BioMath. Evolution By Substitution: Amino Acid Changes Over Time. Student Edition

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1 BioMath volution By ubstitution: mino cid Changes Over Time tudent dition

2 unded by the ational cience oundation, Proposal o. I This material was prepared with the support of the ational cience oundation. owever, any opinions, findings, conclusions, and/or recommendations herein are those of the authors and do not necessarily reflect the views of the. t the time of publishing, all included URs were checked and active. We make every effort to make sure all links stay active, but we cannot make any guaranties that they will remain so. If you find a UR that is inactive, please inform us at info@comap.com. IMC Published by COMP, Inc. in conjunction with IMC, Rutgers University. 205 COMP, Inc. Printed in the U... COMP, Inc. 75 Middlesex Turnpike, uite 3B Bedford, M IB: ront Cover Photograph: P U BRZ BORTORY, PTO-BIOOY B. I RP ITT This work is in the public domain in the United tates because it is a work prepared by an officer or employee of the United tates overnment as part of that person s official duties.

3 volution By ubstitution: mino cid Changes Over Time, or deoxyribonucleic acid, carries the code for life and that code directs the making of proteins that will carry out the organism s functions. Proteins are made from twenty different amino acids and the number and order of those amino acids will determine the properties and function of the protein. ny alterations in the sequence of amino acids may have an effect on the function of the protein. The protein may not function as well, may lose all function, or may possibly function better. It is also possible that the substitution may not affect the function of the protein at all. Mathematical analysis of similar proteins in different organisms based on the sequence of amino acids may give insight into their possible evolutionary history and perhaps even that of the organisms that contain those proteins. uch analysis may also lead to explanations of the mechanisms of evolution, which resulted in the natural selection of these proteins. Unit oals and Objectives oal: tudents will experience the excitement of modern biology from both the biological and mathematical point of view. Objectives: Relate changes and resulting amino acid substitutions to evolution. evelop a deeper understanding of evolution through the study of amino acid substitution and matrix multiplication. oal: tudents will explore the connections between the mathematical and biological sciences. Objectives: Identify the probability for single events. Relate the use of a matrix to the probability for compound events and for events repeated over time. emonstrate a proficiency in multiplying two matrices together and raising a square matrix to a power. Understand the relationship between powers of a matrix and future evolutionary states. oal: tudents will experience how mathematical modeling simulates theoretically behavior of a proposed system. Objectives: Identify state diagrams and their properties. Construct a state diagram to describe changes in a system. volution By ubstitution tudent

4 esson volutionary Relationships Man has grouped organisms based on physical similarities for hundreds of years. cientists have used these similarities to determine evolutionary relationships among organisms. or example, a mouse and a rat have many characteristics in common, more than a mouse shares with a chicken. Based on these observations, a mouse and a rat are more closely related to each other than they are to a chicken; therefore, they share a more recent common ancestor. gain based on similarities, a mouse is more closely related to a chicken than it is to a fish. s more and more information is gathered, a tree can be drawn to show these relationships, as shown in igure.. igure.: Portion of an evolutionary tree. The study of evolutionary relationships raises many questions. iven two organisms, what is their evolutionary relationship? What was their common ancestor like? ow long ago did they diverge from this common ancestor? Biology Background In all living organisms,, or deoxyribonucleic acid, carries the code for life. The code determines the proteins that an organism s cells will make, and proteins carry out the organism s functions. Through a series of complex processes, a segment of called a gene may be read and the message in the gene may be used to build a protein from building blocks called amino acids. There are a total of twenty amino acids used to build proteins in living things (see Table.). protein is a chain of amino acids whose properties are determined by its particular amino acid sequence. In summary, it is differences in that result in different amino acid sequences. volution By ubstitution tudent 2

5 Changing one amino acid in the sequence can result in a protein that does not function as well or may completely destroy the functioning of the protein altogether. Occasionally the change produces a protein that functions better than its predecessor and improves the fitness of the organism. In such cases, natural selection will result in improved survival rates for the organisms with this protein. Ultimately, nature will determine which proteins function best given a particular environment. Molecular biology today offers new ways to compare organisms. Proteins may be sequenced and compared giving a more detailed comparison of organisms. Today s scientist can look for evolutionary relationships based on the sequence of amino acids in proteins rather than looking at bone structure or type of teeth. This unit uses mathematics to examine changes over time in the amino acids that make up proteins. Making the BioMath Connection There are twenty amino acids that may be coded for in. mino acids are all alike in that they have 3 common parts: an amino group (2), a carboxyl group (COO), and an R group all attached to a central carbon. igure.2 shows the characteristic makeup of an amino acid. igure.2: Characteristics of an amino acid. The mino cid table on the next page shows each of the amino acids with its unique R group. R groups have different chemical properties. or example, an R group may make an amino acid polar (charged positive or negative) or nonpolar (uncharged), hydrophobic (repelled by water) or hydrophilic (attracted to water). These differences impact the overall functioning of the protein and how it will fold when made from a long string of amino acids with different properties. ome substitutions in amino acids will have greater effects than others. change from a polar amino acid to another polar amino acid will not affect the protein as much as a change to a nonpolar amino acid. If the change is too great and the protein does not function, then nature would select against that change. One can begin to see how the probability of some selected changes may be greater than other selected changes. The table also shows the common abbreviations used for each amino acid. or example, alanine can be abbreviated la or represented by the capital letter. rginine is volution By ubstitution tudent 3

6 abbreviated rg, but is represented by the letter R since has already been used. This use of letters is universal and recognized by scientists. Table.: mino cids volution By ubstitution tudent 4

7 When two amino acid sequences are compared, it is possible to consider how recently they shared a common ancestor. Mathematically, two sequences can be aligned to determine their evolutionary relationship. mino acids are denoted by a capital letter. The two amino acid sequences below illustrate an alignment between two growth hormone proteins. The top is the partial protein from a domesticated cat and the bottom is from a domesticated dog. MPRCPWPQTPMPRQQTYR MPRCPWPQPMPRQQTYR The alignment below is for the same partial protein from a domesticated chicken and a domesticated dog. s might be expected a dog and a cat share more common amino acids than the dog does with a chicken. If scores were being assigned to show their commonalities, then the dog and cat protein alignment would receive a higher score. MPWP-ITPQTPMPRQTYR MPRCPWPQ-PMPRQQTYR ifferences in like (homologous) proteins are the result of mutations in the of a common, but perhaps unknown, ancestor. s seen in the first alignment, the amino acid in the 4 th position was replaced by another amino acid, but the substitution allowed for functionality of the protein since cats and dogs do quite fine with their growth hormones. In this mathematical model of evolution by examination of amino acids, the assumption is made that amino acids change independently of each other. One evolutionary unit (e.u.) is the average amount of time it takes for % of the amino acids to change. uppose that over a period of one e.u., 3 out of every 000 amino acids change into amino acid R. This is denoted as a probability: P( changes into R) = 3/000 =.003. enerally we talk about probabilities of events so if is the event that changes to R then we say that P() =.003. lso, the chance that a certain change takes place is the probability of its occurrence, often stated as a percentage. In this case.003 becomes 0.3%. s stated earlier, substitutions of dissimilar amino acids are less likely to be acceptable. Based on these ideas a 20 x 20 matrix can be assembled based upon the probabilities that the substitutions can occur over time. Margaret ayhoff developed just such a substitution data matrix in 978. volution By ubstitution tudent 5

8 volution By ubstitution tudent 6 rom Original mino cid dapted from Margaret ayhoff s 978 mutation data matrix for PM. (or a given row, the cell entries give all possible probabilities for that amino acid to change, so each row should add up to.) al Tyr Trp Thr er Pro Phe Met ys eu Ile is ly lu ln Cys sp sn rg la from To Replacement mino cid Y W T P M I Q C R into la R rg sn sp C Cys Q ln lu ly is I Ile eu ys M Met Phe P Pro er T Thr.9976 W Trp Y Tyr al Table.2: ubstitution Matrix

9 or this discussion, the assumption is made that the rate of change remains constant, but that is not always the case. ifferent proteins will have different rates of substitutions of their amino acids depending on their function and how harmful a change may be to that function. s a result, one protein may have an e.u. of 5 million years and another may have an e.u. of 50 million years. Table.2 is based upon the average of many different proteins. Pioneer of Bioinformatics r. Margaret Oakley ayhoff ( ) was a pioneer in the use of computers in chemistry and biology, beginning with her Ph thesis project in 948. er work was multi-disciplinary, and she used her knowledge of chemistry, mathematics, biology and computer science to develop an entirely new field. he is credited today as a founder of the field of Bioinformatics. This field is defined as the use of computers in solving information problems in the life sciences, mainly involving the creation of extensive electronic databases on protein sequences and genomes. r. ayhoff was the first woman in the field of Bioinformatics. he was also the first woman to hold office in the Biophysical ociety, serving first as ecretary and later as President. source: Questions for iscussion. Where in each row does the highest probability occur in the substitution matrix (Table.2)? xplain what the information tells us about amino acid evolution. 2. The probabilities of an amino acid not changing are given along the diagonal. a. What amino acid has the greatest chance of not changing after one evolution unit? xplain how your answer is represented in the matrix. b. ypothesize why this amino acid is not likely to be substituted by another amino acid. volution By ubstitution tudent 7

10 CTIITY - Using the ubstitution Matrix Objective: Use the substitution matrix to examine amino acid relationships and determine probabilities of substitutions. Participants: roups of 2-3 students Materials: andout - ubstitution Matrix andout -3 Using the ubstitution Matrix Worksheet. xamine the substitution matrix. a. etermine the most common amino acid substitution. b. What is the probability that this substitution will occur? c. ook at the amino acid table and compare the two amino acids in this substitution. Why is this change so probable? 2. The probability of amino acid changing into is in one e.u. a. Out of 000, what is the number of s expected to change into after one e.u.? b. The probability of changing into is in one e.u. Out of 000, what is the number of s expected to change into after one e.u.? c. Out of 000 amino acids, which are half and half, how many s should be expected after one e.u.? 3. To actually calculate the probability of a certain change in a protein, it is necessary to know the probability of each amino acid changing into each of the other amino acids. There are 400 probabilities! Why are there 400 of them? 4. In one e.u., what is the probability that: a. changes into? b. changes into R? c. M does not change? If and B are disjoint events (cannot happen at the same time), then the Probability of or B is. 5. ssume that in one e.u. amino acid substitutions are disjoint events. What is the probability that: a. changes into or T? b. changes into or R? 6. What is the probability that does change? volution By ubstitution tudent 8

11 7. What is the probability of C changing into? 8. fter one e.u., does R have a higher probability of changing into a Q or a? 9. What amino acids will C least likely change into? Possible mino cid ubstitutions otice the sum of probabilities in each of the first three rows of the substitution matrix equals. hould this property hold for all the rows in the matrix? Why? ll rows should sum up to. ach amino acid can only change to one new amino acid, at the end of one evolution unit, thus the sum of the probabilities cannot exceed. On the other hand, an amino acid will either change into one of the other 9 amino acids or not change and stay the same. This will exhaust all the possibilities, thus the sum is exactly. If you compute all rows you will notice that most of them do not add up to exactly. This is due to the fact that our accuracy is four decimal points.we took the liberty of rounding the first five rows, so that the sum is exactly. Questions for iscussion 3. mino Puzzle. a. ind three examples of pairs of amino acids that have the following property: the probability of the first changing into the second, after one e.u., equals the probability of the second changing into the first after one e.u. b. ypothesize why this is true. 4. Consider the following matrix as a substitution matrix in a world with only five amino acids. escribe the evolution that takes place in this world. into la rg sn sp Cys from R C la rg R sn sp Cys C If and B are independent events then the Probability of and B is:. 5. iven 000 amino acids, can the number of amino acids expected after two evolution units be determined? ssume substitutions in the first and second e.u. are independent events. volution By ubstitution tudent 9

12 Practice. Referring to the substitution matrix, state the following probabilities: a. changing into Q after one e.u. b. changing into after one e.u. c. Q changing to or after one e.u. d. not changing after one e.u. e. Q changing after one e.u. 2. Which is more likely to occur after one e.u.: changing into I or I changing into? xtension is made of four different bases:, T, C, and. Combinations of three of these bases called triplets code for the 20 different amino acids or signal the end of the protein. The following chart that depicts which triplets code for each amino acid. Use this chart and the substitution matrix to answer the following questions. mino cid Triplet mino cid Triplet lanine () C, CT, CC, C eucine () T, C, T,, C, rginine (R) TCT, TCC, C, CT, C, CC ycine () TTT, TTC sparagine () TT, TT Methionine (M) TC spartic cid () CT, CT Phenylalanine (), Cysteine (C) C, C Proline (P) T,, C, lutamic cid () CTT, CTC erine () T,, C,, TC, TC lutamine (Q) TT, TC Threonine (T) TT, T, TC, T lycine () CCT, CC, CCC, CC Tryptophan CC volution By ubstitution tudent 0

13 (W) istidine () T, T Tyrosine (Y) T, T Isoleucine (I) TT, T, T aline () CT, C, CC, C TOP TT, TC, CT. mino acids are rarely substituted with tryptophan. oes the fact that only one triplet codes for tryptophan support this? xplain. 2. etermine if the fact that only one triplet codes for trytophan affects this rate. Compare the rate of substitution with tryptophan to the rate of substitution with methionine, which also has only one coding triplet. ow do the two compare? oes this comparison support the fact that substitutions with trytophan are rare because it has only one coding triplet? 3. mino acids are more commonly substituted with serine. o the number of triplets that code for serine support this? xplain. 4. rginine has the same number of triplets that code for it as serine does. Is it substituted as often as serine? 5. Based on the answers to a d, does the number of triplets coding for an amino acid appear to be the main factor contributing to its rate of substitution? If not, what other factor(s) may be important in determining the rate of substitution? volution By ubstitution tudent

14 esson 2 Multiple ubstitutions In esson, the problem of monitoring how amino acids change in species was introduced. The natural way to describe that change is by using probabilities--numbers between 0 and that measure how likely (or unlikely) it is that a particular substitution will occur during one e.u. or example, P() = 0.00 means that there is a in 000 chance that changes will result in lanine () being substituted for aspartic acid (). It is also convenient to store those numbers in a two-way table called a matrix, like the data contained in Table.2. In that matrix, the various rows represent the original amino acid and the various columns represent the new amino acid resulting from a substitution. ature is too complex to regulate that these transitions take place one at a time. With twenty amino acids all capable of turning into twenty different amino acids, calculations can be quite difficult. It turns out that matrices are also useful for handling the specific kind of calculations needed for making predictions. This lesson will examine all the possibilities at one time, help you understand exactly what calculation is being made, and allow you to learn how to use a calculator to do the work. xamining a maller Problem In modeling complex situations, it is sometimes convenient to pretend that a simpler set of conditions is present. or now, make the following simplifying assumption: There are only 5 kinds of amino acids: alanine (), arginine (R), asparagine (), aspartic acid () and one of the other kinds of amino acids. Consider the other sixteen amino acids to be a category called other (O). With only 5 kinds of amino acids we can investigate how substitutions occur over e.u. time periods. CTIITY 2- Investigating ubstitutions Objective: Investigate substitutions over e.u. time periods. Materials: andout -4 Investigating ubstitutions Worksheet andout - ubstitution Table. Create a matrix describing the various probabilities relative to the simplified version of this problem. et up a matrix like what is shown in Table 2.. a. ssign the appropriate probabilities for to be replaced by, R, and, in their correct locations in your matrix. volution By ubstitution tudent 2

15 from original to replacement R O R O Table 2.: ubstitution matrix for simplified problem. b. eep in mind that our simplified model uses O to represent any one of the other sixteen amino acids that appear at the end of the first row from Table.2. What is P(O)? ow is that answer calculated? (Be sure to record that answer in your matrix.) c. The sum of all the numbers across the row should be. Why does that make sense in the context of changing amino acids? d. In a similar fashion, fill in the entries for Rows 2, 3 and 4 of your matrix, where the original amino acid was R, or. e. The sum of all the numbers in any column is not, since each row refers to a different initial condition. owever, it is still possible to determine P(O), P(OR), P(O) and P(O) from the information in Table.2, and you can use the property that each row must add up to to find the final entry, P(OO). ill in the entries for Row 5 of your matrix. abeling and Reading a Matrix It is useful to give a matrix a name, so it can be easily referred to. Rather than a long name like substitution matrix, a single bold capital letter is used. or example, the substitution matrix in Table.2 will be named. In referring to individual entries in a matrix, the matrix name and two subscript numbers, one for the row followed by one for the column are used. or example, 3 or 3, has the value ome calculators might require syntax such as [](3,). Questions for iscussion The matrix created in ctivity 2- will be named B.. What is the value of B2 5 and the value of B5 2? 2. Which is bigger: B or B3 3? 3. What is: (B4 3)(B 5)? volution By ubstitution tudent 3

16 tate iagram It is sometimes useful to look at something in a different way, either to understand it better or to provide an alternative explanation. cientists use a picture called a state diagram to visualize the various possibilities for an event. igure 2. contains a portion of the work recorded in matrix B the part that affects amino acid : O R igure 2.: Incomplete state diagram. In igure 2. the curved arrow and its associated number (0.9867) represent the situation in which an alanine () molecule remains an alanine molecule; in other words, no substitution has taken place. The number comes from the fact that P() = Think of igure 2. as describing, What happened to alanine ()? since the arrows are going away from in the diagram. If you start with 50,000 alanines (), how many s, R s, s, s, and O s can be expected at the end of one e.u.? We calculated this by multiplying the initial number of molecules by the percentages along each path. (50000)(0.9867) = molecules of would remain. (50000)(0) = 5 molecules of R would be formed. (50000)(0) = 20 molecules of would be formed. (50000)(0.0006) = 30 molecules of would be formed. (50000)(0.022) = 60 molecules of would become something else other than R, or. ote the sum of new molecules is = volution By ubstitution tudent 4

17 CTIITY 2-2 Calculating Resulting mino cids Objective: Calculate amino acids resulting from substitutions Materials: andout -5: Calculating Resulting mino cids Worksheet andout -4: Investigating ubstitutions (completed) ow, instead of thinking about the various amino acids that might be substituted for alanine (), reverse the process and consider the ways in which a substitution might result in.. raw a state diagram to describe this situation; attach probabilities for each path. 2. uppose you start with molecules of, of R, of, of and 0000 of O. Write a single expression that will compute the number of molecules of expected after one e.u. 3. ow many molecules of can be expected after one e.u.? tart again with molecules of, of R, of, of and 0000 of O. (ote: from a biological point of view, these values are unreasonably large, but are assumed true in order to create whole-number answers for the computation.) Consider the matrix = [ ]. otice that is a x5 matrix and contains the number of each amino acid present in our current situation. The order in which numbers appear in this matrix must be the same as the order the amino acids appear in the substitution matrix used (Table 2., Matrix B), R,,, and O respectively. 4. Use matrices and B to answer the following questions. a. Write a single expression that will compute how many molecules of R are expected after one e.u. (int: drawing a new state diagram may help.) b. ow many molecules of R are expected after one e.u.? c. Which amino acid, or R, underwent more substitutions during that time period? xplain. Matrix Multiplication In working through Question 4, you have actually done part of a matrix multiplication. The matrix multiplication problem B is below: volution By ubstitution tudent 5

18 In working out previous calculations, it is easier to imagine the process of multiplying the two matrices. The first two steps are shown below: Row and Column Row and Column 2 and = = The numbers in are paired with the numbers in the first column of B and multiplied. These 5 products are added to get the expected number for each amino acid. This process is repeated for all 5 amino acids (all 5 columns of B). The results represent the expected number of each amino acid after one e.u. and are recorded in a matrix. B = One reason for using matrices is to organize information. or that purpose, there is no difference between tables and matrices. Matrices allow certain types of computations, but the procedures for these computations can be laborious. The bigger the matrix, the more helpful technology is. Matrix Technology otes ?????? Matrix multiplication operations are much easier to accomplish with the help of technology. You can use either a calculator or a spreadsheet program to assist you. raphing Calculator Most graphing calculators have matrix operations as a built-in function. or example, the TI-84 Plus calculator accesses all matrix operations by pressing the 2 nd and x - keys in??? volution By ubstitution tudent 6

19 sequence. otice that by pressing the 2 nd key you are using the Matrix (or Matrx) command above the x - key to access the Matrix menus. The steps involved may include the following: Create the matrix first (or modify an existing one) elect IT menu, then choose the matrix with which to work. pecify the dimensions (rows and columns) for the matrix. Type in the specific entries for the matrix by highlighting the appropriate cell, typing the number and pressing enter. rom the ome-creen, use the matrix names in calculation expressions like [][B]^5. You generate the matrix name on the ome creen from the same menu (use the 2 nd and x - keys to get there) by pressing a number key when the M menu is highlighted. preadsheets preadsheet programs are also useful for matrix multiplication. The programs may refer to the matrices as arrays. or example, if you enter matrix B as a 5x5 array in J3:7, the matrix B 2 is created using the command MMUT (matrix multiplication). In order to do this, you would highlight a 55 set of cells (where the result will appear), and type the command: =MMUT(J3:7, J3:7) and then press CTR + IT + TR to execute the instruction on the entire matrix at one time. In a similar fashion, successive powers of the substitution matrix can be built to display (or view) the transition probabilities over more e.u. s. Questions for iscussion Using a calculator or spreadsheet that is able to perform matrix computations, create a matrix that contains the original number of amino acids and a substitution matrix B, as shown in igure Interpret what the values of,4 and B,4 mean for this problem. 5. Why does it make no sense to ask for an interpretation of the value of 4, here? 6. Using available technology (calculator or spreadsheet), multiply B. (Be careful; the order makes a big difference!) ow many of each of the, and O molecules are expected after one e.u.? 7. What happens if the order is reversed and you attempt to find B? In order to do matrix multiplication, the number of columns in the first matrix must match the number of rows in the second matrix. The product is another matrix that has the same number of rows as the first matrix, and the same number of columns as the second matrix. This can be summed up easily, using the notation m n to describe the dimensions of a matrix with m rows and n columns. In that case, matrix multiplication is volution By ubstitution tudent 7

20 possible when the matrix of dimension m n is multiplied by a matrix of dimension n p. The dimension of the answer (product) is m p. Practice. specialist is interested in tracking the number of a particular amino acid, glycine (), present in a sample that contains 6000 molecules of and molecules of something else (O). (Refer to Table.2.) a. ow many rows and columns does the specialist s substitution matrix need? b. What substitution matrix describes this situation? c. What does the complete state diagram look like? d. ow many molecules of the amino acid are expected after e.u.? 2. Perform the following matrix multiplications. 0.2 a b. 0 c Use the numbers, 2, 3, 4, 5, 6 (in order from left to right, and from top to bottom) to create a 2 3 matrix called, and a 3 2 matrix called B. a. What is a2,2 b2,? b. or this example, what is B? c. What is B? d. oes the commutative property hold for matrix multiplication? In other words, does B give you the same answer as B? xplain. 4. Perform the given multiplication operations and record each answer a b volution By ubstitution tudent 8

21 c xtension. In iscussion Question 6, the number of each amino acid expected after e.u. was computed, starting with [ ] molecules of each amino acid. The result is: [ ]. What happens if the time interval is extended one more e.u.? (int: Begin with [ ]. Use the 5 5 substitution matrix B one more time, and record the new numbers expected after the second e.u.). 2. Biologists have created three categories of coyote pup, yearling and adult. The following matrix describes the probability of change in a certain coyote population over one year: P = a. xplain what P2 3 means for this problem. b. If the categories pup, yearling and adult are distinct (so you cannot be more than one of them at a time), and are based on age and maturity, explain why some of the entries in the matrix have value 0. c. What would be a realistic reason (within the context of this problem) why P3 3 does not have a value of? d. Would you expect each of the rows of this matrix to add up to? xplain. 3. mallville is made up of three separate (and smaller) regions: OldTown, owntown and ewtown. In any given year, 3% of the people living in OldTown move to owntown 8% of the people living in OldTown move to ewtown % of the people living in owntown move to OldTown 6% of the people living in owntown move to ewtown Once someone lives in ewtown, they never want to move away from there. a. raw a state diagram describing the movement of mallville s population during a one-year time period. volution By ubstitution tudent 9

22 b. Create a matrix that describes the movement of mallville s population during a one-year time period. 4. The partial state diagram below details the probability of having a peanut butter sandwich (PB), a tuna fish sandwich (T) or a grilled cheese sandwich (C) tomorrow, depending on what was eaten today T 0.25 C 0.30 Partial state diagram for lunch options. a. dd in arrows and probabilities to complete the state diagram. PB b. Use the diagram to develop a matrix that describes the same set of conditions. (ssume the rows are what you ate today, and the columns are what you are going to eat tomorrow.) 2 5. escribe how each multiplication affects the contents of the matrix M = a b c d volution By ubstitution tudent 20

23 esson 3 The Power of a Matrix o far, the lessons have shown how proteins change over time by substitution of one amino acid with another amino acid. Probabilities describe the likelihood that the amino acids will undergo a substitution during some established amount of time. s a result, the probabilities can be used to predict how many of each amino acid would be present at the end of a time period, if the original numbers are known. Matrices are a convenient way to keep track of the numbers involved and the computations required. Predicting uture Change t the end of esson 2, the substitution matrix was applied over two successive intervals of time. igure 3. illustrates this idea. times B times B Matrix : Initial alues alues after e.u. alues after 2 e.u. igure 3.: Calculation over two evolutionary units. The substitution matrix B describes the changes that take place over one e.u. To understand the evolution of living things though, it is necessary to describe these changes over many e.u. s. This lesson focuses on using the substitution matrix to examine change over extended periods of time. Matrix Multiplication Review In algebra, when an expression like a x x is used, exponents are used to write the same expression in simplest form as a x 2. or the calculation shown in igure 3., the a would represent the initial amounts of the various amino acids (which we had assumed to be [ ]) and the x would represent the substitution matrix containing all the probabilities. Recall the order of operations have a hierarchy of importance: ) Parentheses, 2) xponentiation, 3) Multiplication or ivision (from left to right) and 4) ddition or ubtraction (from left to right). The same order of operations holds for matrices and so in the expression a x 2 the exponentiation would be done first. When the variables represent individual numbers, the expressions a x x and a x 2 are always possible to compute. This is not always true if a and x are matrices. The problems arise in the dimensions of the matrices. volution By ubstitution tudent 2

24 volution By ubstitution tudent 22 Questions for iscussion. Multiply the following matrices. 2. If a and x are matrices, is it not always possible to compute a x or x 2. To experiment with when this is possible, try multiplying the following matrices. If the multiplication is possible, determine its product; if not, indicate why not? a. b. c. d. e. f. What conclusion can be drawn about squaring a matrix? Tracing mino cid ubstitutions with Matrices Recall that B is the 5 5 matrix containing the probabilities of five amino acids (, R,, and O) undergoing a substitution over one e.u

25 B = We know we can square matrix B because it is a square matrix (same number of rows as columns). B 2 = Recall the initial number of amino acids: = In esson 2, we found B and multiplied B by the result and obtained: ind B 2 using the and the B 2 matrices above. oes your answer match the B B matrix from esson 2? The B 2 matrix describes the likelihood of any of the five amino acids undergoing substitution over two e.u. s. It seems reasonable that powers of the substitution matrix can be used to predict what happens to amino acids over multiple e.u. s. owever, it is not true that each element of the matrix is simply raised to that power. The calculation is much more complex than that. We need to look more closely at where the numbers in B 2 come from to have greater confidence in the process. CTIITY 3- Investigating and Interpreting Powers of Matrices Objective: Investigate and interpret results of matrix multiplication. Materials: Calculator or computer andout -6: Investigating and Interpreting Powers of Matrices Worksheet Use the B 2 matrix to complete this ctivity.. The notation B 2 indicates the entry in Row Column of the matrix B 2 a. What does the number in B 2 represent? b. There are five different pathways that begin with amino acid and end with amino acid after two e.u. s. One of them is R. Using that same notation, what are the other 4 possibilities? 2. It is assumed that each situation involving substitution is independent of each other. or that reason, the probability over a chain of events involves multiplying the various probabilities that describe the links. or example, P(R) = (0)(0) = a. Calculate the probability for each of the remaining four pathways identified in part.b. volution By ubstitution tudent 23

26 b. ach of the five pathways is disjoint. In other words, if you go down one path, you cannot be going down a different path at the same time. Therefore, the probability of any of the events taking place is found by adding together the probabilities of all the paths. Using this addition rule and the answer from part (a), find the probability of starting with amino acid and ending up with amino acid at the end of the second e.u.? 3. The notation B indicates the entry in Row 3 Column 5 of the matrix B 2. a. In the context of amino acids undergoing substitution, what does B mean, and what is its value? b. raw a state diagram for the answer to part a. that represents all possible amino acid substitutions over two e.u. s, and the probabilities of each substitution. c. In the state diagram in part b., identify the five pathways in which amino acid is replaced by amino acid O in 2 e.u. s, and include the expression that calculates the probability for each of those events. d. Write a single expression that calculates B and verify that that this expression generates the same value as recorded in part a. 4. In general, the matrix multiplication B = C is defined in such a way that Ci j is found by multiplying the i th row of by the j th column of B and pairing terms, multiplying them together and then adding all the products. Take another look at our matrix B 2 and how the entries are determined. The square notation B 2 represents the matrix multiplication B B: a. Write down the elements in the first row of B, and the elements in the first column of B. Then identify how the elements of the first row pair up with the elements in the first column when computing B 2. b. Write a single expression that shows how you would compute B 2 according to the definition of matrix multiplication. Compute the value and check if it is the same as in the B 2 matrix. c. What row and column are multiplied together to determine B 2 3 5? d. Write a single expression that shows how to compute B 2 3 5, and then compute the value of the expression. volution By ubstitution tudent 24

27 e. What does the value for B represent? 5. igher powers of the matrix B are used to project farther than 2 e.u. s into the future. or example, probabilities of amino acid substitutions over 5 e.u. s are contained in the matrix B 5. Using this new distribution of the same five amino acids used in previous problems: , how many of each amino acid are expected after 5 e.u. s? 6. Use the 55 substitution matrix B and new initial numbers of amino acids = a. Project from 5 e.u. s to 00 e.u. s into the future to complete the table. etermine how many molecules of each amino acid are expected at the end of each of the following time intervals. Round each number to the nearest integer. Time interval R O 5 e.u. s 0 e.u. s 50 e.u. s 00 e.u. s b. Initially, which of the amino acids had the least and most molecules? re those numbers increasing or decreasing over time? 7. typical graphing calculator cannot compute much higher powers of B than B 00. owever, these limitations are overcome by considering what happens over multiples of hundreds of e.u. s. a. Create a new matrix C whose entries are equal to that of B 00. What is the numeric content of C? b. Interpret the meaning of C in this context. ow does it relate to B 00? c. Predict what the result of calculating C will be. erify this prediction using the available technology. What is the result? 8. In algebraic expressions, (x a ) b x ab 6. or example, ( x 2 ) 3 x. a. xperiment with the matrix B to see if the same property holds for matrices. Compute (B 2 ) 3 and B 6. re these two matrices the same? b. Using matrix C, which contains probabilities for undergoing change over one hundred e.u. s, and the exponent property you verified in part (b), determine the distribution of amino acids expected after 200, 500, 800 and 000 e.u. s. Round each number to the nearest integer. Remember: the initial distribution is matrix : [ ]. int: C 2 = (B 00 ) 2. volution By ubstitution tudent 25

28 Time interval R O 200e.u. s 500 e.u. s 800 e.u. s 000 e.u. s c. Mathematically, does the distribution of amino acids become constant over time? If so, estimate how long it takes to reach a constant distribution (called steady state) and what the final distribution of amino acids will be. d. xamine the contents of the matrix C 0. What can you tell about the distribution of amino acids from looking at this matrix near the steady state time period? teady tate The probability that any amino acid remains the same after one e.u. can be found along the diagonal of the original matrix B. Because those values are approximately equal to, one may think that the relative numbers of amino acids would be fairly stable. Over long periods of time that are typical of evolutionary processes, the system moves toward its own steady state, where substitutions for one amino acid are being offset by the combined substitutions in the others. or example, at steady state, the number of molecules of that change into another amino acid is offset by the number of molecules of other amino acids that change into. fter a system reaches steady state, the same distribution is calculated for any time interval into the future. ystems with long-term behaviors that tend toward a steady state are said to be stable. tability is the state or quality of being resistant to change. Using our example, the number of molecules of stays the same after the system reaches steady state. s long as conditions remain constant, there is good reason to continue using the same probability numbers until the point where steady state is reached. This is a mathematical consequence of the model that has been built, and depends on the assumption that conditions remain constant. owever, if conditions change (and they do!), so will the probabilities in B, and evolution by substitution will continue with these new probabilities. or this reason, there is a limit to the ability of a model to accurately predict the future. Practice ssume that M =. M 2 2. M 3 3. M etermine the matrix for each of the following: 4 volution By ubstitution tudent 26

29 Challenge. ow would you find M 50? M 225? 2. Consider the 5 x 5 matrix B in this lesson representing the amino acids,, R, and O. a. In e.u., how many different ways can amino acid undergo substitution by amino acid R? b. Over the course of 2 e.u. s, how many different ways can amino acid undergo substitution that results in amino acid R? int: One way is R. c. Over the course of 3 e.u. s, how many different ways can amino acid undergo substitution that results in amino acid R? ow is this answer determined? d. Predict the number of different ways amino acid can undergo substitution that results in amino acid R in just 0 e.u. s? ow is this answer determined? xtension efinition: transition matrix is a square matrix whose dimension is determined by the number of discrete outcomes for a dynamic situation. transition matrix contains all the probabilities of going from any specific state to another (or possibly the same one) in some fixed time interval. Because all possible outcomes are described by the probability of that event taking place, each row of the matrix must add up to. efinition: Markov chain consists of two initial conditions: a transition matrix T (whose probabilities are assumed to remain constant over time) and an initial distribution. When combined together by the matrix multiplication operation T n, they represent a chain of outcomes repetitively over n successive time intervals as well as the predicted result at the end of the n th time period.. pizzeria serves up three kinds of pie: pepperoni, salami and cheese. Company records show the following trend: 60% of the time, a customer orders the same type of pizza the next day. 20% of the time, a customer orders one of the other types of pizza the next day. 20% of the time, a customer orders the other remaining type of pizza the next day. a. Create a transition matrix T for this situation. b. The day that the pizzeria opened a new store in a nearby location, they sold 500 pepperoni, 200 salami and 300 cheese pizzas. If the customer preferences remain the same at this new location, how many pizzas do you expect they will sell the next day? c. ow many pizzas do you expect the new store to sell five days after they open? volution By ubstitution tudent 27

30 2. or an automated assembly line, consistent performance is a critical issue. machine that has worked correctly 80% of the time on average is brought in for repair. fter being fixed, whenever it does a job correctly, the machine will do the next job correctly 90% of the time. When it does not do a job correctly, it will do the next job correctly only 70% of the time. We are interested in whether the repair will improve the long-term performance of the machine. a. Create a transition matrix for this situation. b. What percent of the tasks will be successfully completed at the end of one time period? c. What percent of the tasks will be successfully completed after five time periods has elapsed? d. id the repair improve the long-term performance of the machine? xplain. 3. Two rival cable companies, TellyT and addleite, are in hot competition in one town. Researchers found the current yearly conditions as shown in the state diagram: 60% 5% 80% TellyT addleite 30% 0% 0% 50% 5% o Cable ervice 40% a. Create a transition matrix for this situation. b. Currently, 35,000 people subscribe to TellyT, 5,000 to addleite, and 50,000 have no cable service. Under these conditions, what can TellyT expect as the longterm share of the market in this town? c. The marketing director for TellyT determines that an aggressive advertising campaign can influence the people who do not have cable yet. er figures indicate that 80% would go to her company, with 0% going to addleite and 0% remaining without cable. What can TellyT expect as the long-term share of the market if the advertising campaign is started? d. The sales director for TellyT vetoes the advertising campaign. e has his own plan: offer greater services for a slightly reduced rate. e estimates that 80% of the TellyT customers will stay with the company, and only 0% will switch to addleite. If this plan is enacted, what can TellyT expect as the long-term share of the market? volution By ubstitution tudent 28

31 4. Trends in recent national elections are studied intensely as a way to understand how voters might behave in the future. In one such study, the party affiliation of the voters in one state was examined. The result of that study is summarized in the following table (transition matrix): ext lection Current lection Republican emocrat either Republican emocrat either or example, there was a 75% chance that a registered Republican in one election would remain a Republican in the next election, while there was a 5% chance that a Republican would switch to being a emocrat. oes this situation ever reach steady state? If so, what will be the voter distribution along party lines? 5. The cme Rent--Car Company owns and maintains many cars in its business. very car is inspected each week, and assigned a letter (good), (fair) or P(poor), depending on its current condition. cme also keeps track of how those conditions change from one week to the next, with the probabilities given in the following table (transition matrix): ext week P This Week P a. What is the likelihood that a good car will become a poor car over a five-week time period? b. If the cme Rent--Car Company currently has 200 cars that are rated, 400 that are rated and 200 that are rated P, how many of each rating will the company have at the end of the five-week time period? c. oes this situation reach steady state? If so, how many weeks will it take? 6. study was done by professors at Brock University on the land usage of the iagara region in Canada. total of 886 acres were examined in 976, with 24 acres classified as wooded, 340 acres as agricultural and 305 acres as urban. The researchers reported volution By ubstitution tudent 29

32 their data in the table below showing how many acres of each type underwent change, and what it became. and Use in 976 (acres) and Use in 98 (acres) wooded agricultural urban wooded agricultural urban ssume that the development of each acre is independent, and that all decisions on their land use were random (i.e., which acres were affected, and how they were changed). In that case, the percentage of the total land in each category that underwent change is a good estimate of the probability that a particular acre of that type will undergo the corresponding change. a. Create a matrix, similar to the one shown, which contains the various probabilities associated with the change in land use over the period from 976 to 98. To do that, take each row, and divide it by the total number of acres of that type that was present in 976. b. ow many acres of each type of land were remaining at the end of 98? xplain how to use both matrix multiplication and the given table to answer that question. c. ssume the same changes in land use as what the researchers found in their 6-year study continues. ow many acres of each type of land would be remaining at the end of 2005? (ote: that is the end of a thirty-year period from the beginning of the original study.) d. Projecting current demands into the future often leads to bad estimates, since it is unlikely the conditions will remain the same for long. owever, assume that those conditions do not change. Will this situation ever reach steady state? xplain. e. iven typical land use changes over time, do the results from part (d) make sense? xplain. volution By ubstitution tudent 30