Fluctuations effects in population genetics and in protein translation

This thesis is an investigation of the effects of fluctuations in some problems at the interface between statistical physics and biology. In the first chapter we introduce Langevin equations with multiplicative noise, by focusing, in particular, on a generic power-law amplitude of the noise. This equation, depending on the noise exponent (i.e., on the character of the fluctuations), encompasses many known stochastic systems, and we show that the absorbing properties of the boundaries strongly depend on this exponent. The second chapter investigates the role of fluctuations in population genetics, which describes how the biodiversity and the composition of a population evolve in time due to the action of evolutionary forces. In the absence of mutations, every population constituted by a finite number of individuals will eventually lose its biodiversity through a process called fixation. We show that, if the evolutionary force known as balancing selection acts on a subdivided population, the mean time to fixation as a function of the migration rate develops a nonmonotonicity. Furthermore our analysis predicts, in the limit of infinitely many subpopulations, a transition between a phase characterized by the presence of biodiversity in the total population and a phase characterized by its absence. The third chapter deals with the role of fluctuations in protein translation, a crucial and only partly understood step in gene expression and one of the most common biochemical reactions occurring in the cell: the individual triplets of nucleotides (the codons) composing a messenger RNA (mRNA) are translated into amino acids (the units composing the proteins) by the ribosomes. More in detail we address an intriguing question concerning the binding time distribution, i.e., the distribution of the time intervals needed by the ribosome to bind with a transport RNA (tRNA) charged with the correct amino acid. We provide an analytic estimate for this distribution, which deviates from the exponential distribution expected in the absence of fluctuations in the number of charged tRNAs around the ribosome.