Section: New Results

Stochastic Models of Biological Networks

Participants : Renaud Dessalles, Sarah Eugene, Philippe Robert.

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 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 106 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. 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.

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 tend to test the hypothesis usually made that such feedback permits a noise reduction of protein distribution compared to the “open loop” model. We have made the mathematical analysis of the model (using a scaling to be able to have explicit results), it appeared that reduction of variance compared to an “open loop” model is limited: the variance cannot be reduced for more than 50%.

We proposed another possible effect of the feedback loop: the return to equilibrium is faster in the case of a feedback model compared to the open loop model. Such behaviour can be beneficial for the bacteria to change of command for a new level of production of a particular protein (due, for example, to a radical change in the environment) by reducing the respond time to reach this new average. This study has been mainly performed by simulation and it has been shown that the feedback model can go 50% faster than the open loop results. See [13] ..

Transcription-translation model for all proteins

The other model that has been studied integrates the production of all the proteins. Each gene has to be transcribed in mRNA (using RNA-Polymerase molecules) and each mRNA has to be translated in protein (using ribosome molecules). Experiments (as the one from Taniguchi et al. (2010)) have shown that protein production is subject to high variability especially for highly expressed proteins. Our goal is to determine what in the protein production mechanism is responsible for the noise.

We already made simulations that takes into amount of RNA-Polymerases and Ribosomes and that genes and mRNAs sequester these molecules during the whole the time of elongation. This global sharing of Ribosomes/RNA-Polymerases reproduce only a part of the unknown noise experimentally seen. We are developing Python simulations that extends this model and take into account other feature that might be responsible for the noise in protein production. This new simulation will include new features such as:

  • The volume of the cell. We consider it as proportional to the total number of proteins, and will increase as the cell grows. Transcription and translation initiation are then depending on the concentration of respectively free RNA-polymerase and free ribosomes.

  • The division of the cell. At division, all components have an equal chance to go in either one of the two daughter cell.

  • DNA replication. At some point in the cell cycle, the genome duplicates, doubling therefore the copy number of each gene

The simulation parameters will be fit with the data of Taniguchi et al. (2010) and the goal is to compare our result to see if which aspects of the protein production are responsible for the noise of the proteins.

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 polymere, 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.

The second part concerns the study of telomeres. This work is made in collaboration with Zhou Xu, Teresa Teixeira, from IBCP in Paris.

In eukaryotic cells, at each mitosis, chromosomes are shortened, because the DNA polymerase is not able to duplicate one ending of the chromosome. To prevent loss of genetic information- which could be catastrophic for the cell-chromosomes are equipped with telomeres at their endings. These telomeres do not contain any genetic information; they are a repetition of the sequence T-T-A-G-G-G thousands times. At each mitosis, there is therefore a loss of telomere. As it has a finite length, when the telomeres are too short, the cell cannot divide anymore: they enter in replicative senescence. Our model tries to captures the two phases of the shortening of telomeres: first, the initial state of the cells, when the telomerase is still active to repair the telomeres. Second, when the telomerase is inhibited, we try to estimate the senescence threshold, when the replication of the cells stops.