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Register a new userEvaluation of Parallel Tempering to Accelerate Bayesian Parameter Estimation in Systems Biology
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Models of biological systems often have many unknown parameters that must be determined in order for model behavior to match experimental observations. Commonly-used methods for parameter estimation that return point estimates of the best-fit parameters are insufficient when models are high dimensional and under-constrained. As a result, Bayesian methods, which treat model parameters as random variables and attempt to estimate their probability distributions given data, have become popular in systems biology. Bayesian parameter estimation often relies on Markov Chain Monte Carlo (MCMC) methods to sample model parameter distributions, but the slow convergence of MCMC sampling can be a major bottleneck. One approach to improving performance is parallel tempering (PT), a physics-based method that uses swapping between multiple Markov chains run in parallel at different temperatures to accelerate sampling. The temperature of a Markov chain determines the probability of accepting an unfavorable move, so swapping with higher temperatures chains enables the sampling chain to escape from local minima. In this work we compared the MCMC performance of PT and the commonly-used Metropolis-Hastings (MH) algorithm on six biological models of varying complexity. We found that for simpler models PT accelerated convergence and sampling, and that for more complex models, PT often converged in cases MH became trapped in non-optimal local minima. We also developed a freely-available MATLAB package for Bayesian parameter estimation called PTempEst (http://github.com/RuleWorld/ptempest), which is closely integrated with the popular BioNetGen software for rule-based modeling of biological systems.
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Reproducibility and reusability of the results of data-based modeling studies are essential. Yet, there has been -- so far -- no broadly supported format for the specification of parameter estimation problems in systems biology. Here, we introduce PEtab, a format which facilitates the specification of parameter estimation problems using Systems Biology Markup Language (SBML) models and a set of tab-separated value files describing the observation model and experimental data as well as parameters to be estimated. We already implemented PEtab support into eight well-established model simulation and parameter estimation toolboxes with hundreds of users in total. We provide a Python library for validation and modification of a PEtab problem and currently 20 example parameter estimation problems based on recent studies. Specifications of PEtab, the PEtab Python library, as well as links to examples, and all supporting software tools are available at https://github.com/PEtab-dev/PEtab, a snapshot is available at https://doi.org/10.5281/zenodo.3732958. All original content is available under permissive licenses.
Ordinary differential equations (ODE) are widely used for modeling in Systems Biology. As most commonly only some of the kinetic parameters are measurable or precisely known, parameter estimation techniques are applied to parametrize the model to experimental data. A main challenge for the parameter estimation is the complexity of the parameter space, especially its high dimensionality and local minima. Parameter estimation techniques consist of an objective function, measuring how well a certain parameter set describes the experimental data, and an optimization algorithm that optimizes this objective function. A lot of effort has been spent on developing highly sophisticated optimization algorithms to cope with the complexity in the parameter space, but surprisingly few articles address the influence of the objective function on the computational complexity in finding global optima. We extend a recently developed multiple shooting for stochastic systems (MSS) objective function for parameter estimation of stochastic models and apply it to parameter estimation of ODE models. This MSS objective function treats the intervals between measurement points separately. This separate treatment allows the ODE trajectory to stay closer to the data and we show that it reduces the complexity of the parameter space. We use examples from Systems Biology, namely a Lotka-Volterra model, a FitzHugh-Nagumo oscillator and a Calcium oscillation model, to demonstrate the power of the MSS approach for reducing the complexity and the number of local minima in the parameter space. The approach is fully implemented in the COPASI software package and, therefore, easily accessible for a wide community of researchers.
Motivated by applications in systems biology, we seek a probabilistic framework based on Markov processes to represent intracellular processes. We review the formal relationships between different stochastic models referred to in the systems biology literature. As part of this review, we present a novel derivation of the differential Chapman-Kolmogorov equation for a general multidimensional Markov process made up of both continuous and jump processes. We start with the definition of a time-derivative for a probability density but place no restrictions on the probability distribution, in particular, we do not assume it to be confined to a region that has a surface (on which the probability is zero). In our derivation, the master equation gives the jump part of the Markov process while the Fokker-Planck equation gives the continuous part. We thereby sketch a {}``family tree for stochastic models in systems biology, providing explicit derivations of their formal relationship and clarifying assumptions involved.
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.
In the current work we present two generalizations of the Parallel Tempering algorithm, inspired by the so-called continuous-time Infinite Swapping algorithm. Such a method, found its origins in the molecular dynamics community, and can be understood as the limit case of the continuous-time Parallel Tempering algorithm, where the (random) time between swaps of states between two parallel chains goes to zero. Thus, swapping states between chains occurs continuously. In the current work, we extend this idea to the context of time-discrete Markov chains and present two Markov chain Monte Carlo algorithms that follow the same paradigm as the continuous-time infinite swapping procedure. We analyze the convergence properties of such discrete-time algorithms in terms of their spectral gap, and implement them to sample from different target distributions. Numerical results show that the proposed methods significantly improve over more traditional sampling algorithms such as Random Walk Metropolis and (traditional) Parallel Tempering.
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