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 on December 2014.

In procaryots 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 85% 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. All molecules are supposed to move freely into the cytoplasm, it is assumed that the the encounter time between a given entity and its target is exponentially distributed.

Transcription-translation model for all proteins

The first model that has been studied integrates the production of all the proteins. Each gene has to be transcribed in mRNA and each mRNA has to be translated in protein. The transcription step needs a RNA-Polymerase molecule that is sequestered during the time of elongation. Likewise, each mRNA needs a ribosome in order to produce a protein. RNA-Polymerases/Ribosomes are present in limited amount and the genes/mRNAs sequester these molecules during the whole the time of elongation. Finally each mRNA has an exponentially distributed lifetime with an average value of 4 min and the proteins disappear at a rate of one hour, hence simulating the global dilution in the growing bacteria.

This global sharing of Ribosomes/RNA-Polymerases among all proteins induces a general regulation: each gene competing to each other to have access to these common resources. Because of the parameters of affinity (between gene and RNA-Poymerase and between mRNA and ribosome) are specific to each gene, it allows a large range of average protein production but induce some noise, especially for highly expressed proteins.

We developed a Python simulation, and using the biological experiments of Tanichuchi et al. (2010), and we have investigated a biologically coherent range of parameters. By making the simulations, we have been able to reproduce certain aspects of the biological measures, especially for the high amount of noise for well expressed proteins.

Simple feedback model

We have also investigated the production of a single protein, with the transcription and the translation steps, but we also introduced a direct feedback on it: the protein tends to bind on the promoter of its own gene, blocking therefore the transcription. The protein remains on it during an exponential time until its detachment caused by thermal agitation.

The mathematical analysis aims at understanding the nature of the internal noise of the system and to quantify it. We try to determine if, for instance, for the same average protein level, the feedback permits a noise reduction of protein distribution compared to the "open loop" model; or if it rather allows a better efficiency in case of a change of command for a new level of production (due, for example, to a radical change in the environment) by reducing the respond time to reach this new average.

Stochastic Modelling of Protein Polymerization

This is a collaboration with Marie Doumic, Inria MAMBA team.

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 a translation.

The first study we did proposed a simplified stochastic model to analyze this phenomenon. For this model, when the volume gets large, the quantity of polymers has the typical sigmoidal shape. A second order result has also been obtained for this model. We were able to compute the asymptotic distribution of the lag time and express its variance. The parameters of the model have been obtained by using data given by Wei-Feng Xue, University of Kent.

The current project concerns a more sophisticated mathematical model. Indeed, we have added a conformation step: before polymerizing, 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 follows the conformation step: this modelling leads to the study of averaging principles.