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A model checking approach to the parameter estimation of biochemical pathways
 in Computational Methods in Systems Biology
"... Abstract. Model checking has historically been an important tool to verify models of a wide variety of systems. Typically a model has to exhibit certain properties to be classed ‘acceptable’. In this work we use model checking in a new setting; parameter estimation. We characterise the desired beha ..."
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Cited by 35 (7 self)
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Abstract. Model checking has historically been an important tool to verify models of a wide variety of systems. Typically a model has to exhibit certain properties to be classed ‘acceptable’. In this work we use model checking in a new setting; parameter estimation. We characterise the desired behaviour of a model in a temporal logic property and alter the model to make it conform to the property (determined through model checking). We have implemented a computational system called MC2(GA) which pairs a model checker with a genetic algorithm. To drive parameter estimation, the fitness of set of parameters in a model is the inverse of the distance between its actual behaviour and the desired behaviour. The model checker used is the simulationbased Monte Carlo Model Checker for Probabilistic Lineartime Temporal Logic with numerical constraints, MC2(PLTLc). Numerical constraints as well as the overall probability of the behaviour expressed in temporal logic are used to minimise the behavioural distance. We define the theory underlying our parameter estimation approach in both the stochastic and continuous worlds. We apply our approach to biochemical systems and present an illustrative example where we estimate the kinetic rate constants in a continuous model of a signalling pathway. 1
Comparing families of dynamic causal models
 PLoS Comput. Biol
, 2010
"... Mathematical models of scientific data can be formally compared using Bayesian model evidence. Previous applications in the biological sciences have mainly focussed on model selection in which one first selects the model with the highest evidence and then makes inferences based on the parameters of ..."
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Cited by 27 (7 self)
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Mathematical models of scientific data can be formally compared using Bayesian model evidence. Previous applications in the biological sciences have mainly focussed on model selection in which one first selects the model with the highest evidence and then makes inferences based on the parameters of that model. This ‘‘best model’ ’ approach is very useful but can become brittle if there are a large number of models to compare, and if different subjects use different models. To overcome this shortcoming we propose the combination of two further approaches: (i) family level inference and (ii) Bayesian model averaging within families. Family level inference removes uncertainty about aspects of model structure other than the characteristic of interest. For example: What are the inputs to the system? Is processing serial or parallel? Is it linear or nonlinear? Is it mediated by a single, crucial connection? We apply Bayesian model averaging within families to provide inferences about parameters that are independent of further assumptions about model structure. We illustrate the methods using Dynamic Causal Models of brain imaging data.
Accelerating Bayesian inference over nonlinear differential equations with Gaussian processes
 Adv. Neur. Inform. Process
, 2009
"... Identification and comparison of nonlinear dynamical system models using noisy and sparse experimental data is a vital task in many fields, however current methods are computationally expensive and prone to error due in part to the nonlinear nature of the likelihood surfaces induced. We present an ..."
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Cited by 20 (3 self)
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Identification and comparison of nonlinear dynamical system models using noisy and sparse experimental data is a vital task in many fields, however current methods are computationally expensive and prone to error due in part to the nonlinear nature of the likelihood surfaces induced. We present an accelerated sampling procedure which enables Bayesian inference of parameters in nonlinear ordinary and delay differential equations via the novel use of Gaussian processes (GP). Our method involves GP regression over timeseries data, and the resulting derivative and time delay estimates make parameter inference possible without solving the dynamical system explicitly, resulting in dramatic savings of computational time. We demonstrate the speed and statistical accuracy of our approach using examples of both ordinary and delay differential equations, and provide a comprehensive comparison with current state of the art methods. 1
BioBayes: a software package for Bayesian inference in systems biology, Bioinformatics 24
, 2008
"... Motivation: There are several levels of uncertainty involved in the mathematical modelling of biochemical systems. There often may be a degree of uncertainty about the values of kinetic parameters, about the general structure of the model, and about the behaviour of biochemical species which cannot ..."
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Cited by 9 (0 self)
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Motivation: There are several levels of uncertainty involved in the mathematical modelling of biochemical systems. There often may be a degree of uncertainty about the values of kinetic parameters, about the general structure of the model, and about the behaviour of biochemical species which cannot be observed directly. The methods of Bayesian inference provide a consistent framework for modelling and predicting in these uncertain conditions. We present a software package for applying the Bayesian inferential methodology to problems in Systems Biology. Results: Described herein is a software package, BioBayes, which provides a framework for Bayesian parameter estimation and evidential model ranking over models of biochemical systems defined using ordinary differential equations. The package is extensible allowing additional modules to be included by developers. There are no other such packages available which provide this functionality.
ODE parameter inference using adaptive gradient matching with Gaussian processes
 Sixteenth International Conference on Artificial Intelligence and Statistics; AISTATS
, 2013
"... Parameter inference in mechanistic models based on systems of coupled differential equations is a topical yet computationally challenging problem, due to the need to follow each parameter adaptation with a numerical integration of the differential equations. Techniques based on gradient matching, wh ..."
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Cited by 9 (4 self)
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Parameter inference in mechanistic models based on systems of coupled differential equations is a topical yet computationally challenging problem, due to the need to follow each parameter adaptation with a numerical integration of the differential equations. Techniques based on gradient matching, which aim to minimize the discrepancy between the slope of a data interpolant and the derivatives predicted from the differential equations, offer a computationally appealing shortcut to the inference problem. The present paper discusses a method based on nonparametric Bayesian statistics with Gaussian processes due to Calderhead et al. (2008), and shows how inference in this model can be substantially improved by consistently sampling from the joint distribution of the ODE parameters and GP hyperparameters. We demonstrate the efficiency of our adaptive gradient matching technique on three benchmark systems, and perform a detailed comparison with the method in Calderhead et al. (2008) and the explicit ODE integration approach, both in terms of parameter inference accuracy and in terms of computational efficiency. 1
Bayesian Model Comparison and Parameter Inference in Systems Biology Using Nested Sampling
, 2013
"... Inferring parameters for models of biological processes is a current challenge in systems biology, as is the related problem of comparing competing models that explain the data. In this work we apply Skilling’s nested sampling to address both of these problems. Nested sampling is a Bayesian method f ..."
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Cited by 4 (1 self)
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Inferring parameters for models of biological processes is a current challenge in systems biology, as is the related problem of comparing competing models that explain the data. In this work we apply Skilling’s nested sampling to address both of these problems. Nested sampling is a Bayesian method for exploring parameter space that transforms a multidimensional integral to a 1D integration over likelihood space. This approach focusses on the computation of the marginal likelihood or evidence. The ratio of evidences of different models leads to the Bayes factor, which can be used for model comparison. We demonstrate how nested sampling can be used to reverseengineer a system’s behaviour whilst accounting for the uncertainty in the results. The effect of missing initial conditions of the variables as well as unknown parameters is investigated. We show how the evidence and the model ranking can change as a function of the available data. Furthermore, the addition of data from extra variables of the system can deliver more information for model comparison than increasing the data from one variable, thus providing a basis for experimental design.
Towards Automatic Model Comparison An Adaptive Sequential Monte Carlo Approach
, 2013
"... Model comparison for the purposes of selection, averaging and validation is a problem found throughout statistics and related disciplines. Within the Bayesian paradigm, these problems all require the calculation of the posterior probabilities of models within a particular class. Substantial progress ..."
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Cited by 4 (0 self)
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Model comparison for the purposes of selection, averaging and validation is a problem found throughout statistics and related disciplines. Within the Bayesian paradigm, these problems all require the calculation of the posterior probabilities of models within a particular class. Substantial progress has been made in recent years, but there are numerous difficulties in the practical implementation of existing schemes. This paper presents adaptive sequential Monte Carlo (SMC) sampling strategies to characterise the posterior distribution of a collection of models, as well as the parameters of those models. Both a simple product estimator and a combination of SMC and a path sampling estimator are considered and existing theoretical results are extended to include the path sampling variant. A novel approach to the automatic specification of distributions within SMC algorithms is presented and shown to outperform the state of the art in this area. The performance of the proposed strategies is demonstrated via an extensive simulation study making use of the Gaussian mixture model and two challenging realistic examples. Comparisons with state of the art algorithms show that the proposed algorithms are always competitive, and often substantially superior to alternative techniques, at equal computational cost and considerably less applicationspecific implementation effort.
Designing attractive models via automated identification of chaotic and oscillatory dynamical regimes
 Nat. Commun
"... Chaos and oscillations continue to capture the interest of both the scientific and public domains. Yet despite the importance of these qualitative features, most attempts at constructing mathematical models of such phenomena have taken an indirect, quantitative approach, e.g. by fitting models to a ..."
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Cited by 4 (1 self)
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Chaos and oscillations continue to capture the interest of both the scientific and public domains. Yet despite the importance of these qualitative features, most attempts at constructing mathematical models of such phenomena have taken an indirect, quantitative approach, e.g. by fitting models to a finite number of datapoints. Here we develop a qualitative inference framework that allows us to both reverse engineer and design systems exhibiting these and other dynamical behaviours by directly specifying the desired characteristics of the underlying dynamical attractor. This change in perspective from quantitative to qualitative dynamics, provides fundamental and new insights into the properties of dynamical systems. Mathematical modelling requires a combination of experimentation, domain knowledge and, at times, a measure of luck. Beyond the intrinsic challenges of describing complex and complicated phenomena, the difficulty resides at a very fundamental level with the diversity of models that could explain a given set of observations. This is a manifestation of the socalled inverse problem [1], which is encountered whenever we aim to reconstruct a model of
D.: A Hybrid Approach to Piecewise Modelling of Biochemical Systems
 In: PPSN 2012 (12th International Conference on Parallel Problem Solving From Nature
, 2012
"... Abstract. Modelling biochemical systems has received considerable attention over the last decade from scientists and engineers across a number of fields, including biochemistry, computer science, and mathematics. Due to the complexity of biochemical systems, it is natural to construct models of th ..."
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Cited by 4 (2 self)
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Abstract. Modelling biochemical systems has received considerable attention over the last decade from scientists and engineers across a number of fields, including biochemistry, computer science, and mathematics. Due to the complexity of biochemical systems, it is natural to construct models of the biochemical systems incrementally in a piecewise manner. This paper proposes a hybrid approach which applies an evolutionary algorithm to select and compose predefined building blocks from a library of atomic models, mutating their products, thus generating complex systems in terms of topology, and employs a global optimization algorithm to fit the kinetic rates. Experiments using two signalling pathways show that given target behaviours it is feasible to explore the model space by this hybrid approach, generating a set of synthetic models with alternative structures and similar behaviours to the desired ones. 1
Causal network inference using biochemical kinetics
"... Motivation: Network models are widely used as structural summaries of biochemical systems. Statistical estimation of networks is usually based on linear or discrete models. However, the dynamics of these systems are generally nonlinear, suggesting that suitable nonlinear formulations may offer gains ..."
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Motivation: Network models are widely used as structural summaries of biochemical systems. Statistical estimation of networks is usually based on linear or discrete models. However, the dynamics of these systems are generally nonlinear, suggesting that suitable nonlinear formulations may offer gains with respect to network inference and associated prediction problems. Results:We present a general framework for both network inference and dynamical prediction that is rooted in nonlinear biochemical kinetics. This is done by considering a dynamical system based on a chemical reaction graph and associated kinetics parameters. Inference regarding both parameters and the reaction graph itself is carried out within a fully Bayesian framework. Prediction of dynamical behavior is achieved by averaging over both parameters and reaction graphs, allowing prediction even when the underlying reactions themselves are unknown or uncertain. Results, based on (i) data simulated from a mechanistic model of mitogenactivated protein kinase signaling and (ii) phosphoproteomic data from cancer cell lines, demonstrate that nonlinear formulations can yield gains in network inference and permit dynamical prediction in the challenging setting where the reaction graph is unknown. Availability: MATLAB R2014a software is available to download from warwick.ac.uk/chrisoates. Contact: