Mathematical Biosciences Institute, Ohio State University, Columbus, OH, USA

Abstract

Background

A heat shock response model of

Results

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.

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

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

On the other hand, Ball et al.

The multiscale approximation method in _{0} be a fixed constant and choosing a large value for _{0}, for example _{0}=100, we express magnitudes of species numbers and reaction rate constants in terms of powers of _{0} with different scaling exponents. For instance, 1 to 10molecules are expressed as _{0} to 8×_{0}molecules, and 0.0002 sec becomes _{0} is large, we replace _{0}by a large parameter

A specific time scale of interest is expressed in terms of a power of

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}=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

The paper

A chemical reaction network in the heat shock response model of E. coli

**A chemical reaction network in the heat shock response model of E. coli.** A dotted line represents the effect of the species acting as catalysts.

Applying the multiscale approximation method to the heat shock response model of _{
i
} represent the _{23}be addition of species _{2}and _{3}.

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 _{8}from _{2}and _{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 _{2} and _{3} occurs very frequently and _{2}and _{3}play a role as a single “virtual” species rather than separate species. The species numbers of _{23} and _{8}are described as two independent birth processes and the species number of _{7} is governed by conversion. In this time scale, the species number of _{8}is normalized and treated as a continuous variable. The interesting thing is that the behavior of the species _{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

As we see in Figure _{2}and _{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 _{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 _{23}, _{8}, and _{9}, which come from averaging of the species numbers of _{2}and _{6}which fluctuate rapidly in the third time scale. Similarly, the propensity of catalytic degradation of _{8} is not proportional to the number of molecules of _{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 ^{−1}.

Methods

In the next several sections, we apply the multiscale approximation to the heat shock response model of

1. Write a chemical reaction network involving _{0}species and _{0} reactions in the form of

where _{
ik
} and

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

(a) Letting _{
i
}(_{
i
}at time

where

(b) _{
k
}(

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

(a) In the equation for _{
i
}(_{
i
}by ^{
α
}
_{
i
}. In the ^{
γ + ρ
}
_{
k
} in the propensity and replace _{
k
}(

(b) In the equation in Step 3 (a),

(c) In the most reactions, _{
k
}in _{
k
}. In case the

4. Write a set of species balance equations and their time-scale constraints.

(a) Define _{
i
}increases or decreases every time the reaction occurs. Comparing _{
k
}’s for

(b) Time-scale constraints are given as

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

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

(a) the species number _{
i
}and the reaction rate constant

(b) the normalized species number _{
k
}are of order 1;

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

(d) _{
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 _{
i
}’s and _{
k
}’s in the time-scale constraints obtained in Steps 4 and 6, compute an upper bound (denoted as _{0}) for a time-scale exponent. Then, the chosen set of exponents _{
i
}’s and _{
k
}’s can be used for _{0}. For _{0}, select another set of exponents _{
i
}’s and _{
k
}’s using Steps 7 and 8.

9. Using each set of values for _{
i
}’s and _{
k
}’s, identify a natural time scale exponent of each species (denoted as _{
i
} for species _{
i
}) so that _{
i
} satisfies

We collect _{
i
}’s with the same values, whose species are in the same time scales in the approximation.

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

11. For each chosen _{
i
}with _{
i
}=

(a) For _{
i
}>_{
k
}.

(b) If _{
i
}=_{
k
}, the _{
i
}=0, while it is a continuous variable with the limit of its propensity if _{
i
}>0.

(c) There is no _{
i
}<_{
k
}in the equation for species _{
i
}with _{
i
}due to the definition of _{
i
}given in Step 9.

12. In the limiting equation for each species _{
i
}with _{
i
}=_{
j
}appears in the propensities.

(a) If _{
j
}>_{
j
}is its initial value.

(b) If _{
j
}=_{
j
}appears as a variable in the propensities in the limiting equation.

(c) If _{
j
}<_{
j
}is expressed as a function of the limits of the normalized species numbers for _{
i
}with _{
i
}=_{
j
}is obtained by dividing the equation for _{
j
}by

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

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 _{1}, _{2}, _{3}, _{1} + _{2}→_{1} + _{3}→_{1} is larger than that of _{2}but with the same order of magnitude, and that production rate of _{3} is much smaller than those of _{1}and _{2}. Then, _{1}(_{2}(

Results and discussion

Model description

We analyze a heat shock response model of ^{32}factors play an important role in recovery from the stress under the high temperature. ^{32}factors catalyze production of the heat shock proteins such as chaperon proteins and other proteases. In this model, ^{32}-regulated stress protein, and ^{32}-mediated stress response protein.

^{32} factors are in three different forms, free ^{32}protein, ^{32} combined with RNA polymerase (^{32}), and ^{32} combined with a chaperon complex (^{32}-^{32} factors combine with chaperon complexes and form ^{32}-^{32}factors in an inactive form, and ^{32}factors can directly respond to the stress by changing into different forms. When there exist ^{32}factors combined with chaperon complexes, ^{32} factors. Thus, if enough ^{32}-regulated stress proteins are produced, ^{32}factors are degraded.

Not only ^{32}factors, but recombinant proteins also require chaperon complexes to form a complex so that denatured protein can be fixed. Therefore, ^{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 ^{32} factors and recombinant proteins have the same affinity to bind to chaperon complexes. The system is sensitive to the amount and forms of ^{32} factors: a small decrease of ^{32}factors causes a large reduction of production of chaperon complexes and ^{32}-regulated stress proteins, and the ratio of three different forms of ^{32}factors determines system dynamics in the stress response ^{32} factors in each cell is small _{2}, _{3}, and _{7} which are 1, 1, and 7 in Table

_{1}

=

# of _{1}

^{32} mRNA

_{1}(0)

=

10

_{2}

=

# of _{2}

^{32} protein

_{2}(0)

=

1

_{3}

=

# of _{3}

^{32}

_{3}(0)

=

1

_{4}

=

# of _{4}

_{4}(0)

=

93

_{5}

=

# of _{5}

_{5}(0)

=

172

_{6}

=

# of _{6}

_{6}(0)

=

54

_{7}

=

# of _{7}

^{32}-

_{7}(0)

=

7

_{8}

=

# of _{8}

Recombinant protein

_{8}(0)

=

50

_{9}

=

# of _{9}

_{9}(0)

=

0

The model involves 9 species and 18 reactions. Denote _{0} as the number of species and _{0} as the number of reactions. Let _{
i
} at time _{0}. Define a random process which counts the number of times that the

where _{
k
}(_{
k
}’s are independent unit Poisson processes. Therefore, _{
k
}(·) gets large, the moment when _{
ik
}(_{
i
} that are consumed (produced) in the _{
k
}(_{0}-dimensional vector whose _{
ik
}(

That is, species numbers at time

**Reaction**

**Transition**

In Reaction 5, 6, and 7, we assume that the number of molecules of each gene is 1 and that these reactions are effectively unimolecular. Similarly, Reactions 1 and 13 are treated as production from a source.

R1

Recombinant protein synthesis

R2

_{2}→_{3}

Holoenzyme association

R3

_{3}→_{2}

Holoenzyme disassociation

R4

^{32} translation

R5

R6

R7

R8

_{7}→_{2} + _{6}

^{32}-

R9

_{2} + _{6}→_{7}

^{32}-

R10

_{6} + _{8}→_{9}

Recombinant protein-

R11

_{8}→∅

Recombinant protein degradation

R12

_{9}→_{6} + _{8}

Recombinant protein-

R13

^{32} transcription

R14

_{1}→∅

^{32} mRNA decay

R15

^{32} degradation

R16

R17

R18

**Rates**

**Rates**

We convert deterministic rate constants in ^{−15}

4.00×10^{0}

3.62×10^{−4}

7.00×10^{−1}

9.99×10^{−5}

1.30×10^{−1}

4.40×10^{−5}

7.00×10^{−3}

1.40×10^{−5}

6.30×10^{−3}

1.40×10^{−6}

4.88×10^{−3}

1.42×10^{−6}

4.88×10^{−3}

1.80×10^{−8}

4.40×10^{−4}

6.40×10^{−10}

3.62×10^{−4}

7.40×10^{−11}

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, _{
i
} at some time scales depending on _{
i
})^{
γ
} but it shows dependence of _{
k
} be a scaled reaction rate constant for the

approximates the network when the times are of order 1 sec. Denote _{2}and _{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 fast-fluctuating species _{2}, _{3}, and _{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

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 _{2}and _{3} are

In Equation (6), species numbers of _{2}and _{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 ^{−11}) to ^{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 _{0}=100 as a fixed unitless constant used to express the magnitude of the species numbers and the reaction rate constants. Define _{
i
} for _{0} and _{
k
} for _{0} as the scaling exponent for species _{
i
} and for the reaction rate constant _{
k
} is of order 1 and is determined so that _{6}=−1 so that the reaction rate is expressed as _{0} is large, we replace _{0} by ^{
N
}(_{0}. Then, the equation for

where _{0}. Since the numbers of molecules of species are in different orders of magnitude, we scale the number of molecules of the ^{
α
}
_{
i
} and set a normalized species number as

The _{
i
} may have different values for different time scales. Now, we set the initial values as

so that

Next, we scale the propensities of reactions by replacing ^{
β
}
_{
k
}
_{
k
}. For example, consider the 9th reaction term in (6a). Replacing ^{
β
}
_{9}
_{9},

For simplicity, set _{9}=_{9} + _{2} + _{6} and define a scaling exponent in the propensity of the

where _{
ik
} gives the number of molecules of species _{
i
}consumed in the

Dividing (6a) by ^{
α
}
_{2} and (6b) by ^{
α
}
_{3} and scaling the propensities, we get

For each reaction, _{
k
}is given in terms of _{
i
}and _{
k
} in the Additional file

**Supplementary material for “A multiscale approximation in a heat shock response model of E. coli.”** This is a supplementary material of the paper including calculations and tables.

Click here for file

We are interested in dynamics of species numbers _{2} and _{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 _{2}and _{3} are asymptotically equal to non-constant limits. In the late stage of time period, the normalized species numbers of _{2} and _{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 ^{
γ
}

Then, ^{
γ
}. A _{
i
}is the time when

Changing a time variable by replacing ^{
γ
}
_{2}and _{3}after a time change satisfy

where ^{
γ
}in each propensity comes from the change of the time variable. Here, the initial values may depend on _{
i
}for each

Balance conditions

Our goal is to approximate dynamics of the full network in the heat shock response model of _{
i
}’s and _{
k
}’s so that the limit is nonzero finite. For each time period of interest of order _{0}=100, we choose values for scaling exponents so that orders of magnitude of the species number for _{
i
} and the

It is natural to choose _{
k
}’s in monotone decreasing manner in _{
k
}’s holds in each class of reactions. In other words, we choose _{
k
}’s so that

Next, in order to make the normalized specie number _{
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 _{
i
}, we set the balance equation for _{
i
}’s and _{
k
}’s so that the maximal exponent in the propensities of the reactions producing _{
i
} is equal to that in the propensities of the reactions consuming _{
i
}. For example, to obtain a balance equation for species _{2}, we compare the scaling exponents in propensities of reactions involving _{2}using (11a), and set the maximal exponents of production and consumption of _{2} equal. Similarly, using (11b), we set the maximal exponents in the production rates and the consumption rates of _{3} equal. Then, the balance equations for species _{2}and _{3} are

If the maximal orders of magnitudes of production and consumption rates for _{2} are different from each other, the species number of _{2}should be large enough so that a difference between production and consumption of _{
i
} is not noticeable. In other words if _{
i
}’s and _{
k
}’s do not satisfy (12a), _{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, _{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

Inequalities in (14) mean that if maximal production and consumption rates are not balanced either for _{2} or _{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 ^{
u
}
_{2} or ^{
u
}
_{3}. For times later than those of order ^{
u
}
_{2}or ^{
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 _{2} is satisfied.

Even though species balance conditions for _{2}and _{3} are satisfied, the limit of the normalized species numbers for _{2}or _{3} may become degenerate. Consider addition of species _{2}and _{3} as a single virtual species, and compare the collective rates of production and consumption of this species. Recall that _{23} denotes addition of species _{2}and _{3}. Since production of one species is canceled by consumption of the other species, maximal production rate of _{23} may be different from that of _{2}or _{3}. Suppose that the maximal collective rates of production or consumption of _{23} are slower than the maximal production or consumption rates of _{2}and _{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 _{23}can be zero or infinity, even though the species balance conditions for _{2} and _{3} are satisfied. Therefore, we need an additional condition to obtain balance between collective production and consumption rates for _{23}. To obtain a balance equation for _{23}, we unnormalize (11a) and (11b) by multiplying ^{
α
}
_{2} and ^{
α
}
_{3}, respectively. Adding the unnormalized equations for species _{2}and _{3} and dividing it by

Comparing the maximal scaling exponents of production and consumption of _{23} in (15), a balance equation for _{23}is given as

In case (16) is not satisfied, the order of magnitude of the species number for _{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

Similarly to the time-scale constraint in the species balance condition, (18) implies that if maximal collective production and consumption rates for _{23}are not balanced, our choice of values for scaling exponents are valid up to times of order ^{
u
}
_{23}.

We call (16) and (18) the _{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 _{2} and _{3} should be satisfied, since reactions producing _{2}or _{3} may not increase the species number of _{23}. In Table

**Balance equations**

**Time-scale constraints**

In each case, either the balance equation or the time-scale constraint must hold.

_{1}

_{13}=_{14}

_{2}

_{3}

_{4}

_{6}=_{18}

_{5}

_{5}=_{16}

_{6}

_{7}

_{8}

_{9}

_{10}=_{12}

_{2} + _{3} + _{7}

_{4}=_{15}

_{2} + _{3}

_{2} + _{7}

_{6} + _{7} + _{9}

_{7}=_{17}

_{6} + _{7}

_{6} + _{9}

_{8} + _{9}

_{1}=_{11}

Based on species and collective species balance equations in Table _{
i
}’s and _{
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 _{
i
}’s and _{
k
}’s are valid. The time-scale constraint, _{0}, implies that the set of scaling exponents _{
i
}’s and _{
k
}’s chosen is appropriate only up to time whose order of magnitude is equal to ^{
γ
}
_{0}. For the times larger than ^{
γ
}
_{0}), we need to choose a different set of values for the scaling exponents, _{
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 _{
k
}’s for all time scales and may use several sets of _{
i
}’s. A large change of the species numbers in time requires different _{
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 _{1}, _{2}, _{3}, _{8}, and _{9} may depend on the time scale. _{4}, _{5}, _{6}, and _{7} are the same for all time scales.

Before we determine scaling exponents for _{1}, _{2}, and _{3}, we run one realization of stochastic simulation to find ranges of the species numbers in time. Using initial values for species _{1}, _{2}, and _{3}, _{1}(0)=10 and _{2}(0)=_{3}(0)=1 as given in Table _{0}=100, we set _{1}=1 and _{2}=_{3}=0 in the early stage of time period. Plugging in _{
i
}’s and _{
k
}’s in the balance equations for _{2}, _{3}, and _{23}, equality holds in (12a) and (12b) but not in (16). Therefore, (18) gives

Then, the first set of scaling exponents with _{1}=1 and _{2}=_{3}=0 is valid only when _{2}(_{3}(_{2}=_{3}=0 for _{1}=0. Then, (12a) and (12b) are satisfied but not (16). The condition (18) gives _{1}=_{2}=_{3}=0 is valid when _{1}=0 and _{2}=_{3}=1 for _{2}and _{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 _{1}=0 and _{2}=_{3}=1 can be used for

The three sets of values for the scaling exponents chosen are given in the Additional file _{
i
}’s and _{
k
}’s from the ones in the Additional file _{
i
}’s and _{
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

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 _{
i
} is the time period of order ^{
γ
}
_{
i
} when _{
i
}for species _{
i
} is rigorously determined by

where _{
i
} increases every time the reaction occurs. Similarly, _{
i
}decreases every time the reaction occurs. In (19), the left-side term is the maximal order of magnitude of rates of reactions involving _{
i
}and the right-side term is the order of magnitude of the species number for _{
i
}. If times are earlier than those of order ^{
γ
}
_{
i
}(_{
i
}), fluctuations of species number of _{
i
} due to the reactions involving _{
i
}are not noticeable compared to magnitude of the species number of _{
i
}. Then, the species number of _{
i
} is approximated as its initial value. In the times of order ^{
γ
}
_{
i
}(_{
i
}), changes of species number of _{
i
} due to the reactions and the species number of _{
i
} are similar in magnitude and behavior of the species number of _{
i
}is described by its nondegenerate limit. If times are later than those of order ^{
γ
}
_{
i
}(_{
i
}), the species number of _{
i
} fluctuates very rapidly due to the reactions involving _{
i
} compared to the magnitude of the species number of _{
i
}. Then, the averaged behavior of the species number of _{
i
}is approximated by some function of other species numbers. Note that _{
i
} depends on _{
i
}’s and _{
k
}’s, and the time scale of the _{
i
}’s.

All values of _{
i
}’s and _{
k
}’s for three scalings which are used to derive limiting models are given in the Additional file

Consider a model with the first set of scaling exponents including _{1}=1 and _{2}=_{3}=0. Note that the first set of scaling exponents is valid when _{2}=0 and _{
k
}’s for the first scaling to the equation for

When _{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 _{2}. Therefore, _{2}satisfies

and we get _{2}=0. Similarly, we get _{3}=_{8}=0.

Next, we plug _{1}=1 and _{
k
}’s for the first scaling in the equation for

By comparing the maximal scaling exponent in the propensities of all reaction terms in (22) and the scaling exponent for the species number of _{1}, _{1} satisfies

and we get _{1}=2. Similarly, we get _{
i
}>0 for ^{0})=_{1}>0, _{
i
}’s for

To derive the limiting equation for _{2}, we set ^{0}=1 and the species number of _{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 ^{−1} or ^{−2} which are smaller than the species number for _{2}of order 1. Therefore, these reaction terms converge to zero as _{2}=_{3}=0. Then, using

Similarly, we get a limiting model with

Next, consider a model with the second set of scaling exponents including _{1}=_{2}=_{3}=0. Note that the second set of scaling exponents is valid when _{6}, substitute _{6}=0 and _{
k
}’s for the second scaling in the equation for

Comparing the exponents inside and outside of the reaction terms in (24), _{6} satisfies

and we get _{6}=1. Similarly, we get _{7}=_{8}=1, _{
i
}<1 for _{
i
}>1 for _{2}, _{3}, and _{8} through the limiting model when ^{1}) as the second time scale we are interested in, and derive a limiting model for _{6}, _{7}, and _{8} when _{8} is involved in the limiting models for both _{
i
}>1. Thus, in the 12th and 15th reaction terms in (24), ^{
γ−2}=^{−1} for _{6} is of order 1, these reaction terms go to zero as _{6}=_{7}=_{8}=1.

Now, consider the asymptotic behavior of the 7th reaction term in (24) when _{3}<1, _{3}. To get the limit of

The law of large numbers of Poisson processes gives an asymptotic limit of the scaled reaction terms as

where the _{
k
}’s are unit Poisson processes and _{
i
}>0. For example, the 2nd reaction term in (26) divided by

Dividing (26) by

as

We introduce an auxiliary variable to make the limiting model closed and define

Plugging _{2}=_{3}=0 and _{
k
}’s in the second scaling in the equation for

Since _{23} denotes a natural time scale exponent of _{23}, we compare the scaling exponents of _{23}satisfies

and we get _{23}=1. Since these reaction terms have ^{
γ−2}=^{−1} in their propensities when _{23} of order 1, these reaction terms converge to zero as

Adding and subtracting terms in (28) and dividing the equation by −(_{2} + _{3}), we get

as

In (30), note that _{9}(0)=0 as given in Table

Last, consider a model with the third scaling exponents with _{1}=0 and _{2}=_{3}=1. To derive a limiting equation for _{
k
}’s and _{2}=_{3}=1 for the third scaling in the equation for

In (31), the 8th reaction term is asymptotically zero, since the term is of order ^{−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 _{1}=2, _{2},_{6}<2, both _{3}<2. We actually show convergence of the fast-fluctuating species numbers of _{2} and _{3} to some limits in the Additional file

uniformly as

On the other hand, since _{6}<2, _{6}. Using the equation for

Dividing (35) by ^{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

as

which converges to

as

Now, from the equations for

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 _{9}(0)=0 as given in Table

Since _{8}=2,

Using (38) and (42),

From (33) and (43), we get

For more details used in (43) and (44), see Lemma 1.5 and Theorem 2.1 in

Theorem 1

For

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 (_{
i
}) are larger than _{
i
}>_{
i
}<

Remark 2

For

respectively, that is,

For

and

The detailed method to compute conditional equilibrium distributions is given in Section 6 in

Mean value of the random variable with a binomial distribution,

Mean value of the random variable with a Poisson distribution,

Simulation results

Recall that the normalized species numbers after a time change are defined as

Using the limiting models in the three time scales given in (3)-(5), we approximate the species numbers in the full model by unnormalizing the species numbers and applying time change backward as

using a real value _{0}=100 for the parameter. In Figures _{
i
}(

Simulation results when

**Simulation results when ****= 0.** Simulation of the full model (left) and that of approximation using the limiting model (right) when the time is of order

Simulation results when

**Simulation results when **** = 1.** Simulation of the full model (left) and that of approximation using the limiting model (right) when the time is of order _{0}(=100).

Simulation results when

**Simulation results when ****= 2.** Simulation of the full model (left) and that of approximation using the limiting model (right) when the time is of order

Simulation results when

**Simulation results when ****= 2 (continued).** Simulation of the full model (left) and that of approximation using the limiting model (right) when the time is of order _{6}and _{7}.

In Figure _{8}(_{8}. Therefore, consumption of _{8}may not be captured well in the approximation.

In Figure _{8}shown in Figure _{8} in the full model given in Figure _{8}(_{8}. The remaining three species, _{23}, _{6}, and _{7}are approximated by stochastic variables. The increasing species number of _{23} in time and the rapid decrease in species number of _{6}are well captured by the limiting model. The species numbers of _{7}are described by stochastic variables both in the full model and in the limiting model. The behavior of _{7}in two models is not exactly the same, and discrepancy of the mean species numbers of _{7} comes from the approximation of _{4}(_{7}is approximated as a stochastic process decreasing by 1 with the propensity proportional to _{4}(_{7} in the limiting model than that in the full model.

In Figure _{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 _{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 _{1} is random, but its standard deviation is very small. Therefore, in one realization of simulation of the limiting model, behavior of _{1}appears as constant. Since all the remaining variables in the limiting model are governed by the variable for _{1} and they satisfy the stochastic differential equations, evolution of one sample path of the species numbers for _{23}, _{4}, _{5}, _{8}, and _{9} in the limiting model looks like a solution of the system of ordinary differential equations.

In Figure _{6}and _{7}in the full model and their averaged values in the limiting model. Note that the species numbers for _{6} and _{7} do not appear in the limiting model, since their values are approximated in terms of other species numbers. Therefore, the difference between mean species numbers for _{6}and _{7}in the full model and those in the approximation does not affect the error directly. For _{6} plays an important role in the evolution of _{7} gives the conditional mean value for _{7} in the limiting model, we compare the species numbers of _{6}and _{7} in the full model and the approximated averaged values in the limiting model. In Figure

in time. They are stochastic variables determined by the ones in the limiting model with very small fluctuations. Since _{6}and _{7}, _{6}(_{7}(

In Figure _{6} and _{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.

Error estimates

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 _{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 _{0}is replaced by a large parameter

In this section, we estimate an error between the normalized species numbers and their limit given in (5) at the times of 10,000 sec. Define

and denote _{23}(_{4}(_{5}(_{8}(_{9}(^{
T
} as a limit of ^{
N
}(^{
N
}(_{
i
}(

for _{23}, _{4}, _{5}, _{8}, and _{9}.

Remark 3

For ^{
N
}(

where

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 ^{−1/2}. In our approach

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 _{
N
}. Among the species _{23}, _{4}, _{5}, _{8}, and _{9}, the species number of _{23}is scaled with the smallest exponent, and thus noise in the limit of _{
N
}=^{1/2}and the error between the scaled species numbers and their limit is of order _{
N
}and

Conclusions

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.

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.

Competing interests

The author(s) declare that they have no competing interests.

Authors’ contributions

Based on the model of heat shock response of

Acknowledgements

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.