Stochastic Methods in Biology

The main aim of this paper is to develop population dynamics of ‘compensatory neutral mutations’ which are individually deleterious but restore the normal fitness in combination. I shall make use of the diffusion equation method and investigate the problem: how long does it take for such a pair of compensatory mutations to become fixed in a finite population under mutational pressure in the course of evolution.

During the last several years, molecular biology has undergone a second round of remarkable development. This has brought a number of new findings regarding the genetic organization of higher organisms, as distinct from that of lower organisms. One such discovery is the prevalence of multigene families and other repetitive sequences. For about ten years, I have been studying the evolution of multigene families from the standpoint of population genetics. Recently, I extended the analyses to treat repetitive DNA sequences that are dispersed in genomes. Repetitive sequences are characterized by concerted evolution, i.e., gene copies belonging to a family of repeating elements do not evolve independently, but evolve as a set through various molecular interaction mechanisms such as unequal crossing-over, gene conversion and duplicative transposition. By incorporating such interaction mechanisms, the population genetics model of an evolving multigene family becomes very complicated. In this paper, I shall review models and analyses on repetitive gene families.

Kingman [8], [9] introduced the coalescent as a means of describing the genealogy of samples taken from a large evolving haploid population. The coalescent partitions a sample of genes into equivalence classes with respect to an ancestral population some time t into the past; genes in the same equivalence class in the sample have the same ancestors. As t increases the equivalence classes coalesce until, sufficiently far in the past, all individuals in the sample are equivalent, being descended from a common ancestor.

Let X be a countable set and N be a positive integer. X is a collection of colonies. Consider the process of gene frequencies of an allele A₁, which is subjected to the following changes:(a) mutation occurs from A_l into A₂ and from A₂ into A_l with mutation rates u and v, respectively, (b) for any colonies x and z the genes migrate from z to x with migration rates λ_xz, (c) selection occurs in each colony x, where A_l and A₂ have relative fitness 1 + s_x/2, 1 − s_x/2 respectively, and (d) after having reproduced an infinite numbers of offsprings, N individuals are sampled at random within each colony. The process is called the stepping stone model. The process is a Markov chain with state space describing gene frequencies at colonies. Possible values of gene frequencies at each colony are 0, 1/(2N), ..., (2N)/(2N). We say the process is ergodic if the distribution of the process at the n-th time converges to the unique equilibrium measure independently of the initial distribution as n → ∞. It is shown that if s_x ≧ 0 for all x in X or s_x ≦ 0 for all x in X, and u + v ≦ 1 and u, v > 0, the process is ergodic, but assuming u > 0 and v = 0 and X = Z^d the d-dimensional lattice, then there exists a constant s_l < 2 such that the process is ergodic if sup_x∈X s_x < s₁ and is not ergodic if inf_x∈X s_x > s₁. The latter fact shows that under the condition u > 0 = v, the process has a nontrivial equilibrium measure, if s_x is large enough. Assume X = Z¹ and consider the case that sign of s_x is different between x ≧ 0 and x < 0. Some monotonicity properties concerning the equilibrium measure and related estimates are obtained under the condition that the process is ergodic.

We study the diffusion model for a random mating diploid species in population genetics admitting intergroup selection, which was recently proposed by M. Kimura. We first enumerate the stationary distributions for the relative gene frequencies of the altruistic allele and inspect their stability. According to the regions of the parameters, we divide the model into seven cases. In the three of them, there are only ‘trivial’ stationary distributions, i.e. the relative gene frequency is equal to zero with probability one or equal to one. But in the other four cases, we can find one or infinitely many stationary distributions with density functions. Under a mild assumption, we also show that, for any initial distribution, the distribution of the gene frequencies goes to one of the stationary distributions as time goes to infinity. We next study how the data affect the distribution of the gene frequencies. Actually, the moment sequence of the distribution depends on the data monotonically. As a result, we verify one of the main results of M. Kimura, which gives a criterion for predominance of the altruistic allele or the other allele by a simple index D = c/m − 4Ns.

In [4], a diffusion model is constructed for a genetic system in which all alleles are selectively neutral and all mutants are new. The state of this model is the vector of order statistics of the gene frequencies. This reordering of the frequencies is necessary because of the assumption on mutation. Fixing the order of the alleles results in a model in which the sum of the gene frequencies is less than one for all positive time. Unfortunately, reordering makes it virtually impossible to study models with selection using this approach.

We first consider a multi-allelic Markov chain model for which one-step transition consists of three stages — independent reproduction, mutation and random sampling. Taking account of difference among alleles in means and variances of offspring numbers we discuss a diffusion approximation of the Markov chain model both in a finite-allelic case and in a countably infinite-allelic case. This diffusion approximation was derived by Gillespie heuristically in a di-allelic case, and by Sato in a multi-allelic case, neglecting any mutation factor. Our result extends Sato’s one.

We next consider a continuum limit of the alleles space in the diffusion model. The limiting process is then a measure-valued diffusion process. Particularly if the alleles space is one-dimensional and the mutation operator is the Laplacian, we can derive an infinite dimensional stochastic differential equation of which solution defines a probabilitydensity-valued diffusion process.

The diffusion process, which will be discussed in this paper, is closely related to a genetical model which has been investigated by T. Ohta. We will discuss an n locus haploid model in which mutation, gene conversion and random drift are taken into consideration. Since we are concerned only with the average identity probability at different loci on one chromosome, random partitions of the number n determined by chromosomes with n loci should be investigated. The diffusion process describes the time evolution of distributions of the random partitions.

The average probability of types of chromosomes at equilibrium, that is, the mean vector of the stationary distribution can be obtained exactly. It is written in a form similar to the Ewens sampling formula. This result will be applied to calculation of the average identity probability at different loci on one chromosome.

The electrical behavior of neuronal membranes and the role of ion currents have been studied and understood since the landmark 1952 papers of Hodgkin and Huxley. The existence of ionic gates exhibiting stochastic behavior was confirmed over ten years ago with the development of the experimental patch-clamp technique, and the nature and structure of those ionic gates was illuminated dramatically in 1984 with the publication by Noda et al. [1984] of the complete amino acid chain comprising the sodium gating channel for the electric organ of electrophorus electricus. Despite all this experimental progress, and despite the wide interest in gaining a better understanding of the behavior of individual neurons and of systems of neurons, there has been little success in the efforts to develop stochastic mathematical models capable of reflecting and helping to predict neuronal activity. It is a goal of our research to develop such models in order to illuminate the connection between the microkinetic behavior of thousands of gating molecules scattered over the surface membrane of a single isolated neuron subject to a stream of excitatory and inhibitory impulses, and the macrokinetic behavior of that same neuron in generating and propagating action potentials and spike trains. We do not address the interesting questions of how networks of interconnected neurons behave.

Spontaneous activity of a single neuron is considered using the Ornstein-Uhlenbeck process model. The interspike intervals are viewed as the first-passagetimes for the process starting at some initial value through a constant boundary. The moments of the first-passage-time probability density function(p.d.f.), the asymptotic forms of the p.d.f. for large boundary and for large time are obtained.

The electric potential in living cells of paramecium shows large spontaneous fluctuation, which consists of basic fluctuation and spikelike fluctuation. The spikelike fluctuation triggers transient reversal of ciliary beating and causes discontinuous change of the swimming direction. A positive shift of the basic potential increases the probability of the spikelike fluctuation. The spike is generated by opening of electric field-sensitive channels in the cell membrane. A differential equation is proposed to describe the probabilistic behavior of these channels in a fluctuating electric field. The fluctuating field increases the average rates of open-close transitions of channels and shifts the average opening probability towards 1/2. The open-close fluctuation in an assembly of the channels has asymmetry with respect to time reversal. Free energy is continuously consumed for generation of the spikelike fluctuation. The proposed equation which contains fluctuating quantities in exponential terms has a definite physical basis and is useful for the analysis of stochastic processes in a fluctuating field.

Professor Oosawa [1] proposed an ordinary stochastic differential equation arising from the study of fluctuation in living cells. In this report, we consider Gaussian approximations to the above equation by using the averaging methods.

This note gives a brief review of some aspects of epidemic modelling which I have found of interest. The field is now very broad, and it is difficult to provide more than a partial flavor of recent developments. We shall concentrate on three topics: epidemic data fitting, matrix-geometric methods for general epidemics, and the duration of an epidemic in an immigration type model. These will provide a sample of current research problems.

Many diffusion operators appearing in the diffusion approximation to discrete models are degenerate. For example, the diffusion operator ½ x(1-x)(d/dx)² + [u-(u+v)x](d/dx) (0 < x < 1) arises as a diffusion approximation for the Wright-Fisher model with mutation and migration. To obtain an error estimate for such diffusion approximation, it is useful to show the smoothness of solutions of the diffusion equations. Ethier has obtained important results for the smoothness problem, especially in the one-dimensional case. Motivated by his work, we will study the smoothness problem for certain degenerate integro-differential operators appearing in the theory of Markov processes. These operators include diffusion operators and are called Lévy operators (in this note, we treat the case without boundary). Our result can be used to get some limit theorem for stochastic processes with discontinuous paths. We will discuss the generation of the semigroups by one-dimensional Lévy operators and the differentiability preserving properties of the semigroups. Convergence problems for the semigroups are also treated.

In population genetics, it is frequently done to approximate discrete models by diffusion models. From a mathematical point of view, this diffusion approximation problem is usually treated as the problem of convergence of discrete models (suitably normalized and interpolated) to a diffusion process on a bounded closed domain. In many cases it is easy to get from the discrete models an explicit form of the limiting differential operator. But, in order to prove the convergence of the stochastic processes, it is necessary to prove the uniqueness of the diffusion process associated with the differential operator. This uniqueness problem is often a hard mathematical problem due to degeneracy of the operator on the boundary. Sometimes it is helpful to prove certain regularity properties for diffusion processes (not assumed to be unique) associated with the given operator. Especially in the problem of convergence of discrete models without mutation and migration, it is important to show that sample paths of the diffusion processes cannot enter the interior from the boundary.

In this paper we consider this property for diffusion processes on a general bounded closed domain whose behavior in the interior is determined by a given diffusion operator. As an application of our results, we get this property for the diffusion process corresponding to the discrete models without mutation and migration. Consequently its uniqueness follows easily.

Let u(x,t)be the temperature at position x in R ³ and at time t≥0. For N=1,2,3,... let w ₁ ^N ,w ₂ ^N ,...,w _N ^N , be a sequence of points in R ³ and let D ^N be the domain in R ³ defined by where α is a fixed positive constant. The temperature u satisfies the initial-boundary value problem Here f (x) is a smooth, positive function of compact support representing the initial temperature distribution and B _j ^N is the sphere centered at w _j ^N with radius α/N.

The distribution of the generalized sojourn time T for a birth-and-death process up to the first passage time to the state n, starting at the state m (m < n), is considered. The generalized sojourn time is the sum of the sojourn times at the states i, over i = 0,1,...,n−1, with a state-dependent signed weight. A main new result of the paper is that the distribution of T is unimodal. Explicit description of the distribution is given by using exponential distributions on the positive and the negative axis. Bounds of the mode are derived from this description. Other bounds of the modes of general unimodal distributions are given in terms of absolute moments and central absolute moments. Infinite divisibility of T is also proved. These results are extended to generalized sojourn times of diffusion processes, which arise in models of neurobiology and population genetics.

A dynamics for a large population of randomly mating individuals is formulated as a Markov process on a large product space which involves a pairwise interaction between components. Each individual in the population possesses a character (the state of individual). An individual undergoes Markovian change of its character during its life time. Two individuals may make random mating after which they die, leaving two children with new characters. Each individual is a unique member of a lineage at each time. We are interested in the number of lineages which, comprise individuals possessing the character x_i at time t_i for i = 1,…,m, where x_l,…,x_m and t_l,…,t_m are given in advance. If the initial distribution is i.i.d. and the size n of the population is very large, the proportion of this number to n is approximately computed by means of a joint distribution of a nonlinear Markov process (for a single individual in a infinite population). We shall study the law of the error in this approximation and establish a limit theorem about it.