The Site-Frequency Spectrum of Linked Sites

The site-frequency spectrum, representing the distribution of allele frequencies at a set of polymorphic sites, is a commonly used summary statistic in population genetics. Explicit forms of the spectrum are known for both models with and without selection if independence among sites is assumed. The availability of these explicit forms has allowed for maximum likelihood estimation of selection, developed first in the Poisson random field model of Sawyer and Hartl, which is now the primary method for estimating selection directly from DNA sequence data. The independence assumption, which amounts to assume free recombination between sites, is, however, a limiting case for many population genetics models. Here, we extend the site-frequency spectrum theory to consider the case where the sites are completely linked. We use diffusion approximation to calculate the joint distribution of the allele frequencies of linked sites for models without selection and for models with equal coefficient selection. The joint distribution is derived by first constructing Green’s functions corresponding to multiallele diffusion equations. We show that the site-frequency spectrum is highly correlated between frequencies that are complementary (i.e., sum to 1), and the correlation is significantly elevated by positive selection. The results presented here can be used to extend the Poisson random field to allow for estimating selection for correlated sites. More generally, the Green’s function construction should be able to aid in studying the genetic drift of multiple alleles in other cases.     

Mutator Genes and Selection for the Mutation Rate in Bacteria

Gene frequencies in populations of haploid, asexual organisms are described by linear recurrence equations. Several models in which the mutation rate is controlled by one locus and the fitness is controlled at one or more other loci are developed. It is shown that good approximations can be introduced to give explicit solutions for the course of selection in these models. It is shown that a strong non-equilibrium selection for mutator genes is possible even when the presence of such a gene decreases the fitness of an individual. Experiments that corroborate these conclusions are discussed along with the effects of population size that determine the applicability of these results to natural populations.     

Cumulant dynamics of a population under multiplicative selection, mutation, and drift

We revisit the classical population genetics model of a population evolving under multiplicative selection, mutation, and drift. The number of beneficial alleles in a multilocus system can be considered a trait under exponential selection. Equations of motion are derived for the cumulants of the trait distribution in the diffusion limit and under the assumption of linkage equilibrium. Because of the additive nature of cumulants, this reduces to the problem of determining equations of motion for the expected allele distribution cumulants at each locus. The cumulant equations form an infinite dimensional linear system and in an authored appendix Adam Prügel-Bennett provides a closed form expression for these equations. We derive approximate solutions which are shown to describe the dynamics well for a broad range of parameters. In particular, we introduce two approximate analytical solutions: (1) Perturbation theory is used to solve the dynamics for weak selection and arbitrary mutation rate. The resulting expansion for the system's eigenvalues reduces to the known diffusion theory results for the limiting cases with either mutation or selection absent. (2) For low mutation rates we observe a separation of time-scales between the slowest mode and the rest which allows us to develop an approximate analytical solution for the dominant slow mode. The solution is consistent with the perturbation theory result and provides a good approximation for much stronger selection intensities.     

Replicator Equations and the Principle of Minimal Production of Information

Many complex systems in mathematical biology and other areas can be described by the replicator equation. We show that solutions of a wide class of replicator equations minimize the KL-divergence of the initial and current distributions under time-dependent constraints, which in their turn, can be computed explicitly at every instant due to the system dynamics. Therefore, the Kullback principle of minimum discrimination information, as well as the maximum entropy principle, for systems governed by the replicator equations can be derived from the system dynamics rather than postulated. Applications to the Malthusian inhomogeneous models, global demography, and the Eigen quasispecies equation are given.     

栖息地选择的理论与模型

栖息地选择理论和模型的发展经历了两个主要阶段:理想自由分布模型和空间直观的栖息地选择模型。随着对理论模型假设的放宽,近年来产生了越来越多的新模型。通过对栖息地选择过程的分析,提出了栖息地选择的几个关键环节:栖息地偏好、信息获取、行为决策及选择行为。在建立栖息地选择模型的各个关键环节上均存在大量有待解决的问题。目前对栖息地偏好的研究主要为相关分析,栖息地信息获取的过程仍然是一黑箱;对动物在栖息地选择过程中的行为决策以及对其生理状态的影响尚不了解,而解决这些问题需要生态学、生态学及认知学等多个领域的研究结果支持,也有待新的理论及方法加以充实,甚至还需要其他学科的介入。     

Modelling the biological invasion of Carcinus maenas (the European green crab)

This paper proposes a system of integro-difference equations to model the spread of Carcinus maenas, commonly called the European green crab, that causes severe damage to coastal ecosystems. A model with juvenile and adult classes is first studied. Here, standard theory of monotone operators for integro-difference equations can be applied and yields explicit formulas for the asymptotic spreading speeds of the juvenile and adult crabs. A second model including an infected class is considered by introducing a castrating parasite Sacculina carcini as a biological control agent. The dynamics are complicated and simulations reveal the occurrence of periodic solutions and stacked fronts. In this case, only conjectures can be made for the asymptotic spreading speeds because of the lack of mathematical theory for non-monotone operators. This paper also emphasizes the need for mathematical studies of non-monotone operators in heterogeneous environments and the existence of stacked front solutions in biological invasion models.     

Fractional Reproduction-Dispersal Equations and Heavy Tail Dispersal Kernels

Reproduction-Dispersal equations, called reaction-diffusion equations in the physics literature, model the growth and spreading of biological species. Integro-Difference equations were introduced to address the shortcomings of this model, since the dispersal of invasive species is often more widespread than what the classical RD model predicts. In this paper, we extend the RD model, replacing the classical second derivative dispersal term by a fractional derivative of order 1相似文献   

Electromagnetic van Kampen waves

The theory of van Kampen waves in plasma with an arbitrary anisotropic distribution function is developed. The obtained solutions are explicitly expressed in terms of the permittivity tensor. There are three types of perturbations, one of which is characterized by the frequency dependence on the wave vector, while for the other two, the dispersion relation is lacking. Solutions to the conjugate equations allowing one to solve the initial value problem are analyzed.     

Density distribution of calcium-induced alginate gels. A numerical study

Charged polysaccharides often form hydrogels in the presence of cations. In many applications the polymer network density distribution and associated physical properties are of major practical importance. Depending on the detailed conditions, the resulting gel density may vary from fully homogeneous to strongly inhomogeneous. We have established a simple set of coupled chemical reaction–diffusion equations to model the gelling process of calcium-induced alginate gels. The necessary algorithms for numerical solution of the resulting simultaneous parabolic differential equations have been developed both for one-dimensional models and three-ldimensional models with cylindrical or spherical symmetry. The algorithms make use of the Crank–Nicolson implicit finite difference method. The results of the numerical analyses of the gel formation can be divided into several different regimes depending on the physical and chemical parameters of the alginates and the cations. The numerical results are in good agreements with reported experimental results. © 1995 John Wiley & Sons, Inc.     

Nonlinear theory of ion-acoustic waves in an ideal plasma with degenerate electrons

A nonlinear theory is constructed that describes steady-state ion-acoustic waves in an ideal plasma in which the electron component is a degenerate Fermi gas and the ion component is a classical gas. The parameter ranges in which such a plasma can exist are determined, and dispersion relations for ion-acoustic waves are obtained that make it possible to find the linear ion-acoustic velocity. Analytic gas-dynamic models of ion sound are developed for a plasma with the ion component as a cold, an isothermal, or an adiabatic gas, and moreover, the solutions to the equations of all the models are brought to a quadrature form. Profiles of a subsonic periodic and a supersonic solitary wave are calculated, and the upper critical Mach numbers of a solitary wave are determined. For a plasma with cold ions, the critical Mach number is expressed by an explicit exact formula.     

Analytical results on the neutral non-equilibrium allele frequency spectrum based on diffusion theory

The allele frequency spectrum has attracted considerable interest for the simultaneous inference of the demographic and adaptive history of populations. In a recent study, Evans et al. (2007) developed a forward diffusion equation describing the allele frequency spectrum, when the population is subject to size changes, selection and mutation. From the diffusion equation, the authors derived a system of ordinary differential equations (ODEs) for the moments in a Wright–Fisher diffusion with varying population size and constant selection. Here, we present an explicit solution for this system of ODEs with variable population size, but without selection, and apply this result to derive the expected spectrum of a sample for time-varying population size. We use this forward-in-time-solution of the allele frequency spectrum to obtain the backward-in-time-solution previously derived via coalescent theory by Griffiths and Tavaré (1998). Finally, we discuss the applicability of the theoretical results to the analysis of nucleotide polymorphism data.     

Kolmogorov’s Differential Equations and Positive Semigroups on First Moment Sequence Spaces

Spatially implicit metapopulation models with discrete patch-size structure and host-macroparasite models which distinguish hosts by their parasite loads lead to infinite systems of ordinary differential equations. In several papers, a this-related theory will be developed in sufficient generality to cover these applications. In this paper the linear foundations are laid. They are of own interest as they apply to continuous-time population growth processes (Markov chains). Conditions are derived that the solutions of an infinite linear system of differential equations, known as Kolmogorov’s differential equations, induce a C ₀-semigroup on an appropriate sequence space allowing for first moments. We derive estimates for the growth bound and the essential growth bound and study the asymptotic behavior. Our results will be illustrated for birth and death processes with immigration and catastrophes. An erratum to this article can be found at     

Heterogeneity of the spatial distribution of the primordial organic substance as an initial stage of biological evolution

An approach to describe the emergence of dynamics of polymerization/depolymerization of some spatially distributed prebiological structures has been analyzed, and two phases of the development of the system have been identified. In the first phase, polymerization of organic monomers occurs under the influence of external factors, and in the second one depolymerization takes place. Both processes are accompanied by “diffusive mixing” of reaction products. The dynamic equations of the system are presented. Numerical examination of the space nonuniform solution of model equations has shown that, in conditions of low stability of uniform space distribution, these solutions resolve into a number of discrete peaks of nonzero density, which are isolated from each other by free space. Such nonuniform distributions are stable when close to the bifurcation point; yet in other conditions, they can lose their stability, which entails a more pronounced nonuniformity of space dynamics. Thus, interaction of polymerization/depolymerization processes results in chaotic self-organization and leads to origination of complex and inhomogeneous (patchy) spatial structures. These structures in physical space can reflect the emergence of the spatial nonuniformity in prebiological associations, while in the distributive space of characters they can correspond to the initial steps of emergence of the first discrete domains fixed in biological evolution.     

Recolonisation by diffusion can generate increasing rates of spread

Diffusion is one of the most frequently used assumptions to explain dispersal. Diffusion models and in particular reaction-diffusion equations usually lead to solutions moving at constant speeds, too slow compared to observations. As early as 1899, Reid had found that the rate of spread of tree species migrating to northern environments at the beginning of the Holocene was too fast to be explained by diffusive dispersal. Rapid spreading is generally explained using long distance dispersal events, modelled through integro-differential equations (IDEs) with exponentially unbounded (EU) kernels, i.e. decaying slower than any exponential. We show here that classical reaction-diffusion models of the Fisher-Kolmogorov-Petrovsky-Piskunov type can produce patterns of colonisation very similar to those of IDEs, if the initial population is EU at the beginning of the considered colonisation event. Many similarities between reaction-diffusion models with EU initial data and IDEs with EU kernels are found; in particular comparable accelerating rates of spread and flattening of the solutions. There was previously no systematic mathematical theory for such reaction-diffusion models with EU initial data. Yet, EU initial data can easily be understood as consequences of colonisation-retraction events and lead to fast spreading and accelerating rates of spread without the long distance hypothesis.     

A procedure for resolving overlapping curves suitable for use with a microcomputer

To solve the problem of calibration of interference patterns (or scanner traces) with respect to concentration, the true base level in an explicit form is introduced to the sedimentation equilibrium equations. Computer-simulated experiments with homogeneous, two and three high-molecular-weight component heterogeneous solutions have shown that the transformed equations can be solved relative to all linear and nonlinear parameters by means of a linear regression analysis supplemented with a straight-search technique. This enables one to determine, with a satisfactory accuracy, the molecular weights of all components of a heterogeneous solution. Furthermore, assuming that the mass conservation is observed, one may get reasonable information about the initial weight ratio of components. The method is free from any restrictions on the speed selection (“all-speed”), and does not require an auxiliary run for determination of the initial concentration. The latter may be as low as in “high-speed” experiments which, in conjunction with a low centrifugal potential, reduces considerably the nonideality effects and optical distortions all over the cell.     

Diffusion through a membrane: Approach to equilibrium

The transfer of solute through a membrane separating two aqueous solutions is studied with the time-dependent diffusion equation for composite media. By introducing new independent and dependent variables it is shown that the differential equations and boundary conditions can be transformed into a dimensionless form which does not explicitly depend on the diffusivities of the media. Laplace transforms are used to derive explicit solutions for the solute concentration as a function of position and time. It is shown that at large time the concentration approaches the equilibrium distribution exponentially. Explicit results are given for the decay time as a function of the parameters of the system. In addition, an accurate and simplified expression is derived for the decay time for the case of small membrane permeability. The accuracy of the analytic solutions for the concentration profiles is tested by comparing them with numerical results obtained by solving the diffusion equations by the method of finite differences. Excellent agreement is found. Research supported in part by a grant from the National Science Foundation.     

Flow in a two-dimensional collapsible channel with rigid inlet and outlet

This paper examines mainly oscillatory behavior of a fluid-conveying collapsible tube using a two-dimensional flexible channel made of a pair of membranes. The equation of equilibrium of the membrane in a large deflection theory is coupled with the equations of continuity and momentum of an incompressible flow in a one-dimensional flow theory accounting for flow separation. An explicit finite difference method was used to solve the governing equations numerically. According to numerical results, the fluids in the inlet and outlet rigid channels have strong effects on the oscillation of the system. Depending on initial values for the numerical integration, there may exist both a stable static equilibrium and an oscillatory solution for the same parameter values, but only if the external pressure is sufficiently large.     

Uniqueness and Stability of Action Potential Models during Rest, Pacing, and Conduction Using Problem-Solving Environment

Development and application of physiologically detailed dynamic models of the action potential (AP) and Ca²⁺ cycling in cardiac cells is a rapidly growing aspect of computational cardiac electrophysiology. Given the large scale of the nonlinear system involved, questions were recently raised regarding reproducibility, numerical stability, and uniqueness of model solutions, as well as ability of the model to simulate AP propagation in multicellular configurations. To address these issues, we reexamined ventricular models of myocyte AP developed in our laboratory with the following results. 1), Recognizing that the model involves a system of differential-algebraic equations, a procedure is developed for estimating consistent initial conditions that insure uniqueness and stability of the solution. 2), Model parameters that can be used to modify these initial conditions according to experimental values are identified. 3), A convergence criterion for steady-state solution is defined based on tracking the incremental contribution of each ion species to the membrane voltage. 4), Singularities in state variable formulations are removed analytically. 5), A biphasic current stimulus is implemented to completely eliminate stimulus artifact during long-term pacing over a broad range of frequencies. 6), Using the AP computed based on 1-5 above, an efficient scheme is developed for computing propagation in multicellular models.     

Mappings of the plane that simulate prey-predator systems

A prey-predator system that is spatially dependent is built around the familiar logistic equation. The result is a set of coupled integro-differential equations. If at least one of the populations reproduces periodically in a time that is short to other characteristic times, these equations can be transformed to a set of coupled maps, each map representing a spatial point. Various models are developed that include predator ranging as well predator migration via diffusion. In two dimensions the coupling among the spatial points will have the symmetry of the regular polyhedra if the infinite plane is viewed in polar coordinates, or toroidal symmetry if periodic boundary conditions are imposed. Numerical examples are given. Depending on the value of the parameters and on the initial conditions the system can oscillate in a variety of different spatial modes, giving rise to unusual bifurcation portraits, or it can behave chaotically. The solutions show that a patchy distribution of population is more stable against the influences of environmental noise than is a smoothly distributed population. The numerical results in two dimensions are in constrast with the results of a previous study in one dimension.