We develop an efficient method to calculate probabilities of large deviations from the typical behavior (rare events) in reaction-diffusion systems. The method is based on a semiclassical treatment of underlying "quantum" Hamiltonian, encoding the system's evolution. To this end we formulate corresponding canonical dynamical system and investigate its phase portrait. The method is presented for a number of pedagogical examples.

Equilibrium phase transitions are associated with rearrangements of minima of a (Lagrangian) potential. Treatment of nonequilibrium systems requires doubling of degrees of freedom, which may be often interpreted as a transition from the "coordinate" -to the "phase"-space representation. As a result, one has to deal with the Hamiltonian formulation of the field theory instead of the Lagrangian one. We suggest a classification scheme of phase transitions in reaction-diffusion models based on the topology of the phase portraits of corresponding Hamiltonians. In models with an absorbing state such a topology is fully determined by intersecting curves of zero "energy." We identify four families of topologically distinct classes of phase portraits stable upon renormalization group transformations.

The intrinsic stochasticity of gene expression can lead to large variability in protein levels for genetically identical cells. Such variability in protein levels can arise from infrequent synthesis of mRNAs which in turn give rise to bursts of protein expression. Protein expression occurring in bursts has indeed been observed experimentally and recent studies have also found evidence for transcriptional bursting, i.e. production of mRNAs in bursts. Given that there are distinct experimental techniques for quantifying the noise at different stages of gene expression, it is of interest to derive analytical results connecting experimental observations at different levels. In this work, we consider stochastic models of gene expression for which mRNA and protein production occurs in independent bursts. For such models, we derive analytical expressions connecting protein and mRNA burst distributions which show how the functional form of the mRNA burst distribution can be inferred from the protein burst distribution. Additionally, if gene expression is repressed such that observed protein bursts arise only from single mRNAs, we show how observations of protein burst distributions (repressed and unrepressed) can be used to completely determine the mRNA burst distribution. Assuming independent contributions from individual bursts, we derive analytical expressions connecting means and variances for burst and steady-state protein distributions. Finally, we validate our general analytical results by considering a specific reaction scheme involving regulation of protein bursts by small RNAs. For a range of parameters, we derive analytical expressions for regulated protein distributions that are validated using stochastic simulations. The analytical results obtained in this work can thus serve as useful inputs for a broad range of studies focusing on stochasticity in gene expression.

The intrinsic stochasticity of gene expression can lead to large variations in protein levels across a population of cells. To explain this variability, different sources of messenger RNA (mRNA) fluctuations ("Poisson" and "telegraph" processes) have been proposed in stochastic models of gene expression. Both Poisson and telegraph scenario models explain experimental observations of noise in protein levels in terms of "bursts" of protein expression. Correspondingly, there is considerable interest in establishing relations between burst and steady-state protein distributions for general stochastic models of gene expression. In this work, we address this issue by considering a mapping between stochastic models of gene expression and problems of interest in queueing theory. By applying a general theorem from queueing theory, Little's Law, we derive exact relations which connect burst and steady-state distribution means for models with arbitrary waiting-time distributions for arrival and degradation of mRNAs and proteins. The derived relations have implications for approaches to quantify the degree of transcriptional bursting and hence to discriminate between different sources of intrinsic noise in gene expression. To illustrate this, we consider a model for regulation of protein expression bursts by small RNAs. For a broad range of parameters, we derive analytical expressions (validated by stochastic simulations) for the mean protein levels as the levels of regulatory small RNAs are varied. The results obtained show that the degree of transcriptional bursting can, in principle, be determined from changes in mean steady-state protein levels for general stochastic models of gene expression.

The physiochemical determinants of drug-target interactions in the microenvironment of the cell are complex and generally not defined by simple diffusion and intrinsic chemical reactivity. Non-specific interactions of drugs and macromolecules in cells are rarely considered formally in assessing pharmacodynamics. Here, we demonstrate that non-specific interactions lead to very slow incorporation kinetics of DNA binding drugs. We observe a rate of drug incorporation in cell nuclei three orders of magnitude slower than in vitro due to anomalous drug diffusion within cells. This slow diffusion, however, has an advantageous consequence: it leads to virtually irreversible binding of the drug to specific DNA targets in cells. We show that non-specific interactions drive slow drug diffusion manifesting as slow reaction front propagation. We study the effect of non-specific interactions in different cellular compartments by permeabilization of plasma and nuclear membranes in order to pinpoint differential compartment effects on variability in intracellular drug kinetics. These results provide the basis for a comprehensive model of the determinants of intracellular diffusion of small-molecule drugs, their target-seeking trajectories, and the consequences of these processes on the apparent kinetics of drug-target interactions.

Reversible reaction-diffusion systems display anomalous dynamics characterized by a power-law relaxation toward stationarity. In this paper we study in the aging regime the nonequilibrium dynamical properties of some model systems with reversible reactions. Starting from the exact Langevin equations describing these models, we derive expressions for two-time correlation and autoresponse functions and obtain a simple aging behavior for these quantities. The autoresponse function is thereby found to depend on the specific nature of the chosen perturbation of the system.

A protein undergoes conformational dynamics with multiple time scales, which results in fluctuating enzyme activities. Recent studies in single molecule enzymology have observe this "age-old" dynamic disorder phenomenon directly. However, the single molecule technique has its limitation. To be able to observe this molecular effect with real biochemical functions in situ, we propose to couple the fluctuations in enzymatic activity to noise propagations in small protein interaction networks such as zeroth order ultra-sensitive phosphorylation-dephosphorylation cycle. We showed that enzyme fluctuations could indeed be amplified by orders of magnitude into fluctuations in the level of substrate phosphorylation -a quantity widely interested in cellular biology. Enzyme conformational fluctuations sufficiently slower than the catalytic reaction turn over rate result in a bimodal concentration distribution of the phosphorylated substrate. In return, this network amplified single enzyme fluctuation can be used as a novel biochemical "reporter" for measuring single enzyme conformational fluctuation rates.

Regulation of mRNA decay is a critical component of global cellular adaptation to changing environments. The corresponding changes in mRNA lifetimes can be coordinated with changes in mRNA transcription rates to fine-tune gene expression. Current approaches for measuring mRNA lifetimes can give rise to secondary effects due to transcription inhibition and require separate experiments to estimate changes in mRNA transcription rates. Here, we propose an approach for simultaneous determination of changes in mRNA transcription rate and lifetime using regulatory small RNAs (sRNAs) to control mRNA decay. We analyze a stochastic model for coupled degradation of mRNAs and sRNAs and derive exact results connecting RNA lifetimes and transcription rates to mean abundances. The results obtained are then generalized to include nonstoichiometric coupled degradation of sRNAs. Our analysis suggests experimental protocols for determining parameters controlling the efficiency of stoichiometric regulation by small RNAs and for analyzing factors and processes regulating changes in mRNA transcription and decay.