A multiscale approximation in a heat shock response model of E. coli
© Kang; licensee BioMed Central Ltd. 2012
Received: 17 November 2011
Accepted: 7 November 2012
Published: 21 November 2012
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© Kang; licensee BioMed Central Ltd. 2012
Received: 17 November 2011
Accepted: 7 November 2012
Published: 21 November 2012
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 time-scales and model reduction for stochastic reaction networks extended by Kang and Kurtz (2012), we approximate the chemical network in the heat shock response model.
Scaling the species numbers and the rate constants by powers of the scaling parameter, we embed the model into a one-parameter family of models, each of which is a continuous-time 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.
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.
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 . 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 well-mixed and relative fluctuations of small species numbers may play a role in the system dynamics.
The conventional stochastic model for the well-stirred 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 . For models with well-separated time scales, numerous authors suggested stochastic simulation algorithms for biochemical reaction networks by assuming that “fast” subnetworks have reached a “partial equilibrium”  or a “quasi-steady state” . Using these assumptions, the approximate stochastic simulation algorithms involve a reduced number of species or reactions.
On the other hand, Ball et al.  described the state of the biochemical reaction network in the well-stirred 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  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.  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  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 to , 500 to 800molecules are rewritten as 5×N 0 to 8×N 0molecules, and 0.0002 sec becomes . Assuming N 0 is large, we replace N 0by 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 quasi-steady 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 or 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  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 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 8from S 2and S 3.
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 2and S 3play a role as a single “virtual” species rather than separate species. The species numbers of S 23 and S 8are 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 8is 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.
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 2and S 3play 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 Michaelis-Menten 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 2and S 6which 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  and show that the error is of order 10−1.
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.
where ν ik and are nonnegative integers. Rearrange the reactions so that the reaction rate constants are decreasing monotonically as k gets large.
where counts the number of times that the kth reaction occurs up to time t.
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 kth reaction is second-order ( ) with different types of reactants, . When the reactants are two molecules of the same species, .
In the equation in Step 3 (a), .
In the most reactions, is obtained by replacing by κ k in λ k . In case the kth reaction is second-order with reactants of the same species, is replaced by .
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 time-scale 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 and .
the species number X i and the reaction rate constant are approximately of orders and ;
the normalized species number and the scaled reaction rate constant κ k are of order 1;
most of the balance equations obtained in Steps 4 and 6 are satisfied;
β k ’s are monotone decreasing among each class of reactions which have the same number of molecules of reactants.
8. Plugging the chosen values for α i ’s and β k ’s in the time-scale constraints obtained in Steps 4 and 6, compute an upper bound (denoted as γ 0) for a time-scale exponent. Then, the chosen set of exponents α i ’s and β k ’s can be used for γsatisfying γ≤γ 0. For γ>γ 0, select another set of exponents α i ’s and β k ’s using Steps 7 and 8.
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.
For , the kth reaction term converges to zero if α i >γ + ρ k .
If α i =γ + ρ k , the kth reaction term appears as a limit in the limiting equation. The limit of the kth reaction term is discrete if α i =0, while it is a continuous variable with the limit of its propensity if α i >0.
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.
If γ j >γ, the limit of the normalized species number for S j is its initial value.
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.
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 and letting N go to infinity.
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 , 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 2but with the same order of magnitude, and that production rate of S 3 is much smaller than those of S 1and 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.
We analyze a heat shock response model of E. coli developed by Srivastava, Peterson, and Bentley . 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 32factors play an important role in recovery from the stress under the high temperature. σ 32factors 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 32protein, 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 32factors in an inactive form, and σ 32factors can directly respond to the stress by changing into different forms. When there exist σ 32factors combined with chaperon complexes, FtsH catalyzes degradation of σ 32 factors. Thus, if enough σ 32-regulated stress proteins are produced, σ 32factors are degraded.
Species in the heat shock response model of E. coli and their initial values
# of S 1
σ 32 mRNA
# of S 2
σ 32 protein
# of S 3
E σ 32
# of S 4
# of S 5
# of S 6
# of S 7
# of S 8
# of S 9
Reactions in the heat shock response model of E. coli
Recombinant protein synthesis
S 2→S 3
S 3→S 2
σ 32 translation
S 7→S 2 + S 6
S 2 + S 6→S 7
S 6 + S 8→S 9
Recombinant protein-J association
Recombinant protein degradation
S 9→S 6 + S 8
Recombinant protein-J disassociation
σ 32 transcription
σ 32 mRNA decay
σ 32 degradation
Stochastic reaction rate constants in the heat shock response model of E. coli
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.
so that and .
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 and 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 2and S 3 are asymptotically equal to non-constant limits. In the late stage of time period, the normalized species numbers of S 2 and S 3fluctuate very rapidly and their averaged behavior is captured in terms of some function of other species numbers.
Then, gives a normalized species number at the times of order N γ . A natural time scale of S i is the time when has a nonzero finite limit which is not constant and of order 1.
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 γdue 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.
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 2or 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 time-scale 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.
Similarly to the time-scale constraint in the species balance condition, (18) implies that if maximal collective production and consumption rates for S 23are not balanced, our choice of values for scaling exponents are valid up to times of order N u 23.
Balance equations and time-scale constraints for each species and for each collective species chosen
ρ 13=ρ 14
ρ 6=ρ 18
ρ 5=ρ 16
ρ 10=ρ 12
S 2 + S 3 + S 7
ρ 4=ρ 15
S 2 + S 3
S 2 + S 7
S 6 + S 7 + S 9
ρ 7=ρ 17
S 6 + S 7
S 6 + S 9
S 8 + S 9
ρ 1=ρ 11
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 time-scale constraints give a range of γ where the chosen α i ’s and β k ’s are valid. The time-scale 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.
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 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 2and 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 time-scale 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.
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.
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 left-side term is the maximal order of magnitude of rates of reactions involving S i and the right-side 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 ith 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 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.
and we get γ 2=0. Similarly, we get γ 3=γ 8=0.
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, as N→∞. Similarly, 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.
Similarly, we get a limiting model with , , and for γ=0 as given in (3).
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 as N→∞, since γ i >1. Thus, in the 12th and 15th reaction terms in (24), and 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), , , and are asymptotically O(1) and converge to , , and as N→∞ since γ 6=γ 7=γ 8=1.
In (30), note that since X 9(0)=0 as given in Table 1. Limiting equations for and can be derived similarly, and a limiting model with , , , and for γ=1 is given in (4).
uniformly as N→∞.
In (41), note that since X 9(0)=0 as given in Table 1.
For γ=0, converges to the solution of (3) for . For γ=1, converges to the solution of (4) for . In (3), is a discrete process, while is a deterministic process in (4). For γ=2, converges to the solution of (5) for .
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 <γfluctuate 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 slowly-moving variables and describing behavior of the fast-fluctuating variables in terms of slowly-moving 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.
The detailed method to compute conditional equilibrium distributions is given in Section 6 in [].
Mean value of the random variable with a Poisson distribution, Pois(λ), is equal to λ, and we obtain a limit of the averaged values for and as the parameters given in Remark 2.
In Figure 2, we compare the simulation for the full model and for the approximation using the limiting model in the first scaling. The first scaling (γ=0) is for the times of order , and we look at the evolution of mean and standard deviation of the species numbers up to 100 sec. The full model and the limiting model for γ=0 are stochastic, and the limiting model approximates the evolution of statistics of the species numbers quite precisely. As shown in Figure 2(f) overestimates X 8(t), since the limiting model does not include reactions consuming S 8. Therefore, consumption of S 8may not be captured well in the approximation.
In Figure 3, we compare the simulation for the full model and for the approximation using the limiting model in the second scaling. Since the second scaling (γ=1) is for the times of order , we observe the evolution of the species numbers up to 1000 sec. In this time scale, the evolution of S 8shown in Figure 3(h) is approximated by a deterministic variable. The evolution of the species number of S 8 in the full model given in Figure 3(g) is stochastic, but its standard deviation is very small. As in the previous time scale, slightly overestimates X 8(t), since the limiting model does not include any consumptions of S 8. The remaining three species, S 23, S 6, and S 7are approximated by stochastic variables. The increasing species number of S 23 in time and the rapid decrease in species number of S 6are well captured by the limiting model. The species numbers of S 7are described by stochastic variables both in the full model and in the limiting model. The behavior of S 7in two models is not exactly the same, and discrepancy of the mean species numbers of S 7 comes from the approximation of X 4(t) in terms of its initial value. In the limiting model, S 7is approximated as a stochastic process decreasing by 1 with the propensity proportional to . However, X 4(t) increases during the times in [0,1000] sec in the full model, and this difference gives slower decreasing rate of the mean number of S 7 in the limiting model than that in the full model.
In Figure 4 and Figure 5(a)-(d), we compare the simulation for the full model and for the approximation obtained from the limiting model in the third scaling. Since the third scaling (γ=2) is for the times of order , we look at the simulation up to 20,000 sec. In this time scale, the limiting model is stochastic. The species number of S 1 in the limiting model is approximated by a stochastic discrete variable increasing and decreasing by 1, and the remaining species numbers in the limiting model satisfy stochastic equations driven by the stochastic discrete variable . As we have seen in the proof of Theorem 1 in the Additional file 1: Section 5.1, the processes for S 1 in the full model and in the limiting model are exactly the same. Therefore, we use a same series of random numbers, when we simulate the full and limiting models. In Figure 4(b), the process for S 1 is random, but its standard deviation is very small. Therefore, in one realization of simulation of the limiting model, behavior of S 1appears as constant. Since all the remaining variables in the limiting model are governed by the variable for S 1 and they satisfy the stochastic differential equations, evolution of one sample path of the species numbers for S 23, S 4, S 5, S 8, and S 9 in the limiting model looks like a solution of the system of ordinary differential equations.
in time. They are stochastic variables determined by the ones in the limiting model with very small fluctuations. Since and describe averaged behavior of S 6and S 7, X 6(t) and X 7(t) in Figure 5(e) and (g) have more fluctuations than the averaged species numbers in Figure 5(f) and (h).
In Figure 5(e)-(h) there is a discrepancy between the species numbers and their averaged values in the very early time, and the discrepancy comes from a disagreement in initial values of the species numbers in the full model and those of the averaged values in the limiting model. The integrated species numbers for S 6 and S 7 up to times of order 10,000 are supposed to be approximated by the integrated averaged values over the time interval, and the initial difference is due to a boundary layer phenomenon.
In the previous sections, we scaled species numbers and derived their limit to approximate temporal behavior of the species numbers in the full network. Among three limiting models given in (3)-(5), the first two are systems with discrete variables (except for ) which change by integer values. On the other hand, the last one is a hybrid system with both discrete and continuous variables. A discrete variable increases or decreases by one and stochasticity of all other variables comes from how much fluctuates. Since rarely changes at the times of our interest, the rest of the variables in (5) behaves such as a solution of systems of ordinary differential equations. Our choice of the scaling parameter value, N 0=100, is not very large and it is possible that the limiting model does not contain enough fluctuations as much as the full network actually has due to our assumption that N 0is replaced by a large parameter N.
for S 23, S 4, S 5, S 8, and S 9.
The detailed method to compute an error using the central limit theorem is derived in .
Estimating order of magnitude of an error is an analogue of that in van Kampen’s system size expansion . A difference is that in the system size expansion, the system state representing the species numbers is scaled by the system size Ω and noise between the scaled process and its deterministic value is approximated as a random variable of order Ω −1/2. In our approach N is not a system size but a parameter for scaling, and species numbers are scaled by powers of N. Though the limiting model for γ=2 is not deterministic, it is still possible to estimate an error analytically due to the fact that which produces stochasticity in the limiting model is an exact process equal to . Another difference between our approach and van Kampen’s system size expansion is that a subset of species numbers is averaged in terms of other species numbers which appear in the limiting model for γ=2 due to the various scales involved.
Our estimates of the error is also different from diffusion approximations. In the diffusion approximations, the reaction terms centered by their propensities in the stochastic equations for discrete variables of species numbers are approximated in terms of time-changed Brownian motion. On the other hand, the noise term in the error estimates is determined by both the centered reaction terms in the equations for discrete variables and a difference between the discrete variables for the normalized species number and their continuous limit.
To find the asymptotic order of magnitude of , we show convergence of to a nonzero finite limit for some r N . Among the species S 23, S 4, S 5, S 8, and S 9, the species number of S 23is scaled with the smallest exponent, and thus noise in the limit of is determined dominantly by the component . Since is the species number scaled by N, we expect that r N =N 1/2and the error between the scaled species numbers and their limit is of order . For a detailed approach to derive r N and U(t), see more about the central limit theorem in . The fact that all components but the first one in the diffusion term in the equation for U(t) are zero supports the idea that noise is dominantly determined by the error between and . A sketch of the proof of Remark 3 is given in the Additional file 1: Section 6.
We considered a stochastic model for a well-stirred biochemical network with small numbers of molecules for some species. As the biochemical network consists of more species and reactions, network topology becomes more complex and it is harder to analyze. Therefore, how to reduce the biochemical network while preserving its important biochemical features is a very important issue.
In this paper, we applied the multiscale approximation method introduced by Ball et al.  and extended by Kang and Kurtz  to a heat shock response model of E. coli developed by Srivastava et al. . Using the fact that the species numbers and the reaction rate constants in the model vary over several orders of magnitude, we scaled them using a scaling parameter with different exponents both of which contribute to determining the time scales of species. We derived balance conditions for each species and for a subset of linear combinations of species explicitly in this model, and chose appropriate values for the scaling exponents satisfying the balance conditions. Assuming that initial values of the species numbers are positive, satisfying the balance conditions is required to get a nondegenerate limiting model. We assumed that the reaction rate constants do not change in time, while we may use several sets of scaling exponents for the species numbers due to rapid changes in some species numbers in time. In this analysis, we chose three sets of scaling exponents, and they are used to derive limiting models in different time scales.
In each time scale we derived a limiting model, and used it to approximate the species numbers in the full network. In the limiting model, species numbers whose scaling exponents are larger than those of all rates of reactions involving the species are treated as constants, since changes of the species numbers due to the reactions are not noticeable at these times. When the scaling exponent of the species number is smaller than the scaling exponents of the rates of some productions and consumptions of the species and in case the scaling exponents for both kinds of reactions are equal, the scaled species number is averaged out and is approximated in terms of other variables. Therefore, the limiting model includes a subset of species and reactions and network topology in it becomes simpler. We derived the conditional equilibrium distributions of the fast-fluctuating species numbers and studied errors between the scaled species numbers and their limits in the third time scale.
Using the limiting models, we approximated the temporal evolution of species numbers in three time scales. By comparing stochastic simulation of the full model and approximations using the limiting models, we see that the main features of evolution of species numbers are well captured by the limiting models.
The author would like to greatly thank Thomas G. Kurtz for his continuous support and many helpful discussion. This work is an extension of the author’s Ph.D work at the University of Wisconsin, is proceeded while the author held a postdoctoral appointment under Hans G. Othmer at the University of Minnesota, and is completed while the author held a postdoctoral appointment in the Mathematical Biosciences Institute at the Ohio State University. The support provided by three appointments is acknowledged. This research has been supported in part by the National Science Foundation under grant DMS 05-53687, 08-05793, and 09-31642 and the Mathematical Biosciences Institute.
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