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 the reaction times be large compared to the microscopic time, irrespective of the origin - energetic, entropic, cooperative - of the timescale 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.Comment: 4 pages, 4 figures, RevTex

The switching dynamics of the bacterial flagellar motor

Many swimming bacteria are propelled by flagellar motors that stochastically switch between the clockwise and counterclockwise rotation direction. While the switching dynamics are one of the most important characteristics of flagellar motors, the mechanisms that control switching are poorly understood. We present a statistical-mechanical model of the flagellar rotary motor, which consists of a number of stator proteins that drive the rotation of a ring of rotor proteins, which in turn drives the rotation of a flagellar filament. At the heart of our model is the assumption that the rotor protein complex can exist in two conformational states corresponding to the two respective rotation directions, and that switching between these states depends on interactions with the stator proteins. This naturally couples the switching dynamics to the rotation dynamics, making the switch sensitive to torque and speed. Another key element of our model is that after a switching event, it takes time for the load to build up, due to polymorphic transitions of the filament. Our model predicts that this slow relaxation dynamics of the filament, in combination with the load dependence of the switching frequency, leads to a characteristic switching time, in agreement with recent observations.Comment: 7 pages, 6 figures, RevTeX

Fitness in time-dependent environments includes a geometric contribution

Phenotypic evolution implies sequential fixations of new genomic sequences. The speed at which these mutations fixate depends, in part, on the relative fitness (selection coefficient) of the mutant vs. the ancestor. Using a simple population dynamics model we show that the relative fitness in dynamical environments is not equal to the fitness averaged over individual environments. Instead it includes a term that explicitly depends on the sequence of the environments. This term is geometric in nature and depends only on the oriented area enclosed by the trajectory taken by the system in the environment state space. It is 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

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.Comment: 46 pages, 5 figure

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 analyse 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 behaviour 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.Comment: 11 pages, 13 figure

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.Comment: Revised manuscript;35 pages, 4 figures, REVTeX4; to appear in PLoS Computational Biolog

Investigating Rare Events by Transition Interface Sampling

We briefly review simulation schemes for the investigation of rare transitions and we resume the recently introduced Transition Interface Sampling, a method in which the computation of rate constants is recast into the computation of fluxes through interfaces dividing the reactant and product state.Comment: 12 pages, 1 figure, contributed paper to the proceedings of NEXT 2003, Second Sardinian International Conference on News and Expectations in Thermostatistics, 21-28 Sep 2003, Cagliari (Italy

The fidelity of dynamic signaling by noisy biomolecular networks

This is the final version of the article. Available from Public Library of Science via the DOI in this record.Cells live in changing, dynamic environments. To understand cellular decision-making, we must therefore understand how fluctuating inputs are processed by noisy biomolecular networks. Here we present a general methodology for analyzing the fidelity with which different statistics of a fluctuating input are represented, or encoded, in the output of a signaling system over time. We identify two orthogonal sources of error that corrupt perfect representation of the signal: dynamical error, which occurs when the network responds on average to other features of the input trajectory as well as to the signal of interest, and mechanistic error, which occurs because biochemical reactions comprising the signaling mechanism are stochastic. Trade-offs between these two errors can determine the system's fidelity. By developing mathematical approaches to derive dynamics conditional on input trajectories we can show, for example, that increased biochemical noise (mechanistic error) can improve fidelity and that both negative and positive feedback degrade fidelity, for standard models of genetic autoregulation. For a group of cells, the fidelity of the collective output exceeds that of an individual cell and negative feedback then typically becomes beneficial. We can also predict the dynamic signal for which a given system has highest fidelity and, conversely, how to modify the network design to maximize fidelity for a given dynamic signal. Our approach is general, has applications to both systems and synthetic biology, and will help underpin studies of cellular behavior in natural, dynamic environments.We acknowledge support from a Medical Research Council and Engineering and Physical Sciences Council funded Fellowship in Biomedical Informatics (CGB) and a Scottish Universities Life Sciences Alliance chair in Systems Biology (PSS). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript

Thermodynamic formalism for systems with Markov dynamics

The thermodynamic formalism allows one to access the chaotic properties of equilibrium and out-of-equilibrium systems, by deriving those from a dynamical partition function. The definition that has been given for this partition function within the framework of discrete time Markov chains was not suitable for continuous time Markov dynamics. Here we propose another interpretation of the definition that allows us to apply the thermodynamic formalism to continuous time. We also generalize the formalism --a dynamical Gibbs ensemble construction-- to a whole family of observables and their associated large deviation functions. This allows us to make the connection between the thermodynamic formalism and the observable involved in the much-studied fluctuation theorem. We illustrate our approach on various physical systems: random walks, exclusion processes, an Ising model and the contact process. In the latter cases, we identify a signature of the occurrence of dynamical phase transitions. We show that this signature can already be unravelled using the simplest dynamical ensemble one could define, based on the number of configuration changes a system has undergone over an asymptotically large time window.Comment: 64 pages, LaTeX; version accepted for publication in Journal of Statistical Physic

Induction level determines signature of gene expression noise in cellular systems

Noise in gene expression, either due to inherent stochasticity or to varying inter- and intracellular environment, can generate significant cell-to-cell variability of protein levels in clonal populations. We present a theoretical framework, based on stochastic processes, to quantify the different sources of gene expression noise taking cell division explicitly into account. Analytical, time-dependent solutions for the noise contributions arising from the major steps involved in protein synthesis are derived. The analysis shows that the induction level of the activator or transcription factor is crucial for the characteristic signature of the dominant source of gene expression noise and thus bridges the gap between seemingly contradictory experimental results. Furthermore, on the basis of experimentally measured cell distributions, our simulations suggest that transcription factor binding and promoter activation can be modelled independently of each other with sufficient accuracy