Wright–Fisher Model

The Wright–Fisher model is the canonical stochastic mathematical framework in population genetics and evolutionary biology used to describe random genetic drift—the change in allele frequencies across generations driven by finite-population binomial sampling of gametes. Formulated independently in 1930–1931 by the British statistician and geneticist Ronald Fisher (in The Genetical Theory of Natural Selection) and the American evolutionary biologist Sewall Wright (in "Evolution in Mendelian Populations"), the Wright–Fisher model replaces the deterministic infinite-population assumptions of Hardy–Weinberg equilibrium (N → ∞, where allele frequency p remains constant forever in the absence of selection, mutation, or migration) with a discrete-time Markov chain over a finite diploid population of size N (2N gene copies), establishing the exact transition probabilities for heterozygosity decay, allele fixation, time to fixation, and backward-in-time coalescent theory.

🧬 Interactive Population Genetics Simulation: Run live Wright–Fisher binomial drift trajectories across different effective population sizes (N_e) and selection pressures (s) in Allele Drift.

The Idealized Wright–Fisher Population

In its baseline neutral formulation, the diploid Wright–Fisher model specifies five simplifying biological assumptions:

1. Constant Finite Population Size (N): Each generation consists of N diploid hermaphroditic (or monoecious) individuals, corresponding to 2*N* alleles (gene copies) at an autosomal locus (or N gene copies in a haploid population).
2. Discrete, Non-Overlapping Generations (t = 0, 1, 2, …): All parents in generation t reproduce simultaneously and die afterward, being replaced by the N offspring of generation t + 1 (characteristic of annual plants or univoltine insects).
3. Infinite Gamete Pool and Random Sampling With Replacement: In generation t, each adult contributes an equally large number of gametes to a common panmictic gamete pool. If i out of the 2N gene copies in generation t belong to allele A₁ (so the current allele frequency is p_t = i / 2*N*) and the remaining 2*N* − i copies belong to allele A₂ (frequency q_t = 1 − p_t), then generation t + 1 is formed by drawing 2*N* independent Bernoulli trials with success probability p_t from that gamete pool.
4. Selective Neutrality, No Mutation, and No Migration (Baseline Model): Both alleles A₁ and A₂ confer equal biological fitness (s = 0), with mutation rate μ = 0 and migration rate m = 0 (though all three evolutionary forces can be incorporated directly into the sampling probability p_t*).

Exact Binomial Transition Probability P_ij

Under these assumptions, the number of A₁ alleles X_t+1 ∈ {0, 1, 2, …, 2*N*} in generation t + 1 given X_t = i (so p = i / 2*N*) follows the binomial distribution B(2*N*, p):

P_ij = Pr(X_t+1 = j | X_t = i) = [ (2*N*)! / ( j! (2*N* − j)! ) ] · (i / 2*N*)^j · (1 − i / 2*N*)^{2*N* − j}

Because the mean and variance of a binomial random variable j ~ B(2*N*, p) are E[j] = 2*N p* and Var(j) = 2*N p*(1 − p), dividing by 2*N* yields the conditional expectation and single-generation Wright–Fisher drift variance of the allele frequency p_t+1 = j / 2*N*:

• Expected Allele Frequency (Martingale Property): E[p_t+1 | p_t] = p_t (genetic drift has zero directional bias).
• Single-Generation Sampling Variance: Var(p_t+1 | p_t) = p_t (1 − p_t) / (2*N*)

1. Absorbing Boundaries (j = 0 and j = 2*N*) and Ultimate Fixation Probability

In the absence of new mutation (μ = 0), the state space {0, 1, 2, …, 2*N*} of the Wright–Fisher Markov chain possesses two absorbing boundary states:

• Loss (j = 0, p = 0): Setting i = 0 in the binomial formula gives P_0,0 = 1; once allele A₁ is lost from the population, it cannot reappear without mutation.
• Fixation (j = 2*N*, p = 1): Setting i = 2*N* gives P_2*N*,2*N* = 1; once allele A₁ reaches 100% frequency, every individual is homozygous A₁A₁.

Because every interior polymorphic state (1 ≤ i ≤ 2*N* − 1) is transient, a finite neutral population with probability 1 eventually loses all genetic polymorphism and becomes fixed at either p = 0 or p = 1 as t → ∞. By the martingale theorem (E[p_∞] = E[p₀] = p₀ = 1 · Π(p₀) + 0 · [1 − Π(p₀)]), the probability Π(p₀) that a neutral allele starting at initial frequency p₀ is the one that ultimately reaches fixation (p = 1) equals its initial frequency:

Π(p₀) = p₀

For a newly arising neutral mutation present as a single copy in one diploid individual (i = 1 out of 2*N* gene copies), the initial frequency and ultimate fixation probability are both:

Π(new neutral mutation) = p₀ = 1 / (2*N*)

As Motoo Kimura showed in 1968 to found the Neutral Theory of Molecular Evolution, if neutral mutations arise at rate μ₀ per gamete per generation across 2N gene copies, the total number of new neutral mutations entering the population per generation is 2*N μ*₀, and since each has probability 1/(2*N*) of fixing, the long-term rate of neutral molecular substitution (k = 2*N μ*₀ × [1 / (2*N*)] = μ₀) is independent of population size N!

2. Exponential Decay of Expected Heterozygosity (H_t)

Let H_t = 2 p_t (1 − p_t) denote the expected Hardy–Weinberg heterozygosity (the probability that two alleles drawn at random without replacement from generation t are distinct, also termed gene diversity or nucleotide diversity π). When two alleles are drawn at random to form an offspring in generation t + 1, there is a probability 1 / (2*N*) that both alleles are copied from the same parental gene copy in generation t (making them Identical by Descent, IBD, and necessarily homozygous), and a complementary probability 1 − 1 / (2*N*) that they are drawn from two distinct parental gene copies (retaining the previous generation's heterozygosity H_t):

H_t+1 = ( 1 − 1 / [2*N*] ) · H_t

Iterating across t generations from an initial heterozygosity H₀ yields the geometric / exponential decay of genetic diversity under pure drift:

H_t = H₀ · ( 1 − 1 / [2*N*] )^t ≈ H₀ · exp( − t / [2*N*] )

Consequently, genetic diversity decays with a characteristic e-folding time of 2*N* generations (and a half-life of 2*N* ln 2 ≈ 1.386 N generations). Small populations (such as an island founder population or a conservation bottleneck with N = 25) lose half their heterozygosity in just ~35 generations, whereas large populations (N = 10⁵) retain polymorphism for hundreds of thousands of generations.

Why Real Populations Have N_e < N_census

Real biological populations depart from one or more assumptions of the idealized Wright–Fisher model (unequal sex ratios, variance in family size where a few individuals produce most offspring, overlapping generations, or boom-and-bust population bottlenecks). In 1931, Sewall Wright defined the effective population size (N_e) as the size of an idealized Wright–Fisher population that experiences the same rate of heterozygosity decay or allele-frequency drift variance as the actual census population (N):

• Unequal Breeding Sex Ratio (N_m males, N_f females): N_e = 4 N_m N_f / (N_m + N_f).
• Fluctuating Population Size Across T Generations (Bottlenecks): governed by the harmonic mean 1/N_e = (1/T) ∑ (1/N_t), meaning a single population bottleneck dominates long-term genetic drift.
• Non-Poisson Variance in Offspring Number (V_k > 2): N_e = (4*N* − 2) / (V_k + 2).

Kimura's Forward Diffusion Equation and Kingman's Backward Coalescent

Because raising a (2*N* + 1) × (2*N* + 1) binomial transition matrix P to the t-th power is computationally intractable when N is large, two complementary mathematical duals of the Wright–Fisher model form the foundation of modern computational genomics:

1. Forward-in-Time Kimura Diffusion Approximation (1955–1964): Rescaling time in units of 2N_e generations (τ = t / 2*N*_e) as N_e → ∞ converts the discrete Markov chain into the continuous Kolmogorov forward (Fokker–Planck) partial differential equation for the probability density φ(p, t) of allele frequency p under drift, directional selection (s), and mutation (μ, ν):

∂*φ*(p, t) / ∂*t* = − ∂[M(p) φ] / ∂*p* + ½ ∂²[V(p) φ] / ∂*p*²

where M(p) = s p(1 − p) − μ p + ν(1 − p) is the deterministic directional drift velocity and V(p) = p(1 − p) / (2*N*_e) is the stochastic diffusion variance. Solving the backward Kolmogorov equation yields Kimura's famous formula for the fixation probability of a newly arising mutation with additive selection coefficient s:

Π(p₀) = [ 1 − exp(−4 N_e s p₀) ] / [ 1 − exp(−4 N_e s) ]

which shows that natural selection dominates over random genetic drift if and only if |4*N*_e*s*| ≫ 1 (the nearly neutral regime of Tomoko Ohta occurs when |4*N*_e*s*| ≤ 1).

1. Backward-in-Time Kingman Coalescent (1982): Rather than tracking all 2N gene copies forward in time, J. F. C. Kingman traced the genealogy of a sample of k ≪ 2*N* DNA sequences backward in time through the Wright–Fisher pedigree. Any two lineages have probability 1/(2*N*_e) per generation of picking the same parent (coalescing into a common ancestor), and among k(k − 1)/2 pairs the waiting time T_k until the first coalescence follows an exponential distribution with rate k(k − 1) / (4*N*_e) (mean waiting time E[T_k] = 4*N*_e / [k(k − 1)] generations), reaching the Most Recent Common Ancestor (MRCA) of the entire sample in an average of 4*N*_e(1 − 1/k) ≈ 4*N*_e generations.

Why do even beneficial mutations (s > 0) get lost by genetic drift the vast majority of the time when they first arise as a single copy?

When a brand-new beneficial mutation (s > 0, e.g., a +1% fitness advantage, s = 0.01) arises as a single copy in one diploid individual (p₀ = 1 / [2*N*_e]) inside a large population, that single heterozygous carrier could easily fail to transmit the mutant chromosome to its offspring simply due to Mendelian 50:50 coin-flipping or accidental mortality during the first few generations. Substituting p₀ = 1/(2*N*_e) into Kimura's fixation formula (or J. B. S. Haldane's 1927 branching-process derivation) for 2*N*_es ≫ 1 and small s gives an ultimate fixation probability of Π ≈ 2*s*. Thus, a new mutation conferring a 1% selective advantage (s = 0.01) has only a 2*s* = 2% chance of surviving early stochastic drift to reach fixation, and a 98% probability of being lost by pure bad luck within its first ~10 generations! Only after a beneficial allele survives this initial stochastic "establishment threshold" and reaches copy number i ≫ 1/(2*s*) (frequency p ≫ 1/[4*N*_es]) does deterministic logistic selection take over to drive a selective sweep.

How does the Wright–Fisher model differ from the Moran model of genetic drift?

Whereas the Wright–Fisher model assumes discrete, non-overlapping generations in which all 2N gene copies are replaced simultaneously in a single binomial step every generation, the Moran model (introduced by Patrick Moran in 1958) is a continuous-time birth-death Markov chain with overlapping generations: at each elementary time step, one individual is chosen at random to reproduce (duplicating its allele) and one individual is chosen at random to die, changing the allele count i by at most +1, 0, or −1 (a birth-death walk whose exact eigenvalues and fixation times can be solved without diffusion approximations). When 2*N* Moran birth-death steps are equated to 1 Wright–Fisher generation, the Moran model produces the exact same fixation probabilities and coalescent tree topology as the Wright–Fisher model, with exactly twice the drift variance (corresponding to N_e,Moran = ½ N_WF).

Why does a neutral mutation that successfully reaches fixation take an average of 4*N*_e generations to fix?

At first glance, it seems paradoxical that a neutral mutation starting at p₀ = 1/(2*N*_e) has zero directional selection (s = 0) yet reaches 100% frequency (p = 1) in an average of 4*N*_e generations (rather than taking infinitely long). The resolution lies in conditioning on eventual fixation: the vast majority (1 − 1/[2*N*_e]) of neutral mutations drift to extinction (p = 0) very quickly (in an average of ~2 ln(2*N*_e) generations). The tiny fraction (1/[2*N*_e]) of neutral lineages that do happen to reach p = 1 without ever hitting the absorbing boundary at p = 0 must have experienced a sustained lucky streak of upward binomial sampling steps; conditioning the backward Kimura diffusion equation on never hitting p = 0 introduces an effective upward conditioning drift that carries the winning neutral lineage from 1/(2*N*_e) to 1.0 in E[t_fix] = 4*N*_e generations (identical to the expected coalescent time 4*N*_e back to the Most Recent Common Ancestor of the population).

1. Fisher, R. A. (1930). The Genetical Theory of Natural Selection. Clarendon Press, Oxford. https://doi.org/10.5962/bhl.title.27468
2. Wright, S. (1931). "Evolution in Mendelian populations." Genetics, 16(2), 97–159. https://doi.org/10.1093/genetics/16.2.97
3. Kimura, M. (1962). "On the probability of fixation of mutant genes in a population." Genetics, 47(6), 713–719. https://doi.org/10.1093/genetics/47.6.713
4. Kingman, J. F. C. (1982). "The coalescent." Stochastic Processes and their Applications, 13(3), 235–248. https://doi.org/10.1016/0304-4149(82)90011-490011-4)