Stochastic Modelling of self-regulation in the protein production system of bacteria.

This is a collaboration with Vincent Fromion from INRA Jouy-en-Josas, which started in December 2013.

In prokaryotic cells (e.g. E. Coli. or B. Subtilis) the protein production system has to produce in a cell cycle (i.e. less than one hour) more than ${10}^6$ molecules of more than 2500 kinds, each having different level of expression. The bacteria uses more than $67\%$ of its resources to the protein production. Gene expression is a highly stochastic process: bacteria sharing the same genome, in a same environment will not produce exactly the same amount of a given protein. Some of this stochasticity can be due to the system of production itself: molecules, that take part in the production process, move freely into the cytoplasm and therefore reach any target in the cell after some random time; some of them are present in so much limited amount that none of them can be available for a certain time; the gene can be deactivated by repressors for a certain time, etc. We study the integration of several mechanisms of regulation and their performances in terms of variance and distribution. As all molecules tends to move freely into the cytoplasm, it is assumed that the encounter time between a given entity and its target is exponentially distributed.

Models with Cell Cycle

Usually, classical models of protein production do not explicitly represent several aspects of the cell cycle: the volume variations, the division and the gene replication. Yet these aspects have been proposed in literature to impact the protein production. We have therefore proposed a series of “gene-centered” models (that concentrates on the production of only one type of protein) that integrates successively all the aspects of the cell cycle. The goal is to obtain a realistic representation of the expression of one particular gene during the cell cycle. When it was possible, we analytically determined the mean and the variance of the protein concentration using Marked Poisson Point Process framework.

We based our analysis on a simple model where the volume changes across the cell cycle, and where only the mechanisms of protein production (transcription and translation) are represented. The variability predicted by this model is usually assimilated to the “intrinsic noise” (i.e. directly due to the protein production mechanism itself). We then add the random segregation of compounds at division to see its effect on protein variability: at division, every mRNA and every protein has an equal chance to go to either of the two daughter cells. It appears that this division sampling of compounds can add a significant variability to protein concentration. This effect directly depends on the relative variance (Fano factor) of the protein concentration: this effect is stronger as the relative variance is low. The dependence on the relative variance can be explained by considering a simplified model. With parameters deduced from real experimental measures, we estimate that the random segregation of compounds can double the variability of the genes with the lowest relative variance.

Finally, we integrate the gene replication to the model: at some point in the cell cycle, the gene is replicated, hence doubling the transcription rate. We are able to give analytical expressions for the mean and the variance of protein concentration at any moment of the cell cycle; it allows to directly compare the variance with the previous model with division. We show that gene replication has little impact on the protein variability: an environmental state decomposition shows that the part of the variance due to gene replication represents only at most $2\%$ of the total variability predicted by the model.

Finally, we have investigated other possible sources of variability by presenting other simulations that integrate some specific aspects: variability in the production of RNA-polymerases and ribosomes, uncertainty in the division and DNA replication decisions, etc. None of the considered aspects seems to have a significant impact on the protein variability.

Stochastic Modelling of Protein Polymerization

This is a collaboration with Marie Doumic, Inria MAMBA team. The first part of our work focuses on the study of the polymerization of protein. This phenomenon is involved in many neurodegenerative diseases such as Alzheimer's and Prion diseases, e.g. mad cow. In this context, it consists in the abnormal aggregation of proteins. Curves obtained by measuring the quantity of polymers formed in in vitro experiments are sigmoids: a long lag phase with almost no polymers followed by a fast consumption of all monomers. Furthermore, repeating the experiment under the same initial conditions leads to somewhat identical curves up to translation. After having proposed a simple model to explain this fluctuations, we studied a more sophisticated model, closer to the reality. We added a conformation step: before being able to polymerize, proteins have to misfold. This step is very quick and remains at equilibrium during the whole process. Nevertheless, this equilibrium depends on the polymerization which is happening on a slower time scale. The analysis of these models involves stochastic averaging principles.

We have also investigated a more detailed model of polymerisation by considering the the evolution of the number of polymers with different sizes $\left(X_i\left(t\right)\right)$ where $X_i\left(t\right)$ is the number of polymers of size $i$ at time $t$. By assuming that the transitions rates are scaled by a large parameter $N$, it has been shown that, in the limit, the process $\left(X_i^N\left(t\right)\right)$ is converging to the solution of Becker-Döring equations as $N$ goes to infinity. For another model including nucleation, we have given an asymptotic description of the lag time at the first and second order. These results are obtained in particular by proving stochastic averaging theorems.

Central Limit Theorems

Study of the Nucleation Phenomenon

We have investigated a new stochastic model describing the time evolution of a polymerization process. The initial state of the system consists only of isolated monomers. We study the lag time of the polymerization process, that is, the first instant when a fraction of the initial monomers is polymerized, i.e. the fraction of monomers used in the polymers. The mathematical model includes a nucleation property: polymers with a size below some threshold $n_c$, the size of the nucleus, are quickly fragmented into smaller polymers. For a size greater than $n_c$, the fragmentation still occurs but at a smaller rate. A scaling approach is used, by taking the volume $N$ of the system as a scaling parameter. If $n_c\ge 3$, under quite general assumptions on the way polymers are fragmented, we prove a limit theorem for the instant $T^N$ of creation of the first “stable” polymer, i.e. a polymer of size $n_c$. It is proved that the distribution of $T^N/N^{n_c-3}$ converges to an exponential distribution. We also show that, if $n_c\ge 4$, then the lag time has the same order of magnitude as $T^N$ and, if $n_c=3$, it is of the order of $logN$. An original feature of our model is the significant variability (asymptotic exponential distribution) proved for the instants associated to polymerization. This is a well known phenomenon observed in the experiments in biology but it has not been really proved in appropriate mathematical models up to now. The results are proved via a series of (quite) delicate technical estimates for occupations measures on fast time scales associated to the first $n_c$ coordinates of the corresponding Markov process. Extensive Stochastic calculus with Poisson processes, several coupling arguments and classical results from continuous branching processes theory are the main ingredients of the proofs.