## DO CELLS PLAY DICE?

**We are studying the randomness of gene expression in relation to developmental biology and infectiology.**Our mathematical approaches to biological randomness are based on hybrid approximations of discrete, stochastic, chemical reaction netoworks, that are well adapted to the situation when microscopic timescales are slower than mesoscopic ones.

**Transcriptional bursting.** The non correlated, stochastic activity of transcription sites in different cells, represents the intrinsic expression noise. When this noise results from alternating active and inactive states of the promoter, one talks of transcription bursting. The movie shows the transcription activity of HIV-1 promoters in HeLa cells.

**Types of variables and important timescales of stochastic biochemical reactions networks.**** (A) **The partition of variables and of the reactions of a biochemical network. XD are discrete species, whose numbers change seldom by discrete jumps; XC are continuous species, whose numbers change often, quasi-continuously. RD are reactions acting on discrete variables and whose rates depend on discrete variables,RC are reactions acting on continuous variables and whose rates depend on continuous variables. RCD,RDC are reactions acting on discrete and on continuous variables, respectively, and have rates depending on both discrete and continuous variables. Dotted line arrows mean that reaction rates depend on the source variables. Continuous line arrow mean that the source reaction acts on the end variable. **(B) **Typical trajectories and time scales (discreteness time tD, switching time tS ) of continuous and discrete variables.

**Different approximations of stochastic biochemical reactions networks.**

The “exact” model is a completely discrete Markov process described by the chemical master equation. Various approximations can be obtained as asymptotic limits when some parameter is very small or very large from the “exact” model or from approximations.

**References**

- GCP. Innocentini, A. Hodgkinson, F. Antoneli, A. Debussche, O.Radulescu. Pushforward method for piecewise deterministic biochemical simulations. 2020, in review Theoretical Computer Science, Elsevier.
- G. Innocentini, A Hodgkinson, O Radulescu. Time Dependent Stochastic mRNA and Protein Synthesis in Piecewise-deterministic Models of Gene Networks. Frontiers in Physics. (2018) 6, 46.
- A.Crudu, A.Debussche, A.Muller, O.Radulescu, Convergence of stochastic genenetworks to hybrid piecewise deterministic processes, Annals of Applied Probability (2012) 22: 1822-1859.
- M.L. Ferguson, D. Le Coq, M. Jules, S. Aymerich, O.Radulescu, N. Declerck, C.A.Royer. Reconciling molecular regulatory mechanisms with noise patterns of bacterial metabolic promoters in induced and repressed states, Proceedings of the National Academy of Sciences USA (2012) 109: 155.
- O.Radulescu, A.Muller, A.Crudu. Théorèmes limites pour processus de Markov à sauts; Synthèse de résultats et applications en biologie moléculaire, TSI (Technique et Science Informatiques) (2007) 26/3-4 : 443-469.