## Section: New Results

### Stochastic Modeling of Biological Networks

Participants : Emanuele Leoncini, Philippe Robert.

This is a collaboration with Vincent Fromion from INRA Jouy en Josas, which started on October 2010.

The goal is to propose a mathematical model of the production of proteins in prokaryotes. Proteins are biochemical compounds that play a key role in almost all the cell functions and are crucial for cell survival and for life in general. In bacteria the protein production system has to be capable to produce abut 2500 different types of proteins in different proportions (from few dozens for the replication machinery up to 100000 for certain key metabolic enzymes). Bacteria uses more than the 85% of their resources to the protein production, making it the most relevant process in these organisms. Moreover this production system must meet two opposing problems: on one side it must provide a minimal quantity for each protein type in order to ensure the smooth-running of the cell, on the other side an “overproduction policy” for all the proteins is infeasible, since this would impact the global performance of the system and of the bacterium itself.

Gene expression is intrinsically a stochastic process: gene activation/deactivation occurs by means the encounter of polymerase/repressor with the specific gene, moreover many molecules that take part in the protein production act at extremely low concentrations. We have restated mathematically the classical model using Poisson point processes (PPP). This representation, well-known in the field of queueing networks but, as far as we know, new in the gene expression modeling, allowed us to weaken few hypothesis of the existing models, in particular the Poisson hypothesis, which is well-suited in some cases, but that, in some situations, is far from the biological reality as we consider for instance the protein assemblage.

The theoretical environment of Poisson point processes has lead us to propose a new model of gene expression which captures on one side the main mechanisms of the gene expression and on the other side it tries to consider hypothesis that are more significant from a biological viewpoint. In particular we have modeled: gene activation/deactivation, mRNA production and degradation, ribosome attachment on mRNA, protein elongation and degradation.
We have shown how the probability distribution of the protein production and the protein lifetime may have a significant impact on the fluctuations of the number of proteins. We have obtained analytic formulas when the duration of protein assemblage and degradation follows a general probability distribution, i.e. without the Poisson hypothesis.
In particular, by using a PPP representation we have been able to include the deterministic continuous phenomenon of protein degradation, which is the main protein degradation mechanism for stable proteins. We have showed moreover that this more realistic description is surprisingly identical in distribution with the classic assumption of protein degradation by means of a degrading protein (*proteosome*).
We have used our model also to compare the variances resulting by choosing different hypotheses for the probability elongation, in particular we have hypothesize the protein assembly to be deterministic. This assumption is justified because of the elongation step, which consists of a large number of elementary steps, can be described by the sum of exponential steps and the resulting distribution is well approximated by a Gaussian distribution because of the central limit theorem. Under the hypothesis of small variance of the resulting Gaussian distribution, we can assume the elongation step to be deterministic.
The model has showed how, under the previous hypothesis, the variance on the number of proteins is bigger than the classical model with the Poisson hypothesis.

We have developed a C++ stochastic simulator for our general model, which has allowed the computation of variance when it was not possible to derive explicit analytic close formulas and the simulation of some extension of the actual model.

This year we have investigated a mathematical model of the production of proteins in prokaryotic cells. Up to now most of the mathematical used to study these problems concern the production of *one* fixed class of proteins. When several classes of proteins are considered, each class requires in fact a fraction of the common and limited resources of the cell. One has therefore to understand how the allocation of the resources within the cell is done. Due to the fact that the cytoplasm of the cell is a quite disorganized medium where the components of the cell move, the whole production process has an important stochastic component. A model describing the allocation of the ribosomes of the cell to produce proteins is investigated via a Markovian representation. Asymptotic results for the equilibrium and for the transient behavior have been obtained under a scaling procedure and a reasonable biological assumption of saturation, i.e. when resources of the cell are tight. The equilibrium and the transient behavior have been investigated, it has been shown in particullar that, in the limit, the number of free ribosomes converges in distribution to a Poisson distribution whose parameter satisfies a fixed point equation.