like to understand the value of predictive analysis when applied on web analytics data to help improve your understanding relationship between different variables? So register now for our Upcoming Webinar: How to perform predictive analysis on your web analytics tool data. Get More Info & Book Your Seat Now! | http://www.tatvic.com/blog/linear-regression-using-r/ | 13 |
45 | Pre-Algebra, Algebra 1, and Algebra 2 all require you to master new math skills. Do you find solving equations and word problems difficult in Algebra class? Are the exponents, proportions, and variables of Algebra keeping you up at night? Intercepts, functions, and expressions can be confusing to most Algebra students, but a qualified tutor can clear it all up! Our Algebra tutors are experts in math and specialize in helping students like you understand Algebra. If you are worried about an upcoming Algebra test or fear not passing your Algebra class for the term, getting an Algebra tutor will make all the difference.
Pre-algebra - The goal of Pre-algebra is to develop fluency with rational numbers and proportional relationships. Students will: extend their elementary skills and begin to learn algebra concepts that serve as a transition into formal Algebra and Geometry; learn to think flexibly about relationships among fractions, decimals, and percents; learn to recognize and generate equivalent expressions and solve single-variable equations and inequalities; investigate and explore mathematical ideas and develop multiple strategies for analyzing complex situations; analyze situations verbally, numerically, graphically, and symbolically; and apply mathematical skills and make meaningful connections to life's experiences.
Algebra I - The main goal of Algebra is to develop fluency in working with linear equations. Students will: extend their experiences with tables, graphs, and equations and solve linear equations and inequalities and systems of linear equations and inequalities; extend their knowledge of the number system to include irrational numbers; generate equivalent expressions and use formulas; simplify polynomials and begin to study quadratic relationships; and use technology and models to investigate and explore mathematical ideas and relationships and develop multiple strategies for analyzing complex situations.
Algebra II - A primary goal of Algebra II is for students to conceptualize, analyze, and identify relationships among functions. Students will: develop proficiency in analyzing and solving quadratic functions using complex numbers; investigate and make conjectures about absolute value, radical, exponential, logarithmic and sine and cosine functions algebraically, numerically, and graphically, with and without technology; extend their algebraic skills to compute with rational expressions and rational exponents; work with and build an understanding of complex numbers and systems of equations and inequalities; analyze statistical data and apply concepts of probability using permutations and combinations; and use technology such as graphing calculators.
College Algebra – Topics for this course include basic concepts of algebra; linear, quadratic, rational, radical, logarithmic, exponential, and absolute value equations; equations reducible to quadratic form; linear, polynomial, rational, and absolute value inequalities, and complex number system; graphs of linear, polynomial, exponential, logarithmic, rational, and absolute value functions; conic sections; inverse functions; operations and compositions of functions; systems of equations; sequences and series; and the binomial theorem.
No matter the level of the algebra course that the student is taking, we have expert tutors available and ready to help. All of our algebra tutors have a degree in mathematics, science, or a related field (like accounting). We are so confident in our algebra tutors that you can meet with them for free. Just ask your tutoring coordinator about our Meet and Greet program.
Our Tutoring Service
We offer our clients choice when searching for a tutor, and we work with you all the way through the selection process. When you choose to work with one of our tutors, expect quality, professionalism, and experience. We will never offer you a tutor that is not qualified in the specific subject area you request. We will provide you with the degrees, credentials, and certifications each selected tutor holds so that you have the same confidence in them that we do. And for your peace of mind, we conduct a nation-wide criminal background check, sexual predator check and social security verification on every single tutor we offer you. We will find you the right tutor so that you can find success! | http://www.advancedlearners.com/albuquerque/algebra/tutor/find.aspx | 13 |
14 | Not Just Black and White
Not Just Black and White is a science project that teaches kids about color and light. Different colors will appear when you and your kids view spinning black-and-white circles.
What You'll Need:
- White paper
- Black paper
- Black marker
- Knitting needle
- Paper plate
Learn About Not Just Black and White:Step 1: Draw and cut out 3 circles of white paper that are each 5-1/2 inches in diameter. Put a small hole in the center of each circle.
Step 2: Draw and cut out a circle of black paper that is 5-1/2 inches in diameter. Cut the black circle in half. Cut 1 of the halves in half.
Step 3: Use these materials to make several different disks. Glue a black half-circle onto a white circle so that the disk is 1/2 black and 1/2 white. Glue a black quarter-circle onto a white circle so that the disk is 1/4 black and 3/4 white.
Step 4: Using a black marker, divide 1 white disk into 8 pie-wedge shapes. Color some of the pie wedges black, leaving others white.
Step 5: Wrap some tape around the middle of a knitting needle. Put the knitting needle through the middle of a 6-inch paper plate, and push the plate down to rest on the tape.
Step 6: Spin the plate. Be sure it spins smoothly and doesn't wobble. Use this as your spinner. Poke the knitting needle through the hole in the center of 1 disk, and let the disk rest on the paper plate.
Step 7: Spin the plate, and look at the disk as it spins. What colors do you see? Do you see different colors when the disk is spinning quickly or slowly? Spin the other disks to see what colors they produce.
Colors at a Distance is a science project that teaches kids about visual perception. Learn about Colors at a Distance on the next page of science projects for kids: spectrum of colors. | http://tlc.howstuffworks.com/family/science-projects-for-kids-spectrum-of-colors1.htm | 13 |
11 | 3.8 Related Rates
Two variables, perhaps x and y, are both functions of a third variable, time, t, and x and y are related by an equation.
Example A fire has started in a dry, open field and spreads in the form of a circle. The radius of the circle increases at a rate of 6 ft/min. Find the rate at which the fire area is increasing when the radius is 150 ft.
Strategy: Draw and label picture. What are we finding? Name variables and equations involved. (Substitute) and differentiate, then "plug-in" values.
Implicitly differentiate with respect to t. Note: The area and radius are both functions of t.
Given: ft/min and r = 150
Example: A ladder 26 ft long leans against a vertical wall. The foot of the ladder is drawn away from the wall at a rate of 4 ft/s. How fast is the top of the ladder sliding down the wall, when the foot of the ladder is 10 ft from the wall?
Strategy: Draw and label pictures. What are we finding? Name variables and equations involved. (Substitute) and differentiate, then "plug-in" values.
Differentiate implicitly with respect to the variable t. Note: x and y are both functions of t.
Given: Must find y using
Example: Water runs into a conical tank shown at a constant rate of 2 ft3 per minute. The dimensions of the tank are altitude of 12ft and base radius of 6 ft. How fast is the water level rising when the water is 6 feet deep?
Draw a picture. Need both the volume of a cone and similar triangle proportions.
Find: Volume of cone = Similar Triangle:
Example: A spherical balloon is inflated with gas at the rate of 100 ft3/min. Assuming the gas pressure remains constant, how fast is the radius of the balloon increasing when the radius is 3 ft?
Find: Volume of Sphere =
Given: r = 3 Know:
Example: A man 6 ft tall walks at the rate of 5 ft/sec. toward a street light that is 16 ft. above the ground. At what rate is the tip of his shadow moving?
Find: Similar Triangles
Assignment 3.8 pg 186; 1-11 odd, 12 [ans:-1/(20π)], 13,15, 16 [ans: 0.6 m/s]19, 20 [ans: ] 21, 27, 29, 30 [ans: -1/8 rad/s], 31 | http://homepages.ius.edu/MEHRINGE/M215/Fall%2007%20Notesr/Section3.8.htm | 13 |
24 | Dec. 8, 2010 Thirteen billion years ago our universe was dark. There were neither stars nor galaxies; there was only hydrogen gas left over after the Big Bang. Eventually that mysterious time came to an end as the first stars ignited and their radiation transformed the nearby gas atoms into ions. This phase of the universe's history is called the Epoch of Reionization (EoR), and it is intimately linked to many fundamental questions in cosmology. But looking back so far in time presents numerous observational challenges.
Arizona State University's Judd Bowman and Alan Rogers of Massachusetts Institute of Technology have developed a small-scale radio astronomy experiment designed to detect a never-before-seen signal from the early universe during this period of time, a development that has the potential to revolutionize the understanding of how the first galaxies formed and evolved.
"Our goal is to detect a signal from the time of the Epoch of Reionization. We want to pin down when the first galaxies formed and then understand what types of stars existed in them and how they affected their environments," says Bowman, an assistant professor at the School of Earth and Space Exploration in ASU's College of Liberal Arts and Sciences.
Bowman and Rogers deployed a custom-built radio spectrometer called EDGES to the Murchison Radio-astronomy Observatory in Western Australia to measure the radio spectrum between 100 and 200 MHz. Though simple in design -- consisting of just an antenna, an amplifier, some calibration circuits, and a computer, all connected to a solar-powered energy source -- its task is highly complex. Instead of looking for early galaxies themselves, the experiment looks for the hydrogen gas that existed between the galaxies. Though an extremely difficult observation to make, it isn't impossible, as Bowman and Rogers have demonstrated in their paper published in Nature on Dec. 9.
"This gas would have emitted a radio line at a wavelength of 21 cm -- stretched to about 2 meters by the time we see it today, which is about the size of a person," explains Bowman. "As galaxies formed, they would have ionized the primordial hydrogen around them and caused the radio line to disappear. Therefore, by constraining when the line was present or not present, we can learn indirectly about the first galaxies and how they evolved in the early universe." Because the amount of stretching, or redshifting, of the 21 cm line increases for earlier times in the Universe's history, the disappearance of the inter-galactic hydrogen gas should produce a step-like feature in the radio spectrum that Bowman and Rogers measured with their experiment.
Radio measurements of the redshifted 21 cm line are anticipated to be an extremely powerful probe of the reionization history, but they are very challenging. The experiment ran for three months, a rather lengthy observation time, but a necessity given the faintness of the signal compared to the other sources of emission from the sky.
"We carefully designed and built this simple instrument and took it out to observe the radio spectrum and we saw all kinds of astronomical emission but it was 10,000 times stronger than the theoretical expectation for the signal we are looking for," explains Bowman. "That didn't surprise us because we knew that going into it, but it means it's very hard to see the signal we want to see."
The low frequency radio sky is dominated by intense emission from our own galaxy that is many times brighter than the cosmological signal. Add to that the interference from television, FM radio, low earth orbit satellites, and other telecommunications radio transmitters (present even in remote areas like Australia's Outback) and it is a real challenge. Filtering out or subtracting these troublesome foreground signals is a principal focus of instrument design and data analysis techniques. Fortunately, many of the strongest foregrounds have spectral properties that make them possible to separate from the expected EoR signals.
After careful analysis of their observations, Bowman and Rogers were able to show that the gas between galaxies could not have been ionized extremely rapidly. This marks the first time that radio observations have directly probed the properties of primordial gas during the EoR and paves the way for future studies. "We're breaking down barriers to open an entirely new window into the early universe," Bowman says.
The next generation of large radio telescopes is under construction right now to attempt much more sophisticated measurements of the 21 cm line from the EoR. Bowman is the project scientist for one of the telescopes called the Murchison Widefield Array. According to him, the most likely physical picture for the EoR looked like a lot of bubbles that started percolating out from galaxies and then grew together -- but that idea needs to be tested. If lots of galaxies all put out a little bit of radiation, then there would be many little bubbles everywhere and those would grow and eventually merge like a really fizzy and frothy foam. On the other hand, if there were just a few big galaxies that each emitted a lot of radiation then there would have been only a few big bubbles that grew together.
"Our goal, eventually, is to make radio maps of the sky showing how and when reionization occurred. Since we can't make those maps yet, we are starting with these simple experiments to begin to constrain the basic properties of the gas and how long it took for galaxies to change it," explains Bowman. "This will improve our understanding of the large-scale evolution of the universe."
Other social bookmarking and sharing tools:
- Judd D. Bowman, Alan E. E. Rogers. A lower limit of Δz > 0.06 for the duration of the reionization epoch. Nature, 2010; 468 (7325): 796 DOI: 10.1038/nature09601
Note: If no author is given, the source is cited instead. | http://www.sciencedaily.com/releases/2010/12/101208132210.htm | 13 |
16 | Solution to Additional Practice
Unit 1: Analyzing Lines on a Graph
In the graph below, the straight line
S is given by the equation y = c + dx. If the line
shifts from this initial position S0 to a new position of
what must have changed in the equation?
- In this graph, the line has changed
in steepness, which means the slope must have changed.
- In the equation y = c + dx, "d"
is the slope of the line. Since the slope must have
changed, the constant "d"
must have changed. Since S1 is steeper than S0 , "d" must
have increased. and "c" is the y-intercept.
- In the equation y = c + dx,
"c" is the y-intercept. In the
graph, the lines have not been extended
to where they intercept the y-axis, so it is hard to tell
if "c" changed or not. Unless you extend
the lines to the y-axis and can be certain the two lines
both intercept the y-axis in the same place, it is hard to tell if "c" changed
or not, but we can be certain that
"d" did change.
- If you do extend both lines through
the y-axis, you will find they have the same y-intercept,
which means "c" does not change.
If you feel comfortable with this material, move on to the
If you still do not understand this practice, you may need more
review than is offered by this book. You may wish to review Book
I of this series before moving on. | http://cstl.syr.edu/fipse/graphb/unit6/SolnT2Full.html | 13 |
17 | ENGLISH AND LANGUAGE ARTS
Sixth Graders review parts of speech and verb tenses and write detailed reports and compositions. Grammar emphasis is on clauses, phrases and the formulation of good sentences and paragraphs. Oral presentations of reports and research are given with an artistic component. Students practice lengthy recitation of epic poems such as “Horatio at the Bridge” or “Hiawatha.” Class plays usually come from Roman or Medieval history. Biographies are assigned for reports, and readers include: The Bronze Bow, King Arthur legends, and Otto of the Silver Hand.
The seventh grade grammar lessons emphasize different styles of writing, use of an outline, paragraph format, self-editing, organization of compositions, note taking and the development of compound and complex sentences. Creative writing is practiced in the Wish, Wonder and Surprise block. For the first time in an English block, the students are graded on quizzes, tests, essays, artwork, class participation, and timeliness. Poetry continues to be spoken daily, and oral reports are given to the class. The class play is usually placed in the Renaissance or late medieval times. Independent reading with regular book reports gives the students an opportunity to explore different literature. Often choices include The Giver, Education of Little Tree, Midwives Apprentice, Wrinkle in Time and Robin Hood.
Eighth Graders learn to edit their writing, summarize written work, and solidify their grammar skills (passive and active verbs, direct and indirect objects, clauses and phrases, pronouns). The spoken work continues with more oral reports including biographies, modern history and geography. Poetry continues to be a lively part of the main lesson. The class play is often Shakespeare or a modern play with rich use of language. Each individual now begins to understand a point of view and the dramatic themes used in acting. Eighth grade continues with some assigned reading, book reports and short stories such as Dragon Wings, The Master Puppeteer, and Johnny Tremain.
MATHEMATICSThe sixth grade Math curriculum is based on an intense review of previously taught material. This review is done in such a way that there is always something new. A continual theme through the year is the sense of number and the interrelationship between division, fractions, decimals, and percents, with fractions playing the central role. Another theme in sixth grade math is developing good work habits. Weekly homework assignments, organization skills, and keeping a good notebook are emphasized. Percents, business math, and algebraic formulas are introduced in sixth grade as well as drawing geometric figures exactly with Euclidean tools: the compass and the straight edge.
The seventh graders’ introduction to algebra (done in one three-week Main Lesson block) is an important milestone in development of the students’ abstract thinking. This serves as a crucial foundation for studying mathematics in high school. Another central theme for the seventh grade year is ratios, through which p and irrational numbers are introduced. The study of geometry continues with the Euclidean constructions that were introduced in sixth grade, and then moves on to theorems and proofs, culminating in the Pythagorean theorem. The year often ends with the students learning how to calculate the square roots of numbers by hand.
Instead of devoting a large portion of the eighth grade year to algebra in order to get the students “ahead,” the bulk of the material found in a traditional Algebra I course is kept for ninth grade, the year that we feel most students are ripe for algebra. Much of our eighth grade year is dedicated to non-traditional topics, such as number bases, in order to develop abstract thinking, and stereometry (the study of three-dimensional solids) and loci (the study of two-dimensional curves such as the conic sections), in order to develop the capacity of “exact” imagination. The traditional topics covered in eighth grade include volumes, proportions, dimensional analysis, percents and exponential growth.
Middle School Science
In the next three grades, the study of science turns to the lawfulness that comes from cause-and-effect relationships in the physical world. The focus now shifts to a threefold approach to the phenomena: observation, evaluation, and conceptualization. There is an emphasis on the hands-on and visual approaches in the middle school, by doing experiments that speak to the kinesthetic learners and drawings on the board that serve the visual.
In sixth grade, the threefold approach is now applied to electricity, magnetism, optics, acoustics, and heat in physics. Geography expands again, spiraling out to include either Europe (paralleling the study of Rome in history) or South America (as an extension of the North American studies in fifth grade). The polarity between the heights and the depths is explored in the complementary studies of Astronomy and Mineralogy.
In seventh grade, a mathematical approach is applied for the first time to physics content in mechanics, acoustics, electricity, heat and optics. In mechanics, for example, fulcrums are studied by first approaching the phenomena with seesaws and weights, and by identifying levers all around them in their homes and lives, then developing a rule or law. The students then use the rule to predict leverage and mechanical advantage for new arrangements. In chemistry, combustion, the lime cycle, and acids and bases form the content. The transformation of a substance through burning is an important highlight in this course. Nutrition, as well as Physiology, is taught in Main Lesson. In Geography, Africa is studied, continuing the expansion outward from the local to the farther extents of the world.
In eighth grade, Geography either focuses on a study of Asia, or of world religions. In physics, students learn how certain concepts are applied to technology or natural systems. The content areas (heat, light, electricity, acoustics, and mechanics) manifest as convection systems, refraction and lenses, the electric motor, musical instruments, and fluid mechanics and hydraulics. Fats, carbohydrates and proteins are studied in chemistry both in terms of what is happening in their own metabolisms and what can be achieved externally, such as by making personal care products (lip balm, soap, lotion, etc.). In biology, the human anatomy is studied, for example the musculoskeletal and nervous systems, to complement and complete the work done in seventh grade. Eight Graders also study Meteorology.
SOCIAL STUDIES AND HISTORY
Sixth grade history often begins with the life and conquests of Alexander the Great. In two three-week blocks, important highlights of life in the Roman Empire are studied, including the rise of the Empire, the emperors, the Republic, conquests, government, building and construction, barbarian incursions and the fall of the empire. Also included, are the life of Jesus of Nazareth and the influence of Christianity on the Empire. The Sixth Grader is left with a strong impression of all we have inherited from ancient Rome.
Later in the year, a three-week block delves into the life of medieval Europe. This includes, but is not limited to feudalism, peasant life, knighthood and the life of the monasteries. The life of Mohammed and the rise of Islam as a counterforce to Christianity are studied. This naturally brings in the Crusades. Parallels to modern life become evident in this block. The geography of Latin America is the focus this year. Each country is handled much like the states in our study of the U.S., but in one three-week block. Each student will write a report on one of the countries in this region.
Some of the books that may be read during this year to further support these studies may include, The Sword and the Circle, by Rosemary Sutcliff, The Bronze Bow, by Elizabeth George Spear, Otto of the Silver Hand, by Howard Pyle, and Secret of the Andes, by Ann Nolan Clark.
In seventh grade the students study European history from the late Middle Ages through the Renaissance. There are usually three, three or four week Main Lesson blocks. Key biographies of either people who were forerunners of the times or individuals who particularly exemplified a character type from that time are studied in depth. In the Late Middle Ages, Marco Polo, Eleanor of Aquitaine, and Joan of Arc are typical biographies. As the curriculum moves towards the Reformation, the role of the Roman Catholic Church is explored with emphasis on the developments that took place within the church that contributed to the turbulence of the times. Martin Luther is typical of a key biography for this time period. Not only are the changes that took place in the religious/political life studied, but also the explorers in science, art, and world travel. Copernicus, Galileo, Columbus, Magellan, de Vinci, and Michelangelo are some of the fascinating biographies that tell the story of the times. The students deeply immerse themselves in the art of the times through their own reproductions of the work of “the renaissance masters.” The geography of Africa and Europe are covered in seventh grade. Typically, students write a report related to some aspect of a particular country. Some of the books related to history that are read in seventh grade include: Robin Hood, Adam of the Road, and Young Joan.
The eighth grade History curriculum spans the time from Elizabethan England through the modern times, with particular emphasis on the founding of America. First, the social, political, and economic climates in Europe set a stage for the mass migration to the American continent. The Revolutionary War, the Declaration of Independence, and the Constitution of the U.S. are studied in depth through biographies, art, literature, and pertinent readings. The settling of America, including the interaction of the settlers with the Native American people, is explored. Biographies of great Americans, such as, Abraham Lincoln lead the students into the Civil War and the Industrial Revolution. Rockefeller and Carnegie are two major biographies juxtaposed to the life of the common factory worker or miner. Through student presentations on the inventions of the 1900’s, the class is introduced to the genius of the modern world. The students are led through history to the two World Wars as well as the Civil Rights Movement and the biography of Martin Luther King Jr. Geography focuses on the Asian continent. Students continue to write reports on a country or on some aspect of world geography related to commerce. A wide variety of readers can be used in eighth grade depending on the focus of the teacher. Some examples related to history and geography may include: Johnny Tremain, Dragonwings, The Master Puppeteer, and My Brother Sam is Dead.
Studying music gives children an inspiring aesthetic experience while it develops focus, discipline, and social skills. Both singing and playing in ensembles strengthens students’ ability to work as individuals within a group. Middle school students become aware of their individual responsibility to the group as they work together to create a meaningful musical experience. They have many opportunities to perform in concerts, assemblies, and festivals throughout the school year.
Sixth grade students continue to develop their musical skills in choir, band and orchestra. They begin to explore how music developed throughout history by studying and performing music of different styles and eras.
Students continue to participate in choir, band and orchestra classes, bringing musical concepts and skill acquisition together in rehearsing and performing. Seventh graders are introduced to music related to the historical and geographical eras they study—such as the Renaissance and Africa.
In their ongoing musical education, eighth grade students benefit from the opportunity to experience more intense and varied emotions through the music they create together. Study and performance of good music of various styles enhances their aesthetic development and helps them begin to develop musical judgment and an understanding of the profound effects music can have on human beings.
WORLD LANGUAGESMWS offers German and Spanish in Grades 1-12. Grades 1-7 have three lessons each week in blocks. At the end of a block, students switch to the alternate language. In Grade 8 students choose between Spanish and German and continue with this selection in the High School. Beginning in Grade 9 each student has four World Language classes per week. The World Language teachers strive to integrate Morning Lesson topics into the World Language lessons in support of our interdisciplinary approach to teaching.
By Grade 6, writing and reading has become a focal point. Beginning elements of grammar are taught. The language teacher uses dialogues, storytelling, verses, songs, tongue twisters, and small plays during instruction. Throughout these years, the students’ vocabulary comprehension increases,and they are able to say simple descriptive sentences, perform dialogues, and retell simple stories. In Grade 6 German, students use Zusammen Lesen by Roswitha Garff. In Spanish, they use an easy reader called Piratas del Caribe.
In Grades 7 and 8 teachers emphasize the languages’ phonetic structure so that students can read and write correctly. Teachers also place emphasis on listening comprehension and oral competence.
EURYTHMYThe sixth grader has changed physically from the well-proportioned fifth grader into the developing adolescent, often with limbs akimbo. The eurythmy curriculum for this grade is designed to meet the physical and emotional changes that accompany this challenging developmental time. One way to work with these changes is to introduce the orderly forms of geometry, with their accompanying laws such as: the five-pointed star, hexagon, square, and figure eight.. The students use a capacity they are just beginning to develop, cause and effect thinking.
Students learn to listen to and identify the major intervals. They then learned to form the eurythmy gestures for these intervals, forming the gestures for the tonic to the octave, where they must reach upwards, out of the narrow confines of themselves.
Some of the eurythmy elements include the vowel and consonant forms, and mirroring. Copper rod exercises continue, including: the seven-fold, waterfall, spiral, spinning, and tossing. Copper rod exercises help improve the students’ posture, as well as enhance their spatial orientation.
The seventh grade eurythmy curriculum is full of the dark and light aspects in movement that reflect the turbulent emotional climate of the developing adolescent. Humor and drama are key elements in expressing this range. Head and foot gestures are learned as a kind of punctuation to enhance the understanding of poetry and music. The work with the copper rods becomes more challenging. Forms learned in years past become more complicated in their execution, e.g., the figure-eight form and the seven-pointed star.
The 8th grade year reviews forms learned in previous years, but now taken up in new ways, with the students beginning to apply their own understanding in the creation of the forms. They learn the deeper meaning of the gestures for the sounds of the alphabet and create their own poetry to move to. They often perform a story set to eurythmy for the younger students.
HANDWORK AND PRACTICAL ARTSFollowing the lower school years, in Middle School students expand on their skills with increasingly sophisticated and complex projects. Sixth grade brings the opportunity to design and hand-sew an animal. Seventh grade progresses to hand-sewn dolls and doll clothing. In the eighth grade, while students are studying the Industrial Age, the Handwork curriculum involves sewing clothes on a treadle sewing machine.
Middle school students are combined weekly for a double period of Practical Arts. During these classes, mixed-age groups of students rotate through a variety of project-based classes. This provides an opportunity for our middle school students to learn and work together, and encourages greater familiarity among the grades. Through performing, fine and practical arts students deepen and transform experience. Every creation bears the stamp of individuality and expresses the student’s response to the world. The student uses imagination, cognition, and skill to bring each artistic or practical task to fruition. Experiencing this process repeatedly builds confidence for setting and implementing goals later in adult life.
Middle School Practical Arts activities include: watercolor painting in both veil and wet-on-wet technique, needle and wet felting, baking/cooking, batik, pastel drawing, charcoal drawing, figure drawing, mosaics, stained glass, folk dancing, basketry, bead work, metal work, printmaking, pottery, gymnastics, geometric string art, clay work, mountain biking, figure drawing and print making, and Outdoor education skills (gardening, earth-based skills and Winter skills). | http://shiningmountainwaldorf.org/our-program/middle-school/main-lesson-and-subject-classes/ | 13 |
28 | Here's a very simple GNU Make function: it takes three arguments and makes a
'date' out of them by inserting / between the first and second and second and third arguments:
make_date = $1/$2/$3
The first thing to notice is that make_date is defined just like any other GNU Make macro (you must use = and not := for reasons we'll see below).
To use make_date we $(call) it like this:
today = $(call make_date,19,12,2007)
That will result in today containing 19/12/2007.
The macro uses special macros $1, $2, and $3. These macros contain the argument specified in the $(call). $1 is the first argument, $2 the second and so on.
There's no maximum number of arguments, but if you go above 10 then you need parens: you can't write $10 instead of $(10). There's also no minimum number. Arguments that are missing are just undefined and will typically be treated as an empty string.
The special argument $0 contains the name of the function. In the example above $0 is make_date.
Since functions are just macros with some special automatic macros filled in (if you use the $(origin) function on any of the argument macros ($1 etc.) you'll find that they are classed as automatic just like $@), you can use GNU Make built in functions to build up complex functions.
Here's a function that turns every / into a \ in a path"
unix_to_dos = $(subst /,\,$1)
using the $(subst). Don't be worried about the use of / and \ there. GNU Make does very
little escaping and a literal \ is most of the time just a \.
Some argument handling gotchas
When GNU Make is processing a $(call) it starts by splitting the argument list on commas to set $1 etc. The arguments are expanded so that $1 etc. are completely expanded before they are ever referenced (it's as if GNU Make used := to set them). This means that if an argument has a side-effect (such as calling $(shell)) then that side-effect will always occur as soon as the $(call) is executed, even if the argument was never actually used by the function.
One common problem is that if an argument contains a comma the splitting of
arguments can go wrong. For example, here's a simple function that swaps its two arguments:
swap = $2 $1
If you do $(call swap,first,argument,second) GNU Make doesn't have any way to know that the first argument was meant to be first,argument and swap ends up returning argument first instead of second first,argument.
There are two ways around this. You could simply hide the first argument inside a macro. Since GNU Make doesn't expand the arguments until after splitting a comma inside a macro will not cause any confusion:
FIRST := first,argument
SWAPPED := $(call swap,$(FIRST),second)
The other way to do this is to create a simple macro that just contains a comma and use that instead:
c := ,
SWAPPED := $(call swap,first$cargument,second)
Or even call that macro , and use it (with parens):
, := ,
SWAPPED := $(call swap,first$(,)argument,second)
Calling built-in functions
It's possible to use the $(call) syntax with built in GNU Make functions. For example, you could call $(warning) like this:
This is useful because it means that you can pass any function name as an argument to a user-defined function and $(call) it without needing to know if it's built-in or not.
This gives you the ability to created functions that act on functions. The classic functional programming map function (which applies a function to every member of a list returning the resulting list) can be created | http://www.agileconnection.com/article/gnu-make-user-defined-functions | 13 |
19 | Least common multiple
In arithmetic and number theory, the least common multiple (also called the lowest common multiple or smallest common multiple) of two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is divisible by both a and b. If either a or b is 0, LCM(a, b) is defined to be zero.
The LCM of more than two integers is also well-defined: it is the smallest integer that is divisible by each of them.
A multiple of a number is the product of that number and an integer. For example, 10 is a multiple of 5 because 5 × 2 = 10, so 10 is divisible by 5 and 2. Because 10 is the smallest positive integer that is divisible by both 5 and 2, it is the least common multiple of 5 and 2. By the same principle, 10 is the least common multiple of −5 and 2 as well.
What is the LCM of 4 and 6?
Multiples of 4 are:
- 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, ...
and the multiples of 6 are:
- 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, ...
Common multiples of 4 and 6 are simply the numbers that are in both lists:
- 12, 24, 36, 48, 60, 72, ....
So, from this list of the first few common multiples of the numbers 4 and 6, their least common multiple is 12.
When adding, subtracting, or comparing vulgar fractions, it is useful to find the least common multiple of the denominators, often called the lowest common denominator, because each of the fractions can be expressed as a fraction with this denominator. For instance,
where the denominator 42 was used because it is the least common multiple of 21 and 6.
Computing the least common multiple
Reduction by the greatest common divisor
Many school age children are taught the term greatest common factor (GCF) instead of the greatest common divisor(GCD); therefore, for those familiar with the concept of GCF, substitute GCF when GCD is used below.
The following formula reduces the problem of computing the least common multiple to the problem of computing the greatest common divisor (GCD):
This formula is also valid when exactly one of a and b is 0, since gcd(a, 0) = |a|.
Because gcd(a, b) is a divisor of both a and b, it's more efficient to compute the LCM by dividing before multiplying:
This reduces the size of one input for both the division and the multiplication, and reduces the required storage needed for intermediate results (overflow in the a×b computation). Because gcd(a, b) is a divisor of both a and b, the division is guaranteed to yield an integer, so the intermediate result can be stored in an integer. Done this way, the previous example becomes:
Finding least common multiples by prime factorization
The unique factorization theorem says that every positive integer greater than 1 can be written in only one way as a product of prime numbers. The prime numbers can be considered as the atomic elements which, when combined together, make up a composite number.
Here we have the composite number 90 made up of one atom of the prime number 2, two atoms of the prime number 3 and one atom of the prime number 5.
This knowledge can be used to find the LCM of a set of numbers.
Example: Find the value of lcm(8,9,21).
First, factor out each number and express it as a product of prime number powers.
The lcm will be the product of multiplying the highest power of each prime number together. The highest power of the three prime numbers 2, 3, and 7 is 23, 32, and 71, respectively. Thus,
This method is not as efficient as reducing to the greatest common divisor, since there is no known general efficient algorithm for integer factorization, but is useful for illustrating concepts.
This method can be illustrated using a Venn diagram as follows. Find the prime factorization of each of the two numbers. Put the prime factors into a Venn diagram with one circle for each of the two numbers, and all factors they share in common in the intersection. To find the LCM, just multiply all of the prime numbers in the diagram.
Here is an example:
- 48 = 2 × 2 × 2 × 2 × 3,
- 180 = 2 × 2 × 3 × 3 × 5,
and what they share in common is two "2"s and a "3":
- Least common multiple = 2 × 2 × 2 × 2 × 3 × 3 × 5 = 720
- Greatest common divisor = 2 × 2 × 3 = 12
This also works for the greatest common divisor (GCD), except that instead of multiplying all of the numbers in the Venn diagram, one multiplies only the prime factors that are in the intersection. Thus the GCD of 48 and 180 is 2 × 2 × 3 = 12.
A simple algorithm
This method works as easily for finding the LCM of several integers.
Let there be a finite sequence of positive integers X = (x1, x2, ..., xn), n > 1. The algorithm proceeds in steps as follows: on each step m it examines and updates the sequence X(m) = (x1(m), x2(m), ..., xn(m)), X(1) = X. The purpose of the examination is to pick up the least (perhaps, one of many) element of the sequence X(m). Assuming xk0(m) is the selected element, the sequence X(m+1) is defined as
- xk(m+1) = xk(m), k ≠ k0
- xk0(m+1) = xk0(m) + xk0.
In other words, the least element is increased by the corresponding x whereas the rest of the elements pass from X(m) to X(m+1) unchanged.
The algorithm stops when all elements in sequence X(m) are equal. Their common value L is exactly LCM(X). (For a proof and an interactive simulation see reference below, Algorithm for Computing the LCM.)
A method using a table
This method works for any number of factors. One begins by listing all of the numbers vertically in a table (in this example 4, 7, 12, 21, and 42):
The process begins by dividing all of the factors by 2. If any of them divides evenly, write 2 at the top of the table and the result of division by 2 of each factor in the space to the right of each factor and below the 2. If a number does not divide evenly, just rewrite the number again. If 2 does not divide evenly into any of the numbers, try 3.
Now, check if 2 divides again:
Once 2 no longer divides, divide by 3. If 3 no longer divides, try 5 and 7. Keep going until all of the numbers have been reduced to 1.
Now, multiply the numbers on the top and you have the LCM. In this case, it is 2 × 2 × 3 × 7 = 84. You will get to the LCM the quickest if you use prime numbers and start from the lowest prime, 2.
Fundamental theorem of arithmetic
where the exponents n2, n3, ... are non-negative integers; for example, 84 = 22 31 50 71 110 130 ...
Given two integers and their least common multiple and greatest common divisor are given by the formulas
In fact, any rational number can be written uniquely as the product of primes if negative exponents are allowed. When this is done, the above formulas remain valid. Using the same examples as above:
The positive integers may be partially ordered by divisibility: if a divides b (i.e. if b is an integer multiple of a) write a ≤ b (or equivalently, b ≥ a). (Forget the usual magnitude-based definition of ≤ in this section - it isn't used.)
Under this ordering, the positive integers become a lattice with meet given by the gcd and join given by the lcm. The proof is straightforward, if a bit tedious; it amounts to checking that lcm and gcd satisfy the axioms for meet and join. Putting the lcm and gcd into this more general context establishes a duality between them:
- If a formula involving integer variables, gcd, lcm, ≤ and ≥ is true, then the formula obtained by switching gcd with lcm and switching ≥ with ≤ is also true. (Remember ≤ is defined as divides).
The following pairs of dual formulas are special cases of general lattice-theoretic identities.
This identity is self-dual:
Let D be the product of ω(D) distinct prime numbers (i.e. D is squarefree).
where the absolute bars || denote the cardinality of a set.
The LCM in commutative rings
The least common multiple can be defined generally over commutative rings as follows: Let a and b be elements of a commutative ring R. A common multiple of a and b is an element m of R such that both a and b divide m (i.e. there exist elements x and y of R such that ax = m and by = m). A least common multiple of a and b is a common multiple that is minimal in the sense that for any other common multiple n of a and b, m divides n.
In general, two elements in a commutative ring can have no least common multiple or more than one. However, any two least common multiples of the same pair of elements are associates. In a unique factorization domain, any two elements have a least common multiple. In a principal ideal domain, the least common multiple of a and b can be characterised as a generator of the intersection of the ideals generated by a and b (the intersection of a collection of ideals is always an ideal). In principal ideal domains, one can even talk about the least common multiple of arbitrary collections of elements: it is a generator of the intersection of the ideals generated by the elements of the collection.
See also
- Crandall, Richard; Pomerance, Carl (2001), Prime Numbers: A Computational Perspective, New York: Springer, ISBN 0-387-94777-9
- Hardy, G. H.; Wright, E. M. (1979), An Introduction to the Theory of Numbers (Fifth edition), Oxford: Oxford University Press, ISBN 978-0-19-853171-5
- Landau, Edmund (1966), Elementary Number Theory, New York: Chelsea
- Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77-171950
- Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 77-81766 | http://en.wikipedia.org/wiki/Least_common_multiple | 13 |
13 | 2. Sample Prolog Programs
In this chapter we provide several sample Prolog programs. The programs
are given in a progression from fairly simple programs to more complex
programs. The key goals of the presentation are to show several important
methods of knowledge representation in Prolog and the declarative programming
methodology of Prolog.
2.1 Map colorings
This section uses a famous mathematical problem -- that of coloring planar
maps -- to motivate logical representations of facts and rules in Prolog.
The prolog program developed provides a representation for adjacent regions
in a map, and shows a way to represent colorings, and a definition of when
the colorings in conflict; that is, when two adjacent regions have the
same coloring. The section introduces the concept of a semantic program
clause tree -- to motivate the issue of semantics for logic-based programming.
2.2 Two factorial definitions
This section introduces the student to computations of mathematical functions
using Prolog. Various built-in arithmetic operators are discussed. Also
discussed is the concept of a Prolog derivation tree, and how derivation
trees are related to tracings of Prolog.
2.3 Towers of Hanoi puzzle
This famous puzzle is formulated in Prolog. The discussion concerns both
the declarative and the procedural meanings of the program. The program
write puzzle solutions to the screen.
2.4 Loading programs, editing programs
Examples show various ways to load programs into Prolog, and an example
of a program calling a system editor is given. The reader is encouraged
to read sections 3.1 an 3.2 on How Prolog Works before continuing with
2.5 Negation as failure
The section gives an introduction to Prolog's negation-as-failure feature,
with some simple examples. Further examples show some of the difficulties
that can be encountered for programs with negation as failure.
2.6 Tree data and relations
This section shows Prolog operator definitions for a simple tree structure.
Tree processing relations are defined and corresponding goals are studied.
2.7 Prolog lists
This section contains some of the most useful Prolog list accessing and
processing relations. Prolog's primary dynamic structure is the list, and
this structure will be used repeatedly in later sections.
2.8 Change for a dollar
A simple change maker program is studied. The important observation here
is how a Prolog predicate like 'member' can be used to generate choices,
the choices are checked to see whether they solve the problem, and then
backtracking on 'member' generates additional choices. This fundamental
generate and test strategy is very natural in Prolog.
2.9 Map coloring redux
We take another look at the map coloring problem introduced in Section
2.1. This time, the data representing region adjacency is stored in a list,
colors are supplied in a list, and the program generates colorings which
are then checked for correctness.
2.10 Simple I/O
This section discusses opening and closing files, reading and writing of
2.11 Chess queens challenge puzzle.
This familiar puzzle is formulate in Prolog using a permutation generation
program from Section 2.7. Backtracking on permutations produces all solutions.
2.12 Set of answers
Prolog's 'setof' and 'bagof' predicates are presented. An implementation
of 'bagof' using 'assert' and 'retract' is given.
2.13 Truth table maker
This section designs a recursive evaluator for infix Boolean expressions,
and a program which prints a truth table for a Boolean expression. The
variables are extracted from the expression and the truth assignments are
2.14 DFA parser
A generic DFA parser is designed. Particular DFAs are represented as Prolog
2.15 Graph structures and paths
This section designs a path generator for graphs represented using a static
Prolog representation. This section serves as an introduction to and motivation
for the next section, where dynamic search grows the search graph as it
The previous section discussed path generation in a static graph. This
section develops a general Prolog framework for graph searching, where
the search graph is constructed as the search proceeds. This can be the
basis for some of the more sophisticated graph searching techniques in
2.17 Animal identification game
This is a toy program for animal identification that has appeared in several
references in some form or another. We take the opportunity to give a unique
formulation using Prolog clauses as the rule database. The implementation
of verification of askable goals (questions) is especially clean. This
example is a good motivation for expert systems, which are studied in Chapter
2.18 Clauses as data
This section develops a Prolog program analysis tool. The program analyses
a Prolog program to determine which procedures (predicates) use, or call,
which other procedures in the program. The program to be analyzed is loaded
dynamically and its clauses are processed as first-class data.
2.19 Actions and plans
An interesting prototype for action specifications and plan generation
is presented, using the toy blocks world. This important subject is continued
and expanded in Chapter 7.
Prolog Tutorial Contents | http://www.csupomona.edu/~jrfisher/www/prolog_tutorial/2.html | 13 |
10 | talk about the structure of DNA and RNA
Warming up the brain: Nucleic acids are made up of nucleotides, consisting of
bases (purines and pyrimidines), as you probably recall from
your genetics or cell biology class, sugars (ribose or deoxyribose),
and a phosphate backbone.
Remember that we have some rules, called "Watson-Crick" base pairing, by
which adenylate nucleotides can hydrogen bond to thymidylate nucleotides (or
uridylate in RNA), while guanylate nucleotides hydrogen bond to cytidylate nucleotides.
A pairs with T (or U)
Is this all starting to come back to you now? Let's find out.
about the bases
Stop me if you've heard this one...
A guy walks into a bar and says "My name's Chargaff, and 22% of my DNA is "A"
nucleotides. I'll bet anyone that they can't guess what percentage of my DNA is "C"
nucleotides!" You say "I'm thirsty, so I'll take that bet!" and then
Yes, it can be done! Erwin explains, we have double stranded DNA genomes, so if 22%
is "A", then there must also be 22% "T", because every "A"
base will be paired with a "T" base. You with me? So 22%+22%=44% is the
percentage of the DNA that is either "A" or "T". That implies
that the percentage that is "G" or "C" must be whatever is left,
or 100%-44%=56%. Every "G" must be base paired with a "C" and
every "C" must be base paired with a "G", so exactly half of
that 56% must be "C" bases. That is, 28% are "C" bases and 28%
are "G" bases.
Here's a photo gallery (Click for larger images):
red = oxygen, blue
= nitrogen, white = hydrogen, gray = carbon.
What atom does the amber color represent?
The nucleotide bases make up the core
of the double helix, as you can see in the picture below.
This snapshot comes from a site you'll probably want to investigate, an "Interactive
Animated Nonlinear Tutorial"
by Eric Martz, from the Department of Microbiology at the University of Massachusetts-Amherst.
Here's another good site to visit, to learn about the Chime plug-in,
and to study the overall structure of DNA. Chime is pronounced with a hard "K"
sound as in "kind", not a "Ch" sound as in "chair."
You can develop a real "feel" for molecules if you familiarize yourself
with the shareware RasMol (RasMac) program. With this program, you can
inspect crystallographic structures downloaded from Brookhaven National Labs, turning
the molecules on the screen so you can see them from every side and angle. Downloading
instructions are available on
the Web, as are instructions for finding molecules to play with.
If you have the Chime plug-in working, you may be able to see the following two examples,
generated by GLACTONE (http://chemistry.gsu.edu/glactone/). You may also download them directly and use
An AT base pair
hydrogen bonding come to pass?
Well, suppose this is a cherry, and you're going to make chocolate cupcakes with
cherries on top. You make the cake mix, fill the little cupcake holders and bake
the cupcakes. Then you put a cherry on top of each, and whip up a batch of chocolate
icing. Here is one, ready to cover with frosting!
Here's one that was covered well, in fact it was so evenly covered with frosting
that you can no longer see the cherry!
Then, an interesting thing happens. On some of the cupcakes, the chocolate icing
is very thin. It dribbles down onto the cake, leaving the cherry somewhat visible
through the frosting.
It is almost as if the cupcake and the cherry are fighting for the frosting, and
the cupcake is winning!
In fact, sometimes the frosting gets so thin, that there's nothing left to hold the
cherry in place, so it pops out, leaving the frosting still stuck to the cake.
Hmmm... What does this make us think of? Why polar covalent bonds, of course!
You see, some atoms are more electronegative
than others. Oxygen is more electronegative than hydrogen, so in an -OH group, the
oxygen takes more than its fair share of electrons. That's just like the cupcake
taking more than its fair share of frosting. The electrons get very thinly distributed
over the hydrogen and get more thickly distributed over the oxygen.
That gives a partial negative charge to the oxygen and a partial positive charge
to the hydrogen. Why? Because the electron is charged, and if more of it is distributed
in one place, that place will get a bit of charge.
Nitrogen can play the same trick, because it is also more electronegative than hydrogen.
On the other hand, carbon and hydrogen are about the same in electronegativity, so
they share the electrons pretty fairly. There will not be a partial charge on the
carbon, because the electrons are distributed evenly in the bond. The carbon-hydrogen
bond reminds us of the well-frosted cake - all neutrally distributed:
On the other hand, the oxygen-hydrogen and nitrogen-hydrogen bonds remind us of the
thinly-frosted cake, and the thin frosting leads to a "dipole moment",
or partial charge:
difference between DNA and RNA?
DNA contains the sugar deoxyribose while RNA is made with the sugar ribose. It's
just a matter of a single 2' hydroxyl, which deoxyribose doesn't have, and ribose
does have. Of course, you all remember that RNA uses the base uracil instead of thymine
Cytosine naturally has a high rate of deamination to give uracil
Cytosine deamination (i.e. water attacks!)
Uracil in the DNA is a big no no,
and there are specific enzymes called uracil N-glycosylases (from the gene called
ung, about which we'll have much more to say in a later lecture) that excises
the offending deoxyuridylate nucleotide so that it can be replaced. If the uracil
had arisin by deamination, then what will be the nucleotide base across from it?
There will be a G nucleotide across from it, if the mutation just occurred. That's
because the G was paired with the C that deaminated to a U. On the other hand, if
there is a round of DNA replication before the uracil N-glycosylase arrives on the
scene, then there will be an A nucleotide across from the U. That's because the U
will have had a chance to be a template in DNA replication, and U base pairs to A,
If you're an organism that doesn't want
to end up looking like a Teenage
Mutant Ninja Turtle (who as
you may recall, were suffering from the effects of a "retromutagen" that
made them behave like adolescent boys), then you should keep a sharp eye out for
deoxyuridylate nucleotides. The dU should be excised rapidly and replaced with a
C, so that these deamination events do not become "fixed" as a mutation.
Some types of mutations change a pyrimidine
to a different pyrimidine, or a purine to a different purine. We call these transition mutations. If a purine is mutated to a pyrimidine, then
it is a transversion
mutation. So, for example, a mutation
of A to T or C to A would be what? Right! A transversion, and a mutation of A to
G or T to G would be a transition.
Sometimes deoxycytosine is methylated on its "5 position," so what would
happen to the coding content of deoxy-5-methyl-cytosine if it were unlucky enough
to be naturally deaminated?
Deamination of 5-methyl cytosine gives you
Do you know my name?
So you see the problem...the 5-methyl cytosine is deaminated to thymidine. The new
thymidine looks like any other thymidine - it's a mutation! A transition mutation,
because it is a pyrimidine changed to another pyrimidine.
Perhaps that is why there are so few CG dinucleotides in mammalian genomes. CG dinucleotides
are frequently methylated on the C base, so CG may frequently mutate to TG, leaving
CG "under represented". In fact, CG dinucleotides are sometimes associated
with regulatory regions of genes, and we call them "CG islands" because
they are so rare.
about the sugars
Now let's look at the sugar
component of nucleic acids. Remember that ribose and which is deoxyribose?
There is a 5' end
and a 3' end to a nucleic acid. The 5' end frequently has
a phosphate attached, while the 3' end is typically a hydroxyl group. A single strand
of DNA has a "polarity" or "directionality."
It isn't like a piece of string, in which you cannot distinguish one end from the
DNA vs. RNA sugars
Deoxyribose with thymine base
Ribose with uracil base
Study the phosphate at the 5' end
click for larger image
Study the hydroxyl at the 3' end
click for larger image
Synthesize? Degrade? Sit and wait? How
does an enzyme like DNA polymerase Klenow Fragment know what to do next? Well, there
are some general rules of conduct that these enzymes learn in school, and you can
learn them too.
rules of conduct for Klenow and T4 DNA polymerases
1. Remember your base
pairing rules: G goes with C and A goes with T.
2. The 5' ends are strictly
off limits, unless you have your holoenzyme license (and for your information, you
3. There will be no
synthesis without a free 3' end, unless you have your RNA polymerase license (and
for your information, you don't!)
4. There will be no
degradation without a free 3' end, unless you have your endonuclease license (and
for your information, you don't!)
5. There will be no
synthesis without an underlying template, unless you have your terminal transferase
license (and for your information, you don't!). Excess nucleotide substrates is NOT
accepted as an excuse for untemplated additions to the 3' end.
6. Under no circumstances
may you make a synthetic addition to the 5' end (even holoenzymes are not permitted
to do that!). Having a template or substrate available is not an excuse for 3' to
7. There will be no
reconstruction of a broken phosphodiester bond, unless you have your ligase license
(and for your information, you don't!). If you are synthesizing DNA and run into
an obstruction on your template, you must stop and leave the nick unrepaired. You
may not excise the 5' nucleotide that is obstructing your path (see rule 2).
8. If you have no remaining
template, then you must excise the nucleotide at the 3' end (and don't be tempted
to break rule 5!). (repeat rule 8 until it does not apply).
9. If you have been
provided with a free 3' end, a template, and a substrate molecule that is correct,
you must add that nucleotide to the growing end of the strand (i.e. to the 3' end.)
10. If you have a free
3' end and a template, but after waiting for the appropriate number of milliseconds
you are still missing the appropriate nucleotide substrate for the next synthetic
step, you may go back and remove the one preceding nucleotide. Either of rules 9
or 10 may apply thereafter. | http://escience.ws/b572/L1/L1.htm | 13 |
22 | Geometry is heavily tested on the GRE Math section, and a thorough review of geometrical concepts is essential to a high score. Consider the following problem:
“If the length of an edge of a cube X is twice the length of an edge of cube Y, what is the ratio of the volume of cube Y to the volume of cube X?”
The easiest way to solve this is to pick a number for the initial edge length and plug it into the problem. For instance, let’s say cube X is a 4x4x4 cube. Cube X would have a volume of 64. Cube Y would have to be a 2x2x2 cube, since 2 is half of 4, and it would have a volume of 8. The ratio of the volume of cube Y to the volume of cube X would thus be 8 to 64, or 1/8.
However, you really should have known that to begin with. Imagine that cube X had edges that were three times as long as those of Cube Y. Then Cube X would now be a 6x6x6 cube if Cube Y remains a 2x2x2 cube, and the volume ratio would be 8 to 216, or 1/27. Notice something? 8 is 2 ^3, and 27 is 3^3. If the ratio of the sides is 1:4, the ratio of the volumes will be 1:64. If the ratio of the sides is 1:5, the ratio of the volumes will be 1:125. Since these are cubes, you just cube the ratios. 1^3 is 1, and 4^3 is 64; 5^3 is 125. If you know this simple property of the relationship between length and volume, it will take a problem that would take 30 seconds to solve and turn it into a problem that takes 5 seconds to solve. On a timed exam, that could be the difference between getting another, harder question right or wrong. Memorizing these kinds of mathematical facts is something that the GRE test writers expect top scorers to do, and they write the questions so that they can be solved quickly if you know them. It also pays to memorize the squares and cubes of the numbers 1 through 12.
So with cubes, you cube the ratio of the sides. What about squares? If you guessed that you square the ratio of the side lengths in order to get the ratio of the areas, you’d be right, as you can see from a quick demonstration. If the original square has side lengths of 1 and the new square has side lengths of 2, the side ratio is 1:2 and the area ratio is 1:4. If the new square has side lengths of 3, then the side ratio is 1:3 and the area ratio is 1:9. If the new square has side lengths of 4, then the side ratio is 1:4 and the area ratio is 1:16, and so on. Sure enough, you just square the original ratio.
So now you know about cubes and squares, but what about tesseracts? “Tessawhats?” you say? A tesseract is to a cube as a cube is to a square, just as a cube is to a square what a square is to a line. Still confused? Let me explain it this way: say you draw a line a foot long running from east to west. This line only exists in one dimension: east-west. Then, you decide to square it by adding three more lines: two perpendicular to it running north to south and one parallel to it running east to west. This square exists in two dimensions: east-west and north-south. Now you decide to turn the square into a cube by adding lines in the up-down dimension, so that each edge of the original square is now the edge of another square emanating from it. This cube exists in three spatial dimensions: east-west, north-south, and up-down. Now you take this cube you’ve made and decide to square it…in a fourth spacial dimension.
What is this fourth dimension? Who knows. We live in a world in which we experience only three spacial dimensions, so it is impossible for us to imagine what a four dimensional object would look like. That hasn’t stopped mathematicians from naming four-dimensional objects, and this hypercube I’ve just described to you is called a tesseract. As you know, even though a cube is a three dimensional object, it is possible to draw a cube on a piece of paper in only two dimensions by using perspective and all those other artistic illusions. Likewise, some have attempted to render tesseracts in three dimensions in order to give some approximation of what they might look like. Having never seen an actual tesseract, though, you might still find these representations confusing.
In terms of doing calculations, though, tesseracts are simple as can be. For a square with side lengths of 1 and another square with side lengths of 2, the ratio of side lengths is 1:2^1 (since sides are 1 dimensional), or 1:2, and the ratio of areas will be 1:2^2 (since squares are 2 dimensional) or 1:4. For a cube with side lengths of 1 and another cube with side lengths of 2, the ratio of volumes is 1:2^3 (since cubes are 3 dimensional), or 1:8. So, for a tesseract with side lengths of 1 and another tesseract with side lengths of 2, the ratio of hypervolumes(?) is 1:2^4 (since tesseracts are 4 dimensional), or 1:16. It just follows the pattern. Try not to think about it too much.
If you’re having trouble with tesseracts, don’t worry. They’re not on the test. I just wrote about them to mess with your head.
Remember, if you ever want extra help getting ready for the GRE, you can always study with experts like me through Test Masters. Until then, happy studying! | http://www.newgre.org/preparation/sample-math-problem-hip-square-or-cube/ | 13 |
20 | Smooth muscle is responsible for the contractility of hollow organs, such as blood vessels, the gastrointestinal tract, the bladder, or the uterus. Its structure differs greatly from that of skeletal muscle, although it can develop isometric force per cross-sectional area that is equal to that of skeletal muscle. However, the speed of smooth muscle contraction is only a small fraction of that of skeletal muscle.
Structure: The most striking feature of smooth muscle is the lack of visible cross striations (hence the name smooth). Smooth muscle fibers are much smaller (2-10 m in diameter) than skeletal muscle fibers (10-100 m ). It is customary to classify smooth muscle as single-unit and multi-unit smooth muscle (Fig. SM1). The fibers are assembled in different ways. The muscle fibers making up the single-unit muscle are gathered into dense sheets or bands. Though the fibers run roughly parallel, they are densely and irregularly packed together, most often so that the narrower portion of one fiber lies against the wider portion of its neighbor. These fibers have connections, the plasma membranes of two neighboring fibers form gap junctions that act as low resistance pathway for the rapid spread of electrical signals throughout the tissue. The multi-unit smooth muscle fibers have no interconnecting bridges. They are mingled with connective tissue fibers.
Fig. SM1. Single-unit and multi-unit smooth muscle.
Innervation and stimulation: Smooth muscle is primarily under the control of autonomic nervous system, whereas skeletal muscle is under the control of the somatic nervous system. The single-unit smooth muscle has pacemaker regions where contractions are spontaneously and rhythmically generated. The fibers contract in unison, that is the single unit of smooth muscle is syncytial. The fibers of multi-unit smooth muscle are innervated by sympathetic and parasympathetic nerve fibers and respond independently from each other upon nerve stimulation.
Nerve stimulation in smooth muscle causes membrane depolarization, like in skeletal muscle. Excitation, the electrochemical event occurring at the membrane is followed by the mechanical event, contraction. In the case of smooth muscle, this excitation-contraction coupling is termed electromechanical coupling; the link for the coupling is Ca2+ that permeates from the extracellular space into the intracellular water of smooth muscle. There is another excitation mechanism in smooth muscle, which is independent of the membrane potential change; it is based on receptor activation by drugs or hormones followed by muscle contraction. This is termed pharmacomechanical coupling. The link is Ca2+ that is released from an internal source, the sarcoplasmic reticulum.
The role of mechanical events of smooth muscle in the wall of hollow organs is twofold: 1) Its tonic contraction maintains organ dimensions against imposed load. 2) Force development and muscle shortening, like in skeletal muscle.
Myofibril proteins: In general, smooth muscle contains much less protein (~110 mg/g muscle) than skeletal muscle (~200 mg/g). Notable is the decreased myosin content, ~20 mg/g in smooth muscle versus ~80 mg/g in skeletal muscle. On the other hand, the amounts of actin and tropomyosin are the same in both types of muscle. Smooth muscle does not contain troponin, instead of it there are two other thin filament proteins, caldesmon and calponin.
The amino acid sequence of smooth muscle actin is very similar to that of its skeletal muscle counterpart, and it seems likely that their three-dimensional structures are also similar. Smooth muscle actin combines with either smooth or skeletal muscle myosin. However, there is a major difference in the activation of myosin ATPase by actin, smooth muscle myosin has to be phosphorylated for actin-activation to occur.
The size and shape of the smooth muscle myosin molecule is similar to that of the skeletal muscle myosin (Fig. M1). There is a small difference in the light chain composition; out of the four light chains of the smooth muscle myosin two have molecular weight of 20,000 and two of 17,000. The 20,000 light chain is phosphorylatable. Upon phosphorylation of the light chain the actin-activated smooth muscle myosin ATPase increases about 50-fold, to about 0.16 mol ATP hydrolyzed per mol of myosin head per sec, at physiological ionic strength and temperature. (Under the same conditions, the actin-activated skeletal muscle myosin ATPase is 10 -20 mol/mol/sec). The ionic strength dependence of smooth muscle myosin Ca2+-activated ATPase also differs from that of skeletal muscle myosin (Fig. M5), increasing ionic strength increases the smooth muscle myosin ATPase but decreases the skeletal muscle myosin ATPase.
Four smooth muscle specifc myosin heavy chain isoforms are known ( described in Quevillon-Cheruel et al., 1999). Two isoforms (named SMB and SMA) are defined by the presence or the absence of an insert of seven amino acids in the N-terminal globular head region. The two others (SM1 and SM2) differ at their C-termini by 43 versus 9 amino acids. To understand the role of the C-terminal extremities of SM1 and SM2 in smooth muscle thick filament assembly, various fragments of these myosins, such as the rod region, the rod with no tailpiece, or light meromyosins were prepared as recombinant proteins in bacterial cells (Rowner et al., 2001; Quevillon-Cheruel et al.,1999). The results showed that the smooth muscle myosin tailpieces differentially affect filament assembly and suggested that homogeneous thick filaments containing SM1 or SM2 myosin could serve distinct functions within smooth muscle cells.
Although the mechanism of thick filament assembly for purified smooth muscle myosins in vitro has been described, the regulation of thick filament formation in intact muscle is poorly understood. Cross-sectional density of the thick filaments measured electron microscopically in intact airway smooth muscle (Herrera et al., 2002) showed that the density increased substantially (144%) when the muscle was activated. In resting muscle, in the absence of Ca2+, the filament density decreased by 35%. It appears that in smooth muscle filamentous myosin exists in equilibrium with monomeric myosin; activation favors filament formation.
Kathleen Trybus pioneered in expressing and purifying smooth muscle myosin subfragments using the baculovirux /insect cell expression system. This procedure and the methods needed to characterize the new proteins (gel assays, ATPase activity determinations, transient state kinetic parameters, and the vitro motility assay) are described in her review (Trybus, 2000). Studies on engineered smooth muscle myosin and heavy meromyosin showed: the interaction between the regulatory light chain domains on two heads is critical for regulation of smooth muscle myosin (Li et al., 2000; Sweeney et al., 2000), a long, weakly charged actin-binding loop is required for phosphorylation-dependent regulation of smooth muscle myosin (Rovner, 1998), and coiled-coil unwinding at the smooth muscle myosin head-rod junction is required for optimal mechanical performance (Lauzon et al., 2001).
In vitro, both caldesmon and calponin are inhibiting the actin-activated ATPase activity of phosphorylated smooth muscle myosin. In case of calponin, this inhibitory activity is reversed by the binding of Ca2+-calmodulin or by phosphorylation. Calponin is a 34-kDa protein containing binding sites for actin, tropomyosin and Ca2+-calmodulin. Caldesmon is a long, flexible, 87-kDa protein containing binding sites for myosin, as well as actin, tropomyosin, and Ca2+-calmodulin. Electron microscopy and three-dimensional image reconstruction of isolated smooth muscle thin filaments revealed that calponin and caldesmon are located peripherally along the long-pitch actin helix (Hodgkinson et al., 1997; Lehman et al., 1997). The physiological role of caldesmon or calponin is not known.
Phosphorylation and Dephosphorylation of the 20-kDa Myosin Light Chain
Myosin light chain kinase and myosin light chain phosphatase: Smooth muscle (as well as skeletal and cardiac muscle) contains myosin light chain kinase (MLCK), activated by Ca2+-calmodulin, the enzyme which transfers the terminal phosphate group of ATP to serine (and/or threonine) hydroxyl groups of phosphorylatable light chain (LC) according to the following reaction:
LC-OH + MgATP2- ® LC-O-PO32- + MgADP- + H+ (1)
Dephosphorylation is brought about by smooth muscle myosin light chain phosphatase (MLCP) according to the following reaction:
LC-O-PO32- + H2O ® LC-OH + HPO42- (2)
The properties of MLCK are reviewed by Stull et al. (1996) and the properties of MLCP are reviewed by Hartshorne et al. (1998).
It is generally believed that LC phosphorylation-dephosphorylation controls the contraction-relaxation cycle of smooth muscle:
For a long time, research focused on the role of MLCK in smooth muscle contractility, but recently the interest shifted to MLCP. It turned out that MLCP is composed of three subunits: a catalytic subunit of 37-38-kDa of the type 1 phosphatase, a subunit of about 20-kDa whose function is not known, and a larger 110-130-kDa subunit that targets MLCP to myosin. The phosphatase activity of the catalytic subunit is low and it is enhanced significantly by addition of the targeting subunit. Upon phosphorylation of serine and threonine residues in the targeting subunit, its activating effect on the catalytic subunit is lost, and thereby the MLCP holoenzyme is inhibited.
Recent reports (Feng et al., 1999; Kaibuchi et al., 1999; Nagumo et al., 2000; Somlyo and Somlyo, 2000; Sward et al., 2000) indicate that in smooth muscle a Rho-regulated system of MLCP exists. Rho-kinase is the major player in this system, the enzyme phosphorylates the 130-kDa myosin binding subunit of MLCP and thereby inhibits MLCP activity. Due to the antagonism between MLCK and MLCP, inhibition of MLCP results in an increase in the phosphoryl content of LC with concomitant increase in muscle force. Under these conditions, submaximal Ca2+-levels are sufficient for maximal force, a phenomenon called increased Ca2+-sensitivity (Somlyo and Somlyo, 1994). Specific inhibitors for rho-kinase Y-27632 (Feng et al., 1999; Kaibuchi et al., 1999), and HA-1077 (Nagumo et al., 2000; Sward el al., 2000) are available.
MLCP activity can also be inhibited by a 17-kDa myosin phosphatase inhibitor protein, called CPI-17, (Kitazawa et al., 2000) , which inhibits the catalytic subunit of MLCP and the holoenzyme MLCP. Phosphorylation of CPI-17 at Thr38 increases its inhibitory potency 1000-fold. The solution NMR structure of CPI-17 has been determined.(Ohki et al., 2001), it forms a novel four-helix. Phosphorylation of Thr38 induces a conformational change involving displacement of one helix without significant movement of the other three helices. Rho-kinases and PKC are responsible for the phosphorylation of CPI-17.
A rich array of second messengers regulate MLCP activity under physiological and pathological conditions (Solaro, 2000) through phosphorylation of either the targeting subunit of MLCP or CPI-17.
Myosin light chain phosphorylation in intact smooth muscle: 32P-labeling of the muscle is a reliable method for such studies. When a dissected smooth muscle, e.g. artery or a uterine strip, is incubated at 37oC in physiological salt solution containing radioactive inorganic phosphate, the 32P permeates the plasma membrane and enters the intracellular space of the muscle. Through the oxidative phosphorylation mechanism the 32P incorporates into the terminal P group of ATP:
ADP + 32P ® ADP32P
Transfer of the terminal 32P of ATP to LC-OH by MLCK (equation 1) yields the radioactive LC-O-32PO32- species that can be isolated and quantified. The isolation involves two-dimensional (2D) gel electrophoresis and the quantification requires measuring the specific radioactivity of the terminal P of ATP.
Smooth muscle contraction is correlated with LC phosphorylation (reviewed by Bárány and Bárány, 1996c). Fig. SM2 illustrates an experiment: Two carotid arteries were dissected from freshly killed pigs and labeled with 32P. One artery was contracted with KCl for 30 sec then frozen in liquid nitrogen, while the other artery was frozen in the resting state. The arteries were pulverized, washed with perchloric acid to precipitate the muscle proteins and remove 32P-containing phosphate metabolites from the muscle. The washed residue was neutralized with a NaOH solution then dissolved in sodium dodecyl sulfate (SDS). After centrifugation at high speed to remove insoluble particles, the protein content of the supernatant was determined and aliquots of 360 mg protein were subjected to 2D polyacrylamide gel electrophoresis. This procedure separates the proteins according to their charge (pH 4-6) in the first dimension and according to their size (SDS ) in the second dimension. After staining, the profile of the arterial proteins appeared, shown in the upper row of Fig. SM2. LC, is in the lower middle part of the gel, it contains multiple spots. The LC spots were scanned, the staining intensities are shown in the lower row of the Figure. The radioactive spots on the gel were detected by autoradiography, the middle row of Fig. SM2 shows the black spots on the film corresponding to the radioactive spots on the gel.
Visual inspection of the radioactive LC spots in the Figure shows much more radioactivity in LC from the contracting muscle (right) than from the resting muscle (left). One can calculate the incorporation of the 32P-phosphate into LC as follows. First one has to determine the specific radioactivity of the terminal P of ATP from the muscle. The ATP is in the perchloric acid extract of the frozen and pulverized muscle, described before, and Bárány and Bárány (1996c) describe the determination of the specific radioactivity. The next step is the determination of the radioactivity in LC: the gel spots are excised, digested with H2O2, and after the gel is dissolved, radioactivity (counts per minute) is measured. The extent of LC phosphorylation can be calculated from the radioactivity in the LC spots and in the terminal phosphate of ATP, from the total protein applied onto the gel, and from the LC content of the total protein (Bárány and Bárány, 1996c). Such a calculation shows that under conditions of Fig. SM2, the LC of the resting muscle contained 0.25 mol 32P-phosphate/mol LC, whereas the LC of the contracting muscle contained 0.70 mol. Thus, 0.45 mol 32P-phosphate was transferred by MLCK from the terminal phosphate of ADP32P to free LC-OH groups as the result of muscle contraction.
Fig. SM2. Light chain phosphorylation during smooth muscle contraction as studied by 2D gel electrophoresis. (Bárány and Bárány, 1996a, with permission from Biochemistry of Smooth Muscle Contraction, 1996, Academic Press). Left, 32P-labeled porcine carotid arterial muscle was frozen at rest. Right, 32P-labeled porcine carotid arterial muscle was frozen 30 sec after 100 mM KCl challenge. Upper panel shows the Coomassie blue staining pattern of the arterial proteins; middle panel shows the corresponding autoradiograms; bottom panel shows the corresponding densitometric scans of LC.
Isoforms of the 20-kDa myosin light chain: Protein isoforms have the same size but different charge. They are generated either by protein modification or genetic alteration. Protein phosphorylation is the physiological protein modification, because phosphorylation of a protein increases its negative charge. Thus, LC has at least two isoforms, a non-phosphorylated and a phosphorylated one. Genetic alteration changes the amino acid composition of a protein, thereby providing at least two isoforms. For instance, completely dephosphorylated LC exhibits two spots on 2D gels (Fig. SM3) with a percentage distribution of 85% and 15%, corresponding to the major and minor LC isoforms.
Fig. SM3. Myosin light chain isoforms as analyzed by 2D gel electrophoresis. LC was dephosphorylated by homogenizing porcine carotid arteries in 150 mM NaCl and 1 mM EGTA, followed by incubation at 25oC for 2 hours. Top, stained gel, LC spots are numbered as 2 and 4, corresponding to their isoform number. Bottom, densitometric tracing of the LC spots.
Figure SM4 illustrates the formation of LC isoforms as a result of phosphorylation. The major isoform (LCa) when mono-phosphorylated (PLCa) moves into Spot 3, and when it is di-phosphorylated (2PLCa) moves into Spot 2. The same Spot 2 also contains the non-phosphorylated minor isoform (LCb), thus the comigration of the di-phosphorylated LC isoform with the minor isoform makes Spot 2 radioactive. This explains why out of the four LC spots three are phosphorylated. The mono-phosphorylated minor isoform (PLCb) moves into Spot 1, which is the most acidic spot.
Fig. SM4. Scheme for the explanation of four stained and three radioactive LC spots, shown on Fig. SM2. (Bárány and Bárány, 1996a, with permission from Biochemistry of Smooth Muscle Contraction, 1996, Academic Press).
Phosphorylation site: The amino acid sequence of LC exhibits a similarity among LCs from various smooth muscles. Such a conservative sequence suggests a functional significance for the protein. The phosphorylation sites are located at the amino terminal part of the LC molecule, shown in Fig. SM5. Serine 19 is the site that is phosphorylated by MLCK in the intact muscle. Threonine 18 is phosphorylated by MLCK rarely. Beside MLCK, protein kinase C (PKC) also phosphorylates LC; the sites involve Serine 1, Serine 2, and Threonine 9.
Fig. SM5. Phosphorylation sites of LC.
Two-dimensional tryptic peptide mapping: Phosphopeptide maps differentiate MLCK-catalyzed LC phosphorylation from that catalyzed by PKC (Erdodi et al., 1998). Fig. SM6 illustrates the experiment: With ADP32P as a substrate, pure LC was phosphorylated either by MLCK (middle panel), or PKC (right panel). Actomyosin that contains endogenous LC, MLCK, and PKC, was also phosphorylated (left panel). The 32P-LC was isolated by 2D gel electrophoresis, digested by trypsin, and the peptides were separated by 2D peptide mapping. The map of LC phosphorylated by MLCK exhibits four peptides: A, B, both containing serine residues, corresponding to the Ser-19 site, and C, D, both containing threonine, corresponding to the Thr-18 site. When LC is phosphorylated by PKC, the map exhibits two peptides: E, containing serine, corresponding to Ser-1 or Ser-2 site, and F, containing threonine, corresponding to theThr-9 site. When LC is phosphorylated in actomyosin, peptides characteristic for both MLCK and PKC phosphorylation are present.
Fig. SM6. Autoradiograms of 2D phosphopeptide maps of LC tryptic digests.
Fig. SM7. Phosphopeptide maps of LC from K+-contracted muscle versus PDBu-treated muscle.
The role of Ca2+ in light chain phosphorylation: As in skeletal muscle, Ca2+ also plays a central role in the contractility of smooth muscle. In skeletal muscle TN-C is the target of the myoplasmic Ca2+, whereas in smooth muscle Ca2+ activates MLCK. Actually, the Ca2+ complexed to calmodulin is the activator of the enzyme. In agreement with the in vitro studies, intact smooth muscle cease contracting when Ca2+ is omitted from the bathing solution, or when it is complexed with EGTA. Furthermore, inhibitors of calmodulin, such as trifluoperazine or chlorpromazine inhibit smooth muscle contraction.
In the resting muscle there is about 0.1 µM Ca2+, upon stimulation the Ca2+ concentration increases about 100-fold through electromechanical or pharmacomechanical coupling. It is conventional to use fluorescent indicators to follow changes in the intracellular Ca2+ concentration immediately after the stimulation and during the plateau of the mechanical activity. Large variations are reported, depending on the nature of the smooth muscle, the tissue preparation, or the drug used. However, all investigators agree that in order to elicit relaxation the Ca2+ level in the sarcoplasm must be returned near to the resting value. Two mechanisms participate in decreasing the Ca2+ level: 1) The plasma membrane Ca2+ transporting ATPase pumps Ca2+ from the inside into the extracellular space. 2) The sarco(endo)plasmic reticulum Ca2+ transporting ATPase pumps Ca2+ into the SR.
Stretch-induced light chain phosphorylation: As discussed before, smooth muscle can be stimulated electrically or by chemical agents. Here we describe the mechanochemical activation of smooth muscle. Stretching of arterial or uterine muscles induced light chain phosphorylation to the same extent as was observed in muscles contracted by K+ or norepinephrine (Bárány and Bárány, 1996c). Muscles which were stretched 1.6 times their resting length did not develop tension, but contracted normally when the stretch was released and the muscles were allowed to return to their rest length. Importantly, this contraction was spontaneous, indicating that the stretch-induced activation carries all the information necessary for normal contraction. Mobilization of Ca2+ was necessary for the stretch-induced light chain phosphorylation and contraction to occur. When EGTA (the strong Ca2+ complexing agent) was added to the muscle bath both the stretch-induced phosphorylation and the stretch-release-induced tension were inhibited; however, upon removal of EGTA by washings, both processes were fully restored. Treatment of the muscle with chlorpromazine (the calmodulin inhibitor) also abolished both the stretch-induced LC phosphorylation and the stretch-release-induced tension development. These results suggest the presence of mechanosensitive receptors in smooth muscle that are interacting with Ca2+ release channels in SR.
Further comments are warranted on the finding that 1.6 times stretched muscles, which are unable to contract (because there is no overlap between actin and myosin filaments), are able to fully phosphorylate their LC. Accordingly, smooth muscle contraction and LC phosphorylation are not coupled. Time course experiment also demonstrated that LC phosphorylation precedes tension development. Thus, LC phosphorylation plays a role in the activation process but not in the contraction per se. Furthermore, K+-contracted muscle maintains its tension for a prolonged time although its LC becomes dephosphorylated. This is another example for the lack of coupling between phosphate content of LC and contractility of muscle.
Phosphorylation of Heat-Shock Proteins
Low molecular weight heat schock proteins are phosphorylated in smooth muscle: A 27-28-kDa protein is phosphorylated in various intact smooth muscles and smooth muscle cells (reviewed in Bárány and Bárány, (1996c). Cyclic nucleotide-dependent vasorelaxation is associated with the phosphorylation of a 20-kDa heat shock protein, called HSP20 (Beal et al., 1997; Rembold et al., 2000). It was found that HSP20 is an actin-associated protein (Brophy et al.,1999; Rembold et al. 2000) suggesting that smooth muscle relaxation may be brought about by the binding of the phosphorylated HSP20 to the actin filaments.
The binding of an agonist (e.g. norepinephrine or oxytocin) to the surface receptor of smooth muscle induces a signal that spreads from the outside to the inside of the plasma membrane and activates several effectors that ultimately initiate contraction. There are three components of this system that we discuss: 1) Inositol 1,4,5-trisphosphate, 2) G-proteins, 3) Phosphoinositide-specific phospholipase C.
Inositol 1,4,5-trisphosphate: The inositol ring contains six hydroxyl residues, most of them can be phosphorylated by specific kinases. Inositol 1-monophosphate is the constituent of phosphatidylinositol (PI) one of the phospholipids in animal cell membranes. PI 4-kinase and PI (4) P 5-kinase to generate PI (4) P and PI (4,5) P2, respectively, sequentially phosphorylate PI. Inside the cell membrane resides a phosphoinositide specific phospholipase C, one of its hydrolytic product is inositol 1,4,5-trisphosphate (IP3), (see Fig. SM 8).
Fig. SM8. D-myo-inositol 1,4,5-trisphosphate. (Bárány and Bárány, 1996b,with permission from Biochemistry of Smooth Muscle Contraction, 1996, Academic Press). The arrow indicates the site of the ester link with diacylglycerol in phosphatidylinositol. The negative charge of the phosphate group is not indicated.
G-proteins: The guanine nucleotide binding proteins (G-proteins) are heterotrimers consisting of a-, b- and g-subunits. The a-subunits appear to be most diverse and are believed to be responsible for the specificity of the interaction of different G-proteins with their effectors. Fig. SM9 depicts a simple model for the activation of G-proteins. In the basal state, the a-subunit contains bound GDP and association of a- and bg-subunits is highly favored, keeping the G-protein in the inactive form. Stimulation of the G-protein results when it binds GTP rather than GDP. Receptors interact most efficiently with the heterotrimeric form of the G-protein and accelerate activation by increasing the rate of dissociation of GDP and enhancing the association of GTP. Activation of G-protein coupled receptor results in the dissociation of heterotrimeric G-proteins into a-subunits and bg-dimers. Finally, the G-protein a-subunit has an intrinsic hydrolytic activity that slowly converts GTP to GDP and returns the G-protein to its inactive form.
Fig. SM9. Model for the activation of G-proteins. ( Bárány and Bárány, 1996b, with permission from Biochemistry of Smooth Muscle Contraction, 1996, Academic Press).
Phosphoinositide-specific phospholipase C: This term refers to a family of enzymes all specific for the phosphoinositide moiety of the phosphatidylinositol, but differing in their specificity depending on the number of the phosphoryl groups on the inositol ring. The b-, g- and d-isoforms of PI-phospholipase C (PI-PLC) show the greatest specificity for the trisphosphorylated phospholipid (PIP2)). There are two basic mechanisms by which agonists activate PIP2 hydrolysis (Fig. SM10). In case of hormones, neurotransmitters, and certain other agonists, the signal is transduced to b-isozymes of PI-PLC. The upper left row of Fig. SM10 shows the most common pathway for PI-PLCb-isoform activation, initiated by stimulation of a a1-adrenergic receptor (a1-R) with norepinephrine (NE), and involving Gaq-proteins. The lower left row shows the activation of PI-PLC-b isoforms, initiated by acetylcholine (ACH) stimulation of M2-muscarinic receptor (M2-R), and mediated by the b g-subunit of the pertussis toxin-sensitive G-protein (GI). Concerning the other basic activating mechanism, e.g. in the case of growth factors, activation of their receptors results in enhanced tyrosine kinase activity. The right part of Fig. SM10 shows the activation of PI-PLC-g isoforms, initiated by the binding of epidermal growth factor (EGF) to its receptors, and executed by the tyrosine phosphorylation (YP) of PI-PLC-g . In all three examples, the activated PI-PLC hydrolyzes PIP2 to form the messengers IP3 and diacylglycerol (DAG). IP3 releases Ca2+ from the sarcoplasmic reticulum and thereby initiates smooth muscle contraction. DAG activates protein kinase C, the exact result of this activation is not known at the cellular level.
Fig. SM 10. Pathways for activation of PI-PLC isoforms. (Bárány and Bárány, 1996b, with permission from Biochemistry of Smooth Muscle Contraction, 1996, Academic Press). .
The Contractile Event of Smooth Muscle
A scheme for smooth muscle contraction is shown in Fig. SM11. Contraction is initiated by the increase of Ca2+ in the myoplasm; this happens in the following ways:
- Ca2+ may enter from the extracellular fluid through channels in the plasmalemma. These channels open, when the muscle is electrically stimulated or the plasmalemma is depolarized by excess K+.
- Due to agonist induced receptor activation, Ca2+ may be released from the sarcoplasmic reticulum (SR). In this pathway, the activated receptor interacts with a G-protein (G) which in turn activates phospholipase C (PLC). The activated PLC hydrolyzes phosphatidyl inositol bisphosphate; one product of the hydrolysis is inositol 1,4,5-trisphosphate (IP3). IP3 binds to its receptor on the surface of SR, this opens Ca2+ channels and Ca2+ from SR is entering the myoplasm.
- Ca2+ combines with calmodulin (CaM) and the Ca2+ -CaM complex activates MLCK, which in turn phosphorylates LC. The phosphorylated myosin filament combines with the actin filament and the muscle contracts.
Fig. SM11. A scheme for smooth muscle contraction. ( Bárány, 1996, with permission from Biochemistry of Smooth Muscle Contraction, 1996, Academic Press).
Two books (Bárány, 1996; Kao and Carsten, 1997) and a special journal issue (Murphy, 1999) are recommended for further studying the mechanism of smooth muscle contraction.and relaxation.
Monomer (G) to Polymer (F) Transformation of Actin in Smooth Muscle
Mehta and Gunst (1999) and Jones et al (1999) reported the existence of G-actin in smooth muscle, based on the method of DNase I inhibition and phalloidin staining, respectively. Subsequently, Bárány, et al (2001) showed the exchange of the actin-bound nucleotide in intact smooth muscle. This was based on the separation of the actin bound nucleotides from the cytoplasmic nucleotides with 50% ethanol (Fig. SM12).
Fig. SM12. Extraction of
nucleotides and radioactivity from 32P-labeled
arterial smooth muscles (From Bárány et al.,
2001).The percentage of the total absorbance and counts
eluted from the muscles in 8 extractions is shown on the
Fig. SM12. Extraction of nucleotides and radioactivity from 32P-labeled arterial smooth muscles (From Bárány et al., 2001).The percentage of the total absorbance and counts eluted from the muscles in 8 extractions is shown on the ordinate.
The composition of the PCA extract is shown on Fig. SM13.
Fig. SM13. Dowex -1
chromatography of Extracts No. 7 and 8, shown in Fig.
SM12.(From Bárány et al., 2001). Squares
correspond to Counts per ml and triangles correspond to
Fig. SM13. Dowex -1 chromatography of Extracts No. 7 and 8, shown in Fig. SM12.(From Bárány et al., 2001). Squares correspond to Counts per ml and triangles correspond to Absorbance.
In order to quantify the extent of exchange of the actin-bound nucleotide and Pi, one has to determine their specific activity (counts /min/mol nucleotide or Pi) and compare it with those of the specific activities (s.a.) of the gamma- and beta-phosphates of the cytoplasmic ATP and that of PCr (Bárány et al., 2001). With this knowledge one can calculate the percentage exchange for each of the actin components; for instance , the percentage exchange of the actin-bound- ADP is:
(s.a. of actin-ADP/s.a. of beta-P of cytoplasmic ATP) x 100
Fig. SM14 compares the exchange of the actin-bound ADP between smooth and skeletal muscles. The exchange is rapid in smooth muscle, half-time about 15 min, whereas the exchange is slow in skeletal muscle, about 15% in three hours, in agreement with the studies of Martonosi et al., 1960) in live animals.
Fig. SM14. Time course of the
exchange of the actin-bound ADP in smooth (porcine carotid
artery) and skeletal (rat vastus lateralis) muscle. (From
Bárány et al., 2001).
Fig. SM14. Time course of the exchange of the actin-bound ADP in smooth (porcine carotid artery) and skeletal (rat vastus lateralis) muscle. (From Bárány et al., 2001).
Characteristics of the exchange of the actin-bound nucleotide in smooth muscle:
ATP is a prerequisite for the exchange to take place. If ATP synthesis is inhibited by azide or iodoacetamide the exchange is also inhibited. If ATP sysnthesis is reduced, by incubation of the muscles with deoxyglucose, instead of glucose, the exchange is also reduced.
Ca2+ is not required for the exchange, i.e. full exchange is observed in the muscle in the presence of EGTA.
Several smooth muscles, arteries, uteri, urinary bladder, and stomach exhibiited the exchange of the actin-bound nucleotide and phosphate, suggesting that the exchange is a property of every smooth muscle.
Upon contraction of smooth muscle, the exchange of the bound-nucleotide and phosphate decreased and upon relaxation from the contracted state it increased, suggesting that polymerization-deplolymerization of actin is a part of the contraction-relaxation cycle of smooth muscle.
Bárány, M. (1996). Biochemistry of Smooth Muscle Contraction. Academic Press.
Bárány, K. and Bárány, M. (1996a). Myosin light chains. In Biochemistry of Smooth Muscle Contraction (M. Bárány , Ed.), pp. 21-35, Academic Press.
Bárány, M. and Bárány, K. (1996b). Inositol 1,4,5-trisphosphate production. In Biochemistry of Smooth Muscle Contraction (M. Bárány, Ed.), pp. 269-282, Academic Press.
Bárány, M. and Bárány, K. (1996c). Protein phosphorylation during contraction and relaxation. In Biochemistry of Smooth Muscle Contraction (M. Bárány, Ed.), pp. 321-339, Academic Press.
Bárány, M., Barron, J.T., Gu, L., and Bárány, K. (2001). Exchange of the actin-bound nucleotide in intact arterial smooth muscle. J. Biol. Chem., 276, 48398-48403.
Beall, A.C., Kato, K., Goldenring, J.R., Rasmussen, R., and Brophy, C.M. (1997) Cyclic nucleotide-dependent vasorelaxation is associated with the phosphorylation of a small heat shock-related protein. J. Biol. Chem. 272, 11283-11287.
Brophy, C.M., Lamb, S., and Graham, A. (1999). The small heat shock-related protein-20 is an actin-associated protein. J. Vasc. Surg. 29, 326-333.
Erdödi, F., Rokolya, A., Bárány, M., and Bárány, K. (1988). Phosphorylation of the 20,000 dalton myosin light chain isoforms of arterial smooth muscle by myosin light chain kinase and protein kinase C. Arch. Biochem. Biophys. 266, 583-591.
Feng, J., Ito, M., Ichikawa, K., Isaka, N., Nishikawa, M., Hartshorne, D.J., and Nakano, T. (1999). Inhibitory phosphorylation site for rho-associated kinase on smooth muscle myosin phosphatase. J. Biol. Chem. 274, 37385-37390.
Hartshorne, D.J., Ito, M., and Erdödi, F. (1998). Myosin light chain phosphatase: subunit composition, interactions and regulation. J. Muscle Res. Cell Motil. 19, 325-341.
Herrera, A.M., Kuo, K-H., and Seow, C.Y. (2002). Influence of calcium on myosin thick filament formation in intact airway smooth muscle. Am. J. Physiol. Cell Physiol., 282, C310-C316.
Hodgkinson, J.L., el-Mezgueldi, M., Craig, R., Vibert, P., Marston, S.B., and Lehman, W. (1997). 3-D image reconstruction of reconstituted smooth muscle thin filaments containing calponin : visulaization of interactions between F-actin and calponin. J. Mol. Biol., 273, 159-159.
Jones, K.A., Perkins, W.J., Lorenz, R.R., Prakash, Y.S., Sieck, G.C., Warner, D.O. (1999). F-actin stabilization increases tension cost during contraction of permeabilized airway smooth muscles in dog. J.Physiol., 519, 527-538.
Kaibuchi, K., Kuroda, S., and Amano, M. (1999). Regulation of the cytoskeleton and cell adhesion by the rho family GTPases in mammalian cells. Annu. Rev. Biochem. 68, 459-486.
Kao, C.Y. and Carsten, M. E. (1997). Cellular Aspects of Smooth Muscle Function. Cambridge University Press.
Kitazawa, T., Eto, M., Woodsome, T.P., and Brautigan, D.L. (2000). Agonists trigger G protein-mediated activation of the CPI-17 inhibitor phosphoprotein of myosin light chain phosphatase to enhance vascular smooth muscle contractility. J. Biol. Chem., 275, 9897-9900.
Lauzon, A-M., Fagnant, P.M., Warshaw, D.M., and Trybus, K.M. (2001) Coiled-coil unwinding at the smooth muscle myosin head-rod junction is required for optimal mechanical performance. Biophys. J. 80, 1900-1904.
Lehman, W., Vibert, P., Craig, R. (1997). Visualization of caldesmon on smooth muscle thin filaments. J. Mol. Biol., 274, 310-317.
Li, X-D., Saito, J., Ikebe, R., Mabuchi, K., and Ikebe, M. (2000). The interaction between the regulatory light chain domains on two heads is critical for regulation of smooth muscle myosin. Biochemistry, 39, 2254-2260.
Martonosi, A., Gouvea, M.A., and Gergely, J. (1960). Studies on actin. III. G-F transformation of actin and muscular contraction (experiments in vivo). J. Biol. Chem. 235, 1707-1710.
Mehta, D. and Gunst, S.J. (1999). Actin polymerization stimulated by contractile activation regulates force development in canine tracheal smooth muscle. J. Physiol., 519, 820-840.
Murphy, R.A. (1999). Signal transduction in smooth muscle. Reviews of Physiology Biochemistry and Pharmacology. vol.134
Nagumo, H., Sasaki, Y., Ono, Y., Okamoto, H., Seto, M., and Takuwa, Y. (2000). Rho-kinase inhibitor HA-1077 prevents rho-mediated myosin phosphatase inhibition in smooth muscle cells. Am. J. Physiol., 278, C57-C65.
Ohki, S-Y., Eto, M., Kariya, A., Hayano, T., Hayashi, Y., Yazawa, M., Brautigan, D., and Kainosho, M. (2001). Solution NMR structure of the myosin phosphatase inhibitor protein CPI-17 shows phosphorylation-induced conformational changes responsible for activation. J. Mol. Biol. 314, 839-849.
Quevillon-Cheruel, S., Foucault, G., Desmadril, M., Lompre, A-M., and Bechet, J-J. (1999). Role of the C-terminal extremities of the smooth muscle myosin heavy chains: implication for assembly properties. FEBS Letters 454, 303-306.
Rembold, C.M., Foster, D.B., Strauss, J.D., Wingard, C.J., Van Eyk, J.E. (2000). cGMP-mediated phosphorylation of heat shock protein 20 may cause smooth muscle relaxation without myosin light chain dephosphorylation in swine carotid artery. J. Physiol., 524, 865-878.
Rovner, A.S., Fagnant, P.M., Lowey, S, and Trybus, K.M. (2002). The carboxyl-terminal isoforms of smooth muscle myosin heavy chain determine thick filament assembly properties. J. Cell. Biol. 156, 113-124.
Rowner, A.S. (1998). A long, weakly charged actin-binding loop is required for phosphorylation dependent regulation of smooth muscle myosin. J. Biol. Chem. 273, 27939-27944.
Solaro, R.J. (2000). Myosin light chain phosphatase a cinderella of cellular signaling. Circ. Res. 87, 173-175.
Somlyo, A.P. and Somlyo, A.V. (1994). Signal transduction and regulation in smooth muscle. Nature, 372, 231-236.
Somlyo, A.P. and Somlyo, A.V. (2000). Signal transduction by G-proteins, Rho-kinase and protein phosphatase to smooth muscle and non-muscle myosin II. J. Physiol., 522, 177-185.
Stull, J.T., Krueger, J.K., Kamm, K.E., Gao, Z-H., Zhi, G., and Padre, R. (1996). Myosin light chain kinase. In Biochemistry of Smooth Muscle Contraction (M. Bárány, Ed.), pp. 119-130. Academic Press.
Sward, K., Dreja, K., Susnjar, M., Hellstrand, P., Hartshorne, D.J., and Walsh, M.P. (2000). Inhibition of rho-associated kinase blocks agonist induced Ca2+sensitization of myosin phosphorylation and force in guinea pig ileum. J. Physiol. 522, 33-49.
Sweeney, H.L., Chen, L-Q., and Trybus, K.M. (2000). Regulation of asymmetric smooth muscle myosin II molecules. J.B iol. Chem. 275, 41273-41277.
Trybus, K. (2000). Biochemical studies of myosin. METHODS, 22, 327-335.
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16 | Knowing how and where birds migrate and breed is an important part of understanding how and why their numbers increase or decrease over time. However, we don't know much about the exact migratory patterns of most birds. After all, they are one of the most itinerant animals on earth, coming and going from one place to another as regularly as the seasons change. Where are they going? How will they get there? Why do they go? Based on what we currently know about migration, we can assume that they head toward areas where the weather is more conducive to survival and breeding. Now, a new technique to track birds is helping researchers understand an important concept called migratory connectivity.
Migratory connectivity is the degree to which breeding and non-breeding populations of birds are linked to one another; it is the relationship that helps us understand how these interactions contribute to the natural ecology of the animals' habitats. Until recently, the mark-recapture method has been the only technique used to do this. This method includes tagging individual birds and recording where they were recaptured, if they were recaptured at all. Unfortunately, this method was not always successful if the bird was not caught again.
Researchers recently tried a new method to track gray catbirds (Dumetella carolinensis), in addition to the older mark-recapture technique. The scientists fit 13 male and nine female birds with geolocators, which are special devices that resemble tags on the birds' legs and that can record the estimated latitude and longitude of the wearer based on sunlight levels every 10 minutes.
Birds wore these during the breeding and non-breeding seasons from July 2009 through May/June of 2010, when researchers recovered the devices. Only three males and three females were recaptured, so the data from the six geolocators was the only new information that the researchers had to work with. They were able to successfully download this data and use special software to correct and calibrate any errors in the information. They were also aware that the readings were sometimes slightly skewed if the bird had been perching in a shady area rather than in direct sunlight, but despite this, the team was able to get a good impression as to where the birds had been.
Next, the data was compared to previously documented mark-recapture records from 1914 to 2009 in order to provide a wide-range view of migratory connectivity. The data from both were similar, indicating a strong connectivity. It also showed that the gray catbirds that bred in Washington D.C. migrated to Florida and the Caribbean during the winters. In addition, long-term mark recapture data from the U.S. Geological Services Bird Banding Lab indicated that gray catbirds from the Midwest migrate down to Central America during the winters.
Although the data was consistent from both sources, there are limitations to both. The geolocators require proper light so that data is not misrepresented. The birds wearing these devices may also have trouble getting around, as weight and drag are increased, and a piece protrudes unnaturally. This can ultimately affect their survival. The authors did state however, that statistically the recapture and return rates of birds from both the geolocator study and the historical records were about the same. Data collected via mark-recapture techniques only seems to have meaning when the data is collected over a long period of time.
This article summarizes the information in this publication:
Ryder, Thomas B., Fox, James W., and Peter P. Marra. 2011. Estimating Migratory Connectivity of Gray Catbirds (Dumetella carolinensis) Using Geolocator and Mark-Recapture Data. The Auk 128(3):448-453.
Understanding the connectivity between breeding and nonbreeding populations of migratory birds is fundamental to our knowledge of biological phenomena such as population dynamics and dispersal. Moreover, our ability to quantify migratory connectivity has inevitable consequences for both conservation and management of species that utilize distinct geographic locations. Technology is rapidly advancing our ability to track birds throughout the annual cycle and to collect data on the degree of connectivity among breeding and nonbreeding populations. We combined two direct methods, mark–recapture (n = 17) and geolocation (n = 6), to estimate the migratory connectivity of breeding and nonbreeding populations of Gray Catbirds (Dumetella carolinensis). Data from geolocators show that birds breeding in the Mid-Atlantic overwinter in both Cuba and southern Florida. Mark–recapture data supported our geolocator results but also provided a broader spatial perspective by documenting that Mid-Atlantic and Midwestern populations occupy distinct geographic localities during the nonbreeding period. This research underscores the importance of geolocators, as well as other tools, to advance our understanding of migratory connectivity. Finally, our results highlight the potential value of U.S. Geological Survey (USGS) Bird Banding Laboratory mark–recapture data, which are often underutilized in ornithological research.
Teachers, Standards of Learning, as they apply to these articles, are available for each state. | http://nationalzoo.si.edu/scbi/migratorybirds/science_article/default.cfm?id=137 | 13 |
11 | This figure from NASA's Dawn mission shows the varied minerals on the surface of the giant asteroid Vesta in false color. The colors, derived from data obtained by Dawn's visible and infrared mapping spectrometer, have been chosen to emphasize mineral differences on a half-mile (kilometer) scale. Data from the spectrometer also demonstrate that Vesta's surface and subsurface show localized areas of bright and dark hues.
Geological structures at scales of tens of miles (kilometers) often show mineralogical differences. The differences can be seen particularly around craters that are surrounded by ejected material and that have experienced landslides.Oppia Crater is highlighted in the white box.
Colors were assigned to ratios of particular infrared wavelengths to emphasize differences not visible to the human eye. In this color scheme, green shows the relative strength of a particular mineralogical characteristic -- the absorption of pyroxene, an iron- and magnesium-rich mineral. Brighter green signifies a higher relative strength of this band, which indicates chemistry involving pyroxene. On the other hand, reddish colors indicate a different mineral composition.
The data used to create this mosaic were collected in August 2011, at an average altitude of 1,700 miles (2,700 kilometers). The visible and infrared mapping spectrometer data lie over a mosaic made by Dawn's framing camera.
The Dawn mission to Vesta and Ceres is managed by NASA's Jet Propulsion Laboratory, a division of the California Institute of Technology in Pasadena, for NASA's Science Mission Directorate, Washington. UCLA is responsible for overall Dawn mission science. The visible and infrared mapping spectrometer was provided by the Italian Space Agency and is managed by Italy's National Institute for Astrophysics, Rome, in collaboration with Selex Galileo, where it was built.
More information about Dawn is online at http://www.nasa.gov/dawn and http://dawn.jpl.nasa.gov. | http://photojournal.jpl.nasa.gov/catalog/PIA15671 | 13 |
14 | Arithmetic coding actually refers to half of an Arithmetic Coding data compression system.
It has two parts:
- An arithmetic coder
- A data model (e.g., Markovian model)
- There is a new data character (a fixed number of bits per character)
- There is a set of probabilities for each possible character
The larger the range, the less bits it takes to code the character. The smaller the range, the more bits it takes to code the character.
Typically, the model used to code the data changes based on the data input stream contents. This is known as adaptive coding.
An arithmetic encoder takes a string of symbols as input and produces a rational number in the interval [0, 1) as output. As each symbol is processed, the encoder will restrict the output to a smaller interval.
Let N be the number of distinct symbols in the input; let x1, x2 ... xN represent the symbols, and let P1, P2 ... PN represent the probability of each symbol appearing. At each step in the process, the output is restricted to the current interval [y, y+R). Partition this interval into N disjoint subintervals:
- I1 = [y, y + P1R)
- I2 = [y + P1R, y + P1R + P2R)
Note that at each stage, all the possible intervals are pairwise disjoint. Therefore a specific sequence of symbols produces exactly one unique output range, and the process can be reversed.
Since arithmetic encoders are typically implemented on binary computers, the actual output of the encoder is generally the shortest sequence of bits representing the fractional part of a rational number in the final interval.
Suppose our entire input string contains M symbols: then xi appears exactly PiM times in the input. Therefore, the size of the final interval will be
However, IBM and other companies own patents in the United States and other countries on algorithms essential for implementing an arithmetic encoder. But are those patent holders willing to license the patents royalty-free for use in open-source software? | http://www.encyclopedia4u.com/a/arithmetic-coding.html | 13 |
25 | August 30 - By observing the collision of two distant galaxies, scientists say that they now have direct evidence of dark matter's existence. For decades scientists have proposed the existence of dark matter as an explanation for how galaxies rotate at their observed velocities.
Dark matter emits no light and can only be detected by how it interacts with ordinary matter through gravity. One of the ways dark matter can be detected is by a phenomenon called gravitational lensing, which occurs when an object's gravitational field distorts light from background galaxies. However, dark matter is often embedded in galaxies, making it difficult to isolate the lensing it causes.
Researchers were able to directly detect dark matter by observing the collision between an enormous cluster of galaxies and a smaller galaxy cluster more than 3 billion light years away. The team reasoned that when the galaxies hit each other, the vast volumes of gas in each would slow down, but the dark matter would continue to speed along. Images from NASA's Chandra X-ray Observatory, the Hubble Space Telescope, and other instruments showed gravitational lensing in an area where there was no visible matter, indicating the presence of dark matter.
There are several National Research Council reports dealing with dark matter. Connecting Quarks with the Cosmos: Eleven Science Questions for the New Century examines 11 questions that need to be and can be answered in the next decade, including "what is the nature of dark matter and energy." Astronomy and Astrophysics in the New Millennium recommends further research into dark matter and into developing dark matter detectors. Revealing the Hidden Nature of Space and Time: Charting the Course for Elementary Particle Physics affirms how particle physics research is necessary to maintain the United States' position as a scientific world leader and recommends several frontiers for further research, including dark matter.
of News and Public Information
Science in the Headlines
|Copyright © 2006. National Academy of Sciences. All rights reserved. 500 Fifth St. N.W., Washington, D.C. 20001.| | http://www.nationalacademies.org/printer/headlines/20060830.html | 13 |
16 | Language is a means of communication. By using a language people can communicate with each other. Using a language is not as simply as we thought because there is a set of rules that must be followed, which is called Grammar. Actually grammar is used to mean the structure of a language. It is an essential part of the use of language process, both in spoken and written language. Based on Digital Library of PETRA University, the grammar of a language is a description of the ways in which the language uses patterns of structure to convey the meaning. It would be impossible to learn language effectively without knowing the grammar, because grammar helps learners to identify grammatical forms, which serve to enhance and sharpen the expression of meaning.
Having a good grammar system of a language, learners will be helpful in delivering their ideas, messages and feelings either to the listeners or readers. Language without grammar would be disorganized and causes some communicative problems, like grammatical errors in writing. Hence, learners need to know the grammatical system of language they can communicate with others in order to transfer the message properly.
In order to use a language well, learners should learn the rules of a language or to know how they work. They cannot avoid errors because errors mostly occur in learning process. It happens because they use different forms to deliver their ideas, fellings or messages so they need considerable amount of time to be able to master the target language well. Besides, by making errors, learners will build their new knowledge to use the target language as Littlewood stated that making errors during studying the second language can be considered as a means of building learners’ abilities because they can learn something from making errors (Littlewood, 1992)
According to Robert and Freida in Yulianti’s thesis (1972: 154), learning English is not easy, the language learners may have difficulties. The difficulties that are encountered by every student will vary according to his / her native language. Because of these, there will be errors that can be found in their learning.
These errors will influence their communication. Therefore, it is important to analyze the errors because by learning the errors there are many advantages such as (a) a device which the learner uses in order to learn ( Selinker in Soesanti’s thesis, 1992 : 150 ), (b) to fully grasp and understand the nature of errors, and (c) instead of just being able to recognize errors, the learners are now able to explain the rules and correct the errors ( Mei Lin Ho, 2003 : 1).
The errors usually occur in the productive skills, speaking and writing, but to analyze the errors in productive skills in short time is not easy. It takes much time, money, and requires a high ability of an analyst. Therefore, the writer decided to analyze only the grammatical errors in students’ writing.
The writer chooses the students of Writing IV class as the subject of the research because they are expected to make writings which are correct in grammar, so it is important to know whether the students make grammatical errors or not and what kind of grammatical errors that students make. The writer hopes the result of the research will be useful; not only for the students of Writing IV class, but also for the lectures.
The grammatical error that will be analyze are subject and verb; verb agreement, tense, and form; pronoun agreement, and reference.
1.2 Research Problem
The central problem of this research is: “What grammatical errors are made by the students taking Writing IV class at the English Department Academic Year 2009 / 2010?”
1.3 Objective Of The Study
Based on the problem above, this research intends to find out the grammatical errors which are made by students of Writing IV class at the English Department in their writing of argumentative essay academic year test 2009 / 2010.
1.4 Significances Of The Study
This research has significances as follows:
1. To help teachers of the English Department, by giving them an important contribution in the English teaching process which is part of grammar they should pay attention to.
2. To help students, by giving valuable input about errors they encounter and how to overcome them.
3. It hopes that this thesis will help other researchers to do the some related researches in deeper, further and better techniques.
The scope of this study is the grammatical errors made by students taking Writing IV class at the English department in their three assignments of argumentative essay Academic Year 2009 / 2010.
The errors which the researcher will analyze are only the errors which include in the following three categories of problem areas. Those are:
1. Subject and verb
e.g. There is some glasses on the table
2. Verb agreement, tense, and form
e.g. I will coming soon.
3. Pronoun, agreement, and reference
e.g. Julie likes the flower. He will buy it.
1.6 Definitions of Key Term
• Error is a part of conversation or a composition that deviates from some selected norm of mature language performance.
• Error analysis is identifying, clasifying errors of a foreign language and giving solution.
• Grammatical errors are errors which happen in writing
• Students of English Department Academic year 2007 regular A are the students who were registered in 2007 in English Department and particularly taking the course of Writing IV in their fifth semester.
• English Department is one of the departments in the Faculty of Teachers Training and Education of the Lambung Mangkurat University, Banjarmasin which is located on Jl. H. Hasan Bashri Kayu Tangi Banjarmasin.
REVIEW OF LTERATURE
2.1 The Nature of writing
2.1.1 Definition of writing
According to Cohen and Riel in Yulianti’s thesis (1989), writing as a communicative act, a way of sharing observations, information, thought, or ideas with others. Meanwhile, Bryne in Yulianti’s thesis (1979) defined writing is transforming our thoughts into language. In other words, writing is transforming our thoughts into language. In other words, writing can be defined as a way of communication by transforming observations, information, thought, or ideas into language, so it can be shared with others. Also, Bryne (1979) added that it is neither easy nor spontaneous; it requires conscious mental effort. Writing is not only just transforming our thought or idea in written form but also it relays to the process of monitoring any single words or features that we have written and the process of rereading and revising our writing.
Voss and Keene (1992:2-3) write why we should bother with writing and purposes for writing as follows:
1. writng is a way of thinking and learning. Writing gives unique opportunities of explore ideas and enquire information. By writing, we come to know subjects well and make them our own.
2. Writing is a way of discovering. The act of writing allows us to make unexpected connections among ideas and language
3. Writing create reading. Writing create permanent, visible record of our ideas for others to read and ponder. Writing is powerfull means of communication for reading information and shapes human thought.
4. Writing ability is needed by educated people. Our skill writing is often considered to reflect our level of education.
Purpose for writing:
- To express yourself
- To provide information for your reader
- To persuade your reader
- To create a literary work
In Wikipedia’s website, it is stated that according to William Caslon, writing may refer to two activities:
1. The inscribing of characters on a medium, with the intention of forming words and other constructs that represent language or record information.
2. The creation of material to be conveyed trough written language (there are some expectation; for example, the use of a type writer to recard language is generally called typing, rather than writing).
Therefore, there are some writing components that should be considered by a writer before he begins to write because without considering the components we will not produce a good writing.
According to Raimes (1983), there are eight writing components that should be considered by a writer in order to produce good writing. The components are ;
1. Grammar : rules of verbs, agreement, pronouns.
2. Mechanics : handwriting, spelling, punctuation.
3. Organization : paragraphs, topics, and supports,
cohesion and unity.
4. word choice : vocabulary and idiom.
5. Purpose : reason for writing.
6. Audience : reader(s).
7. The writer’s purposes : getting ideas, getting started, writing
8. Content : relevance, clarity, originality, logic.
In order to get good result of writing, the writer should consider them in writing a paragraph or an essay.
Definition of a Paragraph and an Essay
A paragraph is a basic unit of organization in writing in which a group of related sentences develops one main idea. It can consist of one sentence or as long as ten sentences. However the number of sentences is unimportant but it should be long enough to develop the main idea clearly (Oshima and Ann Houge. 1999:16). A paragraph consists of several related sentences that develop one unit of content. A paragraph may stand alone as a brief work, but usually it functions as a part of a longer piece of writing (Dornan and Dawe. 1987:244).
A paragraph consists of one topic sentence and some support sentences. Some paragraph can create an essay, because an essay consists of some general statement and a thesis statement. Also, there is a concluding paragraph which concludes the main points in the body of the essay.
An essay is a piece of writing several paragraphs. It consists more than main idea, so it needs more than one paragraph to cover the ideas (Oshima and Houge. 1999:100)
In this research, the writer will analyze essays writing of students for their three writing assignments in argumentative essay academic year 2009 / 2010. They are required to write argumentative essays with the topics that have been prepared by the lectures. They developed the topics become essays writing.
2.2 The Nature of Error
2.2.1 Definition of Error
An error is different from mistake, so we have to be careful to differentiate. According to Yulianti (2007: 9):
- A mistake is a performance error, which is either a random guess or a ‘slip’, i.e. a failure to utilize a known system correctly.
- An error is a noticeable deviation from the adult grammar of a native speaker, reflecting the inter language competence of the learner.
She also clearly differentiated a mistake from an error. She stated:
- A mistake is a slip that a learner can self-correct.
- An error is what a learner can not self-correct.
From those definitions above, the writer concludes that a mistake is just a slip that the learner forgets the right form. While, an error is a deviation which is made by the learner because he does not know the rule and he /she will make it repetitively.
The Sources of Error Occurrence
The sources of error occurrence according to Ancker (2000: 1):
(1) Interference from the native language
The learner may assume that the target language and his native language are similar. Then, he will over generalize the rules of his native language and the target language.
(2) An incomplete knowledge of the target language
Because of the incomplete knowledge, the learner may make guesses. When he has something that he doesn’t know, he may guess what it should be there. Lengo (1995:1) added that foreign language learners commit errors largely because of the paucity of their knowledge of the target language whereas deviant forms produced by native speakers are dismissed as slips of the tongue or slips of the pen.
(3) The complexity of the target language
Certain aspects in English are difficult for some learners, it may be caused the rules of their native language are quite different from English and even more complex than their native language.
2.2.3 The Benefits of Analyzing Errors
Errors are normal and unavoidable during the learning process as Richard (1974: 95) mentioned that no one could learn without making errors. Meanwhile, Lengo (2003: 1) mentioned that errors are believed to be an indicator of the learners’ stages in their target language development. So, it is important to analyze the errors because there are many benefits in analyzing the errors, such as:
(1) a device which the learner uses in order to learn ( Selinker in Soesanti’s thesis, 1992: 150 )
(2) to fully grasp and understand the nature of the errors made, and
(3) instead of just being able to explain the rules and correct the errors ( Mei Lin Ho, 2003 : 1 ).
Grammar can be defined as a set of shared assumptions about how language works (Yulianti 2007:11). The assessment whether the learners have mastered some grammatical points should not be based on their ability to state the rules of grammar, but on their ability to use the grammatical points to share their ideas, emotions, feelings, or observations with other people. Especially in the context of the teaching English in Indonesia, the teaching of grammar should be integrated in the development of the four language skills.
Knowing about how grammar works is to understand more about how grammar is used and misused (Yulianti.2007:12). It means that there is a possibility of error occurrence in students learning. In this research, the term of error in grammar will be called a grammatical error. The writer has chosen only three catagories or problem areas in grammatical errors, they are:
1. Subject and verb
In a sentence, there are at least one subject and one verb. The subject may be a noun, a pronoun, and the predicate may be a verb or to be.
Some types of errors that might appear in this category are:
a. Subject missing
e.g., From the text above, can be concluded that book is important.
It should be: from the text above, it can conclude that book is important
b. Simple predicate missing be
e.g., Water very important for human being.
It should be: Water is very important for human being.
c. Wrong simple predicate missing be
e.g., There are student in the library.
It should be: There is student in the library.
d. Superfluous be
e.g. John and Taylor are do their homework.
It should be: John and Taylor do their homework.
2. Verb agreement, tense, and form.
Every sentence has at least one verb. It indicates number of the subject, the tense, etc wherever it stands in a sentence.
a. Misinformation of passive form
e.g., Andi was borrow it two days ago.
It shoul be: Andi was borrowed it two days ago
b. Passive order, but active form
e.g., The wedding will held tomorrow.
It should be: the wedding will be hold tomorrow
c. Active order, but passive form
e.g., The police is caught by the thief.
It should be: the police caught the thief
d. Misinformation of the next verbal word
e.g., We will coming soon
it should be: we will come soon.
e. The verb comes after the subject
e.g., Jane look at herself in a mirror.
It should be : jane looks at herself in a mirror
f. A form of have/ has
e.g., She have a book.
It should be: she has a book
g. A form of do / does
e.g., Andi do not know the rules
it should be: andi doesn’t know the rules.
3. Pronoun form, agreement, reference
Pronoun is a word that used to replace noun in a sentence or a paragraph. So, there is no repetition for the noun that may bore the audience, that is, the reader or the listener.
The example of the error that might appear in this area is:
e.g., He borrows the books. It will be returned soon.
It should be: he borrows teh books. They will be returned
METHODOLOGY OF RESEARCH
3.1 The Research Method
This research uses a descriptive method to describe the grammatical errors in students’ writing for the final test made by the students taking Writing IV class at English Department academic year 2009 / 2010.
3.2 The Research Variable
The variable of this research is the grammatical errors which occur in the students’ writing for assignments of argumentative essay.
3.3 Data Sources
The population of this research is the students of Regular A who take Writing IV class at English Department academic year 2009 / 2010. The total numbers of the student are about 30 students.
According to Suharsimi Arikunto (2002:1200), untuk sekedar ancer-ancer maka apabila subjeknya kurang dari 100, maka lebih baik dianbil semua. Therefore, from the 30 students of writing IV class, the writer takes the entire student as samples.
3.4 Technique of Data Collection
The data which is used in this research is from the students’ writings of all English Department students taking Writing IV class for three assignments of argumentative essay academic year 2009 / 2010. In order to collect the data, the writer asks the lecturers of Writing IV class for their permission. Then, the writer borrows them to make the copies.
3.5 Technique of Data Analysis
The technique which is used in analyzing the data is qualitative. The data will be classified into three categories of problem areas: subject and verb; verb agreement, tense, and form; pronoun form, agreement, and reference. In which the first category is divided into four types of errors: surrogate subject missing; simple predicate missing; wrong simple predicate missing; superfluous. The second is divided into fives types of errors: misinformation of passive form; passive order but active form, active order but passive form; misinformation of verb after modal; the verb comes after the subject; a from of have / has; a form of do / does. And the third is only one type of errors: wrong pronouns.
3.6 Method of Drawing Conclusion
The writer uses inductive method in making final conclusion. The conclusion is from the data analysis as the result of the research and the answer of the problem.
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17 | Topics covered: Exceptions to Lewis structure rules; Ionic bonds
Instructor: Catherine Drennan, Elizabeth Vogel Taylor
Lecture Notes (PDF - 1.1MB)
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PROFESSOR: OK, let's get started here. Go ahead and take 10 more seconds on the clicker question, which probably looks all too familiar at this point, if you went to recitation yesterday. All right, and let's see how we do here.
OK. So, let's talk about this for one second. So what we're asking here, if we can settle down and listen up, is which equations can be used if we're talking about converting wavelength to energy for an electron. Remember, the key word here is electron. This might look familiar to the first part of problem one on the exam, and problem one on the exam is what tended to be the huge problem on the exam. I think over 2/3 of you decided on the exam to use this first equation, e equals h c over wavelength.
So I just want to reiterate one more time, why can we not use this equation if we're talking about an electron? C. OK, good, good, I'm hearing it. So the answer is c. What you need to do is you need to ask yourself if you're trying to convert from wavelength to energy for an electron, and you are tempted, because we are all tempted to use this equation, and if you were tempted, say, does an electron travel at the speed of light? And if the answer is no, an electron does not travel at the speed of light, light travels at the speed of light, then you want to stay away from using this equation. And I know how tempting it is to do that, but we have other equations we can use -- the DeBroglie wavelength, and this is just a combination of energy equals 1/2 m v squared, and the definition of momentum, so we can combine those things to get it.
You might be wondering why I'm telling you this now, you've already -- if you've lost points on that, lost the points on it, and what I'm saying to you is if there are parts of exam 1 that you did not do well on, you will have a chance to show us again that you now understand that material on the final. One quarter of the final is going to be exam 1 material, and what that means is when we look at your grade at the end of the semester, and we take a look at what you got on exam 1, and you're right at that borderline, and we say well, what happened, did they understand more at the end of the semester, did the concepts kind of solidify over the semester? And if they did and if you showed us that they did, then you're going to get bumped up into that next grade category.
So keep that in mind as you're reviewing the exam, sometimes if things don't go as well as you want them to, the temptation is just to put that exam away forever and ever. But the reality is that new material builds on that material, and specifically exam 1 a, question 1 a that deals with converting wavelength to energy for an electron. I really want you guys know this and to understand it, so I can guarantee you that you will see this on the final. Specifically, question 1, part a. You will see something very, very similar to this on the final. If you are thinking about 1 thing to go back and study on exam 1, 1 a is a really good choice for that. This is important to me, so you're going to see it on the final.
So if you have friends that aren't here, you might want to mention it to them, or maybe not, maybe this is your reward for coming to class, which is fine with me as well.
All right. So I want to talk a little bit about exam 1. I know most you picked up your examine in recitation. If you didn't, any extra exams can be picked up in the Chemistry Education office, that's room 2204.
So, the class average for the exam was a 68%, which is actually a strong, solid average for an exam 1 grade in the fall semester of 511-1. What we typically see is something right in this range, either ranging from the 50's for an exam 1 average to occasionally getting into the 70's, but most commonly what we've seen for exam 1 averages is 60, 61 -- those low 60's. So in many ways, seeing this 68 here, this is a great sign that we are off to a good start for this semester. And I do want to address, because I know many of you, this is only your second exam at MIT, and perhaps you've never gotten an exam back that didn't start with a 90 or start with an 80 in terms of the grades. So one thing you need to keep in mind is don't just look at the number grade. The reason that we give you these letters grade categories is that you can understand what it actually means, what your exam score actually says in terms of how we perceive you as understanding the material.
So, for example, and this is the same categories that were shared in recitation, so I apologize for repeating, but I know sometimes when you get an exam back, no more information comes into your head except obsessing over the exam, so I'm just going to say it one more time, and that is between 88 and 100, so that's 20% of you got an A. This is just absolutely fantastic, you really nailed this very hard material and these hard questions on the exam where you had to both use equations and solve problems, but also understand the concept in order to get yourself started on solving the problem.
The same with the B, the B range was between 69 and 87 -- anywhere in between those ranges, you've got a B, some sort of B on the exam. So again, if you're in the A or the B category here, this is really something to be proud of, you really earned these grades. You know these exams, our 511-1 exams, we're not giving you points here, there are no give me, easy points, you earned every single one of these points. So, A and B here, these are refrigerator-worthy grades, hang those up in your dorm. This is something to feel good about.
All right. So, for those of you that got between a 51 and a 68, this is somewhere in the C range. For some people, they feel comfortable being in the C range, other people really do not like being in this range. We understand that, there is plenty of room up there with the A's and the B's. You are welcome to come up to these higher ranges starting with the next exam. And what I want to tell you if you are in the C range, and this is not a place that you want to be in, anyone that's got below the class average, so below a 68 -- or a 68 or below, is eligible for free tutoring, and I put the website on the front page of your notes. This means you get a one-on-one tutor paid for by the Chemistry Department to help you if it's concepts you're not quite up on, if it's exam strategy that you need to work on more. Whatever it is that you need to work on, we want to help you get there.
So, if you have a grade that you're not happy with, that you're feeling upset or discouraged about, please, I'm happy to talk to all of you about your grades individually. You can come talk to me, bring your exam, and we'll go over what the strategy should be in terms of you succeeding on the next exam. You can do the same thing with all of your TAs are more than happy to meet with each and every one of you. And then in addition to that, we can set you up with a tutor if you are in the C range or below, in terms of this first exam.
All right. So 44 to 50, this is going to be in the D range. And then anything below a 44 is going to be failing on this exam. And also keep in mind, for those of you that are freshman, you need at least a C to pass the class. So, if you did get a D or an F on the first exam, you are going to need to really evaluate why that happened and make some changes, and we're absolutely here to help you do that. So the real key is identifying where the problem is -- is it with understanding the concepts, are you in a study group that's dragging you along but you're not understanding? Do you kind of panic when you get in the exam? There are all sorts of scenarios we can talk about and we want to talk about them with you.
Seriously, even if we had a huge range in this exam from 17 to 100, if you're sitting there and you're the 17, and actually there's more than 1 so don't feel alone, if you're a 17 or you're a 20, it's not time to give up, it's not time to drop the class and say I'm no good at chemistry, I can't do this. You still can, this is your first couple of exams, certainly your first in this class, potentially one of your first at MIT, so there's tons of room to improve from here on out. This is only 100 points out of 750. So, the same thing goes if you did really well, you still have 650 other points that you need to deal with. So, make sure you don't just rest on your high score from this first exam.
So, OK, so that's pretty much what I wanted to say about the exam, and in terms of there's tons of resources if things didn't work out quite as you wanted. If you feel upset in any way, please come and talk to me. We want you to love chemistry and feel good about your ability to do it. Nobody get into MIT by mistake, so you all deserve to be sitting here, and you all can pass this class and do well in it, so we can help you get there no matter what. You all absolutely can do this.
And then one more time, to reiterate, in case anyone missed it, 1 a, make sure you understand that, I feel like that's important. And actually all of 1 -- I really feel like the photoelectric effect is important for understanding all of these energy concepts. So, as you go on in this class, make sure you don't go on before you go back and make sure you understand that problem.
All right, so let's move on to material for exam 2 now, and we're already three lectures into exam 2 material. And I do want to say that in terms of 511-1, what tends to happen is the exam scores go up and up and up, in terms of as we go from exam 1, to exam 2, to exam 3. One of these reasons is we are building on material, the other reason is you'll be shocked at how much better you are at taking an exam just a few weeks from now. So this will be on, starting with the Lewis structures, so go back in your notes -- if this doesn't sound familiar, if you spent too much time -- or not too much time, spent a lot of time studying exam 1 and didn't move on here.
Today we're going to talk about the breakdown of the octet rule. Cases where we don't have eight electrons around our Lewis structures, then we'll move on to talking about ionic bonds. We had already talked about covalent bonds, and then we talked about Lewis structures, which describe the electron configuration in covalent bonds. So now let's think about the other extreme of ionic bonds, and then we'll talk about polar covalent bonds to end, if we get there or will start with that in class on Monday.
Also, public service announcement for all of you, voter registration in Massachusetts, which is where we are, is on Monday, the deadline if you want to register to vote. There's some websites up there that can guide you through registering and also can guide you through, if you need an absentee ballot for your home state. And I actually saw, and I saw a 5.111 student manning, there's some booths around MIT that will register you or get you an absentee ballot. So, the deadline's coming soon, so patriotic duty, I need to remind you of that as your chemistry teacher -- chemistry issues are important in politics as well. So make sure you get registered to vote.
I just remembered one more announcement, too, that I did want to mention, some of you may have friends in 511-2 and have heard their class average for exam 1. And I want to tell you, this happens every year, their average was 15 points higher than our average. Last year, their average was 15 points higher than our average. This is for exam 1. This is what tends to happen to 511-2 grades as the exam goes on. This is what happens to 511-1. You guys are in a good spot. Also, I want to point out that what's not important is just that number grade, but also the letter that goes with it.
So, for example, if you got a 69 in this class on this exam, that's a B minus. If you got a 69 on your exam in 511-2, that's a D, you didn't pass the exam. So keep that in mind when your friend might have gotten a higher number grade than you and you know you understand the similar material just as well. Similarly, an 80 in this class on the exam was a B plus, a very high B. An 80 in that class is going to be a C. So, just don't worry so much about exactly where that average lies, you really want to think about what the letter grade means. OK, I've said enough. I just -- I hate to see people discouraged, and I know that a few people have been feeling discouraged, so that's my long-winded explanation of exam 1 grades.
All right. So, let's move on with life though, so talking about the breakdown of the octet rule. The first example where we're going to see a breakdown is any time we have an odd number of valence electrons. This is probably the easiest to explain and to think about, because if we have an odd number that means that we can't have our octet rule, because our octet rule works by pairing electrons. And if we have an odd number, we automatically have an odd electron out.
So, if we look at an example, the methyl radical, we can first think about how we draw the Lewis structure -- we draw the skeletal structure here. And then what we're going to do is add up our valence electrons -- we have 3 times 1 for the hydrogen atoms, carbon has 4 valence electrons, so we have a total of 7. If we want to fill all of our valence shells in each of these atoms, we're going to need a total of 14 electrons. So, what we see we're left with is that we have 7 bonding electrons. So we can fill in 6 of those straightforward here, because we know that we need to make 3 different bonds. And now we're left over with 1 electron, we can't make a bond.
So, what we'll do is carbon does not have an octet yet. We can't get it one, but we can do the best we can and help it out with adding that extra electron onto the carbon atom, so that at least we're getting as close as possible to filling our octets.
This is what we call a radical species or a free radical. Free radical or radical species is essentially any type of a molecule that has this unpaired electron on one of the atoms. This might look really strange, we're used to seeing octets. But you'll realize, if you calculate the formal charge on this molecule, that it's not the worst situation ever for carbon. At least it's formal charge is zero, even if it doesn't have -- it would rather have an extra bond and have a full octet. But it's not the worst scenario that we can imagine. But still, radicals tend to be incredibly reactive because they do want to fill that octet.
So, what happens when you have a radical is it tends to react with the first thing that it runs into, especially highly reactive radicals that are not stabilized in some other way, which you'll tend to talk about it organic chemistry -- how you can stabilize radicals.
So the term free radical should sound familiar to you, whether you've heard it in chemistry before, or you haven't heard it in chemistry, but maybe have heard it, I don't know, commercials for facial products or other things. People like to talk about free radicals, and they're sort of the hero that gets rid of free radicals, which are antioxidants. So you hear in a lot of different creams or products or vitamins that they have antioxidants in them, which get rid of free radicals. The reason you would want to get rid of free radicals is that free radicals can damage DNA, so they're incredibly reactive. It makes sense that if they hit a strand of DNA, they're going to react with the DNA, you end up breaking the strands of DNA and causing DNA damage.
So, this is actually what happens in aging because we have a lot of free radicals in our body. We can introduce them artificially, for example, cigarette smoke has a lot of really dangerous free radicals that get into the cells in your lungs, which damage your lung DNA, which can cause lung cancer. But also, all of us are living and breathing, which means we're having metabolism go on in our body, which means that as we use oxygen and as we metabolize our food, we are actually producing free radicals as well. So it's kind of a paradox because we need them because they are a natural by-product of these important processes, but then they can go on and damage cells, which is what kind of is causing aging and can lead to cancer.
We have enzymes in our body that repair damage that is done by free radicals, that will put the strands of DNA back together. And we also have antioxidants in our body. So, you might know that, for example, very brightly colored fruit is full of antioxidants, they're full of chemicals that will neutralize free radicals. Lots of vitamins are also antioxidants, so we have vitamin A on the top there and vitamin E.
So, the most common thing we think of when we think of free radicals is very reactive, bad for your body, causes DNA damage. But the reality is that free radicals are also essential for life. So this is kind of interesting to think about. And, for example, certain enzymes or proteins actually use free radicals in order to carry out the reactions that they carry out in your body. So, for example, this is a picture or a snapshot of a protein, this is a crystal structure of ribonucleotide reductase is what it's called. It's an enzyme that catalyzes the reaction of an essential step in both DNA synthesis and also DNA repair, and it requires having radicals within its active site in order to carry out the chemistry.
So, this is kind of a neat paradox, because radicals damage DNA, but in order to repair your DNA, you need certain enzymes, and those enzymes require different types of free radicals. So, free radicals are definitely very interesting, and once we get -- or hopefully you will get into organic chemistry at some point and get to really think about what they do in terms of a radical mechanism.
We can think about radicals that are also more stable, so let's do another example with the molecule nitric acid. So we can again, draw the skeleton here, and just by looking at it we might not know it's a radical, but as we start to count valence electrons, we should be able to figure it out very quickly, because what we have is 11 valence electrons. We need 16 electrons to have full octets. So, we're left with 5 bonding electrons. We put a double bond in between our nitrogen and our oxygen, so what we're left over with is this single bonding electron, and we'll put that on the nitrogen here. And I'll explain why we put it on the nitrogen and not the oxygen in just a minute.
But what we find is then once we fill in the rest of the valence electrons in terms of lone pairs, this is the structure that we get. And if you add up all of the formal charges on the nitrogen and on the oxygen, what you'll see is they're both 0. So if you happen to try drawing this structure and you put the lone pair on oxygen and then you figured out the formal charge and saw that you had a split charge, a plus 1 and a minus 1, the first thing you might want to try is putting it on the other atom, and once you did that you'd see that you had a better structure with no formal charge.
I have to mention what nitric oxide does, because it's a very interesting molecule. Don't get it confused with nitrous oxide, which is happy gas, that's n o 2. This is nitric oxide, and it's actually much more interesting than nitrous oxide. It's a signaling molecule in your body, it's one of the very few signaling molecules that is a gas, and obviously, it's also a radical. What happens with n o is that it's produced in the endothelium of your blood vessels, so the inner lining of your blood vessels, and it signals for smooth muscle that line your blood vessels to relax, which causes vasodilation , and by vasodilation, I just mean a widening of the blood vessels. So, n o signals for your blood vessels to get wider and allow more blood to flow through. And if you think about what consequences this could have, in terms of places where they have high altitude, so they have lower oxygen levels, do you think that they produce more or less and n o their body? More? Yeah, it turns out they do produce more. The reason they produce more is that they want to have more blood flowing through their veins so that they can get more oxygenated blood into different parts of their body.
N o is also a target in the pharmaceutical industry. A very famous one that became famous I guess over 10 years ago now, and this is from a drug that actually targets one of n o's receptors, and this drug has the net effect of vasodilation or widening of blood vessels in a certain area in the body. So this is viagra, some of you may be familiar, I think everyone's heard of viagra. Now you know how viagra works. Viagra breaks down, or it inhibits the breakdown of n o's binding partner in just certain areas, not everywhere in your body. So, in those areas, what happens is you get more n o signaling, you get more vasodilation, you get increased blood flow. So that's a little bit of pharmacology for you here today.
All right, so let's talk about one more example in terms of the breakdown of the octet rule with radicals. Let's think about molecular oxygen. So let's go ahead and quickly draw this Lewis structure. We have o 2. The second thing we need to do is figure out valence electrons. 6 plus 6, so we would expect to see 12. For a complete octet we would need 8 electrons each, so 16. So in terms of bonding electrons, what we have is 4 bonding electrons. So, we can go ahead and fill those in as a double bond between the two oxygens.
So, what we end up having left, and this would be step six then because five was just filling in that, is 12 minus 4, so we have 8 lone pair electrons left. So we can just fill it in to our oxygens like this.
All right, so using everything we've learned about Lewis structures, we here have the structure of molecular oxygen. And I just want to point out for anyone that gets confused, when we talk about oxygen as an atom, that's o, but molecular oxygen is actually o 2, the same for molecular hydrogen, for example.
All right, so let's look at what the actual Lewis structure is for molecular oxygen, and it turns out that actually we don't have a double bond, we have a single bond, and we have two radicals. And any time we have two radicals, we talk about what's called a biradical. And while using this exception to the Lewis structure rule, to the octet rule for odd numbers of valence electrons can clue us into the fact that we have a radical, there's really no way for us to use Lewis structures to predict when we have a biradical, right, because we would just predict that we would get this Lewis structure here.
So, when I first introduced Lewis structures, I said these are great, they're really easy to use and they work about 90% of the time. This falls into that 10% that Lewis structures don't work for us. It turns out, in order to understand that this is the electron configuration for o 2, we need to use something called molecular orbital theory, and just wait till next Wednesday and we will tell you what that is, and we will, in fact, use it for oxygen. But until that point, I'll just tell you that molecular orbital theory takes into account quantum mechanics, which Lewis theory does not. So that's why, in fact, there are those 10% of cases that Lewis structures don't work for.
All right, the second case of exceptions to the octet rule are when we have octet deficient molecules. So basically, this means we're going to have a molecule that's stable, even though it doesn't have a complete octet. And these tend to happen in group 13 molecules, and actually happen almost exclusively in group 13 molecules, specifically with boron and aluminum. So, any time you see a Lewis structure with boron or aluminum, you want to just remember that I should look out to make sure that these might have an incomplete octet, so look out for that when you see those atoms.
So, let's look at b f 3 as our example here. And what we see for b f 3 is the number of valence electrons that we have are 24, because the valence number of electrons for boron is 3, and then 3 times 7 for each fluorine. For total filled octets we need 32, so that means we need 8 bonding electrons. So, let's assign two to each bond here, and then we're going to have two extra bonding electrons, so let's just arbitrarily pick a fluorine to give a double bond to. And then we can fill in the lone pair electrons, we have 16 left over. So thinking about what the formal charge is, if we want to figure out the formal charge for the boron here, what we're talking about is the valence number for boron, which is 3, minus because there are no lone pairs, minus 1/2 of 8 because there are eight shared electrons. We get a formal charge of minus 1.
What is our formal charge since we learned this on Monday for thinking about the double bonded fluorine in boron? So, look at your notes and look at the fluorine that has a double bond with it, and I want you to go ahead and tell me what that formal charge should be.
All right, let's take 10 more seconds on that. OK, so 49%. So, let's go look back at the notes, we'll talk about why about 50% of you are right, and 50% need to review, which I totally understand you haven't had time to do yet, your formal charge rules from Monday's class, there were other things going on. But let's talk about how we figure out formal charge. Formal charge is just the number of valence electrons you have. So fluorine has 7. You should be able to look at a periodic table and see that fluorine has seven. What we subtract from that is the number of lone pair electrons, and there are four lone pair electrons on this double bonded fluorine, so it's minus 4. Then we subtract 1/2 of the shared electrons. Well we have a double bond with boron here, so we have a total of 4 shared electrons. And when we do the subtraction here, what we end up with is a formal charge plus 1 on the double bonded fluorine.
Without even doing a calculation, what do you think that the formal charge should be on you single bonded fluorines? Good. OK, it should be and it is 0. The reason it's zero in terms of calculating it is 7 minus 6 lone pair electrons minus 1/2 half of 2 shared electrons is 0. The reason that you all told me, I think, and I hope, is that you know that the formal charge on individual atoms has to equal the total charge on the molecule. So if we already have a minus 1 and a plus 1, and we know we have no charge in the molecule, and we only have one type of atom left to talk about, that formal charge had better be 0.
OK. So this looks pretty good in terms of a Lewis structure, we figured out our formal charges. These also look pretty good, too, we don't have too much charge separation. But what actually it turns out is that if you experimentally look at what type of bonds you have, it turns out that all three of the b f bonds are equal in length, and they all have a length that would correspond to a single bond. So, experimentally, we know we have to throw out this Lewis structure here, we have some more information, let's think about how this could happen.
So this could happen, for example, is if we take this two of the electrons that are in the b f double bond and we put it right on to the fluorine here, so now we have all single bonds. And let's think about what the formal charge situation would be in this case here. What happens here is now we would have a formal charge of on the boron, we'd have a formal charge of on all of the fluorine molecules as well. So, it turns out that actually looking at formal charge, even though the first case didn't look too bad, this case actually looks a lot better. We have absolutely no formal charge separation whatsoever. It turns out again, boron and aluminum, those are the two that you want to look out for. They can be perfectly happy without a full octet, they're perfectly happy with 6 instead of 8 in terms of electrons in their valence shell. So that is our exception the number two.
We have one more exception and this is a valence shell expansion, and this can be the hardest to look out for, students tend to forget to look for this one, but it's very important as well, because there are a lot of structures that are affected for this . And this is only applicable if we're talking about a central atom that has an n value or a principle quantum number that's equal to or greater than three. What happens when we have n that's equal to or greater to three, is that now, in addition to s orbitals and p orbitals, what else do we have available to us? D orbitals, great. So what we see is we have some empty d orbitals, which means that we can have more than eight electrons that fit around that central atom.
If you're looking to see if this is going to happen, do you think this would happen with a large or small central atom? So think of it in terms of just fitting. We've got to fit more than 8 electrons around here. Yeah, so it's going to be, we need to have a large central atom in order for this to take place. Literally, we just need to fit everything around is probably the easiest way to think about it. And what happens is it also tends to have small atoms that it's bonded to. Again, just think of it in terms of all fitting in there.
So, let's take an example p c l 5. The first example is the more straightforward example, because let's start to draw the Lewis structure, and what we see is that phosphorous has five chlorines around it. So we already know if we want to form five bonds we've broken our octet rule. But let's go through and figure this out and see how that happens.
What we know is we need 40 valence electrons, we have those -- 5 from the phosphorous, and we have 7 from each of the chlorine atoms. If we were to fill out all of those octets, that would be 48 electrons. So what we end up with when we do our Lewis structure calculation is that we only have 8 bonding electrons available to us. So we can fill those in between the phosphorous and the chlorine, those 8 bonding electrons.
So, this is obviously a problem. To make 5 p c l bonds we need 10 shared electrons, and we know that that's the situation because it's called p c l 5 and not p c l 4, so we can go right ahead and add in that extra electron pair. So we've used up 10 for bonding, so that means what we have left is 30 lone pair electrons, and I would not recommend filling all of these in your notes right now, you can go back and do that, but just know the rest end up filling up the octets for all of the chlorines.
So, in this first case where you actually need to make more than for bonds, you will immediately know you need to use this exception to the Lewis structure octet rule, but sometimes it won't be as obvious. So, let's look at c r o 4, the 2 minus version here, so a chromate ion, and if we draw the skeletal structure, we have four things that the chromate needs to bond to.
So, let's do the Lewis structure again. When we figure out the valence electrons, we have total, we have 6 from the chromium, we have 6 from each of the different oxygens, and where did this 2 come from? Yup, the negative charge. So, remember, we have 2 extra electrons hanging out in our molecule, so we need to include those. We have a total of 32. 40 are needed to fill up octets. So again, we have 8 bonding electrons available, so we can go ahead and fill these in between each of the bonds. What happens is that we then have 24 lone pair electrons left, and we can fill those in like this. And the problem comes now when we figure out the formal charge.
So, when we do that what we find is that the chromium has a formal charge of plus 1, and that each of the oxygens has a total charge of minus 1. So we actually have a bit of charge separation here. Without even doing a calculation, what is the total charge of these that are added up? OK, it's minus 2, that's right. We know that the total charge of each of the formal charges has to add up to minus 2, because that's the charge in our molecule. We can also just calculate it -- the chromate gives us a plus 2, then we have 4 times minus 1 for each of the oxygens, so we have a minus 2.
So, we have some charge separation here, and in some cases, if we're not at n equals 3 or higher, there's really nothing we can do about it, this would be the best structure we can do. But since we have these d orbitals available, we can use them, and it turns out that experimentally this is what's found, that the length and the strength are not single bonds, but they're actually something between a single bond and a double bond.
So how do we get a 1 and 1/2 bond, for example, what's the term that let's us do that? Resonance. That's right. So that's exactly what's happening here. So, if we went ahead and drew this structure here where we have now two double bonds and two single bonds, that would be in resonance with another structure where we have two double bonds instead to these two oxygens, and now, single bonds to these two oxygens. We can actually also have several other resonance structures as well. Remember, the definition of a resonance structure is where all the atoms stay the same, but what we can do is move around the electrons -- we're moving around those extra two electrons that can be in double bonds.
So, why don't you tell me how many other resonance structures you would expect to see for this chromate ion? All right, let's take 10 more seconds on this.
All right. This is good. I know this is a real split response, but the right answer is the one that is indicated in the graph here that it's four. This takes a little bit of time to get used to thinking about all the different Lewis structures you can have. So, you guys should all go back home if you can't see it immediately right now and try drawing out those four other Lewis structures, for chromate, there are four others. You'll probably get a chance to literally do this example in recitation where you draw out all four, but it's even better to make sure you understand it before you get to that point. So, we can go back to the class notes.
So it turns out there's four other Lewis structures, so basically just think about all the other different combinations where you can have single and double bonds, and when you draw those out, you end up with four. So, for every single one of these Lewis structures, we could figure out what the formal charges are, and what we would find is that it's on the chromium, it's for the double bonded oxygens, and it's going to be negative 1 for the single bonded oxygens.
So, what you can see is that in this situation, we end up having less formal charge separation, and that's what we're looking for, that's the more stable structure. So any time you can have an expanded octet -- an expanded valence shell, where you have n is equal to or greater than 3, and by expanding and adding more electrons into that valence shell, you lower the charge separation, you want to do that.
I also want to point out, I basically said there's 6 different ways we can draw this in terms of drawing all the resonance structures. You might be wondering if you have to figure out the formal charge for each structure individually, and the answer is no, you can pick any single structure and the formal charges will work out the same. So, for example, if you pick this structure and your friend picks this structure, you'll both get the right answer that there's just the negative 1 on the oxygens and no other formal charges in the molecule.
All right. So those are the end of our exceptions to the octet rule for Lewis structures, that's everything we're going to say about Lewis structures. And remember, that when we talk about Lewis structures, what they tell us is the electron configuration in covalent bonds, so that valence shell electron configuration. So we talked a lot about covalent bonds before we got into Lewis structures, and then how to represent covalent bonds by Lewis structures.
So now I'll say a little bit about ionic bonds, which are the other extreme, and when you have an ionic bond, what you have now is a complete transfer of either one or many electrons between two atoms. So the key word for covalent bond was electron sharing, the key word for ionic bonds is electron transfer. And the bonding between the two atoms ends up resulting from an attraction that we're very familiar with, which is the Coulomb or the electrostatic attraction between the negatively charged and the positively charged ions.
So let's take an example. The easiest one to think about is where we have a negative 1 and a positive 1 ion. So this is salt, n a c l -- actually lots of things are call salt, but this is what we think of a table salt. So, let's think about what we have to do if we want the form sodium chloride from the neutral sodium and chlorine atoms. So, the first thing that we're going to need to do is we need to convert sodium into sodium plus.
What does this process look like to you? Is this one of those periodic trends, perhaps? Can anyone name what we're looking at here? Exactly, ionization energy. So, if we're going to talk about the energy difference here, what we're going to be talking about is the ionization energy, or the energy it takes to rip off an electron from sodium in order to form the sodium plus ion. So, we can just put right here, that's 494 kilojoules per mole.
The next thing that we want to look at is chlorine, so in terms of chlorine we need to go to chlorine minus, so we actually need to add an electron. This is actually the reverse of one of the periodic trends we talked about. Which trend is that this is the reverse of? Electron affinity, right. Because if we go backwards we're saying how badly does chlorine want to grab an electron? Chlorine wants to do this very badly, and it turns out the electron affinity for chlorine is huge, it's 349 kilojoules per mole, but remember, we're going in reverse, so we need to talk about it as negative 349 kilojoules per mole.
So if we talk about the sum of what's happening here, what we need to do is think about going from the neutrals to the ions, so we can just add those two energies together, and what we end up with is plus 145 kilojoules per mole, in order to go from neutral sodium in chlorine to the ions.
So, the problem here is that we have to actually put energy into our system, so this doesn't seem favorable, right. What's favorable is when we actually get energy out and our energy gets lower, but what we're saying here is that we actually need to put in energy. So another way to say this is this process actually requires energy. It does not emit energy, it does not give off excess energy, it requires energy.
So, we need to think about how can we solve this problem in terms of thinking about ionic bonds, and the answer is Coulomb attraction. So there's one more force that we need to talk about, and that is when we talk about the attraction between the negatively and the positively charged ions, such that we form sodium chloride. So this process here has a delta energy, a change in energy of negative 589 kilojoules per mole. So that's huge, we're giving off a lot of energy by this attraction. So if we add up the net energy for all of this process, all we need to do is add negative 589 to plus 145. So what we end up getting is the net energy change is going to be negative 444 kilojoules per mole, so you can see that, in fact, it is very favorable for neutral sodium and neutral chloride to form sodium chloride in an ionic bond. And the net increase then, is a decrease in energy.
So, I just gave you the number in terms of what that Coulomb potential would be in attraction, but we can I easily calculate it as well using this equation here where the energy is equal to the charge on each of the ions, and this is just multiplied by the value of charge for an electron divided by 4 pi epsilon nought times r, are r is just the distance in terms of the bond length we could talk about.
So, let's calculate and make sure that I didn't tell you a false number here. Let's say we do the calculation with the bond length that we've looked up, which is 2 . 3 6 angstroms for the bond length between sodium and chloride. So we should be able to figure out the Coulombic attraction for this.
So, if we talk about the energy of attraction, we need to multiply plus 1, that's the charge on the sodium, times minus 1, the charge on the chlorine, times the charge in an electron, 1 . 6 2 times 10 the negative 19 Coulombs, and that's all divided by 4 pi, and then I've written out epsilon nought in your notes, so I won't write it on the board. And then r, so r is going to be 2 . 3 6 and times -- what is angstrom, everyone? Yup, 10 to the negative 10. So 10 to the negative 10 meters. So, if we do this calculation here, what we end up with is negative 9.774 times 10 to the negative 19 joules.
So that's what we have in terms of our energy. That does not look the same as what we saw -- yup, do you have a question?
PROFESSOR: OK. Luckily, although, I did not write it in my own notes, I did it when I put in my calculator, thank you. So you need to square this value here and then you should get this value right here, negative 9.77.
All right, so what we need to do though is convert from joules into kilojoules per mole, because that's what we were using. So if we multiply that number there by kilojoules per mole -- or excuse me, first kilojoules per joule, so we have 1,000 joules in every kilojoule. And then we multiply that by Avagadro's number, 6.022 times 10 to the 23 per mole. What we end up with is negative 589 kilojoules per mole. So this is that same Coulombic attraction that we saw in the first place.
So, notice that you will naturally get out a negative charge here, remember negative means an attractive force in this case, because you have the plus and the minus 1 in here. So we should be able to easily do that calculation, and what we end up getting matches up with what I just told you, luckily, and thank you for catching the square, that's an important part in getting the right answer. So, experimentally then, what we find is that the change in energy for this reaction is negative 444 kilojoules per mole.
If we look experimentally what we see, it's actually a little bit different, it's negative 411 kilojoules per mole. So, in terms of this class, this is the method that we're going to use, and we're going to say this gets us close enough such that we can make comparisons and have a meaningful conversations about different types of ionic bonds and the attraction between them.
But let's think about where this discrepancy comes from, and before I do that I want to point out, one term we use a lot is change in energy for a reaction where, for example, you break a bond. Remember that the negative of the change in energy is what's called delta e sub d. We first saw this when we first introduced the idea of covalent bonds. Do you remember what this term here means, delta e sub d? A little bit and some no's, which this was pre-exam, I understand, you still need to review those notes, it's dissociation energy. So you get a negative energy out by breaking the bond. The dissociation energy means how much energy that bond is worth in terms of strength, so it's the opposite of the energy you get out of breaking the bond -- or excuse me, the energy that you get out of forming the bond. It's the amount of energy you need to put in to break the bond is dissociation energy. It takes this much energy to dissociate your bond, excuse me.
All right. So, let's take a look here at our predictions, so I just put them both ways so we don't get confused. The dissociation energy is 444. The change in energy for forming the bond is negative 444. We made the following approximations, which explain why, in fact, we got a different experimental energy, if we look at that.
The first thing is that we ignored any repulsive interactions. If you think about salt, it's not just two single atoms that you are talking about. It's actually in a whole network or whole lattice of other molecules, so you actually have some other chlorines around that are going to be having repulsive interactions with our chlorine that we're talking about. We're going to ignore those, make the approximation that those don't matter, at this point, in these calculations. And the result for that is that we end up with a larger dissociation energy than the experimental value. That's because the bond is going to be a little bit more broken than it was in our calculation, because we do have these repulsive interactions.
The other thing that we did is that we treated both sodium and the chlorine as point charges. And this is what actually allowed us to make this calculation and calculate the Coulomb potential so easily, we just treated them as if they're point charges. We're ignoring quantum mechanics in this -- this is sort of the class where we ignore quantum mechanics, we ignored it for Lewis structures, we're ignoring it here. We will be back to paying a lot of attention to quantum mechanics in lecture 14 when we talk about MO theory, but for now, these are approximations, these are models where we don't take it into consideration. And I think you'll agree that we come reasonably close such that we'll be able to make comparisons between different kinds of ionic bonds.
All right. So, the last thing I want to introduce today is talking about polar covalent bonds. We've now covered the two extremes. One extreme is complete total electron sharing -- if we have a perfectly covalent bond, we have perfect sharing. The other is electron transfer in terms of ionic bonds. So when we talk about a polar covalent bond, what we're now talking about is an unequal sharing of electrons between two atoms.
So, this is essentially something we've seen before, we just never formally talked about what we would call it. This is any time you have a bond forming between two non-metals that have different electronegativities, so, for example, hydrogen choride, h c l. The electronegativity for hydrogen is 2.2, for chlorine it's 3.2. And in general, what we say is we consider a difference in terms of a first approximation if the difference in electronegativity is more than 0. 5, so this is on the Pauling electronegativity scale. So what we end up having is we sort of have a kind of, and what we call it is a partial negative charge on the chlorine, and a partial positive charge in the hydrogen. The reason we have that is because the chlorine's more electronegative, it wants to pull more of that shared electron density to itself. If it has more electron density, it's going to have a little bit of a negative charge and the hydrogen's going to be left with a little bit of a positive charge.
So, we can compare this, for example to, molecular hydrogen where they're going to have that complete sharing, so there's not going to be a delta plus or a delta minus, delta is going to be equal to zero on each of the atoms. They are completely sharing their electrons.
And we can also explain this in another way by talking about a dipole moment where we have a charged distribution that results in this dipole, this electric dipole. And we talk about this using the term mu, which is a measurement of what the dipole is. A dipole is always written in terms of writing an arrow from the positive charge to the negative charge. In chemistry, we are always incredibly interested in what the electrons are doing, so we tend to pay attention to them in terms of arrows. Oh, the electrons are going over to the chlorine, so we're going to draw our arrow toward the chlorine atom.
So, we measure this here, so mu is equal to q times r, the distance between the two. And q, that charge is just equal to the partial negative or the partial positive times the charge on the electron. So this is measured in Coulomb meters, you won't ever see a measurement of electronegativity in Coulomb meters -- we tend to talk about it in terms of debye or 1 d, or sometimes there's no units at all, so the d is just assumed, and it's because 1 debye is just equal to this very tiny number of Coulomb meters and it's a lot easier to work with debye's here.
So, when we talk about polar molecules, we can actually extend our idea of talking about polar bonds to talking about polar molecules. So, actually let's start with that on Monday. So everyone have a great weekend. | http://ocw.mit.edu/courses/chemistry/5-111-principles-of-chemical-science-fall-2008/video-lectures/lecture-12/ | 13 |
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