Claudia	Cianci - Università di Firenze & INFN	#	
Non Gaussian corrections and finite size effects in a model of 
autocatalytic reactions	#
The cell is a structural and 
functional unit, the building block of any living systems. 
Cells are constituted by a tiny membrane, made of lipid 
bilayer, which encloses a finite volume and protects the 
genetic material stored inside. Modern cells with their 
complex machineries, result from evolution of ancient 
supposedly minimalistic cell entity, the protocell. It is 
customarily believed that autocatalytic reactions might have 
be at play in primordial protocell. The shared view is that 
protocell's volume might have been occupied by interacting 
families of replicators, organized in autocatalytic cycles. A 
chemical reaction is called autocatalytic if one of the 
reaction products is itself a catalyst for the chemical 
reaction. Even if only a small amount of the catalyst is 
present, the reaction may start off slowly, but will quickly 
speed up once more catalyst is produced. If the reactant is 
not replaced, the process will again slow down producing the 
typical sigmoid shape for the concentration of the product. 
All this is for a single chemical reaction, but of greater 
interest is the case of many chemical reactions, where one or 
more reactions produce a catalyst for some of the other 
reactions. Then the whole collection of constituents is called 
an autocatalytic set. The study of the dynamical evolution of 
interacting species of homologous quantities defines the field 
of population dynamics, which finds particularly important 
applications within the realm of life science. Population is 
indeed a technical term which is referred to various, 
completely distinct fields of applications. The classical 
deterministic approach to population dynamics relies on 
characterizing quantitatively the densities of species through 
a system of ordinary differential equations which incorporate 
the specific interactions being at play. As opposed to this 
formulation, a different (stochastic) level of modeling can be 
invoked which instead focuses on the individual-based 
description. This amounts to characterizing the microscopic 
dynamics via explicit rules governing the interactions among 
individuals and with the surrounding environment. The 
stochasticity is intrinsic to the systems and stems from the 
microscopic finiteness of the investigated medium. Remarkably, 
inherent demographic perturbations might induce regular 
behaviours at the macroscopic level, emerging as a spontaneous 
colletive self-organized phenomenon.In this paper, we 
investigate the stochastic dynamics of a complex network of 
autocatalytic reactions, within a spatially bounded domain, so 
to mimick a primordial cell back at the orgin of life [1,2].  
The role of stochastic fluctuations is elucidated through the 
use of the van Kampen system-size expansion and shown to 
induce regular oscillations in time of in the concentration 
amount [3]. Corrections beyond the Gaussian  approximation are 
analytically computed within the van Kampen operative ansatz. 
An extended Fokker-Planck equation is obtained and the moments 
of the multivariate non Gaussian distribution of fluctuations 
quantified. The theory predictions are challenged versus 
direct stochastic simulations and  shown to return an 
excellent agreement. Possible implications of our findings as 
concerns protocells origin and evolution are addressed