Introductory seminar on mathematical population genetics

Transcription

1 Exercises Sheets Introductory seminar on mathematical population genetics WS 20/202 Kristan Schneider, Ada Akerman Ex Assume a single locus with alleles A and A 2 Denote the frequencies of the three (unordered genotypes) A A, A A 2, and A 2 A 2, in consecutive generations by X, 2Y, Z, and X, 2Y, Z, respectively Prove that X = X 2Y = 2Y Z = Z Y 2 = XZ (Characterization of Hardy-Weinberg proportions) In the following consider character that is determined by a single locus with two alleles A and A 2 in a random-mating population The measurements of the genotypes A A, A A 2, and A 2 A 2, are respectively m, m 2, m 22 m, σ 2 σa 2, σ2 D, denote mean measurement, variance, additive genetic variance, and dominance variance, respectively Ex 2 Calculate σ 2 D Ex 3 Show that the expression S = x 2 (m 2β ) 2 + 2x( x)(m 2 β β 2 ) 2 + ( x) 2 (m 22 m) 2 is minimized for β = xm + ( x)m 2 m 2 () β 2 = xm 2 + ( x)m 22 m 2 (weighted least square fit of m, m 2, m 22 by values 2β, β + β 2, 2β 2 ) Can you find an interpretation for ()? Ex 4 Calculate m, σ 2 A, σ2 D, σ2, for for the cases m = 2α, m 2 = α +α 2, m 22 = 2α 2 (no dominance in character), and m = m 2 m 22 (A is (completely) dominant)! Ex 5 Calculate σ 2, σ 2 A, σ2 D, for m = 2β, m 2 = β + β 2, m 22 = 2β 2, where β, β 2 given by ()! Ex 6 Assume m = 0, m 2 = h, m 22 = Calculate m, σ 2, σa 2, σ2 D, for h = 0, 2 0, Try to use Mathematica! Ex 7 Show that the correlation of full siblings is given by (Hint: use the table from the lecture!) corr(son, son) = σ2 A 2σ 2 + σ2 D 4σ 2 Ex 8 Show that the correlation of uncle and nephew is given by corr(uncle, nephew) = σ2 D 4σ 2 Ex 9 Show that the correlation of double first cousins is given by corr(double first cousins) = σ2 A 4σ 2 + σ2 D 6σ , and x =

2 In the following consider the one locus, two allele selection model Ex 0 Prove that the map x x is monotonic, ie, dx dx 0 Ex Use the above to conclude that every trajectory (x(t)) t N converges monotonically to one of the equilibria Ex 2 Prove that an equilibrium ˆx is (locally) stable if dx dx < and unstable if ˆx dx dx > ˆx Ex 3 Show that the (boundary) maxima/minima of w = w(x) in [0, ] coincide with the satble/unstable equilibria Ex 4 Mean fitness increases theorem (MFIT): Use the above to show w w and w = w if and only if x = x, i,e, at equilibrium Alternative way: show w 2 (w w) 0 Ex 5 Plot the trajectory x(t) for various values of x(0), s and h with Mathematica Proceed as follows: Define the map f[x ]=x and use the commands l=nestlist[f, x0, Tmax] and ListPlot[l] Discuss the observations! There are plenty of ways to derive the trajectory x(t) in Mathematica Can you find an alternative way as outlined below, for instance using Do, While, or For-loops Ex 6 Assume the frequency of A raises from x to x 2 Then the time is approximated by x 2 x sx( x) ( x + h( 2x) )dx Use Mathematica to check how accurate this approximations is for various values of x, x 2, s and h Do you see a trend? Try to explain it! Ex 7 Consider the simple model of non-random mating (which can be interpreted as selfing or alternatively as inbreeding or as assortative mating) Show that lim X (t) = x t 2H, lim X 2(t) = H, t and lim X 22 (t) = x 4( f)x( x) t 2H, where H = 2 f Ex 8 Derive the mutation-selection equation for two alleles Measure allele frequencies among zygotes, before selection occurs This is followed by the production of germ cells during which mutation occurs, and the formation of zygotes Discuss why it is convenient to measure allele frequencies among zygotes Ex 9 Implement the Wright-Fisher model in Mathematica, and explore several different values for N and X(0) Proceed as follows: Define the map f[x0 ]=RandomInteger[BinomialDistribution [M,x0/M]] and use the commands l=nestlist[f,x0,tmax] and ListPlot[l] Discuss the observations! Allele A will go to fixation or de out Try to estimate the probability that A dies out for given 2N and X(0) Ex 20 Show by induction that the solution of the haploid selection model given by x i (t) = x i(0)wi t n x j (0)wj t j= (i =, n) (2) 2

3 Ex 2 Show that the haploid selection model transforms into a linear system of difference equations by substituting x by x, and x i by x i x ((i=2, n)) Show this way that the solution is given by (2) Ex 22 Consider the pairwise interaction model (PIM) with two alleles, ie, w ij = 2 k,l= α ij,klx k x l Assume the interaction coefficients are given by the matrix What are the stable and unstable equilibria? Do all trajectories converge to an equilibrium? Does the mean fitness increases theorem (MFIT) hold? Do the local maxima and minima of w correspond to equilibria? (Hint: consider the map x x, and regard w as a function in x) Ex 23 Assuming finitely many equilibria, how many equilibria can exist for the pairwise interaction model with two alleles? Can you construct an example with the maximum number of equilibria? Ex 24 Consider the PIM with the following (symmetric) interaction coefficients Prove for this example that the MFIT holds and that the stable and unstable equilibria correspond to local maxima and minima of w, respectively Ex 25 Consider the PIM with asymmetric interaction coefficients Prove for this example that the MFIT holds and that the stable and unstable equilibria correspond to local maxima and minima of w, respectively Ex 26 Consider the PIM with asymmetric interaction coefficients Prove that w is decreasing along all trajectories Ex 27 Derive the equations for the two-locus two-allele selection model Use the following selection cycle: Gamete frequencies are measured among zygotes (newborns) Because of random mating, the frequency of genotype A i B j /A k B l is the product of the respective gamete frequencies Then selection act on genotypes through differential viabilities Then mating and recombination occurs Discuss why this life cycle makes sense 3

4 Ex 28 Derive for the additive selection model (a ij, and b ij, are the effects of A i A j, and B i B j, respectively), that the mean fitness is given by w = a + b, where a and b are the single locus mean fitnesses Ex 29 Find an example in where the mean fitness increases theorem does not hold (Hint: use multiplicative fitnesses, large fitness differences and a small recombination rate) Try to solve the problem with Mathematica Try to plot a trajectory along which w is not increasing use x 4 = x x 2 x 3, to represent the trajectory in R 3 Can you plot the trajectory also in the thetraeder? Ex 30 Show the relation w 2 D = D [ W W 4 + x 2 x 3 (a a 22 a 2 2)(b b 22 b 2 2) ra 2 b 2 w ] (3) for the multiplicative selection model Ex 3 Show that the term in the brackets in (3) is 0 (Hint: use the fact 0 < r 2 ) If it is too difficult, let Mathematica help you (command Reduce for versions 60 and higher, or InequalitySolve for older versions) Ex 32 Show that a product equilibrium ˆx = ˆp ˆq, ˆx 2 = ˆp ˆq 2,ˆx 3 = ˆp 2ˆq,ˆx 4 = ˆp 2ˆq 2, is in linkage equilibrium a 2 a 22 Ex 33 Use Mathematica to show that the product equilibrium with ˆp = 2a 2 a a 22 and ˆq = b 2 b 22 2b 2 b b 22 of the additive selection model is also an equilibrium for the multiplicative selection model Ex 34 Assume the haploid selection model with two alleles, and fitnesses w = + s and w 2 = How long does it take for the beneficial allele A to increase in frequency from ε to ε 2 Ex 35 Suppose the hitchhiking model and assume the beneficial mutation p 0 occurs initially in a single copy, ie, p 0 small Use Mathematica, for given p 0 and s to plot Q t and H e (t) := E ( 2Q t ( Q t ) ) E ( ) as a function of r for different t, ie, use Mathematica to study how the 2R 0 ( R 0 ) hitchhiking effect changes with time Assume, eg, p 0 = 0 8, 0 6, 0 4, and s = 00, s = 005, and s = 0 Use the time estimate from Ex 34 to get an idea for the when the equilibrium heterozygosity is approximately changed Study in particular the change in the early and late phase of the sweep Ex 36 For the hitchhiking model derive R t, and plot it as a function of r using Mathematica Ex 37 Derive the relative expected heterozygosity at the neutral locus among deleterious genotypes for the hitchhiking model, ie, E ( 2R t ( R t ) ) E ( ) Is it independent of the initial distribution of 2R 0 ( R 0 ) R 0 (assuming A initially occurred in a single copy)? Plot it using Mathematica Ex 38 In the hitchhiking model the heterozygosity is given by 2q t ( q t ), where q t is the frequency of the neutral allele B at time t Derive the expected relative heterozygosity E ( 2q t ( q t ) ) E ( 2R 0 ( R 0 ) ) (assuming A initially occurred in a single copy)? Plot it using Mathematica At what time will the sweep become apparent? Thin about what the results imply for incomplete sweeps, ie, selective sweeps that are currently happening! 4

5 Ex 39 Consider the island migration model with two alleles, and fitnesses w = + s, w 2 = + hs, w 22 = s (0 < s <, h, m (0, )) Show that the equilibria are given by: where ν = m/s x ± = [ + 3h + m( h) ± ] ( h) 4h 2 ( + m) 2 + 8h(ν + m), (4) Ex 40 Plot the solutions (4) with Mathematica as a function of µ := ν between (a) h < h 0 and (b) h 0 h with h 0 = +m 3 m for given h Distinguish Ex 4 Derive the stability properties of the two allele island migration model Define µ = and µ 2 = 8h ( h) 2 (+m) 2 +8hm Show that x = 0 is stable for µ < µ and unstable otherwise Furthermore show that for h < h 0 : (i) x exists for µ 2 µ, furthermore it is stable, (ii) x + exists for µ 2 µ < µ, and is unstable, and for h h 0 (iii) x exists for µ µ, furthermore it is stable (iv) x + does not exist +h( m) Ex 42 Show that the linearized dynamics in the two-allele soft-selection migration model near x = (0,, 0) are given by x = Qx, where Q = (q i,j ), with q ij = m ij v j with v j = w i,22 w i,2 Ex 43 Show that the linearized dynamics in the two-allele hard-selection migration model near x = (0,, 0) are given by x = Rx, where R = (r i,j ), with r ij = P c jw j,2 m ji k c kw k,22 m ki Ex 44 Let A and B be n n-matrices Show that the matrices AB and BA have the same eigenvalues Show further, if λ 0 is an eigenvalue of AB, then the corresponding Eigenspaces of lambda for the matrices AB and BA have the same dimension, ie, dim E(AB, λ) = dim E(BA, λ) Is this also true for λ = 0? (Hint: assume λ is an eigenvalue for AB with eigenvector x Be careful in the case λ = 0) 5