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# Markovian approaches to modeling intracellular reaction processes with molecular memory

Edited by Hong Qian, University of Washington, Seattle, WA, and accepted by Editorial Board Member Curtis G. Callan Jr. October 11, 2019 (received for review August 13, 2019)

## Significance

Modeling intracellular processes has long relied on the markovian assumption. However, as soon as a reactant interacts with its environment, molecular memory definitely exists and its effects cannot be neglected. Since the Markov theory cannot translate directly to modeling and analysis of nonmarkovian processes, this leads to many significant challenges. We develop a formulation, namely the stationary generalized chemical-master equation, to model intracellular processes with molecular memory. This formulation converts a nonmarkovian question to a markovian one while keeping the stationary probabilistic behavior unchanged. Both a stationary generalized Fokker–Planck equation and a generalized linear noise approximation are further developed for the fast evaluation of fluctuations. These formulations can have broad applications and may help us discover new biological knowledge.

## Abstract

Many cellular processes are governed by stochastic reaction events. These events do not necessarily occur in single steps of individual molecules, and, conversely, each birth or death of a macromolecule (e.g., protein) could involve several small reaction steps, creating a memory between individual events and thus leading to nonmarkovian reaction kinetics. Characterizing this kinetics is challenging. Here, we develop a systematic approach for a general reaction network with arbitrary intrinsic waiting-time distributions, which includes the stationary generalized chemical-master equation (sgCME), the stationary generalized Fokker–Planck equation, and the generalized linear-noise approximation. The first formulation converts a nonmarkovian issue into a markovian one by introducing effective transition rates (that explicitly decode the effect of molecular memory) for the reactions in an equivalent reaction network with the same substrates but without molecular memory. Nonmarkovian features of the reaction kinetics can be revealed by solving the sgCME. The latter 2 formulations can be used in the fast evaluation of fluctuations. These formulations can have broad applications, and, in particular, they may help us discover new biological knowledge underlying memory effects. When they are applied to generalized stochastic models of gene-expression regulation, we find that molecular memory is in effect equivalent to a feedback and can induce bimodality, fine-tune the expression noise, and induce switch.

- chemical-master equation
- biochemical-reaction system
- nonmarkovian reaction kinetics
- gene-expression noise
- molecular memory

Quantitative understanding of the dynamics of single living cells is one of the main goals of modern molecular-systems biology. Traditionally, modeling of intracellular biochemical processes is based on the markovian hypothesis, i.e., the stochastic motion of the reactants is uninfluenced by previous states, only by the current state. This memoryless property implies that markovian reaction kinetics can be described by poissonian processes with constant rates, which are characterized by exponential waiting-time distributions (1, 2). The mathematical tractability of markovian reaction processes enables great simplifications in problem formulation, leading to important successes in the description of many intracellular processes (1⇓⇓–4). These studies revealed that the characteristic parameters in the waiting-time distributions, as well as the complex properties of the intracellular process, can have a strong impact on the reaction kinetics.

However, intracellular reaction processes are not necessarily markovian but may be nonmarkovian. First, as a general rule, the dynamics of a given reactant resulting from its interactions with the environment cannot be described as a markovian process since this interaction can create “molecular memory” characterized by nonexponential waiting-time distributions. Second, the reduction of multistep reactions into a single step one may lead to a nonmarkovian process (5). Third, synthesis of a macromolecule would involve several small single- or multimolecular reaction steps (referring to Fig. 1*A*), creating a memory between individual events (6⇓⇓–9). This is evidenced by the fact that inactive phases of the promoter involving the prolactin gene in a mammalian cell are differently distributed, showing strong memory (10). More generally, the complex control process of gene expression, which would involve several repressors, transcription factors (TFs), and mediators as well as chromatin remodeling or changes in supercoiling, can generate nonexponential time intervals between transcription windows. Indeed, molecular memory, which can result in nonmarkovian (or nonpoissonian) reaction kinetics, has been confirmed by the increasing availability of time-resolved data on different kinds of interactions (10⇓⇓⇓⇓⇓–16).

The extensive existence of molecular memory raises important yet unsolved questions: e.g., how does this memory affect nonmarkovian reaction kinetics? In what way and how accurately do life forms achieve the order required to develop and sustain their lives from the disordered reaction events? These questions were partially addressed in a seminal paper (6) wherein Pedraza and Paulsson analyzed a queuing model of stochastic gene expression with molecular memory, which was further generalized and analyzed (7, 17, 18). Recently, Park et al. (19) analyzed another class of queuing models of stochastic gene expression with molecular memory and presented the chemical fluctuation theorem, which provides an accurate relationship between the environment-coupled chemical dynamics of gene expression and gene-expression variability. These analyzed models are based on queuing theory, which is useful in treating simple single-molecular reaction networks such as birth–death processes, but it seems difficult to extend it to complex bi- or multimolecular reactions, which are a characteristic of many intracellular reaction networks. On the other hand, the continuous time random walk (CTRW) framework provides a different modeling way for stochastic reaction processes and has been extensively used in modeling and analysis of stochastic processes (20⇓–22). Despite this, we still lack a general theory for modeling and analysis of general reaction networks with intrinsic waiting-time distributions. Since the sources of nonmarkovianity and network topology can jointly influence the behavior of the whole biochemical system, the reliable tools and mathematical machinery of the Markov theory do not translate directly to modeling and analysis of nonmarkovian reaction processes, leading to many significant challenges. It is needed to develop new methods to characterize nonmarkovian reaction kinetics.

In recent years, the interest in nonmarkovian processes within scientific communities has been blossoming, from the viewpoints of experimental observations (10⇓⇓⇓⇓⇓–16), mathematical modeling (5⇓–7, 17⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–30), and numerical simulation (30⇓⇓⇓⇓⇓⇓⇓–38). However, because of difficulties in treating nonmarkovian reaction kinetics, only a few exact master equations are known in the literature, most of which are rather formal, or concerning specific systems or stochastic processes (6, 7, 17⇓⇓–20, 39, 40). The aim of this paper is to fill this gap by introducing useful formulations for a general biochemical-reaction system with molecular memory. The basic idea is that we first construct a topology-equivalent, memoryless reaction network by introducing an effective transition rate (ETR) for each reaction in the original nonmarkovian system and then establish an effective master equation for the equivalent reaction network, whose stationary equation is termed as a stationary generalized chemical-master equation (sgCME). The sgCME exactly captures the stationary probabilistic behavior of the original nonmarkovian system. In other words, the original nonmarkovian question is converted into a markovian one (Fig. 1*B*), greatly simplifying analysis of nonmarkovian processes. Based on the sgCME, we further develop another 2 useful techniques for studying stochastic reaction processes on networks: stationary generalized Fokker–Planck equation (sgFPE) and generalized linear noise approximation (gLNA), each enabling fast yet effective evaluation of fluctuations, including the effect of molecular memory.

We demonstrate the power of the above 3 techniques by analyzing 4 examples: a generalized model of constitutive gene expression, a generalized model of gene self-regulation, a generalized ON-OFF model, and a generalized model of genetic toggle switch, each considering molecular memory. These examples are chosen so that analytical or numerical approximations of solutions to the sgCMEs can easily be obtained, and these approximations are then used to clarify the origins of nonmarkovianity and fluctuations in each case and to trace the effects of different parameters on the stochastic properties of the systems. The results reveal the importance of the effect of molecular memory on reaction kinetics.

## Methods

### A Theoretical Framework for Reaction Processes on a Network.

Before presenting our sgCME, let us give the representation of a general reaction network in terms of intrinsic waiting-time distributions. In order to cast the dynamics of N different species that participate in L different reactions (numbered by i with

Note that

If all of the reaction events happen in a markovian (or memoryless) manner, then

### sgCME.

#### Effective transition rates.

For convenience, let

Now, we introduce a memory function for reaction i, denoted by

Interestingly, we find that the limit *SI Appendix* for details)

which is termed as the ETR for reaction i, where *SI Appendix* for details). If waiting-time distributions for all of the reactions are exponential, the nonmarkovian reaction network reduces to a markovian reaction one. Therefore, ETRs are extensions of reaction-propensity functions. However, different from function *SI Appendix* also gives the explicit expressions of ETRs in the case of environmental perturbations or external noise that can lead to stochastic reaction delays.

#### Effective CME.

For the above (nonmarkovian) reaction network, if we let

By the final value theorem (42), we know that if the limit *SI Appendix*):

which is called the sgCME, where *C*). Eq. 4 with Eq. 2 is one of the main results of this paper.

On the other hand, we can use

which is called an effective CME for the original nonmarkovian reaction system, where “effective” will be interpreted below.

Notably, the stationary equation corresponding to Eq. 5 is exactly the same as Eq. 4 except for notation, implying that the stationary probabilistic behavior of the original nonmarkovian reaction system is exactly the same as that of the constructed markovian reaction network. In this sense, the original nonmarkovian problem is converted to a markovian one. However, there would exist differences in dynamic probability behavior between the 2 reaction networks, referring to Fig. 2*C*, which also verifies that the stationary distribution indeed exists even in the presence of molecular memory. Despite this difference, Eq. 5 provides a way for studying complex nonmarkovian reaction kinetics.

### Generalized Linear Noise Approximation.

In analysis of reaction networks, an extensively used technique is the linear noise approximation (LNA) (1, 2). Here, we derive a generalized LNA (gLNA) for the above general reaction system with arbitrary waiting-time distributions.

First, the rate equations corresponding to the constructed-above markovian reaction network read

where

determines the steady state of the deterministic system described by Eq. 6, which is denoted by

Then, we adopt the Ω-expansion method (1, 2) to derive an algebraic equation for covariance matrix. Write *SI Appendix*, we can derive the following Lyapunov matrix equation:

where

We point out that the above analysis framework is convenient for both clarifying the origins of nonmarkovianity (including fluctuations) and tracing the effects of different parameters on the stochastic properties of the underlying systems. *SI Appendix* provides details for examples of analysis by the gLNA. We point out that a slow-scale LNA (ssLNA) has been developed, which describes a class of nonmarkovian systems stemming from timescale separation (43). Specifically, starting with a markovian system composed of fast and slow species, the authors of that paper derived the LNA for the nonmarkovian system, which describes only the slow species (the observables).

### sgFPE.

As an effective approximation of the CME in some situations, the Fokker–Planck equation (FPE) has extensively been used (1, 2, 44, 45), mainly because the latter is more easily analyzed and can often provide more intuitive understanding of a biochemical system than the former. However, the FPE has not been established in the presence of molecular memory. Here, we derive a sgFPE for a general nonmarkovian reaction network with arbitrary waiting-time distributions.

First, although Eq. 3 holds for the discrete variables, it also holds for the corresponding continuous variables. Second, Taylor-expanding the CME in the case of continuous variables to the second-order term yields

Multiplying s on both sides of Eq. 9, letting

In the next section, we will use Eq. 10 to analyze generalized stochastic models of gene expression and obtain some interesting results on the effect of molecular memory.

In the following section, we will apply the above general theory to 4 gene models: a generalized model of constitutive gene expression, a generalized model of gene self-regulation, a generalized ON-OFF model, and a generalized model of genetic toggle switch, where by “generalized,” we mean that each model considers molecular memory or nonmarkovianity. Then, we discover biological knowledge, e.g., molecular memory is in effect equivalent to a feedback and can induce bimodality, fine-tune the expression noise, and induce switch. (The related data will be available from the corresponding author upon request.)

## Results

### The Effect of Molecular Memory Is Equivalent to the Introduction of a Feedback.

Understanding how a gene is turned on at a mechanistic level has been one of the big challenges in molecular biology and has received extensive attention over decades. Identifying the actual sequence of events during gene expression and establishing the method of recruitment have turned out to be a surprisingly difficult task (46). Here, we introduce a generalized model to mimic complex biochemical processes underlying gene expression, referring to Fig. 2*A*, where the proteins are assumed to be produced instantaneously after messenger RNAs (mRNAs) are produced. Fig. 2*B* is a schematic representation of transitions between protein states with time. Let

First, consider the case without regulation but with molecular memory. Two waiting-time distributions are set as *C* demonstrates that the stationary protein distribution indeed exists even in the presence of molecular memory. Hereafter, we will vary memory index (

As pointed out above, the stationary probabilistic behavior of the original nonmarkovian reaction system is exactly the same as that of the constructed markovian reaction network, but there would exist differences in dynamic probability behavior between the 2 networks. Fig. 2*C* shows that 2 dynamic distributions are different at the initial stage, but this difference gradually reduces and finally disappears with time. Fig. 2*E* demonstrates how molecular memory (i.e., *D* shows that

In order to obtain analytical results, we consider the case of

### Molecular Memory Can Induce Bimodality.

In the markovian case, gene self-regulating systems have been extensively studied, and some analytical results have been obtained (49⇓⇓–52). However, gene self-regulating processes are in general nonmarkovian as pointed out in the introduction, raising the question of how nonmarkovianity impacts gene-product distributions. To address this question, let us consider the following gene model:

Similar to the case of no regulation, 2 effective transition rates, *SI Appendix* for details)

Numerical results are demonstrated in Fig. 3, where Fig. 3 *A* and *C* shows how the number of the most probable protein molecules obtained by a statistical method depends on memory index L, whereas Fig. 3 *B* and *D* demonstrates that the stationary distribution predicted by Eq. 11 can well approximate that obtained by solving the sgCME. Note that *A* and *C*, the shadowed areas represent that bimodality exists. Also note that the number of the most probable protein molecules shown in Fig. 3 *A* and *C* is in agreement with that predicted by a deterministic system (*SI Appendix*). From Fig. 3 *B* and *D*, we observe that bimodal protein distribution exists only for moderately large L with

### Molecular Memory Can Fine-Tune the Gene Expression Noise.

Transcription is a key step in gene expression. Biochemical processes associated with transcription often involve a variety of TFs, which regulate the promoter kinetics. For bacterial cells, promoters can exist in a surprisingly large number of regulatory states, e.g., the PRM promoter of phage lambda in *Escherichia* *coli* is regulated by 2 different TFs binding to 2 sets of 3 operators that can be brought together by looping out the intervening DNA, and, as a result, the number of regulatory states of the PRM promoter is up to 128 (54). In contrast, eukaryotic promoters are more complex, involving nucleosomes competing with or being removed by TFs (55). In addition to the conventional regulation by TFs, the eukaryotic promoters can also be epigenetically regulated via histone modifications (56⇓–58), and such regulation may lead to very complex promoter structures (59). Given this complexity, we introduce intrinsic waiting-time distributions to model promoter kinetics. Specifically, assume that the switch times from OFF to ON and vice versa follow gamma distributions given, respectively, by *A*.

Let *SI Appendix*. Moreover, the analytical expression of *SI Appendix* for details). Fig. 4 shows numerical results, where we vary memory index

From this figure, we observe that the mRNA mean is monotonically decreasing in memory index *B*) or in memory index *D*) if the other memory indices are set as 1, but monotonically increasing in memory index *C*) or in memory index *E*) if the other memory indices are set as 1. Fig. 4*B* shows that memory index *C*–*E* demonstrates that the other 3 memory indices can reduce the mRNA noise. In a word, Fig. 4 indicates that molecular memory plays an unneglectable role in gene expression.

### Molecular Memory Can Induce Switch.

Recall that a toggle-switch network (Fig. 5*A*) can model the cross-repression between the determinants of different cellular states, which can result in a definite choice between 2 outcomes (60⇓–62). Conventional models of genetic toggle switch consider exponential waiting-time distributions. However, the expression of a gene in general involves a multistep process. Indeed, transcriptional repressor monomer (A or B) binds first to dimers and then to specific DNA sequences near the promoter, repressing the production of transcriptional repressor monomer (B or A). This multistep process can lead to nonexponential waiting times, creating a memory between individual reaction events. Here, we consider a generalized model of genetic toggle switch, which is schematically shown in Fig. 5*A* with 4 reactions listed in Fig. 5*B*, where

Numerical results are demonstrated in Fig. 5 *C*–*H*, where Fig. 5 *C*–*E* corresponds to the case of exponential waiting times, whereas Fig. 5 *F*–*H* to the case of nonexponential waiting times. We observe that if the waiting times for synthesis of protein A and B follow exponential distributions (i.e., if we set *C* and D. However, if the waiting times for synthesis of protein A and B follow nonexponential distributions (e.g., if we set *F* and *G*. To examine the time dependence of the populations of 2 proteins in a single cell, we first perform stochastic simulations with a numerical algorithm (see *SI Appendix* for details) and then calculate the difference between the levels of proteins A and B. Numerical results are shown in Fig. 5 *E* and *H*. Comparing Fig. 5*E* with Fig. 5*H*, we find that 2 switching states occur only in the case of nonexponential waiting times or molecular memory. Thus, we conclude from Fig. 5 that molecular memory can induce bimodal distributions in the toggle-switch model depicted in Fig. 5*A* or Fig. 5*B*.

## Discussion

Previous studies of biochemical-reaction processes on networks are mainly based on markovian (i.e., memoryless) hypothesis. However, as soon as a reactant interacts with its environment, the effect of molecular memory cannot be neglected. We have derived an exact sgCME, an sgLNA, and an sgFPE for a general biochemical-reaction network with molecular memory characterized by nonexponential waiting-time distributions. These derived equations allow one to retain analytical and/or numerical tractability, being general in scope, and thus are of a potential applicability in a wide variety of problems that transcend pure physics applications. The derived sgCME is particularly useful in finding stationary distributions in a number of nonmarkovian biochemical systems, as demonstrated in this article. Analysis of stochastic gene expression examples has indicated that the sgCME can help us find new biological knowledge, e.g., the effect of molecular memory is equivalent to the introduction of a feedback, and molecular memory can induce bimodality, although the distribution is not bimodal in the corresponding markovian case. The power of the sgCME can be enhanced by analyzing other examples, such as nonmarkovian random walks and diffusion on networks (21, 63⇓–65) and nonmarkovian open quantum systems (66).

Our general theory can reproduce some known results for queuing models of biological processes. First, recall that Pedraza and Paulsson (6) analyzed a *SI Appendix*, we have used the above theory to reproduce this formula. Second, environmental perturbations or external noise, which is often inevitable in cellular processes, can be modeled with time delay (20). We have derived the analytical expressions of effective transition rates and further established the corresponding gLNA (see *SI Appendix* for details). Functionally, this gLNA may be analogous to the chemical fluctuation theorem for

Our theoretical framework can also be used in the inference of the structure and parameters involved in system modes for a broad class of nonmarkovian biochemical-reaction processes on networks. For example, the structure of gene promoters and their kinetics, which would be complex due to, e.g., TF regulation, can be inferred based on experimental data. In fact, we can first infer the key parameter k in the Erlang waiting-time distribution from experimental data, since it can represent the number of small, difficultly specified reaction steps involved in transitions from ON to OFF states or vice versa, implying that the promoter structure can be determined. Then, we can use the standard method (e.g., the maximum likelihood estimation) to infer the values of other parameters from the experimental data, such as the mean switching rates between ON and OFF states, the mean transcription or translational rate. These inferred kinetic parameters in turn determine promoter kinetics and gene-expression dynamics. Furthermore, the sgCME can be used in the analysis of the corresponding stationary probabilistic behavior. In a word, we expect that our analytical framework will be of use for studying a variety of phenomena in biological and physical sciences and, indeed, in other areas where individual-based models with general waiting-time distributions and/or delayed interactions are relevant.

In the realistic world, “non-Markov is the rule, Markov is the exception,” as remarked by N. G. van Kampen (67). A stochastic process (i.e., the biological phenomenon evolving in time) may be or may not be markovian, depending on the variables used to describe it. If all of the variables are observable or measurable, the process is markovian. In general, however, this is impractical and even impossible. Therefore, most of real stochastic processes we observe are nonmarkovian. To model real stochastic processes with some unobservable variables, many different methods of modeling have been proposed, e.g., queuing models (6, 7, 17⇓–19), delay models (5, 28), Langevin equations with color noise (68), and CTRW models (20⇓–22, 26). Correspondingly, some simulation algorithms have also been developed, e.g., those based on general renewal processes (30⇓⇓–33) and the one by introducing some exogenously reaction channels (38). These approaches, despite their own advantages, have finite applications, e.g., queuing models are inconvenient to treating bi- or multimolecular reaction networks.

Finally, we point out that CTRWs used in our theory incorporate the timing of move, where a random walker waits between 2 moves for a duration that independently follows an intrinsic waiting-time distribution. In other words, the move events are generated by a renewal process. On the other hand, CTRWs can be categorized into the 2 classes of active CTRWs and passive CTRWs, depending on whether a random walker actively initializes them as it travels or passively follows states when available (21). In active CTRWs, the interevent time of a state is reinitialized when a random walker lands on it. In passive CTRWs to which queuing models correspond, however, the interevent time of a state is not reset, and the waiting time depends on the last activation time. Usually, active CTRWs generates interevent times from a given PDF, more suitable to the analytical study of random processes, whereas passive CTRWs use interevent times observed in real data, less favorable to model and analyze. Based on the active CTRW framework, we have established a set of theories for a general biochemical network with arbitrary intrinsic waiting-time distributions. Our analysis of generalized birth and death processes based on the passive CTRW framework (*SI Appendix*) has provided a general thinking by establishing the relationship between the active and passive CTRWs.

## Acknowledgments

T.S. was supported by National Natural Science Foundation of China (Grants 11932019, 11775314, and 91530320). J.J. was supported by National Natural Science Foundation of China (Grants 11475273 and 11631005), Science and Technology Program of Guangzhou (Grant 201707010117), and Guangdong Key Research and Development Project 2019B0233002.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: mcszhtsh{at}mail.sysu.edu.cn.

Author contributions: J.Z. and T.Z. designed research, performed research, contributed new reagents/analytic tools, analyzed data, and wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission. H.Q. is a guest editor invited by the Editorial Board.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1913926116/-/DCSupplemental.

- Copyright © 2019 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

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