Damian Wójtowicz

A DISCRETE MODEL OF EVOLUTION OF SMALL PARALOG FAMILIES

We introduce and analyze a simple probabilistic model of genome evolution. It is based on three fundamental evolutionary events: gene loss, duplication and accumulated change. We are mainly interested in asymptotic size distribution of small paralogous gene families in a genome. This is motivated by previous works which consisted in fitting the available genomic data into, what is called, paralog distributions. This formalism is described as a discrete-time Markov chain. The formulas for equilibrium paralog family sizes are derived. Moreover, we show that when probabilities of gene removal and duplication are small and close to each other, then the resulting distribution is close to logarithmic distribution. Some empirical results for microbial genomes are presented.

A Case Study of Genome Evolution: From Continuous to Discrete Time Model

We introduce and analyse a simple model of genome evolution. It is based on two fundamental evolutionary events: gene loss and gene duplication. We are mainly interested in asymptotic distributions of gene families in a genome. This is motovated by previous work which consisted in fitting the available genomic data into, what is called paralog distributions. Two approaches are presented in this paper: continuous and discrete time models. A comparison of them is presented too – the asymptotic distribution for the continuous time model can be seen as a limit of the discrete time distributions, when probabilities of gene loss and gene duplication tend to zero. We view this paper as an intermediate step towards mathematically settling the problem of characterizing the shape of paralog distribution in bacterial genomes.

On genome evolution with accumulated change and innovation

We introduce and analyse a simple discrete probabilistic model of genome evolution. It is based on four fundamental evolutionary events: gene duplication, loss, change and innovation, and it is called DLCI model. This is the first such model rigorously analysed. The focus of the paper is around the size distribution of gene families. The formulas for equilibrium gene family sizes are derived showing that they follow a logarithmic distribution. We consider also a disjoint union of DLCI models and we present the result of this study. Some empirical results for microbial genomes are presented.

Almost FPRAS for lattice models of protein folding

The provably efficient randomized approximation scheme which evaluates the partition function for the wide class of lattice models of protein prediction is presented. We propose to apply the idea of self-testing algorithms introduced recently in [8]. We consider the protein folding process which is simplified to a self-avoiding walk on a lattice. The power of a simplified approach is in its ability to search the conformation space, to train the search parameters and to test basic assumptions about the nature of the protein folding process. Our main theoretical results are formulated in the general setting, i.e. we do not assume any specific lattice model. For the simulation study we have chosen the HP model on the FCC lattice.