Abstract Background Stochastic simulation of gene networks by Markov processes has important applications in molecular biology. The complexity of exact simulation algorithms scales with the number of discrete jumps to be performed. Approximate schemes reduce the computational time by reducing the number of simulated discrete events. Also, answering important questions about the relation between network topology and intrinsic noise generation and propagation should be based on general mathematical results. These general results are difficult to obtain for exact models. Results We propose a unified framework for hybrid simplifications of Markov models of multiscale stochastic gene networks dynamics. We discuss several possible hybrid simplifications, and provide algorithms to obtain them from pure jump processes. In hybrid simplifications, some components are discrete and evolve by jumps, while other components are continuous. Hybrid simplifications are obtained by partial Kramers-Moyal expansion <abbrgrp><abbr bid="B1">1</abbr><abbr bid="B2">2</abbr><abbr bid="B3">3</abbr></abbrgrp> which is equivalent to the application of the central limit theorem to a sub-model. By averaging and variable aggregation we drastically reduce simulation time and eliminate non-critical reactions. Hybrid and averaged simplifications can be used for more effective simulation algorithms and for obtaining general design principles relating noise to topology and time scales. The simplified models reproduce with good accuracy the stochastic properties of the gene networks, including waiting times in intermittence phenomena, fluctuation amplitudes and stationary distributions. The methods are illustrated on several gene network examples. Conclusion Hybrid simplifications can be used for onion-like (multi-layered) approaches to multi-scale biochemical systems, in which various descriptions are used at various scales. Sets of discrete and continuous variables are treated with different methods and are coupled together in a physically justified approach.</p

Crudu, Alina

Debussche, Arnaud

Radulescu, Ovidiu

English

PubMed

International audienceStochastic simulation of gene networks by Markov processes has important applications in molecular biology. The complexity of exact simulation algorithms scales with the number of discrete jumps to be performed. Approximate schemes reduce the computational time by reducing the number of simulated discrete events. Also, answering important questions about the relation between network topology and intrinsic noise generation and propagation should be based on general mathematical results. These general results are difficult to obtain for exact models. We propose a unified framework for hybrid simplifications of Markov models of multiscale stochastic gene networks dynamics. We discuss several possible hybrid simplifications, and provide algorithms to obtain them from pure jump processes. In hybrid simplifications, some components are discrete and evolve by jumps, while other components are continuous. Hybrid simplifications are obtained by partial Kramers-Moyal expansion 123 which is equivalent to the application of the central limit theorem to a sub-model. By averaging and variable aggregation we drastically reduce simulation time and eliminate non-critical reactions. Hybrid and averaged simplifications can be used for more effective simulation algorithms and for obtaining general design principles relating noise to topology and time scales. The simplified models reproduce with good accuracy the stochastic properties of the gene networks, including waiting times in intermittence phenomena, fluctuation amplitudes and stationary distributions. The methods are illustrated on several gene network examples.Hybrid simplifications can be used for onion-like (multi-layered) approaches to multi-scale biochemical systems, in which various descriptions are used at various scales. Sets of discrete and continuous variables are treated with different methods and are coupled together in a physically justified approach

INRIA a CCSD electronic archive server

Additional material for the paper Hybrid stochastic simplifi-cations for multiscale gene networksAlina Crudu 1, Arnaud Debussche 2,3 , Ovidiu Radulescu ∗1,31 IRMAR UMR CNRS 6625 Université de Rennes1, Campus de Beaulieu, 35042 Rennes, France2 ENS Cachan, Antenne de Bretagne Avenue Robert Schumann, 35170, BRUZ, France3 INRIA-IRISA, Campus de Beaulieu, 35042 Rennes, FranceEmail: Alina Crudu - alina.crudu@univ-rennes1.fr ; Arnaud Debussche - arnaud.debussche@bretagne.ens-cachan.fr; OvidiuRadulescu ∗- ovidiu.radulescu@univ-rennes1.fr;∗Corresponding author1 Numerical scheme to calculate the steady probability distribution for Stein’smodelThe infinitesimal generator readsAf(x) = −αxf ′(x) + λ(f(x+ a)− f(x)).The adjoint of the infinitesimal generator can be found from the relation∫∞0p(x)(Af(x))dx =∫∞0(A∗p(x))f(x)dx with the condition p(x) = 0, for all x ≤ 0. It follows thatA∗p(x) = αxp′(x) + αp(x) + λ(p(x− a)− p(x)).The steady probability distribution is the solution of the following differential equation with delaysA∗p(x) = 0.with p(x) = 0, for all x ≤ 0.In order to solve this equation numerically on [0,∞[ we reduce the problem to an equivalent one defined onthe bounded set [0, 1] and use the method of finite differences to approximate derivatives.First, we use the substitutionp(x) = g(z), z =x1 + x,1that transforms the domain of the problem from [0,∞[ to the interval [0, 1[ and the equation to solvebecomesαg(z) + αz(1− z)g′(z) + λ[g(z − a(1− z)1− a(1− z))− g(z)]= 0,for all z ∈]0, 1[.Thus, we use an equidistant discretisation of [0, 1] (zi = ip, p = 1/n, n ∈ N∗).We get the following systemαi(1− ip) [g((i+ 1)p)− g(ip)] + αg(ip) + λ[g(i− ap (1− ip)1− a(1− ip)p)− g(ip)]= 0,for all 0 ≤ i ≤ n-1. We let Xi = g(ip) and get the following linear system[−αi(1− ip) + a− λ]Xi + αi(1− ip)Xi+1 + λXj = 0,for all i ≤ n− 1 (where j is chosen to be the positive integer closest to i−ap (1−ip)1−a(1−ip) ).2 The numerical method to estimate stationary distributionsThe stationary distributions of various pure jump and piece-wise deterministic processes were estimated bya Monte Carlo method. We consider a fixed time horizon TMAX. First, a trajectory is simulated usingthe PDP algorithm (see main text). Then, the initial part of the trajectory between 0 and TMIN isdiscarded. The choice of the equilibration time TMIN should be larger than the slowest relaxation time ofthe process. We also assume ergodicity. This hypothesis guarantees uniqueness of the steady probabilitydistribution and also allows us to estimate the steady probability distribution by using the part of thetrajectory between [TMIN, TMAX].The truncated trajectory of a pure jump process is a series of times (jump instants)t0 = TMIN < t2 < . . . < tN = TMAX and a corresponding series of states x0, . . . , xN . Considering thatthe process is constant between ti and ti+1, the steady probability density function can be estimated as ahistogram of the values xi with weights ti+1 − ti.For piece-wise deterministic processes, the assumption that the process is constant between two successivejump instants is no longer satisfied. In this case, the interval between two jump instants is sampled by thedeterministic solver. Considering that the time step δt = ti+1 − ti is small enough, a linear approximationis suitable for the histogram, corresponding to the trapezium rule: xi should be replaced by (xi + xi+1)/2.2Sometimes, we choose to resample the deterministic pieces of the trajectories, using uniform discretisationand spline interpolation.3 Justification of averaging resultsIn order to obtain averaged hybrid approximations from the master equation we generally need twoparameters. A large parameter V (representing volume) is used to rescale large molecule numbers as inKramers-Moyal or Ω expansions. A small parameter ε (representing ratio of fast to slow timescales) is usedto separate fast dynamics within cycles and slow dynamics between cycles. In the most general case,Kramers-Moyal and averaging can be performed in one step starting with the master equation anddeveloping with respect to 1/V and ε. This is a singular perturbation problem because the dynamics atε > 0, 1/V > 0 is not an uniform approximation of the dynamics at ε = 0, 1/V = 0. Arbitrarily largedifferences between the two dynamics occur if we wait long enough. For instance, weakly coupled cycleshave a different long time dynamics and reach a different steady state than totally uncoupled cycles.We can consider the following cases:Case 1 : Fast discrete cycles producing continuous species. There are fast super-reactions of the type 1that act on discrete species γDi 6= 0 for some i ∈ S1. In this case there is only one small parameterη = 1/V .Case 2 : Fast discrete cycles. In this case there are some fast transitions between discrete species. Thesmall parameter is ε.Case 3 : A combination of the above two cases. There are two small parameters, η and ε.We discuss only the first two cases.Case 1 The conditions defining the Case 1 mean that some rapid cycles change in the same time discreteand continuous variables. The discrete species change within a finite set of values, and remain in smallnumbers. The continuous species can be produced continuously, in large numbers. This case is similar tothe QSSA SPA-Ω expansion in [1]. Our discrete species are the quasi-stationary species in [1]. Thedifference is that here we consider rapid cycling (implying production and consumption) of discrete species,while in [1] only rapid consumption is considered (meaning that γDi < 0 for i ∈ S1. To simplify, weconsider that all reactions in RDC are of this type (RDC = S1 and γDi 6= 0 for all reactions in RDC).3Let xc = ηXc be the concentrations of the continuous species. Let X2D be the discrete species that areaffected by super-reactions of the type 1 and X1D the remaining discrete species.We start from the following master equation:∂p∂t(X1D, X2D, xc, t) =∑i∈RCη−1vi(xc − ηγCi )p(X1D, X2D, xc − ηγCi , t)− η−1p(X1D, X2D, xc, t)∑i∈VCvi(xc)++∑i∈RDVi(X1D − γD,1i , X2D − γD,2i )p(X1D − γD,1i , X2D − γD,2i , xc, t)− p(X1D, X2D, xc, t)∑i∈RDVi(X1D, X2D)+∑i∈RDCη−1vi(X1D, X2D − γD,2i , xc − ηγCi )p(X1D, X2D − γD,2i , xc − ηγCi , t)−− η−1p(X1D, X2D, xc, t)∑i∈RDCvi(X1D, X2D, xc)(1)We develop p as a power series in η:p(X1D, X2D, xc, t) = p0(X1D, X2D, xc, t) + ηp1(X1D, X2D, xc, t) + . . .This series will provide the large time solution of the master equation. Using a Taylor expansion of (1) weobtain at various orders:At order η−1∑i∈RDCvi(X1D, X2D − γD,2i , xc)p0(X1D, X2D − γD,2i , xc, t)− p0(X1D, X2D, xc, t)∑i∈RDCvi(X1D, X2D, xc) = 0This means that at X1D, xc, t fixed, p0 is the steady-state distribution of the process defined by the fastreactions RDC . Now, we consider that this process is ergodic, meaning that starting from any state we canreach any other state in finite time. The reaction mechanism RDC should be made of one or severalinterconnected cycles. This condition ensures the uniqueness of the steady state distribution and we canwrite that:p0(X1D, X2D, xc, t) = ψ(X1D, xc, t)ρ(X2D) (2)where ρ(X2D) is the unique steady state distribution of the fast process and ψ(X1D, xc, t) is thetime-dependent distribution of the remaining variables.At order η04∂p0∂t= −∇xc .[χc(xc)p0]++∑i∈RDVi(X1D − γD,1i , X2D − γD,2i )p0(X1D − γD,1i , X2D − γD,2i , xc, t)− p0(X1D, X2D, xc, t)∑i∈RDVi(X1D, X2D)−∇xc .[∑i∈RDCvi(X1D, X2D − γD,2i , xc)γCi p(X1D, X2D − γD,2i , xc, t)]++∑i∈RDCvi(X1D, X2D − γD,2i , xc)p1(X1D, X2D − γD,2i , xc, t)− p1(X1D, X2D, xc, t)∑i∈RDCvi(X1D, X2D, xc)(3)where χc(xc) =∑i∈RC viγCi .By using (2) and by summing (3) with respect to all possible values of X2D we obtain:∂ψ∂t= −∇xc .[(χc(xc) + χ̄DC(X1D, xc))ψ(X1D, xc, t)]++∑i∈RDV̄i(X1D − γD,1i )ψ(X1D − γD,1i , xc, t)− ψ(X1D, xc, t)∑i∈RDV̄i(X1D)(4)where V̄i(X1D) =∑X2DVi(X1D, X2D)ρ(X2D), χ̄DC(X1D, xc) =∑X2D∑i∈RDC vi(X1D, X2D, xc)γCi are averagedrates.Eq.(4) shows that the zeroth order approximation is an averaged PDP.Case 2 We do not need to distinguish here between discrete and continuous species. The reader couldimagine the situation when there are only discrete species. We consider that there are fast reactions R2and slow reactions R1. We consider that Vi = ε−1vi for i ∈ R2. Like for Case 2 we consider two types ofspecies. X2D are the species that are affected by fast reactions and X1D the remaining species.This case can be treated exactly in the same way as the preceding case. We start from the following masterequation:∂p∂t(X1D, X2D, t) =∑i∈R1Vi(X1D − γD,1i , X2D − γD,2i )p(X1D − γD,1i , X2D − γD,2i , t)−− p(X1D, X2D, t)∑i∈R1Vi(X1D, X2D) +∑i∈R2ε−1vi(X1D − γD,1i , X2D − γD,2i )p(X1D − γD,1i , X2D − γD,2i , t)−− p(X1D, X2D, t)∑i∈R2ε−1vi(X1D, X2D)(5)With the same assumption on the ergodicity of the process defined by fast reactions, we havep = ψ(X1D, t)ρ(X2D) + εp1(X1D, X2D, t). In the zero-th approximation, ρ(X2D) is the steady state distributionof the fast mechanism and ψ(X1D, t) represents the dynamics of an averaged mechanism:5∂ψ∂t(X1D, t) =∑i∈R1V̄i(X1D − γD,1i )ψ(X1D − γD,1i , t)− ψ(X1D, t)∑i∈R1V̄i(X1D) (6)where V̄i(X1D) =∑X2DVi(X1D, X2D)ρ(X2D) are averaged rates.When the mechanism R2 is made of a single cycle A21 → A22 → ...A2m → A21, the steady state distributioncan be easily calculated. This distribution is parametrised by the value of the total mass of the cycleN =∑mi=1X2i , which is a slow variable. Consider total separation of the reaction constants of the cycleand that last step is limiting. Then, mass is concentrated with probability 1−∑i 6=m klim/ki close to one tothe beginning of the limiting step and with small probabilities klim/ki at the beginning of step i 6= m. Theaverage of X2j with respect to the steady distribution ρ is X̄2j = Nklim/kj . Then the average rate of thereaction A2j → B of reaction constant k where B is exterior to the cycle is V̄j = kX̄2j = Nkklim/kj .4 Justification of the singular switching resultsConsider the following PDP with singular switching. The flow function is:χ(XD, x) = χ0(x) +1εχ1(XD, x)whereχ0(x) =∑i∈RCγCi vi(x), χ1(XD, x) =∑i∈S3γCi vi(XD, x),and the jump intensity isλ(XD, x) =∑i∈RD\R−DVi(XD) +1ε∑j∈R−Dvj(XD).We shall obtain the limit process by two methods. The first method employs a scaling argument, and thesecond method employs the expansion of the hybrid Fokker-Planck equation.The state of the system is (x,XD, X ′D) where x are continuous species, XD are discrete species, and X′Dare discrete species substrates of of rapid reactions of the type S3 that are responsible for the breakage. Tosimplify the calculations we consider that variables X ′D take only two values X′D ∈ {0, 1}. If X ′D = 1species x are rapidly produced by reactions S3, and if X ′D = 0 production stops. Once in the stateX ′D = 1, the system rapidly switches back to X′D = 0 by fast reactions R−D .6Scaling argument Let us consider the process in a state (XD, X ′D = 0, x). With intensity Vi(XD) it jumpsto the state (XD + γDi , X′D = 1, x), where it stays only for a very short time. In this state, the continuousvariables are submitted to the fast dynamics:dxεdt=1εχ1(XD + γDi , xε), xε(0) = xduring the short random time τε that satisfiesP [τε > t] = exp[−1εvj(XD + γDi )t] (7)then jumps to the state XD + γDi + γDj , X′D = 0 by a reaction j ∈ R−D .Let Φ(s;x,XD) be the solution of the equationdΦds= χ1(Φ, XD), Φ(0) = xPassing to short timescales s = t/ε, we can easily show that the variation of the continuous variablestarting from x and during the time τε is:∆xε = xε(τε)− x = Φ(s̃;x,XD + γDi )− xwhere the random time s̃ satisfiesP [s̃ > s] = exp[−vj(XD + γDi )s]Keeping the random variable ∆xε as jump (breakage) of the continuous variable and noticing from (7) thatτε → 0 (almost surely), we obtain the following generator of the limit hybrid process:A f(XD, x) = χ0(x).∇xf +∑i∈RD\(R−D∪R+D)Vi(XD)[f(XD + γi, x)− f(XD, x)]++∑i∈R+DVi(XD)∑j∈R−D∫ ∞0[f(XD + γDi + γDj ,Φ(s;x,XD + γDi ))− f(XD, x)]ρij(s)ds(8)where ρij(s) = vj(XD + γDi ) exp[−vj(XD + γDi )s] is a probability density satisfying∫∞0ρij(s)ds = 1.From the generator we can obtain the hybrid Fokker-Planck equation (using the adjoint of the generator).In the adjoint, we should replace trajectories starting from x to trajectories arriving in x, thus changing s7into −s and renormalizing the density to take into account the phase space volume transformation. Thus,the hybrid Fokker-Planck equation reads:∂p∂t(XD, x, t) = −∇x.(χ0(x)p(XD, x, t))++∑i∈RD\(R−D∪R+D)p(XD − γi, x, t)Vi(XD − γi)− p(XD, x, t)∑i∈RD\(R−D∪R+D)∑Vi(XD)++∑i∈R+D∑j∈R−DVi(XD − γDi − γDj )∫ ∞0p(XD − γDi − γDj , φ(−s;x,XD − γDj ), t)ρ′j(s)ds−− p(XD, x, t)∑i∈R+DVi(XD)(9)where ρ′j(s) = vj(XD − γDj ) exp[−∫ s0(λ−(XD − γDj ) +∇x.χ1(XD − γDj , φ(−s′;x,XD − γDj )))ds′].Fokker-Planck equation argument To simplify formulae, we consider with no loss of generality thatRD = R−D ∪R+D.The hybrid Fokker-Planck equation for the singular switching process reads:∂p∂t(XD, 1, x, t) = −∇x.[p(XD, 1, x, t)(χ0(x) +1εχ1(XD, x))] +∑i∈R+Dp(XD − γDi , 0, x, t)Vi(XD − γDi )−− 1ελ−(XD)p(XD, 1, x, t)∂p∂t(XD, 0, x, t) = −∇x.[(p(XD, 0, x, t)χ0(x)] +1ε∑j∈R−Dp(XD − γDj , 1, x, t)vj(XD − γDj , 1)−− λ+(XD)p(XD, 0, x, t)(10)whereλ−(XD) =∑j∈R−Dvj , λ+(XD) =∑j∈R−DVj .Consider that p(XD, X ′D, x, t) = p0(XD, X′D, x, t) + εp1(XD, X′D, x, t) + . . .Then we Taylor expand the Fokker-Planck equations (10) and we obtain:At order ε−1p0(XD, 1, x, t) = 0At order ε08−∇x[p1(XD, 1, x, t)χ1(XD, x)] +∑i∈R+Dp0(XD − γDi , 0, x, t)Vi(XD − γDi )− λ−(XD)p1(XD, 1, x, t) = 0∂p0∂t(XD, 0, x, t) = −∇x[p0(XD, 0, x, t)χ0(x)] +∑j∈R−Dp1(XD − γDj , 1, x, t)vj(XD − γDj , 1)−− λ+(XD)p0(XD, 0, x, t)(11)From the first of the Eqs.11 we obtaindp1(∗, 1,Φ(−s;x, ∗∗), t)ds=[λ−(∗) +∇x.χ1(XD,Φ(−s;x, ∗∗))]p1(∗, 1,Φ(−s;x, ∗∗), t)−−∑i∈R+Dp0(∗ − γDi , 0,Φ(−s;x, ∗∗), t)Vi(∗ − γDi )(12)The linear differential equation (12) has the solution:p1(∗, 1, x, t) = p1(∗, 1,Φ(0;x, ∗∗), t) ==∫ ∞0∑i∈R+Dp0(∗ − γDi , 0,Φ(−s;x, ∗∗), t)Vi(∗ − γDi )exp[−∫ s0(λ−(∗) +∇.χ1(XD,Φ(−s′;x, ∗∗)))ds′]ds.(13)By replacing (13) in the second of the Eqs.11 we obtain again Eq.(9).References1. Mastny E, Haseltine E, Rawlings J: Two classes of quasi-steady-state model reductions forstochastic kinetics. The Journal of Chemical Physics 2007, 127:094106.9

Hybrid stochastic simplifications for multiscale gene networks

Alina Crudu

Arnaud Debussche

Ovidiu Radulescu

Crossref

HAL-CentraleSupelec

HAL-Rennes 1

Springer - Publisher Connector

A: Extracting macroscopic dynamics: model problems and algorithms. Nonlinearity

A: Limitation and averaging for deterministic and stochastic biochemical reaction networks.

Approaches to complexity reduction in a systems biology research environment (SYCAMORE).

Approximation par un processus de diffusion, des oscillations, autour d'une valeur moyenne, d'un processus de Markov de saut pur.

Averaging of time-varying differential equations revisited.

Bagchi A: Modeling stochastic hybrid systems. System Modeling and Optimization

Binomial leap methods for simulating stochastic chemical kinetics.

Bistability and switching in the lysis/lysigeny genetic regulatory network of bacteriophage λ. J Theor bio

Brownian motion in a field of force and the diffusion model of chemical reactions. Physica A

Crudu A: Théorèmes limites pour des processus de Markov à sauts. Synthèse des resultats et applications en biologie moleculaire. Technique et Science Informatique

Discrete-time dynamic systems arising from singularly perturbed Markov chains. Nonlinear Analysis of Theory Methods and Applications

Dynamic and static limitation in multiscale reaction networks, revisited.

Dynamic partitioning for hybrid simulation of the bistable HIV-1 transactivation network. Bioinformatics

Elements of the Theory

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

Exact stochastic simulation of coupled chemical reactions.

Falcke M: Hybrid Stochastic and Deterministic Simulations of Calcium Blips.

General stochastic hybrid systems: Modelling and optimal control.

HH: Stochastic Kinetic Analysis of Developmental Pathway Bifurcation in Phage λ-infected Escherichia Coli Cells. Genetics

Huisinga W: Adaptive simulation of hybrid stochastic and deterministic models for biochemical systems.

Huu TN: Aggregation of variables and applications to population dynamics.

I: Graded and binary responses in stochastic gene expressions. Phys Biol

Improved leap-size selection for accelerated stochastic simulation.

IV: Invariant manifolds for physical and chemical kinetics,

Kurtz TG: Markov Processes

Lilienbaum A: Robust simplifications of multiscale biochemical networks.

Limit theorems for sequences of jump Markov processes approximating ordinary differential processes.

Linking stochastic dynamics to population distribution: an analytical framework of gene expression. Physical review letters

Markov Models and Optimization London:

Markov processes and differential equations: asymptotic problems Basel: Birkhauser;

Mitropolski YA: Asymptotic Methods in the Theory of Nonlinear Oscillations

Multiscale Hy3S: hybrid stochastic simulation for supercomputers.

N: Stochastic processes in physics and chemistry third edition.

Neuronal variability: noise or part of the signal? Nature Reviews Neuroscience

Noise-based switches and amplifiers for gene expression. Proc Natl Acad Sci USA

O: Hybrid weak limits and averaging for multiscale stochastic gene networks.

On/Off Storage Systems with State-Dependent Input, Output and Switching Rates.

Oudenaarden A: Heritable Stochastic Switching Revealed by Single-Cell Genealogy. Plos Biology

Oudenaarden A: Regulation of noise in the expression of a single gene. Nature Genet

Radulescu O: Dynamical robustness of biological networks with hierarchical distribution of time scales.

Rempala G: Asymptotic analysis of multiscale approximations to reaction networks. Ann Appl Probab

Sawant A: On a computational approach for the approximate dynamics of averaged variables in nonlinear ODE systems: Toward the derivation of constitutive laws of the rate type.

Shivashankar G: Stochastic simulations of the origins and implications of long-tailed distributions in gene expression.

Shivashankar G: Stochastic simulations of the origins and implications of long-tailed distributions in gene expression. PNAS

Siggia E: Intrinsic and extrinsic contributions to stochasticity in gene expression.

Simulation of genetic networks modelled by piecewise deterministic Markov processes. Systems Biology,

Singularly Perturbed Markov Chains: Convergence and Aggregation.

Some models of neuronal variability.

Statistical Fluctuations in Autocatalytic Reactions.

Stochastic hybrid models: An overview.

Stochastic Processes and Statistical Physics.

Stochastic processes in the neurosciences Philadelphia: Society for Industrial and Applied Mathematics;

Stochasticity in gene expression: from theories to phenotypes. Nature Rev Genet

Summing up the noise in gene networks.

Supplementary chapters to the theory of ordinary differential equations

Swishchuk A: Semi-Markov Random Evolutions Dordrecht:

Tapscott SJ: Modeling stochastic gene expression: Implications for haploinsufficiency.

The Chemical Langevin equation.

The Effect of Transcription and Translation Initiation Frequencies on the Stochastic Fluctuations in Prokaryotic Gene Expression.

The Fokker-Planck equation: Methods of Solution and Applications

Theoretical and experimental analysis of the phage lambda genetic switch implies missing levels of cooperativity.

Two classes of quasi-steadystate model reductions for stochastic kinetics.

Verhulst F: Averaging methods in nonlinear dynamical systems

W: Exact simulation of hybrid stochastic and deterministic models for biochemical systems. Research Report RR-5435, INRIA

Watanabe S: Stochastic differential equations and diffusion processes

Xie X: Stochastic protein expression in individual cells at the single molecule level. Nature

Xie XS: Probing Gene Expression in Live Cells, One Protein Molecule at a Time. Science

http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2761401

Hybrid stochastic simplifications for multiscale gene networks

Abstract

Similar works

Full text

Available Versions

INRIA a CCSD electronic archive server

Crossref

HAL-CentraleSupelec

HAL-Rennes 1

Springer - Publisher Connector