Practical limits for reverse engineering of dynamical systems: a statistical analysis of sensitivity and parameter inferability in systems biology models

Deep reinforcement learning for optimal experimental design in biology

The field of optimal experimental design uses mathematical techniques to determine experiments that are maximally informative from a given experimental setup. Here we apply a technique from artificial intelligence-reinforcement learning-to the optimal experimental design task of maximizing confidence in estimates of model parameter values. We show that a reinforcement learning approach performs favourably in comparison with a one-step ahead optimisation algorithm and a model predictive controller for the inference of bacterial growth parameters in a simulated chemostat. Further, we demonstrate the ability of reinforcement learning to train over a distribution of parameters, indicating that this approach is robust to parametric uncertainty.

Bayesian parameter estimation for dynamical models in systems biology

Dynamical systems modeling, particularly via systems of ordinary differential equations, has been used to effectively capture the temporal behavior of different biochemical components in signal transduction networks. Despite the recent advances in experimental measurements, including sensor development and '-omics' studies that have helped populate protein-protein interaction networks in great detail, modeling in systems biology lacks systematic methods to estimate kinetic parameters and quantify associated uncertainties. This is because of multiple reasons, including sparse and noisy experimental measurements, lack of detailed molecular mechanisms underlying the reactions, and missing biochemical interactions. Additionally, the inherent nonlinearities with respect to the states and parameters associated with the system of differential equations further compound the challenges of parameter estimation. In this study, we propose a comprehensive framework for Bayesian parameter estimation and complete quantification of the effects of uncertainties in the data and models. We apply these methods to a series of signaling models of increasing mathematical complexity. Systematic analysis of these dynamical systems showed that parameter estimation depends on data sparsity, noise level, and model structure, including the existence of multiple steady states. These results highlight how focused uncertainty quantification can enrich systems biology modeling and enable additional quantitative analyses for parameter estimation.

A dynamically structured matrix population model for insect life histories observed under variable environmental conditions

Various environmental drivers influence life processes of insect vectors that transmit human disease. Life histories observed under experimental conditions can reveal such complex links; however, designing informative experiments for insects is challenging. Furthermore, inferences obtained under controlled conditions often extrapolate poorly to field conditions. Here, we introduce a pseudo-stage-structured population dynamics model to describe insect development as a renewal process with variable rates. The model permits representing realistic life stage durations under constant and variable environmental conditions. Using the model, we demonstrate how random environmental variations result in fluctuating development rates and affect stage duration. We apply the model to infer environmental dependencies from the life history observations of two common disease vectors, the southern (Culex quinquefasciatus) and northern (Culex pipiens) house mosquito. We identify photoperiod, in addition to temperature, as pivotal in regulating larva stage duration, and find that carefully timed life history observations under semi-field conditions accurately predict insect development throughout the year. The approach we describe augments existing methods of life table design and analysis, and contributes to the development of large-scale climate- and environment-driven population dynamics models for important disease vectors.

Model comparison via simplicial complexes and persistent homology

In many scientific and technological contexts, we have only a poor understanding of the structure and details of appropriate mathematical models. We often, therefore, need to compare different models. With available data we can use formal statistical model selection to compare and contrast the ability of different mathematical models to describe such data. There is, however, a lack of rigorous methods to compare different models a priori. Here, we develop and illustrate two such approaches that allow us to compare model structures in a systematic way by representing models as simplicial complexes. Using well-developed concepts from simplicial algebraic topology, we define a distance between models based on their simplicial representations. Employing persistent homology with a flat filtration provides for alternative representations of the models as persistence intervals, which represent model structure, from which the model distances are also obtained. We then expand on this measure of model distance to study the concept of model equivalence to determine the conceptual similarity of models. We apply our methodology for model comparison to demonstrate an equivalence between a positional-information model and a Turing-pattern model from developmental biology, constituting a novel observation for two classes of models that were previously regarded as unrelated.

Interrogating theoretical models of neural computation with emergent property inference

A cornerstone of theoretical neuroscience is the circuit model: a system of equations that captures a hypothesized neural mechanism. Such models are valuable when they give rise to an experimentally observed phenomenon -- whether behavioral or a pattern of neural activity -- and thus can offer insights into neural computation. The operation of these circuits, like all models, critically depends on the choice of model parameters. A key step is then to identify the model parameters consistent with observed phenomena: to solve the inverse problem. In this work, we present a novel technique, emergent property inference (EPI), that brings the modern probabilistic modeling toolkit to theoretical neuroscience. When theorizing circuit models, theoreticians predominantly focus on reproducing computational properties rather than a particular dataset. Our method uses deep neural networks to learn parameter distributions with these computational properties. This methodology is introduced through a motivational example of parameter inference in the stomatogastric ganglion. EPI is then shown to allow precise control over the behavior of inferred parameters and to scale in parameter dimension better than alternative techniques. In the remainder of this work, we present novel theoretical findings in models of primary visual cortex and superior colliculus, which were gained through the examination of complex parametric structure captured by EPI. Beyond its scientific contribution, this work illustrates the variety of analyses possible once deep learning is harnessed towards solving theoretical inverse problems.

Identifiability analysis for stochastic differential equation models in systems biology

Mathematical models are routinely calibrated to experimental data, with goals ranging from building predictive models to quantifying parameters that cannot be measured. Whether or not reliable parameter estimates are obtainable from the available data can easily be overlooked. Such issues of parameter identifiability have important ramifications for both the predictive power of a model, and the mechanistic insight that can be obtained. Identifiability analysis is well-established for deterministic, ordinary differential equation (ODE) models, but there are no commonly adopted methods for analysing identifiability in stochastic models. We provide an accessible introduction to identifiability analysis and demonstrate how existing ideas for analysis of ODE models can be applied to stochastic differential equation (SDE) models through four practical case studies. To assess structural identifiability, we study ODEs that describe the statistical moments of the stochastic process using open-source software tools. Using practically motivated synthetic data and Markov chain Monte Carlo methods, we assess parameter identifiability in the context of available data. Our analysis shows that SDE models can often extract more information about parameters than deterministic descriptions. All code used to perform the analysis is available on Github.

Optimally Designed Model Selection for Synthetic Biology

Modeling parts and circuits represents a significant roadblock to automating the Design-Build-Test-Learn cycle in synthetic biology. Once models are developed, discriminating among them requires informative data, computational resources, and skills that might not be readily available. The high cost entailed in model discrimination frequently leads to subjective choices on the selected structures and, in turn, to suboptimal models. Here, we outline frequentist and Bayesian approaches to model discrimination. We ranked three candidate models of a genetic toggle switch, which was adopted as a test case, according to the support from in vivo data. We show that, in each framework, efficient model discrimination can be achieved via optimally designed experiments. We offer a dynamical-systems interpretation of our optimization results and investigate their sensitivity to key parameters in the characterization of synthetic circuits. Our approach suggests that optimal experimental design is an effective strategy to discriminate between competing models of a gene regulatory network. Independent of the adopted framework, optimally designed perturbations exploit regions in the input space that maximally distinguish predictions from the competing models.

Multi-model and network inference based on ensemble estimates: avoiding the madness of crowds

Recent progress in theoretical systems biology, applied mathematics and computational statistics allows us to compare the performance of different candidate models at describing a particular biological system quantitatively. Model selection has been applied with great success to problems where a small number-typically less than 10-of models are compared, but recent studies have started to consider thousands and even millions of candidate models. Often, however, we are left with sets of models that are compatible with the data, and then we can use ensembles of models to make predictions. These ensembles can have very desirable characteristics, but as I show here are not guaranteed to improve on individual estimators or predictors. I will show in the cases of model selection and network inference when we can trust ensembles, and when we should be cautious. The analyses suggest that the careful construction of an ensemble-choosing good predictors-is of paramount importance, more than had perhaps been realized before: merely adding different methods does not suffice. The success of ensemble network inference methods is also shown to rest on their ability to suppress false-positive results. A Jupyter notebook which allows carrying out an assessment of ensemble estimators is provided.

Computer-Aided Whole-Cell Design: Taking a Holistic Approach by Integrating Synthetic With Systems Biology

Computer-aided design (CAD) for synthetic biology promises to accelerate the rational and robust engineering of biological systems. It requires both detailed and quantitative mathematical and experimental models of the processes to (re)design biology, and software and tools for genetic engineering and DNA assembly. Ultimately, the increased precision in the design phase will have a dramatic impact on the production of designer cells and organisms with bespoke functions and increased modularity. CAD strategies require quantitative models of cells that can capture multiscale processes and link genotypes to phenotypes. Here, we present a perspective on how whole-cell, multiscale models could transform design-build-test-learn cycles in synthetic biology. We show how these models could significantly aid in the design and learn phases while reducing experimental testing by presenting case studies spanning from genome minimization to cell-free systems. We also discuss several challenges for the realization of our vision. The possibility to describe and build whole-cells in silico offers an opportunity to develop increasingly automatized, precise and accessible CAD tools and strategies.

Phenotypic selection through cell death: stochastic modelling of O-6-methylguanine-DNA methyltransferase dynamics

Glioblastoma (GBM) is the most aggressive malignant primary brain tumour with a median overall survival of 15 months. To treat GBM, patients currently undergo a surgical resection followed by exposure to radiotherapy and concurrent and adjuvant temozolomide (TMZ) chemotherapy. However, this protocol often leads to treatment failure, with drug resistance being the main reason behind this. To date, many studies highlight the role of O-6-methylguanine-DNA methyltransferase (MGMT) in conferring drug resistance. The mechanism through which MGMT confers resistance is not well studied-particularly in terms of computational models. With only a few reasonable biological assumptions, we were able to show that even a minimal model of MGMT expression could robustly explain TMZ-mediated drug resistance. In particular, we showed that for a wide range of parameter values constrained by novel cell growth and viability assays, a model accounting for only stochastic gene expression of MGMT coupled with cell growth, division, partitioning and death was able to exhibit phenotypic selection of GBM cells expressing MGMT in response to TMZ. Furthermore, we found this selection allowed the cells to pass their acquired phenotypic resistance onto daughter cells in a stable manner (as long as TMZ is provided). This suggests that stochastic gene expression alone is enough to explain the development of chemotherapeutic resistance.

Parameter inference in dynamical systems with co-dimension 1 bifurcations

Dynamical systems with intricate behaviour are all-pervasive in biology. Many of the most interesting biological processes indicate the presence of bifurcations, i.e. phenomena where a small change in a system parameter causes qualitatively different behaviour. Bifurcation theory has become a rich field of research in its own right and evaluating the bifurcation behaviour of a given dynamical system can be challenging. An even greater challenge, however, is to learn the bifurcation structure of dynamical systems from data, where the precise model structure is not known. Here, we study one aspects of this problem: the practical implications that the presence of bifurcations has on our ability to infer model parameters and initial conditions from empirical data; we focus on the canonical co-dimension 1 bifurcations and provide a comprehensive analysis of how dynamics, and our ability to infer kinetic parameters are linked. The picture thus emerging is surprisingly nuanced and suggests that identification of the qualitative dynamics-the bifurcation diagram-should precede any attempt at inferring kinetic parameters.

Comprehensive Review of Models and Methods for Inferences in Bio-Chemical Reaction Networks

The key processes in biological and chemical systems are described by networks of chemical reactions. From molecular biology to biotechnology applications, computational models of reaction networks are used extensively to elucidate their non-linear dynamics. The model dynamics are crucially dependent on the parameter values which are often estimated from observations. Over the past decade, the interest in parameter and state estimation in models of (bio-) chemical reaction networks (BRNs) grew considerably. The related inference problems are also encountered in many other tasks including model calibration, discrimination, identifiability, and checking, and optimum experiment design, sensitivity analysis, and bifurcation analysis. The aim of this review paper is to examine the developments in literature to understand what BRN models are commonly used, and for what inference tasks and inference methods. The initial collection of about 700 documents concerning estimation problems in BRNs excluding books and textbooks in computational biology and chemistry were screened to select over 270 research papers and 20 graduate research theses. The paper selection was facilitated by text mining scripts to automate the search for relevant keywords and terms. The outcomes are presented in tables revealing the levels of interest in different inference tasks and methods for given models in the literature as well as the research trends are uncovered. Our findings indicate that many combinations of models, tasks and methods are still relatively unexplored, and there are many new research opportunities to explore combinations that have not been considered-perhaps for good reasons. The most common models of BRNs in literature involve differential equations, Markov processes, mass action kinetics, and state space representations whereas the most common tasks are the parameter inference and model identification. The most common methods in literature are Bayesian analysis, Monte Carlo sampling strategies, and model fitting to data using evolutionary algorithms. The new research problems which cannot be directly deduced from the text mining data are also discussed.

Proteolytic dynamics of human 20S thymoproteasome

An efficient immunosurveillance of CD8⁺ T cells in the periphery depends on positive/negative selection of thymocytes and thus on the dynamics of antigen degradation and epitope production by thymoproteasome and immunoproteasome in the thymus. Although studies in mouse systems have shown how thymoproteasome activity differs from that of immunoproteasome and strongly impacts the T cell repertoire, the proteolytic dynamics and the regulation of human thymoproteasome are unknown. By combining biochemical and computational modeling approaches, we show here that human 20S thymoproteasome and immunoproteasome differ not only in the proteolytic activity of the catalytic sites but also in the peptide transport. These differences impinge upon the quantity of peptide products rather than where the substrates are cleaved. The comparison of the two human 20S proteasome isoforms depicts different processing of antigens that are associated to tumors and autoimmune diseases.

A climate-driven and field data-assimilated population dynamics model of sand flies

Sand flies are responsible for the transmission of leishmaniasis, a neglected tropical disease claiming more than 50,000 lives annually. Leishmaniasis is an emerging health risk in tropical and Mediterranean countries as well as temperate regions in North America and Europe. There is an increasing demand for predicting population dynamics and spreading of sand flies to support management and control, yet phenotypic diversity and complex environmental dependence hamper model development. Here, we present the principles for developing predictive species-specific population dynamics models for important disease vectors. Based on these principles, we developed a sand fly population dynamics model with a generic structure where model parameters are inferred using a surveillance dataset collected from Greece and Cyprus. The model incorporates distinct life stages and explicit dependence on a carefully selected set of environmental variables. The model successfully replicates the observations and demonstrates high predictive capacity on the validation dataset from Turkey. The surveillance datasets inform about biological processes, even in the absence of laboratory experiments. Our findings suggest that the methodology can be applied to other vector species to predict abundance, control dispersion, and help to manage the global burden of vector-borne diseases.

Component Characterization in a Growth-Dependent Physiological Context: Optimal Experimental Design

Synthetic biology design challenges have driven the use of mathematical models to characterize genetic components and to explore complex design spaces. Traditional approaches to characterization have largely ignored the effect of strain and growth conditions on the dynamics of synthetic genetic circuits, and have thus confounded intrinsic features of the circuit components with cell-level context effects. We present a model that distinguishes an activated gene’s intrinsic kinetics from its physiological context. We then demonstrate an optimal experimental design approach to identify dynamic induction experiments for efficient estimation of the component’s intrinsic parameters. Maximally informative experiments are chosen by formulating the design as an optimal control problem; direct multiple-shooting is used to identify the optimum. Our numerical results suggest that the intrinsic parameters of a genetic component can be more accurately estimated using optimal experimental designs, and that the choice of growth rates, sampling schedule, and input profile each play an important role. The proposed approach to coupled component–host modelling can support gene circuit design across a range of physiological conditions.

Quantifying the value of surveillance data for improving model predictions of lymphatic filariasis elimination

Background

Mathematical models are increasingly being used to evaluate strategies aiming to achieve the control or elimination of parasitic diseases. Recently, owing to growing realization that process-oriented models are useful for ecological forecasts only if the biological processes are well defined, attention has focused on data assimilation as a means to improve the predictive performance of these models.

Methodology and principal findings

We report on the development of an analytical framework to quantify the relative values of various longitudinal infection surveillance data collected in field sites undergoing mass drug administrations (MDAs) for calibrating three lymphatic filariasis (LF) models (EPIFIL, LYMFASIM, and TRANSFIL), and for improving their predictions of the required durations of drug interventions to achieve parasite elimination in endemic populations. The relative information contribution of site-specific data collected at the time points proposed by the WHO monitoring framework was evaluated using model-data updating procedures, and via calculations of the Shannon information index and weighted variances from the probability distributions of the estimated timelines to parasite extinction made by each model. Results show that data-informed models provided more precise forecasts of elimination timelines in each site compared to model-only simulations. Data streams that included year 5 post-MDA microfilariae (mf) survey data, however, reduced each model’s uncertainty most compared to data streams containing only baseline and/or post-MDA 3 or longer-term mf survey data irrespective of MDA coverage, suggesting that data up to this monitoring point may be optimal for informing the present LF models. We show that the improvements observed in the predictive performance of the best data-informed models may be a function of temporal changes in inter-parameter interactions. Such best data-informed models may also produce more accurate predictions of the durations of drug interventions required to achieve parasite elimination.

Significance

Knowledge of relative information contributions of model only versus data-informed models is valuable for improving the usefulness of LF model predictions in management decision making, learning system dynamics, and for supporting the design of parasite monitoring programmes. The present results further pinpoint the crucial need for longitudinal infection surveillance data for enhancing the precision and accuracy of model predictions of the intervention durations required to achieve parasite elimination in an endemic location.

Although parasite transmission models offer powerful tools for predicting the impacts of interventions, there is growing realization that these models can be useful for this purpose only if their governing biological processes are well defined. Recently, model-data assimilation has been applied to address this problem and improve the performance of process-oriented models for ecological forecasting. Here, we developed an analytical framework that allowed the sequential coupling of the three existing lymphatic filariasis (LF) models with longitudinal infection monitoring data collected in field sites undergoing mass drug administrations (MDAs) to examine the relative value of such data for parameterizing these models and for improving their predictions of the required durations of drug interventions to break parasite transmission. We found that data-informed models provided more precise and reliable forecasts of elimination timelines in the study sites compared to model-only predictions, and that data collected up to 5 years post-MDA reduced each model’s predictive uncertainty most. We also found that this improved performance may be intriguingly related to temporal changes in system dynamics. Our results underscore the significance of sequential model-data fusion for enhancing the understanding of LF transmission dynamics, design of surveillance, and generation of reliable model predictions for management decision making.

Anti-angiogenic drug scheduling optimisation with application to colorectal cancer

Bevacizumab (bvz) is a first choice anti-angiogenic drug in oncology and is primarily administered in combination with chemotherapy. It has been hypothesized that anti-angiogenic drugs enhance efficacy of cytotoxic drugs by "normalizing" abnormal tumor vessels and improving drug penetration. Nevertheless, the clinical relevance of this phenomenon is still unclear with several studies over recent years suggesting an opposing relationship. Herein, we sought to develop a new computational tool to interrogate anti-angiogenic drug scheduling with particular application in the setting of colorectal cancer (CRC). Specifically, we have employed a mathematical model of vascular tumour growth which interrogates the impact of anti-angiogenic treatment and chemotherapeutic treatment on tumour volume. Model predictions were validated using CRC xenografts which underwent treatment with a clinically relevant combinatorial anti-angiogenic regimen. Bayesian model selection revealed the most appropriate term for capturing the effect of treatments on the tumour size, and provided insights into a switch-like dependence of FOLFOX delivery on the tumour vasculature. Our experimental data and mathematical model suggest that delivering chemotherapy prior to bvz may be optimal in the colorectal cancer setting.

Control mechanisms for stochastic biochemical systems via computation of reachable sets

Controlling the behaviour of cells by rationally guiding molecular processes is an overarching aim of much of synthetic biology. Molecular processes, however, are notoriously noisy and frequently nonlinear. We present an approach to studying the impact of control measures on motifs of molecular interactions that addresses the problems faced in many biological systems: stochasticity, parameter uncertainty and nonlinearity. We show that our reachability analysis formalism can describe the potential behaviour of biological (naturally evolved as well as engineered) systems, and provides a set of bounds on their dynamics at the level of population statistics: for example, we can obtain the possible ranges of means and variances of mRNA and protein expression levels, even in the presence of uncertainty about model parameters.

How to deal with parameters for whole-cell modelling

Dynamical systems describing whole cells are on the verge of becoming a reality. But as models of reality, they are only useful if we have realistic parameters for the molecular reaction rates and cell physiological processes. There is currently no suitable framework to reliably estimate hundreds, let alone thousands, of reaction rate parameters. Here, we map out the relative weaknesses and promises of different approaches aimed at redressing this issue. While suitable procedures for estimation or inference of the whole (vast) set of parameters will, in all likelihood, remain elusive, some hope can be drawn from the fact that much of the cellular behaviour may be explained in terms of smaller sets of parameters. Identifying such parameter sets and assessing their behaviour is now becoming possible even for very large systems of equations, and we expect such methods to become central tools in the development and analysis of whole-cell models.

Multitype Network-Guided Target Controllability in Phenotypically Characterized Osteosarcoma: Role of Tumor Microenvironment

This study highlights the relevance of network-guided controllability analysis as a precision oncology tool. Target controllability through networks is potentially relevant to cancer research for the identification of therapeutic targets. With reference to a recent study on multiple phenotypes from 22 osteosarcoma (OS) cell lines characterized both in vitro and in vivo, we found that a variety of critical proteins in OS regulation circuits were in part phenotype specific and in part shared. To generalize our inference approach and match cancer phenotypic heterogeneity, we employed multitype networks and identified targets in correspondence with protein sub-complexes. Therefore, we established the relevance for diagnostic and therapeutic purposes of inspecting interactive targets, namely those enriched by significant connectivity patterns in protein sub-complexes. Emerging targets appeared with reference to the OS microenvironment, and relatively to small leucine-rich proteoglycan members and D-type cyclins, among other collagen, laminin, and keratin proteins. These described were evidences shared across all phenotypes; instead, specific evidences were provided by critical proteins including IGFBP7 and PDGFRA in the invasive phenotype, and FGFR3 and THBS1 in the colony forming phenotype.

A large-scale stochastic spatiotemporal model for Aedes albopictus-borne chikungunya epidemiology

Chikungunya is a viral disease transmitted to humans primarily via the bites of infected Aedes mosquitoes. The virus caused a major epidemic in the Indian Ocean in 2004, affecting millions of inhabitants, while cases have also been observed in Europe since 2007. We developed a stochastic spatiotemporal model of Aedes albopictus-borne chikungunya transmission based on our recently developed environmentally-driven vector population dynamics model. We designed an integrated modelling framework incorporating large-scale gridded climate datasets to investigate disease outbreaks on Reunion Island and in Italy. We performed Bayesian parameter inference on the surveillance data, and investigated the validity and applicability of the underlying biological assumptions. The model successfully represents the outbreak and measures of containment in Italy, suggesting wider applicability in Europe. In its current configuration, the model implies two different viral strains, thus two different outbreaks, for the two-stage Reunion Island epidemic. Characterisation of the posterior distributions indicates a possible relationship between the second larger outbreak on Reunion Island and the Italian outbreak. The model suggests that vector control measures, with different modes of operation, are most effective when applied in combination: adult vector intervention has a high impact but is short-lived, larval intervention has a low impact but is long-lasting, and quarantining infected territories, if applied strictly, is effective in preventing large epidemics. We present a novel approach in analysing chikungunya outbreaks globally using a single environmentally-driven mathematical model. Our study represents a significant step towards developing a globally applicable Ae. albopictus-borne chikungunya transmission model, and introduces a guideline for extending such models to other vector-borne diseases.

The Limitations of Model-Based Experimental Design and Parameter Estimation in Sloppy Systems

We explore the relationship among experimental design, parameter estimation, and systematic error in sloppy models. We show that the approximate nature of mathematical models poses challenges for experimental design in sloppy models. In many models of complex biological processes it is unknown what are the relevant physical mechanisms that must be included to explain system behaviors. As a consequence, models are often overly complex, with many practically unidentifiable parameters. Furthermore, which mechanisms are relevant/irrelevant vary among experiments. By selecting complementary experiments, experimental design may inadvertently make details that were ommitted from the model become relevant. When this occurs, the model will have a large systematic error and fail to give a good fit to the data. We use a simple hyper-model of model error to quantify a model’s discrepancy and apply it to two models of complex biological processes (EGFR signaling and DNA repair) with optimally selected experiments. We find that although parameters may be accurately estimated, the discrepancy in the model renders it less predictive than it was in the sloppy regime where systematic error is small. We introduce the concept of a sloppy system–a sequence of models of increasing complexity that become sloppy in the limit of microscopic accuracy. We explore the limits of accurate parameter estimation in sloppy systems and argue that identifying underlying mechanisms controlling system behavior is better approached by considering a hierarchy of models of varying detail rather than focusing on parameter estimation in a single model.

Sloppy models are often unidentifiable, i.e., characterized by many parameters that are poorly constrained by experimental data. Many models of complex biological systems are sloppy, which has prompted considerable debate about the identifiability of parameters and methods of selecting optimal experiments to infer parameter values. We explore how the approximate nature of models affects the prospect for accurate parameter estimates and model predictivity in sloppy models when using optimal experimental design. We find that sloppy models may no longer give a good fit to data generated from “optimal” experiments. In this case, the model has much less predictive power than it did before optimal experimental selection. We use a simple hyper-model of model error to quantify the model’s discrepancy from the physical system and discuss the potential limits of accurate parameter estimation in sloppy systems.

Inferring extrinsic noise from single-cell gene expression data using approximate Bayesian computation

Background

Gene expression is known to be an intrinsically stochastic process which can involve single-digit numbers of mRNA molecules in a cell at any given time. The modelling of such processes calls for the use of exact stochastic simulation methods, most notably the Gillespie algorithm. However, this stochasticity, also termed “intrinsic noise”, does not account for all the variability between genetically identical cells growing in a homogeneous environment.

Despite substantial experimental efforts, determining appropriate model parameters continues to be a challenge. Methods based on approximate Bayesian computation can be used to obtain posterior parameter distributions given the observed data. However, such inference procedures require large numbers of simulations of the model and exact stochastic simulation is computationally costly.

In this work we focus on the specific case of trying to infer model parameters describing reaction rates and extrinsic noise on the basis of measurements of molecule numbers in individual cells at a given time point.

Results

To make the problem computationally tractable we develop an exact, model-specific, stochastic simulation algorithm for the commonly used two-state model of gene expression. This algorithm relies on certain assumptions and favourable properties of the model to forgo the simulation of the whole temporal trajectory of protein numbers in the system, instead returning only the number of protein and mRNA molecules present in the system at a specified time point. The computational gain is proportional to the number of protein molecules created in the system and becomes significant for systems involving hundreds or thousands of protein molecules.

Conclusions

We employ this simulation algorithm with approximate Bayesian computation to jointly infer the model’s rate and noise parameters from published gene expression data. Our analysis indicates that for most genes the extrinsic contributions to noise will be small to moderate but certainly are non-negligible.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-016-0324-x) contains supplementary material, which is available to authorized users.

Bridging Mechanistic and Phenomenological Models of Complex Biological Systems

The inherent complexity of biological systems gives rise to complicated mechanistic models with a large number of parameters. On the other hand, the collective behavior of these systems can often be characterized by a relatively small number of phenomenological parameters. We use the Manifold Boundary Approximation Method (MBAM) as a tool for deriving simple phenomenological models from complicated mechanistic models. The resulting models are not black boxes, but remain expressed in terms of the microscopic parameters. In this way, we explicitly connect the macroscopic and microscopic descriptions, characterize the equivalence class of distinct systems exhibiting the same range of collective behavior, and identify the combinations of components that function as tunable control knobs for the behavior. We demonstrate the procedure for adaptation behavior exhibited by the EGFR pathway. From a 48 parameter mechanistic model, the system can be effectively described by a single adaptation parameter τ characterizing the ratio of time scales for the initial response and recovery time of the system which can in turn be expressed as a combination of microscopic reaction rates, Michaelis-Menten constants, and biochemical concentrations. The situation is not unlike modeling in physics in which microscopically complex processes can often be renormalized into simple phenomenological models with only a few effective parameters. The proposed method additionally provides a mechanistic explanation for non-universal features of the behavior.

Large-Scale Modelling of the Environmentally-Driven Population Dynamics of Temperate Aedes albopictus (Skuse)

The Asian tiger mosquito, Aedes albopictus, is a highly invasive vector species. It is a proven vector of dengue and chikungunya viruses, with the potential to host a further 24 arboviruses. It has recently expanded its geographical range, threatening many countries in the Middle East, Mediterranean, Europe and North America. Here, we investigate the theoretical limitations of its range expansion by developing an environmentally-driven mathematical model of its population dynamics. We focus on the temperate strain of Ae. albopictus and compile a comprehensive literature-based database of physiological parameters. As a novel approach, we link its population dynamics to globally-available environmental datasets by performing inference on all parameters. We adopt a Bayesian approach using experimental data as prior knowledge and the surveillance dataset of Emilia-Romagna, Italy, as evidence. The model accounts for temperature, precipitation, human population density and photoperiod as the main environmental drivers, and, in addition, incorporates the mechanism of diapause and a simple breeding site model. The model demonstrates high predictive skill over the reference region and beyond, confirming most of the current reports of vector presence in Europe. One of the main hypotheses derived from the model is the survival of Ae. albopictus populations through harsh winter conditions. The model, constrained by the environmental datasets, requires that either diapausing eggs or adult vectors have increased cold resistance. The model also suggests that temperature and photoperiod control diapause initiation and termination differentially. We demonstrate that it is possible to account for unobserved properties and constraints, such as differences between laboratory and field conditions, to derive reliable inferences on the environmental dependence of Ae. albopictus populations.

Identification of potential COPD genes based on multi-omics data at the functional level

A novel systematic approach MMMG (Methylation–MicroRNA–MRNA–GO) to identify potential COPD genes and their classifying performance evaluation.

Clustering reveals limits of parameter identifiability in multi-parameter models of biochemical dynamics

Background

Compared to engineering or physics problems, dynamical models in quantitative biology typically depend on a relatively large number of parameters. Progress in developing mathematics to manipulate such multi-parameter models and so enable their efficient interplay with experiments has been slow. Existing solutions are significantly limited by model size.

Results

In order to simplify analysis of multi-parameter models a method for clustering of model parameters is proposed. It is based on a derived statistically meaningful measure of similarity between groups of parameters. The measure quantifies to what extend changes in values of some parameters can be compensated by changes in values of other parameters. The proposed methodology provides a natural mathematical language to precisely communicate and visualise effects resulting from compensatory changes in values of parameters. As a results, a relevant insight into identifiability analysis and experimental planning can be obtained. Analysis of NF- κB and MAPK pathway models shows that highly compensative parameters constitute clusters consistent with the network topology. The method applied to examine an exceptionally rich set of published experiments on the NF- κB dynamics reveals that the experiments jointly ensure identifiability of only 60 % of model parameters. The method indicates which further experiments should be performed in order to increase the number of identifiable parameters.

Conclusions

We currently lack methods that simplify broadly understood analysis of multi-parameter models. The introduced tools depict mutually compensative effects between parameters to provide insight regarding role of individual parameters, identifiability and experimental design. The method can also find applications in related methodological areas of model simplification and parameters estimation.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-015-0205-8) contains supplementary material, which is available to authorized users.

Accelerated Sensitivity Analysis in High-Dimensional Stochastic Reaction Networks

Existing sensitivity analysis approaches are not able to handle efficiently stochastic reaction networks with a large number of parameters and species, which are typical in the modeling and simulation of complex biochemical phenomena. In this paper, a two-step strategy for parametric sensitivity analysis for such systems is proposed, exploiting advantages and synergies between two recently proposed sensitivity analysis methodologies for stochastic dynamics. The first method performs sensitivity analysis of the stochastic dynamics by means of the Fisher Information Matrix on the underlying distribution of the trajectories; the second method is a reduced-variance, finite-difference, gradient-type sensitivity approach relying on stochastic coupling techniques for variance reduction. Here we demonstrate that these two methods can be combined and deployed together by means of a new sensitivity bound which incorporates the variance of the quantity of interest as well as the Fisher Information Matrix estimated from the first method. The first step of the proposed strategy labels sensitivities using the bound and screens out the insensitive parameters in a controlled manner. In the second step of the proposed strategy, a finite-difference method is applied only for the sensitivity estimation of the (potentially) sensitive parameters that have not been screened out in the first step. Results on an epidermal growth factor network with fifty parameters and on a protein homeostasis with eighty parameters demonstrate that the proposed strategy is able to quickly discover and discard the insensitive parameters and in the remaining potentially sensitive parameters it accurately estimates the sensitivities. The new sensitivity strategy can be several times faster than current state-of-the-art approaches that test all parameters, especially in "sloppy" systems. In particular, the computational acceleration is quantified by the ratio between the total number of parameters over the number of the sensitive parameters.

Modelling Anopheles gambiae s.s. Population Dynamics with Temperature- and Age-Dependent Survival

Climate change and global warming are emerging as important threats to human health, particularly through the potential increase in vector- and water-borne diseases. Environmental variables are known to affect substantially the population dynamics and abundance of the poikilothermic vectors of disease, but the exact extent of this sensitivity is not well established. Focusing on malaria and its main vector in Africa, Anopheles gambiae sensu stricto, we present a set of novel mathematical models of climate-driven mosquito population dynamics motivated by experimental data suggesting that in An. gambiae, mortality is temperature and age dependent. We compared the performance of these models to that of a "standard" model ignoring age dependence. We used a longitudinal dataset of vector abundance over 36 months in sub-Saharan Africa for comparison between models that incorporate age dependence and one that does not, and observe that age-dependent models consistently fitted the data better than the reference model. This highlights that including age dependence in the vector component of mosquito-borne disease models may be important to predict more reliably disease transmission dynamics. Further data and studies are needed to enable improved fitting, leading to more accurate and informative model predictions for the An. gambiae malaria vector as well as for other disease vectors.

Causal drift, robust signaling, and complex disease

The phenotype of many regulatory circuits in which mutations can cause complex, polygenic diseases is to some extent robust to DNA mutations that affect circuit components. Here I demonstrate how such mutational robustness can prevent the discovery of genetic disease determinants. To make my case, I use a mathematical model of the insulin signaling pathway implicated in type 2 diabetes, whose signaling output is governed by 15 genetically determined parameters. Using multiple complementary measures of a parameter's importance for this phenotype, I show that any one disease determinant that is crucial in one genetic background will be virtually irrelevant in other backgrounds. In an evolving population that drifts through the parameter space of this or other robust circuits through DNA mutations, the genetic changes that can cause disease will vary randomly over time. I call this phenomenon causal drift. It means that mutations causing disease in one (human or non-human) population may have no effect in another population, and vice versa. Causal drift casts doubt on our ability to infer the molecular mechanisms of complex diseases from non-human model organisms.

A generalised enzyme kinetic model for predicting the behaviour of complex biochemical systems

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We propose the dQSSA model as a novel way of modelling complex biological networks.

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No low enzyme concentration assumption, covering more biological settings.

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Reduces the number of parameters, which simplifies optimisation.

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dQSSA was validated both in silico and in vitro.

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Both biochemical and signalling pathways can be modelled accurately and simply.

Quasi steady-state enzyme kinetic models are increasingly used in systems modelling. The Michaelis Menten model is popular due to its reduced parameter dimensionality, but its low-enzyme and irreversibility assumption may not always be valid in the in vivo context. Whilst the total quasi-steady state assumption (tQSSA) model eliminates the reactant stationary assumptions, its mathematical complexity is increased. Here, we propose the differential quasi-steady state approximation (dQSSA) kinetic model, which expresses the differential equations as a linear algebraic equation. It eliminates the reactant stationary assumptions of the Michaelis Menten model without increasing model dimensionality. The dQSSA was found to be easily adaptable for reversible enzyme kinetic systems with complex topologies and to predict behaviour consistent with mass action kinetics in silico. Additionally, the dQSSA was able to predict coenzyme inhibition in the reversible lactate dehydrogenase enzyme, which the Michaelis Menten model failed to do. Whilst the dQSSA does not account for the physical and thermodynamic interactions of all intermediate enzyme-substrate complex states, it is proposed to be suitable for modelling complex enzyme mediated biochemical systems. This is due to its simpler application, reduced parameter dimensionality and improved accuracy.

Patient-specific driver gene prediction and risk assessment through integrated network analysis of cancer omics profiles

Extensive and multi-dimensional data sets generated from recent cancer omics profiling projects have presented new challenges and opportunities for unraveling the complexity of cancer genome landscapes. In particular, distinguishing the unique complement of genes that drive tumorigenesis in each patient from a sea of passenger mutations is necessary for translating the full benefit of cancer genome sequencing into the clinic. We address this need by presenting a data integration framework (OncoIMPACT) to nominate patient-specific driver genes based on their phenotypic impact. Extensive in silico and in vitro validation helped establish OncoIMPACT's robustness, improved precision over competing approaches and verifiable patient and cell line specific predictions (2/2 and 6/7 true positives and negatives, respectively). In particular, we computationally predicted and experimentally validated the gene TRIM24 as a putative novel amplified driver in a melanoma patient. Applying OncoIMPACT to more than 1000 tumor samples, we generated patient-specific driver gene lists in five different cancer types to identify modes of synergistic action. We also provide the first demonstration that computationally derived driver mutation signatures can be overall superior to single gene and gene expression based signatures in enabling patient stratification and prognostication. Source code and executables for OncoIMPACT are freely available from http://sourceforge.net/projects/oncoimpact.

Topological sensitivity analysis for systems biology

Mathematical models of natural systems are abstractions of much more complicated processes. Developing informative and realistic models of such systems typically involves suitable statistical inference methods, domain expertise, and a modicum of luck. Except for cases where physical principles provide sufficient guidance, it will also be generally possible to come up with a large number of potential models that are compatible with a given natural system and any finite amount of data generated from experiments on that system. Here we develop a computational framework to systematically evaluate potentially vast sets of candidate differential equation models in light of experimental and prior knowledge about biological systems. This topological sensitivity analysis enables us to evaluate quantitatively the dependence of model inferences and predictions on the assumed model structures. Failure to consider the impact of structural uncertainty introduces biases into the analysis and potentially gives rise to misleading conclusions.

Mouse hair cycle expression dynamics modeled as coupled mesenchymal and epithelial oscillators

The hair cycle is a dynamic process where follicles repeatedly move through phases of growth, retraction, and relative quiescence. This process is an example of temporal and spatial biological complexity. Understanding of the hair cycle and its regulation would shed light on many other complex systems relevant to biological and medical research. Currently, a systematic characterization of gene expression and summarization within the context of a mathematical model is not yet available. Given the cyclic nature of the hair cycle, we felt it was important to consider a subset of genes with periodic expression. To this end, we combined several mathematical approaches with high-throughput, whole mouse skin, mRNA expression data to characterize aspects of the dynamics and the possible cell populations corresponding to potentially periodic patterns. In particular two gene clusters, demonstrating properties of out-of-phase synchronized expression, were identified. A mean field, phase coupled oscillator model was shown to quantitatively recapitulate the synchronization observed in the data. Furthermore, we found only one configuration of positive-negative coupling to be dynamically stable, which provided insight on general features of the regulation. Subsequent bifurcation analysis was able to identify and describe alternate states based on perturbation of system parameters. A 2-population mixture model and cell type enrichment was used to associate the two gene clusters to features of background mesenchymal populations and rapidly expanding follicular epithelial cells. Distinct timing and localization of expression was also shown by RNA and protein imaging for representative genes. Taken together, the evidence suggests that synchronization between expanding epithelial and background mesenchymal cells may be maintained, in part, by inhibitory regulation, and potential mediators of this regulation were identified. Furthermore, the model suggests that impairing this negative regulation will drive a bifurcation which may represent transition into a pathological state such as hair miniaturization.

The hair cycle represents a complex process of particular interest in the study of regulated proliferation, apoptosis and differentiation. While various modeling strategies are presented in the literature, none attempt to link extensive molecular details, provided by high-throughput experiments, with high-level, system properties. Thus, we re-analyzed a previously published mRNA expression time course study and found that we could readily identify a sizeable subset of genes that was expressed in synchrony with the hair cycle itself. The data is summarized in a dynamic, mathematical model of coupled oscillators. We demonstrate that a particular coupling scheme is sufficient to explain the observed synchronization. Further analysis associated specific expression patterns to general yet distinct cell populations, background mesenchymal and rapidly expanding follicular epithelial cells. Experimental imaging results are presented to show the localization of candidate genes from each population. Taken together, the results describe a possible mechanism for regulation between epithelial and mesenchymal populations. We also described an alternate state similar to hair miniaturization, which is predicted by the oscillator model. This study exemplifies the strengths of combining systems-level analysis with high-throughput experimental data to obtain a novel view of a complex system such as the hair cycle.

In silico model-based inference: a contemporary approach for hypothesis testing in network biology

Inductive inference plays a central role in the study of biological systems where one aims to increase their understanding of the system by reasoning backwards from uncertain observations to identify causal relationships among components of the system. These causal relationships are postulated from prior knowledge as a hypothesis or simply a model. Experiments are designed to test the model. Inferential statistics are used to establish a level of confidence in how well our postulated model explains the acquired data. This iterative process, commonly referred to as the scientific method, either improves our confidence in a model or suggests that we revisit our prior knowledge to develop a new model. Advances in technology impact how we use prior knowledge and data to formulate models of biological networks and how we observe cellular behavior. However, the approach for model-based inference has remained largely unchanged since Fisher, Neyman and Pearson developed the ideas in the early 1900s that gave rise to what is now known as classical statistical hypothesis (model) testing. Here, I will summarize conventional methods for model-based inference and suggest a contemporary approach to aid in our quest to discover how cells dynamically interpret and transmit information for therapeutic aims that integrates ideas drawn from high performance computing, Bayesian statistics, and chemical kinetics.

Modelling proteasome and proteasome regulator activities

Proteasomes are key proteases involved in a variety of processes ranging from the clearance of damaged proteins to the presentation of antigens to CD8+ T-lymphocytes. Which cleavage sites are used within the target proteins and how fast these proteins are degraded have a profound impact on immune system function and many cellular metabolic processes. The regulation of proteasome activity involves different mechanisms, such as the substitution of the catalytic subunits, the binding of regulatory complexes to proteasome gates and the proteasome conformational modifications triggered by the target protein itself. Mathematical models are invaluable in the analysis; and potentially allow us to predict the complex interactions of proteasome regulatory mechanisms and the final outcomes of the protein degradation rate and MHC class I epitope generation. The pioneering attempts that have been made to mathematically model proteasome activity, cleavage preference variation and their modification by one of the regulatory mechanisms are reviewed here.

Model selection in systems biology depends on experimental design

Experimental design attempts to maximise the information available for modelling tasks. An optimal experiment allows the inferred models or parameters to be chosen with the highest expected degree of confidence. If the true system is faithfully reproduced by one of the models, the merit of this approach is clear - we simply wish to identify it and the true parameters with the most certainty. However, in the more realistic situation where all models are incorrect or incomplete, the interpretation of model selection outcomes and the role of experimental design needs to be examined more carefully. Using a novel experimental design and model selection framework for stochastic state-space models, we perform high-throughput in-silico analyses on families of gene regulatory cascade models, to show that the selected model can depend on the experiment performed. We observe that experimental design thus makes confidence a criterion for model choice, but that this does not necessarily correlate with a model's predictive power or correctness. Finally, in the special case of linear ordinary differential equation (ODE) models, we explore how wrong a model has to be before it influences the conclusions of a model selection analysis.

Different models of the same process represent distinct hypotheses about reality. These can be decided between within the framework of model selection, where the evidence for each is given by their ability to reproduce a set of experimental data. Even if one of the models is correct, the chances of identifying it can be hindered by the quality of the data, both in terms of its signal to measurement error ratio and the intrinsic discriminatory potential of the experiment undertaken. This potential can be predicted in various ways, and maximising it is one aim of experimental design. In this work we present a computationally efficient method of experimental design for model selection. We exploit the efficiency to consider the implications of the realistic case where all models are more or less incorrect, showing that experiments can be chosen that, considered individually, lead to unequivocal support for opposed hypotheses.

Kinetic models in industrial biotechnology - Improving cell factory performance

An increasing number of industrial bioprocesses capitalize on living cells by using them as cell factories that convert sugars into chemicals. These processes range from the production of bulk chemicals in yeasts and bacteria to the synthesis of therapeutic proteins in mammalian cell lines. One of the tools in the continuous search for improved performance of such production systems is the development and application of mathematical models. To be of value for industrial biotechnology, mathematical models should be able to assist in the rational design of cell factory properties or in the production processes in which they are utilized. Kinetic models are particularly suitable towards this end because they are capable of representing the complex biochemistry of cells in a more complete way compared to most other types of models. They can, at least in principle, be used to in detail understand, predict, and evaluate the effects of adding, removing, or modifying molecular components of a cell factory and for supporting the design of the bioreactor or fermentation process. However, several challenges still remain before kinetic modeling will reach the degree of maturity required for routine application in industry. Here we review the current status of kinetic cell factory modeling. Emphasis is on modeling methodology concepts, including model network structure, kinetic rate expressions, parameter estimation, optimization methods, identifiability analysis, model reduction, and model validation, but several applications of kinetic models for the improvement of cell factories are also discussed.

Bayesian model comparison and parameter inference in systems biology using nested sampling

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 multi-dimensional integral to a 1D integration over likelihood space. This approach focuses 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 reverse-engineer 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.

Reverse engineering and identification in systems biology: strategies, perspectives and challenges

The interplay of mathematical modelling with experiments is one of the central elements in systems biology. The aim of reverse engineering is to infer, analyse and understand, through this interplay, the functional and regulatory mechanisms of biological systems. Reverse engineering is not exclusive of systems biology and has been studied in different areas, such as inverse problem theory, machine learning, nonlinear physics, (bio)chemical kinetics, control theory and optimization, among others. However, it seems that many of these areas have been relatively closed to outsiders. In this contribution, we aim to compare and highlight the different perspectives and contributions from these fields, with emphasis on two key questions: (i) why are reverse engineering problems so hard to solve, and (ii) what methods are available for the particular problems arising from systems biology?

A framework for parameter estimation and model selection from experimental data in systems biology using approximate Bayesian computation

As modeling becomes a more widespread practice in the life sciences and biomedical sciences, researchers need reliable tools to calibrate models against ever more complex and detailed data. Here we present an approximate Bayesian computation (ABC) framework and software environment, ABC-SysBio, which is a Python package that runs on Linux and Mac OS X systems and that enables parameter estimation and model selection in the Bayesian formalism by using sequential Monte Carlo (SMC) approaches. We outline the underlying rationale, discuss the computational and practical issues and provide detailed guidance as to how the important tasks of parameter inference and model selection can be performed in practice. Unlike other available packages, ABC-SysBio is highly suited for investigating, in particular, the challenging problem of fitting stochastic models to data. In order to demonstrate the use of ABC-SysBio, in this protocol we postulate the existence of an imaginary reaction network composed of seven interrelated biological reactions (involving a specific mRNA, the protein it encodes and a post-translationally modified version of the protein), a network that is defined by two files containing 'observed' data that we provide as supplementary information. In the first part of the PROCEDURE, ABC-SysBio is used to infer the parameters of this system, whereas in the second part we use ABC-SysBio's relevant functionality to discriminate between two different reaction network models, one of them being the 'true' one. Although computationally expensive, the additional insights gained in the Bayesian formalism more than make up for this cost, especially in complex problems.

Data-driven quantification of the robustness and sensitivity of cell signaling networks

Robustness and sensitivity of responses generated by cell signaling networks has been associated with survival and evolvability of organisms. However, existing methods analyzing robustness and sensitivity of signaling networks ignore the experimentally observed cell-to-cell variations of protein abundances and cell functions or contain ad hoc assumptions. We propose and apply a data-driven maximum entropy based method to quantify robustness and sensitivity of Escherichia coli (E. coli) chemotaxis signaling network. Our analysis correctly rank orders different models of E. coli chemotaxis based on their robustness and suggests that parameters regulating cell signaling are evolutionary selected to vary in individual cells according to their abilities to perturb cell functions. Furthermore, predictions from our approach regarding distribution of protein abundances and properties of chemotactic responses in individual cells based on cell population averaged data are in excellent agreement with their experimental counterparts. Our approach is general and can be used to evaluate robustness as well as generate predictions of single cell properties based on population averaged experimental data in a wide range of cell signaling systems.