Exchange of stability as a function of system size in a nonequilibrium system.

In equilibrium systems with short-ranged interactions, the relative stability of different thermodynamic states generally does not depend on system size (as long as this size is larger than the interaction range). Here, we use a large deviations approach to show that, in contrast, different states can exchange stability as system size is varied in a driven, bistable reaction-diffusion system. This striking effect is related to a shift from a spatially uniform to a nonuniform transition state and should generically be possible in a wide range of nonequilibrium physical and biological systems.

Fitness in time-dependent environments includes a geometric phase contribution.

Phenotypic evolution implies sequential rise in frequency of new genomic sequences. The speed of the rise depends, in part, on the relative fitness (selection coefficient) of the mutant versus the ancestor. Using a simple population dynamics model, we show that the relative fitness in dynamical environments is not equal to the geometric average of the fitness over individual environments. Instead, it includes a term that explicitly depends on the sequence of the environments. For slowly varying environments, this term depends only on the oriented area enclosed by the trajectory taken by the system in the environment state space. It is closely related to the well-studied geometric phases in classical and quantum physical systems. We discuss possible biological implications of these observations, focusing on evolution of novel metabolic or stress-resistant functions.

Speeding up Evolutionary Search by Small Fitness Fluctuations.

We consider a fixed size population that undergoes an evolutionary adaptation in the weak mutation rate limit, which we model as a biased Langevin process in the genotype space. We show analytically and numerically that, if the fitness landscape has a small highly epistatic (rough) and time-varying component, then the population genotype exhibits a high effective diffusion in the genotype space and is able to escape local fitness minima with a large probability. We argue that our principal finding that even very small time-dependent fluctuations of fitness can substantially speed up evolution is valid for a wide class of models.

Reciprocal sign epistasis is a necessary condition for multi-peaked fitness landscapes.

Having multiple peaks within fitness landscapes critically affects the course of evolution, but whether their presence imposes specific requirements at the level of genetic interactions remains unestablished. Here we show that to exhibit multiple fitness peaks, a biological system must contain reciprocal sign epistatic interactions, which are defined as genetic changes that are separately unfavorable but jointly advantageous. Using Morse theory, we argue that it is impossible to formulate a sufficient condition for multiple peaks in terms of local genetic interactions. These findings indicate that systems incapable of reciprocal sign epistasis will always possess a single fitness peak. However, reciprocal sign epistasis should be pervasive in nature as it is a logical consequence of specificity in molecular interactions. The results thus predict that specific molecular interactions may yield multiple fitness peaks, which can be tested experimentally.

Spatio-temporal correlations can drastically change the response of a MAPK pathway.

Multisite covalent modification of proteins is omnipresent in eukaryotic cells. A well-known example is the mitogen-activated protein kinase (MAPK) cascade where, in each layer of the cascade, a protein is phosphorylated at two sites. It has long been known that the response of a MAPK pathway strongly depends on whether the enzymes that modify the protein act processively or distributively. A distributive mechanism, in which the enzyme molecules have to release the substrate molecules in between the modification of the two sites, can generate an ultrasensitive response and lead to hysteresis and bistability. We study by Green's Function Reaction Dynamics (GFRD), a stochastic scheme that makes it possible to simulate biochemical networks at the particle level in time and space, a dual phosphorylation cycle in which the enzymes act according to a distributive mechanism. We find that the response of this network can differ dramatically from that predicted by a mean-field analysis based on the chemical rate equations. In particular, rapid rebindings of the enzyme molecules to the substrate molecules after modification of the first site can markedly speed up the response and lead to loss of ultrasensitivity and bistability. In essence, rapid enzyme-substrate rebindings can turn a distributive mechanism into a processive mechanism. We argue that slow ADP release by the enzymes can protect the system against these rapid rebindings, thus enabling ultrasensitivity and bistability.

The switching dynamics of the bacterial flagellar motor.

Many bacteria are propelled by flagellar motors that stochastically switch between the clockwise and counterclockwise rotation direction. Although the switching dynamics is one of their most important characteristics, the mechanisms that control it are poorly understood. We present a statistical-mechanical model of the bacterial flagellar motor. At its heart is the assumption that the rotor protein complex, which is connected to the flagellum, can exist in two conformational states and that switching between these states depends on the interactions with the stator proteins, which drive the rotor. This couples switching to rotation, making the switch sensitive to torque and speed. Another key element is that after a switch, it takes time for the load to build up, due to conformational transitions of the flagellum. This slow relaxation dynamics of the filament leads, in combination with the load dependence of the switching frequency, to a characteristic switching time, as recently observed. Hence, our model predicts that the switching dynamics is not only controlled by the chemotaxis-signaling network, but also by mechanical feedback of the flagellum.

Homogeneous nucleation under shear in a two-dimensional Ising model: cluster growth, coalescence, and breakup.

We compute rates and pathways for nucleation in a sheared two-dimensional Ising model with Metropolis spin flip dynamics using forward flux sampling (FFS). We find a peak in the nucleation rate at intermediate shear rate. We analyze the origin of this peak using modified shear algorithms and committor analysis. We find that the peak arises from an interplay between three shear-mediated effects: Shear-enhanced cluster growth, cluster coalescence, and cluster breakup. Our results show that complex nucleation behavior can be found even in a simple driven model system. This work also demonstrates the use of FFS for simulating rare events, including nucleation, in nonequilibrium systems.

Regulatory control and the costs and benefits of biochemical noise.

Experiments in recent years have vividly demonstrated that gene expression can be highly stochastic. How protein concentration fluctuations affect the growth rate of a population of cells is, however, a wide-open question. We present a mathematical model that makes it possible to quantify the effect of protein concentration fluctuations on the growth rate of a population of genetically identical cells. The model predicts that the population's growth rate depends on how the growth rate of a single cell varies with protein concentration, the variance of the protein concentration fluctuations, and the correlation time of these fluctuations. The model also predicts that when the average concentration of a protein is close to the value that maximizes the growth rate, fluctuations in its concentration always reduce the growth rate. However, when the average protein concentration deviates sufficiently from the optimal level, fluctuations can enhance the growth rate of the population, even when the growth rate of a cell depends linearly on the protein concentration. The model also shows that the ensemble or population average of a quantity, such as the average protein expression level or its variance, is in general not equal to its time average as obtained from tracing a single cell and its descendants. We apply our model to perform a cost-benefit analysis of gene regulatory control. Our analysis predicts that the optimal expression level of a gene regulatory protein is determined by the trade-off between the cost of synthesizing the regulatory protein and the benefit of minimizing the fluctuations in the expression of its target gene. We discuss possible experiments that could test our predictions.

Eliminating fast reactions in stochastic simulations of biochemical networks: a bistable genetic switch.

In many stochastic simulations of biochemical reaction networks, it is desirable to "coarse grain" the reaction set, removing fast reactions while retaining the correct system dynamics. Various coarse-graining methods have been proposed, but it remains unclear which methods are reliable and which reactions can safely be eliminated. We address these issues for a model gene regulatory network that is particularly sensitive to dynamical fluctuations: a bistable genetic switch. We remove protein-DNA and/or protein-protein association-dissociation reactions from the reaction set using various coarse-graining strategies. We determine the effects on the steady-state probability distribution function and on the rate of fluctuation-driven switch flipping transitions. We find that protein-protein interactions may be safely eliminated from the reaction set, but protein-DNA interactions may not. We also find that it is important to use the chemical master equation rather than macroscopic rate equations to compute effective propensity functions for the coarse-grained reactions.

Reaction coordinates for the flipping of genetic switches.

We present a detailed analysis, based on the forward flux sampling simulation method, of the switching dynamics and stability of two models of genetic toggle switches, consisting of two mutually repressing genes encoding transcription factors (TFs); in one model (the exclusive switch), the two transcription factors mutually exclude each other's binding, while in the other model (general switch), the two TFs can bind simultaneously to the shared operator region. We assess the role of two pairs of reactions that influence the stability of these switches: TF-TF homodimerization and TF-DNA association/dissociation. In both cases, the switch flipping rate increases with the rate of TF dimerization, while it decreases with the rate of TF-operator binding. We factorize the flipping rate k into the product of the probability rho(q*) of finding the system at the dividing surface (separatrix) between the two stable states, and a kinetic prefactor R. In the case of the exclusive switch, the rate of TF-operator binding affects both rho(q*) and R, while the rate of TF dimerization affects only R. The general switch displays a higher flipping rate than the exclusive switch, and both TF-operator binding and TF dimerization affect k, R, and rho(q*). To elucidate this, we analyze the transition state ensemble. For the exclusive switch, the transition state ensemble is strongly affected by the rate of TF-operator binding, but unaffected by varying the rate of TF-TF binding. Thus, varying the rate of TF-operator binding can drastically change the pathway of switching, while changing the rate of dimerization changes the switching rate without altering the mechanism. The switching pathways of the general switch are highly robust to changes in the rate constants of both TF-operator and TF-TF binding, even though these rate constants do affect the flipping rate; this feature is unique for nonequilibrium systems.

Signal detection, modularity, and the correlation between extrinsic and intrinsic noise in biochemical networks.

We present an expression for the power spectrum of the output signal of a biochemical network, which reveals that the reactions that allow a network to detect biochemical signals, induce correlations between the extrinsic noise of the input signals and the intrinsic noise of the reactions that form the network. We show that anticorrelations between the extrinsic and intrinsic noise enhance the robustness of zero-order ultrasensitive networks to biochemical noise. We discuss the consequences for a modular description of noise transmission using the mitogen-activated protein kinase cascade.

Diffusion of transcription factors can drastically enhance the noise in gene expression.

We study by Green's Function Reaction Dynamics the effect of the diffusive motion of repressor molecules on the noise in mRNA and protein levels for a gene that is under the control of a repressor. We find that spatial fluctuations due to diffusion can drastically enhance the noise in gene expression. After dissociation from the operator, a repressor can rapidly rebind to the DNA. Our results show that the rebinding trajectories are so short that, on this timescale, the RNA polymerase (RNAP) cannot effectively compete with the repressor for binding to the promoter. As a result, a dissociated repressor molecule will on average rebind many times, before it eventually diffuses away. These rebindings thus lower the effective dissociation rate, and this increases the noise in gene expression. Another consequence of the timescale separation between repressor rebinding and RNAP association is that the effect of spatial fluctuations can be described by a well-stirred, zero-dimensional, model by renormalizing the reaction rates for repressor-DNA (un) binding. Our results thus support the use of well-stirred, zero-dimensional models for describing noise in gene expression. We also show that for a fixed repressor strength, the noise due to diffusion can be minimized by increasing the number of repressors or by decreasing the rate of the open complex formation. Lastly, our results emphasize that power spectra are a highly useful tool for studying the propagation of noise through the different stages of gene expression.

Exact results for noise power spectra in linear biochemical reaction networks.

We present a simple method for determining the exact noise power spectra and related statistical properties for linear chemical reaction networks. The method is applied to reaction networks which are representative of biochemical processes such as gene expression. We find, for example, that a post-translational modification reaction can reduce the noise associated with gene expression. Our results also indicate how to coarse grain networks by the elimination of fast reactions. In this context we have discovered a breakdown of the sum rule which relates the noise power spectrum to the total noise. The breakdown can be quantified by a sum rule deficit, which is found to be universal, and can be attributed to the high-frequency noise in the fast reactions.

Topological methods for searching barriers and reaction paths.

We present a family of algorithms for the fast determination of reaction paths and barriers in phase space and the computation of the corresponding rates. The method requires that reaction times be large compared to the microscopic time, irrespective of the origin--energetic, entropic, cooperative--of the time scale separation. It lends itself to temperature cycling as in simulated annealing and to activation-relaxation routines. The dynamics is ultimately based on supersymmetry methods used years ago to derive Morse theory. Thus, the formalism automatically incorporates all relevant topological information.