Approximate maximum likelihood estimation for stochastic chemical kinetics

Comprehensive Review of Models and Methods for Inferences in Bio-Chemical Reaction Networks

The key processes in biological and chemical systems are described by networks of chemical reactions. From molecular biology to biotechnology applications, computational models of reaction networks are used extensively to elucidate their non-linear dynamics. The model dynamics are crucially dependent on the parameter values which are often estimated from observations. Over the past decade, the interest in parameter and state estimation in models of (bio-) chemical reaction networks (BRNs) grew considerably. The related inference problems are also encountered in many other tasks including model calibration, discrimination, identifiability, and checking, and optimum experiment design, sensitivity analysis, and bifurcation analysis. The aim of this review paper is to examine the developments in literature to understand what BRN models are commonly used, and for what inference tasks and inference methods. The initial collection of about 700 documents concerning estimation problems in BRNs excluding books and textbooks in computational biology and chemistry were screened to select over 270 research papers and 20 graduate research theses. The paper selection was facilitated by text mining scripts to automate the search for relevant keywords and terms. The outcomes are presented in tables revealing the levels of interest in different inference tasks and methods for given models in the literature as well as the research trends are uncovered. Our findings indicate that many combinations of models, tasks and methods are still relatively unexplored, and there are many new research opportunities to explore combinations that have not been considered-perhaps for good reasons. The most common models of BRNs in literature involve differential equations, Markov processes, mass action kinetics, and state space representations whereas the most common tasks are the parameter inference and model identification. The most common methods in literature are Bayesian analysis, Monte Carlo sampling strategies, and model fitting to data using evolutionary algorithms. The new research problems which cannot be directly deduced from the text mining data are also discussed.

Moment-Based Parameter Estimation for Stochastic Reaction Networks in Equilibrium

Calibrating parameters is a crucial problem within quantitative modeling approaches to reaction networks. Existing methods for stochastic models rely either on statistical sampling or can only be applied to small systems. Here, we present an inference procedure for stochastic models in equilibrium that is based on a moment matching scheme with optimal weighting and that can be used with high-throughput data like the one collected by flow cytometry. Our method does not require an approximation of the underlying equilibrium probability distribution and, if reaction rate constants have to be learned, the optimal values can be computed by solving a linear system of equations. We discuss important practical issues such as the selection of the moments and evaluate the effectiveness of the proposed approach on three case studies.

Generalized method of moments for estimating parameters of stochastic reaction networks

BACKGROUND

Discrete-state stochastic models have become a well-established approach to describe biochemical reaction networks that are influenced by the inherent randomness of cellular events. In the last years several methods for accurately approximating the statistical moments of such models have become very popular since they allow an efficient analysis of complex networks.

RESULTS

We propose a generalized method of moments approach for inferring the parameters of reaction networks based on a sophisticated matching of the statistical moments of the corresponding stochastic model and the sample moments of population snapshot data. The proposed parameter estimation method exploits recently developed moment-based approximations and provides estimators with desirable statistical properties when a large number of samples is available. We demonstrate the usefulness and efficiency of the inference method on two case studies.

CONCLUSIONS

The generalized method of moments provides accurate and fast estimations of unknown parameters of reaction networks. The accuracy increases when also moments of order higher than two are considered. In addition, the variance of the estimator decreases, when more samples are given or when higher order moments are included.

Experimental Design for Stochastic Models of Nonlinear Signaling Pathways Using an Interval-Wise Linear Noise Approximation and State Estimation

BACKGROUND

Computational modeling is a key technique for analyzing models in systems biology. There are well established methods for the estimation of the kinetic parameters in models of ordinary differential equations (ODE). Experimental design techniques aim at devising experiments that maximize the information encoded in the data. For ODE models there are well established approaches for experimental design and even software tools. However, data from single cell experiments on signaling pathways in systems biology often shows intrinsic stochastic effects prompting the development of specialized methods. While simulation methods have been developed for decades and parameter estimation has been targeted for the last years, only very few articles focus on experimental design for stochastic models.

METHODS

The Fisher information matrix is the central measure for experimental design as it evaluates the information an experiment provides for parameter estimation. This article suggest an approach to calculate a Fisher information matrix for models containing intrinsic stochasticity and high nonlinearity. The approach makes use of a recently suggested multiple shooting for stochastic systems (MSS) objective function. The Fisher information matrix is calculated by evaluating pseudo data with the MSS technique.

RESULTS

The performance of the approach is evaluated with simulation studies on an Immigration-Death, a Lotka-Volterra, and a Calcium oscillation model. The Calcium oscillation model is a particularly appropriate case study as it contains the challenges inherent to signaling pathways: high nonlinearity, intrinsic stochasticity, a qualitatively different behavior from an ODE solution, and partial observability. The computational speed of the MSS approach for the Fisher information matrix allows for an application in realistic size models.

Parameter identification using stochastic simulations reveals a robustness in CD95 apoptotic response

A number of mathematical models of apoptosis generated recently allowed us to understand intrinsic mechanisms of life/death decisions in a cell. Nevertheless, the parameters for the mathematical models are often experimentally difficult to obtain and there is an emerging need for the development of efficient approaches for parameter estimation. In this study we suggest a new method for parameter estimation, which is based on stochastic simulations and can be used when the number of molecules in the system is small. Our approach comprised the following steps: we start from the selection of parameters that lead to a good ordinary differential equation (ODE) fit. We continued by carrying out stochastic simulations for each of these parameters. Comparing the correlation structure of these simulations with the data, we finally could identify the best parameter set. The method was applied for a model of CD95-induced apoptosis, the new best identified parameters fit well to the experimental data. The best parameter set allowed us to get new insights into CD95 apoptosis regulation and can be applied for the comprehensive analysis of other signaling networks. The modeling approach allowed us to get new insights into network regulation, in particular, to identify robustness in CD95 apoptotic response. Taken together, this new method provides valuable predictions and can be applied for the analysis of other signaling networks.