The mathematical methods of population genetics theory
characterize quantitatively the gene distribution dynamics in
evolving populations [1-3]. There are two types of models:
deterministic and stochastic. Deterministic models are based on
the approximation of an infinitely large population size. In this
case the fluctuations of gene frequencies (in a gene
distribution) can be neglected and the population dynamics can be
described in terms of the mean gene frequencies. The stochastic
models describe the probabilistic processes in finite size
populations. Here we review very briefly the main equations and
mathematical methods of population genetics by considering
the most representative examples.
## Deterministic models

Let's consider a population
of diploid^{1)} organisms with several alleles^{2)}
*A*_{1} , *A*_{2 },..., *A*_{K}
in some locus^{3)}. We assume that the organism fitness
is determined mainly by the considered locus.* *Designating
the number of organisms and the fitness of the gene pair *A*_{i}
*A*_{j }by *n*_{ij} and *W*_{ij}, respectively, we can introduce the genotype and gene
frequencies *P*_{ij} and *P*_{i },
as well as the mean gene fitnesses *W*_{i} in
accordance with the expressions [1,2,4]:
*P*_{ij} = *n*_{ij
}/*n* , |
*P*_{i }= S _{j }*P*_{ij}*
*, |
*W*_{i}* *=*P*_{i}^{-1}
S _{j }*W*_{ij
}*P*_{ij} , |
(1) |

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 panmictic^{4)} 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]:

*P*_{ij}* *=*P*_{i}*
P*_{j} , |
*i, j *= 1,...,*K* . |
(2) |

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]:

*dP*_{i}* /dt *=
*W*_{i}* P*_{i} -
<*W> P*_{i} - S _{j }*u*_{ji
}*P*_{i} + S _{j }*u*_{ij
}*P*_{j}* *,* * |
*i *= 1,...,*K *,* * |
(3) |

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:

*dP*_{i}* /dt *=
*W*_{i}* P*_{i} -
<*W> P*_{i} , |
*i *= 1,...,*K .* |
(4) |

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]:

*d*<*W>/dt *= 2 S _{i} *P*_{i}*
*(_{ }*W*_{i}* *-
<*W>*)^{2}* .* |
(5) |

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

Deterministic models provide
effective methods for evolving population description. However,
they use the approximation of an infinitely large population size,
which is too strong for many real cases. To overcome this
limitation, the probabilistic methods of population genetics
were developed [1,3,4,6]. These methods include the analysis by
means of Markov chains (especially, by using the generating
functions) [4,7], and the diffuse approximation [1,3,4,6].
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
*A*_{1} and *A*_{2} 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 *A*_{1} 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]:

*¶j**/**¶**t** *= -* ¶** *(*M*_{d}_{ X }*j** *)*/**¶**X** *+ (1/2)* ¶* ^{2}(*V*_{d}_{ X }*j*_{ })*/**¶**X*^{2} , |
(6) |

*¶j**/**¶**t *=* M*_{d}_{ P }*¶j**/**¶**P* + (1/2)*V*_{d}_{ P}* ¶** *^{2}*j**/**¶**P*^{2} , |
(7) |

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 *A*_{1} and *A*_{2}.
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* A*_{1}
and *A*_{2} are equal to 1 and 1-*s*,
respectively and 2) the gene contributions to the fitnesses of
the gene pairs *A*_{1} *A*_{1},
*A*_{1} *A*_{2}, and *A*_{2}
*A*_{2} 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]:

*M*_{d}_{
X} = *sX*(1-*X*) ,* * |
*M*_{d}_{
P} = *sP*(1-*P*) , |
*V*_{d }_{X}
= *X*(1-*X*)*/*2*n* , |
*V*_{d }_{P
}= *P*(1-*P*)*/*2*n* . |
(8) |

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

*¶j**/**¶**t *= (1/4*n*)* ¶** *^{2}[*X*(1-*X*)*j*]*/**¶**X*^{2} . |
(9) |

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 (*A*_{1} or*
A*_{2}) 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 2*n*
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 *A*_{1 }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**/*2*n* (here *u*(*P*)
= *j** dX, *where* dX*
= 1*/*2*n* is the minimal frequency change step in
population, see also [3] for more rigorous consideration). Using
this approximation and combining (7), (8), we obtain:

*s*_{ }*du**
/**dP *+ (1*/*4*n*)* d*^{2}*u
**/**dP*^{2} = 0 . |
(10) |

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

*u*(*P*) = [1 - *exp* (-
4*nsP*)] [1 - *exp* (- 4*ns*)]^{-1}
. |
(11) |

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

## Conclusion

The mathematical models of population
genetics describe the gene frequency distributions in evolving
populations. The deterministic methods are used to analyze the
mean frequency dynamics; the stochastic methods take into account
the fluctuations, which are due to the finite population size.
**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.

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