Parameter estimation of dynamic biological network models using integrated fluxes

Localization of Noise in Biochemical Networks

Noise, or uncertainty in biochemical networks, has become an important aspect of many biological problems. Noise can arise and propagate from external factors and probabilistic chemical reactions occurring in small cellular compartments. For species survival, it is important to regulate such uncertainties in executing vital cell functions. Regulated noise can improve adaptability, whereas uncontrolled noise can cause diseases. Simulation can provide a detailed analysis of uncertainties, but parameters such as rate constants and initial conditions are usually unknown. A general understanding of noise dynamics from the perspective of network structure is highly desirable. In this study, we extended the previously developed law of localization for characterizing noise in terms of (co)variances and developed noise localization theory. With linear noise approximation, we can expand a biochemical network into an extended set of differential equations representing a fictitious network for pseudo-components consisting of variances and covariances, together with chemical species. Through localization analysis, perturbation responses at the steady state of pseudo-components can be summarized into a sensitivity matrix that only requires knowledge of network topology. Our work allows identification of buffering structures at the level of species, variances, and covariances and can provide insights into noise flow under non-steady-state conditions in the form of a pseudo-chemical reaction. We tested noise localization in various systems, and here we discuss its implications and potential applications. Results show that this theory is potentially applicable in discriminating models, scanning network topologies with interesting noise behavior, and designing and perturbing networks with the desired response.

Time Integrated Flux Analysis: Exploiting the Concentration Measurements Directly for Cost-Effective Metabolic Network Flux Analysis

Background: Flux analyses, such as Metabolic Flux Analysis (MFA), Flux Balance Analysis (FBA), Flux Variability Analysis (FVA) or similar methods, can provide insights into the cellular metabolism, especially in combination with experimental data. The most common integration of extracellular concentration data requires the estimation of the specific fluxes (/rates) from the measured concentrations. This is a time-consuming, mathematically ill-conditioned inverse problem, raising high requirements for the quality and quantity of data. Method: In this contribution, a time integrated flux analysis approach is proposed which avoids the error-prone estimation of specific flux values. The approach is adopted for a Metabolic time integrated Flux Analysis and (sparse) time integrated Flux Balance/Variability Analysis. The proposed approach is applied to three case studies: (1) a simulated bioprocess case studying the impact of the number of samples (experimental points) and measurements’ noise on the performance; (2) a simulation case to understand the impact of network redundancies and reaction irreversibility; and (3) an experimental bioprocess case study, showing its relevance for practical applications. Results: It is observed that this method can successfully estimate the time integrated flux values, even with relatively low numbers of samples and significant noise levels. In addition, the method allows the integration of additional constraints (e.g., bounds on the estimated concentrations) and since it eliminates the need for estimating fluxes from measured concentrations, it significantly reduces the workload while providing about the same level of insight into the metabolism as classic flux analysis methods.

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.

Application of Parameter Optimization to Search for Oscillatory Mass-Action Networks Using Python

Biological systems can be described mathematically to model the dynamics of metabolic, protein, or gene-regulatory networks, but locating parameter regimes that induce a particular dynamic behavior can be challenging due to the vast parameter landscape, particularly in large models. In the current work, a Pythonic implementation of existing bifurcation objective functions, which reward systems that achieve a desired bifurcation behavior, is implemented to search for parameter regimes that permit oscillations or bistability. A differential evolution algorithm progressively approximates the specified bifurcation type while performing a global search of parameter space for a candidate with the best fitness. The user-friendly format facilitates integration with systems biology tools, as Python is a ubiquitous programming language. The bifurcation–evolution software is validated on published models from the BioModels Database and used to search populations of randomly-generated mass-action networks for oscillatory dynamics. Results of this search demonstrate the importance of reaction enrichment to provide flexibility and enable complex dynamic behaviors, and illustrate the role of negative feedback and time delays in generating oscillatory dynamics.

A dynamic model of lignin biosynthesis in Brachypodium distachyon

BACKGROUND

Lignin is a crucial molecule for terrestrial plants, as it offers structural support and permits the transport of water over long distances. The hardness of lignin reduces plant digestibility by cattle and sheep; it also makes inedible plant materials recalcitrant toward the enzymatic fermentation of cellulose, which is a potentially valuable substrate for sustainable biofuels. Targeted attempts to change the amount or composition of lignin in relevant plant species have been hampered by the fact that the lignin biosynthetic pathway is difficult to understand, because it uses several enzymes for the same substrates, is regulated in an ill-characterized manner, may operate in different locations within cells, and contains metabolic channels, which the plant may use to funnel initial substrates into specific monolignols.

RESULTS

We propose a dynamic mathematical model that integrates various datasets and other information regarding the lignin pathway in Brachypodium distachyon and permits explanations for some counterintuitive observations. The model predicts the lignin composition and label distribution in a BdPTAL knockdown strain, with results that are quite similar to experimental data.

CONCLUSION

Given the present scarcity of available data, the model resulting from our analysis is presumably not final. However, it offers proof of concept for how one may design integrative pathway models of this type, which are necessary tools for predicting the consequences of genomic or other alterations toward plants with lignin features that are more desirable than in their wild-type counterparts.

The best models of metabolism

Biochemical systems are among of the oldest application areas of mathematical modeling. Spanning a time period of over one hundred years, the repertoire of options for structuring a model and for formulating reactions has been constantly growing, and yet, it is still unclear whether or to what degree some models are better than others and how the modeler is to choose among them. In fact, the variety of options has become overwhelming and difficult to maneuver for novices and experts alike. This review outlines the metabolic model design process and discusses the numerous choices for modeling frameworks and mathematical representations. It tries to be inclusive, even though it cannot be complete, and introduces the various modeling options in a manner that is as unbiased as that is feasible. However, the review does end with personal recommendations for the choices of default models. WIREs Syst Biol Med 2017, 9:e1391. doi: 10.1002/wsbm.1391 For further resources related to this article, please visit the WIREs website.

Identification of microbiota dynamics using robust parameter estimation methods

The compositions of in-host microbial communities (microbiota) play a significant role in host health, and a better understanding of the microbiota's role in a host's transition from health to disease or vice versa could lead to novel medical treatments. One of the first steps toward this understanding is modeling interaction dynamics of the microbiota, which can be exceedingly challenging given the complexity of the dynamics and difficulties in collecting sufficient data. Methods such as principal differential analysis, dynamic flux estimation, and others have been developed to overcome these challenges. Despite their advantages, these methods are still vastly underutilized in fields such as mathematical biology, and one potential reason for this is their sophisticated implementation. While this paper focuses on applying principal differential analysis to microbiota data, we also provide comprehensive details regarding the derivation and numerics of this method and include a functional implementation for readers' benefit. For further validation of these methods, we demonstrate the feasibility of principal differential analysis using simulation studies and then apply the method to intestinal and vaginal microbiota data. In working with these data, we capture experimentally confirmed dynamics while also revealing potential new insights into the system dynamics.

Parameter estimation in large-scale systems biology models: a parallel and self-adaptive cooperative strategy

Background

The development of large-scale kinetic models is one of the current key issues in computational systems biology and bioinformatics. Here we consider the problem of parameter estimation in nonlinear dynamic models. Global optimization methods can be used to solve this type of problems but the associated computational cost is very large. Moreover, many of these methods need the tuning of a number of adjustable search parameters, requiring a number of initial exploratory runs and therefore further increasing the computation times.

Here we present a novel parallel method, self-adaptive cooperative enhanced scatter search (saCeSS), to accelerate the solution of this class of problems. The method is based on the scatter search optimization metaheuristic and incorporates several key new mechanisms: (i) asynchronous cooperation between parallel processes, (ii) coarse and fine-grained parallelism, and (iii) self-tuning strategies.

Results

The performance and robustness of saCeSS is illustrated by solving a set of challenging parameter estimation problems, including medium and large-scale kinetic models of the bacterium E. coli, bakerés yeast S. cerevisiae, the vinegar fly D. melanogaster, Chinese Hamster Ovary cells, and a generic signal transduction network.

The results consistently show that saCeSS is a robust and efficient method, allowing very significant reduction of computation times with respect to several previous state of the art methods (from days to minutes, in several cases) even when only a small number of processors is used.

Conclusions

The new parallel cooperative method presented here allows the solution of medium and large scale parameter estimation problems in reasonable computation times and with small hardware requirements. Further, the method includes self-tuning mechanisms which facilitate its use by non-experts. We believe that this new method can play a key role in the development of large-scale and even whole-cell dynamic models.

Electronic supplementary material

The online version of this article (doi:10.1186/s12859-016-1452-4) contains supplementary material, which is available to authorized users.

Network-Oriented Approaches to Anticancer Drug Response

A long-standing paradigm in drug discovery has been the concept of designing maximally selective drugs to act on individual targets considered to underlie a disease of interest. Nonetheless, although some drugs have proven to be successful, many more potential drugs identified by the "one gene, one drug, one disease" approach have been found to be less effective than expected or to cause notable side effects. Advances in systems biology and high-throughput in-depth genomic profiling technologies along with an analysis of the successful and failed drugs uncovered that the prominent factor to determine drug sensitivity is the intrinsic robustness of the response of biological systems in the face of perturbations. The complexity of the molecular and cellular bases of systems responses to drug interventions has fostered an increased interest in systems-oriented approaches to drug discovery. Consonant with this knowledge of the multifactorial mechanistic basis of drug sensitivity and resistance is the application of network-based approaches for the identification of molecular (multi-)feature signatures associated with desired (multi-)drug phenotypic profiles. This chapter illustrates the principal network analysis and inference techniques which have found application in systems-oriented drug design and considers their benefits and drawbacks in relation to the nature of the data produced by network pharmacology.

Inverse problem studies of biochemical systems with structure identification of S-systems by embedding training functions in a genetic algorithm

An efficient inverse problem approach for parameter estimation, state and structure identification from dynamic data by embedding training functions in a genetic algorithm methodology (ETFGA) is proposed for nonlinear dynamical biosystems using S-system canonical models. Use of multiple shooting and decomposition approach as training functions has been shown for handling of noisy datasets and computational efficiency in studying the inverse problem. The advantages of the methodology are brought out systematically by studying it for three biochemical model systems of interest. By studying a small-scale gene regulatory system described by a S-system model, the first example demonstrates the use of ETFGA for the multifold aims of the inverse problem. The estimation of a large number of parameters with simultaneous state and network identification is shown by training a generalized S-system canonical model with noisy datasets. The results of this study bring out the superior performance of ETFGA on comparison with other metaheuristic approaches. The second example studies the regulation of cAMP oscillations in Dictyostelium cells now assuming limited availability of noisy data. Here, flexibility of the approach to incorporate partial system information in the identification process is shown and its effect on accuracy and predictive ability of the estimated model are studied. The third example studies the phenomenological toy model of the regulation of circadian oscillations in Drosophila that follows rate laws different from S-system power-law. For the limited noisy data, using a priori information about properties of the system, we could estimate an alternate S-system model that showed robust oscillatory behavior with predictive abilities.

Identification of Metabolic Pathway Systems

The estimation of parameters in even moderately large biological systems is a significant challenge. This challenge is greatly exacerbated if the mathematical formats of appropriate process descriptions are unknown. To address this challenge, the method of dynamic flux estimation (DFE) was proposed for the analysis of metabolic time series data. Under ideal conditions, the first phase of DFE yields numerical representations of all fluxes within a metabolic pathway system, either as values at each time point or as plots against their substrates and modulators. However, this numerical result does not reveal the mathematical format of each flux. Thus, the second phase of DFE selects functional formats that are consistent with the numerical trends obtained from the first phase. While greatly facilitating metabolic data analysis, DFE is only directly applicable if the pathway system contains as many dependent variables as fluxes. Because most actual systems contain more fluxes than metabolite pools, this requirement is seldom satisfied. Auxiliary methods have been proposed to alleviate this issue, but they are not general. Here we propose strategies that extend DFE toward general, slightly underdetermined pathway systems.

Effective Estimation of Dynamic Metabolic Fluxes Using (13)C Labeling and Piecewise Affine Approximation: From Theory to Practical Applicability

The design of microbial production processes relies on rational choices for metabolic engineering of the production host and the process conditions. These require a systematic and quantitative understanding of cellular regulation. Therefore, a novel method for dynamic flux identification using quantitative metabolomics and ¹³C labeling to identify piecewise-affine (PWA) flux functions has been described recently. Obtaining flux estimates nevertheless still required frequent manual reinitalization to obtain a good reproduction of the experimental data and, moreover, did not optimize on all observables simultaneously (metabolites and isotopomer concentrations). In our contribution we focus on measures to achieve faster and robust dynamic flux estimation which leads to a high dimensional parameter estimation problem. Specifically, we address the following challenges within the PWA problem formulation: (1) Fast selection of sufficient domains for the PWA flux functions, (2) Control of over-fitting in the concentration space using shape-prescriptive modeling and (3) robust and efficient implementation of the parameter estimation using the hybrid implicit filtering algorithm. With the improvements we significantly speed up the convergence by efficiently exploiting that the optimization problem is partly linear. This allows application to larger-scale metabolic networks and demonstrates that the proposed approach is not purely theoretical, but also applicable in practice.

Design Principles as a Guide for Constraint Based and Dynamic Modeling: Towards an Integrative Workflow

During the last 10 years, systems biology has matured from a fuzzy concept combining omics, mathematical modeling and computers into a scientific field on its own right. In spite of its incredible potential, the multilevel complexity of its objects of study makes it very difficult to establish a reliable connection between data and models. The great number of degrees of freedom often results in situations, where many different models can explain/fit all available datasets. This has resulted in a shift of paradigm from the initially dominant, maybe naive, idea of inferring the system out of a number of datasets to the application of different techniques that reduce the degrees of freedom before any data set is analyzed. There is a wide variety of techniques available, each of them can contribute a piece of the puzzle and include different kinds of experimental information. But the challenge that remains is their meaningful integration. Here we show some theoretical results that enable some of the main modeling approaches to be applied sequentially in a complementary manner, and how this workflow can benefit from evolutionary reasoning to keep the complexity of the problem in check. As a proof of concept, we show how the synergies between these modeling techniques can provide insight into some well studied problems: Ammonia assimilation in bacteria and an unbranched linear pathway with end-product inhibition.

REDEMPTION: reduced dimension ensemble modeling and parameter estimation

Summary: Here, we present REDEMPTION (Reduced Dimension Ensemble Modeling and Parameter estimation), a toolbox for parameter estimation and ensemble modeling of ordinary differential equations (ODEs) using time-series data. For models with more reactions than measured species, a common scenario in biological modeling, the parameter estimation is formulated as a nested optimization problem based on incremental parameter estimation strategy. REDEMPTION also includes a tool for the identification of an ensemble of parameter combinations that provide satisfactory goodness-of-fit to the data. The functionalities of REDEMPTION are accessible through a MATLAB user interface (UI), as well as through programming script. For computational speed-up, REDEMPTION provides a numerical parallelization option using MATLAB Parallel Computing toolbox.

Availability and implementation: REDEMPTION can be downloaded from http://www.cabsel.ethz.ch/tools/redemption.

Contact:rudi.gunawan@chem.ethz.ch