 Research article
 Open Access
 Published:
A multiscale approximation in a heat shock response model of E. coli
BMC Systems Biology volume 6, Article number: 143 (2012)
Abstract
Background
A heat shock response model of Escherichia coli developed by Srivastava, Peterson, and Bentley (2001) has multiscale nature due to its species numbers and reaction rate constants varying over wide ranges. Applying the method of separation of timescales and model reduction for stochastic reaction networks extended by Kang and Kurtz (2012), we approximate the chemical network in the heat shock response model.
Results
Scaling the species numbers and the rate constants by powers of the scaling parameter, we embed the model into a oneparameter family of models, each of which is a continuoustime Markov chain. Choosing an appropriate set of scaling exponents for the species numbers and for the rate constants satisfying balance conditions, the behavior of the full network in the time scales of interest is approximated by limiting models in three time scales. Due to the subset of species whose numbers are either approximated as constants or are averaged in terms of other species numbers, the limiting models are located on lower dimensional spaces than the full model and have a simpler structure than the full model does.
Conclusions
The goal of this paper is to illustrate how to apply the multiscale approximation method to the biological model with significant complexity. We applied the method to the heat shock response model involving 9 species and 18 reactions and derived simplified models in three time scales which capture the dynamics of the full model. Convergence of the scaled species numbers to their limit is obtained and errors between the scaled species numbers and their limit are estimated using the central limit theorem.
Background
Stochasticity may play an important role in biochemical systems. For example, stochasticity may be beneficial to give variability in gene expression, to produce population heterogeneity, and to adjust or respond to fluctuations in environment [1]. We are interested in local dynamics of biochemical networks involving some species with a small number of molecules so that the system is assumed to be wellmixed and relative fluctuations of small species numbers may play a role in the system dynamics.
The conventional stochastic model for the wellstirred biochemical network is based on the chemical master equation. The chemical master equation governs the evolution of the probability density of species numbers and is expressed as the balanced equation between influx and outflux of the probability density. When the biochemical network involves many species or bimolecular reactions, it is rarely possible to obtain an exact solution of the master equation in a closed form. Instead of searching for the solution of the master equation, stochastic simulation algorithms are used to obtain the temporal evolution of the species numbers. For example, Gillespie’s Stochastic Simulation Algorithm (SSA, or the direct method) is well known [2, 3] and provides a realization of the exact trajectory of the sample path for the species numbers. As the biochemical network has more species and reactions, SSA becomes computationally expensive and more efficient algorithms were suggested by many authors [4–6]. The detailed review of stochastic simulation methods, stochastic approximations, and hybrid simulation methods is given in [7]. For models with wellseparated time scales, numerous authors suggested stochastic simulation algorithms for biochemical reaction networks by assuming that “fast” subnetworks have reached a “partial equilibrium” [6] or a “quasisteady state” [4]. Using these assumptions, the approximate stochastic simulation algorithms involve a reduced number of species or reactions.
On the other hand, Ball et al. [8] described the state of the biochemical reaction network in the wellstirred system directly using stochastic equations for species numbers, and suggested an approximation of the reaction network via limiting models derived using different scalings for the species numbers and for the reaction rate constants. Kang and Kurtz [9] extended this multiscale approximation method and gave a systematic way to obtain limiting models in the time scales of interest. Conditions are given to help identify appropriate values for a set of scaling exponents which determine the time scale of each species and reaction. Using this method, nonstationary behavior of biochemical systems can be analyzed. Moreover, application of the method is flexible in the sense that the method does not require the exact parameter values but gives approximations valid for a range of parameter values. More recently, Crude et al. [10] also proposed a reduction method to derive simplified models with preserving stochastic properties and with key parameters using averaging and hybrid simplification.
The multiscale approximation method in [9] requires consideration of magnitude of both species numbers and rate constants of the reactions involving the corresponding species. When a moderately fast reaction involves two species, one with a small number of molecules and the other with a large number of molecules, the effects of this reaction on these species are different. Net molecule changes of species with large numbers due to the reaction is less noticeable than those of species with small numbers. Therefore, though the same reaction governs these species, their time scales may be different from each other. Letting N_{0} be a fixed constant and choosing a large value for N_{0}, for example N_{0}=100, we express magnitudes of species numbers and reaction rate constants in terms of powers of N_{0} with different scaling exponents. For instance, 1 to 10molecules are expressed as 1\times {N}_{0}^{0} to 10\times {N}_{0}^{0}\text{molecules}, 500 to 800molecules are rewritten as 5×N_{0} to 8×N_{0}molecules, and 0.0002 sec becomes 2\times {N}_{0}^{2}sec. Assuming N_{0} is large, we replace N_{0}by a large parameter N and stochastic equations for species numbers are expressed in terms of N. Then, N is an analogue of 1/ε where ε is a small parameter in perturbation theory.
A specific time scale of interest is expressed in terms of a power of N, and its exponent contributes to reaction rates due to change of variables in time. For each species (or linear combination of species), we compare a power of N for the species number and those for reaction rates involving this species. Consider a case when the power for the species number is larger than those for the rates of all reactions where the species is involved. Then net molecule changes due to the reactions are not large enough to be noticeable in this time scale, and the species number is approximated as constant. Next, consider a case when the power for the species number is smaller than those for some reaction rates involving the species. In this case, the species number fluctuates very rapidly due to the fast reactions in this time scale, and the averaged behavior of the species number can be described in terms of other species numbers. The method of averaging is similar to approximation of one variable in terms of others using a quasisteady state assumption. Last, when the power for the species number is equal to those for the rates of reactions where the species is involved, the scaled species number is approximated by a nondegenerate limit describing nonstationary behavior of the species number in the specific time scale of interest. The limit could be described in various kinds of variables: a continuous time Markov chain, a deterministic model given by a system of ordinary differential equations, or a hybrid model with both discrete and continuous variables. Since some of the scaled species numbers are approximated as constants or the averaged behavior of some species numbers is expressed in terms of other variables, dimension of species in the approximation of the biochemical network is reduced.
In the multiscale approximation method, scaling exponents for species numbers and for reaction rate constants are not uniquely determined, since the choice of values for the exponents is flexible. For example, 0.005 sec can be expressed as 0.5\times {N}_{0}^{1} or 5\times {N}_{0}^{1.5} when N_{0}=100. The goal in this method is to find an appropriate set of scaling exponents to obtain a nondegenerate limit of the scaled species numbers. Orders of magnitude of species numbers in the propensities affect reaction rates, and reaction rates contribute to determining rates of net molecule changes of the species involved in the reactions. Since species numbers and reaction rates interact, it is not easy to determine scaling exponents for all species numbers and reaction rate constants so that the limits of the scaled species numbers become balanced.
Kang and Kurtz [9] introduced balance conditions for the scaling exponents, which help to determine values for a set of exponents. The key idea in these conditions is that for each species (or linear combination of species) the maximum of scaling exponents in the rates of the reactions where this species is produced should be the same as that in the rates of the reactions where this species is consumed, i.e. maximal production and consumption rates of the species should be balanced in the order of magnitude. In case the maximums of scaling exponents for productions and consumptions are not balanced for some species, an increase or decrease of the scaled species number can be described by its limit during a certain time period. However after this time period, the scaled species number will either become zero or blow up to infinity. Therefore, if some of the scaled species numbers are not balanced due to a difference between orders of magnitude of production and consumption rates, the chosen scaling is valid up to a certain time scale. After this time scale, we need to choose different values for scaling exponents. In each time scale of interest we derive a limiting model including a subset of species and reactions, which is used to approximate the state of the full reaction network. The multiscale approximation method is applicable in case some of reaction rates are not known accurately, since the chosen scaling is applicable in some ranges of the parameters. Therefore, based on the behavior of the limiting models, we may be able to estimate behavior for a range of parameter values without performing a huge number of stochastic simulations.
The paper [9] included several simple examples of biochemical networks involving two to four species, and derived limiting models in each time scale of interest. To apply this method, more scaling exponents must be determined as the biochemical network involves more species or reactions. Therefore, it is challenging to apply the method to complex biochemical systems and to determine appropriate values for scaling exponents so that the corresponding limiting models preserve important dynamical features of the full system. One of the goals of this paper is to illustrate how to apply this method to an example with significant complexity. In this paper, using a significantly complicated biochemical network, we derive limiting models, show convergence of the scaled species numbers to their limit, and estimate the error analytically between the scaled species numbers and their limit. We analyze a heat shock response model of Escherichia coli (E. coli) developed by Srivastava, Peterson, and Bentley in [11]. The model involves 9 species and 18 reactions with significant complexity as shown in Figure 1, and it has various time scales due to wide ranges of species numbers and reaction rate constants. Because of various scales involved, this model has been used as an example to show accuracy of the stochastic simulation algorithms which are developed to increase computational efficiency using the multiscale nature of the chemical reaction network [12, 13]. Another version of a heat shock response model of E. coli is studied in [6] using an accelerated SSA that also exploits the multiscale nature of the system.
Applying the multiscale approximation method to the heat shock response model of E. coli, we derive limiting models in three time scales of our interests, which approximate the full network given in Figure 1. Denote ∅ as species we are not interested in. Let S_{ i } represent the i th species and S_{23}be addition of species S_{2}and S_{3}. A→B denotes a reaction where one molecule of species A is converted to one molecule of species B. In the early stage of time period of order 1 sec, we obtain the following reduced network:
The reduced network in the early stage has very simple structure without any bimolecular reactions, and all reactions involved are either production from a source or conversion. Moreover, the reduced network is well separated into two due to independence of S_{8}from S_{2}and S_{3}.
In the medium stage of time period of order 100 sec, the full network is reduced to
where a species over the arrow accelerates or inhibits the corresponding reaction. The reaction does not change this species number, but the propensity of the corresponding reaction is a function of this species number. In this time scale, conversion between S_{2} and S_{3} occurs very frequently and S_{2}and S_{3}play a role as a single “virtual” species rather than separate species. The species numbers of S_{23} and S_{8}are described as two independent birth processes and the species number of S_{7} is governed by conversion. In this time scale, the species number of S_{8}is normalized and treated as a continuous variable. The interesting thing is that the behavior of the species S_{8} which rapidly increases in time is well approximated in both first and second time scales.
In the late stage of time period of order 10,000 sec, we get a reduced network with more species involved than those in the previous time scales. However, the reduced network is still much simpler than the full network in Figure 1. At this time scale, we get
As we see in Figure 1, the full network involves reactions with more than two reactants or products. However, all reactions in the reduced network at the times of order 10,000 sec consist of either production or degradation of each species, though most of the species (6 species out of 9) are involved in the reduced model. As in the medium stage of time period, S_{2}and S_{3}play a role as a single species. In the early and medium stages of time period propensities are in a form following the law of mass action, while in the late stage of time period the propensity for degradation of S_{23} is a nonlinear function of the species numbers similar to the reaction rate appearing in the MichaelisMenten approximation for an enzyme reaction. The nonlinear function involves the species numbers of S_{23}, S_{8}, and S_{9}, which come from averaging of the species numbers of S_{2}and S_{6}which fluctuate rapidly in the third time scale. Similarly, the propensity of catalytic degradation of S_{8} is not proportional to the number of molecules of S_{8}.
In the late stage of time period of order 10,000 sec, we study the error between the scaled species numbers and their limit analytically using the central limit theorem derived in [14] and show that the error is of order 10^{−1}.
Methods
In the next several sections, we apply the multiscale approximation to the heat shock response model of E. coli and derive the limiting models. The multiscale approximation method is described in terms of the following steps so that the method can be applied to the general cases.

1.
Write a chemical reaction network involving s _{0}species and r _{0} reactions in the form of
\sum _{i=1}^{{s}_{0}}{\nu}_{\mathit{ik}}{S}_{i}\to \sum _{i=1}^{{s}_{0}}{\nu}_{\mathit{ik}}^{\prime}{S}_{i},\phantom{\rule{2em}{0ex}}k=1,\cdots \phantom{\rule{0.3em}{0ex}},{r}_{0},where ν_{ ik } and {\nu}_{\mathit{ik}}^{\prime} are nonnegative integers. Rearrange the reactions so that the reaction rate constants are decreasing monotonically as k gets large.

2.
Derive a system of stochastic equations for species numbers.

(a)
Letting X _{ i }(t) be the number of molecules of species S _{ i }at time t, the corresponding stochastic equation is
{X}_{i}\left(t\right)={X}_{i}\left(0\right)+\sum _{k=1}^{{r}_{0}}{R}_{k}^{t}\left({\lambda}_{k}\right(X\left)\right)({\nu}_{\mathit{ik}}^{\prime}{\nu}_{\mathit{ik}}),\phantom{\rule{1em}{0ex}}i=1,\cdots {s}_{0},where{R}_{k}^{t}(\xb7) counts the number of times that the k th reaction occurs up to time t.

(b)
λ _{ k }(x) is determined by a stochastic version of mass action kinetics, and is expressed as a product of the rate constant and the numbers of molecules of reactants. If the k th reaction is secondorder (\sum _{i=1}^{{s}_{0}}{\nu}_{\mathit{ik}}=2) with different types of reactants, {\lambda}_{k}\left(x\right)={\kappa}_{k}^{\prime}{x}_{p}{x}_{q}. When the reactants are two molecules of the same species, {\lambda}_{k}\left(x\right)={\kappa}_{k}^{\prime}{x}_{p}({x}_{p}1).

(a)

3.
Derive a system of stochastic equations for the normalized species numbers after a time change, Z ^{N,γ}(t).

(a)
In the equation for X _{ i }(t) obtained in Step 2 (a), replace X _{ i }by {Z}_{i}^{N,\gamma} and divide reaction terms by N ^{α} _{ i }. In the k th reaction term, put N ^{γ + ρ} _{ k } in the propensity and replace λ _{ k }(X) by {\widehat{\lambda}}_{k}\left({Z}^{N,\gamma}\right). Then, we have

(b)
In the equation in Step 3 (a), {\rho}_{k}={\beta}_{k}+\sum _{j=1}^{{s}_{0}}{\alpha}_{j}{\nu}_{\mathit{jk}}.

(c)
In the most reactions, {\widehat{\lambda}}_{k} is obtained by replacing {\kappa}_{k}^{\prime} by κ _{ k }in λ _{ k }. In case the k th reaction is secondorder with reactants of the same species, {\lambda}_{k}\left(x\right)={\kappa}_{k}^{\prime}{x}_{p}({x}_{p}1) is replaced by {\widehat{\lambda}}_{k}\left(z\right)={\kappa}_{k}{z}_{p}({z}_{p}{N}^{{\alpha}_{p}}).
\begin{array}{l}{Z}_{i}^{N,\gamma}\left(t\right)={Z}_{i}^{N,\gamma}\left(0\right)+{N}^{{\alpha}_{i}}\sum _{k=1}^{{r}_{0}}{R}_{k}^{t}\left({N}^{\gamma +{\rho}_{k}}{\widehat{\lambda}}_{k}\right({Z}^{N,\gamma}\left)\right)\\ \phantom{\rule{5em}{0ex}}\times ({\nu}_{\mathit{ik}}^{\prime}{\nu}_{\mathit{ik}}),\phantom{\rule{1em}{0ex}}i=1,\cdots {s}_{0}.\end{array}

(a)

4.
Write a set of species balance equations and their timescale constraints.

(a)
Define {\Gamma}_{i}^{+} and {\Gamma}_{i}^{} as subsets of reactions where the species number of S _{ i }increases or decreases every time the reaction occurs. Comparing ρ _{ k }’s for k\in {\Gamma}_{i}^{+} and those for k\in {\Gamma}_{i}^{}, set the balance equations as

(b)
Timescale constraints are given as
\underset{k\in {\Gamma}_{i}^{+}}{max}{\rho}_{k}=\underset{k\in {\Gamma}_{i}^{}}{max}{\rho}_{k},\phantom{\rule{2em}{0ex}}i=1,\cdots \phantom{\rule{0.3em}{0ex}},{s}_{0}.\gamma \le \underset{k\in {\Gamma}_{i}^{+}\cup {\Gamma}_{i}^{}}{max}{\rho}_{k},\phantom{\rule{2em}{0ex}}i=1,\cdots \phantom{\rule{0.3em}{0ex}},{s}_{0}.

(a)

5.
Find a minimum set of linear combinations of species whose maximum of collective production (or consumption) rates may be different from that of one of any species. We construct a minimum set of linear combinations of species by selecting a linear combination of species if any reaction term involving the species consisting of the linear combination is canceled in the equation for the linear combination of species.

6.
For each selected linear combination of species, write a collective species balance equation and its timescale constraint. They are obtained similarly to the ones in Step 4 using subsets of reactions where the number of molecules of linear combinations of species either increases or decreases instead of using {\Gamma}_{i}^{+} and {\Gamma}_{i}^{}.

7.
Select a large value for N _{0}and choose an appropriate set of α _{ i }’s and β _{ k }’s so that

(a)
the species number X _{ i }and the reaction rate constant {\kappa}_{k}^{\prime} are approximately of orders {N}_{0}^{{\alpha}_{i}} and {N}_{0}^{{\beta}_{k}};

(b)
the normalized species number {Z}_{i}^{N,\gamma} and the scaled reaction rate constant κ _{ k }are of order 1;

(c)
most of the balance equations obtained in Steps 4 and 6 are satisfied;

(d)
β _{ k }’s are monotone decreasing among each class of reactions which have the same number of molecules of reactants.

(a)

8.
Plugging the chosen values for α _{ i }’s and β _{ k }’s in the timescale constraints obtained in Steps 4 and 6, compute an upper bound (denoted as γ _{0}) for a timescale exponent. Then, the chosen set of exponents α _{ i }’s and β _{ k }’s can be used for γ satisfying γ≤γ _{0}. For γ>γ _{0}, select another set of exponents α _{ i }’s and β _{ k }’s using Steps 7 and 8.

9.
Using each set of values for α _{ i }’s and β _{ k }’s, identify a natural time scale exponent of each species (denoted as γ _{ i } for species S _{ i }) so that γ _{ i } satisfies
\underset{k\in {\Gamma}_{i}^{+}\cup {\Gamma}_{i}^{}}{max}({\gamma}_{i}+{\rho}_{k})\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\alpha}_{i},\phantom{\rule{2em}{0ex}}i=1,\cdots \phantom{\rule{0.3em}{0ex}},{s}_{0}.We collect γ_{ i }’s with the same values, whose species are in the same time scales in the approximation.

10.
Modify α _{ i }’s and β _{ k }’s so that the conditions in Step 7 are satisfied and that γ _{ i }’s are divided into appropriate number of values, which gives the number of time scales, N ^{γ}=N ^{γ} _{ i }, we are interested in.

11.
For each chosen γ, derive a limiting equation for each species S _{ i }with γ _{ i }=γ. Using the stochastic equation obtained in Step 3 (a), we let N go to infinity.

(a)
For k\in {\Gamma}_{i}^{+}\cup {\Gamma}_{i}^{}, the k th reaction term converges to zero if α _{ i }>γ + ρ _{ k }.

(b)
If α _{ i }=γ + ρ _{ k }, the k th reaction term appears as a limit in the limiting equation. The limit of the k th reaction term is discrete if α _{ i }=0, while it is a continuous variable with the limit of its propensity if α _{ i }>0.

(c)
There is no k satisfying α _{ i }<γ + ρ _{ k }in the equation for species S _{ i }with γ=γ _{ i }due to the definition of γ _{ i }given in Step 9.

(a)

12.
In the limiting equation for each species S _{ i }with γ _{ i }=γ, we approximate propensities in the reaction terms. Suppose that the normalized species number for S _{ j }appears in the propensities.

(a)
If γ _{ j }>γ, the limit of the normalized species number for S _{ j }is its initial value.

(b)
If γ _{ j }=γ, the limit of the normalized species number for S _{ j }appears as a variable in the propensities in the limiting equation.

(c)
If γ _{ j }<γ, the limit of the normalized species number for S _{ j }is expressed as a function of the limits of the normalized species numbers for S _{ i }with γ _{ i }=γ. The function for S _{ j }is obtained by dividing the equation for S _{ j }by {N}^{\underset{k\in {\Gamma}_{j}^{+}\cup {\Gamma}_{j}^{}}{max}(\gamma +{\rho}_{k}){\alpha}_{j}} and letting N go to infinity.

(a)

13.
If a limiting model is not closed, consider limiting equations for some linear combinations of species selected in Step 5 whose natural time scale exponents are equal to the chosen γ.
The method for multiscale approximation described above can be applied to general chemical reaction networks containing different scales in species numbers and reaction rate constants. We can apply the method in case the rates of chemical reactions are determined by law of mass action and when there is no species whose number is either zero or infinity at all times. As given in [9], in the reaction network involving ∅→S_{1}, ∅→S_{2}, ∅→S_{3}, S_{1} + S_{2}→∅, and S_{1} + S_{3}→∅, convergence of the limit for the scaled species numbers may not be guaranteed at some time scales. Suppose that production rate of S_{1} is larger than that of S_{2}but with the same order of magnitude, and that production rate of S_{3} is much smaller than those of S_{1}and S_{2}. Then, X_{1}(t) may blow up to infinity and X_{2}(t) may go to zero at some time scales. In this case, the method is not applicable.
Results and discussion
Model description
We analyze a heat shock response model of E. coli developed by Srivastava, Peterson, and Bentley [11]. The heat shock response model gives a simplified mechanism occurring in the E. coli to respond to high temperature. Heat causes unfolding, misfolding, or aggregation of proteins, and cells overcome the heat stress by producing heat shock proteins, which refold or degrade denatured proteins. In E. coli, σ^{32}factors play an important role in recovery from the stress under the high temperature. σ^{32}factors catalyze production of the heat shock proteins such as chaperon proteins and other proteases. In this model, J denotes a chaperon complex, FtsH represents a σ^{32}regulated stress protein, and GroEL is a σ^{32}mediated stress response protein.
σ^{32} factors are in three different forms, free σ^{32}protein, σ^{32} combined with RNA polymerase (E σ^{32}), and σ^{32} combined with a chaperon complex (σ^{32}J). Under the normal situation without stress, most of the σ^{32} factors combine with chaperon complexes and form σ^{32}J. A chaperon complex J keeps σ^{32}factors in an inactive form, and σ^{32}factors can directly respond to the stress by changing into different forms. When there exist σ^{32}factors combined with chaperon complexes, FtsH catalyzes degradation of σ^{32} factors. Thus, if enough σ^{32}regulated stress proteins are produced, σ^{32}factors are degraded.
Not only σ^{32}factors, but recombinant proteins also require chaperon complexes to form a complex so that denatured protein can be fixed. Therefore, σ^{32}factors and recombinant proteins compete to bind chaperon complexes, and different levels of binding affinity of recombinant proteins to chaperon complexes change the evolution of the system state. In the model, we assume that σ^{32} factors and recombinant proteins have the same affinity to bind to chaperon complexes. The system is sensitive to the amount and forms of σ^{32} factors: a small decrease of σ^{32}factors causes a large reduction of production of chaperon complexes and σ^{32}regulated stress proteins, and the ratio of three different forms of σ^{32}factors determines system dynamics in the stress response [11]. The total initial number of molecules of σ^{32} factors in each cell is small [11] (also see initial values for S_{2}, S_{3}, and S_{7} which are 1, 1, and 7 in Table 1), and the stochastic model is appropriate to be considered.
The model involves 9 species and 18 reactions. Denote s_{0} as the number of species and r_{0} as the number of reactions. Let X(t) be a state vector whose i th component represents the number of molecules of species S_{ i } at time t for i=1,⋯,s_{0}. Define a random process which counts the number of times that the k th reaction occurs by time t as
where λ_{ k }(X) is the propensity of the k th reaction and the Y_{ k }’s are independent unit Poisson processes. Therefore, {R}_{k}^{t}(\xb7) is a nonnegative integervalued random process increasing by 1. As λ_{ k }(·) gets large, the moment when {R}_{k}^{t}\left({\lambda}_{k}\right(\xb7\left)\right) increases becomes more frequent. Let ν_{ ik }({\nu}_{\mathit{ik}}^{\prime}) be the number of molecules of S_{ i } that are consumed (produced) in the k th reaction. Define ν_{ k }({\nu}_{k}^{\prime}) as an s_{0}dimensional vector whose i th component is ν_{ ik }({\nu}_{\mathit{ik}}^{\prime}). Then, X(t) is given as
That is, species numbers at time t are expressed in terms of their initial values and sum of the number of times that each reaction occurs multiplied by net molecule changes in the corresponding reaction. In our model, the system of equations are derived using a set of reactions in Table 2 as:
{\kappa}_{k}^{\prime} represents the stochastic reaction rate constant for the k th reaction, and their values from [11] are given in Table 3.
We derive the limiting models in three time scales, which approximate a full network in a certain time period involving a subset of species and reactions. In what follows, {Z}_{i}^{\gamma} is a limit of the scaled species number of S_{ i } at some time scales depending on γ, and as γ gets larger the times are in the later stage. Note that the exponent γ in {Z}_{i}^{\gamma} does not imply (Z_{ i })^{γ} but it shows dependence of {Z}_{i}^{\gamma} on γ. Let κ_{ k } be a scaled reaction rate constant for the k th reaction. In the first time scale (when the times are in the early stage), the subnetwork governed by
approximates the network when the times are of order 1 sec. Denote {Z}_{23}^{1} as the limit of the addition of the scaled species numbers for S_{2}and S_{3}. In the second time scale (when the times are in the medium stage), the subnetwork governed by
approximates the network at the times of order 100 sec. In the third time scale, set the limit of the averaged scaled species numbers of fastfluctuating species S_{2}, S_{3}, and S_{6} as
When the times are in a late stage, the subnetwork governed by
approximates the network at the times of order 10,000 sec. Detailed derivation is given in the later sections. Note that it is possible to identify different numbers of time scales depending on the scaling of the species numbers and reaction rate constants. In the heat shock response model of E. coli, it is possible to obtain approximate models with two or four time scales. However, if the number of time scales are too many, the limiting model in each time scale may involve one species and a few number of reactions and the model in this case may not be interesting to consider.
Derivation of the scaled models
The stochastic equations given in Equations (2) describe temporal evolution of the species numbers. For example, the equations for species S_{2}and S_{3} are
In Equation (6), species numbers of S_{2}and S_{3} are determined by the times when reactions occur and by the number of times that reactions happen. On the other hand, reaction time and frequency are determined by propensities which are some functions of species numbers. Therefore, reaction rates and species numbers interact one another. Reaction rates vary from O(10^{−11}) to O(1) as we see in Table 3, and species numbers in this model are from O(1) to O(10^{4}) as we see later in the simulation of the full network. We express each species number and rate constant in terms of powers of a common number with different weights on exponents. Define N_{0}=100 as a fixed unitless constant used to express the magnitude of the species numbers and the reaction rate constants. Define α_{ i } for i=1,⋯,s_{0} and β_{ k } for k=1,⋯,r_{0} as the scaling exponent for species S_{ i } and for the reaction rate constant {\kappa}_{k}^{\prime}. We express the reaction rate constants in a form of {N}_{0}^{{\beta}_{k}}{\kappa}_{k} where κ_{ k } is of order 1 and is determined so that {\kappa}_{k}^{\prime}={N}_{0}^{{\beta}_{k}}{\kappa}_{k}. For example, we have {\kappa}_{6}^{\prime}=4.88\times 1{0}^{3} and we can choose β_{6}=−1 so that the reaction rate is expressed as {\kappa}_{6}^{\prime}=0.488\times {N}_{0}^{{\beta}_{6}}. Assuming that N_{0} is large, we replace N_{0} by N and express the process as {X}_{i}^{N}\left(t\right) to show dependence of the species numbers on N. Note that {X^{N}(t)} is a family of processes depending on N and {X}_{i}^{N}\left(t\right)={X}_{i}\left(t\right) when N=N_{0}. Then, the equation for {X}_{3}^{N} is given as
where {X}_{3}^{N}\left(0\right) is defined later so that {X}_{3}^{N}\left(0\right)={X}_{3}\left(0\right) when N=N_{0}. Since the numbers of molecules of species are in different orders of magnitude, we scale the number of molecules of the i th species by N^{α}_{ i } and set a normalized species number as
The i th species number may have different orders of magnitude at different times so α_{ i } may have different values for different time scales. Now, we set the initial values as
so that {X}_{i}^{{N}_{0}}\left(0\right)={X}_{i}\left(0\right) and \underset{N\to \infty}{lim}{Z}_{i}^{N}\left(0\right)=\underset{N\to \infty}{lim}{N}^{{\alpha}_{i}}{X}_{i}^{N}\left(0\right)={N}_{0}^{{\alpha}_{i}}{X}_{i}\left(0\right).
Next, we scale the propensities of reactions by replacing {X}_{i}^{N} by {N}^{{\alpha}_{i}}{Z}_{i}^{N} and replacing {\kappa}_{k}^{\prime} by N^{β}_{ k }κ_{ k }. For example, consider the 9th reaction term in (6a). Replacing {\kappa}_{9}^{\prime} by N^{β}_{9}κ_{9}, {X}_{2}^{N} by {N}^{{\alpha}_{2}}{Z}_{2}^{N}, and {X}_{6}^{N} by {N}^{{\alpha}_{6}}{Z}_{6}^{N}, the 9th reaction term becomes
For simplicity, set ρ_{9}=β_{9} + α_{2} + α_{6} and define a scaling exponent in the propensity of the k th reaction term as
where \alpha ={({\alpha}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{\alpha}_{{s}_{0}})}^{T} and {\nu}_{k}={({\nu}_{1k},\cdots \phantom{\rule{0.3em}{0ex}},{\nu}_{{s}_{0}k})}^{T}. Here, ν_{ ik } gives the number of molecules of species S_{ i }consumed in the k th reaction. Then, (8) is rewritten as
Dividing (6a) by N^{α}_{2} and (6b) by N^{α}_{3} and scaling the propensities, we get
For each reaction, ρ_{ k }is given in terms of α_{ i }and β_{ k } in the Additional file 1: Table S1.
We are interested in dynamics of species numbers {Z}_{2}^{N}\left(t\right) and {Z}_{3}^{N}\left(t\right) in various stages of time period. In the early stage of time period, normalized species numbers of S_{2} and S_{3} are very close to their scaled initial values, since these species numbers have not changed yet. In the medium stage of time period, the normalized species numbers of S_{2}and S_{3} are asymptotically equal to nonconstant limits. In the late stage of time period, the normalized species numbers of S_{2} and S_{3}fluctuate very rapidly and their averaged behavior is captured in terms of some function of other species numbers.
We want to express the time scale of each species in terms of power of N. First, we express order of magnitude of a specific time period of interest as a power of N with a time scale exponent γ. Applying a time change by replacing t by N^{γ}t in {Z}_{i}^{N}\left(t\right), we define a variable for the normalized species numbers after a time change as
Then, {Z}_{i}^{N,\gamma}\left(t\right) gives a normalized species number at the times of order N^{γ}. A natural time scale of S_{ i }is the time when {Z}_{i}^{N,\gamma}\left(t\right) has a nonzero finite limit which is not constant and of order 1.
Changing a time variable by replacing t by N^{γ}t in (9a) and (9b), the normalized species numbers of S_{2}and S_{3}after a time change satisfy
where N^{γ}in each propensity comes from the change of the time variable. Here, the initial values may depend on γ, since we can choose different values for α_{ i }for each γ due to changes in order of magnitude of species numbers in time. The stochastic equations after scaling and a time change for all species are given in the Additional file 1: Section 1.
Balance conditions
Our goal is to approximate dynamics of the full network in the heat shock response model of E. coli in specific times of interest in terms of simplified subnetworks preserving significant biological features. In each time period of interest, we obtain a nondegenerate limiting model which is not equal to zero and does not blow up to infinity. In this section, we introduce balance conditions which help us to choose appropriate values for the scaling exponents α_{ i }’s and β_{ k }’s so that the limit is nonzero finite. For each time period of interest of order {N}_{0}^{\gamma} where N_{0}=100, we choose values for scaling exponents so that orders of magnitude of the species number for S_{ i } and the k th reaction rate constant are about {N}_{0}^{{\alpha}_{i}} and {N}_{0}^{{\beta}_{k}}, respectively. That is,
It is natural to choose β_{ k }’s in monotone decreasing manner in k, since {\kappa}_{k}^{\prime}’s are in monotone decreasing order as shown in Table 3. In Table 3, the production rates from a source are the rates per second. The unimolecular reaction rates are the rates per molecule per second while the bimolecular reaction rates are the rates per a pair of molecules per second. Since the reaction rates are expressed in different units, we separate rate constants into three classes based on the number of reactants and assume that monotonicity of β_{ k }’s holds in each class of reactions. In other words, we choose β_{ k }’s so that
Next, in order to make the normalized specie number {Z}_{i}^{N,\gamma}\left(t\right) balanced, it is required that the rates of production and consumption of S_{ i }should be in the same order of magnitude. If the order of magnitude of production rate is larger than that of consumption, the normalized species number asymptotically goes to infinity. In the opposite case, the normalized species number asymptotically becomes zero. Therefore, for each species S_{ i }, we set the balance equation for α_{ i }’s and β_{ k }’s so that the maximal exponent in the propensities of the reactions producing S_{ i } is equal to that in the propensities of the reactions consuming S_{ i }. For example, to obtain a balance equation for species S_{2}, we compare the scaling exponents in propensities of reactions involving S_{2}using (11a), and set the maximal exponents of production and consumption of S_{2} equal. Similarly, using (11b), we set the maximal exponents in the production rates and the consumption rates of S_{3} equal. Then, the balance equations for species S_{2}and S_{3} are
If the maximal orders of magnitudes of production and consumption rates for S_{2} are different from each other, the species number of S_{2}should be large enough so that a difference between production and consumption of S_{ i } is not noticeable. In other words if α_{ i }’s and β_{ k }’s do not satisfy (12a), α_{2}should be at least as large as the scaling exponents located in all reaction terms in (11a) to prevent the limit becoming zero or blowing up to infinity. Similarly, in case (12b) is not satisfied, α_{3} should be at least as large as the scaling exponents located in the reaction terms in (11b) to prevent the limit becoming zero or blowing up to infinity.
Solving (13) for γ, we get the following timescale constraints:
Inequalities in (14) mean that if maximal production and consumption rates are not balanced either for S_{2} or S_{3}, the chosen set of values for scaling exponents can be used to approximate the dynamics of the full network up to times of order N^{u}_{2} or N^{u}_{3}. For times later than those of order N^{u}_{2}or N^{u}_{3}, we need to choose another set of values for scaling exponents based on the balance equations. We call the balance equation and the timescale constraint for each species as the species balance condition. If either (12a) or (??) is satisfied, we say that the species balance condition for S_{2} is satisfied.
Even though species balance conditions for S_{2}and S_{3} are satisfied, the limit of the normalized species numbers for S_{2}or S_{3} may become degenerate. Consider addition of species S_{2}and S_{3} as a single virtual species, and compare the collective rates of production and consumption of this species. Recall that S_{23} denotes addition of species S_{2}and S_{3}. Since production of one species is canceled by consumption of the other species, maximal production rate of S_{23} may be different from that of S_{2}or S_{3}. Suppose that the maximal collective rates of production or consumption of S_{23} are slower than the maximal production or consumption rates of S_{2}and S_{3}. Also, suppose that the maximal collective rates of production and consumption of the complex have different orders of magnitude. Then, a limit of the normalized species number of S_{23}can be zero or infinity, even though the species balance conditions for S_{2} and S_{3} are satisfied. Therefore, we need an additional condition to obtain balance between collective production and consumption rates for S_{23}. To obtain a balance equation for S_{23}, we unnormalize (11a) and (11b) by multiplying N^{α}_{2} and N^{α}_{3}, respectively. Adding the unnormalized equations for species S_{2}and S_{3} and dividing it by {N}^{max({\alpha}_{2},{\alpha}_{3})}, we get
Comparing the maximal scaling exponents of production and consumption of S_{23} in (15), a balance equation for S_{23}is given as
In case (16) is not satisfied, the order of magnitude of the species number for S_{23} should be larger than those of collective production and consumption rates so that a difference between production and consumption is not noticeable. This gives
Solving (??) for γ, we get
Similarly to the timescale constraint in the species balance condition, (18) implies that if maximal collective production and consumption rates for S_{23}are not balanced, our choice of values for scaling exponents are valid up to times of order N^{u}_{23}.
We call (16) and (18) the collective species balance condition for S_{23}, that is, either (16) or (18) must hold. The species balance conditions for all species and the collective species balance conditions for all positive linear combinations of species should be satisfied to obtain a nondegenerate limit of {Z}_{i}^{N,\gamma} (Condition 3.2 in [9]). Condition 3.2 can be reduced by Lemma 3.43.8 and Remark 3.9 in [9]. A key idea is to find a minimum subset of linear combinations of species so that production of one species is canceled by consumption of the other species when we combine the species. In that case, maximal collective production rate of the linear combination of the species may be different from that of each species. Therefore, species balance conditions may not imply the collective species balance condition for the linear combination of the species. For example, a collective species balance condition for addition of S_{2} and S_{3} should be satisfied, since reactions producing S_{2}or S_{3} may not increase the species number of S_{23}. In Table 4, we choose linear combinations of species whose collective species balance conditions may not be satisfied by the species balance conditions. For other linear combinations of species, their collective species balance conditions are derived from the ones in Table 4. Satisfying all balance conditions in Table 4 guarantees satisfying balance conditions for all positive linear combination of species, and these conditions help to identify scaling exponents which give a nondegenerate limit of the normalized species numbers in the heat shock response model of E. coli. In most cases satisfying balance conditions gives nondegenerate limiting models in the times of interest, but we can still find counter examples as given in the last paragraph in the section for methods.
Based on species and collective species balance equations in Table 4, we choose appropriate values for α_{ i }’s and β_{ k }’s so that most of the balance equations are satisfied. If some of the balance equations are not satisfied, corresponding timescale constraints give a range of γ where the chosen α_{ i }’s and β_{ k }’s are valid. The timescale constraint, γ≤γ_{0}, implies that the set of scaling exponents α_{ i }’s and β_{ k }’s chosen is appropriate only up to time whose order of magnitude is equal to N^{γ}_{0}. For the times larger than O(N^{γ}_{0}), we need to choose a different set of values for the scaling exponents, α_{ i }’s. Assuming that reaction rate constants do not change in time and that the species numbers vary in time, we in general use one set of β_{ k }’s for all time scales and may use several sets of α_{ i }’s. A large change of the species numbers in time requires different α_{ i }’s in different time scales. For the heat shock model we identify three different time scales as we will see in the section of limiting models in three time scales, and α_{1}, α_{2}, α_{3}, α_{8}, and α_{9} may depend on the time scale. α_{4}, α_{5}, α_{6}, and α_{7} are the same for all time scales.
Before we determine scaling exponents for S_{1}, S_{2}, and S_{3}, we run one realization of stochastic simulation to find ranges of the species numbers in time. Using initial values for species S_{1}, S_{2}, and S_{3}, X_{1}(0)=10 and X_{2}(0)=X_{3}(0)=1 as given in Table 1 and using N_{0}=100, we set {X}_{1}\left(t\right)\approx O\left(100\right)=O\left({N}_{0}^{{\alpha}_{1}}\right), {X}_{2}\left(t\right)\approx O\left(1\right)=O\left({N}_{0}^{{\alpha}_{2}}\right), and {X}_{3}\left(t\right)\approx O\left(1\right)=O\left({N}_{0}^{{\alpha}_{3}}\right) with α_{1}=1 and α_{2}=α_{3}=0 in the early stage of time period. Plugging in α_{ i }’s and β_{ k }’s in the balance equations for S_{2}, S_{3}, and S_{23}, equality holds in (12a) and (12b) but not in (16). Therefore, (18) gives
Then, the first set of scaling exponents with α_{1}=1 and α_{2}=α_{3}=0 is valid only when γ≤0. Next, based on the fact that X_{2}(t)≈O(10) and X_{3}(t)≈O(10) in the medium stage of time period, we choose α_{2}=α_{3}=0 for γ>0. At this stage of time period, we set {X}_{1}\left(t\right)=O\left(10\right)\approx O\left({N}_{0}^{{\alpha}_{1}}\right) with α_{1}=0. Then, (12a) and (12b) are satisfied but not (16). The condition (18) gives γ≤1, and the second set of scaling exponents with α_{1}=α_{2}=α_{3}=0 is valid when γ≤1. Finally, we set α_{1}=0 and α_{2}=α_{3}=1 for γ>1 based on the fact that the numbers of molecules of S_{2}and S_{3} grow in time and are of order 100. Then, (12a), (12b), and (16) are all satisfied, and the third set of scaling exponents with α_{1}=0 and α_{2}=α_{3}=1 can be used for γ>1.
The three sets of values for the scaling exponents chosen are given in the Additional file 1: Table S4. With chosen values for the scaling exponents, we check whether each balance equation is satisfied and give a timescale constraint in the Additional file 1: Table S6 in case the balance equation is not satisfied. Different choices of α_{ i }’s and β_{ k }’s from the ones in the Additional file 1: Table S4 give different limiting models. As long as the chosen values for α_{ i }’s and β_{ k }’s satisfy balance conditions, the limiting model will describe nontrivial behavior of the species numbers which are nonzero and finite in the specific time of interest.
Limiting models in three time scales
In the heat shock response model of E. coli, we identify a time scale of interest using the chosen set of scaling exponents and derive a limiting model which approximates dynamics of the full chemical reaction network. Each limiting model involves a subset of species and reactions, and gives features of the full network during the time interval of interest.
To identify a time scale involving a limiting model with interesting dynamics (nondegenerate), we first need to determine a natural time scale of each species. Recall that a natural time scale of species S_{ i } is the time period of order N^{γ}_{ i } when {Z}_{i}^{N,{\gamma}_{i}}\left(t\right) is of order 1. The natural time scale exponent γ_{ i }for species S_{ i } is rigorously determined by
where Γ i + denotes the collection of reactions where the species number of S_{ i } increases every time the reaction occurs. Similarly, Γ i− is the subset of reactions where the species number of S_{ i }decreases every time the reaction occurs. In (19), the leftside term is the maximal order of magnitude of rates of reactions involving S_{ i }and the rightside term is the order of magnitude of the species number for S_{ i }. If times are earlier than those of order N^{γ}_{ i }(γ<γ_{ i }), fluctuations of species number of S_{ i } due to the reactions involving S_{ i }are not noticeable compared to magnitude of the species number of S_{ i }. Then, the species number of S_{ i } is approximated as its initial value. In the times of order N^{γ}_{ i }(γ=γ_{ i }), changes of species number of S_{ i } due to the reactions and the species number of S_{ i } are similar in magnitude and behavior of the species number of S_{ i }is described by its nondegenerate limit. If times are later than those of order N^{γ}_{ i }(γ>γ_{ i }), the species number of S_{ i } fluctuates very rapidly due to the reactions involving S_{ i } compared to the magnitude of the species number of S_{ i }. Then, the averaged behavior of the species number of S_{ i }is approximated by some function of other species numbers. Note that γ_{ i } depends on α_{ i }’s and β_{ k }’s, and the time scale of the i th species may change if we use several sets of α_{ i }’s.
All values of α_{ i }’s and ρ_{ k }’s for three scalings which are used to derive limiting models are given in the Additional file 1: Table S4. The equations for normalized species numbers and the equation for {Z}_{23}^{N,\gamma} which are used later in this section are given in the Additional file 1: Section 1 and Section 2, respectively. When we derive limiting models in three time scales, boundedness of the normalized species numbers is required. For first two time scales, we define stopping times so that the normalized species numbers are bounded up to those times. For the last time scale, we proved stochastic boundedness of some normalized species numbers in a finite time interval. For more details, see Additional file 1: Section 5.
Consider a model with the first set of scaling exponents including α_{1}=1 and α_{2}=α_{3}=0. Note that the first set of scaling exponents is valid when γ≤0 based on the time scale constraints given in Table 4. Substituting α_{2}=0 and ρ_{ k }’s for the first scaling to the equation for {Z}_{2}^{N,\gamma} given in (11a), we have
When γ=γ_{2}, the maximal scaling exponent in the propensities of all reaction terms in (20) should be equal to the scaling exponent for the species number of S_{2}. Therefore, γ_{2}satisfies
and we get γ_{2}=0. Similarly, we get γ_{3}=γ_{8}=0.
Next, we plug α_{1}=1 and ρ_{ k }’s for the first scaling in the equation for {Z}_{1}^{N,\gamma} and get
By comparing the maximal scaling exponent in the propensities of all reaction terms in (22) and the scaling exponent for the species number of S_{1}, γ_{1} satisfies
and we get γ_{1}=2. Similarly, we get γ_{ i }>0 for i=4,5,6,7,9. Among all natural time scale exponents of species, we choose the smallest one, γ=0, and set t∼O(N^{0})=O(1) as the first time scale we are interested in. Since γ_{1}>0, {Z}_{1}^{N,0}\left(t\right)\to {Z}_{1}^{0}\left(0\right) as N→∞. Similarly, {Z}_{i}^{N,0}\left(t\right)\to {Z}_{i}^{0}\left(0\right)={N}_{0}^{{\alpha}_{i}}{X}_{i}\left(0\right) for i=4,5,6,7,9 as N→∞. To sum up, in this time scale with γ=0, the species numbers of S_{ i }’s for i=1,4,5,6,7,9 change more slowly than other species numbers, and the species numbers with slow time scales are approximated as constant.
To derive the limiting equation for S_{2}, we set γ=0 in (20). Since the 2nd, 3rd, and 4th reaction terms have propensities with N^{0}=1 and the species number of S_{2} is of order 1, these reaction terms converge to nonzero limits in the limiting equation. On the other hand, the propensities of the 5th, 6th, 7th, 8th and 9th reaction terms are of order N^{−1} or N^{−2} which are smaller than the species number for S_{2}of order 1. Therefore, these reaction terms converge to zero as N→∞ at least in the finite time interval. In the 2nd and 3rd reaction terms in (20), {Z}_{2}^{N,0}\left(s\right)\to {Z}_{2}^{0}\left(s\right) and {Z}_{3}^{N,0}\left(s\right)\to {Z}_{3}^{0}\left(s\right) as N→∞ since γ_{2}=γ_{3}=0. Then, using {Z}_{1}^{N,0}\left(s\right)\to {Z}_{1}^{0}\left(0\right) as N→∞, the limit of {Z}_{2}^{N,0} satisfies
Similarly, we get a limiting model with {Z}_{2}^{0}, {Z}_{3}^{0}, and {Z}_{8}^{0} for γ=0 as given in (3).
Next, consider a model with the second set of scaling exponents including α_{1}=α_{2}=α_{3}=0. Note that the second set of scaling exponents is valid when γ≤1 based on the time scale constraints given in Table 4. To determine the natural time scale of S_{6}, substitute α_{6}=0 and ρ_{ k }’s for the second scaling in the equation for {Z}_{6}^{N,\gamma}, and we have
Comparing the exponents inside and outside of the reaction terms in (24), γ_{6} satisfies
and we get γ_{6}=1. Similarly, we get γ_{7}=γ_{8}=1, γ_{ i }<1 for i=2,3, and γ_{ i }>1 for i=1,4,5,9. We already get the temporal behavior of species numbers of S_{2}, S_{3}, and S_{8} through the limiting model when γ=0. Thus, we set t∼O(N^{1}) as the second time scale we are interested in, and derive a limiting model for S_{6}, S_{7}, and S_{8} when γ=1. Note that species S_{8} is involved in the limiting models for both γ=0 and γ=1, since we use different sets of scaling exponents in these models. For i=1,4,5,9{Z}_{i}^{N,1}\left(t\right)\to {Z}_{i}^{1}\left(0\right) as N→∞, since γ_{ i }>1. Thus, in the 12th and 15th reaction terms in (24), {Z}_{9}^{N,1}\left(s\right)\to {Z}_{9}^{1}\left(0\right) and {Z}_{4}^{N,1}\left(s\right)\to {Z}_{4}^{1}\left(0\right) as N→∞. Since the propensities of the 8th, 9th, and 17th reaction terms in (24) are of order N^{γ−2}=N^{−1} for γ=1 and the species number of S_{6} is of order 1, these reaction terms go to zero as N→∞. In the 10th and 15th reaction terms in (24), {Z}_{6}^{N,1}\left(s\right), {Z}_{7}^{N,1}\left(s\right), and {Z}_{8}^{N,1}\left(s\right) are asymptotically O(1) and converge to {Z}_{6}^{1}\left(s\right), {Z}_{7}^{1}\left(s\right), and {Z}_{8}^{1}\left(s\right) as N→∞ since γ_{6}=γ_{7}=γ_{8}=1.
Now, consider the asymptotic behavior of the 7th reaction term in (24) when γ=1. Since γ_{3}<1, {Z}_{3}^{N,1}\left(t\right) fluctuates very much, and there exists no functional limit as N→∞. However, \underset{0}{\overset{t}{\int}}{Z}_{3}^{N,1}\left(s\right)\phantom{\rule{0.3em}{0ex}}\mathit{ds} still converges, which gives the averaged behavior of the normalized species number of S_{3}. To get the limit of \underset{0}{\overset{t}{\int}}{Z}_{3}^{N,1}\left(s\right)\phantom{\rule{0.3em}{0ex}}\mathit{ds}, we plug the second set of scaling exponents in the equation for {Z}_{3}^{N,\gamma} and obtain
The law of large numbers of Poisson processes gives an asymptotic limit of the scaled reaction terms as
where the Y_{ k }’s are unit Poisson processes and α_{ i }>0. For example, the 2nd reaction term in (26) divided by N is approximated as
Dividing (26) by N and using the law of large numbers for Poisson processes, we get
as N→∞.
We introduce an auxiliary variable to make the limiting model closed and define
Plugging α_{2}=α_{3}=0 and ρ_{ k }’s in the second scaling in the equation for {Z}_{23}^{N,\gamma}, we get
Since {Z}_{23}^{N,{\gamma}_{23}}\left(t\right)\sim O\left(1\right) where γ_{23} denotes a natural time scale exponent of S_{23}, we compare the scaling exponents of N in the reaction terms in (29) and the scaling exponent of N outside the reaction terms. Then γ_{23}satisfies
and we get γ_{23}=1. Since these reaction terms have N^{γ−2}=N^{−1} in their propensities when γ=1, which is smaller than the species number for S_{23} of order 1, these reaction terms converge to zero as N→∞. Using {Z}_{1}^{N,1}\left(s\right)\to {Z}_{1}^{1}\left(0\right), the limit of {Z}_{23}^{N,1} satisfies
Adding and subtracting terms in (28) and dividing the equation by −(κ_{2} + κ_{3}), we get
as N→∞, and this is used to obtain the limit of the 7th reaction term in (24). Letting N→∞, the limiting equation for {Z}_{6}^{N,1} is given as
In (30), note that {R}_{12}^{t}\left({\kappa}_{12}{Z}_{9}^{1}\right(0\left)\right)=0 since X_{9}(0)=0 as given in Table 1. Limiting equations for {Z}_{7}^{N,1} and {Z}_{8}^{N,1} can be derived similarly, and a limiting model with {Z}_{23}^{1}, {Z}_{6}^{1}, {Z}_{7}^{1}, and {Z}_{8}^{1} for γ=1 is given in (4).
Last, consider a model with the third scaling exponents with α_{1}=0 and α_{2}=α_{3}=1. To derive a limiting equation for {Z}_{23}^{N,2}, we plug ρ_{ k }’s and α_{2}=α_{3}=1 for the third scaling in the equation for {Z}_{23}^{N,\gamma} and get
In (31), the 8th reaction term is asymptotically zero, since the term is of order N^{−1}. Using the law of large numbers for Poisson processes in (27), the 4th and the 9th terms in (31) are asymptotically equal to
Since γ_{1}=2, {Z}_{1}^{N,2}\left(s\right)\to {Z}_{1}^{2}\left(s\right) as N→∞. On the other hand, since γ_{2},γ_{6}<2, both {Z}_{2}^{N,2}\left(s\right) and {Z}_{6}^{N,2}\left(s\right) in (32) fluctuate rapidly and we must identify the averaged limit. {Z}_{3}^{N,2} is also averaged, since γ_{3}<2. We actually show convergence of the fastfluctuating species numbers of S_{2} and S_{3} to some limits in the Additional file 1: Section 5.1. For any ε>0 and for any t such that \epsilon <t\le {\tau}_{\infty}^{2},
uniformly as N→∞.
On the other hand, since γ_{6}<2, \underset{0}{\overset{t}{\int}}{Z}_{6}^{N,2}\left(s\right)\phantom{\rule{0.3em}{0ex}}\mathit{ds} converges to a limit which gives averaged behavior of the normalized species number of S_{6}. Using the equation for {Z}_{6}^{N,\gamma}, we get
Dividing (35) by N^{2}, using the law of large numbers for Poisson processes in (27), and using the stochastic boundedness of the propensities of the 8th, 9th, 15th, and 17th reaction terms in the finite time interval shown in the Additional file 1: Section 5.1, we get
as N→∞. Therefore, a difference between the 10th and 12th reaction terms is approximated in terms of
which converges to {\int}_{0}^{t}{\kappa}_{7}{\stackrel{\u0304}{Z}}_{3}^{2}\left(s\right)\phantom{\rule{0.3em}{0ex}}\mathit{ds} from (34). Therefore, we get
as N→∞.
Now, from the equations for {Z}_{8}^{N,\gamma} and {Z}_{9}^{N,\gamma}, we get
Using the law of large numbers of Poisson processes in (27), the reaction terms in (39a) and (39b) are asymptotically equal to
Using (40a), (40b), and (38), the limiting equations of (39a) and (39b) are given as
In (41), note that {Z}_{9}^{2}\left(0\right)=0 since X_{9}(0)=0 as given in Table 1.
Since {Z}_{8}^{N,2}\left(0\right)>0 and balance conditions are satisfied, {Z}_{8}^{N,2}\left(t\right)\ne 0 in the finite time interval. Since γ_{8}=2,
Using (38) and (42), {\int}_{0}^{t}{Z}_{6}^{N,2}\left(s\right)\phantom{\rule{0.3em}{0ex}}\mathit{ds} is averaged as
From (33) and (43), we get
For more details used in (43) and (44), see Lemma 1.5 and Theorem 2.1 in [15]. Finally, we get the limiting equation of {Z}_{23}^{N,2} as
Theorem 1
For γ=0, \{{Z}_{2}^{N,0},{Z}_{3}^{N,0},{Z}_{8}^{N,0}\} converges to the solution of (3) for t\in [0,{\tau}_{\infty}^{0}). For γ=1, \{{Z}_{23}^{N,1},{Z}_{6}^{N,1},{Z}_{7}^{N,1},{Z}_{8}^{N,1}\} converges to the solution of (4) for t\in [0,{\tau}_{\infty}^{1}). In (3), {Z}_{8}^{0} is a discrete process, while {Z}_{8}^{1} is a deterministic process in (4). For γ=2, \{{Z}_{1}^{N,2},{Z}_{23}^{N,2},{Z}_{4}^{N,2},{Z}_{5}^{N,2},{Z}_{8}^{N,2},{Z}_{9}^{N,2}\} converges to the solution of (5) for t\in [0,{\tau}_{\infty}^{2}).
Conditional equilibrium distributions
In the previous section, we derived limiting models in three different time scales. Except for the subset of species in the limiting model, the remaining species are approximated as constants in the first time scale, since their natural time scale exponents (γ_{ i }) are larger than γ=0, i.e., species with γ_{ i }>γ=0 did not start to fluctuate at these times yet. In the second and third time scales, there are subsets of species whose natural time scale exponents are smaller than γ=1 and 2, respectively. Normalized species numbers with γ_{ i }<γ fluctuate very rapidly at these times and their averaged behavior is approximated in terms of other variables which converge to a nondegenerate limit. For those species, the normalized species numbers do not converge to a limit in a functional sense, but still we can find a limit in a probabilistic sense (i.e. convergence in distribution) and their distribution. Conditioned on the normalized species numbers which converge to a nondegenerate limit in the time scale of interest, we can find the conditional equilibrium (or the local averaging) distributions of species numbers whose natural time scale exponents are smaller than the time scale exponents of interests. Conditioning on the normalized species numbers which converge to a nondegenerate limit is similar to fixing slowlymoving variables and describing behavior of the fastfluctuating variables in terms of slowlymoving variables treating them as constants. In the next remark, we give a conditional equilibrium distribution of the subset of species with natural time scale exponents smaller than γ=1 and γ=2.
Remark 2
For γ=1, for each t>0, \left({Z}_{2}^{N,1}\left(t\right),{Z}_{3}^{N,1}\left(t\right)\right) converges in distribution to \left({\widehat{Z}}_{2}^{1}\left(t\right),{\widehat{Z}}_{3}^{1}\left(t\right)\right) such that \left({\widehat{Z}}_{2}^{1}\left(t\right),{\widehat{Z}}_{3}^{1}\left(t\right)\right) conditioned on {Z}_{23}^{1}\left(t\right) has a binomial distribution with parameter
respectively, that is,
For γ=2, for each t>0, \left({Z}_{6}^{N,2}\left(t\right),{Z}_{7}^{N,2}\left(t\right)\right) converges in distribution to \left({\widehat{Z}}_{6}^{2}\left(t\right),{\widehat{Z}}_{7}^{2}\left(t\right)\right) where {\widehat{Z}}_{6}^{2}\left(t\right) and {\widehat{Z}}_{7}^{2}\left(t\right) are independent Poisson distributed random variables with parameters
and