Mathematical Methods of Population Genetics

where n is the population size, index i refers to the class of organisms {A_i A_j}_{j=1,2,..., K} , which contain the gene A_i. The population is supposed to be a panmictic⁴⁾ one: during reproduction the new gene combinations are chosen randomly throughout in the whole population. For panmictic populations the Hardy-Weinberg principle can be approximately applied [1]:

Eqs. (2) implies, that during mating the genotypes are formed proportionally to the corresponding gene frequencies.

The evolutionary dynamics of the population in terms of the gene frequencies P_i can be described by the following differential equations [1,2,4]:

where t is time, <W> = S _ij W_ij P_ij is the mean fitness in a population; u_ij is the mutation rate of the transition A_j --> A_i, u_ii =0 (i, j = 1,..., K). The first term in the right side of Eqs. (3) characterizes the selection of the organisms in accordance with their fitnesses, the second term takes into account the condition S _i P_i = 1, the third and fourth terms describe the mutation transitions.

Note that similar equations are used in the quasispecies model (for the deterministic case) [5].

Neglecting the mutations, we can analyze the dynamics of genes in the population by means of the equations:

Using (1), (2), (4), one can deduce (under the condition that the values W_ij are constant), that the rate of increase for the mean fitness is proportional to the fitness variance V = S _i P_i ( W_i - <W>)² [1,3]:

In accordance with (4), (5), the mean fitness <W> always increases, until an equilibrium state (dP_i /dt = 0) is reached.

The equation (5) characterizes quantitatively The Fundamental Theorem of Natural Selection (R.A.Fisher, 1930), which in our case can be formulated as follows [3]:

In a sufficiently large panmictic population, where the organisms' fitness is determined by one locus and the selection pressure parameters are defined by the constant values W_ij, the mean fitness in a population increases, reaching a stationary value in some genetic equilibrium state. The increase rate of the mean fitness is proportional to the fitness variance; it becomes zero in an equilibrium state.

The described model is a simple example of the deterministic approach. The wide spectrum of analogous models, which describe the different particularities, concerning several gene loci, age and female/male distributions in a population, inbreeding, migrations, subdivisions of populations, were developed and investigated, especially in connection with concrete genetic data interpretations [1,3,4].

Stochastic models

Below we sketch the main equations and some examples of the diffuse approximation. This approximation provides a non-trivial and effective method of population genetics.

We consider a population of diploid organisms with two alleles A₁ and A₂ in a certain locus. The population size n is supposed to be finite, but sufficiently large, so that the gene frequencies can be described by continuous values. We also suppose that the population size n is constant.

Let's introduce the function j = j (X,t|P,0) , which characterizes the probability density of the frequency X of the gene A₁ at the time moment t under condition that the initial frequency (at t = 0) of this gene is equal to P. Under the assumption that the changes of the gene frequencies at one generation are small, the populations dynamics can be described approximately by the following partial differential equations [1,3,4]:

where M_{d X} , M_{d P} and V_{d X} , V_{d P} are the mean values and the variances of the changes of the frequencies X, P during one generation; time unit is equal to one generation. Eq. (6) is the forward Kolmogorov differential equation (in physics it is called the Fokker-Planck equation); Eq. (7) is the backward Kolmogorov differential equation.

The first terms in the right sides of Eqs. (6), (7) describe a systematic selection pressure, which is due to the fitness difference of the genes A₁ and A₂. The second terms characterize the random drift of the frequencies, which is due to the fluctuations in the finite size population.

Using Eq. (6), one can determine the time evolution of the gene frequency distribution, Eq. (7) provides the means to estimate the probabilities of gene fixation.

Assuming that 1) the fitnesses of gene A₁ and A₂ are equal to 1 and 1-s, respectively and 2) the gene contributions to the fitnesses of the gene pairs A₁ A₁, A₁ A₂, and A₂ A₂ are additive, one can obtain, that the values M_{d X} , M_{d P} and V_{d X} , V_{d P} are determined by the following expressions [1,3,4]:

If the evolution is purely neutral (s = 0), Eq. (6) takes the form:

This equation was solved analytically by M.Kimura [1,6]. The solution is rather complex. The main results can be summarized as follows: 1) only one gene (A₁ or A₂) is fixed in the final population, 2) the typical transition time from the initial gene frequency distribution to the final one is of the order of 2n generations. Note that these results agree with the results of a simple neutral evolution game.

Using Eq. (7), we can estimate the probability of the fixation of the gene A₁ in the final population u(P). Considering the infinite time asymptotic, for the final population we can set ¶j/¶t = 0. The probability to be found can be approximated by the value [1]: u(P) = j/2n (here u(P) = j dX, where dX = 1/2n is the minimal frequency change step in population, see also [3] for more rigorous consideration). Using this approximation and combining (7), (8), we obtain:

Solving this simple equation for the natural boundary conditions: u (1) = 1, u (0) = 0, we obtain the probability of gene A₁ fixation in a final population [1,3,6]:

This expression shows, that if 4ns << 1, the neutral gene fixation takes place: u(P) » P, if 4ns >> 1, the advantageous gene A₁ is selected: u(P) » 1; the population size n_c ~ (4s)^-1 is the boundary value, demarcating "neutral" and "selective" regions.

Conclusion

Glossary:

¹⁾ Diploid organism: An individual having two chromosome sets in each of its cells.

²⁾ Allele: One of the different forms of a gene that can exist at a single locus.

³⁾ Gene locus: The specific place on a chromosome where a gene is located.

⁴⁾ Panmictic population: Random-mating population.

References:

1. J.F. Crow, M. Kimura. "An introduction to population genetics theory". New York etc, Harper & Row. 1970.

2. T. Nagylaki. "Introduction to theoretical population genetics ". Berlin etc, Springer Verlag. 1992.

3. Yu.M. Svirezhev, V.P. Pasekov. "Fundamentals of mathematical evolutionary genetics". Moscow, Nauka. 1982 (In Russian), Dordrecht, Kluwer Academic Publishers, 1990.

4. P.A.P. Moran. "The statistical processes of evolutionary theory", Oxford, Clarendon Press, 1962.

5. M. Eigen. Naturwissenshaften. 1971. Vol.58. P. 465. M. Eigen, P. Schuster. The Hypercycle: A principle of natural selforganization, Springer, Berlin, 1979

6. M. Kimura. "The neutral theory of molecular evolution". Cambridge Un-ty Press. 1983.

7. S. Karlin. "A first course in stochastic processes". New York, London, Academic Press. 1968.

Fisher R. A. The Genetical Theory of Natural Selection, 2nd edition, Dover Publications, New York, 1958.