How trait distributions evolve in populations with parametric heterogeneity.

We consider the problem of determining the time evolution of a trait distribution in a mathematical model of non-uniform populations with parametric heterogeneity. This means that we consider only heterogeneous populations in which heterogeneity is described by an individual specific parameter that differs in general from individual to individual, but does not change with time for the whole lifespan of this individual. Such a restriction allows obtaining a number of simple and yet important analytical results.

Stable coevolutionary regimes for genetic parasites and their hosts: you must differ to coevolve.

Background: Genetic parasites are ubiquitous satellites of cellular life forms most of which host a variety of mobile genetic elements including transposons, plasmids and viruses. Theoretical considerations and computer simulations suggest that emergence of genetic parasites is intrinsic to evolving replicator systems.

Results: Using methods of bifurcation analysis, we investigated the stability of simple models of replicator-parasite coevolution in a well-mixed environment.

Non-linearity and heterogeneity in modeling of population dynamics.

The study of population growth reveals that the behaviors that follow the power law appear in numerous biological, demographical, ecological, physical and other contexts. Parabolic models appear to be realistic approximations of real-life replicator systems, while hyperbolic models were successfully applied to problems of global demography and appear relevant in quasispecies and hypercycle modeling. Nevertheless, it is not always clear why non-exponential growth is observed empirically and what possible origins of the non-exponential models are.

Pseudo-chaotic oscillations in CRISPR-virus coevolution predicted by bifurcation analysis.

Background: The CRISPR-Cas systems of adaptive antivirus immunity are present in most archaea and many bacteria, and provide resistance to specific viruses or plasmids by inserting fragments of foreign DNA into the host genome and then utilizing transcripts of these spacers to inactivate the cognate foreign genome. The recent development of powerful genome engineering tools on the basis of CRISPR-Cas has sharply increased the interest in the diversity and evolution of these systems. Comparative genomic data indicate that during evolution of prokaryotes CRISPR-Cas loci are lost and acquired via horizontal gene transfer at high rates.

Parabolic replicator dynamics and the principle of minimum Tsallis information gain.

Background: Non-linear, parabolic (sub-exponential) and hyperbolic (super-exponential) models of prebiological evolution of molecular replicators have been proposed and extensively studied. The parabolic models appear to be the most realistic approximations of real-life replicator systems due primarily to product inhibition. Unlike the more traditional exponential models, the distribution of individual frequencies in an evolving parabolic population is not described by the Maximum Entropy (MaxEnt) Principle in its traditional form, whereby the distribution with the maximum Shannon entropy is chosen among all the distributions that are possible under the given constraints.

On the asymptotic behaviour of the solutions to the replicator equation.

Selection systems and the corresponding replicator equations model the evolution of replicators with a high level of abstraction. In this paper, we apply novel methods of analysis of selection systems to the replicator equations. To be suitable for the suggested algorithm, the interaction matrix of the replicator equation should be transformed; in particular, the standard singular value decomposition allows us to rewrite the replicator equation in a convenient form.

The universal distribution of evolutionary rates of genes and distinct characteristics of eukaryotic genes of different apparent ages.

The evolutionary rates of protein-coding genes in an organism span, approximately, 3 orders of magnitude and show a universal, approximately log-normal distribution in a broad variety of species from prokaryotes to mammals. This universal distribution implies a steady-state process, with identical distributions of evolutionary rates among genes that are gained and genes that are lost. A mathematical model of such process is developed under the single assumption of the constancy of the distributions of the propensities for gene loss (PGL).

Population models with singular equilibrium.

A class of models of biological population and communities with a singular equilibrium at the origin is analyzed; it is shown that these models can possess a dynamical regime of deterministic extinction, which is crucially important from the biological standpoint. This regime corresponds to the presence of a family of homoclinics to the origin, so-called elliptic sector. The complete analysis of possible topological structures in a neighborhood of the origin, as well as asymptotics to orbits tending to this point, is given.

Mathematical modeling of tumor therapy with oncolytic viruses: effects of parametric heterogeneity on cell dynamics.

Background: One of the mechanisms that ensure cancer robustness is tumor heterogeneity, and its effects on tumor cells dynamics have to be taken into account when studying cancer progression. There is no unifying theoretical framework in mathematical modeling of carcinogenesis that would account for parametric heterogeneity.

Results: Here we formulate a modeling approach that naturally takes stock of inherent cancer cell heterogeneity and illustrate it with a model of interaction between a tumor and an oncolytic virus.

Biological applications of the theory of birth-and-death processes.

In this review, we discuss applications of the theory of birth-and-death processes to problems in biology, primarily, those of evolutionary genomics. The mathematical principles of the theory of these processes are briefly described. Birth-and-death processes, with some straightforward additions such as innovation, are a simple, natural and formal framework for modeling a vast variety of biological processes such as population dynamics, speciation, genome evolution, including growth of paralogous gene families and horizontal gene transfer and somatic evolution of cancers.

Mathematical modeling of tumor therapy with oncolytic viruses: regimes with complete tumor elimination within the framework of deterministic models.

Background: Oncolytic viruses that specifically target tumor cells are promising anti-cancer therapeutic agents. The interaction between an oncolytic virus and tumor cells is amenable to mathematical modeling using adaptations of techniques employed previously for modeling other types of virus-cell interaction.

Results: A complete parametric analysis of dynamic regimes of a conceptual model of anti-tumor virus therapy is presented.

Modeling genome evolution with a diffusion approximation of a birth-and-death process.

Motivation: In our previous studies, we developed discrete-space birth, death and innovation models (BDIMs) of genome evolution. These models explain the origin of the characteristic Pareto distribution of paralogous gene family sizes in genomes, and model parameters that provide for the evolution of these distributions within a realistic time frame have been identified. However, extracting the temporal dynamics of genome evolution from discrete-space BDIM was not technically feasible.

Mathematical modeling of evolution of horizontally transferred genes.

We describe a stochastic birth-and-death model of evolution of horizontally transferred genes in microbial populations. The model is a generalization of the stochastic model described by Berg and Kurland and includes five parameters: the rate of mutational inactivation, selection coefficient, invasion rate (i.e.

Gene family evolution: an in-depth theoretical and simulation analysis of non-linear birth-death-innovation models.

Background: The size distribution of gene families in a broad range of genomes is well approximated by a generalized Pareto function. Evolution of ensembles of gene families can be described with Birth, Death, and Innovation Models (BDIMs). Analysis of the properties of different versions of BDIMs has the potential of revealing important features of genome evolution.

Simple stochastic birth and death models of genome evolution: was there enough time for us to evolve?

Motivation: The distributions of many genome-associated quantities, including the membership of paralogous gene families can be approximated with power laws. We are interested in developing mathematical models of genome evolution that adequately account for the shape of these distributions and describe the evolutionary dynamics of their formation.

Results: We show that simple stochastic models of genome evolution lead to power-law asymptotics of protein domain family size distribution.

The structure of the protein universe and genome evolution.

Despite the practically unlimited number of possible protein sequences, the number of basic shapes in which proteins fold seems not only to be finite, but also to be relatively small, with probably no more than 10,000 folds in existence. Moreover, the distribution of proteins among these folds is highly non-homogeneous -- some folds and superfamilies are extremely abundant, but most are rare. Protein folds and families encoded in diverse genomes show similar size distributions with notable mathematical properties, which also extend to the number of connections between domains in multidomain proteins.

Birth and death of protein domains: a simple model of evolution explains power law behavior.

Background: Power distributions appear in numerous biological, physical and other contexts, which appear to be fundamentally different. In biology, power laws have been claimed to describe the distributions of the connections of enzymes and metabolites in metabolic networks, the number of interactions partners of a given protein, the number of members in paralogous families, and other quantities. In network analysis, power laws imply evolution of the network with preferential attachment, i.