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. These models, called Birth, Death and Innovation Models (BDIM), represent a special class of balanced birth-and-death processes, in which domain duplication and deletion rates are asymptotically equal up to the second order. The simplest, linear BDIM shows an excellent fit to the observed distributions of domain family size in diverse prokaryotic and eukaryotic genomes. However, the stochastic version of the linear BDIM explored here predicts that the actual size of large paralogous families is reached on an unrealistically long timescale. We show that introduction of non-linearity, which might be interpreted as interaction of a particular order between individual family members, allows the model to achieve genome evolution rates that are much better compatible with the current estimates of the rates of individual duplication/loss events.     

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.     

Intravascular repair in extended arterial occlusive lesions (a preliminary communication)

Subintimal recanalization as a procedure of percutaneous vascular repair is under study now. The Regional Clinical Hospital has performed 10 procedures of subintimal angioplasty in patients with extended total occlusions of iliac and femoral arteries. All the patients had presented with critical ischemia of the lower extremities with the mean humeromalleolar index of 0.24+/-0.08. A technical success was achieved in 8 patients in whom the mean humeromalleolar index was 0.37+/-0.09. There were no procedure-associated complications.     

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., rate of arrival of a novel sequence from outside of the recipient population), within-population horizontal transmission ("infection") rate, and population size. The model of Berg and Kurland included four parameters, namely, mutational inactivation, selection coefficient, population size, and "infection." However, the effect of "infection" was disregarded in the interpretation of the results, and the overall conclusion was that horizontally acquired sequences can be fixed in a population only when they confer a substantial selective advantage onto the recipient and therefore are subject to strong positive selection. Analysis of the present model in different domains of parameter values shows that, as long as the rate of within-population horizontal transmission is comparable to the mutational inactivation rate and there is even a low rate of invasion, horizontally acquired sequences can be fixed in the population or at least persist for a long time in a substantial fraction of individuals in the population even when they are neutral or slightly deleterious. The available biological data strongly suggest that intense within-population and even between-populations gene flows are realistic for at least some prokaryotic species and environments. Therefore, our modeling results are compatible with the notion of a pivotal role of horizontal gene transfer in the evolution of prokaryotes.     

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. We first analyze what appears to be the simplest imaginable system of this type, one in which the parasite evolves during the replication of the host genome through a minimal mutation that renders the genome of the emerging parasite incapable of producing the replicase but able to recognize and recruit it for its own replication. This model has only trivial or “semi-trivial”, parasite-free equilibria: an inefficient parasite is outcompeted by the host and dies off, whereas an efficient one pushes the host out of existence, leading to the collapse of the entire system. We show that stable host-parasite coevolution (a non-trivial equilibrium) is possible in a modified model where the parasite is qualitatively distinct from the host replicator in that the replication of the parasite depends solely on the availability of the host but not on the carrying capacity of the environment.

Conclusions

We analytically determine the conditions for stable coevolution of genetic parasites and their hosts coevolution in simple mathematical models. It is shown that the evolutionary dynamics of a parasite that initially evolves from the host through the loss of the ability to replicate autonomously must substantially differ from that of the host, for a stable host-parasite coevolution regime to be established.

     

Scale‐free networks in biology: new insights into the fundamentals of evolution?*

Scale-free network models describe many natural and social phenomena. In particular, networks of interacting components of a living cell were shown to possess scale-free properties. A recent study((1)) compares the system-level properties of metabolic and information networks in 43 archaeal, bacterial and eukaryal species and claims that the scale-free organization of these networks is more conserved during evolution than their content.     

From Experiment to Theory: What Can We Learn from Growth Curves?

Finding an appropriate functional form to describe population growth based on key properties of a described system allows making justified predictions about future population development. This information can be of vital importance in all areas of research, ranging from cell growth to global demography. Here, we use this connection between theory and observation to pose the following question: what can we infer about intrinsic properties of a population (i.e., degree of heterogeneity, or dependence on external resources) based on which growth function best fits its growth dynamics? We investigate several nonstandard classes of multi-phase growth curves that capture different stages of population growth; these models include hyperbolic–exponential, exponential–linear, exponential–linear–saturation growth patterns. The constructed models account explicitly for the process of natural selection within inhomogeneous populations. Based on the underlying hypotheses for each of the models, we identify whether the population that it best fits by a particular curve is more likely to be homogeneous or heterogeneous, grow in a density-dependent or frequency-dependent manner, and whether it depends on external resources during any or all stages of its development. We apply these predictions to cancer cell growth and demographic data obtained from the literature. Our theory, if confirmed, can provide an additional biomarker and a predictive tool to complement experimental research.     

Preventing the Tragedy of the Commons Through Punishment of Over-Consumers and Encouragement of Under-Consumers

The conditions that can lead to the exploitative depletion of a shared resource, i.e., the tragedy of the commons, can be reformulated as a game of prisoner’s dilemma: while preserving the common resource is in the best interest of the group, over-consumption is in the interest of each particular individual at any given point in time. One way to try and prevent the tragedy of the commons is through infliction of punishment for over-consumption and/or encouraging under-consumption, thus selecting against over-consumers. Here, the effectiveness of various punishment functions in an evolving consumer-resource system is evaluated within a framework of a parametrically heterogeneous system of ordinary differential equations (ODEs). Conditions leading to the possibility of sustainable coexistence with the common resource for a subset of cases are identified analytically using adaptive dynamics; the effects of punishment on heterogeneous populations with different initial composition are evaluated using the reduction theorem for replicator equations. Obtained results suggest that one cannot prevent the tragedy of the commons through rewarding of under-consumers alone—there must also be an implementation of some degree of punishment that increases in a nonlinear fashion with respect to over-consumption and which may vary depending on the initial distribution of clones in the population.     

On the Asymptotic Behaviour of the Population Number

We consider the stochastic model of an asexual population in which the number of couples formed in some generation is random variable depending on the number of individuals in that generation only. The conditions of convergence were obtained almost everywhere and in mean square of the normalized number of individuals in the n-th generation. These results may be considered as the generalization of some known statements about the models constructed on the basis of the branching processses theory.