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 [1] [2].

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 [3]

∂ 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 [4] and the Random Time-Change (RTC) algorithm [17] [18] based on the RTC representation of Kurtz [19].

The Chemical Master Equation (1) is computationally intractable for the majority of biochemical systems encountered in applications. To deal with this difficulty, Gillespie [4] [5] proposed an exact Monte Carlo strategy for the Chemical Master Equation, referred to as the stochastic simulation algorithm (SSA).

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. [18] and the Coupled Finite-Difference (CFD) scheme proposed by Anderson [17]. For further information on the CFD method, using the random time change representation [19] of the Markov process X ( t ) , we refer the reader to [17]. The CRN method is presented below. Notice that the CRN and CFD techniques are based on exact Monte Carlo simulation schemes for the Chemical Master Equation.

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 [18]. To estimate the sensitivity of the mean E ( X ( t , c ) ) , with respect to a parameter c, two systems will be considered: the unperturbed system X ( t , c ) and the perturbed counterpart X ( t , c + h ) . The CRN method applies Gillespie’s algorithm with the same string of common of random numbers to generate the coupled perturbed and unperturbed paths, corresponding to the parameters c + h and c, respectively. The use of a common random number sequence reduces the variance of the estimator, thus increases its accuracy for the same number of sample paths. The CRN algorithm consists of the following steps:

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 [17]. The CRN algorithm is a popular technique for sensitivity estimation, being efficient and easy to implement. The parametric sensitivity estimators discussed above is utilized to simplify complex stochastic models of homogeneous biochemical networks. Alternatively, a finite-difference scheme may be employed for estimating sensitivities by approximate Monte Carlo simulation techniques [20] [21], such as adaptive tau-leaping methods; these strategies are efficient on moderately stiff to very stiff models.

The infectious disease model is a simple example of a two species reaction network [23]. The reaction channels of the infectious disease model and their rate parameters are given in Table 1. The model is simulated on the interval [ 0,10 ] with initial conditions S 1 ( 0 ) = 20 and S 2 ( 0 ) = 40 .

The parametric sensitivities with respect to c₂ are estimated using the CRN and the CFD methods on 10,000 sample paths. Figure 1 depicts the evolution in time of the fully normalized sensitivities of the species S₁ and S₂, with respect to the kinetic parameter c₂. 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 c₂ are very small. Thus, reaction R₂ is a good candidate for elimination, and this is validated by solving the optimization problem.

The full and reduced (without reaction R₂) models are simulated with the SSA on 10,000 trajectories. The means and standard deviations of the molecular counts of S₁ and S₂ as functions of time, for the full and reduced models, are plotted in Figure 2 and Figure 3, respectively. The match between the numerical results for the reduced, in which reaction R₂ is deleted, and the full models is excellent in each case.

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 [24].

The EGFR model reactions and their rate parameter values are given in Table 2. The molecular species and their initial molecular counts are listed in Table 3. Assume a cellular volume of 3 × 10⁻¹² 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 [25].

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, c₂₁, is displayed in Figure 4. The set identified at this step constitutes an initial selection of reactions which are good candidates

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 Figure 5 and Figure 6, respectively. Similar results are obtained for the remaining species (not shown, for brevity). Notice the excellent agreement between the results produced with the reduced (with reactions R 6 , R 19 , R 20 , R 21 , R 22 , R 34 and R 35 turned off) and the full systems. Consequently, the proposed model reduction strategy provides very good results on this large size, stiff biochemical reaction network.

The final numerical experiment is performed on the gemcitabine (2,2-difluorodeoxycytidine, dFdC) biochemical network, which is a critical, real-world model [26] [27] designed to study the mechanisms of resistance to gemcitabine effectiveness. Gemcitabine is a chemotherapy drug, commonly administered for the

treatment of non-small cell lung cancer, breast, pancreatic and prostate cancers [26]. Reactions 1, 2, 3 and 4 illustrate how the gemcitabine is carried into cells. Then, in reaction 5, it is phosphorylated by the enzyme (dCK) to become the gemcitabine monophosphate (dFdC-MP). The species dFdC-MP is phosphorylated to its diphosphorylated form (dFdC-DP) in reactions 7 and 8. The triphosphorylated form (dFdC-TP) is created in reactions 9 and 10. Gemcitabine is metabolized to 2,2-difluorodeoxycytidine (dFdU), which can be phosphorylated to its diphosphate (dfdU-DP) and triphosphate (dFdU-TP), these processes are depicted in reactions 11 - 16. Reaction 19 is the DNA incorporation of dFdC-TP, while reaction 21 indicates the synthesis of CDP. Reaction 23 is nucleoside diphosphate kinase. The inhibition of dCK by binding to dCTP is presented in the 25-th reaction.

This is a large model, consisting of 22 species undergoing 29 reactions. The reactions and their rate parameters are listed in Table 4. The initial molecular counts of all species are zero, except for dCK = 1000, RR = 1000, dCMPD = 1000, CDP = 2000 and dFdC-out = 10,000. The model is studied on the time interval [ 0,20 ] .

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 Figure 7 and Figure 8, we compare the plots of the time-evolution of the the means and standard deviations for the molecular counts predicted by the full and the reduced model. In the reduced network, the following reactions were eliminated: R 2 , R 3 , R 12 , R 14 , R 16 and R 18 . As with the previous models, the proposed reduction technique generates a very accurate simplified model, when compared to the original one. The Gemcitabine model exhibits stiffness and has many reactions and species. Hence, it is quite challenging to simulate and analyze in its original formulation. In conclusion, the proposed simplifying strategy performs very well on this complex and computationally demanding biochemical model.