One indication of the relevance of stochastic laws in biology is the fact that genetically similar cells can behave in very different ways. This suggests an analogy with statistical physics, in which stochastic laws govern the behaviour of large ensembles of particles. Biological randomness has important theoretical and practical consequences in medicine and may be responsible for the resistance of pathogens or cancer to drug treatment. Modeling biological randomness is now a major field in theoretical biology and has been approached with methods adapted from physics. Like in physics, biological systems are governed by hierarchical processes involving variables with different timescales. Contrary to most physical processes, biological systems can exhibit inversions in the relationship between timescale and hierarchical rank. As a consequence, microscopic fluctuations can be transmitted to and even amplified by the phenotype. 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.


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