Set-base dynamical parameter estimation and model invalidation for biochemical reaction networks

Background  

Mathematical modeling and analysis have become, for the study of biological and cellular processes, an important complement to experimental research. However, the structural and quantitative knowledge available for such processes is frequently limited, and measurements are often subject to inherent and possibly large uncertainties. This results in competing model hypotheses, whose kinetic parameters may not be experimentally determinable. Discriminating among these alternatives and estimating their kinetic parameters is crucial to improve the understanding of the considered process, and to benefit from the analytical tools at hand.     

Global parameter estimation methods for stochastic biochemical systems

Background  

The importance of stochasticity in cellular processes having low number of molecules has resulted in the development of stochastic models such as chemical master equation. As in other modelling frameworks, the accompanying rate constants are important for the end-applications like analyzing system properties (e.g. robustness) or predicting the effects of genetic perturbations. Prior knowledge of kinetic constants is usually limited and the model identification routine typically includes parameter estimation from experimental data. Although the subject of parameter estimation is well-established for deterministic models, it is not yet routine for the chemical master equation. In addition, recent advances in measurement technology have made the quantification of genetic substrates possible to single molecular levels. Thus, the purpose of this work is to develop practical and effective methods for estimating kinetic model parameters in the chemical master equation and other stochastic models from single cell and cell population experimental data.     

Accelerated maximum likelihood parameter estimation for stochastic biochemical systems

ABSTRACT: BACKGROUND: A prerequisite for the mechanistic simulation of a biochemical system is detailed knowledge of its kinetic parameters. Despite recent experimental advances, the estimation of unknown parameter values from observed data is still a bottleneck for obtaining accurate simulation results. Many methods exist for parameter estimation in deterministic biochemical systems; methods for discrete stochastic systems are less well developed. Given the probabilistic nature of stochastic biochemical models, a natural approach is to choose parameter values that maximize the probability of the observed data with respect to the unknown parameters, a.k.a. the maximum likelihood parameter estimates (MLEs). MLE computation for all but the simplest models requires the simulation of many system trajectories that are consistent with experimental data. For models with unknown parameters, this presents a computational challenge, as the generation of consistent trajectories can be an extremely rare occurrence. RESULTS: We have developed Monte Carlo Expectation-Maximization with Modified Cross-Entropy Method (MCEM2): an accelerated method for calculating MLEs that combines advances in rare event simulation with a computationally efficient version of the Monte Carlo expectation-maximization (MCEM) algorithm. Our method requires no prior knowledge regarding parameter values, and it automatically provides a multivariate parameter uncertainty estimate. We applied the method to five stochastic systems of increasing complexity, progressing from an analytically tractable pure-birth model to a computationally demanding model of yeast-polarization. Our results demonstrate that MCEM2 substantially accelerates MLE computation on all tested models when compared to a stand-alone version of MCEM. Additionally, we show how our method identifies parameter values for certain classes of models more accurately than two recently proposed computationally efficient methods. CONCLUSIONS: This work provides a novel, accelerated version of a likelihood-based parameter estimation method that can be readily applied to stochastic biochemical systems. In addition, our results suggest opportunities for added efficiency improvements that will further enhance our ability to mechanistically simulate biological processes.     

Experimental design for parameter estimation of gene regulatory networks

Systems biology aims for building quantitative models to address unresolved issues in molecular biology. In order to describe the behavior of biological cells adequately, gene regulatory networks (GRNs) are intensively investigated. As the validity of models built for GRNs depends crucially on the kinetic rates, various methods have been developed to estimate these parameters from experimental data. For this purpose, it is favorable to choose the experimental conditions yielding maximal information. However, existing experimental design principles often rely on unfulfilled mathematical assumptions or become computationally demanding with growing model complexity. To solve this problem, we combined advanced methods for parameter and uncertainty estimation with experimental design considerations. As a showcase, we optimized three simulated GRNs in one of the challenges from the Dialogue for Reverse Engineering Assessment and Methods (DREAM). This article presents our approach, which was awarded the best performing procedure at the DREAM6 Estimation of Model Parameters challenge. For fast and reliable parameter estimation, local deterministic optimization of the likelihood was applied. We analyzed identifiability and precision of the estimates by calculating the profile likelihood. Furthermore, the profiles provided a way to uncover a selection of most informative experiments, from which the optimal one was chosen using additional criteria at every step of the design process. In conclusion, we provide a strategy for optimal experimental design and show its successful application on three highly nonlinear dynamic models. Although presented in the context of the GRNs to be inferred for the DREAM6 challenge, the approach is generic and applicable to most types of quantitative models in systems biology and other disciplines.     

Response to temporal parameter fluctuations in biochemical networks

Metabolic response coefficients describe how variables in metabolic systems, like steady state concentrations, respond to small changes of kinetic parameters. To extend this concept to temporal parameter fluctuations, we define spectral response coefficients that relate Fourier components of concentrations and fluxes to Fourier components of the underlying parameters. It is also straightforward to generalize other concepts from metabolic control theory, such as control coefficients with their summation and connectivity theorems. The first-order response coefficients describe forced oscillations caused by small harmonic oscillations of single parameters: they depend on the driving frequency and comprise the phases and amplitudes of the concentrations and fluxes. Close to a Hopf bifurcation, resonance can occur: as an example, we study the spectral densities of concentration fluctuations arising from the stochastic nature of chemical reactions. Second-order response coefficients describe how perturbations of different frequencies interact by mode coupling, yielding higher harmonics in the metabolic response. The temporal response to small parameter fluctuations can be computed by Fourier synthesis. For a model of glycolysis, this approximation remains fairly accurate even for large relative fluctuations of the parameters.     

Proximate parameter tuning for biochemical networks with uncertain kinetic parameters

It is commonly the case in biochemical modelling that we have knowledge of the qualitative 'structure' of a model and some measurements of the time series of the variables of interest (concentrations and fluxes), but little or no knowledge of the model's parameters. This is, then, a system identification problem, that is commonly addressed by running a model with estimated parameters and assessing how far the model's behaviour is from the 'target' behaviour of the variables, and adjusting parameters iteratively until a good fit is achieved. The issue is that most of these problems are grossly underdetermined, such that many combinations of parameters can be used to fit a given set of variables. We introduce the constraint that the estimated parameters should be within given bounds and as close as possible to stated nominal values. This deterministic 'proximate parameter tuning' algorithm turns out to be exceptionally effective, and we illustrate its utility for models of p38 signalling, of yeast glycolysis and for a benchmark dataset describing the thermal isomerisation of alpha-pinene.     

A hybrid approach for efficient and robust parameter estimation in biochemical pathways

Developing suitable dynamic models of biochemical pathways is a key issue in Systems Biology. Predictive models for cells or whole organisms could ultimately lead to model-based predictive and/or preventive medicine. Parameter estimation (i.e. model calibration) in these dynamic models is therefore a critical problem. In a recent contribution [Moles, C.G., Mendes, P., Banga, J.R., 2003b. Parameter estimation in biochemical pathways: a comparison of global optimisation methods. Genome Res. 13, 2467-2474], the challenging nature of such inverse problems was highlighted considering a benchmark problem, and concluding that only a certain type of stochastic global optimisation method, Evolution Strategies (ES), was able to solve it successfully, although at a rather large computational cost. In this new contribution, we present a new integrated optimisation methodology with a number of very significant improvements: (i) computation time is reduced by one order of magnitude by means of a hybrid method which increases efficiency while guaranteeing robustness, (ii) measurement noise (errors) and partial observations are handled adequately, (iii) automatic testing of identifiability of the model (both local and practical) is included and (iv) the information content of the experiments is evaluated via the Fisher information matrix, with subsequent application to design of new optimal experiments through dynamic optimisation.     

An evolutionary computing approach for parameter estimation investigation of a model for cholera

We consider the problem of using time-series data to inform a corresponding deterministic model and introduce the concept of genetic algorithms (GA) as a tool for parameter estimation, providing instructions for an implementation of the method that does not require access to special toolboxes or software. We give as an example a model for cholera, a disease for which there is much mechanistic uncertainty in the literature. We use GA to find parameter sets using available time-series data from the introduction of cholera in Haiti and we discuss the value of comparing multiple parameter sets with similar performances in describing the data.     

Parameter estimation using Simulated Annealing for S-system models of biochemical networks

MOTIVATION: High-throughput technologies now allow the acquisition of biological data, such as comprehensive biochemical time-courses at unprecedented rates. These temporal profiles carry topological and kinetic information regarding the biochemical network from which they were drawn. Retrieving this information will require systematic application of both experimental and computational methods. RESULTS: S-systems are non-linear mathematical approximative models based on the power-law formalism. They provide a general framework for the simulation of integrated biological systems exhibiting complex dynamics, such as genetic circuits, signal transduction and metabolic networks. We describe how the heuristic optimization technique simulated annealing (SA) can be effectively used for estimating the parameters of S-systems from time-course biochemical data. We demonstrate our methods using three artificial networks designed to simulate different network topologies and behavior. We then end with an application to a real biochemical network by creating a working model for the cadBA system in Escherichia coli. AVAILABILITY: The source code written in C++ is available at http://www.engg.upd.edu.ph/~naval/bioinformcode.html. All the necessary programs including the required compiler are described in a document archived with the source code. SUPPLEMENTARY INFORMATION: Supplementary material is available at Bioinformatics online.     

Dynamic Optimization with Particle Swarms (DOPS): a meta-heuristic for parameter estimation in biochemical models

Background

Mathematical modeling is a powerful tool to analyze, and ultimately design biochemical networks. However, the estimation of the parameters that appear in biochemical models is a significant challenge. Parameter estimation typically involves expensive function evaluations and noisy data, making it difficult to quickly obtain optimal solutions. Further, biochemical models often have many local extrema which further complicates parameter estimation. Toward these challenges, we developed Dynamic Optimization with Particle Swarms (DOPS), a novel hybrid meta-heuristic that combined multi-swarm particle swarm optimization with dynamically dimensioned search (DDS). DOPS uses a multi-swarm particle swarm optimization technique to generate candidate solution vectors, the best of which is then greedily updated using dynamically dimensioned search.

Results

We tested DOPS using classic optimization test functions, biochemical benchmark problems and real-world biochemical models. We performed \(\mathcal {T}\) = 25 trials with \(\mathcal {N}\) = 4000 function evaluations per trial, and compared the performance of DOPS with other commonly used meta-heuristics such as differential evolution (DE), simulated annealing (SA) and dynamically dimensioned search (DDS). On average, DOPS outperformed other common meta-heuristics on the optimization test functions, benchmark problems and a real-world model of the human coagulation cascade.

Conclusions

DOPS is a promising meta-heuristic approach for the estimation of biochemical model parameters in relatively few function evaluations. DOPS source code is available for download under a MIT license at http://www.varnerlab.org.

     

捕获网法及其参数估计

介绍了一种采用标记-重捕获技术估计动物种群密度的捕获网抽样方法。在回顾该方法国内外研究进展的基础上,分析了由于该法参数估算过程复杂和缺乏相关的处理计算软件对其应用的限制,提出了捕获网抽样方法参数估计的表格化处理技术。并以美国New Mexico附近白脚鼠的种群密度调查数据分析为例,介绍了嵌入DPS电子工作表的捕获网抽样参数估计表格化处理过程;最后就捕获网抽样参数估计的研究重点提出一些看法和展望。     

Statistical methods for model comparison in parameter estimation problems for distributed systems

In this note we outline some recent results on the development of a statistical testing methodology for inverse problems involving partial differential equation models. Applications to several problems from biology are presented. The statistical tests, which are in the spirit of analysis of variance (ANOVA), are based on asymptotic distributional results for estimators and residuals in a least squares approach.Research supported in part under grants NSF MCS 8504316, NASA NAG-1-517, and AFOSRF-49620-86-C-0111. Part of this research was carried out while the first author was a visiting scientist at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA, which is operated under NASA contracts NASI-18107 and NASI-18605     

Reduced-order modelling of biochemical networks: application to the GTPase-cycle signalling module

Biochemical systems embed complex networks and hence development and analysis of their detailed models pose a challenge for computation. Coarse-grained biochemical models, called reduced-order models (ROMs), consisting of essential biochemical mechanisms are more useful for computational analysis and for studying important features of a biochemical network. The authors present a novel method to model-reduction by identifying potentially important parameters using multidimensional sensitivity analysis. A ROM is generated for the GTPase-cycle module of m1 muscarinic acetylcholine receptor, Gq, and regulator of G-protein signalling 4 (a GTPase-activating protein or GAP) starting from a detailed model of 48 reactions. The resulting ROM has only 17 reactions. The ROM suggested that complexes of G-protein coupled receptor (GPCR) and GAP--which were proposed in the detailed model as a hypothesis--are required to fit the experimental data. Models previously published in the literature are also simulated and compared with the ROM. Through this comparison, a minimal ROM, that also requires complexes of GPCR and GAP, with just 15 parameters is generated. The proposed reduced-order modelling methodology is scalable to larger networks and provides a general framework for the reduction of models of biochemical systems.     

Graphical approach to model reduction for nonlinear biochemical networks

Model reduction is a central challenge to the development and analysis of multiscale physiology models. Advances in model reduction are needed not only for computational feasibility but also for obtaining conceptual insights from complex systems. Here, we introduce an intuitive graphical approach to model reduction based on phase plane analysis. Timescale separation is identified by the degree of hysteresis observed in phase-loops, which guides a "concentration-clamp" procedure for estimating explicit algebraic relationships between species equilibrating on fast timescales. The primary advantages of this approach over Jacobian-based timescale decomposition are that: 1) it incorporates nonlinear system dynamics, and 2) it can be easily visualized, even directly from experimental data. We tested this graphical model reduction approach using a 25-variable model of cardiac β(1)-adrenergic signaling, obtaining 6- and 4-variable reduced models that retain good predictive capabilities even in response to new perturbations. These 6 signaling species appear to be optimal "kinetic biomarkers" of the overall β(1)-adrenergic pathway. The 6-variable reduced model is well suited for integration into multiscale models of heart function, and more generally, this graphical model reduction approach is readily applicable to a variety of other complex biological systems.     

A robust methodology for kinetic model parameter estimation for biocatalytic reactions

Effective estimation of parameters in biocatalytic reaction kinetic expressions are very important when building process models to enable evaluation of process technology options and alternative biocatalysts. The kinetic models used to describe enzyme‐catalyzed reactions generally include several parameters, which are strongly correlated with each other. State‐of‐the‐art methodologies such as nonlinear regression (using progress curves) or graphical analysis (using initial rate data, for example, the Lineweaver‐Burke plot, Hanes plot or Dixon plot) often incorporate errors in the estimates and rarely lead to globally optimized parameter values. In this article, a robust methodology to estimate parameters for biocatalytic reaction kinetic expressions is proposed. The methodology determines the parameters in a systematic manner by exploiting the best features of several of the current approaches. The parameter estimation problem is decomposed into five hierarchical steps, where the solution of each of the steps becomes the input for the subsequent step to achieve the final model with the corresponding regressed parameters. The model is further used for validating its performance and determining the correlation of the parameters. The final model with the fitted parameters is able to describe both initial rate and dynamic experiments. Application of the methodology is illustrated with a case study using the ω‐transaminase catalyzed synthesis of 1‐phenylethylamine from acetophenone and 2‐propylamine. © 2012 American Institute of Chemical Engineers Biotechnol. Prog., 2012     

Targeted maximum likelihood estimation of the parameter of a marginal structural model

Targeted maximum likelihood estimation is a versatile tool for estimating parameters in semiparametric and nonparametric models. We work through an example applying targeted maximum likelihood methodology to estimate the parameter of a marginal structural model. In the case we consider, we show how this can be easily done by clever use of standard statistical software. We point out differences between targeted maximum likelihood estimation and other approaches (including estimating function based methods). The application we consider is to estimate the effect of adherence to antiretroviral medications on virologic failure in HIV positive individuals.