Parameter inference for discretely observed stochastic kinetic models using stochastic gradient descent

Abstract Background Stochastic effects can be important for the behavior of processes involving small population numbers, so the study of stochastic models has become an important topic in the burgeoning field of computational systems biology. However analysis techniques for stochastic models have tended to lag behind their deterministic cousins due to the heavier computational demands of the statistical approaches for fitting the models to experimental data. There is a continuing need for more effective and efficient algorithms. In this article we focus on the parameter inference problem for stochastic kinetic models of biochemical reactions given discrete time-course observations of either some or all of the molecular species. Results We propose an algorithm for inference of kinetic rate parameters based upon maximum likelihood using stochastic gradient descent (SGD). We derive a general formula for the gradient of the likelihood function given discrete time-course observations. The formula applies to any explicit functional form of the kinetic rate laws such as mass-action, Michaelis-Menten, etc. Our algorithm estimates the gradient of the likelihood function by reversible jump Markov chain Monte Carlo sampling (RJMCMC), and then gradient descent method is employed to obtain the maximum likelihood estimation of parameter values. Furthermore, we utilize flux balance analysis and show how to automatically construct reversible jump samplers for arbitrary biochemical reaction models. We provide RJMCMC sampling algorithms for both fully observed and partially observed time-course observation data. Our methods are illustrated with two examples: a birth-death model and an auto-regulatory gene network. We find good agreement of the inferred parameters with the actual parameters in both models. Conclusions The SGD method proposed in the paper presents a general framework of inferring parameters for stochastic kinetic models. The method is computationally efficient and is effective for both partially and fully observed systems. Automatic construction of reversible jump samplers and general formulation of the likelihood gradient function makes our method applicable to a wide range of stochastic models. Furthermore our derivations can be useful for other purposes such as using the gradient information for parametric sensitivity analysis or using the reversible jump samplers for full Bayesian inference. The software implementing the algorithms is publicly available at http://cbcl.ics.uci.edu/sg

Inference for Ecological Dynamical Systems: A Case Study of Two Endemic Diseases

A Bayesian Markov chain Monte Carlo method is used to infer parameters for an open stochastic epidemiological model: the Markovian susceptible-infected-recovered (SIR) model, which is suitable for modeling and simulating recurrent epidemics. This allows exploring two major problems of inference appearing in many mechanistic population models. First, trajectories of these processes are often only partly observed. For example, during an epidemic the transmission process is only partly observable: one cannot record infection times. Therefore, one only records cases (infections) as the observations. As a result some means of imputing or reconstructing individuals in the susceptible cases class must be accomplished. Second, the official reporting of observations (cases in epidemiology) is typically done not as they are actually recorded but at some temporal interval over which they have been aggregated. To address these issues, this paper investigates the following problems. Parameter inference for a perfectly sampled open Markovian SIR is first considered. Next inference for an imperfectly observed sample path of the system is studied. Although this second problem has been solved for the case of closed epidemics, it has proven quite difficult for the case of open recurrent epidemics. Lastly, application of the statistical theory is made to measles and pertussis epidemic time series data from 60 UK cities

Rapid, Automated Determination of Reaction Models and Kinetic Parameters

We herein report a novel kinetic modelling methodology whereby identification of the correct reaction model and kinetic parameters is conducted by an autonomous framework combined with transient flow measurements to enable comprehensive process understanding with minimal user input. An automated flow chemistry platform was employed to initially conduct linear flow-ramp experiments to rapidly map the reaction profile of three processes using transient flow data. Following experimental data acquisition, a computational approach was utilised to discriminate between all possible reaction models as well as identify the correct kinetic parameters for each process. Species that are known to participate in the process (starting materials, intermediates, products) are initially inputted by the user prior to flow ramp experiments, then all possible model candidates are compiled into a model library based on their potential to occur after mass balance assessment. Parallel computational optimisation then evaluates each model by algorithmically altering the kinetic parameters of the model to allow convergence of a simulated kinetic curve to the experimental data provided. Statistical analysis then determines the most likely reaction model based on model simplicity and agreement with experimental data. This automated approach to gaining full process understanding, whereby a small number of data-rich experiments are conducted, and the kinetics are evaluated autonomously, shows significant improvements on current industrial optimisation techniques in terms of labour, time and overall cost. The computational approach herein described can be employed using data from any set of experiments and the code is open-source

Parameter inference for discretely observed stochastic kinetic models using stochastic gradient descent

Background: Stochastic effects can be important for the behavior of processes involving small population numbers, so the study of stochastic models has become an important topic in the burgeoning field of computational systems biology. However analysis techniques for stochastic models have tended to lag behind their deterministic cousins due to the heavier computational demands of the statistical approaches for fitting the models to experimental data. There is a continuing need for more effective and efficient algorithms. In this article we focus on the parameter inference problem for stochastic kinetic models of biochemical reactions given discrete time-course observations of either some or all of the molecular species. Results: We propose an algorithm for inference of kinetic rate parameters based upon maximum likelihood using stochastic gradient descent (SGD). We derive a general formula for the gradient of the likelihood function given discrete time-course observations. The formula applies to any explicit functional form of the kinetic rate laws such as mass-action, Michaelis-Menten, etc. Our algorithm estimates the gradient of the likelihood function by reversible jump Markov chain Monte Carlo sampling (RJMCMC), and then gradient descent method is employed to obtain the maximum likelihood estimation of parameter values. Furthermore, we utilize flux balance analysis and show how to automatically construct reversible jump samplers for arbitrary biochemical reaction models. We provide RJMCMC sampling algorithms for both fully observed and partially observed time-course observation data. Our methods are illustrated with two examples: a birth-death model and an auto-regulatory gene network. We find good agreement of the inferred parameters with the actual parameters in both models. Conclusions: The SGD method proposed in the paper presents a general framework of inferring parameters for stochastic kinetic models. The method is computationally efficient and is effective for both partially and fully observed systems. Automatic construction of reversible jump samplers and general formulation of the likelihood gradient function makes our method applicable to a wide range of stochastic models. Furthermore our derivations can be useful for other purposes such as using the gradient information for parametric sensitivity analysis or using the reversible jump samplers for full Bayesian inference. The software implementing the algorithms is publicly available at http://cbcl.ics.uci.edu/sg

Prospects for Declarative Mathematical Modeling of Complex Biological Systems

Declarative modeling uses symbolic expressions to represent models. With such expressions one can formalize high-level mathematical computations on models that would be difficult or impossible to perform directly on a lower-level simulation program, in a general-purpose programming language. Examples of such computations on models include model analysis, relatively general-purpose model-reduction maps, and the initial phases of model implementation, all of which should preserve or approximate the mathematical semantics of a complex biological model. The potential advantages are particularly relevant in the case of developmental modeling, wherein complex spatial structures exhibit dynamics at molecular, cellular, and organogenic levels to relate genotype to multicellular phenotype. Multiscale modeling can benefit from both the expressive power of declarative modeling languages and the application of model reduction methods to link models across scale. Based on previous work, here we define declarative modeling of complex biological systems by defining the operator algebra semantics of an increasingly powerful series of declarative modeling languages including reaction-like dynamics of parameterized and extended objects; we define semantics-preserving implementation and semantics-approximating model reduction transformations; and we outline a "meta-hierarchy" for organizing declarative models and the mathematical methods that can fruitfully manipulate them

Parameter inference for stochastic biological models

PhD ThesisParameter inference is the field concerned with estimating reliable model parameters from data. In recent years there has been a trend in the biology community toward single cell technologies such as fluorescent flow cytometry, transcriptomics and mass cytometry: providing a rich array of stochastic time series and temporal distribution data for analysis. Deterministically, there are a wide range of parameter inference and global optimisation techniques available. However, these do not always scale well to non-deterministic (i.e., stochastic) settings — whereby the temporal evolution of the system can be described by a chemical master equation for which the solution is nearly always intractable, and the dynamic behaviour of a system is hard to predict. For systems biology, the inference of stochastic parameters remains a bottleneck for accurate model simulation. This thesis is concerned with the parameter inference problem for stochastic chemical reaction networks. Stochastic chemical reaction networks are most frequently modelled as a continuous time discretestate Markov chain using Gillespie’s stochastic simulation algorithm. Firstly, I present a new parameter inference algorithm, SPICE, that combines Gillespie’s algorithm with the cross-entropy method. The cross-entropy method is a novel approach for global optimisation inspired from the field of rare-event probability estimation. I then present recent advances in utilising the generalised method of moments for inference, and seek to provide these approaches with a direct stochastic simulation based correction. Subsequently, I present a novel use of a recent multi-level tau-leaping approach for simulating population moments efficiently, and use this to provide a simulation based correction to the generalised method of moments. I also propose a new method for moment closures based on the use of Padé approximants. The presented algorithms are evaluated on a number of challenging case studies, including bistable systems — e.g., the Schlögl System and the Genetic Toggle Switch — and real experimental data. Experimental results are presented using each of the given algorithms. We also consider ‘realistic’ data — i.e., datasets missing model species, multiple datasets originating from experiment repetitions, and datasets containing arbitrary units (e.g., fluorescence values). The developed approaches are found to be viable alternatives to existing state-ofthe-art methods, and in certain cases are able to outperform other methods in terms of either speed, or accuracyNewcastle/Liverpool/Durham BBSRC Doctoral Training Partnership for financial suppor