Optimal enumeration of state space of finitely buffered stochastic molecular networks and exact computation of steady state landscape probability
- Youfang Cao^{1, 2} and
- Jie Liang^{1, 3}Email author
https://doi.org/10.1186/1752-0509-2-30
© Cao and Liang; licensee BioMed Central Ltd. 2008
Received: 12 February 2008
Accepted: 29 March 2008
Published: 29 March 2008
Abstract
Background
Stochasticity plays important roles in many molecular networks when molecular concentrations are in the range of 0.1 μ M to 10n M (about 100 to 10 copies in a cell). The chemical master equation provides a fundamental framework for studying these networks, and the time-varying landscape probability distribution over the full microstates, i.e., the combination of copy numbers of molecular species, provide a full characterization of the network dynamics. A complete characterization of the space of the microstates is a prerequisite for obtaining the full landscape probability distribution of a network. However, there are neither closed-form solutions nor algorithms fully describing all microstates for a given molecular network.
Results
We have developed an algorithm that can exhaustively enumerate the microstates of a molecular network of small copy numbers under the condition that the net gain in newly synthesized molecules is smaller than a predefined limit. We also describe a simple method for computing the exact mean or steady state landscape probability distribution over microstates. We show how the full landscape probability for the gene networks of the self-regulating gene and the toggle-switch in the steady state can be fully characterized. We also give an example using the MAPK cascade network. Data and server will be available at URL: http://scsb.sjtu.edu.cn/statespace.
Conclusion
Our algorithm works for networks of small copy numbers buffered with a finite copy number of net molecules that can be synthesized, regardless of the reaction stoichiometry, and is optimal in both storage and time complexity. The algorithm can also be used to calculate the rates of all transitions between microstates from given reactions and reaction rates. The buffer size is limited by the available memory or disk storage. Our algorithm is applicable to a class of biological networks when the copy numbers of molecules are small and the network is closed, or the network is open but the net gain in newly synthesized molecules does not exceed a predefined buffer capacity. For these networks, our method allows full stochastic characterization of the mean landscape probability distribution, and the steady state when it exists.
Background
Networks of interacting biomolecules are at the heart of the regulation of cellular processes, and stochasticity plays important roles in many networks, including those responsible for gene regulation, protein synthesis, and signal transduction [1–5]. The stochasticity originates intrinsically from the small copy numbers of the molecular species in a cell, which frequently occur when molecular concentrations are in the range of 0.1 μ M to 1n M (typically from about 100 to 10 copies in a cell) [2, 6]. For example, the regulation of transcriptions depends on the binding of often a few proteins to a promoter site; the synthesis of protein peptides on ribosome involves a small copy number of molecules; and patterns of cell differentiation depend on initial small copy number events. In these biological processes, fluctuations due to the stochastic behavior intrinsic in low copy number events play important roles.
The importance of stochasticity in cellular functions is well recognized. Studies of network models show that stochasticity is important for magnifying signal, sharpening discrimination, and inducing multistability [4, 7–13]. Understanding the stochastic nature and its consequences for cellular processes involving molecular species of small copy numbers in a network is an important problem.
A fundamental framework for studying the full stochasticity is the chemical master equation [14, 15]. Under this framework, the combination of copy numbers of molecular species defines the microscopic state of the molecular interactions in the network. By treating microscopic states of reactants explicitly, linear and nonlinear reactions (such as synthesis, degradation, dimeric binding, and multimerization) can all be effectively modeled as transitions between microstates, with transition rates determined by the physical properties of the molecules and the cell environment. The probability distribution or potential landscape [16–18] over these microstates and its time-evolving behavior provide a full description of the properties of a stochastic molecular network.
However, it is challenging to study a realistic system that involves a nontrivial number of species of small copy numbers. Analytical solutions of the chemical master equation exist only for very simple cases, such as self-regulating genes [19], and the toggle-switch network under certain restrictions [8, 18]. Instead of solving the master equation, a widely used method is to carry out Monte Carlo simulations using the Gillespie algorithm [14]. This method generates samples from multiple runs of simulation, and statistics properties are calculated from the simulation trajectories, which provide characterizations of the network [13, 14, 20, 21]. This approach has found wide applications, although it cannot guarantee a full account of stochasticity, as this method usually does not generate an exhaustive number of trajectories that cover all possible locations in the probability landscape. In addition, as Monte Carlo simulations follow high probability paths, it is especially challenging to sample adequately rare and critical events that may be important in determining cellular fate. It is also difficult to determine whether a simulation is extensive enough to obtain accurate statistics. The amount of computation necessary to obtain an accurate result may be too large to be completed in a reasonable amount of time, especially when the time scales of the various react ions involved are very different [8]. To address these issues, Gillespie, Petzold, and colleagues further developed numerical methods for speeding up the stochastic simulation [20, 21]. Munsky and Khammash developed a method to approximate the solution of chemical master equation by projecting the whole state space of the system to a finite space [22]. Samant and Vlachos developed a multiscale Monte Carlo method for stiff systems where partial equilibrium occurs [23]. An alternative approach is to approximate the master equation using, for example, Fokker-Planck or Langevin equations [15]. These are obtained by adding stochastic terms (often Gaussian) to a deterministic equation [12, 18, 24]. Salis and Kaznessis improved the stochastic simulation method by partitioning the system into components with fast and slow reactions. The fast reactions are approximated by the Langevin equations, and the slow reactions are analyzed by stochastic Monte Carlo simulations [25].
A complete identification and characterization of the space of the microstates is a prerequisite for obtaining the full landscape probability distribution of a network. However, the state space of a network currently cannot be fully characterized in general. There is neither closed-form solution, nor computational algorithm describing the full state space. In this paper, we study the problem of enumerating the state space of a molecular network with small copy numbers of molecular species.
A naive method is to predefine the maximum copy number of the reactants, and bound the state space by the product of the maximum numbers. However, the size of state spaces estimated by this naive approach will be inflated to enormity. For example, if there are 16 species, and there is a total a maximum of 33 molecules in the whole system, this naive method does not take into consideration of the details of the network, and the state space will be estimated to be in the order of (33 + 1)^{16} = 3.19 × 10^{24} states. This naive method is intrinsically inefficient: There may be many states which may never occur. For some states, no reactions may occur and therefore are not needed. For others, no reactions can lead to them under the specified initial condition. An alternative approach is carrying out simulation. One can simply follow explicitly simulated reaction events to whatever microstates of copy numbers the system reaches. However, this approach cannot guarantee that all reachable states will be explored, therefore cannot guarantee full characterization of rare events.
In this study, we develop an optimal algorithm that gives full description of the state space and the set of transitions. Our method works for networks of small copy numbers under the condition that the net gain in newly synthesized molecules in the network does not exceed a predefined finite number. Our algorithm is optimal in both memory requirement and in time complexity. All states reachable from a given initial condition will be accounted for by our method, and no irrelevant states will be included. All possible transitions will be recorded, and no infeasible transitions will be ever attempted. As a result, our algorithm can generate the full state-transition matrix under the framework of the chemical master equation. This matrix is compact without any redundant information. It is also of the minimal size. In addition, the computational time is optimal, up to a constant. We also describe how to obtain the mean landscape probability distribution over the enumerated state space of a network, which is the same as the landscape distribution of the steady state when it exists.
This paper is organized as follows. We first describe how our method can be applied to the simple examples of a self-regulating gene, a toggle-switch network, and the more complex example of the MAPK network. This is followed by conclusion and discussion. We finally describe the technical details of the algorithm for enumerating the space of microstates, and introduce a simple method for computing the steady state landscape probability distribution.
Results and Discussion
Molecular network models
We apply our algorithm to three network models: the self-regulating gene, the small toggle-switch network, and the MAPK cascade network.
Self-regulating gene
We first calculate the state spaces. We use the same initial condition of 1 copy of unbound gene, 0 copies of transcription factor and bound gene, and set the buffer size to allow different copy numbers of protein transcription factor to be synthesized. As there is only one copy of the gene in this model [18], the size of the state space increases with the copy number of the protein transcription factor that can be synthesized. Our results show that when the buffer capacity takes the value of 100, 1,000, and 10,000, the size of the state space is 201, 2,001, and 20,001, respectively. In this model, the size of the state space scales linearly with the copy number of the protein synthesized. In biological condition, the copy number of a transcription factor rarely exceeds 100.
Our results and the results of Schultz et al obtained from multiple runs of Gillespie simulations are identical [18]. As pointed out in [18], the self-repressing and the self-activating genes can have overall similar distributions. This can be explained by the fact that the combined synthesis rate of the protein s_{0} + s_{1} = 60 is the same in both cases (front profile and back profile in Fig 2). Closer examination shows that in the case of the self-repressing gene network (s_{0} = 10 and s_{1} = 50, front profile), the first peak of probability at smaller copy number of the free protein is lower, and the second peak at higher copy number is larger when compared to the distribution of the self-activating gene (s_{0} = 50, and s_{1} = 10, Fig 2, back profile). That is, the self-repressing network has a higher probability in producing more free proteins than the self-activating network. This can be explained by the difference between the protein-DNA binding rate b and unbinding rate u. In this model network, unbinding rate u = d/10 is 25 times greater than the binding rate b = d/250. As a result, this gene is more likely to stay in the unbound state. Since the self-repressing network has a higher synthesis rate in unbound state (s_{1} = 50 > s_{0} = 10), it will produce more free proteins. This results in an overall slightly higher probability for larger number of free proteins for self-repressing network. This small difference in probability distribution is also observed in [18]. As pointed out previously in [18], when both synthesis rates are equal (s_{0} = s_{1} = 50), the binding state transition do not change the synthesis/degradation process, and the network is a simple birth/death process, with a Gaussian probability distribution for protein number centered at s_{0} = s_{1} (Fig 2, middle profile).
Toggle switch
We first calculate the state spaces under the initial condition of 1 copy of unbound gene A, 1 copy of unbound gene B, 0 copies of bound gene A and bound gene B, and 0 copies of their protein products. We set the buffer size to different copies of total protein A and protein B combined that can be synthesized. When the buffer capacity is 20, the size of the state space is 764. At buffer capacity of 200, 400, and 800 copies of proteins, the size increases to 79,604, 319,204, and 1,278,404, respectively.
It is clear that a toggle switch has four different states, corresponding to the "on/on", "on/off", "off/on" and "off/off" states. At the chosen parameter condition, the toggle/switch exhibits clear bi-stability, namely, it has high probabilities for the "on/off" and "off/on" states, but has a low probability for the "on/on" state. The "off/off" state is severely suppressed. Our results are identical with the results of Schultz et al obtained from multiple runs of Gillespie simulations [18].
MAPK network
MAPK cascade network plays important role in signal transduction. Here our purpose is to explore how to apply our algorithm to more realistic network model. Our goal in this paper is not to study the the stochastic nature and the dynamic behavior of MAPK network.
Abbreviations of the molecular species in the MAPK network.
Num. | Abbrev. | Description |
---|---|---|
1 | M | ERK, extracellular signal- regulated kinase |
2 | MpY | ERK with Y phosphorylated |
3 | MpT | ERK with T phosphorylated |
4 | Mpp | ERK with dual phosphorylated |
5 | MEK | ERK kinase |
6 | MKP3 | ERK phosphatase |
7 | MpY_MEK | Binding of MpY and MEK |
8 | MpT_MEK | Binding of MpT and MEK |
9 | M_MEK _Y | Binding of M and MEK at Y site |
10 | M_MEK_T | Binding of M and MEK at T site |
11 | Mpp_MKP3 | Binding of Mpp and MKP3 |
12 | MpY_MKP3 | Binding of MpY and MKP3 |
13 | MpT_MKP3_Y | Binding of MpT and MKP3 at Y |
14 | MpT_MKP3_T | Binding of MpT and MKP3 at T |
15 | M_MKP3_T | Binding of M and MKP3 at T site |
16 | M_MKP3_Y | Binding of M and MKP3 at Y site |
Simple initial conditions
It is clear that different molecular species in this model affect the size of the state space differently. Increasing the copy number of M-MEK-Y, M-MEK-T, and Mpp-MKP3 molecules (species 9, 10 and 11, in bold fonts in Table 1) lead to large state spaces (size 888, 030 at 20 copies, Fig 6), while the initial conditions of 20 copies of any other species result in modest state spaces. For example, species 7, 8, 15 and 16 when given 20 copies have a state-space size of 231. For species 1–6 (M, MpY, MpT, Mpp, MEK, MKP3), no reactions can occur at these initial conditions, and the state space contains only the the initial state.
The state space for each of the 320 initial conditions can be computed within one minute. We further found that when any of S_{9}, S_{10}, or S_{11} has an initial copy of 28 and all others 0 copies, the state spaces increases to 6,724,520, and the computing time also increase, although all can be computed within 10 minutes on a Linux workstation.
Biological initial conditions
We further calculate sizes of the state spaces with several biologically plausible initial conditions, in which species M, MEK and MKP3 are all given an equal number of i copies, while all the other species start with zero copies. We increase i from 1 to 11. These initial conditions correspond to a total number ranging from 3 × 1 = 3 copies to 3 × 11 = 33 copies of molecules of three species in the network. The size of the state space increases with the copy numbers. When there are 1 copy of M, MEK, and MKP3 each, the size of the state space is 14. For 5, 10, and 11 copies of M, MEK, and MKP3 each, the size increases to 8,568, 1,144,066, and 2,496,144, respectively. The computation of the state space at i = 10 and i = 11 requires 156 seconds and 589 seconds of CPU time on a Linux desktop machine, respectively.
Steady state distribution
We compute the steady state probability distributions of the microstates of the MAPK network at the initial condition of 10 copies each of M, MEK and MKP3. That is, we obtain the exact steady state density function of different microstates of all possible 1,144,066 combinations of different copy numbers of the 16 molecular species in the MAPK network. The computation is efficient. At this initial condition, the dimension of the Markovian transition matrix M is 1, 144, 066 × 1, 144, 066, with 14, 574, 406 number of non-zero elements. It takes 1,341 seconds (about 23 minutes) of CPU time to compute the steady state probability distribution on a Linux workstation.
Conclusion
Stochasticity plays important roles in molecular networks for processes involving small copy numbers of molecules. Models of molecular networks based on macroscopic reaction rates and coupled ordinary differential equations are not applicable in these cases, as they can only model high concentrations of interacting molecules with negligible fluctuations.
The stochastic nature of molecular interactions at low copy numbers can be fully characterized if the time-varying landscape probability distribution on all of the microstates of a molecular network can be computed. This is a difficult task, as the state space of the combination of the copy numbers of molecular species needs to be explicitly enumerated, the probability distribution over these microstates and changes of this distribution across many decades of time scale need to be fully computed.
In this study, we have developed an algorithm to enumerate the state space of a molecular network of small copy numbers with a buffer containing a finite number of molecules that can be synthesized. It can also be used to find all possible transitions between states, and to compute the transition rates between these states. We also demonstrate how to obtain the steady state probability distribution based on the enumerated states when it exists.
Our example of the toggle-switch network shows that this method can be used to study the rise of important network properties such as bistability. The enumeration of the full state space of the MAPK cascade network at various initial conditions demonstrate that our method can be used to study a realistic network of nontrivial size, which is more complicated than the simple networks that are often studied for full stochasticity. Although naively the state space at the initial condition of each of 11 copies of unphosphorylated, uniphosphorylated, and biphosphorylated ERK kinase might be as high as (33 + 1)^{16} = 3.19 × 10^{24}, a truly astronomical size, our method showed that the relevant space is only about 2.50 × 10^{6}, which is amenable for computation using a desktop computer.
Our method is applicable to study various carefully constructed model network systems. It complements the Monte Carlo simulation method, as it can be used to characterize the full probability landscape of networks with enumerable state space. For example, it will allow the calculation of the probabilities of the occurrence of rare and critical events. For theoretical studies, one can predefine a fixed number of net molecules that can be synthesized, and investigate the nature of the landscape probability distribution. This is similar to the studies of semi-grand canonical ensemble in statistical physics [28]. Exact characterization of probability landscape is useful, as most network studies are based on stochastic simulation, and relative little is known at the level of the full stochastic landscape probability distribution, even for simple toy systems. For example, analytical solutions to the simple toggle switch model is known only when the model parameters follows the restrictions of small noise and fast transition [8, 18]. We believe our method can be used to study well designed model systems beyond self-regulating genes and simple toggle switches, and the exact results obtained will be helpful for understanding the basic properties and design principles governing stochastic networks. A useful analogy to illustrate the utility of such model studies can be found in the field of protein folding, where a large number of studies using simple short chain HP lattice models revealed remarkable insights about how complex proteins fold [29–36].
Our method can also be applied to more realistic biological networks, such as the MAPK network model, which is a closed system according to the annotated BioModels database [27]. Such closed systems could arise when one focus on a submodule of a larger network. For the majority of realistic networks which are open systems, an important determining factor of the applicability of our method is the limit of the capacity of a buffer, which has to be greater than the maximum copy number of the net gain in protein molecules that can be synthesized. In a cell, this maximum number is determined by the life time of the cell, and the net synthesis rate of protein molecules. The latter depends on both protein synthesis and degradation rates. A simple approach is to estimate the net number of protein molecules that can be synthesized during the life time of a cell. For example, the lifespan of an E. coli cell is about 30 minutes [37]. Estimation based on the rate limiting processes of transcription initialization and elongation indicate that the protein synthesis rate ranges from 0.0077/s (for the C1 protein) [2, 38] to 0.0534/s for the Cro protein [2, 39, 40] in the lambda phage system. Their degradation rates are about 0.0007/s and 0.0025/s, respectively [2]. This suggests that a useful bound of the copy number of newly synthesized molecules for studying the lambda switch network system could be in the order of 150–200 copies under reasonable initial conditions. Naturally, the exact number will depend on the details of the chosen network model and the parameter values. For example, models of cells under stress with retarded synthetic rates may require a relatively small buffer capacity.
In this study, we have described a method to compute the steady state landscape probability distribution. Steady state distribution is of general interests when it exists, as has been shown in previous studies [17, 18]. For realistic network, another approach is to compute the time-dependent dynamic change of landscape probability distribution, using techniques such as those used in [36]. We will describe this approach in more details in future studies.
As the number of molecular species and their copy numbers increase, the state space will eventually become prohibitively large for explicit computation even with an optimal algorithm. In these cases, our method can be used to select important states and to control error bounds at a specific tolerance for developing approximation methods, an approach well demonstrated in [22].
Methods
The Algorithm
Suppose we have a model of a biological network, which contains m molecular species and can have n reactions. Given an initial condition, namely, the copy numbers of each of the m molecular species, we aim to calculate all states that the biological system can reach starting from this initial condition, under the condition that the net number of molecules that can be synthesized does not exceed a predefined limit. These states collectively constitute the state space of the network under this initial condition.
Formally, we have a model of a biological network N= ( M, R), with m + 1 number of molecular species: M= (M_{1},...M_{m+1}), and n reactions: R= {R_{1},..., R_{ n }}. Here m of the species are from the network. A buffer of predefined capacity is used to represent a pool of virtual molecules for open systems, from which synthesis reactions can generate new molecules, and to which degradation reactions can deposit molecules removed from the network. We use the m + 1-th species to represent this buffer pool. The combinations of copy numbers of all molecular species S= (c_{1},...,c_{ m }, c_{m+1}) form the microstate of the system, where c_{m+1}denotes the number of net new molecules that can still be synthesized at this state. A reaction can involve an arbitrary number (≥ 1 and ≤ m) of molecular species as reactants and/or products, with any arbitrary positive integer coefficient (i.e., arbitrary stoichiometry). Synthesis reaction is allowed to occur only if the buffer pool is not exhausted, namely, only if c_{m+1}> 0. The set of all possible states S that can be reached from an initial condition following these rules constitute the state space of the system: $\mathcal{X}$ = {S}. The set of allowed transitions is T= {t_{ ij }}. We are given with an initial condition: ${S}^{t=0}=({c}_{1}^{0},{c}_{2}^{0},\mathrm{...},{c}_{m}^{0},{c}_{m+1}^{0})$, where ${c}_{i}^{0}$ is the initial copy number of the i-th molecular species at time t = 0, and ${c}_{m+1}^{0}=B$ is the predefined buffer size. The maximum copy number of net gain in newly synthesized molecules of the system is restricted by this constant B. Our aim is to enumerate the state space $\mathcal{X}$ under this given initial condition.
The algorithm is written as Algorithm 1 (see Appendix). It performs the following computation: After initialization, we start with the initial state S^{t = 0}. We examine each reaction in turn to determine if this reaction can occur for the current state. If so, and if the buffer is not exhausted, we generate the state that this reaction leads to. If the newly generated state was not encountered before, we add it to our collection of states for the state space, and declare it as a new state. We repeat this for all new states, which is maintained by a stack data structure. This terminates when all new states are exhausted.
In this algorithm, a stack data structure is used. Description of the stack data structure can be found in computer science textbooks such as [41]. A stack is used here to store individual states. These states are "Push"ed onto the stack: If we encounter a previously unseen state, we create it and push it onto the stack so further calculations on this state can be carried out at a later stage. We use the "Pop" operation to obtain a state previously stored on the stack to carry out these calculations. In this case, we pop a state to examine what reactions can occur and what other states these reactions can lead to.
We can compute the transition coefficient {a_{i, j}} between two microstates S_{ i }and S_{ j }using Algorithm 2 (see Appendix) following the approach outlined in references [14, 18, 22]. We give further details in later sections on how this is done for the three networks studied here.
Correctness and optimality
The state space and the transitions under a given initial condition can be considered as a directed graph G= ($\mathcal{X}$, T), in which vertice are the state vectors, i.e., the set of reachable states $\mathcal{X}$, or the m + 1-tuples of copy numbers of the m + 1 molecular species, including the buffer. Edges are the set of allowed transitions T between the states, i.e., reactions connecting two state vertice. Two vertice S_{ i }∈ $\mathcal{X}$ and S_{ j }∈ $\mathcal{X}$ are connected by a directed edge t_{i, j}∈ T if and only if S_{ i }can be transformed to S_{ j }through a reaction. Any reachable state can be transformed from the initial state by one or more steps of reactions, and the directed graph G is a connected graph.
Our algorithm implicitly generates this graph G. Because the set of reactions R is finite, G has a finite tree-width at any finite steps away from the initial condition. Assume the algorithm will not terminate in finite steps. Since in this algorithm each state is only visited no more than twice, G must have an unlimited depth. That is, there must exist a path p in the graph G that starts from the initial state and extends to infinite. Therefore p must contain an infinite number of different states. This is impossible for any given initial condition, as each molecular species has a limited initial copy number, and the size of the buffer limits the number of new molecules that can be synthesized in open systems. The algorithm therefore must terminate.
This algorithm gives correct answers, assuming that the newly synthesized molecules does not exceed the predefined buffer capacity. This is because all states visited in the algorithm can be reached from the initial condition, and all visited states is actually reached as each is brought to by a chemical reaction. In addition, all reachable states will be visited, as the algorithm test at each state all possible reactions, and will only terminates when all new states are exhausted. It is easy to see all possible transitions between states will be recorded.
The time complexity of our algorithm is optimal. Since only unseen state will be pushed onto the stack, every state is pushed and popped at most once, and each state will be generated/visited at most twice before it is popped from the stack. As access to each state and to push/pop operations take O(1) time, the total time required for the stack operations is O(|$\mathcal{X}$|). As the algorithm examines each of the n reactions for each reached state, the complexity of total time required is O(n|$\mathcal{X}$|), where n is usually a modest constant (e.g. < 50). Based on the same argument, it is also easy to see that the algorithm is optimal in storage, as only valid states and valid transitions are recorded.
Computing mean and steady state probability distribution
We can calculate the expected landscape probability distribution over the microstates, namely, the exact mean density function of different microstates of copy numbers in the network. It is the same as the steady state probability distribution function if the steady state exists. Instead of calculating the time trajectories of changes in the probability distribution and wait until it reaches equilibrium, we use a simpler approach applicable to networks in which a steady state exists. Following Kachalo et al [36], we obtain the Markovian state transition matrix M from the reaction rate matrix A: M= I+ A·Δt, where I is the identity matrix, and Δt is the discrete time unit and is chosen to be 1. The probability distribution function P of the microstates can be obtained by solving the equation P= MP. The calculation of the steady state distribution P is not sensitive to the precise choice of the discrete time unit Δt. The steady state distribution corresponds to the eigenvector of M with eigenvalue of 1. We use the Arnoldi method implemented in the software ARPACK to compute the steady state distribution P[42].
Computing transition coefficients
The transition coefficient between different states connected by a reaction is calculated by multiplying the intrinsic rate of this reaction with the reaction order dependent combination number of copies of reactants in the "before" state [14]. We provide more details using examples from the three networks.
Self-regulating gene
enables the transition of the system from the microstate i to j. This reaction denotes the degradation of the protein molecule at an intrinsic rate of d. The stoichiometry of this reaction dictates that the copy number of protein n_{p, j}in the "after" state j is one less than the copy number n_{p, i}in the "before" state i. From the reaction formula, the transition coefficient a_{i, j}for the matrix A is calculated as:
a_{i, j}= d·n_{p, i}.
Recall that since a microstate is uniquely determined by the combination of copy numbers of all molecular species, n_{p, i}therefore is known as a state attribute.
the transition coefficient connecting the "before" state i to the "after" state j can be computed as:
a_{i, j}= b·n_{p, i}·n_{g, i},
where b is the intrinsic reaction rate, n_{p, i}is the protein copy number at state i, and n_{g, i}is the copy number of gene in state i, which is 1, as we assume there is only one copy of the gene in this network model.
we have a_{ ij }= s_{1}·n_{g, i}.
We have described how to compute the transition coefficient for all reactions in the represser gene network. In Algorithm 2, we can compute the transition coefficient a_{i, j}based on the formula of the reaction leading from state i to state j.
Toggle switch
the transition coefficient a_{i, j}can be computed as
a_{i, j}= b·n_{gB, i}·n_{pA, i}·(n_{pA, i}- 1)/2,
where b is the intrinsic reaction rate, n_{pA, i}is the copy number of protein A in state i, and n_{gB, i}is the copy number of unbound gene B, which takes the value of 0 or 1. For this third order reaction, the number of possible ways of choosing two protein molecules from n_{pA, i}copies is $\left(\begin{array}{c}{n}_{pA,i}\\ 2\end{array}\right)={n}_{pA,i}\cdot ({n}_{pA,i}-1)/2$. Transition coefficients for the other reactions in this network can be computed similarly following this reaction and the reactions described earlier for the represser gene network.
MAPK network
If the "before" state i is transformed to the "after" state j by one step of this reaction, the corresponding transition coefficient a_{i, j}can be computed as
a_{i, j}= b_{14}·n_{M, i}·n_{MKP 3, i}
where b_{14} is the intrinsic reaction rate, n_{M, i}and n_{MKP 3, i}are the copy numbers of M and MKP3 molecules in state i, respectively. The other transition coefficients in this network can be computed similarly using the intrinsic reaction rates given in Fig 5 and the copy numbers of reactants determined by the "before" state i.
Appendix
Algorithm 1 State Enumerator(M, R, B)
Network model: N← {M, R};
Initial condition: ${S}^{t=0}\leftarrow \{{c}_{1}^{0},{c}_{2}^{0},\mathrm{...},{c}_{m}^{0}\}$; Set the value of buffer capacity: ${c}_{m+1}^{0}\leftarrow B$;
Initialize the state space and the set of transitions: $\mathcal{X}$ ← ∅; T← ∅;
Stack ST ← ∅; Push(ST, S^{t = 0}); StateGenerated ← FALSE
while ST ≠ ∅ do
S_{ i }← Pop (ST);
for k = 1 to n do
if reaction R_{ k }occurs under condition S_{ i }then
if reaction R_{ k }is a synthetic reaction and generates u_{ k }new molecules then
c_{m+1}← c_{ m+1 }- u_{ k }
if c_{m+1}≥ 0 then
Generate state S(i, R_{ k }) that is reached by following reaction R_{ k }from S_{ i };
StateGenerated ← TRUE
end if
else
if reaction R_{ k }is a degradation reaction and breaks down u_{ k }molecules then
c_{m+1}← c_{m+1}+ u_{ k }
end if
Generate state S(i, R_{ k }) that is reached by following reaction R_{ k }from S_{ i };
StateGenerated ← TRUE
end if
if (StateGenerated = TRUE) and (S(i, R_{ k }) ∉ $\mathcal{X}$) then
$\mathcal{X}$ ← $\mathcal{X}$ ∪ S(i, R_{ k });
Push(ST, S(i, R_{ k }));
T← T∪ ${t}_{{S}_{i},S(i,{R}_{k})}$;
a_{i, j}← Transition Coefficient(S_{ i }, S(i, R_{ k }), R_{ k })
end if
end if
end for
end while
Output $\mathcal{X}$, T and A= {a_{i, j}}.
Algorithm 2 Transition Coefficient(S_{ i }, S_{ j }, R_{ k })
Read in reaction rate parameters for R_{ k }
Retrieve the copy numbers of molecular species occurring in the reaction formula of R_{ k }from the state vector S_{ i }
Compute the combination copy numbers of each reactant molecular species
Compute transition coefficient a_{i, j}based on the reaction rate parameters for R_{ k }, and the combination copy numbers.
Output a_{i, j}.
Declarations
Acknowledgements
We thank Drs. Bhaskar DasGupta, Ming Lin, Michael Samoilov, and Hsiao-Mei Lu for helpful discussions. This work is supported by a phase II 985 Project (Sub-Project:T226208001) at Shanghai Jiao Tong University, Shanghai, China.
Authors’ Affiliations
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