## Modeling of birth-death and diffusion processes in biological complex environments

##### Abstract

This thesis is centered on the theory of stochastic processes and their applications in biological systems characterized by a complex environment. Three case studies have been modeled by the use of the three fundamental tools of stochastic processes:
the master equation (ME), the stochastic differential equation (SDE) and the partial differential equation (PDE). The choice of an approach respect to another is determined also by the nature of the problem, i.e. the scale at which we are interested to observe the system, micro- meso- or macro-scopic. The principal approach here applied to deal with complexity is the characterization of the system by means of probability distributions describing each a parameter of the model or the introduction of fractional order derivatives to include non-local and memory effects maintaining the linearity in the equations. Different mathematical methods have been applied to obtain analytical solutions of the three original models proposed, related in particular to the theory of Laplace, Fourier and Mellin transform. In Chapter 1 we briefly review the theory of stochastic processes to introduce the topics presented in the following chapters. Birth-death processes are fundamental in modeling of population dynamics, as the characterization of relative species abundance (RSA) in ecology. Models derived from ecological community studies have been also used to describe the evolution of genomic elements, and in particular, the dynamics of transposable elements. In Chapter 2 we derive a birth-death process master equation to test if Long Interspersed Elements (LINEs) can be modeled according to the neutral theory of biodi versity. According to this theory, the structure of the collection of LINE subfamilies would be the result of stochastic drift, as opposed to differences in ecological traits between subfamilies. Our results show that although the neutral model fits well the overall LINE distribution in humans, significant deviations from it can be observed by stratifying LINE subfamilies by age groups. This suggests that at specific times during the evolution of the mammalian genome multiple concurrently active LINE subfamilies might have been in direct competition. We further investigated how this competition could have been shaped by the LINE 5’UTR structure and by the chromatin landscape. Dealing with biological systems (but not only), the diffusion process is one of the most important topics for a physicist. Brownian and anomalous diffusions are widely observed in nature and studied by the use of both phenomenological and founding models, the last ones trying to explain the origin of the anomaly in the system under study. One of the key concepts when speaking about anomalous diffusion is the complexity of the system itself, independently of the particular approach of the model in use. Biological systems are complex and stochastic systems by definition at any level: from gene expression and motion of molecules inside the cell till the ecological description of individuals and their dispersal. However, thanks to physics, it is well known that this complexity does not arise from something that is complex at any level. If the problem is decomposed in smaller and smaller bricks it is possible to see that complexity arises at a meso-macroscopic scale from the same fundamental interactions treaten by fundamental physics and that stochasticity itself
is generated by the complexity of the system observed. Following this idea, it seems reasonable that anomalous diffusion can be read in terms of a superposition of simpler processes. In Chapter 3 we derive a model of anomalous diffusion based on a
Langevin approach in which anomalous behavior arise in the asymptotic intermediate state as a consequence of the heterogeneity of the system, from the superposition of Ornstein-Uhlenback processes. Anomalous diffusive behavior can be also described by the fractional generalizations of diffusion equation by the introduction of fractional derivatives. Fractional derivatives are non-local integral operators that generalize the standard integer derivative, suitable to describe systems in which memory and non-local effects are observed. In Chapter 4 we propose an extension of the cable equation, useful to describe anomalous diffusion phenomena as the signal conduction in spiny dendrites, by introducing a Caputo time fractional derivative. The same generalization can be derived within the continuous time random walk framework, building the model as a superposition of Markovian processes, each characterized by its own timescale generated by the random geometry of the system. The same model can be also derived from a generalized grey Brownian motion in which is introduced a non-stationary distribution of length scales. The fundamental solutions of the most common boundary problems are derived by the application of the Efros theorem of
Laplace transforms and written in terms of Wright special functions.