Probabilistic sensitivity analysis of biochemical reaction systems

Development of a sampling-based global sensitivity analysis workflow for multiscale computational cancer models

There are two challenges that researchers face when performing global sensitivity analysis (GSA) on multiscale 'in silico' cancer models. The first is increased computational intensity, since a multiscale cancer model generally takes longer to run than does a scale-specific model. The second problem is the lack of a best GSA method that fits all types of models, which implies that multiple methods and their sequence need to be taken into account. In this study, the authors therefore propose a sampling-based GSA workflow consisting of three phases - pre-analysis, analysis and post-analysis - by integrating Monte Carlo and resampling methods with the repeated use of analysis of variance; they then exemplify this workflow using a two-dimensional multiscale lung cancer model. By accounting for all parameter rankings produced by multiple GSA methods, a summarised ranking is created at the end of the workflow based on the weighted mean of the rankings for each input parameter. For the cancer model investigated here, this analysis reveals that extracellular signal-regulated kinase, a downstream molecule of the epidermal growth factor receptor signalling pathway, has the most important impact on regulating both the tumour volume and expansion rate in the algorithm used.

Reverse engineering gene regulatory networks: coupling an optimization algorithm with a parameter identification technique

Background

To infer gene regulatory networks from time series gene profiles, two important tasks that are related to biological systems must be undertaken. One task is to determine a valid network structure that has topological properties that can influence the network dynamics profoundly. The other task is to optimize the network parameters to minimize the accumulated discrepancy between the gene expression data and the values produced by the inferred network model. Though the above two tasks must be conducted simultaneously, most existing work addresses only one of the tasks.

Results

We propose an iterative approach that couples parameter identification and parameter optimization techniques, to address the two tasks simultaneously during network inference. This approach first identifies the most influential parameters against internal perturbations; this identification is based on sensitivity measurements. Then, a hybrid GA-PSO optimization method infers parameters in accordance with their criticalities. The proposed approach has been applied to several datasets, including subsets of the SOS DNA repair system in E. coli, the Rat central nervous system (CNS), and the protein glycosylation system of yeast S. cerevisiae. The result and analysis show that our approach can infer solutions to satisfy both the requirements of network structure and network behavior.

Conclusions

Network structure is an important though challenging issue to address in inferring sophisticated networks with biological details. In need of prior structural knowledge, we turn to measure parameter sensitivity instead to account for the network structure in an indirect way. By developing an integrated approach for considering both the network structure and behavior in the inference process, we can successfully infer critical gene interactions as well as valid time expression profiles.

Inferring robust gene networks from expression data by a sensitivity-based incremental evolution method

Background

Reconstructing gene regulatory networks (GRNs) from expression data is one of the most important challenges in systems biology research. Many computational models and methods have been proposed to automate the process of network reconstruction. Inferring robust networks with desired behaviours remains challenging, however. This problem is related to network dynamics but has yet to be investigated using network modeling.

Results

We propose an incremental evolution approach for inferring GRNs that takes network robustness into consideration and can deal with a large number of network parameters. Our approach includes a sensitivity analysis procedure to iteratively select the most influential network parameters, and it uses a swarm intelligence procedure to perform parameter optimization. We have conducted a series of experiments to evaluate the external behaviors and internal robustness of the networks inferred by the proposed approach. The results and analyses have verified the effectiveness of our approach.

Conclusions

Sensitivity analysis is crucial to identifying the most sensitive parameters that govern the network dynamics. It can further be used to derive constraints for network parameters in the network reconstruction process. The experimental results show that the proposed approach can successfully infer robust GRNs with desired system behaviors.

Reducing experimental variability in variance-based sensitivity analysis of biochemical reaction systems

Sensitivity analysis is a valuable task for assessing the effects of biological variability on cellular behavior. Available techniques require knowledge of nominal parameter values, which cannot be determined accurately due to experimental uncertainty typical to problems of systems biology. As a consequence, the practical use of existing sensitivity analysis techniques may be seriously hampered by the effects of unpredictable experimental variability. To address this problem, we propose here a probabilistic approach to sensitivity analysis of biochemical reaction systems that explicitly models experimental variability and effectively reduces the impact of this type of uncertainty on the results. The proposed approach employs a recently introduced variance-based method to sensitivity analysis of biochemical reaction systems [Zhang et al., J. Chem. Phys. 134, 094101 (2009)] and leads to a technique that can be effectively used to accommodate appreciable levels of experimental variability. We discuss three numerical techniques for evaluating the sensitivity indices associated with the new method, which include Monte Carlo estimation, derivative approximation, and dimensionality reduction based on orthonormal Hermite approximation. By employing a computational model of the epidermal growth factor receptor signaling pathway, we demonstrate that the proposed technique can greatly reduce the effect of experimental variability on variance-based sensitivity analysis results. We expect that, in cases of appreciable experimental variability, the new method can lead to substantial improvements over existing sensitivity analysis techniques.

Identification of Critical Molecular Components in a Multiscale Cancer Model Based on the Integration of Monte Carlo, Resampling, and ANOVA

To date, parameters defining biological properties in multiscale disease models are commonly obtained from a variety of sources. It is thus important to examine the influence of parameter perturbations on system behavior, rather than to limit the model to a specific set of parameters. Such sensitivity analysis can be used to investigate how changes in input parameters affect model outputs. However, multiscale cancer models require special attention because they generally take longer to run than does a series of signaling pathway analysis tasks. In this article, we propose a global sensitivity analysis method based on the integration of Monte Carlo, resampling, and analysis of variance. This method provides solutions to (1) how to render the large number of parameter variation combinations computationally manageable, and (2) how to effectively quantify the sampling distribution of the sensitivity index to address the inherent computational intensity issue. We exemplify the feasibility of this method using a two-dimensional molecular-microscopic agent-based model previously developed for simulating non-small cell lung cancer; in this model, an epidermal growth factor (EGF)-induced, EGF receptor-mediated signaling pathway was implemented at the molecular level. Here, the cross-scale effects of molecular parameters on two tumor growth evaluation measures, i.e., tumor volume and expansion rate, at the microscopic level are assessed. Analysis finds that ERK, a downstream molecule of the EGF receptor signaling pathway, has the most important impact on regulating both measures. The potential to apply this method to therapeutic target discovery is discussed.

Thermodynamically consistent model calibration in chemical kinetics

Background

The dynamics of biochemical reaction systems are constrained by the fundamental laws of thermodynamics, which impose well-defined relationships among the reaction rate constants characterizing these systems. Constructing biochemical reaction systems from experimental observations often leads to parameter values that do not satisfy the necessary thermodynamic constraints. This can result in models that are not physically realizable and may lead to inaccurate, or even erroneous, descriptions of cellular function.

Results

We introduce a thermodynamically consistent model calibration (TCMC) method that can be effectively used to provide thermodynamically feasible values for the parameters of an open biochemical reaction system. The proposed method formulates the model calibration problem as a constrained optimization problem that takes thermodynamic constraints (and, if desired, additional non-thermodynamic constraints) into account. By calculating thermodynamically feasible values for the kinetic parameters of a well-known model of the EGF/ERK signaling cascade, we demonstrate the qualitative and quantitative significance of imposing thermodynamic constraints on these parameters and the effectiveness of our method for accomplishing this important task. MATLAB software, using the Systems Biology Toolbox 2.1, can be accessed from http://www.cis.jhu.edu/~goutsias/CSS lab/software.html. An SBML file containing the thermodynamically feasible EGF/ERK signaling cascade model can be found in the BioModels database.

Conclusions

TCMC is a simple and flexible method for obtaining physically plausible values for the kinetic parameters of open biochemical reaction systems. It can be effectively used to recalculate a thermodynamically consistent set of parameter values for existing thermodynamically infeasible biochemical reaction models of cellular function as well as to estimate thermodynamically feasible values for the parameters of new models. Furthermore, TCMC can provide dimensionality reduction, better estimation performance, and lower computational complexity, and can help to alleviate the problem of data overfitting.

Thermodynamically consistent Bayesian analysis of closed biochemical reaction systems

Background

Estimating the rate constants of a biochemical reaction system with known stoichiometry from noisy time series measurements of molecular concentrations is an important step for building predictive models of cellular function. Inference techniques currently available in the literature may produce rate constant values that defy necessary constraints imposed by the fundamental laws of thermodynamics. As a result, these techniques may lead to biochemical reaction systems whose concentration dynamics could not possibly occur in nature. Therefore, development of a thermodynamically consistent approach for estimating the rate constants of a biochemical reaction system is highly desirable.

Results

We introduce a Bayesian analysis approach for computing thermodynamically consistent estimates of the rate constants of a closed biochemical reaction system with known stoichiometry given experimental data. Our method employs an appropriately designed prior probability density function that effectively integrates fundamental biophysical and thermodynamic knowledge into the inference problem. Moreover, it takes into account experimental strategies for collecting informative observations of molecular concentrations through perturbations. The proposed method employs a maximization-expectation-maximization algorithm that provides thermodynamically feasible estimates of the rate constant values and computes appropriate measures of estimation accuracy. We demonstrate various aspects of the proposed method on synthetic data obtained by simulating a subset of a well-known model of the EGF/ERK signaling pathway, and examine its robustness under conditions that violate key assumptions. Software, coded in MATLAB^®, which implements all Bayesian analysis techniques discussed in this paper, is available free of charge at http://www.cis.jhu.edu/~goutsias/CSS%20lab/software.html.

Conclusions

Our approach provides an attractive statistical methodology for estimating thermodynamically feasible values for the rate constants of a biochemical reaction system from noisy time series observations of molecular concentrations obtained through perturbations. The proposed technique is theoretically sound and computationally feasible, but restricted to quantitative data obtained from closed biochemical reaction systems. This necessitates development of similar techniques for estimating the rate constants of open biochemical reaction systems, which are more realistic models of cellular function.

A comparison of approximation techniques for variance-based sensitivity analysis of biochemical reaction systems

Background

Sensitivity analysis is an indispensable tool for the analysis of complex systems. In a recent paper, we have introduced a thermodynamically consistent variance-based sensitivity analysis approach for studying the robustness and fragility properties of biochemical reaction systems under uncertainty in the standard chemical potentials of the activated complexes of the reactions and the standard chemical potentials of the molecular species. In that approach, key sensitivity indices were estimated by Monte Carlo sampling, which is computationally very demanding and impractical for large biochemical reaction systems. Computationally efficient algorithms are needed to make variance-based sensitivity analysis applicable to realistic cellular networks, modeled by biochemical reaction systems that consist of a large number of reactions and molecular species.

Results

We present four techniques, derivative approximation (DA), polynomial approximation (PA), Gauss-Hermite integration (GHI), and orthonormal Hermite approximation (OHA), for analytically approximating the variance-based sensitivity indices associated with a biochemical reaction system. By using a well-known model of the mitogen-activated protein kinase signaling cascade as a case study, we numerically compare the approximation quality of these techniques against traditional Monte Carlo sampling. Our results indicate that, although DA is computationally the most attractive technique, special care should be exercised when using it for sensitivity analysis, since it may only be accurate at low levels of uncertainty. On the other hand, PA, GHI, and OHA are computationally more demanding than DA but can work well at high levels of uncertainty. GHI results in a slightly better accuracy than PA, but it is more difficult to implement. OHA produces the most accurate approximation results and can be implemented in a straightforward manner. It turns out that the computational cost of the four approximation techniques considered in this paper is orders of magnitude smaller than traditional Monte Carlo estimation. Software, coded in MATLAB^®, which implements all sensitivity analysis techniques discussed in this paper, is available free of charge.

Conclusions

Estimating variance-based sensitivity indices of a large biochemical reaction system is a computationally challenging task that can only be addressed via approximations. Among the methods presented in this paper, a technique based on orthonormal Hermite polynomials seems to be an acceptable candidate for the job, producing very good approximation results for a wide range of uncertainty levels in a fraction of the time required by traditional Monte Carlo sampling.