Guilherme C. P. Innocentini

Multimodality and flexibility of stochastic gene expression.

We consider a general class of mathematical models for stochastic gene expression where the transcription rate is allowed to depend on a promoter state variable that can take an arbitrary (finite) number of values. We provide the solution of the master equations in the stationary limit, based on a factorization of the stochastic transition matrix that separates timescales and relative interaction strengths, and we express its entries in terms of parameters that have a natural physical and/or biological interpretation. The solution illustrates the capacity of multiple states promoters to generate multimodal distributions of gene products, without the need for feedback. Furthermore, using the example of a three states promoter operating at low, high, and intermediate expression levels, we show that using multiple states operons will typically lead to a significant reduction of noise in the system. The underlying mechanism is that a three-states promoter can change its level of expression from low to high by passing through an intermediate state with a much smaller increase of fluctuations than by means of a direct transition.

Protein synthesis driven by dynamical stochastic transcription

In this manuscript, we propose a mathematical framework to couple transcription and translation in which mRNA production is described by a set of master equations, while the dynamics of protein density is governed by a random differential equation. The coupling between the two processes is given by a stochastic perturbation whose statistics satisfies the master equations. In this approach, from the knowledge of the analytical time-dependent distribution of mRNA number, we are able to calculate the dynamics of the probability density of the protein population.

Analytic framework for a stochastic binary biological switch.

We propose and solve analytically a stochastic model for the dynamics of a binary biological switch, defined as a DNA unit with two mutually exclusive configurations, each one triggering the expression of a different gene. Such a device has the potential to be used as a memory unit for biological computing systems designed to operate in noisy environments. We discuss a recent implementation of this switch in living cells, the recombinase addressable data (RAD) module. In order to understand the behavior of a RAD module we compute the exact time-dependent joint distribution of the two expressed genes starting in one state and evolving to another asymptotic state. We consider two operating regimes of the RAD module, a fast and a slow stochastic switching regime. The fast regime is aggregative and produces unimodal distributions, whereas the slow regime is separative and produces bimodal distributions. Both regimes can serve to prepare pure memory states when all cells are expressing the same gene. The slow regime can also separate mixed states by producing two subpopulations, each one expressing a different gene. Compared to the genetic toggle switch based on positive feedback, the RAD module ensures more rapid memory operations for the same quality of the separation between binary states. Our model provides a simplified phenomenological framework for studying RAD memory devices and our analytic solution can be further used to clarify theoretical concepts in biocomputation and for optimal design in synthetic biology.

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

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-determinisitic 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.

Analytic framework for stochastic binary biological switches

We propose an analytic solution for the stochastic dynamics of a binary biological switch, defined as a DNA unit with two mutually exclusive configurations, each one triggering the expression of a different gene. Such a device could be used as a memory unit for biological computing systems designed to operate in noisy environments. We discuss a recent implementation of an exclusive switch in living cells, the recombinase addressable data (RAD) module. In order to understand the behavior of a RAD module we compute the exact time dependent distributions of the two expressed genes starting in one state and evolving to another asymptotic state. We consider two operating regimes of the RAD module: fast and slow stochastic switching. The fast regime is “aggregative” and produces unimodal distributions, whereas the slow regime is “separative” and produces bimodal distributions. Both regimes can serve to prepare pure memory states when all cells are expressing the same gene. The slow regime can also separate mixed states by producing two sub-populations each one expressing a different gene. Our model provides a simplified, general phenomenological framework for studying biological memory devices and our analytic solution can be further used to clarify theoretical concepts in bio-computation and for optimal design in synthetic biology.

Comment on "Steady-state fluctuations of a genetic feedback loop: an exact solution" [J. Chem. Phys. 137, 035104 (2012)].

The comment presents the complete steady state solution of the model introduced on ”Steady-state fluctuations of a genetic feedback loop: An exact solution” [Grima et al. J. Chem. Phys. 137, 035104 (2012)]. A closed form for the normalization constant is obtained and hence the explicit calculation of the moments as functions of the parameters is possible. We discuss the meaning of an exact solution to a differential equation and the construction of a model to the understanding of a phenomenon.