Time Dependent Stochastic mRNA and Protein Synthesis in Piecewise-Deterministic Models of Gene Networks

Abstract

We discuss piecewise-deterministic approximations of gene networks dynamics. These approximations capture in a simple way the stochasticity of gene expression and the propagation of expression noise in networks and circuits. By using partial omega expansions, piecewise deterministic approximations can be formally derived from the more commonly used Markov pure jump processes (chemical master equation). We are interested in time dependent multivariate distributions that describe the stochastic dynamics of the gene networks. This problem is difficult even in the simplified framework of piecewise-deterministic processes. We consider three methods to compute these distributions: the direct Monte-Carlo; the numerical integration of the Liouville-master equation; and the push-forward method. This approach is applied to multivariate fluctuations of gene expression, generated by gene circuits. We find that stochastic fluctuations of the proteome and, much less, those of the transcriptome can discriminate between various circuit topologies.

Distribution Estimation for Probabilistic Loops

“…In addition, we also crafted three new examples, listed in the last three entries of the first column of Table 1, as follows: (i) line 5 of Table 1 specifies a PP loop approximating a uniform distribution; (ii) line 6 of Table 1 refers to the Vasicek model of Fig. 1; and (iii) the last line of Table 1 lists an example encoding a piece-wise deterministic process (PDP) modeling gene circuits based on [22]. In particular, the PDP model we consider can be used to estimate the distribution of protein x and the mRNA levels y in a gene; our PP encoding of this PDP model is given in Fig.…”

Stationary distributions and metastable behaviour for self-regulating proteins with general lifetime distributions

“…We use a Hill-type response (25) with a 0 = 0.3, a 1 = 1.6, H = 4, and Ω shown in panel captions. (26) is reduced to that of determining the residence times T ± . Previous largedeviation and WKB analyses of the one-dimensional model [18,23,24] provide an Arrhenius-type formula…”

“…The dynamics of these processes is therefore well described by stochastic Markov processes in continuous time with discrete state space [15,22,42]. While few-component or linear-kinetics systems [16] allow for exact analysis, in more complex system one often uses approximative methods [12], such as moment closure [4], linear-noise approximation [3,9], hybrid formulations [25,26,33], and multi-scale techniques [38,39].…”

“…in which the lumped states and ⊕ correspond to the basins of attractions of the two stable fixed points; T − and T + are the respective residence times. The mixture weights p − (t) and p + (t) in (26) are identified with the probabilities of the lumped states in (28); these satisfy the Chapman-Kolmogorov equations…”

Black Swans of CRISPR: Stochasticity and Complexity of Genetic Regulation

“…Such tuning may expectedly comprise some combination of amplifying, damping, or splitting of stochastic processes in pathways; the ultimate goals of these processes may be controlling anything from the average number of molecules of neurotransmitter released from a synaptic bouton in response to an action potential, to the macroscopic vagaries of heterochrony in the evolution of metamorphosis and other life history traits. The function of the mGRN is thus understood, but the complexity of its internal mechanisms is stupendous: for instance, phenomena of signal transduction cascades and network crosstalk, and the mixed dynamics of chromatin remodeling, transcription factor binding, cis‐regulatory sequences, alternative mRNA splicing—amongst myriad others—will all act in parallel …”