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Accurate analytic solution of chemical master equations for gene regulation networks in a single cell
Abstract
Studying gene regulation networks in a single cell is an important, interesting, and hot research topic of molecular biology. Such process can be described by chemical master equations (CMEs). We propose a Hamilton-Jacobi equation method with finite-size corrections to solve such CMEs accurately at the intermediate region of switching, where switching rate is comparable to fast protein production rate. We applied this approach to a model of self-regulating proteins [H. Ge et al., Phys. Rev. Lett. 114, 078101 (2015)] and found that as a parameter related to inducer concentration increases the probability of protein production changes from unimodal to bimodal, then to unimodal, consistent with phenotype switching observed in a single cell.
... The calculation of the fixation probability is a rather difficult mathematical problem even for the two allele case on fluctuating fitness landscape if one considers a large initial number of new mutants. In this case, the Hamilton-Jacobi equation method with finite population size corrections should be used, as has been done in Ref. 33 for the Moran model on fluctuating landscape, see also Ref. 34. We can apply our method to the case when there is some memory in the choice of the environment type. ...
... A conventional method to solve these models has been suggested in Ref. 31. Later, the calculation of the finite size corrections has been performed in Ref. 32. For further research, we suggest an open mathematical problem: the proper choice of the solution branches for the steady-state case, as well as to clarify different phases in the dynamics. ...
... In the general case, more than one persister subpopulation may coexist and, thus, a multimodal killing curve is observed (Balaban et al. 2004). When studying persistence, two aspects are particularly interesting, the first one being pertaining to tolerance, and the second is specific to persistence: (1) the molecular mechanism(s) that enables tolerant bacteria to survive, and (2) the mathematical principle that generates heterogeneity in the population (Ackermann 2015), for example, nonlinear mechanisms leading to bimodality by amplifying stochasticity (Tsimring 2014;Huang et al. 2018). ...
In this chapter, we describe the experimental evolution of antibiotic tolerance and persistence under antibiotic treatments and how these phenomena can speed up the subsequent evolution of resistance. The first two parts are dedicated to defining the difference between antibiotic resistance, tolerance, and persistence with qualitative definitions and quantitative metrics. The third part describes experimental observations of the evolution of tolerance and persistence under antibiotic treatments. The fourth part shows that tolerance and persistence speed up the evolution of antibiotic resistance. In each part, mathematical subsections can be skipped by the reader without losing the qualitative understanding of the effects.
... Not surprisingly, mechanisms linked to tolerance, such as dormancy (see definitions in Box 3), reduced metabolism and ATP levels, have also been identified in persistence 9 . Therefore, when studying persistence, two mechanisms are of interest, and the first one overlaps with tolerance research whereas the second is specific to persistence: (1) the molecular mechanism of tolerance that enables the persister bacteria to survive, for example, a reduction in their metabolism, and (2) the mechanism that generates heterogeneity in the population 17 , for example, nonlinear mechanisms leading to bimodality by amplifying stochasticity 18,19 . Finally, several persister subpopulations may coexist; therefore, a multimodal killing curve may occur. ...
In Figure 2b, the minimal duration for killing (MDK) 99% of tolerant cells was erroneously labelled as MDK99.99 instead of MDK99. This has now been corrected in all versions of the Review. The publisher apologizes to the authors and to readers for this error.
... Not surprisingly, mechanisms linked to tolerance, such as dormancy (see definitions in Box 3), reduced metabolism and ATP levels, have also been identified in persistence 9 . Therefore, when studying persistence, two mechanisms are of interest, and the first one overlaps with tolerance research whereas the second is specific to persistence: (1) the molecular mechanism of tolerance that enables the persister bacteria to survive, for example, a reduction in their metabolism, and (2) the mechanism that generates heterogeneity in the population 17 , for example, nonlinear mechanisms leading to bimodality by amplifying stochasticity 18,19 . Finally, several persister subpopulations may coexist; therefore, a multimodal killing curve may occur. ...
Increasing concerns about the rising rates of antibiotic therapy failure and advances in single-cell analyses have inspired a surge of research into antibiotic persistence. Bacterial persister cells represent a subpopulation of cells that can survive intensive antibiotic treatment without being resistant. Several approaches have emerged to define and measure persistence, and it is now time to agree on the basic definition of persistence and its relation to the other mechanisms by which bacteria survive exposure to bactericidal antibiotic treatments, such as antibiotic resistance, heteroresistance or tolerance. In this Consensus Statement, we provide definitions of persistence phenomena, distinguish between triggered and spontaneous persistence and provide a guide to measuring persistence. Antibiotic persistence is not only an interesting example of non-genetic single-cell heterogeneity, it may also have a role in the failure of antibiotic treatments. Therefore, it is our hope that the guidelines outlined in this article will pave the way for better characterization of antibiotic persistence and for understanding its relevance to clinical outcomes.
Gene expression is a very complex process and involves many small biochemical reaction steps, resulting in a non-Markovian discrete stochastic process due to molecular memory between individual reactions. At present, this process is successfully investigated by generalized chemical master equation models. However, these models do not consider the role of feedback networks in gene expression. How the interaction between feedbacks and molecular memory affects gene expression still remains not well understood. Here, we establish generalized chemical master equation models of gene expression with positive and negative feedbacks. Assuming that the process of producing proteins follows an Erlang probability distribution, we obtain the analytical expression for this model in a steady state, as well as the measure of the noise of protein numbers. We further find that molecular memory competes with the positive feedback in suppressing the noise of the protein number. For our model with a negative feedback, molecular memory can strengthen the intensity of suppressing this noise. These interesting results imply that molecular memory are as important as the feedbacks to affect gene expression.
In this paper, we investigate some novel results on H∞ state estimation for genetic regulatory networks (GRNs) with time-varying delays and Lévy-type noise. The objective of present study is to design the estimator for the considered GRN to analyze the concentrations of mRNAs and proteins through the measured available outputs. The sufficient conditions for stochastic stability of error system is derived by constructing an appropriate Lyapunov-Krasovskii functional (LKF) together with a free-weighing matrix technique and Kunita’s estimate. Further, the results are extended to study H∞ state estimation results. Finally, a physical example of transcriptional regulator is considered to show the effectiveness and feasibility of the proposed estimation scheme.
Evolution on changing fitness landscapes (seascapes) is an important problem in evolutionary biology. We consider the Moran model of finite population evolution with selection in a randomly changing, dynamic environment. In the model, each individual has one of the two alleles, wild type or mutant. We calculate the fixation probability by making a proper ansatz for the logarithm of fixation probabilities. This method has been used previously to solve the analogous problem for the Wright-Fisher model. The fixation probability is related to the solution of a third-order algebraic equation (for the logarithm of fixation probability). We consider the strong interference of landscape fluctuations, sampling, and selection when the fixation process cannot be described by the mean fitness. Such an effect appears if the mutant allele has a higher fitness in one landscape and a lower fitness in another, compared with the wild type, and the product of effective population size and fitness is large. We provide a generalization of the Kimura formula for the fixation probability that applies to these cases. When the mutant allele has a fitness (dis-)advantage in both landscapes, the fixation probability is described by the mean fitness.
We propose a modification of the Crow-Kimura and Eigen models of biological molecular evolution to include a mutator gene that causes both an increase in the mutation rate and a change in the fitness landscape. This mutator effect relates to a wide range of biomedical problems. There are three possible phases: mutator phase, mixed phase and non-selective phase. We calculate the phase structure, the mean fitness and the fraction of the mutator allele in the population, which can be applied to describe cancer development and RNA viruses. We find that depending on the genome length, either the normal or the mutator allele dominates in the mixed phase. We analytically solve the model for a general fitness function. We conclude that the random fitness landscape is an appropriate choice for describing the observed mutator phenomenon in the case of a small fraction of mutators. It is shown that the increase in the mutation rates in the regular and the mutator parts of the genome should be set independently; only some combinations of these increases can push the complex biomedical system to the non-selective phase, potentially related to the eradication of tumors.
Evolutionary games are used in various fields stretching from economics to biology. In most of these games a constant payoff matrix is assumed, although some works also consider dynamic payoff matrices. In this article we assume a possibility of switching the system between two regimes with different sets of payoff matrices. Potentially such a model can qualitatively describe the development of bacterial or cancer cells with a mutator gene present. A finite population evolutionary game is studied. The model describes the simplest version of annealed disorder in the payoff matrix and is exactly solvable at the large population limit. We analyze the dynamics of the model, and derive the equations for both the maximum and the variance of the distribution using the Hamilton–Jacobi equation formalism.
Stochasticity in gene expression can give rise to fluctuations in protein
levels and lead to phenotypic variation across a population of genetically
identical cells. Recent experiments indicate that bursting and feedback
mechanisms play important roles in controlling noise in gene expression and
phenotypic variation. A quantitative understanding of the impact of these
factors requires analysis of the corresponding stochastic models. However, for
stochastic models of gene expression with feedback and bursting, exact
analytical results for protein distributions have not been obtained so far.
Here, we analyze a model of gene expression with bursting and feedback
regulation and obtain exact results for the corresponding protein steady-state
distribution. The results obtained provide new insights into the role of
bursting and feedback in noise regulation and optimization. Furthermore, for a
specific choice of parameters, the system studied maps on to a two-state
biochemical switch driven by a bursty input noise source. The analytical
results derived thus provide quantitative insights into diverse cellular
processes involving noise in gene expression and biochemical switching.
Approximations based on moment-closure (MA) are commonly used to obtain
estimates of the mean molecule numbers and of the variance of fluctuations in
the number of molecules of chemical systems. The advantage of this approach is
that it can be far less computationally expensive than exact stochastic
simulations of the chemical master equation. Here we numerically study the
conditions under which the MA equations yield results reflecting the true
stochastic dynamics of the system. We show that for bistable and oscillatory
chemical systems with deterministic initial conditions, the solution of the MA
equations can be interpreted as a valid approximation to the true moments of
the CME, only when the steady-state mean molecule numbers obtained from the
chemical master equation fall within a certain finite range. The same validity
criterion for monostable systems implies that the steady-state mean molecule
numbers obtained from the chemical master equation must be above a certain
threshold. For mean molecule numbers outside of this range of validity, the MA
equations lead to either qualitatively wrong oscillatory dynamics or to
unphysical predictions such as negative variances in the molecule numbers or
multiple steady-state moments of the stationary distribution as the initial
conditions are varied. Our results clarify the range of validity of the MA
approach and show that pitfalls in the interpretation of the results can only
be overcome through the systematic comparison of the solutions of the MA
equations of a certain order with those of higher orders.
The eikonal approximation (instanton technique) is applied to the problem of large fluctuations of the number of species in spatially homogeneous chemical reactions with the probability density distribution described by a master equation. For both autocatalytic and nonautocatalytic reactions, the analysis of the distribution about a stable stationary state and of the transitions between coexisting stable states comes, to logarithmic accuracy, to the analysis of Hamiltonian dynamics of an auxiliary dynamical system. The latter can be done explicitly in a few cases, including one-species systems, systems with detailed balance, and systems close to the bifurcation points where the number of the stable states changes. In the last case, the fluctuations display universal features, and, for saddle-node bifurcation points, the logarithm of the probability of escape from the metastable state (per unit time) is proportional to the distance to the bifurcation point (in the parameter space) raised to the power 3/2. We compare the eikonal approximation for the stationary distribution of a master equation to Monte Carlo numerical solutions for two chemical two-variable systems with multiple stationary states, where none of the cited restrictions exists. For one of the systems in the pattern of optimal paths we observe caustics emanating from the saddle point.
Moment approximation methods are gaining increasing attention for their use in the approximation of the stochastic kinetics of chemical reaction systems. In this paper we derive a general moment expansion method for any type of propensities and which allows expansion up to any number of moments. For some chemical reaction systems, more than two moments are necessary to describe the dynamic properties of the system, which the linear noise approximation is unable to provide. Moreover, also for systems for which the mean does not have a strong dependence on higher order moments, moment approximation methods give information about higher order moments of the underlying probability distribution. We demonstrate the method using a dimerisation reaction, Michaelis-Menten kinetics and a model of an oscillating p53 system. We show that for the dimerisation reaction and Michaelis-Menten enzyme kinetics system higher order moments have limited influence on the estimation of the mean, while for the p53 system, the solution for the mean can require several moments to converge to the average obtained from many stochastic simulations. We also find that agreement between lower order moments does not guarantee that higher moments will agree. Compared to stochastic simulations, our approach is numerically highly efficient at capturing the behaviour of stochastic systems in terms of the average and higher moments, and we provide expressions for the computational cost for different system sizes and orders of approximation. We show how the moment expansion method can be employed to efficiently quantify parameter sensitivity. Finally we investigate the effects of using too few moments on parameter estimation, and provide guidance on how to estimate if the distribution can be accurately approximated using only a few moments.
Genetic feedback loops in cells break detailed balance and involve bimolecular reactions; hence, exact solutions revealing the nature of the stochastic fluctuations in these loops are lacking. We here consider the master equation for a gene regulatory feedback loop: a gene produces protein which then binds to the promoter of the same gene and regulates its expression. The protein degrades in its free and bound forms. This network breaks detailed balance and involves a single bimolecular reaction step. We provide an exact solution of the steady-state master equation for arbitrary values of the parameters, and present simplified solutions for a number of special cases. The full parametric dependence of the analytical non-equilibrium steady-state probability distribution is verified by direct numerical solution of the master equations. For the case where the degradation rate of bound and free protein is the same, our solution is at variance with a previous claim of an exact solution [J. E. M. Hornos, D. Schultz, G. C. P. Innocentini, J. Wang, A. M. Walczak, J. N. Onuchic, and P. G. Wolynes, Phys. Rev. E 72, 051907 (2005), and subsequent studies]. We show explicitly that this is due to an unphysical formulation of the underlying master equation in those studies.
Moment-closure approximations have in recent years become a popular means to estimate the mean concentrations and the variances and covariances of the concentration fluctuations of species involved in stochastic chemical reactions, such as those inside cells. The typical assumption behind these methods is that all cumulants of the probability distribution function solution of the chemical master equation which are higher than a certain order are negligibly small and hence can be set to zero. These approximations are ad hoc and hence the reliability of the predictions of these class of methods is presently unclear. In this article, we study the accuracy of the two moment approximation (2MA) (third and higher order cumulants are zero) and of the three moment approximation (3MA) (fourth and higher order cumulants are zero) for chemical systems which are monostable and composed of unimolecular and bimolecular reactions. We use the system-size expansion, a systematic method of solving the chemical master equation for monostable reaction systems, to calculate in the limit of large reaction volumes, the first- and second-order corrections to the mean concentration prediction of the rate equations and the first-order correction to the variance and covariance predictions of the linear-noise approximation. We also compute these corrections using the 2MA and the 3MA. Comparison of the latter results with those of the system-size expansion shows that: (i) the 2MA accurately captures the first-order correction to the rate equations but its first-order correction to the linear-noise approximation exhibits the wrong dependence on the rate constants. (ii) the 3MA accurately captures the first- and second-order corrections to the rate equation predictions and the first-order correction to the linear-noise approximation. Hence while both the 2MA and the 3MA are more accurate than the rate equations, only the 3MA is more accurate than the linear-noise approximation across all of parameter space. The analytical results are numerically validated for dimerization and enzyme-catalyzed reactions.
Solutions of the master equation are approximated using a hierarchy of models based on the solution of ordinary differential equations: the macro- scopic equations, the linear noise approximation and the moment equa- tions. The advantage with the approximations is that the computational work with deterministic algorithms grows as a polynomial in the number of species instead of an exponential growth with conventional methods for the master equation. The relation between the approximations is inves- tigated theoretically and in numerical examples. The solutions converge to the macroscopic equations when a parameter measuring the size of the system grows. A computational criterion is suggested for estimating the accuracy of the approximations. The numerical examples are models for the migration of people, in population dynamics and in molecular biology.
Protein and messenger RNA (mRNA) copy numbers vary from cell to cell in isogenic bacterial populations. However, these molecules
often exist in low copy numbers and are difficult to detect in single cells. We carried out quantitative system-wide analyses
of protein and mRNA expression in individual cells with single-molecule sensitivity using a newly constructed yellow fluorescent
protein fusion library for Escherichia coli. We found that almost all protein number distributions can be described by the gamma distribution with two fitting parameters
which, at low expression levels, have clear physical interpretations as the transcription rate and protein burst size. At
high expression levels, the distributions are dominated by extrinsic noise. We found that a single cell’s protein and mRNA
copy numbers for any given gene are uncorrelated.
We investigate the phenomenon of extinction of a long-lived self-regulating stochastic population, caused by intrinsic (demographic) noise. Extinction typically occurs via one of two scenarios depending on whether the absorbing state n=0 is a repelling (scenario A) or attracting (scenario B) point of the deterministic rate equation. In scenario A the metastable stochastic population resides in the vicinity of an attracting fixed point next to the repelling point n=0 . In scenario B there is an intermediate repelling point n=n1 between the attracting point n=0 and another attracting point n=n2 in the vicinity of which the metastable population resides. The crux of the theory is a dissipative variant of WKB (Wentzel-Kramers-Brillouin) approximation which assumes that the typical population size in the metastable state is large. Starting from the master equation, we calculate the quasistationary probability distribution of the population sizes and the (exponentially long) mean time to extinction for each of the two scenarios. When necessary, the WKB approximation is complemented (i) by a recursive solution of the quasistationary master equation at small n and (ii) by the van Kampen system-size expansion, valid near the fixed points of the deterministic rate equation. The theory yields both entropic barriers to extinction and pre-exponential factors, and holds for a general set of multistep processes when detailed balance is broken. The results simplify considerably for single-step processes and near the characteristic bifurcations of scenarios A and B.
We consider the switching rate of a metastable reaction scheme, which includes reactions with arbitrary steps, e.g., kA<-->(k+r)A (both forward and reverse reaction steps are allowed to happen). Employing a WKB approximation, controlled by a large system size, we evaluate both the exponent and the preexponential factor for the rate. The results are illustrated on a number of examples.
Selection in a time-periodic environment is modeled via the two-player
replicator dynamics. For sufficiently fast environmental changes, this is
reduced to a multi-player replicator dynamics in a constant environment. The
two-player terms correspond to the time-averaged payoffs, while the three and
four-player terms arise from the adaptation of the morphs to their varying
environment. Such multi-player (adaptive) terms can induce a stable
polymorphism. The establishment of the polymorphism in partnership games
[genetic selection] is accompanied by decreasing mean fitness of the
population.
Multistability, the capacity to achieve multiple internal states in response to a single set of external inputs, is the defining characteristic of a switch. Biological switches are essential for the determination of cell fate in multicellular organisms, the regulation of cell-cycle oscillations during mitosis and the maintenance of epigenetic traits in microbes. The multistability of several natural and synthetic systems has been attributed to positive feedback loops in their regulatory networks. However, feedback alone does not guarantee multistability. The phase diagram of a multistable system, a concise description of internal states as key parameters are varied, reveals the conditions required to produce a functional switch. Here we present the phase diagram of the bistable lactose utilization network of Escherichia coli. We use this phase diagram, coupled with a mathematical model of the network, to quantitatively investigate processes such as sugar uptake and transcriptional regulation in vivo. We then show how the hysteretic response of the wild-type system can be converted to an ultrasensitive graded response. The phase diagram thus serves as a sensitive probe of molecular interactions and as a powerful tool for rational network design.
A fraction of a genetically homogeneous microbial population may survive exposure to stress such as antibiotic treatment.
Unlike resistant mutants, cells regrown from such persistent bacteria remain sensitive to the antibiotic. We investigated
the persistence of single cells of Escherichia coli with the use of microfluidic devices. Persistence was linked to preexisting heterogeneity in bacterial populations because
phenotypic switching occurred between normally growing cells and persister cells having reduced growth rates. Quantitative
measurements led to a simple mathematical description of the persistence switch. Inherent heterogeneity of bacterial populations
may be important in adaptation to fluctuating environments and in the persistence of bacterial infections.
The method to calculate probabilities of large deviations from the typical behavior in reaction-diffusion systems was described. The method is based on the semiclassical treatment of an underlying Hamiltonian which encodes the system's evolution. It was observed that the process of evolution is a consequence of the interplay of mutation and selection on a population of organisms. It was found that the probability of the rare event is proportional to the exponentiated action along the classical trajectory.
Different proteins have different expression levels. It is unclear to what extent these expression levels are optimized to their environment. Evolutionary theories suggest that protein expression levels maximize fitness, but the fitness as a function of protein level has seldom been directly measured. To address this, we studied the lac system of Escherichia coli, which allows the cell to use the sugar lactose for growth. We experimentally measured the growth burden due to production and maintenance of the Lac proteins (cost), as well as the growth advantage (benefit) conferred by the Lac proteins when lactose is present. The fitness function, given by the difference between the benefit and the cost, predicts that for each lactose environment there exists an optimal Lac expression level that maximizes growth rate. We then performed serial dilution evolution experiments at different lactose concentrations. In a few hundred generations, cells evolved to reach the predicted optimal expression levels. Thus, protein expression from the lac operon seems to be a solution of a cost-benefit optimization problem, and can be rapidly tuned by evolution to function optimally in new environments.
An exact steady-state solution of the stochastic equations governing the behavior of a gene regulated by a self-generated proteomic atmosphere is presented. The solutions depend on an adiabaticity parameter measuring the relative rate of DNA-protein unbinding and protein degradation. The steady-state solution reveals deviations from the commonly used Ackers et al approximation based on the equilibrium law of mass action, allowing anticooperative behavior in the "nonadiabatic" limit of slow binding and unbinding rates. Noise from binding and unbinding events dominates the shot noise of protein synthesis and degradation up to quite high values of the adiabaticity parameter.
The intrinsic stochastic effects in chemical reactions, and particularly in biochemical networks, may result in behaviors significantly different from those predicted by deterministic mass action kinetics (MAK). Analyzing stochastic effects, however, is often computationally taxing and complex. The authors describe here the derivation and application of what they term the mass fluctuation kinetics (MFK), a set of deterministic equations to track the means, variances, and covariances of the concentrations of the chemical species in the system. These equations are obtained by approximating the dynamics of the first and second moments of the chemical master equation. Apart from needing knowledge of the system volume, the MFK description requires only the same information used to specify the MAK model, and is not significantly harder to write down or apply. When the effects of fluctuations are negligible, the MFK description typically reduces to MAK. The MFK equations are capable of describing the average behavior of the network substantially better than MAK, because they incorporate the effects of fluctuations on the evolution of the means. They also account for the effects of the means on the evolution of the variances and covariances, to produce quite accurate uncertainty bands around the average behavior. The MFK computations, although approximate, are significantly faster than Monte Carlo methods for computing first and second moments in systems of chemical reactions. They may therefore be used, perhaps along with a few Monte Carlo simulations of sample state trajectories, to efficiently provide a detailed picture of the behavior of a chemical system.
We apply the spectral method, recently developed by the authors, to calculate the statistics of a reaction-limited multistep birth-death process, or chemical reaction, that includes as elementary steps branching A-->2A and annihilation 2A-->0 . The spectral method employs the generating function technique in conjunction with the Sturm-Liouville theory of linear differential operators. We focus on the limit when the branching rate is much higher than the annihilation rate and obtain accurate analytical results for the complete probability distribution (including large deviations) of the metastable long-lived state and for the extinction time statistics. The analytical results are in very good agreement with numerical calculations. Furthermore, we use this example to settle the issue of the "lacking" boundary condition in the spectral formulation.
We consider an evolution model, in which the mutation rates depend on the structure of population: the mutation rates from lower populated sequences to higher populated sequences are reduced. We have applied the Hamilton–Jacobi equation method to solve the model and calculate the mean fitness. We have found that the modulated mutation rates, directed to increase the mean fitness.
Waddington's epigenetic landscape is an abstract metaphor frequently used to represent the relationship between gene activity and cell fates during development. Over the past few years, it has become a useful framework for interpreting results from single-cell transcriptomics experiments. It has led to the proposal that, during fate transitions, cells experience smooth, continuous progressions of global transcriptional activity, which can be captured by (pseudo)temporal dynamics. Here, focusing strictly on the fate decision events, we suggest an alternative view: that fate transitions occur in a discontinuous, stochastic manner whereby signals modulate the probability of the transition events.
We present analytical results for long-term growth rates of structured populations in randomly fluctuating environments, which we apply to predict how cellular response networks evolve. We show that networks which respond rapidly to a stimulus will evolve phenotypic memory exclusively under random (i.e., nonperiodic) environments. We identify the evolutionary phase diagram for simple response networks, which we show can exhibit both continuous and discontinuous transitions. Our approach enables exact analysis of diverse evolutionary systems, from viral epidemics to emergence of drug resistance.
Multiple phenotypic states often arise in a single cell with different gene-expression states that undergo transcription regulation with positive feedback. Recent experiments show that, at least in E. coli, the gene state switching can be neither extremely slow nor exceedingly rapid as many previous theoretical treatments assumed. Rather, it is in the intermediate region which is difficult to handle mathematically. Under this condition, from a full chemical-master-equation description we derive a model in which the protein copy number, for a given gene state, follows a deterministic mean-field description while the protein-synthesis rates fluctuate due to stochastic gene state switching. The simplified kinetics yields a nonequilibrium landscape function, which, similar to the energy function for equilibrium fluctuation, provides the leading orders of fluctuations around each phenotypic state, as well as the transition rates between the two phenotypic states. This rate formula is analogous to Kramers' theory for chemical reactions. The resulting behaviors are significantly different from the two limiting cases studied previously.
We consider the dynamics of the particle influences by driving force, Gaussian and Poisson noises, which is a modification of the model by Friedman et al (2006 Phys. Rev. Lett. 97 168302). We study the relative integro-differential Fokker–Planck equation (IDFPE) for the probability density distribution and obtain the exact steady state solution of the IDFPE, which is consistent with both analytical and numerical solutions of the IDFPE in the limit of large time t. As in the case of zero Gaussian noise, there is a threshold value of the level of Poissonian noise, related to the intensity of the driving force: below the threshold the maximum of distribution is at the origin; nevertheless, even weak Gaussian noise prevents the formation of a singularity at the maximum point. Above the threshold value, the Gaussian noise cannot shift the maximum to the origin, but makes it less pronounced.
We consider the dynamics in infinite population evolution models with a general symmetric fitness landscape. We find shock waves, i.e., discontinuous transitions in the mean fitness, in evolution dynamics even with smooth fitness landscapes, which means that the search for the optimal evolution trajectory is more complicated. These shock waves appear in the case of positive epistasis and can be used to represent punctuated equilibria in biological evolution during long geological time scales. We find exact analytical solutions for discontinuous dynamics at the large-genome-length limit and derive optimal mutation rates for a fixed fitness landscape to send the population from the initial configuration to some final configuration in the fastest way.
possible states of the cell was specified by snapshots which described each species as "present" or "absent" at a given time. The cell dynamics was e a vastly simplified, stepwise process. Each compound's presence at time step t+1 depends on the presence or absence of a few other compounds at time t. Thus for example compound A would be present at time t+1, if B and C were present at time t, but D was absent. The entire biological cell was described by listing rules like this one for all the N compounds in the cell. Two parameters would describe the average properties of the biological cell, the number of compounds, N, and the average number of precursors, K, whose presence or absence would determine the formation of a given compound. Of course, one could make up a huge number of networks which would fit the description just given. Different networks would be specified by giving different connections and different rules for the formation of each particular chemical compound. The job of finding the rules for a given biological system could be expected to be immense, providing a job more than one generation of biochemists and biologists. Kauffman was unwilling to wait. Instead he built upon work of Paul Erdos and *** networks V1.0
The potential of the one-dimension master equation with both the one-step and the two-step processes is obtained in the leading order of the system-size Omega. Bibtex entry for this abstract Preferred format for this abstract (see Preferences) Find Similar Abstracts: Use: Authors Title Abstract Text Return: Query Results Return items starting with number Query Form Database: Astronomy Physics arXiv e-prints
In order to solve the master equation by a systematic approximation method, an expansion in powers of some parameter is needed. The appropriate parameter is the reciprocal size of the system, defined as the ratio of intensive and extensive variables. The lowest approximation yields the phenomenological law for the approach to equilibrium. The next approximation determines the mean square of the fluctuations about the phenomenological behavior. In equilibrium this approximation has the form of a linear Fokker–Planck equation. The higher approximations describe the effect of the non-linearity on the fluctuations, in particular on their spectral density. The method is applied to three examples: density fluctuations, Alkemade's diode, and Rayleigh's piston. The relation to the expansion recently given by Siegel is also discussed.
The stochastic dynamics of gene expression is often described by highly abstract models involving only the key molecular actors DNA, RNA, and protein, neglecting all further details of the transcription and translation processes. One example of such models is the "gene gate model," which contains a minimal set of actors and kinetic parameters, which allows us to describe the regulation of a gene by both repression and activation. Based on this approach, we formulate a master equation for the case of a single gene regulated by its own product-a transcription factor-and solve it exactly. The obtained gene product distributions display features of mono- and bimodality, depending on the choice of parameters. We discuss our model in the perspective of other models in the literature.
We investigate the multi-chain version of the Chemical Master Equation, when
there are transitions between different states inside the long chains, as well
as transitions between (a few) different chains. In the discrete version, such
a model can describe the connected diffusion processes with jumps between
different types. We apply the Hamilton-Jacobi equation to solve some aspects of
the model. We derive exact {(in the limit of infinite number of particles)}
results for the dynamic of the maximum of the distribution and the variance of
distribution.
While ordinary differential equations (ODEs) form the conceptual framework for modelling many cellular processes, specific situations demand stochastic models to capture the influence of noise. The most common formulation of stochastic models for biochemical networks is the chemical master equation (CME). While stochastic simulations are a practical way to realise the CME, analytical approximations offer more insight into the influence of noise. Towards that end, the two-moment approximation (2MA) is a promising addition to the established analytical approaches including the chemical Langevin equation (CLE) and the related linear noise approximation (LNA). The 2MA approach directly tracks the mean and (co)variance which are coupled in general. This coupling is not obvious in CME and CLE and ignored by LNA and conventional ODE models. We extend previous derivations of 2MA by allowing (a) non-elementary reactions and (b) relative concentrations. Often, several elementary reactions are approximated by a single step. Furthermore, practical situations often require the use of relative concentrations. We investigate the applicability of the 2MA approach to the well-established fission yeast cell cycle model. Our analytical model reproduces the clustering of cycle times observed in experiments. This is explained through multiple resettings of M-phase promoting factor (MPF), caused by the coupling between mean and (co)variance, near the G2/M transition.
Microscopic biological processes have extraordinary complexity and variety at the sub-cellular, intra-cellular, and multi-cellular levels. In dealing with such complex phenomena, conceptual and theoretical frameworks are crucial, which enable us to understand seemingly different intra- and inter-cellular phenomena from unified viewpoints. Decision-making is one such concept that has attracted much attention recently. Since a number of cellular behavior can be regarded as processes to make specific actions in response to external stimuli, decision-making can cover and has been used to explain a broad range of different cellular phenomena [Balázsi et al. (Cell 144(6):910, 2011), Zeng et al. (Cell 141(4):682, 2010)]. Decision-making is also closely related to cellular information-processing because appropriate decisions cannot be made without exploiting the information that the external stimuli contain. Efficiency of information transduction and processing by intra-cellular networks determines the amount of information obtained, which in turn limits the efficiency of subsequent decision-making. Furthermore, information-processing itself can serve as another concept that is crucial for understanding of other biological processes than decision-making. In this work, we review recent theoretical developments on cellular decision-making and information-processing by focusing on the relation between these two concepts.
Gene expression originates from individual DNA molecules within living cells. Like many single-molecule processes, gene expression and regulation are stochastic, that is, sporadic in time. This leads to heterogeneity in the messenger-RNA and protein copy numbers in a population of cells with identical genomes. With advanced single-cell fluorescence microscopy, it is now possible to quantify transcriptomes and proteomes with single-molecule sensitivity. Dynamic processes such as transcription-factor binding, transcription and translation can be monitored in real time, providing quantitative descriptions of the central dogma of molecular biology and the demonstration that a stochastic single-molecule event can determine the phenotype of a cell.
Using a Hamilton-Jacobi equation approach, we obtain analytic equations for steady-state population distributions and mean fitness functions for Crow-Kimura and Eigen-type diploid biological evolution models with general smooth hypergeometric fitness landscapes. Our numerical solutions of diploid biological evolution models confirm the analytic equations obtained. We also study the parallel diploid model for the simple case of recombination and calculate the variance of distribution, which is consistent with numerical results.
Single-cell measurements and lineage-tracing experiments are revealing that phenotypic cell-to-cell variability is often the result of deterministic processes, despite the existence of intrinsic noise in molecular networks. In most cases, this determinism represents largely uncharacterized molecular regulatory mechanisms, which places the study of cell-to-cell variability in the realm of molecular cell biology. Further research in the field will be important to advance quantitative cell biology because it will provide new insights into the mechanisms by which cells coordinate their intracellular activities in the spatiotemporal context of the multicellular environment.
The genetic circuits that regulate cellular functions are subject to stochastic fluctuations, or 'noise', in the levels of their components. Noise, far from just a nuisance, has begun to be appreciated for its essential role in key cellular activities. Noise functions in both microbial and eukaryotic cells, in multicellular development, and in evolution. It enables coordination of gene expression across large regulons, as well as probabilistic differentiation strategies that function across cell populations. At the longest timescales, noise may facilitate evolutionary transitions. Here we review examples and emerging principles that connect noise, the architecture of the gene circuits in which it is present, and the biological functions it enables. We further indicate some of the important challenges and opportunities going forward.
Cell lineage commitment and differentiation are governed by a complex gene regulatory network. Disruption of these processes by inappropriate regulatory signals and by mutational rewiring of the network can lead to tumorigenesis. Cancer cells often exhibit immature or embryonic traits and dysregulated developmental genes can act as oncogenes. However, the prevailing paradigm of somatic evolution and multi-step tumorigenesis, while useful in many instances, offers no logically coherent reason for why oncogenesis recapitulates ontogenesis. The formal concept of "cancer attractors", derived from an integrative, complex systems approach to gene regulatory network may provide a natural explanation. Here we present the theory of attractors in gene network dynamics and review the concept of cell types as attractors. We argue that cancer cells are trapped in abnormal attractors and discuss this concept in the light of recent ideas in cancer biology, including cancer genomics and cancer stem cells, as well as the implications for differentiation therapy.
By monitoring fluorescently labeled lactose permease with single-molecule sensitivity, we investigated the molecular mechanism of how an Escherichia coli cell with the lac operon switches from one phenotype to another. At intermediate inducer concentrations, a population of genetically identical cells exhibits two phenotypes: induced cells with highly fluorescent membranes and uninduced cells with a small number of membrane-bound permeases. We found that this basal-level expression results from partial dissociation of the tetrameric lactose repressor from one of its operators on looped DNA. In contrast, infrequent events of complete dissociation of the repressor from DNA result in large bursts of permease expression that trigger induction of the lac operon. Hence, a stochastic single-molecule event determines a cell's phenotype.
The influence of fluctuations in molecule numbers on genetic control circuits has received considerable attention. The consensus has been that such fluctuations will make regulation less precise. In contrast, it has more recently been shown that signal fluctuations can sharpen the response in a regulated process by the principle of stochastic focusing (SF) (, Proc. Natl. Acad. Sci. USA. 97:7148-7153). In many cases, the larger the fluctuations are, the sharper is the response. Here we investigate how fluctuations in repressor or corepressor numbers can improve the control of gene expression. Because SF is found to be constrained by detailed balance, this requires that the control loops contain driven processes out of equilibrium. Some simple and realistic out-of-equilibrium steps that will break detailed balance and make room for SF in such systems are discussed. We conclude that when the active repressors are controlled by corepressor molecules that display large ("coherent") number fluctuations or when corepressors can be irreversibly removed directly from promoter-bound repressors, the response in gene activity can become significantly sharper than without intrinsic noise. A simple experimental design to establish the possibility of SF for repressor control is suggested.
Biochemical networks in single cells can display large fluctuations in molecule numbers, making mesoscopic approaches necessary for correct system descriptions. We present a general method that allows rapid characterization of the stochastic properties of intracellular networks. The starting point is a macroscopic description that identifies the system's elementary reactions in terms of rate laws and stoichiometries. From this formulation follows directly the stationary solution of the linear noise approximation (LNA) of the Master equation for all the components in the network. The method complements bifurcation studies of the system's parameter dependence by providing estimates of sizes, correlations, and time scales of stochastic fluctuations. We describe how the LNA can give precise system descriptions also near macroscopic instabilities by suitable variable changes and elimination of fast variables.
Organisms in fluctuating environments must constantly adapt their behavior to survive. In clonal populations, this may be achieved through sensing followed by response or through the generation of diversity by stochastic phenotype switching. Here we show that stochastic switching can be favored over sensing when the environment changes infrequently. The optimal switching rates then mimic the statistics of environmental changes. We derive a relation between the long-term growth rate of the organism and the information available about its fluctuating environment.