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Biochemical systems have numerous practical applications, in particular to the study of critical intracellular processes. Frequently, biochemical kinetic models depict cellular processes as systems of chemical reactions. Many biological processes in a cell are inherently stochastic, due to the existence of some low molecular amounts. These stochastic fluctuations may have a great effect on the biochemical system’s behaviour. In such cases, stochastic models are necessary to accurately describe the system’s dynamics. Biochemical systems at the cellular level may entail many species or reactions and their mathematical models may be non-linear and with multiple scales in time. In this work, we provide a numerical technique for simplifying stochastic discrete models of well-stirred biochemical systems, which ensures that the main properties of the original system are preserved. The proposed technique employs sensitivity analysis and requires solving an optimization problem. The numerical tests on several models of practical interest show that our model reduction strategy performs very well.

Modelling and simulation of cellular processes are subjects of significant interest in fields such as Computational and Systems Biology. Biological processes at the level of a single cell are commonly portrayed as systems of biochemical reactions. The dynamics of various biochemically reacting systems may exhibit random fluctuations, due to some molecular species which exist in low amounts. These random fluctuations could have critical implications in biology [

Biochemical processes at the cellular level may entail many biochemical species undergoing a significant number of reactions. The mathematical models of such biochemical networks have high dimension, and thus their numerical simulation is quite demanding. Moreover, numerous biochemical systems evolve on multiple time scales, leading to stiffness. Stiffness is a challenge for numerical simulations. A large number of the species and/or reactions of some biochemical systems encountered in applications imply that their mathematical models have a significant number of parameters. Usually, these parameters cannot be measured accurately. For many biochemical networks, it is difficult to determine the parts of the system which are critical in determining their behaviour. Therefore, it is critical to design accurate and robust strategies to simplify these complex biochemical systems, which maintain the key properties of the original network. The reduced models will be easier to analyze and simulate numerically. In addition, they can be utilized to predict and control the behaviour of the system. Furthermore, the simplified models have a lower number of parameters, which thus become easier to determine. Reduction techniques for deterministic models of biochemical systems include species and reaction lumping methods [

This paper proposes a new technique for reducing the complexity of stochastic discrete models of homogeneous biochemical networks. The discrete stochastic model under consideration is the Chemical Master Equation. The Chemical Master Equation accurately describes the dynamics of a wide range of well-stirred, realistic biochemical systems. The reduced reaction mechanisms generated with the new technique are easier to understand and manipulate and have fewer parameters.

Local sensitivity analysis quantifies the variations in the system’s behaviour caused by small changes in its parameters. Our method relies on estimating (local) parametric sensitivities of the Chemical Master Equation model. These parametric sensitivities are approximated using the Coupled Finite Difference (CFD) scheme [

The paper is organized as follows. In Section 2, we present the background on stochastic discrete models of homogeneous biochemical systems, and methods to simulate and estimate parametric sensitivities for these models. In Section 3, we propose a new model reduction strategy for the Chemical Master Equation. The advantages of the proposed strategy are illustrated on three critical models arising in applications, namely the infection, the epidermal growth factor receptor signalling pathway and the gemcitabine biochemical systems, in Section 4.

The evolution in time of homogeneous biochemical systems is governed by the Chemical Master Equation. This model has been successfully utilized to study vital biological processes, such as gene expression and regulation [

Consider N biochemical species S 1 , ⋯ , S N interacting through M chemical reactions, R 1 , ⋯ , R M . Assume that the system is homogeneous, and at constant volume and thermal equilibrium. It is described by the state vector:

X ( t ) = [ X 1 ( t ) , X 2 ( t ) , ⋯ , X N ( t ) ] T

where X i ( t ) indicates the amount of S i molecules at time t. The biochemical system state, X ( t ) , is a Markov process continuous in time and discrete in space. To each reaction R j it corresponds a state change vector ν j ≡ ( ν 1 j , ⋯ , ν N j ) T , with ν i j representing the number of S i molecules consumed or produced when a reaction R j fires. The matrix ν = { ν i j } 1 ≤ i ≤ N ,1 ≤ j ≤ M , where N is the number of reactants and M is the number of reactions, represents the stoichiometric matrix. In addition, a propensity function a j ( x ) is used for describing a reaction R j . By definition, a j ( x ) d t is the probability of one reaction R j firing during [ t , t + d t ] , provided that at time t the system was in state x .

Each propensity function follows the mass-action kinetics; thus, for the first order reaction S m → c j products , it is a j ( X ( t ) ) = c j X m ( t ) , and for the second order reaction S m + S n → c j products , with m ≠ n , the propensity is a j ( X ( t ) ) = c j X m ( t ) X n ( t ) . Lastly, the propensity of S m + S m → c j products may be expressed as a j ( X ( t ) ) = c j X m ( t ) ( X m ( t ) − 1 ) / 2 .

For t ≥ t 0 , let P ( x , t | x 0 , t 0 ) be the conditional probability that the state vector at time t is X ( t ) = x , given that at time t 0 it was X ( t 0 ) = x 0 . Then, P ( x , t | x 0 , t 0 ) satisfies the Chemical Master Equation [

∂ P ( x , t | x 0 , t 0 ) ∂ t = ∑ j = 1 M [ a j ( x − ν j ) P ( x − ν j , t | x 0 , t 0 ) − a j ( x ) P ( x , t | x 0 , t 0 ) ] , (1)

where ν j is the state-change vector for reaction R j . The Chemical Master Equation is a refined stochastic discrete model of homogeneous biochemical systems. It is worth mentioning that, generally, the CME is a high dimensional model. Exact Monte Carlo techniques for the Chemical Master Equation model were developed in the literature. They include Gillespie’s algorithm [

The Chemical Master Equation (1) is computationally intractable for the majority of biochemical systems encountered in applications. To deal with this difficulty, Gillespie [

Gillespie’s algorithm proceeds as follows:

1) Initialize t ← t 0 , X ( t 0 ) ← x 0 .

2) At time t, evaluate

a s u m ( X ( t ) ) : = ∑ k = 1 M a k ( X ( t ) ) .

3) Draw a unit uniform random number r 1 , and take

τ = [ 1 / a s u m ( X ( t ) ) ] ln ( 1 / r 1 ) .

4) Draw a unit uniform random number r 2 and denote by j the smallest integer satisfying

∑ k = 1 j a k ( X ( t ) ) > r 2 a s u m ( X ( t ) ) .

5) Update X ( t + τ ) ← X ( t ) + ν j and t ← t + τ .

6) Return to step 2 or else stop.

Sensitivity analysis is a key tool for investigating the robustness properties of a model with respect to perturbations in its parameters, for estimating these parameters or for guiding model reduction. Models of biochemically reacting systems depend on several parameters, such as the reaction rate parameters and the initial conditions. In many instances, the values of several such parameters are not available or cannot be evaluated accurately. For this reason, it is critical to investigate how variations in model parameters influence the behaviour of the system. Local sensitivity analysis examines how the system dynamics is influenced by small changes in model parameters. A model is sensitive with respect to one of its parameters if the behaviour of the system varies significantly with minor perturbations in that parameter; else it is robust with respect to variations in that parameter. Parametric sensitivities help determine which components of the biochemical network control the model’s behaviour. If a reaction is such that some parametric sensitivities with respect to its kinetic parameter are large, then it is considered important.

For a deterministic system, the parametric sensitivity of the state vector, X ( t , c ) , with respect to a parameter c can be computed by S = ∂ X ( t , c ) / ∂ c . Estimating parametric sensitivities of a stochastic discrete model can be a daunting task. In this work, we employ finite-difference approximations of the local parametric sensitivities for the Chemical Master Equation model. The sample paths necessary for the finite-difference estimations are generated with Monte Carlo simulations. In the discrete stochastic setting, if E is the expected value, f is a function of interest, X ( t , c ) represents the system state at time t, corresponding to the parameter c, then a finite-difference estimation of the sensitivity of E ( f ( X ( t , c ) ) ) with respect to c is

S ≈ E [ f ( X ( t , c + h ) ) ] − E [ f ( X ( t , c ) ) ] h . (2)

Here h is a small perturbation of the parameter c. In what follows, c represents a kinetic rate parameter.

Let X ( n ) ( t , c + h ) and X ( n ) ( t , c ) be trajectories generated by Monte Carlo simulations, for the parameters c + h and c, respectively. Here n is the trajectory index ( 1 ≤ n ≤ L ) and L is the total number of trajectories. In this case, a finite-difference numerical approximation of the sensitivity, utilizing Monte Carlo simulations, is

S ≈ 1 L ∑ n = 1 L 1 h [ f ( X ( n ) ( t , c + h ) ) − f ( X ( n ) ( t , c ) ) ] . (3)

It is widely known that variance reduction strategies allow more precise estimations of the sensitivity, with a smaller numbers of simulations. To reduce the variance of the estimator, finite-difference sensitivity estimators couple the perturbed and unperturbed trajectories. Among the most effective and accurate finite-difference parametric estimators for stochastic discrete models of biochemical systems are the Common Random Number (CRN) method due to Rathinam et al. [

Common Random Number (CRN) The common random number technique is utilized for estimating parametric sensitivities of stochastic discrete biochemical kinetic models. The CRN estimator has a low variance [

1) Take c = c j . For each n ≤ L , with L being the number of sample paths,

2) Generate an SSA path for the unperturbed system (for parameter c), with a sequence of independent uniform (0, 1) random numbers, X ( n ) ( t , c ) .

3) Generate an SSA path for the perturbed system (for parameter c + h ), with the same sequence of independent uniform (0, 1) random number, X ( n ) ( t , c + h ) .

4) Calculate s ( n ) ( t ) = ( f ( X ( n ) ( t , c + h ) ) − f ( X ( n ) ( t , c ) ) ) / h .

5) End loop over n.

6) Compute the mean of { s ( n ) ( t ) } n ≤ L .

Remark that the perturbation h should be small enough to decrease the truncation error associated with finite-difference approximations of derivatives, but large enough to increase the convergence rate of the sensitivity estimation [

In this section, we introduce a new procedure to simplify stochastic discrete biochemical networks, which are modelled with the Chemical Master Equation. The goal of this procedure is to determine a reduced biochemical reaction system which retains the essential features of the full system, such as its stability features, nonlinear behaviour and physical interpretation of the elementary reactions. The reduced biochemical model will be faster to solve numerically and easier to investigate and understand. Local sensitivity analysis is applied to uncover the unimportant reactions of the full biochemical network. Also, it seeks to find the essential reactions, which should be retained in a simplified model. Further, our strategy extends the global approach for continuous deterministic chemical system reduction proposed in [

Initially, a local sensitivity analysis is performed for the stochastic discrete model of the original biochemical system. Of interest are the kinetic parameters, c = c i . Finite-difference methods, such as the CFD or the CRN algorithms, are employed to estimate the local parametric sensitivities. This allows a preselection of the crucial reactions in the network, those for which the molecular amounts of all, or of the important species, are very sensitive with respect to small variations in the associated rate parameter. Let { R i } i ∈ I c r , for some subset I c r of indexes { 1, ⋯ , M } , be these crucial reactions. These reaction channels should be kept in the system. Moreover, the sensitivity analysis also helps determine the reactions for which small perturbations in their rate parameters cause insignificant changes in the overall behaviour of the system. More precisely, the kinetic parameter c i of such a reaction satisfies

| c i E ( X k ( t , c i ) ) ∂ ∂ c i E ( X k ( t , c i ) ) | ≪ 1 ,

for all the quantities of interest X k . These reactions could be unimportant, and may be considered for deletion.

Stability is a desired property of a simplified reaction network and a reduction technique based on local sensitivity analysis of the original system alone cannot guarantee it. As a consequence, we shall use a global approach to ensure that our strategy leads to a stable reduced system. For this, we consider the following continuous optimization problem for deterministic models (see also [

min ‖ x − z ‖ suchthat x ′ = ν F ( x ) , x ( 0 ) = x 0 z ′ = ν D F ( z ) , z ( 0 ) = x 0 , 0 ≤ t ≤ T l ≤ ∑ i = 1 M d i ≤ u , 0 ≤ d i ≤ 1 , for 1 ≤ i ≤ M , g ( d 1 , ⋯ , d M ) ≤ r . (4)

Note that x and z are the state vectors of the original and the reduced biochemical systems, respectively, modelled with the reaction rate equations. Also, ν is the stoichiometric matrix, F ( ⋅ ) represents the vector of propensities and D is a diagonal matrix, with d i being the i-th row diagonal entry. Observe that d i = 1 if the reaction is kept in the reduced system and d i = 0 , if it is removed.

Here l and u are the simplified model’s lower and upper bounds for the total reaction count, and r > 0 is a small constant. Also, g denotes a non-linear function which becomes 0 when d i are either 0 or 1, for all 1 ≤ i ≤ M . We consider the following nonlinear function

g = ∑ i = 1 M ( d i − d i 2 ) β , (5)

with β ≥ 2 being a parameter. In our implementation, β is set to 2. By relaxing the condition

g ( d 1 , ⋯ , d M ) = 0 to g ( d 1 , ⋯ , d M ) ≤ r ,

the discrete optimization is simplified to a continuous optimization problem. Once a solution of the continuous optimization problem is found, the coefficients d i are rounded to 0 if they are very small; otherwise they are rounded to 1. This result may yield a local rather than a global minimum. Still, for models with many reactions, i.e., a large parameter space, a local minimum would be considered a satisfactory solution for the simplifying strategy. Also, the choice of the parameters (e.g., u, r), would affect how closely the dynamics of the reduced system matches that of the original version. There is a trade-off between improving the accuracy of the simplified mechanism and eliminating many reactions. To obtain a simpler optimization problem, after we identified the important reactions, { R i } i ∈ I c r , and we assigned their d i = 1 , we solve (4) over a smaller set of parameters, { d j } j ∉ I c r .

Remark that the optimization problem is formulated in a continuous deterministic framework. Empirically, we found that posing an optimization problem for the reaction rate equations, rather than for the much more challenging Chemical Master Equation model, yields an accurate reduction mechanism for biochemical systems with a low to moderate level of noise. Moreover, this procedure can handle efficiently large biochemical networks as well.

For solving the optimization problem (4), we utilize GEKKO [

In this section, we illustrate the advantages of the proposed model reduction technique on three systems of biochemical reactions: one is a low size but key test problem, while the other two are high-dimension biochemical networks, with applications to cancer research. In each case, sensitivity analysis is performed to identify which parts of the network do not play a significant role in its overall behaviour. More importantly, the analysis finds the reactions which are critical for the dynamics of the biochemical system and set their d i = 1 . After this preselection, the global optimization strategy is employed to select the reactions which can be removed from the system. Finally, the reduced and full models are simulated with Gillespie’s algorithm over a large ensemble of trajectories. The results are used for comparing the evolution of the mean and standard deviation of the molecular amounts for each species, in the original and simplified models.

The infectious disease model is a simple example of a two species reaction network [

The parametric sensitivities with respect to c_{2} are estimated using the CRN and the CFD methods on 10,000 sample paths. _{1} and S_{2}, with respect to the kinetic parameter c_{2}. The approximations obtained with the two finite-difference sensitivity estimators, the CFD and the CRN algorithms, are in good agreement. We note that, for i = 1 and 2, the normalized sensitivities with respect to c_{2} are very small. Thus, reaction R_{2} is a good candidate for elimination, and this is validated by solving the optimization problem.

The full and reduced (without reaction R_{2}) models are simulated with the SSA on 10,000 trajectories. The means and standard deviations of the molecular counts of S_{1} and S_{2} as functions of time, for the full and reduced models, are plotted in _{2} is deleted, and the full models is excellent in each case.

Reaction channel | Reaction rate | |
---|---|---|

R_{1} | S 1 → ∅ | c 1 = 2.0 |

R_{2} | S 2 → ∅ | c 2 = 0.1 |

R_{3} | ∅ → S 1 | c 3 = 25 |

R_{4} | ∅ → S 2 | c 4 = 75 |

R_{5} | S 1 + S 2 → S 1 + S 1 | c 5 = 0.05 |

The epidermal growth factor receptor signalling pathway participates in cell differentiation and proliferation. The role of the epidermal growth factor in cancer cell proliferation has recently been the subject of intense research. The biochemical reaction network which models the EGFR signalling pathway consists of 23 molecular species undergoing 47 reaction channels [

The EGFR model reactions and their rate parameter values are given in ^{−12} liters, with a nano-Mole concentration = 1800 (i.e., 6.023 × 10 + 23 × 3.00 × 10 − 12 × 10 − 9 ) molecules per cell. For computational purposes, we shall consider molecular amounts rather than concentrations of the species. This requires rescaling the rate parameters. The problem is solved numerically on the time interval [ 0,100 ] . Remark that the EGFR model is stiff, given that it spans multiple time-scales; indeed, some reactions are slow others are fast. Stiffness is a major challenge for numerical simulation [

Initially, we use finite-difference approximations of the sensitivity by employing the CRN scheme with 2000 trajectories. We identify a set of reactions with small normalized parametric sensitivities for all molecular species. A sample plot of the evolution in time of the normalized sensitivities of various species with respect to one of the parameters, c_{21}, is displayed in

Reaction channel | Reaction rate^{a} | |
---|---|---|

R_{1} | EGF + R → Ra | 3.0E−3 |

R_{2} | Ra → EGF + R | 6.0E−2 |

R_{3} | Ra + Ra → R2 | 1.0E−2 |

R_{4} | R2 → Ra + Ra | 1.0E−1 |

R_{5} | R2 → RP | 1.0 |

R_{6} | RP → R2 | 1.0E−2 |

R_{7} | RP → R2 [ MM ] | 4.5E+2, 5.0E+1 |

R_{8} | RP + PLCg → RPL | 6.0E−2 |

R_{9} | RPL → RP + PLCg | 2.0E−1 |

R_{10} | RPL → RPLP | 1.0 |

R_{11} | RPLP → RPL | 5.0E−2 |

R_{12} | RPLP → RP + PLCgP | 3.0E−1 |

R_{13} | RP + PLCgP → RPLP | 6.0E−2 |

R_{14} | PLCgP → PLCg [ MM ] | 1.0, 1.0E+2 |

R_{15} | RP + Grb → RG | 3.0E−3 |

R_{16} | RG → RP + Grb | 5.0E−2 |

R_{17} | RG + SOS → RGS | 1.0E−2 |

R_{18} | RGS → RG + SOS | 6.0E−2 |

R_{19} | RGS → RP + GS | 3.0E−2 |

R_{20} | RP + GS → RGS | 4.5E−3 |

R_{21} | GS → Grb + SOS | 1.5E−3 |

R_{22} | Grb + SOS → GS | 1.0E−4 |

R_{23} | RP + Shc → RSh | 6.0E−2 |

R_{24} | RSh → RP + Shc | 6.0E−1 |

R_{25} | RSh → RShP | 6.0 |

R_{26} | RShP → RSh | 6.0E−2 |

R_{27} | RShP → RP + ShP | 3.0E−1 |

R_{28} | RP + ShP → RShP | 9.0E−4 |

R_{29} | ShP → Shc [ MM ] | 1.7, 3.4E+2 |

R_{30} | RShP + Grb → RShG | 3.0E−3 |

R_{31} | RShG → RShP + Grb | 1.0E−1 |

R_{32} | RShG → RP + ShG | 3.0E−1 |

R_{33} | RP + ShG → RShG | 9.0E−4 |

R_{34} | RShG + SOS → RShGS | 1.0E−2 |

R_{35} | RShGS → RShG + SOS | 4.5E−3 |
---|---|---|

R_{36} | RShGS → RP + ShGS | 1.2E−1 |

R_{37} | RP + ShGS → RShGS | 2.4E−4 |

R_{38} | ShP + Grb → ShG | 3.0E−3 |

R_{39} | ShG → ShP + Grb | 1.0E−1 |

R_{40} | ShG + SOS → ShGS | 3.0E−2 |

R_{41} | ShGS → ShG + SOS | 6.0E−2 |

R_{42} | ShGS → GS + ShP | 1.0E−1 |

R_{43} | GS + ShP → ShGS | 2.0E−2 |

R_{44} | RShP + GS → RShGS | 9.0E−3 |

R_{45} | RShGS → RShP + GS | 4.29E−2 |

R_{46} | PLCgP → PLCgPI | 1.0 |

R_{47} | PLCgPI → PLCgP | 3.0E−2 |

Sid | Species | N |
---|---|---|

1 | EGF | 23,040,183 |

2 | R | 335 |

3 | Ra | 11,774 |

4 | R2 | 9514 |

5 | RP | 1360 |

6 | R-PL | 59 |

7 | R-PLP | 91 |

8 | R-G | 947 |

9 | R-G-S | 300 |

10 | R-Sh | 23 |

11 | R-ShP | 618 |

12 | R-Sh-G | 195 |

13 | R-Sh-G-S | 124 |

14 | G-S | 1776 |

15 | ShP | 152,296 |

16 | Sh-G | 56,545 |

17 | PLCg | 1195 |

18 | PLCgP | 2160 |

19 | PLCg-I | 185,357 |

20 | Grb | 32,547 |

21 | Shc | 2634 |

22 | SOS | 4689 |

23 | Sh-G-S | 52,301 |

for deletion. Given the large dimension of the EGFR model, we also carry a sensitivity analysis to choose several reactions which are very important. We assign the value of their coefficients as d i = 1 . At this step, we selected a group of 17 such reactions from a total of 47 reactions. This reduces the dimension of the optimization problem. Subsequently, we solve the optimization problem (4) to find a reduced biochemical model. We used the following parameter values: l = 2 , u = 42 and r close to 1. As a result, the following reactions are selected for elimination R 6 , R 19 , R 20 , R 21 , R 22 , R 34 and R 35 .

Finally, we run Gillespie’s algorithm over 2000 trajectories for the original and the reduced model and compare the output. The means and standard deviations of the molecular amounts of various species, computed for the full and reduced models, are plotted as functions of time in

The final numerical experiment is performed on the gemcitabine (2,2-difluorodeoxycytidine, dFdC) biochemical network, which is a critical, real-world model [

treatment of non-small cell lung cancer, breast, pancreatic and prostate cancers [

This is a large model, consisting of 22 species undergoing 29 reactions. The reactions and their rate parameters are listed in

We begin by performing a local sensitivity analysis of the Gemcitabine model. To approximate its parametric sensitivities, we apply the common reaction number (CRN) algorithm with 1000 trajectories. At this step, we select some reactions which local sensitivity analysis predicts to be possible choices for deletion. These reactions are then considered an initial collection, when solving the optimization problem for this system. The model has many parameters and finding the global minimum in its large-dimensional parameter space is a challenge. Therefore, we use sensitivity analysis to determine a set of reactions deemed critical in driving the behaviour of the system. For this model, we selected 14 important reactions (and set their d i = 1 ) out of a total of 29. Also, we choose the following parameter values for the optimizer: l = 2 and u = 23 and r ≈ 1 . Solving the optimization problem suggests that reactions R 2 , R 3 , R 12 , R 14 , R 16 and R 18 may be eliminated without a significant impact on the accuracy and stability properties of the model.

To validate the solution of the optimizer, we apply the SSA with 1000 runs to simulate both the reduced (resulted after turning off reactions R 2 , R 3 , R 12 , R 14 , R 16 and R 18 ) and the original biochemical network. In

Reaction channel | Reaction Rates | |
---|---|---|

R_{1} | dFdC-out → k 1 dFdC | k 1 = 99.7234 |

R_{2} | dFdC → k 2 dFdC-out | k 2 = 2.61675 E − 3 |

R_{3} | dFdC-out → k 3 dFdU | k 3 = 4.72336 E − 5 |

R_{4} | dFdU → k 4 dFdU-out | k 4 = 5.08194 E − 1 |

R_{5} | dFdC + dCK → k 5 dFdC-MP + dCK | k 5 = 1.04994 E − 2 |

R_{6} | dFdC-MP → k 6 dFdC | k 6 = 8.75208 E − 1 |

R_{7} | dFdC-MP → k 7 dFdC-DP | k 7 = 23.7162 |

R_{8} | dFdC-DP → k 8 dFdC-MP | k 8 = 2.12216 |

R_{9} | dFdC-DP → k 9 dFdC-TP | k 9 = 25.2037 |

R_{10} | dFdC-TP → k 10 dFdC-DP | k 10 = 14.4908 |

R_{11} | dFdU + dCK → k 11 dFdU-MP + dCK | k 11 = 9.68 E − 3 |

R_{12} | dFdU-MP → k 12 dFdU | k 12 = 5.60415 E − 4 |

R_{13} | dFdU-MP → k 13 dFdU-DP | k 13 = 0.7844 |

R_{14} | dFdU-DP → k 14 dFdU-MP | k 14 = 4.20541 E − 2 |

R_{15} | dFdU-DP → k 15 dFdU-TP | k 15 = 1.64322 |

R_{16} | dFdU-TP → k 16 dFdU-DP | k 16 = 9.05139 E − 4 |

R_{17} | dFdC → k 17 dFdU | k 17 = 4.76746 E − 3 |

R_{18} | dFdC-MP + dCMPD → k 18 dFdU-MP + dCMPD | k 18 = 4.559 E − 5 |

R_{19} | dFdC-TP → k 19 dFdC-TP-DNA | k 19 = 0.544456 |

R_{20} | dFdU-TP → k 20 dFdU-TP-DNA | k 20 = 7.37496 E − 3 |

R_{21} | ∅ → k 21 CDP | k 21 = 100 |

R_{22} | CDP + RR → k 22 dCDP + RR | k 22 = 0.5 |

R_{23} | dCDP → k 23 dCTP | k 23 = 252.037 |

R_{24} | dCTP → k 24 CTP-DNA | k 24 = 5 |

R_{25} | dCTP + dCK → k 25 dCTP-dCK | k 25 = 1.0 E − 4 |

R_{26} | dCTP-dCK → k 26 dCTP + dCK | k 26 = 0.1 |

R_{27} | dFdC-DP + RR → k 27 dFdC-DP-RR | k 27 = 1.0 E − 4 |

R_{28} | dFdC-TP + dCMPD → k 28 dFdC-TP-dCMPD | k 28 = 100 |

R_{29} | dFdC-TP-dCMPD → k 29 dFdC-TP + dCMPD | k 29 = 0.1 |

In the present work, we proposed a numerical procedure for model reduction of homogeneous, discrete stochastic biochemical networks. The dynamics of these networks are governed by the widely used model of the Chemical Master Equation. This model has a diverse range of critical practical applications. Numerous biochemical networks arising in practice are complex, consisting of a large number of species subject to many reaction channels. In addition, the network of interactions of some components of the biochemical system may be quite complicated. Furthermore, biochemical processes present at the cellular level typically entail slow and fast reactions, meaning that their mathematical models are stiff. Stiffness and/or high dimensionality constitute significant challenges for numerical simulation and analysis of these models.

The model reduction procedure developed in this paper utilizes sensitivity analysis of stochastic discrete models of biochemical systems and requires solving a nonlinear optimization problem. The procedure is expected to have an excellent performance on a broad class of biochemical systems, with moderate levels of noise, and various degrees of stiffness. We tested the reduction methodology described above on three realistic biochemical networks, two of these having applications to cancer research. The numerical experiments carried out on these models show that the reduced biochemical network retains the properties of the original version while preserving its overall behaviour.

This work was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada (NSERC).

The authors declare no conflicts of interest regarding the publication of this paper.

Gholami, S. and Ilie, S. (2021) Reducing Stochastic Discrete Models of Biochemical Networks. Applied Mathematics, 12, 449-469. https://doi.org/10.4236/am.2021.125031