Transient Bimodality Research Articles

Characterizing a Propionate Sensor in E.coli Nissle 1917.

Short-chain fatty acids (SCFAs) are commonly found in the large intestine, but generally not in the small intestine, and influence microbiome composition and host physiology. Thus, synthetic biologists are interested in developing engineered probiotics capable of in situ detection of SCFAs as biogeography or disease sensors. One SCFA, propionate, is both sensed and consumed by E.coli. Here, we utilize the E.coli transcription factor PrpR, sensitive to the propionate-derived metabolite (2S,3S)-2-methylcitrate, and its cognate promoter PprpBCDE to detect extracellular propionate with the probiotic chassis bacterium E.coli Nissle 1917. We identify that PrpR-PprpBCDE displays stationary phase leakiness and transient bimodality, and we explain these observations through evolutionary rationales and deterministic modeling, respectively. Our results will help researchers build biogeographically sensitive genetic circuits.

Cavity-field distribution in multiphoton Jaynes-Cummings resonances

We calculate the cavity-field distribution in the Wigner representation for the two-photon resonance of the weakly driven Jaynes-Cummings (JC) oscillator in its strong-coupling limit. Using an effective four-level system, we analytically demonstrate the presence of steady-state and transient bimodality which breaks azimuthal symmetry in phase space. The two steady-state peaks are located at opposite positions and do not correspond to the two-photon amplitude of the driven transition. The developing bimodality is portrayed in parallel with the evolution of the intensity correlation function for the forwards-scattered photons, before being finally contrasted to the few-photon steady-state and transient phase-space profiles for the cavity mode in the JC model driven on resonance.

Stochastic model of memristor based on the length of conductive region

We propose a stochastic model of a voltage controlled bipolar memristive system, which includes the properties of widely used dynamic SPICE models and takes into account the fluctuations inherent in memristors. The proposed model is described by rather simple equations of Brownian diffusion, does not require significant computational resources for numerical modeling, and allows obtaining the exact analytical solutions in some cases. The noise-induced transient bimodality phenomenon, arising under resistive switching, was revealed and investigated theoretically and experimentally in a memristive system, by finding a quite good qualitatively agreement between theory and experiment. Based on the proposed model, the mathematical apparatus of Markov processes for the first passage time of the boundaries can be used to analyse the temporal characteristics of resistive switching.

Stochastic time-dependent enzyme kinetics: Closed-form solution and transient bimodality.

We derive an approximate closed-form solution to the chemical master equation describing the Michaelis-Menten reaction mechanism of enzyme action. In particular, assuming that the probability of a complex dissociating into an enzyme and substrate is significantly larger than the probability of a product formation event, we obtain expressions for the time-dependent marginal probability distributions of the number of substrate and enzyme molecules. For delta function initial conditions, we show that the substrate distribution is either unimodal at all times or else becomes bimodal at intermediate times. This transient bimodality, which has no deterministic counterpart, manifests when the initial number of substrate molecules is much larger than the total number of enzyme molecules and if the frequency of enzyme-substrate binding events is large enough. Furthermore, we show that our closed-form solution is different from the solution of the chemical master equation reduced by means of the widely used discrete stochastic Michaelis-Menten approximation, where the propensity for substrate decay has a hyperbolic dependence on the number of substrate molecules. The differences arise because the latter does not take into account enzyme number fluctuations, while our approach includes them. We confirm by means of a stochastic simulation of all the elementary reaction steps in the Michaelis-Menten mechanism that our closed-form solution is accurate over a larger region of parameter space than that obtained using the discrete stochastic Michaelis-Menten approximation.

Dynamical phase diagram of an auto-regulating gene in fast switching conditions.

While the steady-state behavior of stochastic gene expression with auto-regulation has been extensively studied, its time-dependent behavior has received much less attention. Here, under the assumption of fast promoter switching, we derive and solve a reduced chemical master equation for an auto-regulatory gene circuit with translational bursting and cooperative protein-gene interactions. The analytical expression for the time-dependent probability distribution of protein numbers enables a fast exploration of large swaths of the parameter space. For a unimodal initial distribution, we identify three distinct types of stochastic dynamics: (i) the protein distribution remains unimodal at all times; (ii) the protein distribution becomes bimodal at intermediate times and then reverts back to being unimodal at long times (transient bimodality); and (iii) the protein distribution switches to being bimodal at long times. For each of these, the deterministic model predicts either monostable or bistable behavior, and hence, there exist six dynamical phases in total. We investigate the relationship of the six phases to the transcription rates, the protein binding and unbinding rates, the mean protein burst size, the degree of cooperativity, the relaxation time to the steady state, the protein mean, and the type of feedback loop (positive or negative). We show that transient bimodality is a noise-induced phenomenon that occurs when the protein expression is sufficiently bursty, and we use a theory to estimate the observation time window when it is manifested.

A moment-convergence method for stochastic analysis of biochemical reaction networks.

Traditional moment-closure methods need to assume that high-order cumulants of a probability distribution approximate to zero. However, this strong assumption is not satisfied for many biochemical reaction networks. Here, we introduce convergent moments (defined in mathematics as the coefficients in the Taylor expansion of the probability-generating function at some point) to overcome this drawback of the moment-closure methods. As such, we develop a new analysis method for stochastic chemical kinetics. This method provides an accurate approximation for the master probability equation (MPE). In particular, the connection between low-order convergent moments and rate constants can be more easily derived in terms of explicit and analytical forms, allowing insights that would be difficult to obtain through direct simulation or manipulation of the MPE. In addition, it provides an accurate and efficient way to compute steady-state or transient probability distribution, avoiding the algorithmic difficulty associated with stiffness of the MPE due to large differences in sizes of rate constants. Applications of the method to several systems reveal nontrivial stochastic mechanisms of gene expression dynamics, e.g., intrinsic fluctuations can induce transient bimodality and amplify transient signals, and slow switching between promoter states can increase fluctuations in spatially heterogeneous signals. The overall approach has broad applications in modeling, analysis, and computation of complex biochemical networks with intrinsic noise.

Weiss mean-field approximation for multicomponent stochastic spatially extended systems.

We develop a mean-field approach for multicomponent stochastic spatially extended systems and use it to obtain a multivariate nonlinear self-consistent Fokker-Planck equation defining the probability density of the state of the system, which describes a well-known model of autocatalytic chemical reaction (brusselator) with spatially correlated multiplicative noise, and to study the evolution of probability density and statistical characteristics of the system in the process of spatial pattern formation. We propose the finite-difference method for the numerical solving of a general class of multivariate nonlinear self-consistent time-dependent Fokker-Planck equations. We illustrate the accuracy and reliability of the method by applying it to an exactly solvable nonlinear Fokker-Planck equation (NFPE) for the Shimizu-Yamada model [Prog. Theor. Phys. 47, 350 (1972)] and nonlinear Fokker-Planck equation [Desai and Zwanzig, J. Stat. Phys. 19, 1 (1978)] obtained for a nonlinear stochastic mean-field model introduced by Kometani and Shimizu [J. Stat. Phys. 13, 473 (1975)]. Taking the problems indicated above as an example, the accuracy of the method is compared with the accuracy of Hermite distributed approximating functional method [Zhang et al., Phys. Rev. E 56, 1197 (1997)]. Numerical study of the NFPE solutions for a stochastic brusselator shows that in the region of Turing bifurcation several types of solutions exist if noise intensity increases: unimodal solution, transient bimodality, and an interesting solution which involves multiple "repumping" of probability density through bimodality. Additionally, we study the behavior of the order parameter of the system under consideration and show that the second type of solution arises in the supercritical region if noise intensity values are close to the values appropriate for the transition from bimodal stationary probability density for the order parameter to the unimodal one.

Complex Dynamics Based on a Quorum: Decision‐Making Process by Cockroaches in a Patchy Environment

AbstractIn the absence of complex communication and a global knowledge of the environment, cockroaches are able to assess the availability of resources and to reach a consensual decision: the group aggregates in a single resting site. We show that the aggregation dynamics and the collective shelter selection of cockroaches are influenced by their social context as, unlike single individuals, groups of cockroaches are more likely to respond to environmental heterogeneities. The decision of individuals to stay under a shelter relies on the modulation of their resting time, according to the perception of two local cues: (1) the shelters luminosity and (2) the number of congeners. This study on the cockroach species Periplaneta americana highlights a shelter‐selection mechanism based on an amplification process resulting from the interactions between congeners. This mechanism leads to complex spatiotemporal aggregation dynamics characterized by transient bimodality, bifurcation patterns (shelter selection) and the existence of a quorum size in the settlement behaviour of the cockroaches. Finally, we discuss the generic aspect for other gregarious species of the collective decision‐making process demonstrated for cockroaches.

Transient Bimodality and Catastrophic Jumps in Innovation Diffusion

A new phenomenon in innovation diffusion that exhibits a transient bimodality familiar in physical sciences is predicted in this paper for the first time on the basis of an extended Bass model. This accounts for the population heterogeneity, with the parameters characterizing the ldquoword of mouthrdquo and ldquomass mediardquo processes as random. A theoretical framework based on a nonlinear random differential equation is thus introduced to develop a stochastic model for new product diffusion with parametric uncertainty. The analytical investigation of the model establishes the existence of transient bimodality, which manifests through a cusp catastrophe. Illustrations based on analytical and simulation studies are presented. Empirical validation of the proposed model based on well-known data sets where the Bass model is known to fit very well has been carried out.

Stochastic effects in a thermochemical system with Newtonian heat exchange.

We develop a mesoscopic description of stochastic effects in the Newtonian heat exchange between a diluted gas system and a thermostat. We explicitly study the homogeneous Semenov model involving a thermochemical reaction and neglecting consumption of reactants. The master equation includes a transition rate for the thermal transfer process, which is derived on the basis of the statistics for inelastic collisions between gas particles and walls of the thermostat. The main assumption is that the perturbation of the Maxwellian particle velocity distribution can be neglected. The transition function for the thermal process admits a continuous spectrum of temperature changes, and consequently, the master equation has a complicated integro-differential form. We perform Monte Carlo simulations based on this equation to study the stochastic effects in the Semenov system in the explosive regime. The dispersion of ignition times is calculated as a function of system size. For sufficiently small systems, the probability distribution of temperature displays transient bimodality during the ignition period. The results of the stochastic description are successfully compared with those of direct simulations of microscopic particle dynamics.

Transient bimodality of nonequilibrium states in monostable systems with noise

We discuss a new type of monostable potential profiles for which the phenomenon of transient bimodality occurs in dynamic systems. It is shown that transient bimodality occurs if the potential profile describing the dynamic system has a linear part. A new method for calculating the lifetime of the transient bimodality is proposed. We show that the lifetime of transient bimodality for the considered potential profiles increases in proportion to the fluctuation intensity.

Numerical method for the nonlinear Fokker-Planck equation

A practical method based on distributed approximating functionals ~DAFs! is proposed for numerically solving a general class of nonlinear time-dependent Fokker-Planck equations. The method relies on a numerical scheme that couples the usual path-integral concept to the DAF idea. The high accuracy and reliability of the method are illustrated by applying it to an exactly solvable nonlinear Fokker-Planck equation, and the method is compared with the accurate K-point Stirling interpolation formula finite-difference method. The approach is also used successfully to solve a nonlinear self-consistent dynamic mean-field problem for which both the cumulant expansion and scaling theory have been found by Drozdov and Morillo @Phys. Rev. E 54, 931 ~1996!# to be inadequate to describe the occurrence of a long-lived transient bimodality. The standard interpretation of the transient bimodality in terms of the ‘‘flat’’ region in the kinetic potential fails for the ‘

Solution of nonlinear Fokker-Planck equations.

A finite-difference method for solving a general class of linear and nonlinear time-dependent Fokker-Planck equations, which is based on a K-point Stirling interpolation formula, is suggested. It has a fifth-order convergence in time and a 2Kth-order convergence in space and allows one to achieve a given level of accuracy with a slow (or even without) increase in the number of grid points. The most appealing features of the method are perhaps that it is norm conserved, and equilibrium preserving in the sense that every equilibrium solution of the analytic equations is also an equilibrium solution of the discretized equations. The method is applied to a nonlinear stochastic mean-field model introduced by Kometani and Shimizu [J. Stat. Phys. 13, 473 (1983)], which exhibits a phase transition. The results are compared with those obtained with other methods that rely on not too well controlled approximations. Our finite-difference scheme permits us to establish the region of validity and the limitations of those approximations. The nonlinearity of the system is found to be an obstacle for the application of Suzuki's scaling ideas, which are known to be suitable for linear problems. But what is most remarkable is that this nonlinearity allows for transient bimodality in a globally monostable case, even though there is no ``flat'' region in the potential. \textcopyright{} 1996 The American Physical Society.

Stochastic resonance and transient bimodality

The phenomenon of stochastic resonance is predicted in a monostable system displaying transient bimodality during the evolution. The signal-to-noise ratio as a function of the noise intensity calculated within the rate equation theory with many simplifying assumptions exhibits a promising peak. The numerical simulations are also reported.

Transient bimodality in turbulence-1-turbulence-2 transition in electrohydrodynamic convection in nematic liquid crystals.

In electrohydrodynamic convection, the turbulence-1--turbulence-2 (T1--T2) transition is observed when the applied voltage is very high. The transient property at the T1--T2 transition is studied through the evolution of the probability distribution of the area of T2. At a voltage close to the T1--T2 threshold the distribution transiently shows double peaks, i.e., a transient bimodality.

Relaxation from a marginal state in optical bistability. Anomalous fluctuations and first-passage time statistics

We show that it is possible to characterize anomalous fluctuations in transients from marginal states using first-passage time statistics. The maximum of the fluctuations is found to occur in the median of the first-passage time distribution. The value of this maximum is given in terms of the initial and final state. This characterization of anomalous fluctuations is related to the phenomenon of transient bimodality. We analyze the existence of such a bimodality in terms of an adimensional parameter. Our results are substantiated by numerical simulations. Applications to the Bonifacio-Lugiato model of absorptive bistability are presented.

Transient bimodality in interacting particle systems

We consider a system of spins which have values ±1 and evolve according to a jump Markov process whose generator is the sum of two generators, one describing a spin-flipGlauber process, the other aKawasaki (stirring) evolution. It was proven elsewhere that if the Kawasaki dynamics is speeded up by a factor e−2, then, in the limit e → 0 (continuum limit), propagation of chaos holds and the local magnetization solves a reaction-diffusion equation. We choose the parameters of the Glauber interaction so that the potential of the reaction term in the reaction-diffusion equation is a double-well potential with quartic maximum at the origin. We assume further that for each e the system is in a finite interval ofZ with e−1 sites and periodic boundary conditions. We specify the initial measure as the product measure with 0 spin average, thus obtaining, in the continuum limit, a constant magnetic profile equal to 0, which is a stationary unstable solution to the reaction-diffusion equation. We prove that at times of the order e−1/2 propagation of chaos does not hold any more and, in the limit as e → 0, the state becomes a nontrivial superposition of Bernoulli measures with parameters corresponding to the minima of the reaction potential. The coefficients of such a superposition depend on time (on the scale e−1/2) and at large times (on this scale) the coefficient of the term corresponding to the initial magnetization vanishes (transient bimodality). This differs from what was observed by De Masi, Presutti, and Vares, who considered a reaction potential with quadratic maximum and no bimodal effect was seen, as predicted by Broggi, Lugiato, and Colombo.

Intrinsic randomness and spontaneous symmetry-breaking in explosive systems

The effect of fluctuations of either thermodynamic or environmental origin on ignition in explosive systems is analyzed, with special emphasis on thermal explosion. A simple model due to Semenov is first analyzed in the zero-dimensional approximation. It is shown that the ignition times exhibit a wide dispersion, which at the level of the probability distribution of temperature shows up as a transient bimodality. Next, an extension to a spatially distributed system is developed. It is shown that fluctuations induce unexpected symmetry-breaking phenomena, reflected by a considerable dispersion of the position of the first “hot spot” initiated in the system.

The adiabatic thermal explosion in a small system: Comparison of the stochastic approach with the molecular dynamics simulation

A new model of the adiabatic thermal explosion with biomolecular reactionX+X⇄A+A is proposed. The parameters of reaction are chosen such that the stochastic description predicts a transient bimodality of behavior. For the same system the nondeterministic character of the time evolution is confirmed by a molecular dynamics simulation.

Theory of scaling for fluctuations in thermal explosion conditions

AbstractWe demonstrate that fluctuations in the transient bimodality regime for gaseous thermal explosions scale with the inverse of an effective size parameter Ω. The ratio between this parameter and the volume V is obtained analytically for a nonturbulent heat conduction regime. We show that as the system departs from the laminar flow limit towards a turbulent regime, the fluctuations are enhanced since Ω decreases. We also prove that as V tends to zero, the ratio Ω/V tends to one, an intuitively expected result. The theory is inspired and confirmed by experiments on the decomposition of ethyl azide. Within the induction period, we use a corrected parabolic approximation for the potential determining the drift term in the Fokker‐Planck equation. The maximum occurs at the ignition temperature. The correction arises from size effects (Ω/V ≠ 1) due to an inhomogeneous distribution of temperature. The parabolic approximation is valid only within the induction period, until the fluctuations have macroscopic consequences. Our study reveals that the following anomalies bear a common origin that can be explained quantitatively if the correction is introduced: a) Discrepancy between the experimental and theoretical value for the Frank‐Kamenetsky constant. Early work by O.K. Rice on the gaseous thermal explosion of ethyl azide indicates that this discrepancy is enhanced when the size of the vessel increases. — b) Reduction of the induction period, or, equivalently, contraction of the time‐scale for the transient bimodality during which heat accumulates in the system. This anomaly is accompanied by an enhancement of the growth rate for the Gaussian width of the probability density distribution for the temperature due to fluctuations. This phenomenon will be interpreted by the observer as a premature ignition.