Efficient parametric analysis of the chemical master equation through model order reduction
- Steffen Waldherr^{1}Email author and
- Bernard Haasdonk^{2}
https://doi.org/10.1186/1752-0509-6-81
© Waldherr and Haasdonk; licensee BioMed Central Ltd. 2012
Received: 3 January 2012
Accepted: 18 May 2012
Published: 2 July 2012
Abstract
Background
Stochastic biochemical reaction networks are commonly modelled by the chemical master equation, and can be simulated as first order linear differential equations through a finite state projection. Due to the very high state space dimension of these equations, numerical simulations are computationally expensive. This is a particular problem for analysis tasks requiring repeated simulations for different parameter values. Such tasks are computationally expensive to the point of infeasibility with the chemical master equation.
Results
In this article, we apply parametric model order reduction techniques in order to construct accurate low-dimensional parametric models of the chemical master equation. These surrogate models can be used in various parametric analysis task such as identifiability analysis, parameter estimation, or sensitivity analysis. As biological examples, we consider two models for gene regulation networks, a bistable switch and a network displaying stochastic oscillations.
Conclusions
The results show that the parametric model reduction yields efficient models of stochastic biochemical reaction networks, and that these models can be useful for systems biology applications involving parametric analysis problems such as parameter exploration, optimization, estimation or sensitivity analysis.
Keywords
Background
The chemical master equation (CME) is the most basic mathematical description of stochastic biomolecular reaction networks[1, 2]. The CME is a generally infinite-dimensional linear differential equation. It characterizes the temporal development of the probabilities that the network is in any of its possible configurations, where the different configurations are characterized by the molecular copy numbers of the network’s chemical species.
Due to its infinite dimension, the CME is usually not directly solvable, not even with numerical methods. A recent breakthrough in the numerical treatment of the CME was the establishment of the finite state projection (FSP) method by Munsky and Khammash[3]. They showed that it is possible to compute a good approximation to the real solution by projecting the CME to a suitable finite subdomain of the network’s state space, and solving the resulting finite-dimensional linear differential equation on that domain. Nevertheless, the FSP approach still yields very high-dimensional models which are computationally expensive to simulate, even for small biochemical networks. The efficient simulation of the CME is an area of active research, and recently other simulation methods have been developed that can also be used for larger networks[4, 5].
Despite this progress, the direct simulation of the CME remains a computational bottleneck for common model analysis tasks in systems biology. It is especially problematic for tasks which require the repeated simulation of the model using different parameter values, for example identifiability analysis, parameter estimation, or model sensitivity analysis. Thereby, while a single or a few evaluations of a CME model with the FSP or other approaches may still be computationally feasible, the necessity of many repeated simulations will quickly render higher-level analysis tasks infeasible.
Mathematical methods that approximate the behaviour of a high-dimensional original model through a low-dimensional reduced model are a common way to deal with complex models. Especially for linear differential equations, model order reduction is a well established field and several methods to compute reduced order models are available[6]. Note that the step of generating a reduced model is usually computationally more expensive than a single or even a few simulations of the original high-dimensional model. But the simulation of the resulting reduced models is frequently orders of magnitude faster than the solution of the original model. So, model reduction is worth the effort if many repeated simulations are to be expected. Unfortunately, for analysis tasks which require the repeated model simulation with different parameters, classical model reduction methods are not helpful. With these methods, the reduced model depends on specific parameter values in the original model, and the reduction needs to be redone for different parameter values. Thus, for the mentioned analysis tasks, the model reduction process would have to be repeated for each new parameter value, and no gain in computational efficiency would typically be possible. While classical model reduction techniques have been applied to the CME in the past[7], they are not so suitable for parametric analysis tasks.
Fortunately, model reduction methods where parameters from the original model are retained as adjustable parameters also in the reduced model are now being developed. These methods allow to compute a reduced model which uses the same parameters as the original model, and where the reduced model can directly be simulated with any choice of parameter values[8–11].
The purpose of this paper is to introduce the application of these parametric model reduction methods to finite-state approximations of the chemical master equation, and to show possible usage scenarios of such an approach. The structure is as follows. In the following section, we introduce some background and notation concerning the modelling of chemical reaction networks and parametric model order reduction. We also show how the parametric model order reduction methods can in fact be applied to the CME. Afterwards, we apply the reduction technique on two reaction network models and corresponding parametric analysis tasks.
Methods
We start with some preparatory background on the chemical master equation (CME) and parametric model order reduction. This serves in particular to fix the notation used throughout the remainder of the article. Then the application of parametric model order reduction to the CME is introduced.
The chemical master equation
Reversible reactions can always be written in the form (1) by splitting the forward and reverse path into two separate irreversible reactions.
for${x}_{i}\in {\mathbb{N}}_{0}$, i = 1,…,n. As a short-hand notation for (3), we write p(t,x), with$x\in {\mathbb{N}}_{0}^{n}$.
To avoid needlessly complicated cases, we assume v_{ j }≠ v_{ k }for j ≠ k.
for$x\in {\mathbb{N}}_{0}^{n}$. The CME (7) is subject to an initial condition p(t_{0},x) = p_{0}(x) for$x\in {\mathbb{N}}_{0}^{n}$.
Despite being linear, the CME is hard to solve numerically. This is due to the problem that the state space is for most systems infinite-dimensional, since all possible states$x\in {\mathbb{N}}_{0}^{n}$ of the reaction network (1) must in general be considered. Instead of directly solving the CME (7), a number of alternative approaches to study the stochastic dynamics of biochemical reaction networks have been suggested. The most common approach is to generate a simulated realization of the stochastic process described by the reaction network (1), using for example the Gillespie algorithm[13]. In this approach, the probabilities p(t x) for the possible system states are obtained from many simulated realizations. However, since this requires a large number of realizations, it is computationally expensive.
We will frequently omit the parameter dependence of the solution (and other parametric quantities). Hence the solution P(t), as abbreviation of P(t θ), of (10) is an approximation to the solution p(t x) of the orginal CME on the domain Ω. Munsky and Khammash[3] have also derived an upper bound on the error between the solution P(t) computed via the FSP, and the solution of the original CME p(t x) on Ω.
i.e. C = (x^{(1)},…,x^{(d)}) with p = n.
The basic motivation for the model reduction presented here is that we are interested in parametric analysis of the model, where the model (10) has to be solved many times with different values for the parameters θ. Due to the typical high dimensions of the matrix A(θ), already a single simulation is computationally expensive, and analysis tasks requiring many repeated simulations are often computationally infeasible. Thus, the primary goal is to derive a reduced model which is rapidly solvable and provides an approximation$\u0177\left(t\right)$ to the output y(t), potentially without any consideration of the original state vector P(t).
Order reduction of parametric models
Model order reduction of parametric problems is a very active research field in systems theory, engineering and applied mathematics. We refer to[8, 10, 11] and references therein for more information on the topic.
Here, we apply the reduction technique for parametric problems presented in[9] adopted to our notation. It is based on two biorthogonal global projection matrices$V,W\in {\mathbb{R}}^{d\times r}$ with r ≪ d and W^{ T }V = Id, where r is the dimension of the reduced model. The matrix V is assumed to span a space that approximates the system state variation for all parameters and times. The construction of such matrices will be detailed in the next subsection.
The gain of computational efficiency in repeated simulations comes from a separation of the simulation task into a computationally expensive “offline” phase and a computationally cheap “online” phase. In the offline phase, suitable projection matrices V and W are computed without fixing specific parameter values. With the projection matrices, a reduced model with the same free parameters as the original model is computed. In the online phase, the reduced model is simulated with the actually chosen parameter values, which is typically several orders of magnitude faster than the simulation of the original model. For analysis tasks with repeated simulations, only the online phase has to be repeated for different choices of the parameter values, yielding an overall gain in computational efficiency.
Decomposition in parametric and non-parametric part
From the reduced state P_{ r }(t), an approximate state for the full system can be reconstructed at any desired time by$\widehat{P}\left(t\right)=V{P}_{r}\left(t\right)$. Also the difference between the approximated output$\u0177\left(t\right)$ and the output y(t) of the original model can be bounded by so called error estimators. A-posteriori error bounds for the reduced systems as considered here are given in[9].
Basis generation
Different methods for the computation of the projection bases V and W exist. In systems theory, methods like balanced truncation, Hankel-norm approximation or moment matching are applied, that approximate the input-output behaviour of a linear time-invariant system[6]. The resulting reduced models can be applied for varying input signals. Extensions to parametric problems exist, e.g.[8, 11]. As we do not have varying inputs in the problem studied here, we consider snapshot-based approaches to be more suitable. This means, the projection bases are constructed by solution snapshots, i.e. special solutions computed for selected parameter values.
The POD-Greedy procedure which is given in the pseudo-code below, starts with an arbitrary orthonormal initial basis${V}_{{N}_{0}}\in {\mathbb{R}}^{d\times {N}_{0}}$ and performs an incremental basis extension. The algorithm repeatedly identifies the currently worst resolved parameter (a), orthogonalizes the corresponding full trajectory with the current reduced space (b), computes a POD of the error trajectory (c), and inserts the dominant mode into the basis (d).
- 1.
N := N _{0}
- 2.
while ${\epsilon}_{N}:={\text{max}}_{\theta \in {\mathcal{P}}_{\mathit{\text{train}}}}\Delta (\theta ,{V}_{N})>{\epsilon}_{\mathit{\text{tol}}}$
- (a)
${\theta}^{\ast}:=\text{arg}{\text{max}}_{\theta \in {\mathcal{P}}_{\mathit{\text{train}}}}\Delta (\theta ,{V}_{N})$
- (b)
$E\left(t\right):=P(t,{\theta}^{\ast})-{V}_{N}\left({V}_{N}^{T}P\right(t,{\theta}^{\ast}\left)\right)$
- (c)
v _{N + 1} := POD(E)
- (d)
V _{N + 1} := [V _{ N },v _{N + 1}]
- (e)
N := N + 1
- (a)
- 3.
end while
Note that the algorithm is implemented such that the simulation of the full model, yielding P(t,θ) in (19), is only performed once for each θ in the training set${\mathcal{P}}_{\mathit{\text{train}}}$.
For concluding the basis generation, we set W := V. This satisfies the biorthogonality condition W^{ T }V = Id, as V has orthonormal columns by construction. In practice the time-integrals in (18) are realized by a finite sampling of the time interval.
A theoretical underpinning for the POD-Greedy algorithm has recently been provided by the analysis of convergence rates[19]. This is based on the approximation-theoretical notion of the Kolmogorov n-width${d}_{N}\left(\mathcal{F}\right)$ of a given set$\mathcal{F}\subset {\mathbb{R}}^{d}$, which quantifies how well the set can be approximated by arbitrary N-dimensional linear subspaces of${\mathbb{R}}^{d}$. The convergence statement for the case of exponential convergence then can be summarized as follows: If the set of solutions$\mathcal{F}:=\left\{P\right(t,\theta \left)\right|t\in [0,T],\theta \in \mathcal{P}\}\subset {\mathbb{R}}^{d}$ is compact and has an exponentially decaying Kolmogorov n-width${d}_{N}\left(\mathcal{F}\right)\le M{e}^{-a{N}^{\alpha}}$ for some M a α > 0 and all$N\in \mathbb{N}$, then the error sequence${\left({\epsilon}_{N}\right)}_{N\in \mathbb{N}}$ generated by the POD-Greedy procedure (cf. the definition in Step 2. in the pseudo code) also decays with an exponential rate,${\epsilon}_{N}\le \mathit{\text{CM}}{e}^{-c{N}^{\beta}}$ with suitable constants β c C > 0 depending on M,a,α. Thus, if the set of solutions can be approximated by linear subspaces with an exponentially decaying error term, then the POD-Greedy algorithm will in fact find an approximation with an exponentially decaying error term, though possibly with suboptimal parameters in the error bound.
Extensions of the POD-Greedy algorithm exist, e.g. allowing more than one mode per extension step, performing adaptive parameter and time-interval partitioning, or enabling training-set adaptation[15, 16, 20].
Reduced models of the parametrized chemical master equation
In this section, we describe how to apply the reduction method for parametrized models presented in the previous section to FSP models for the chemical master equation.
More generally, such a decomposition is also possible if reaction rate propensities can be decomposed into the product of two terms, with the first term depending on parameters only, and the second term on molecule numbers only. This case is for example encountered when the temperature-dependance of the reaction rate constant is relevant, and the temperature T is a variable parameter in the Arrhenius equation$\theta =A{e}^{\frac{-{E}_{A}}{\mathit{\text{RT}}}}$. Since the output matrix C and the initial condition P_{0} are usually not depending on parameters in this framework, a decomposition of C and P_{0} is not considered.
The situation is more difficult for reaction propensities involving for example rational terms with parameters in the denominator. The denominator parameters can not be included in the reduced order model by the decomposition outlined in (20) and (21). If variations in these parameters are however not relevant to the planned analysis, then they can be set to their nominal value, and the decomposition can directly be done as described above. Alternatively, approximation steps can be performed, such as Taylor series expansion or empirical interpolation[22], that generate an approximating parameter-separable expansion.
Results
In this section, we present the study of two example networks with the proposed model reduction method. With these examples, the applicability of the reduced modeling approach especially for analysis tasks requiring repeated simulations with different parameter values is illustrated. The first network is a bistable genetic toggle switch, where cells may switch randomly between two states, based on the model in[23]. For this network, the problem of parameter estimation with a reduced model is studied. The second network is a second-order genetic oscillator, based on[24], where we perform a sensitivity analysis over a wide parameter range.
Parameter estimation in a genetic toggle switch model
Network description
The follicle switch model
Reaction | Stoichiometry v_{ j } | Propensity ν_{ j } |
---|---|---|
Production of X_{1} | (1,0)^{T} | ${u}_{1}({k}_{1}+\frac{{V}_{1}{x}_{2}^{3}}{{M}_{1}^{3}+{x}_{2}^{3}})$ |
Degradation of X_{1} | (−1,0)^{T} | u _{1} x _{1} |
Production of X_{2} | (0,1)^{T} | ${u}_{2}\left(\frac{{V}_{2}{x}_{1}^{3}}{{M}_{2}^{3}+{x}_{1}^{3}}\right)$ |
Degradation of X_{2} | (0,−1)^{T} | u _{2} x _{2} |
Parameters for the follicle switch model
k _{1} | V _{1} | M _{1} | u _{1} | V _{2} | M _{2} | u _{2} |
---|---|---|---|---|---|---|
4 | 75 | 25 | $0.01\frac{1}{\text{min}}$ | 75 | 25 | $0.01\frac{1}{\text{min}}$ |
In[23], this network was shown to describe a bistable switch with two probability peaks, one close to x^{(off)} = (0,0)^{T} and the other close to${x}^{\left(\mathit{\text{on}}\right)}={({V}_{1},{V}_{2})}^{\mathrm{T}}$.
In the study[23], only the lower probability peak was of interest. Here, we are interested in the transition of the system from x^{(off)} to x^{(on)}. Therefore, the system is truncated to a rectangle$\stackrel{\u0304}{\Omega}:=\{0,\dots ,150\}\times \{0,\dots ,150\}$ such that${x}^{\left(\mathit{\text{on}}\right)},{x}^{\left(\mathit{\text{off}}\right)}\in \stackrel{\u0304}{\Omega}$, yielding a good approximation in the finite state projection to the infinite-dimensional chemical master equation.
where A^{[i]}, i = 1,…,5 are of dimension 151^{2} × 151^{2} = 22801 × 22801.
For the parametric model reduction, we consider only variations in the parameters u_{1} and u_{2}. These influence both the steady state level of gene activity in the on-state as well as the switching kinetics and are thus of high biological significance in the model. Hence we set$\theta :={({u}_{1},\phantom{\rule{1em}{0ex}}{u}_{2})}^{T}\in {[0.005,\phantom{\rule{1em}{0ex}}0.02]}^{2}$ as the parametric domain$\mathcal{P}$. As final time we choose T = 10^{7} which corresponds to a time range of approximately 19 years, i.e. about three times the half-life time of the off-state estimated in[23].
As typical simulation time for a single trajectory of the full system, we obtain 98.2 seconds on a IBM Lenovo 2.53 GHz Dual Core Laptop.
Basis generation
We generated a reduced basis with the POD-Greedy algorithm, where the training set was chosen as the vertices of a mesh with 9^{2} logarithmically equidistant parameter values over the parameter domain$\mathcal{P}$. We set${\epsilon}_{\mathit{\text{tol}}}=1{0}^{-12}$ as target accuracy. We use the projection error as error measure, hence precompute the 81 trajectories for construction of the reduced basis. As initial basis we set N_{0} = 1 and${V}_{{N}_{0}}:={P}_{0}$ using the parameter independent initial condition.
The final reduced model of dimension 33 can then be simulated in 0.135 seconds, corresponding to a computational speedup factor of more than 700.
Parameter estimation
We exemplify a possible application of the reduced order model in parameter estimation, where we assume that a distorted output y(t) as the expected values E[x_{1}] is available from population-averaged measurements. The task is to estimate the parameter values u_{1} and u_{2} from such a noisy measurement.
In such an optimization problem, typically many forward simulations are required for adjusting$\u0177$ to the measurement. This is a particular beneficial scenario for reduced order models, as these simulations can be computed rapidly.
In order to gain a deeper insight into the optimization problem (25), we plot the values of the error functional J(θ) over the parameter domain (middle of Figure4). Using the reduced model, the computation of the required 21^{2} = 441 trajectories is realized in less than one minute. This would be a significant computational effort when using a non-reduced model.
From the cost function plot, we observe a narrow area of parameters which seem to produce a similar output as the reference parameter θ_{ ref }. This shows that the two model parameters are not simultaneously identifiable from the considered output, and indicates that there may exist a functional dependence between the parameters u_{1} and u_{2} such that the model yields similar outputs y(t).
Assuming a functional dependence of u_{1}and u_{2}we now consider the 1-dimensional optimization problem along the line u_{2} = u_{2,ref} = 0.01. We would like to recover u_{1}from the optimization problem. The corresponding value of the cost function is J(θ_{ ref }) = 3330.68, indicating a significant contribution of the noise. This restricted optimization problem is well conditioned and the optimization with a standard active set algorithm by MATLAB’s command fmincon yields the estimated parameter θ_{ est } := (u_{1,est},0.01) with u_{1,est} = 0.0100204, using 27 evaluations of the cost function. This accounts to a relative error in the u_{1}value of 0.204%, hence excellent recovery. We refrain from plotting the recovered output$\u0177(t,{\theta}_{\mathit{\text{est}}})$ as it is visually indiscriminable from the output in the left of Figure4. Interestingly, the optimization target value J(θ_{ est }) = 3329.56 implies J(θ_{ est }) < J(θ_{ ref }), which may stem from a slight approximation error in the reduced model or from the effects of the measurement noise.
The right plot in Figure4 illustrates another application of reduced parametric models: We incorporated the model in an interactive graphical user interface in RBmatlab, a matlab package for model order reduction, available for download athttp://www.morepas.org. This allows interactive parameter variations and instantaneous simulation response.
Sensitivity analysis in a stochastic oscillator
Network description
The oscillator model
Reaction | Stoichiometry v_{ j } | Propensity ν_{ j } |
---|---|---|
Production of X_{1} | (1,0)^{T} | $\frac{{k}_{1}{s}^{2}}{{k}_{2}s+{x}_{2}}$ |
Degradation of X_{1} | (−1,0)^{T} | k _{3} x _{1} |
Production of X_{2} | (0,1)^{T} | ${k}_{4}s+\frac{{k}_{5}{x}_{2}^{2}{x}_{1}}{{k}_{6}{s}^{2}+{x}_{2}^{2}}$ |
Degradation of X_{2} | (0,−1)^{T} | k _{7} x _{2} |
Parameters for the oscillator model
k _{1} | k _{2} | k _{3} | k _{4} | k _{5} | k _{6} | k _{7} | s |
---|---|---|---|---|---|---|---|
$15\frac{1}{\mathrm{s}}$ | 0.2 | $1\frac{1}{\mathrm{s}}$ | $10\frac{1}{\mathrm{s}}$ | $100\frac{1}{\mathrm{s}}$ | 6.5 | $100\frac{1}{\mathrm{s}}$ | 10 |
The network model in Table4 shows oscillations only in a stochastic description. The deterministic model has a unique asymptotically stable equilibrium point, but in a stochastic model, fluctuations may push the molecular numbers beyond a certain threshold, inducing a dynamical response along a slow manifold, which corresponds to one oscillatory period[24]. Depending on the noise level, such responses will be initiated more or less often, corresponding to a more or less regular oscillatory pattern.
The system is truncated to the rectangle$\stackrel{\u0304}{\Omega}:=\{0,\dots ,300\}\times \{0,\dots ,300\}$, which contains the relevant system states for the parameter ranges of interest.
The time scale of interest for the model in (26) is for 0 ≤ t ≤ T = 6. At the end of the interval, the probability distribution seems to approach a steady state.
Basis generation
with${A}_{r}^{\left[3\right]}={V}^{\mathrm{T}}{A}^{\left[3\right]}V\in {\mathbb{R}}^{109\times 109}$ and${A}_{r}^{\left[o\right]}={V}^{\mathrm{T}}\left({k}_{1}{A}^{\left[1\right]}+{k}_{3}{A}^{\left[2\right]}+{k}_{5}{A}^{\left[4\right]}+{k}_{7}{A}^{\left[5\right]}\right)V\in {\mathbb{R}}^{109\times 109}$. Note that since only k_{4} has been varied in the reduction process, the other parameters are no longer present as parameters in the reduced model, but just take their nominal values. While the same basis V could be used to construct another reduced model where all parameters are retained, it is unlikely that this other model will be a good approximation of the original one for varying values of the other parameters.
Sensitivity analysis of the oscillation amplitude
with T = 6 the final time of the simulation. The results are shown in Figure6 and show a clear decay of oscillatory amplitude for increasing values of k_{4}. Due to the significant time savings from the reduced model, this sensitivity curve can be computed with a high resolution.
To evaluate the quality of the reduced model, we also computed the probability (29) using the original model (26) at two points within the considered interval for the parameter k_{4}. As shown in Figure6, the results from the original model are in perfect agreement with the predictions from the reduced model at these points. Since the points at which the original model was evaluated in this experiment were not part of the training set (shown as triangles on the parameter axis in Figure6), this shows that it is in fact possible to extrapolate the reduced model to parameter values that were not used to construct the basis.
Conclusions
In this paper, we have introduced the application of parametric model reduction methods to finite-state approximations of the chemical master equation. We have also presented two case studies where these methods are applied to CME models of different networks in order to make parametric analysis tasks computationally efficient. By this, it has become clear that parametric model reduction methods are a very useful tool for the analysis of stochastic biochemical reaction network described by the CME.
Especially analysis tasks where many repeated simulations of a network with different parameter values are required can profit significantly from parametric model reduction. This includes for example sensitivity analysis or parameter optimization tasks such as identifiability analysis or estimation. Moreover, the significant speedup of the simulation for the reduced model allows an interactive exploration of the network’s dynamics within the parameter space within a suitable graphical user interface.
This contribution is just a first step in the application of parametric model reduction methods to the CME. One particularly important aspect that we have not discussed here is the computation of error estimates for certifying that the simulation output of the reduced model is within some tolerance of the corresponding simulation output of the original model. To maintain computational efficiency, the error estimation should be done without actually simulating the original model. Error estimation methods have been developed for parametric model reduction of generic models[9], but tighter estimates could likely be obtained by taking into account the special structure of the CME models. Recent work for example refined the previous generic error bounds for stable models[25].
Authors contributions
SW and BH conceived of the study, performed the study, and wrote the manuscript. Both authors read and approved the final manuscript.
Declarations
Acknowledgements
We thank Wolfgang Halter for programming support in the oscillator case study. The authors would like to thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology at the University of Stuttgart. BH also acknowledges the Baden-Württemberg Stiftung gGmbH for funding. This work was also supported by the German Research Foundation (DFG) within the funding programme Open Access Publishing.
Authors’ Affiliations
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