Transcriptional regulation is an inherently noisy process. The origins of this stochastic behavior can be traced to the random transitions among the discrete chemical states of operators that control the transcription rate and to finite number fluctuations in the biochemical reactions for the synthesis and degradation of transcripts. We develop stochastic models to which these random reactions are intrinsic and a series of simpler models derived explicitly from the first as approximations in different parameter regimes. This innate stochasticity can have both a quantitative and qualitative impact on the behavior of gene-regulatory networks. We introduce a natural generalization of deterministic bifurcations for classification of stochastic systems and show that simple noisy genetic switches have rich bifurcation structures; among them, bifurcations driven solely by changing the rate of operator fluctuations even as the underlying deterministic system remains unchanged. We find stochastic bistability where the deterministic equations predict monostability and vice-versa. We derive and solve equations for the mean waiting times for spontaneous transitions between quasistable states in these switches.

Genetically identical cells exposed to the same environmental conditions can show significant variation in molecular content and marked differences in phenotypic characteristics. This variability is linked to stochasticity in gene expression, which is generally viewed as having detrimental effects on cellular function with potential implications for disease. However, stochasticity in gene expression can also be advantageous. It can provide the flexibility needed by cells to adapt to fluctuating environments or respond to sudden stresses, and a mechanism by which population heterogeneity can be established during cellular differentiation and development.

Mitochondria, bacteria and chloroplasts use the free energy stored in transmembrane ion gradients to manufacture ATP by the action of ATP synthase. This enzyme consists of two principal domains. The asymmetric membrane-spanning F0 portion contains the proton channel, and the soluble F1 portion contains three catalytic sites which cooperate in the synthetic reactions. The flow of protons through F0 is thought to generate a torque which is transmitted to F1 by an asymmetric shaft, the coiled-coil gamma-subunit. This acts as a rotating 'cam' within F1, sequentially releasing ATPs from the three active sites. The free-energy difference across the inner membrane of mitochondria and bacteria is sufficient to produce three ATPs per twelve protons passing through the motor. It has been suggested that this proton motive force biases the rotor's diffusion so that F0 constitutes a rotary motor turning the gamma shaft. Here we show that biased diffusion, augmented by electrostatic forces, does indeed generate sufficient torque to account for ATP production. Moreover, the motor's reversibility-supplying torque from ATP hydrolysis in F1 converts the motor into an efficient proton pump-can also be explained by our model.

SUMMARY Many cells undergo symmetry-breaking polarization toward a randomly oriented “front” in the absence of spatial cues. In budding yeast, such polarization involves a positive feedback loop that enables amplification of stochastically arising clusters of polarity factors. Previous mathematical modeling suggested that, if more than one cluster were amplified, the clusters would compete for limiting resources and the largest would “win,” explaining why yeast cells always make one and only one bud. Here, using imaging with improved spatiotemporal resolution, we show the transient coexistence of multiple clusters during polarity establishment, as predicted by the model. Unexpectedly, we also find that initial polarity factor clustering is oscillatory, revealing the presence of a negative feedback loop that disperses the factors. Mathematical modeling predicts that negative feedback would confer robustness to the polarity circuit and make the kinetics of competition between polarity factor clusters relatively insensitive to polarity factor concentration. These predictions are confirmed experimentally.

The ability to construct synthetic gene networks enables experimental investigations of deliberately simplified systems that can be compared to qualitative and quantitative models. If simple, well-characterized modules can be coupled together into more complex networks with behaviour that can be predicted from that of the individual components, we may begin to build an understanding of cellular regulatory processes from the 'bottom up'. Here we have engineered a promoter to allow simultaneous repression and activation of gene expression in Escherichia coli. We studied its behaviour in synthetic gene networks under increasingly complex conditions: unregulated, repressed, activated, and simultaneously repressed and activated. We develop a stochastic model that quantitatively captures the means and distributions of the expression from the engineered promoter of this modular system, and show that the model can be extended and used to accurately predict the in vivo behaviour of the network when it is expanded to include positive feedback. The model also reveals the counterintuitive prediction that noise in protein expression levels can increase upon arrest of cell growth and division, which we confirm experimentally. This work shows that the properties of regulatory subsystems can be used to predict the behaviour of larger, more complex regulatory networks, and that this bottom-up approach can provide insights into gene regulation.

Summary Background Many cells are remarkably proficient at tracking very shallow chemical gradients, despite considerable noise from stochastic receptor-ligand interactions. Motile cells appear to undergo a biased random walk: spatial noise in receptor activity may determine the instantaneous direction, but because noise is spatially unbiased it is filtered out by time-averaging, resulting in net movement up-gradient. How non-motile cells might filter out noise is unknown. Results Using yeast chemotropic mating as a model, we demonstrate that a polarized patch of polarity regulators “wanders” along the cortex during gradient tracking. Computational and experimental findings suggest that actin-directed membrane traffic contributes to wandering by diluting local polarity factors. The pheromone gradient appears to bias wandering via interactions between receptor-activated Gβγ and polarity regulators. Artificially blocking patch wandering impairs gradient tracking. Conclusions We suggest that the polarity patch undergoes an intracellular biased random walk that enables noise filtering by time-averaging, allowing non-motile cells to track shallow gradients.

G protein-coupled receptor signaling is dynamically regulated by multiple feedback mechanisms, which rapidly attenuate signals elicited by ligand stimulation, causing desensitization. The individual contributions of these mechanisms, however, are poorly understood. Here, we use an improved fluorescent biosensor for cAMP to measure second messenger dynamics stimulated by endogenous ␤ 2 -adrenergic receptor (␤ 2 AR) in living cells. ␤ 2 AR stimulation with isoproterenol results in a transient pulse of cAMP, reaching a maximal concentration of ϳ10 M and persisting for less than 5 min. We investigated the contributions of cAMP-dependent kinase, G protein-coupled receptor kinases, and ␤-arrestin to the regulation of ␤ 2 AR signal kinetics by using small molecule inhibitors, small interfering RNAs, and mouse embryonic fibroblasts. We found that the cAMP response is restricted in duration by two distinct mechanisms in HEK-293 cells: G protein-coupled receptor kinase (GRK6)-mediated receptor phosphorylation leading to ␤-arrestin mediated receptor inactivation and cAMP-dependent kinase-mediated induction of cAMP metabolism by phosphodiesterases. A mathematical model of ␤ 2 AR signal kinetics, fit to these data, revealed that direct receptor inactivation by cAMPdependent kinase is insignificant but that GRK6/␤-arrestin-mediated inactivation is rapid and profound, occurring with a halftime of 70 s. This quantitative system analysis represents an important advance toward quantifying mechanisms contributing to the physiological regulation of receptor signaling.Tachyphylaxis, or desensitization, denoting the attenuation of a biological response to sustained or repeated intervention, is a pervasive phenomenon in physiological systems. For G protein-coupled receptors (GPCRs) 7 (or, more broadly, seventransmembrane receptors), desensitization occurs through molecular mechanisms that can profoundly limit further stimulation of downstream signals, either through direct receptor inactivation or inhibition of downstream signaling. At the physiological level, we refer to this general loss of responsiveness as desensitization; we refer to the more specific case of direct inhibition of receptor molecules as "receptor inactivation." At the level of the receptor, GPCR signals represent a dynamic balance between ligand-stimulated activities, such as G protein coupling, and negative feedback mechanisms, such as receptor phosphorylation and ␤-arrestin recruitment (1). An agonist's efficacy is determined by the balance between these activities and is limited by the kinetics of receptor inactivation. Despite intensive research into the molecular mechanisms of desensitization by GPCR inactivation, the relative contributions of these mechanisms are largely unknown. One reason for this is that, until recently, there have been few techniques that directly measure receptor signaling in real time. Rather, data showing desensitization and receptor inactivation are usually based on either physiological assessments, such as hemodynamic parameters, whi...

We present a numerical algorithm that is well suited for the study of biomolecular transport processes. In the algorithm a continuous Markov process is discretized as a jump process and the jump rates are derived from local solutions of the continuous system. Consequently, the algorithm has two advantages over standard numerical methods: (1) it preserves detailed balance for equilibrium processes, (2) it is able to handle discontinuous potentials. The formulation of the algorithm also allows us to calculate the effective diffusion coefficient or, equivalently, the randomness parameter. We provide several simple examples of how to implement the algorithm. All the MATLAB functions files needed to reproduce the results presented in the article are available from www.amath.unc.edu/Faculty/telston/ matlab functions. r