Tests for cycling in a signalling pathway

Modeling cell population dynamics

 Quantitative modeling is quickly becoming an integral part of biology, due to the ability of mathematical models and computer simulations to generate insights and predict the behavior of living systems. Single-cell models can be incapable or misleading for inferring population dynamics, as they do not consider the interactions between cells via metabolites or physical contact, nor do they consider competition for limited resources such as nutrients or space. Here we examine methods that are commonly used to model and simulate cell populations. First, we cover simple models where analytic solutions are available, and then move on to more complex scenarios where computational methods are required. Overall, we present a summary of mathematical models used to describe cell population dynamics, which may aid future model development and highlights the importance of population modeling in biology.

Formulation, construction and analysis of kinetic models of metabolism: A review of modelling frameworks

Kinetic models are critical to predict the dynamic behaviour of metabolic networks. Mechanistic kinetic models for large networks remain uncommon due to the difficulty of fitting their parameters. Recent modelling frameworks promise new ways to overcome this obstacle while retaining predictive capabilities. In this review, we present an overview of the relevant mathematical frameworks for kinetic formulation, construction and analysis. Starting with kinetic formalisms, we next review statistical methods for parameter inference, as well as recent computational frameworks applied to the construction and analysis of kinetic models. Finally, we discuss opportunities and limitations hindering the development of larger kinetic reconstructions.

Dynamic system classifier

Stochastic differential equations describe well many physical, biological, and sociological systems, despite the simplification often made in their derivation. Here the usage of simple stochastic differential equations to characterize and classify complex dynamical systems is proposed within a Bayesian framework. To this end, we develop a dynamic system classifier (DSC). The DSC first abstracts training data of a system in terms of time-dependent coefficients of the descriptive stochastic differential equation. Thereby the DSC identifies unique correlation structures within the training data. For definiteness we restrict the presentation of the DSC to oscillation processes with a time-dependent frequency ω(t) and damping factor γ(t). Although real systems might be more complex, this simple oscillator captures many characteristic features. The ω and γ time lines represent the abstract system characterization and permit the construction of efficient signal classifiers. Numerical experiments show that such classifiers perform well even in the low signal-to-noise regime.

Validation and selection of ODE based systems biology models: how to arrive at more reliable decisions

Background

Most ordinary differential equation (ODE) based modeling studies in systems biology involve a hold-out validation step for model validation. In this framework a pre-determined part of the data is used as validation data and, therefore it is not used for estimating the parameters of the model. The model is assumed to be validated if the model predictions on the validation dataset show good agreement with the data. Model selection between alternative model structures can also be performed in the same setting, based on the predictive power of the model structures on the validation dataset. However, drawbacks associated with this approach are usually under-estimated.

Results

We have carried out simulations by using a recently published High Osmolarity Glycerol (HOG) pathway from S.cerevisiae to demonstrate these drawbacks. We have shown that it is very important how the data is partitioned and which part of the data is used for validation purposes. The hold-out validation strategy leads to biased conclusions, since it can lead to different validation and selection decisions when different partitioning schemes are used. Furthermore, finding sensible partitioning schemes that would lead to reliable decisions are heavily dependent on the biology and unknown model parameters which turns the problem into a paradox. This brings the need for alternative validation approaches that offer flexible partitioning of the data. For this purpose, we have introduced a stratified random cross-validation (SRCV) approach that successfully overcomes these limitations.

Conclusions

SRCV leads to more stable decisions for both validation and selection which are not biased by underlying biological phenomena. Furthermore, it is less dependent on the specific noise realization in the data. Therefore, it proves to be a promising alternative to the standard hold-out validation strategy.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-015-0180-0) contains supplementary material, which is available to authorized users.

Optimal experiment design for model selection in biochemical networks

Background

Mathematical modeling is often used to formalize hypotheses on how a biochemical network operates by discriminating between competing models. Bayesian model selection offers a way to determine the amount of evidence that data provides to support one model over the other while favoring simple models. In practice, the amount of experimental data is often insufficient to make a clear distinction between competing models. Often one would like to perform a new experiment which would discriminate between competing hypotheses.

Results

We developed a novel method to perform Optimal Experiment Design to predict which experiments would most effectively allow model selection. A Bayesian approach is applied to infer model parameter distributions. These distributions are sampled and used to simulate from multivariate predictive densities. The method is based on a k-Nearest Neighbor estimate of the Jensen Shannon divergence between the multivariate predictive densities of competing models.

Conclusions

We show that the method successfully uses predictive differences to enable model selection by applying it to several test cases. Because the design criterion is based on predictive distributions, which can be computed for a wide range of model quantities, the approach is very flexible. The method reveals specific combinations of experiments which improve discriminability even in cases where data is scarce. The proposed approach can be used in conjunction with existing Bayesian methodologies where (approximate) posteriors have been determined, making use of relations that exist within the inferred posteriors.

Parameter trajectory analysis to identify treatment effects of pharmacological interventions

The field of medical systems biology aims to advance understanding of molecular mechanisms that drive disease progression and to translate this knowledge into therapies to effectively treat diseases. A challenging task is the investigation of long-term effects of a (pharmacological) treatment, to establish its applicability and to identify potential side effects. We present a new modeling approach, called Analysis of Dynamic Adaptations in Parameter Trajectories (ADAPT), to analyze the long-term effects of a pharmacological intervention. A concept of time-dependent evolution of model parameters is introduced to study the dynamics of molecular adaptations. The progression of these adaptations is predicted by identifying necessary dynamic changes in the model parameters to describe the transition between experimental data obtained during different stages of the treatment. The trajectories provide insight in the affected underlying biological systems and identify the molecular events that should be studied in more detail to unravel the mechanistic basis of treatment outcome. Modulating effects caused by interactions with the proteome and transcriptome levels, which are often less well understood, can be captured by the time-dependent descriptions of the parameters. ADAPT was employed to identify metabolic adaptations induced upon pharmacological activation of the liver X receptor (LXR), a potential drug target to treat or prevent atherosclerosis. The trajectories were investigated to study the cascade of adaptations. This provided a counter-intuitive insight concerning the function of scavenger receptor class B1 (SR-B1), a receptor that facilitates the hepatic uptake of cholesterol. Although activation of LXR promotes cholesterol efflux and -excretion, our computational analysis showed that the hepatic capacity to clear cholesterol was reduced upon prolonged treatment. This prediction was confirmed experimentally by immunoblotting measurements of SR-B1 in hepatic membranes. Next to the identification of potential unwanted side effects, we demonstrate how ADAPT can be used to design new target interventions to prevent these.

A driving ambition of medical systems biology is to advance our understanding of molecular processes that drive the progression of complex diseases such as Type 2 Diabetes and cardiovascular disease. This insight is essential to enable the development of therapies to effectively treat diseases. A challenging task is to investigate the long-term effects of a treatment, in order to establish its applicability and to identify potential side effects. As such, there is a growing need for novel approaches to support this research. Here, we present a new computational approach to identify treatment effects. We make use of a computational model of the biological system. The model is used to describe the experimental data obtained during different stages of the treatment. To incorporate the long-term/progressive adaptations in the system, induced by changes in gene and protein expression, the model is iteratively updated. The approach was employed to identify metabolic adaptations induced by a potential anti-atherosclerotic and anti-diabetic drug target. Our approach identifies the molecular events that should be studied in more detail to establish the mechanistic basis of treatment outcome. New biological insight was obtained concerning the metabolism of cholesterol, which was in turn experimentally validated.

Applications of analysis of dynamic adaptations in parameter trajectories

Metabolic profiling in combination with pathway-based analyses and computational modelling are becoming increasingly important in clinical and preclinical research. Modelling multi-factorial, progressive diseases requires the integration of molecular data at the metabolome, proteome and transcriptome levels. Also the dynamic interaction of organs and tissues needs to be considered. The processes involved cover time scales that are several orders of magnitude different. We report applications of a computational approach to bridge the scales and different levels of biological detail. Analysis of dynamic adaptations in parameter trajectories (ADAPTs) aims to investigate phenotype transitions during disease development and after a therapeutic intervention. ADAPT is based on a time-dependent evolution of model parameters to describe the dynamics of metabolic adaptations. The progression of metabolic adaptations is predicted by identifying necessary dynamic changes in the model parameters to describe the transition between experimental data obtained during different stages. To get a better understanding of the concept, the ADAPT approach is illustrated in a theoretical study. Its application in research on progressive changes in lipoprotein metabolism is also discussed.

Parameter uncertainty in biochemical models described by ordinary differential equations

Improved mechanistic understanding of biochemical networks is one of the driving ambitions of Systems Biology. Computational modeling allows the integration of various sources of experimental data in order to put this conceptual understanding to the test in a quantitative manner. The aim of computational modeling is to obtain both predictive as well as explanatory models for complex phenomena, hereby providing useful approximations of reality with varying levels of detail. As the complexity required to describe different system increases, so does the need for determining how well such predictions can be made. Despite efforts to make tools for uncertainty analysis available to the field, these methods have not yet found widespread use in the field of Systems Biology. Additionally, the suitability of the different methods strongly depends on the problem and system under investigation. This review provides an introduction to some of the techniques available as well as gives an overview of the state-of-the-art methods for parameter uncertainty analysis.

ABC-SysBio--approximate Bayesian computation in Python with GPU support

MOTIVATION

The growing field of systems biology has driven demand for flexible tools to model and simulate biological systems. Two established problems in the modeling of biological processes are model selection and the estimation of associated parameters. A number of statistical approaches, both frequentist and Bayesian, have been proposed to answer these questions.

RESULTS

Here we present a Python package, ABC-SysBio, that implements parameter inference and model selection for dynamical systems in an approximate Bayesian computation (ABC) framework. ABC-SysBio combines three algorithms: ABC rejection sampler, ABC SMC for parameter inference and ABC SMC for model selection. It is designed to work with models written in Systems Biology Markup Language (SBML). Deterministic and stochastic models can be analyzed in ABC-SysBio.

AVAILABILITY

http://abc-sysbio.sourceforge.net

Parameter estimation and model selection in computational biology

A central challenge in computational modeling of biological systems is the determination of the model parameters. Typically, only a fraction of the parameters (such as kinetic rate constants) are experimentally measured, while the rest are often fitted. The fitting process is usually based on experimental time course measurements of observables, which are used to assign parameter values that minimize some measure of the error between these measurements and the corresponding model prediction. The measurements, which can come from immunoblotting assays, fluorescent markers, etc., tend to be very noisy and taken at a limited number of time points. In this work we present a new approach to the problem of parameter selection of biological models. We show how one can use a dynamic recursive estimator, known as extended Kalman filter, to arrive at estimates of the model parameters. The proposed method follows. First, we use a variation of the Kalman filter that is particularly well suited to biological applications to obtain a first guess for the unknown parameters. Secondly, we employ an a posteriori identifiability test to check the reliability of the estimates. Finally, we solve an optimization problem to refine the first guess in case it should not be accurate enough. The final estimates are guaranteed to be statistically consistent with the measurements. Furthermore, we show how the same tools can be used to discriminate among alternate models of the same biological process. We demonstrate these ideas by applying our methods to two examples, namely a model of the heat shock response in E. coli, and a model of a synthetic gene regulation system. The methods presented are quite general and may be applied to a wide class of biological systems where noisy measurements are used for parameter estimation or model selection.

Parameter estimation is a key issue in systems biology, as it represents the crucial step to obtaining predictions from computational models of biological systems. This issue is usually addressed by “fitting” the model simulations to the observed experimental data. Such approach does not take the measurement noise into full consideration. We introduce a new method built on the combination of Kalman filtering, statistical tests, and optimization techniques. The filter is well-known in control and estimation theory and has found application in a wide range of fields, such as inertial guidance systems, weather forecasting, and economics. We show how the statistics of the measurement noise can be optimally exploited and directly incorporated into the design of the estimation algorithm in order to achieve more accurate results, and to validate/invalidate the computed estimates. We also show that a significant advantage of our estimator is that it offers a powerful tool for model selection, allowing rejection or acceptance of competing models based on the available noisy measurements. These results are of immediate practical application in computational biology, and while we demonstrate their use for two specific examples, they can in fact be used to study a wide class of biological systems.

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 analyze 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 techniques are introduced and employed to explore different hypothesis about the JAK-STAT signaling pathway.

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

MOTIVATION

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.

RESULTS

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.

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

Approximate Bayesian computation (ABC) methods can be used to evaluate posterior distributions without having to calculate likelihoods. In this paper, we discuss and apply an ABC method based on sequential Monte Carlo (SMC) to estimate parameters of dynamical models. We show that ABC SMC provides 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.

Putting the pieces together in diabetes research: towards a hierarchical model of whole-body glucose homeostasis

Type 2 diabetes is one of the most widespread and rapidly spreading diseases world-wide and has been subject of extensive research efforts. However, understanding the molecular basis of the disease is increasing piecemeal and a consensus regarding the overall picture of normal metabolic regulation and malfunction in diabetes has not emerged. A systems biology approach, combining mathematical modelling with simultaneous high-throughput measurements, can be of considerable help. On the whole-body level, this has been done in pharmacokinetic and pharmacodynamic models, which recently have started to mature into more physiologically realistic organ-based models. At the other end of the spectrum, detailed models for crucial cellular processes are starting to mature into complete modules that potentially can be fitted into such whole-body organ-based models. The result of such a merge is a multi-level hierarchical model, which is a model type that has been common in technical systems. In this review, we report and exemplify some of the recent progress that has been made to achieve such a hierarchical model, and we argue why it is only through such a model that a complete picture of diabetes mellitus can be obtained.

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.