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Michael P. H. Stumpf

Publications and source records attributed to Michael P. H. Stumpf.

34 records · Page 2Linked to original sources
arXiv2012

Information Geometry and Sequential Monte Carlo

This paper explores the application of methods from information geometry to the sequential Monte Carlo (SMC) sampler. In particular the Riemannian manifold Metropolis-adjusted Langevin algorithm (mMALA) is adapted for the transition kernels in SMC. Similar to its function in Markov chain Monte Carlo methods, the mMALA is a fully adaptable kernel which allows for efficient sampling of high-dimensional and highly correlated parameter spaces. We set up the theoretical framework for its use in SMC with a focus on the application to the problem of sequential Bayesian inference for dynamical systems as modelled by sets of ordinary differential equations. In addition, we argue that defining the sequence of distributions on geodesics optimises the effective sample sizes in the SMC run. We illustrate the application of the methodology by inferring the parameters of simulated Lotka-Volterra and Fitzhugh-Nagumo models. In particular we demonstrate that compared to employing a standard adaptive random walk kernel, the SMC sampler with an information geometric kernel design attains a higher level of statistical robustness in the inferred parameters of the dynamical systems.

stat.ME↗
arXiv2012

On optimality of kernels for approximate Bayesian computation using sequential Monte Carlo

Approximate Bayesian computation (ABC) has gained popularity over the past few years for the analysis of complex models arising in population genetic, epidemiology and system biology. Sequential Monte Carlo (SMC) approaches have become work horses in ABC. Here we discuss how to construct the perturbation kernels that are required in ABC SMC approaches, in order to construct a set of distributions that start out from a suitably defined prior and converge towards the unknown posterior. We derive optimality criteria for different kernels, which are based on the Kullback-Leibler divergence between a distribution and the distribution of the perturbed particles. We will show that for many complicated posterior distributions, locally adapted kernels tend to show the best performance. In cases where it is possible to estimate the Fisher information we can construct particularly efficient perturbation kernels. We find that the added moderate cost of adapting kernel functions is easily regained in terms of the higher acceptance rate. We demonstrate the computational efficiency gains in a range of toy-examples which illustrate some of the challenges faced in real-world applications of ABC, before turning to two demanding parameter inference problem in molecular biology, which highlight the huge increases in efficiency that can be gained from choice of optimal models. We conclude with a general discussion of rational choice of perturbation kernels in ABC SMC settings.

stat.CO↗
arXiv2012

Optimizing Threshold - Schedules for Approximate Bayesian Computation Sequential Monte Carlo Samplers: Applications to Molecular Systems

The likelihood-free sequential Approximate Bayesian Computation (ABC) algorithms, are increasingly popular inference tools for complex biological models. Such algorithms proceed by constructing a succession of probability distributions over the parameter space conditional upon the simulated data lying in an $ε$--ball around the observed data, for decreasing values of the threshold $ε$. While in theory, the distributions (starting from a suitably defined prior) will converge towards the unknown posterior as $ε$ tends to zero, the exact sequence of thresholds can impact upon the computational efficiency and success of a particular application. In particular, we show here that the current preferred method of choosing thresholds as a pre-determined quantile of the distances between simulated and observed data from the previous population, can lead to the inferred posterior distribution being very different to the true posterior. Threshold selection thus remains an important challenge. Here we propose an automated and adaptive method that allows us to balance the need to minimise the threshold with computational efficiency. Moreover, our method which centres around predicting the threshold - acceptance rate curve using the unscented transform, enables us to avoid local minima - a problem that has plagued previous threshold schemes.

stat.CO↗
arXiv2012

A parameter-free model discrimination criterion based on steady-state coplanarity

We describe a novel procedure for deciding when a mass-action model is incompatible with observed steady-state data that does not require any parameter estimation. Thus, we avoid the difficulties of nonlinear optimization typically associated with methods based on parameter fitting. The key idea is to use the model equations to construct a transformation of the original variables such that any set of steady states of the model under that transformation lies on a common plane, irrespective of the values of the model parameters. Model rejection can then be performed by assessing the degree to which the transformed data deviate from coplanarity. We demonstrate our method by applying it to models of multisite phosphorylation and cell death signaling. Although somewhat limited at present, our work provides an important first step towards a parameter-free framework for data-driven model selection.

q-bio.QM↗
arXiv2011

Dependence of MAPK mediated signaling on Erk isoforms and differences in nuclear shuttling

The mitogen activated protein kinase (MAPK) family of proteins is involved in regulating cellular fate activities such as proliferation, differentiation and apoptosis. Their fundamental importance has attracted considerable attention on different aspects of the MAPK signaling dynamics; this is particularly true for the Erk/Mek system, which has become the canonical example for MAPK signaling systems. Erk exists in many different isoforms, of which the most widely studied are Erk1 and Erk2. Until recently, these two kinases were considered equivalent as they differ only subtly at the sequence level; however, these isoforms exhibit radically different trafficking between cytoplasm and nucleus. Here we use spatially resolved data on Erk1/2 to develop and analyze spatio-temporal models of these cascades; and we discuss how sensitivity analysis can be used to discriminate between mechanisms. We are especially interested in understanding why two such similar proteins should co-exist in the same organism, as their functional roles appear to be different. Our models elucidate some of the factors governing the interplay between processes and the Erk1/2 localization in different cellular compartments, including competition between isoforms. This methodology is applicable to a wide range of systems, such as activation cascades, where translocation of species occurs via signal pathways. Furthermore, our work may motivate additional emphasis for considering potentially different roles for isoforms that differ subtly at the sequence level.

q-bio.CB↗
arXiv2011

Designing Attractive Models via Automated Identification of Chaotic and Oscillatory Dynamical Regimes

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 data-points. 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.

stat.AP↗
arXiv2011

Considerate Approaches to Achieving Sufficiency for ABC model selection

For nearly any challenging scientific problem evaluation of the likelihood is problematic if not impossible. Approximate Bayesian computation (ABC) allows us to employ the whole Bayesian formalism to problems where we can use simulations from a model, but cannot evaluate the likelihood directly. When summary statistics of real and simulated data are compared --- rather than the data directly --- information is lost, unless the summary statistics are sufficient. Here we employ an information-theoretical framework that can be used to construct (approximately) sufficient statistics by combining different statistics until the loss of information is minimized. Such sufficient sets of statistics are constructed for both parameter estimation and model selection problems. We apply our approach to a range of illustrative and real-world model selection problems.

stat.CO↗
arXiv2011

Decomposing Noise in Biochemical Signalling Systems Highlights the Role of Protein Degradation

The phenomena of stochasticity in biochemical processes have been intriguing life scientists for the past few decades. We now know that living cells take advantage of stochasticity in some cases and counteract stochastic effects in others. The source of intrinsic stochasticity in biomolecular systems are random timings of individual reactions, which cumulatively drive the variability in outputs of such systems. Despite the acknowledged relevance of stochasticity in the functioning of living cells no rigorous method have been proposed to precisely identify sources of variability. In this paper we propose a novel methodology that allows us to calculate contributions of individual reactions into the variability of a system's output. We demonstrate that some reactions have dramatically different effects on noise than others. Surprisingly, in the class of open conversion systems that serve as an approximate model of signal transduction, the degradation of an output contributes half of the total noise. We also demonstrate the importance of degradation in other relevant systems and propose a degradation feedback control mechanism that has the capability of an effective noise suppression. Application of our method to some well studied biochemical systems such as: gene expression, Michaelis-Menten enzyme kinetics, and the p53 system indicates that our methodology reveals an unprecedented insight into the origins of variability in biochemical systems. For many systems an analytical decomposition is not available; therefore the method has been implemented as a Matlab package and is available from the authors upon request.

q-bio.QM↗
arXiv2011

Bayesian design of synthetic biological systems

Here we introduce a new design framework for synthetic biology that exploits the advantages of Bayesian model selection. We will argue that the difference between inference and design is that in the former we try to reconstruct the system that has given rise to the data that we observe, while in the latter, we seek to construct the system that produces the data that we would like to observe, i.e. the desired behavior. Our approach allows us to exploit methods from Bayesian statistics, including efficient exploration of models spaces and high-dimensional parameter spaces, and the ability to rank models with respect to their ability to generate certain types of data. Bayesian model selection furthermore automatically strikes a balance between complexity and (predictive or explanatory) performance of mathematical models. In order to deal with the complexities of molecular systems we employ an approximate Bayesian computation scheme which only requires us to simulate from different competing models in order to arrive at rational criteria for choosing between them. We illustrate the advantages resulting from combining the design and modeling (or in-silico prototyping) stages currently seen as separate in synthetic biology by reference to deterministic and stochastic model systems exhibiting adaptive and switch-like behavior, as well as bacterial two-component signaling systems.

q-bio.MN↗
arXiv2010

Simulation-based model selection for dynamical systems in systems and population biology

Computer simulations have become an important tool across the biomedical sciences and beyond. For many important problems several different models or hypotheses exist and choosing which one best describes reality or observed data is not straightforward. We therefore require suitable statistical tools that allow us to choose rationally between different mechanistic models of e.g. signal transduction or gene regulation networks. This is particularly challenging in systems biology where only a small number of molecular species can be assayed at any given time and all measurements are subject to measurement uncertainty. Here we develop such a model selection framework based on approximate Bayesian computation and employing sequential Monte Carlo sampling. We show that our approach can be applied across a wide range of biological scenarios, and we illustrate its use on real data describing influenza dynamics and the JAK-STAT signalling pathway. Bayesian model selection strikes a balance between the complexity of the simulation models and their ability to describe observed data. The present approach enables us to employ the whole formal apparatus to any system that can be (efficiently) simulated, even when exact likelihoods are computationally intractable.

q-bio.QM↗
arXiv2009

Parameter inference and model selection in signaling pathway models

To support and guide an extensive experimental research into systems biology of signaling pathways, increasingly more mechanistic models are being developed with hopes of gaining further insight into biological processes. In order to analyse these models, computational and statistical techniques are needed to estimate the unknown kinetic parameters. This chapter reviews methods from frequentist and Bayesian statistics for estimation of parameters and for choosing which model is best for modeling the underlying system. Approximate Bayesian Computation (ABC) techniques are introduced and employed to explore different hypothesis about the JAK-STAT signaling pathway.

q-bio.QM↗
arXiv2009

Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems

Approximate Bayesian computation methods can be used to evaluate posterior distributions without having to calculate likelihoods. In this paper we discuss and apply an approximate Bayesian computation (ABC) method based on sequential Monte Carlo (SMC) to estimate parameters of dynamical models. We show that ABC SMC gives information about the inferability of parameters and model sensitivity to changes in parameters, and tends to perform better than other ABC approaches. The algorithm is applied to several well known biological systems, for which parameters and their credible intervals are inferred. Moreover, we develop ABC SMC as a tool for model selection; given a range of different mathematical descriptions, ABC SMC is able to choose the best model using the standard Bayesian model selection apparatus.

stat.CO↗
arXiv2005

Sampling properties of random graphs: the degree distribution

We discuss two sampling schemes for selecting random subnets from a network: Random sampling and connectivity dependent sampling, and investigate how the degree distribution of a node in the network is affected by the two types of sampling. Here we derive a necessary and sufficient condition that guarantees that the degree distribution of the subnet and the true network belong to the same family of probability distributions. For completely random sampling of nodes we find that this condition is fulfilled by classical random graphs; for the vast majority of networks this condition will, however, not be met. We furthermore discuss the case where the probability of sampling a node depends on the degree of a node and we find that even classical random graphs are no longer closed under this sampling regime. We conclude by relating the results to real {\it E.coli} protein interaction network data.

cond-mat.stat-mech↗
arXiv2005

Probability Models for Degree Distributions of Protein Interaction Networks

The degree distribution of many biological and technological networks has been described as a power-law distribution. While the degree distribution does not capture all aspects of a network, it has often been suggested that its functional form contains important clues as to underlying evolutionary processes that have shaped the network. Generally, the functional form for the degree distribution has been determined in an ad-hoc fashion, with clear power-law like behaviour often only extending over a limited range of connectivities. Here we apply formal model selection techniques to decide which probability distribution best describes the degree distributions of protein interaction networks. Contrary to previous studies this well defined approach suggests that the degree distribution of many molecular networks is often better described by distributions other than the popular power-law distribution. This, in turn, suggests that simple, if elegant, models may not necessarily help in the quantitative understanding of complex biological processes.\

q-bio.MN↗
arXiv2005

Comparative Analysis of the Saccharomyces cerevisiae and Caenorhabditis elegans Protein Interaction Network

Protein interaction networks aim to summarize the complex interplay of proteins in an organism. Early studies suggested that the position of a protein in the network determines its evolutionary rate but there has been considerable disagreement as to what extent other factors, such as protein abundance, modify this reported dependence. We compare the genomes of Saccharomyces cerevisiae and Caenorhabditis elegans with those of closely related species to elucidate the recent evolutionary history of their respective protein interaction networks. Interaction and expression data are studied in the light of a detailed phylogenetic analysis. The underlying network structure is incorporated explicitly into the statistical analysis. The increased phylogenetic resolution, paired with high-quality interaction data, allows us to resolve the way in which protein interaction network structure and abundance of proteins affect the evolutionary rate. We find that expression levels are better predictors of the evolutionary rate than a protein's connectivity. Detailed analysis of the two organisms also shows that the evolutionary rates of interacting proteins are not sufficiently similar to be mutually predictive. It appears that meaningful inferences about the evolution of protein interaction networks require comparative analysis of reasonably closely related species. The signature of protein evolution is shaped by a protein's abundance in the organism and its function and the biological process it is involved in. Its position in the interaction networks and its connectivity may modulate this but they appear to have only minor influence on a protein's evolutionary rate.

q-bio.MN↗