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# Introduction to Population Genetics Biol 519

WSU

GPA 3.72

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This 17 page Class Notes was uploaded by Marge Schiller on Thursday September 17, 2015. The Class Notes belongs to Biol 519 at Washington State University taught by Staff in Fall. Since its upload, it has received 47 views. For similar materials see /class/205996/biol-519-washington-state-university in Biology at Washington State University.

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Date Created: 09/17/15

Coarse Notes Population Genetics NUMERICAL SIMULATIONS OF SELECTION USING A SPREADSHEET Can use spreadsheet programs like Excel to execute repetitive calculations like the recursion equations we have seen in this course In other words spreadsheets can be used as a simple programming language 0 Purpose of this demonstration show how to do this using one locus selection as a working example with one of the more widely used spreadsheet programs Microsoft Excel Note many of the commands and procedures in this demonstration are very useful for other purposes like analyzing data keeping track of grades alphabetizing budgeting etc 0 Some Useful Terms and Commands cell location The location of a cell is given by a number row and letter column combination For example cell B9 is in the 9th row of the 2nd column column B function Fills cell with the result of a calculation Functions always start with For example to compute 1234 type 1234 in a cell and press return enter 0 Excel has a wide variety of built in functions that are useful for data analysis xed and absolute cell references References to specified cell locations 0 Relative references are used for a variables A relative reference refers to a cell that is a fixed relative location from the current active cell When a cell reference is entered by selecting a cell Excel automatically assumes you want a relative reference 0 Absolute references are used for fixed parameters An absolute reference refers to a fixed cell location in the spreadsheet To create an absolute reference either place the cursor to the right of the cell location and type command t or type in front of the row and column names For example B9 is a absolute reference to the 9th cell of column B fill down Use this to iterate your computations Select the current cell and those below that you want to fill Type command d or select fill down from the edit menu The cells below will be filled with the contents of the topmost cell including adjusted relative cell references Fill right ll upfill left are used similarly NOTE copy and paste also adjust cell references and can be used for the same purpose as fill II 18 Coarse Notes Population Genetics freeze panes Use to freeze column and or row headings To freeze column headings only select cell just below the first column heading and then choose freeze panes from the window menu 0 To freeze row headings only select cell just to the right of the first row heading 0 To freeze row and column headings select cell in the first row and column 0 Tutorial Selection at a Haploid Diallelic Locus Goal compute frequency of A for 50 generations given a selection coefficients and initial value ofp using the equation Ap pq Steps 1 Enter parameter value for selection coefficient 3 eg set s 01 2 Set up column for the generation numbers 0 Enter 0 for generation 0 0 Enter the formula 2 click cell above 1 to generate next generation Important use relative reference since generation is a variable Simply click cell to enter its location instead of typing it into the formula Press return enter 0 Select the previous cell and the next 49 cells below choose fill down from the edit menu Note fill down enters the correct relative reference 2 1 cell above for each cell 3 In column B enter the initial value ofp e g setp 02 4 In column C enter the formula for q Type 1 click cell to the left Hint Type in formula then replace letters with their respective cell locations 5 In column D enter the formula for Ap II 19 Coarse Notes Population Genetics 0 Important use an absolute reference for s 6 In the next row column B enter the formula for p p Ap 4 0 Type 2 click cell aboveclick cell for Ap in previous r0w 7 Use fill down to fill in the remaining row entries 0 Note Fill Down gives the correct relative cell locations and keeps the absolute cell locations fixed 8 Starting with the second row use fill down to fill in the rows for the remaining generations 0 Useful to freeze row headings when examining results With this program in place it is easy to explore at the effects different selection coefficients different initial values of p etc 0 Change a parameter or initial value and Excel updates the rest Can even graph the results if desired 0 Select cells containing information to be graphed then use Insert Chart and the Chart Wizard 0 Over and Underdominance in Fitness at a Diallelic Diploid Locus l for various values 1 p s q t of the selection coefficients tand s and initial frequencies of A In this example use Excel to iterate the equation Ap pq parameters 3 t 0 variables p 61 AP Compare results of iteration with predicted equilibrium 3 ts t 0 Exploring the Adaptive Landscape and Fisher s Fundamental Theorem Use Excel to follow changes inp and W II 20 Coarse Notes Population Genetics Utilize the general formula for selection at a d1alle11c d1p101d locus A1 pq A a w Where WA prA qua Wu pra qwm and 2 2 W I7 WAA2quAa q Wm pWA qw l39 parameters wAA WA Wm W 0 variables p q WA W a Compare actual value of AW W39 W with the predicted value based on Fisher s Fundamental Theorem of Natural Selection AW z varx IW 2pxi qaW where aAwA w andaawa w 11 21 Coarse Notes Population Genetics MIGRATION ROLES OF MIGRATION IN EVOLUTION READING Hedrick pp 499 522 Introduces novel genetic variation into populations Tends to homogenize gene frequencies in different populations 0 Sets the spatial scale for evolution Opposes local adaptation Migration with an evolutionary impact Gene Flow 1f39 39 39I J 1 1 1 and dispersal o r Migrants have no effect on evolution unless their genes are incorporated into a population 0 A OneIsland Model The simplest model of migration Continent Island Two alleles A and a Letp 2 frequency of A on island A fraction m of the island gene pool emigrates from the continent where p m the frequency of A is pa 5 gt gt A fraction 1 m of alleles on the island originated on the island The continent is too vast to be influenced by migration from the island gt pc is constant Then the frequency of A on the island changes according to p 1 m p mpg Coarse Notes Population Genetics At equilibrium set 7 p 0 Solving forp gives 3 pa Rate of approach to equilibrium Rewrite evolutionary equation as p 13 p p 1 mp mp p 1mp p 1 mXp 13 Conclusions 1 At equilibrium both populations have the same allele frequencies 2 Rate of approach to equilibrium 3 pc is determined by the migration rate m 0 General Models of Migration Same conclusions as one island model hold Exceptions however do exist 0 For example consider two m 1 populations with different A allele frequencies that switch locations each generation 0 The populations will obviously K never homogenize because m 1 there s no real exchange of genes Remark Have implicitly assumed gene frequencies differ in different locations How could this be History Genetic drift Selection favors different alleles in different locations Coarse Notes Population Genetics MIGRATION AND DRIFT Migration introduces novel genetic variation into local populations Drift removes local genetic variation Which for dominates One answer Wright s Island Model I Consider a large number of islands each with a population of size N 2N alleles per locus Each generation every island exchanges a fraction m of its gametes with a w sized Migrant Pool migrant pool to which all islands contribute gamates f f i i 0Assumeinfinite isoalleles model C m 0 Let Prpair or randomly drawn gametes on a typical island are IBD in generation I average within island homozygosity By the same logic used when studying mutation drift balance 121 1 1 A 1 0 At e u111br1um z 7 q 1 f f 1 4Nm expression resembles that describing diversity maintained by mutation amp drift with 0 4Nu replaced by 4Nm If 4Nm lt 1 Local homozygosity is substantial drift dominates migration 0 If 4Nm gt 1 Local diversity heterozygosity is substantial migration dominates drift Note 1 4Nmgt 1 same as 2Nmgt 12 Coarse Notes Population Genetics gt Migration dominates drift if at least one migrant gamete is exchanged every other generation Conclusion is independent of m the rate of gene flow Why Note 2 Recall from discussion of F statistics is AvgHSJ z 1 since j is the average local homozygosity and there is no additional inbreeding Also HT 2 1 Prpair of randomly chosen gametes from entire population are IBD 1 01 gtF 7 A 1 HT 1 1M where M 4Nm Suggests way to estimate rate of migration from F ST 1 FST FST 39 M Careful estimate requires lots of assumptions island model equilibrium etc to be valid MIGRATION AND SELECTION Oneisland model with selection A favored on island a fixed on continent pg 2 0 A is dominant Fitnesses on island I Genotype I AA I Ad I ad I I Fitness I 1 I 1 I 1 s I 0 Life Cycle Ly Utes 591mmquot aduito migmi m ametes m immim zygotes p P p p 9 1 0 After selection before migration p pl 2 q S Coarse Notes Population Genetics p1 m 0 After migration amp reproduction p 1 mp m0 1 2 q S 0 To find any equilibria set 17 p 1 Solving forp gives 3 1 lms Require 0 s 1351 0 This occurs only when m lt s 0 Otherwise f7 0 Now assume A is recessive Fitnesses on island I Genotype I AA I Ad I ad I I Fitness I 1 I 1 s I 1 s I 9 1 0 After selection before migration 7 M 1 sq1 p 0 After migration amp reproduction p 1 mp m0 W 1 sq1 p 0 To find equilibria set 7 p and solve forp stable Get cubic equation for f s up to 3 possible solutions f7 0 is always an equilibrium since pa 0 p unstable There are two polymorphic equilibria when s gt 4m assuming m is small 0 one equilibrium is stable the m other is unstable Graphically 0 Implications V5 Coarse Notes Population Genetics If recessive selection is strong enough to maintain A in the face of migration A will spread only if it s initially sufficiently frequent enough Otherwise it will be lost n general unless locally advantageous allele is completely dominant it must reach a threshold frequency to persist If an allele persists it won t be found at a low frequency Historical quotaccidentsquot play a role Identical patches will evolve differently if they differ in initial allele frequency 0 The Levene Model Q What happens when a population is made up of a group of distinct subpopulation patches with different selection pressures occurring in each and migration between locations A Depends on geography population structure Natural populations fall somewhere between the following two extremes Unrestricted migration 0 Restricted migration A simple model of unrestricted migration was presented in 1953 by H Levene Assumptions of Levene s 1953 model 0 n patches in which different patterns of selection occur 0 Frequency of A among gm is p 0 After fertilization diploid ygotes colonize the different patches at random Important this implies that the zygotes within patches are in H W proportions ith patch makes up a fraction ci of the environment Fitnesses in the ith patch I GenotypeI AA I Ad I ad I FitnessI WAAi I wAai I wmi I V 6 Coarse Notes Population Genetics 0 Random mating between patches Individuals from different localities form a single mating gamete pool Why study the Levene model Captures essential features of spatially subdivided population m is mathematcially tractable Is a reasonable representation of certain natural systems as well Back to modelHow many gametes does each patch contribute to the gamete pool 0 Two extremes 1 Hard selection due to Dempster 1955 0 Patch contributes gametes in proportion to the fraction of survivors ie patches with higher fitness contribute disproportionately more Implies population size is not regulated within patches 2 So selection 0 Each patch contributes fixed number of gametes to the mating pool regardless of local fitnesses Number of reproducing adults from each patch is the same from one generation to the next Implies population size is regulated within each patch 9 soft selection 99 A schematic comparison 6 6 between soft and hard env1ronment gamete pool selectlon assum1ng c1 02 2 environment gamete pool Levene model with hard selection constant number of zygotes Assumes contribution of genotype from patch i to the gamete pool is proportional to it39s fitness in that patch wgmtypei X frequency of i patch in environment 0 ie total number of survivors of that genotype in patch i 0C ciwgenotypei V 7 Coarse Notes Population Genetics 0 Overall fitness of genotype in population is its average fitness over patches 7 For example mean fitness ofAA WM gainAAquot 1 Likewise for Ad and aa 0 Consider changes in the frequency p of A in the gamete pool w w W 39 P PW PEA Where WA pWAA qWAa and w w 2 2 w 17 WM 2quA q Wm Looks just like selection with constant fitnesses WM WM Wm 0 Consequences An allele will spread if it has the highest arithmetic mean fitness across patches Selection will maintain a stable polymorphism if heterozygotes have the greatest arithmetic mean fitness across patches 0 For example consider two equally sized patches 61 62 05 Fitness in patch AA Ad ad 1 0 075 1 2 1 075 0 Average 05 075 05 0 Selection maximizes arithmetic mean fitness across environments Levene model with soft selection constant number of adults Within each patch selection operates as usual Fitness in patch i AA Aa After selection frequency of A in patch i is Pwiq1 WAG P W pzw 2pqlt1gtq2v p W Density regulation occurs independently in each patch Survivors contribute to gamete pool in proportion to the size 2 relative proportion of adults of the patch 0 Coarse Notes Population Genetics 1M q p 2 trip l EQP i i pwi2pqqvi Equilibrium set 7 p and solve for p 0 Results in polynomial of degree 2n 1 in p gt as many as 2n 1 equilibria are possible Mathematically too difficult to find all these Alternative protected polymorphism analysis 0 Near 0 2 i l P a P 1 Vi P P 7 where v 2 cii x V 0 Note that p gtp ie Ap gt 0 whenever 1 gt1 ltgt 7 lt1 ie whenever the harmonic mean fitness of aa homozygotes lt mean fitness of heterozygotes is the harmonic mean fitness of aa homozygotes Likewise nearp 1 q 0 q gt q whenever M lt 1 Conclude protected polymorphism occurs with soft selection whenever there is harmonic mean overdominance in fitness across patches M lt 1 gt 7 Bottom linegs for soft selection Harmonic mean fitness across patches is the relevant fitness measure if p z 0 or 1 0 Turns out however that selection maximizes geometric mean fitness Hard versus Soft Selection 0 Conditions exist in which an allele will increase under soft selection but not hard selection Ie polymorphisms can be maintained under a broader range of conditions with soft selection versus hard selection Intuitively follows because under soft selection individuals compete selectively only against quotpatch matesquot With hard selection all compete Coarse Notes Population Genetics Mathematically follows because harmonic mean is never larger than the n arithmetic mean 7 s 7 2 civi Q Why does soft selection seem hard density regulation intense local competition while hard selection seems soft little competition no density regulation A It all depends on your Viewpoint genetic vs demographic inchHm BinWINE body size body size Soft selection top 50 in each patch Hard selectiOH top 50 selected selected regardless of patch Coarse NOTES Population Genetics SELECTION INTRODUCTION TO SELECTION READING Hedrick pp 113 169 204 235 next several lectures General Comments 0 What is selection 0 Elements of Adaptive Evolution 1 Development 2 Ecology 3 Population Genetics I Asexual inheritance Two clones A and a with numbers N A Na Clone A similarly for clone a viability 2 VA ecundity f A absolute fitness 2 WA VA NA WANA N t t39 ex genera 10H N WQNQ Another view genotype freguencies N 0 fre uenc ofA A p q y NANa I W 0 Next generation 17 NA NAWA pWA or p Ap N N NAWANaWa anm W A Ll W OWhatisWpWAan population mean tness average of WA and weighted by frequencies of A and a Coarse NOTES Population Genetics Yet another view Rate of evolution PW 1pWAWp A W WA pWA I pm W 1 W W ApEppV p or W W W W App1ppq Rate of evolution is product of 1 Selection 2 Genetics inheritance DIGRESSION quotAbsolute vs Relative fitnessquot Suppose WA and W are both divided by 2 WA WAZ Wu WuZ Mean fitness is halved W pillA an pWA2 qWa2 WZ Rate of gene frequency change is not affected gt WAIW 120 m 120 p AP p1 p Conclude Only ratio of WA and W contributes to gene frequency change mplication Only relative fitnesses needed to predict genotype frequency change Eg Can use W as astandard WA 2 WAWa wa E WlWa 1 0 NOTE Can go from WA gt wA butn0t wA gt WA Number vs frequency NQNA Nam Evolution within populations is better described by p than N 0 Only need relative fitnesses to follow p m w W so changes in Nwill be ignored IV 2 Coarse NOTES Population Genetics 0 Selection Coef cients Can write ratio WA Wa as 1 1 S WAw usquot is called the selection coefficient of a 0 s ranges from 1 to OC 0 Using this notation 1 1 s A 1S or p 10611 Appq S S Selection coefficients in the real world 0 Famous Example Biston belularia peppered moth Examples of Strong Selection DDT resistance in Drosophila San Jose scale Anopholes mosquitoes antibiotic resistance in bacteria pathogenesis of AIDS 0 Typical selection coefficients Newly arisen mutations in nature IV 3

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