From Genetics to Mathematics

The following sections are included:

• Preface

• Contents

We present the basic knowledge on the structure of molecules coding the genetic information, mechanisms of transfer of this information from DNA to proteins and phenomena connected with replication of DNA. In particular, we describe the differences of mutational pressure connected with replication of the leading and lagging DNA strands. We show how the asymmetric replication of DNA affects the structure of genomes, positions of genes, their function and amino acid composition. Results of Monte-Carlo simulations of evolution of protein coding sequences show a specific role of genetic code in minimizing the effect of nucleotide substitutions on the amino acid composition of proteins. The results of simulations are compared with the results of analyses of genomic and proteomic data bases. This chapter is considered as an introduction to further chapters where chromosomes with genes represented by nucleotide sequences were replaced by bitstrings with single bits representing genes.

We describe modeling the population evolution using the Penna model based on Monte-Carlo method. Individuals in the populations are represented by their diploid genomes. Genes expressed after the minimum reproduction age are under a weaker selection pressure and accumulate more mutations than those expressed before the minimum reproduction age. The generated gradient of defective genes determines the ageing of individuals and age-structured populations are very similar to the natural, sexually reproducing populations. The genetic structure of a population depends on the way how the random death affects the population. The improvement of the medical care and healthier life style are responsible for the increasing of the life expectancy of humans during the last century. Introducing a noise into the relations between the genotype, phenotype, and environment, it is possible to simulate some other effects, like the role of immunological systems and a mother care. One of the most interesting results is the evolution of sex chromosomes. Placing the male sex determinants on one chromosome of a pair of sex chromosomes is enough to condemn it for shrinking if the population is panmictic (random-mating is assumed). If males are indispensable for taking care of their offspring and have to be faithful to their females, the male sex chromosome does not shrink.

Intragenomic recombination (crossover) is a very important evolutionary mechanism. The crossover events are not evenly distributed along the natural chromosomes. Monte-Carlo simulations revealed that frequency of recombinations decides about the strategy of chromsomes' and genomes' evolution. In large panmictic populations, under high recombination rate the Darwinian purifying selection operates keeping the fraction of defective genes at the relatively low level. In small populations and under low recombination rate the strategy of complementing haplotypes seems to be more advantageous. Switching between the two strategies has a character of phase transition - it depends on inbreeding coefficient and crossover rate. The critical recombination rate depends also on the size of chromosome. It is also possible, that in one genome some chromosomes could be under complementing while some other under purifying selection. Such situation stabilizes genome evolution and reproduction strategy. It seems that this phenomenon can be responsible for the positive correlation between kinship and fecundity, recently found in the Icelander population. When large population is forced to enter the complementing strategy, the phenomenon of sympatric speciation is observed.

The aim of these lectures is to give a survey of mathematical models and methods of population dynamics. The main subject will be structured models which describe the distribution of population with respect to some parameters such as age, size, maturity or proliferative state of cells. Some models are general, e.g. age-structured model can be applied to human population as well as to a cell population but others will concern specific populations e.g. the population of erythrocytes. We concentrate on models connected with genetics: the Penna model and models describing telomere shortening, distribution of genes in a genome, stochastic gene expression and the cell-cycle.

Some mathematical models of time evolution of the age-structured biological populations have been discussed. It has been shown that the genetic information represented by the distribution of the defective genes in the population under consideration can be included into the deterministic models which use ordinary differential equations or delay differential equations. The results for the equilibrium genetic populations have been compared with the equilibrium populations in the Penna model of genetic evolution. It has been stressed the difference between the models which use ordinary differential equations and delay differential equations.

Recently, computational modeling became a very important research tool that enables us to study problems that for decades evaded scientific analysis. Evolutionary systems are certainly examples of such problems: they are composed of many units that might reproduce, diffuse, mutate, die, or in some cases for example communicate. These processes might be of some adaptive value, they influence each other and occur on various time scales. That is why such systems are so difficult to study. In this paper we briefly review some computational approaches, as well as our contributions, to the evolution of ecosystems and language. We start from Lotka-Volterra equations and the modeling of simple two-species prey-predator systems. Such systems are canonical example for studying oscillatory behavior in competitive populations. Then we describe various approaches to study long-term evolution of multi-species ecosystems. We emphasize the need to use models that take into account both ecological and evolutionary processes. Recently we introduced a simple model of this kind, and its behavior is briefly summarized. In this multi-species prey-predator system, competition of predators for preys and space results in evolutionary cycling. We suggest that such a behavior of the model might correspond to long-term periodic changes of the biodiversity of the Earth ecosystem as predicted by Raup and Sepkoski. Finally, we address the problem of the emergence and development of language. It is becoming more and more evident that any theory of language origin and development must be consistent with darwinian principles of evolution. Consequently, a number of techniques developed for modeling evolution of complex ecosystems are being applied to the problem of language. We briefly review some of these approaches. We also discuss the behavior of a recently introduced evolutionary version of the naming-game model. In this model communicating agents reach linguistic coherence via a bio-linguistic transition which is due to the coupling of evolutionary and linguistic abilities of agents.

The following sections are included:

• Biological Glossary

• Author Index

• Subject Index