A new formulation of two-time correlation functions of Markov chains applied to gene networks

Derivation of Markov processes that violate detailed balance

Time-reversal symmetry of the microscopic laws dictates that the equilibrium distribution of a stochastic process must obey the condition of detailed balance. However, cyclic Markov processes that do not admit equilibrium distributions with detailed balance are often used to model systems driven out of equilibrium by external agents. I show that for a Markov model without detailed balance, an extended Markov model can be constructed, which explicitly includes the degrees of freedom for the driving agent and satisfies the detailed balance condition. The original cyclic Markov model for the driven system is then recovered as an approximation at early times by summing over the degrees of freedom for the driving agent. I also show that the widely accepted expression for the entropy production in a cyclic Markov model is actually a time derivative of an entropy component in the extended model. Further, I present an analytic expression for the entropy component that is hidden in the cyclic Markov model.

A derivation of the master equation from path entropy maximization

The master equation and, more generally, Markov processes are routinely used as models for stochastic processes. They are often justified on the basis of randomization and coarse-graining assumptions. Here instead, we derive nth-order Markov processes and the master equation as unique solutions to an inverse problem. We find that when constraints are not enough to uniquely determine the stochastic model, an nth-order Markov process emerges as the unique maximum entropy solution to this otherwise underdetermined problem. This gives a rigorous alternative for justifying such models while providing a systematic recipe for generalizing widely accepted stochastic models usually assumed to follow from the first principles.

Stochastic expression dynamics of a transcription factor revealed by single-molecule noise analysis

Gene expression is inherently stochastic; precise gene regulation by transcription factors is important for cell-fate determination. Many transcription factors regulate their own expression, suggesting that autoregulation counters intrinsic stochasticity in gene expression. Using a new strategy, cotranslational activation by cleavage (CoTrAC), we probed the stochastic expression dynamics of cI, which encodes the bacteriophage λ repressor CI, a fate-determining transcription factor. CI concentration fluctuations influence both lysogenic stability and induction of bacteriophage λ. We found that the intrinsic stochasticity in cI expression was largely determined by CI expression level irrespective of autoregulation. Furthermore, extrinsic, cell-to-cell variation was primarily responsible for CI concentration fluctuations, and negative autoregulation minimized CI concentration heterogeneity by counteracting extrinsic noise and introducing memory. This quantitative study of transcription factor expression dynamics sheds light on the mechanisms cells use to control noise in gene regulatory networks.

Analytical calculation of protein production distributions in models of clustered protein expression

A theoretical framework is presented, which derives chemical master equations for the number of protein molecules produced in a given time window, It is applied to derive analytical solutions that describe protein production distributions for the random bursting model (with an exponential or geometric burst-size distribution) and the clustering model. This distribution is experimentally observable using recently developed, single-molecule gene expression experiments. Furthermore, intrinsic stochasticity in a gene's expression can be calculated from protein production distributions using a new, time-dependent noise curve analysis. Different models of gene expression are compared with respect to their protein production distributions and intrinsic stochasticity, revealing the effects of molecular memory and burstlike expression on fluctuations in gene expression. It is distinct from and provides major advantages over measurements of steady-state concentrations.