A MARKOV PROCESS OF GENE FREQUENCY CHANGE IN A

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1 A MARKOV PROCESS OF GENE FREQUENCY CHANGE IN A GEOGRAPHICALLY STRUCTURED POPULATION TAKE0 MARUYAMA National Instituie of Genetics, Mishima, Japan and Cenier for Demographic and Population Genetics, University of Texas at Houston, Texas Manuscript received February 9, 1972 Revised copy received September 26, 1973 ABSTRACT A Markov process (chain) of gene frequency change is derived for a geographically-structured model of a population. The population consists of colonies which are connected by migration. Selection operates in each colony independently. It is shown that there exists a stochastic clock that transforms the originally complicated process of gene frequency change to a random walk which is independent of the geographical structure of the population. The time parameter is a local random time that is dependent on the sample path. In fact, if the alleles are selectively neutral, the time parameter is exactly equal to the sum of the average local genetic variation appearing in the population, and otherwise they are approximately equal. The Kolmogorov forward and backward equations of the process are obtained. As a limit of large population size, a diffusion process is derived. The transition probabilities of the Markov chain and of the diffusion process are obtained explicitly. Certain quantities of biological interest are shown to be independent of the population structure. The quantities are the fixation probability of a mutant, the sum of the average local genetic variation and the variation summed over the generations in which the gene frequency in the whole population assumes a specified value. HE stochastic model of gene frequency change put forward by FISHER (1930) WRIGHT (1931 and later) has become increasingly important in population genetics. For reviews, see MORAN (1962); KIMURA (1964); EWENS (1969); WRIGHT (1969); CROW and KIMURA (1970); CAVALLI-SFORZA and BODMER (1971); KIMURA and OHTA (1971). However, most of the theory has dealt with random mating populations, while hardly any natural population meets this assumption. When we take into consideration the effect of geographical structure which prevents random mating, the mathematics becomes very difficult. WRIGHT ( 1931, 1943, 1951 ) pioneered in the analysis of structured populations. Later MALBCOT (1948, 1967) developed a new model and contributed greatly in this area. KIMURA and WEISS (1964), and BODMER and CAVALLI- SFORZA (1968) have developed still new models. Although different models and different methods are used, the essential point in these works is to investigate the genetic similarity between different localities. WRIGHT (1931,1937,1938) and KIMURA (1955) have dealt with the stochastic Genetm 76: February, 1974

2 368 T. MARUYAMA process of gece frequency change and of gene frequency distribution in a panmictic population of finite size. WRIGHT obtained the stationary distribution when mutation in both directions is assumed, and KIMURA obtained the time-dependent distribution of gene frequency change under the assumption of no mutation. EWENS (1963) obtained the mean of the sojourn time, that is, the duration of time during which the population assumes a particular gene frequency. Although the geographical structure of the population may not necessarily alter significantly the theory based on the assumption of random mating, it is certainly worthwhile to investigate the situation. The purposes of this paper is to develop a Markov process of gene frequency change in a special case and to investigate its properties. Throughout this paper, we consider one locus where two alleles, A and a, are segregating. The organism is haploid. The fitnesses of A and a are 1 + s and 1 respectively, and the fitnesses are constant in time and in the geographical structure. No mutation or migration from the outside occurs during the time under consideration. 2. DERIVATION OF THE MARKOV PROCESS The population consists of colonies. Migration occurs and connects colonies genetically. Each individual has a negative exponential life time distribution with unit mean. When an individual dies, it is immediately replaced by a new individual born in the same colony. This assumption implies that the total population size is constant. We denote the size of the total population by N. When an individual dies, each preexisting individual in the colony has a chance of being chosen to reproduce which is proportional to that individual's fitness. This is a geographically-structured version of the MORAN (1958) model. Let Nt,k be the size of colony IC at time t and Mt be the number of A genes in the entire population at time t. Then the random variable {Mt} is a stochastic process whose state space is [0, 1,2,..., NI. This process with time parameter t can be very complicated, but we will introduce a new time measure which is sample-path-dependent local time based on an additive functional and which makes the process into a Markov process, actually a random walk. First, note that, in any short time interval, At, two or more death-birth events occur with probability of order (At) z. We may therefore assume that the random variable Mt changes by at most one at any moment. Now consider a particular moment when a birth-death occurs, and ask what is the probability, given Mt = i immediately before this event, that the value of Mt changes from i to if+ 1 through this event? Let ik be the number of A genes in colony k at this moment, i.e., i = z ik. k The probability that the death-birth occurs in colony k whose size is Nt,k, is Nt,k/N, and given this, the probability, qi, +1, that one a in the colony dies and ' I it is replaced by one A, 'is The probability, qik,i,-;, that one A dies and replaced by one a, is

3 STRUCTURED POPULATION MARKOV PROCESS 369 The probability, qib,{&, that no change in the number of A occurs, is Therefore the probability, H (t), that the value of Mt changes through the deathbirth event is H(t) can be defined for all t. We assume that migration takes place instantaneously and it is independent of death-birth events. The value of H(t) changes discontinuously as migration or a death-birth event occurs, and otherwise H(t) remains constant. For example, assume that the population consists of two colonies, say 1 and 2, and that each of the two colonies has two A genes and two a genes and s = 0. Under this situation, H(t) = (2/8) (4 x 2 ~2 X 2/ X 2 X 2/16) = x. Now suppose that one A gene moves from colony 1 to colony 2, then H(t) = (2/8) (3 X 1 x 2/9 + 5 x 3 x 2/25) = 7/15. It is important that the effects of migration and of the population structure are incorporated in H(t). Note that the quantity in the square bracket is the probability that two randomly chosen gametes from a colony with replacement are different types. We call this quantity the local genetic variation. Therefore, H(t) is equal to the average local genetic variation if the alleles are selectively neutral, and otherwise they are approximately equal. We now ask, if Mt is changed, what is the probability that Mt is changed from i to i + l? This conditional probability is equal to Through simple algebra, this becomes Ifs -- 2+s. Similarly, the conditional probability that Mt is changed from i to i - 1 is 1 2+s We should note two properties here: (1) these conditional probabilities are independent of state i; (2) they are independent of the geographical structure of the population. Property (1) was shown by MORAN (1959, 1962) "for a panmictic case, and it was used to obtain the ultimate fixation probability of a mutant gene.

4 3 70 T. MARUYAMA Consider a short time interval At. Then the probability of occurrence of one death-birth event is NAt + o(4t). Let Rt be the probability that Mt is unchanged for time interval t. then The first term on the right side of the above equation is the probability that Mt remains unchanged until time t and no death-birth occurs in (t, t + At). The second term is the probability that Mt remains until t and one death-birth occurs in (t, t '+ At) but Mt is unchanged. From this equation we have and therefore drt --NH(t), Rtdt- Et t e -xj H(t)dt (2-2) 0 Consider the stochastic process as a collection of sample path {U}, and let where w indicates a particular sample path and H(w, t) is the same quantity as H([) of (2-1) associated with sample path.u. We assume that no part of the population is completely separated from the other. Consider this T(0,t) = T as a new time measure. The T is a local time based on the functional H(t) of the original process. Because the T depends on the history of the sample path, it is often called a stochastic clock. For a full account of this kind of time substitution, the reader is referred to ITO and MCKFAN (1965, chapters 2,5, and 6). From here on, we shall use the T as the time parameter and construct a Markov process. Let Mr be the number of A genes in the entire population at time T. The M7 is different from the Mt and it is not to be confused with the M,. Let RI be the probability that M7 is unchanged for time interval 7. Then, from (2-2) together with (2-3), we have R 7 = e-n7. (2-4) This process Mr is characterized as follows: (1) The state space is [0, 1,2,... A'] ; (2) The sojourn time at each state at each visit is exponentially distributed with mean 1/N, provided 0 < i < N. This follows from (2-4): (3) The conditional probability that Mr is changed from i to i + i is (1 + s)/(2 + s), and the conditional probability that M is changed from i to i - 1 is 1/(2 4- s), provided 0 < i < N. These conditional probabilities are independent of the state and of the geographical structure of the population. Let be the probability, given that Mr is i at time 0, that it is j at time T. From the properties (1)-(3) above, we have

5 STRUCTURED POPULATION MARKOV PROCESS 371 q..= z q q,. for O < ~ < T, 7,%,1 r&&k &h,~ and M7 is a Markov process. Then we have, for small AT, q7-47,irj = e-n*,q7,i,j + (1- e-n4t) +o(ar), O<i,j< N chqr,i-l,j + W,,i+l,j 1 where A = (1 4- s)/(2 4- s) and p = 1/(2 + s). From this equation, we derive dqr,i,j --- -N[I-q7,i,j+Xq7,i-l,j++q,,i+l,j I, o<i,i<n (2-5) dr the Kolmogorov backward equation. Using the additive functional (2-3), this equation can be also derived from DYNKIN S formula, (cf. DYNKIN 1965 vol. 1, p. 133). We can derive the Kolmogorov forward equation similarly: These differential equations characterize the Markov process, and all the information concerning the process can be obtained from these equations. It is often convenient to express the system of equation in matrix notation. We define the following matrices: Q, = [q,,i,jl (N-WN-~) 0 < i, i < N ; A= A-IpO A O h - l (N-l)S (N-1) where x = (1*+ s)/(2:+ s) and p = 1/(2 + s). Then the system of differential equations (2-5) becomes and (2-6) becomes where AS is the transpose of A. dq NAQ, dt (2-7) -- dq --NASQ (2-8) dt

6 3 752 T. MARUYAMA We shall next derive a diffusion process as a limit for large N. The backward equations (2-5) can be rewritten as dqr,i,j ~ - - ds 2 N[q7,i-i,j - 2q7,i,j + qr,i+i,j 1 Let T be a new time measure such that NT = T, and Ns = S where we assume S is a constant. Let qn (T, i, i -) i = q... Then (2-9) can be written as N N 7,573 Therefore as N + a, this difference equation converges to - --mi, %m(t,x,y) at 2 8x2 2 2X - 1 8% (T x Y ) + s a%j(t,%y). (2-10) The derivation of this equation is not mathematically rigorous, but rigorous derivation is possible (cf. KAC 1947). The Markov process governed by (2-10) is a Brownian motion. 3. SOME SOLUTIONS Based on the stochastic clock, whose time-rate depends on the total amount of the average local genetic variation which has appeared in the sample path, we have derived a Markov process of the gene frequency change. It is governed by differential equations (2-5),(2-6): (2-7),(2-8),or(2-10). Here we shall deal with (2-7) and (2-lo), and investigate biologically interesting questions. These equations together with definitions (2-1) and (2-3) tell us that, if we measure the time by 7 the stochastic process becomes a Markov process that is independent of the geographical structure of the population and furthermore, that it is a random walk (or approximately a Brownian motion) on the finite interval [ 1,2,..., N - I], (or (0,l)) with exit boundaries at both ends. A standard procedure in dealing with equation (2-7) is to obtain a complete spectral analysis of matrix A. The right eigenvectors of A are the column vectors - 37rk P N-1, (N-l)7rk * (A) sin ), k=1,2,..., N-I. E.l N P (3-1)

7 STRUCTURED POPULATION MARKOV PROCESS 373 where a* indicates the transpose of any row vector a. It can be easily shown that where Now let - 2 dk = (($) sin, h N-l/N (+) sin N Aek = hkek xk h,=-112~rpcos-. (3-2) N rrk (YP sin (N - 1)xk. 2xk, (5) ), k=l,2,..., N-I. These row vectors are the left eigenvectors of A, i.e., dha = Xkdk 3xk 3/2 sin,..., where the eigenvalues are the same as (3-2). Note that {ek}, or {d,}, themselves are not orthogonal systems, but {ek} and {d,} are biorthogonal, i.e., where Skk = 1 and Skl = 0 if k f 1. With these eigenvectors and eigenvalues, we can easily obtain the fundamental solution of the process of equation (2-7), that is the transition probability: where 0 indicates the direct product of two vectors. This can be written as The fundamental solutions of (2-10) are m qm(t, 2, y) = 2e-S(~9)/2 2 e-%((k29+wz))t/4 sin kxx sin kxy. k=1 The Markov process governed by qa (T, 5, y) is independent of the geographical structure and it is a Brownian motion. The ultimate fixation probability, ui, of allele A in the whole population is given by the solution of equation Au=O where U = (ul, u2,..., un-l) *, in which * indicates the transpose:

8 3 74 T. MARUYAMA where i is the initial number of A in the whole population. This is the same as MORAN (1959) in which the formula was derived for a panmictic population. Therefore the fixation probability is independent of the geographical structure proposed in this paper. If we replace si and sn of (3-3) by Sx and S respectively, the right side in (3-3) is the solution of the differential equation with boundary conditions f (0) = 0 and f (1) = 1 ( KIMURA 1962). The independence of the fixation probability here was derived under the assumption of constant size of the total population and local random mating, but importantly the structure of the population was not assumed to be fixed. Even if population structure depends on the genetic constitution, as would occur if genetically similar individuals tend to live closer to one another, or the structure depends on the gene frequency, the fixation probability is unaltered. This independence of the fixation probability was suggested by MARUYAMA (1970), based on an approximation. However it was not clear whether the independence is exact or an approximation. The first exit time of the process from [I, 2,..., N - 11 is the solution of equation Affl=O where 1 = (1,1,..., 1) * and 0 = (0, 0, -.., O)*. The i-th entry, fi, of f is the mean duration of time, measured by T, until allele A is fixed in the population or A is lost from the population. (For the first exit time in the case of a panmictic population, the reader is referred I to EWENS 1963.) If we let f (-) fi and let N become large while S = Ns stays N constant, f (5) becomes the solution of differential equation 1 d2f S df l=o: 2 dx2 2 dx The biological meaning of fi is that it is the sum of the average local genetic variation that appears in the population while the two alleles are segregating, provided that initially there are i A genes. This is exact if s = 0 and is approximately so if s# 0, (cf. (2-3)). The second moment, f i2), of the first exit time measured by T is the solution of Af@) + 2f = 0

9 STRUCTURED POPULATION MARKOV PROCESS 3 75 where f is the mean obtained above and f(z) = ( f?), fi2),..., fe!l). The higher moments are similarly obtained: AfW) + nf(n-1) = 0. In the above calculations, we included both kinds of sample paths in which A is established in the whole population and in which A is lost. We can make a distinction between the two kinds, and calculate the quantities for those paths in which A is fixed, (or in which A is lost). Let f i be the mean exit time in the path in which A is fixed. Then g = (ufl, uzfz,. *., un-lj -l) *, is the solution of equation Ag+u=O where U is the vector of the fixation probabilities, U+ Biologically, f+ is the expectation of the sum of the average local genetic variation given that A is established in the population. The solution is uj sin- 1 N-1 N-1 A7. irk. irk sm-, foro<i<n where ui is the fixation probability given in (3-3). The higher moments are the solution of Ag( ) f ng(n-1) = 0 where g(o) = U the fixation probability vector. As N becomes large, while S = Ns remains constant, fl ($) = fi and U (&) = ui satisfy the differential equation Finally we shall obtain the sojourn time at a given state, or in a given range of gene frequency. For 0 < i, j < N, let ai5 be the sojourn time, measured by 7, at state i while A and a are segregating in the population, provided the process starts from state i. Let Q1ij be the sojourn time among the paths in which the fixation of A occurs. Let = [air] (N-l) (N-l) and a1 = [alijui] (N-l) (N-l). Then these matrices are and where U = [64+i]. a = - A-1 Ql = - A- U

10 376 T. MARUYAMA As N becomes large, aij z- (1- e-zs(n-i) ) (1- elzsi) if j>i s( 1 - (3-5) 1.3 == 2i(N - j ) N if j>i a.. = ~- 2j(N - i) 23 if j<i N Formula (3-6) is a well-known result for a panmictic population, (cf. EWENS i i 1963). If we y) -) = aij, this satisfies the following differ- N'N entia1 equation wherewhere 6 (.) is the Dirac delta function. For large N, alii can be obtained as the solution of Approximation formulas (3-4), (3-5), and (3-6) were obtained in MARUYAMA (1972), where differential equation (2-10) and the method described in DYNKIN (1965, vol. 2, pp ) were used. Before concluding this paper, I would like to emphasize that the sojourn time, measured by the number of generations, the time until fixation of an allele, the distribution of the gene frequencies and the rate of decay of genetic variability, depends upon the population structure. I thank the referees for many criticism and suggestions, and for correcting the English. I also thank PROFESSOR T. Umo of the Mathematics Department, University of Tokyo, and PROFESSOR T. G. KURTZ of University of Wisconsin, for helpful conversation connected with the stochastic clock used in the paper. This work was supported in part by U.S. Public Health Service Research Grants GM and GM LITERATURE CITED BODMER, W. F. and L. L. CAVALLI-SFORW, 1968 A migration matrix model for the study of random genetic drift. Genetics 59:

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