Identification of a Metabolic Reaction Network from Time-Series Data of Metabolite Concentrations

Recent development of high-throughput analytical techniques has made it possible to qualitatively identify a number of metabolites simultaneously. Correlation and multivariate analyses such as principal component analysis have been widely used to analyse those data and evaluate correlations among the metabolic profiles. However, these analyses cannot simultaneously carry out identification of metabolic reaction networks and prediction of dynamic behaviour of metabolites in the networks. The present study, therefore, proposes a new approach consisting of a combination of statistical technique and mathematical modelling approach to identify and predict a probable metabolic reaction network from time-series data of metabolite concentrations and simultaneously construct its mathematical model. Firstly, regression functions are fitted to experimental data by the locally estimated scatter plot smoothing method. Secondly, the fitted result is analysed by the bivariate Granger causality test to determine which metabolites cause the change in other metabolite concentrations and remove less related metabolites. Thirdly, S-system equations are formed by using the remaining metabolites within the framework of biochemical systems theory. Finally, parameters including rate constants and kinetic orders are estimated by the Levenberg–Marquardt algorithm. The estimation is iterated by setting insignificant kinetic orders at zero, i.e., removing insignificant metabolites. Consequently, a reaction network structure is identified and its mathematical model is obtained. Our approach is validated using a generic inhibition and activation model and its practical application is tested using a simplified model of the glycolysis of Lactococcus lactis MG1363, for which actual time-series data of metabolite concentrations are available. The results indicate the usefulness of our approach and suggest a probable pathway for the production of lactate and acetate. The results also indicate that the approach pinpoints a probable strong inhibition of lactate on the glycolysis pathway.     

Integrating intracellular dynamics using CompuCell3D and Bionetsolver: applications to multiscale modelling of cancer cell growth and invasion

In this paper we present a multiscale, individual-based simulation environment that integrates CompuCell3D for lattice-based modelling on the cellular level and Bionetsolver for intracellular modelling. CompuCell3D or CC3D provides an implementation of the lattice-based Cellular Potts Model or CPM (also known as the Glazier-Graner-Hogeweg or GGH model) and a Monte Carlo method based on the metropolis algorithm for system evolution. The integration of CC3D for cellular systems with Bionetsolver for subcellular systems enables us to develop a multiscale mathematical model and to study the evolution of cell behaviour due to the dynamics inside of the cells, capturing aspects of cell behaviour and interaction that is not possible using continuum approaches. We then apply this multiscale modelling technique to a model of cancer growth and invasion, based on a previously published model of Ramis-Conde et al. (2008) where individual cell behaviour is driven by a molecular network describing the dynamics of E-cadherin and β-catenin. In this model, which we refer to as the centre-based model, an alternative individual-based modelling technique was used, namely, a lattice-free approach. In many respects, the GGH or CPM methodology and the approach of the centre-based model have the same overall goal, that is to mimic behaviours and interactions of biological cells. Although the mathematical foundations and computational implementations of the two approaches are very different, the results of the presented simulations are compatible with each other, suggesting that by using individual-based approaches we can formulate a natural way of describing complex multi-cell, multiscale models. The ability to easily reproduce results of one modelling approach using an alternative approach is also essential from a model cross-validation standpoint and also helps to identify any modelling artefacts specific to a given computational approach.     

Thermodynamic Constraints in Kinetic Modeling: Thermodynamic‐Kinetic Modeling in Comparison to Other Approaches

Kinetic models of reaction networks may easily violate the laws of thermodynamics and the principle of detailed balance. In large network models, the constraints that are imposed by these laws are particularly difficult to address. This hinders modeling of biochemical reaction networks. Thermodynamic‐kinetic modeling is a method that provides a thermodynamically sound and formally appealing way for deriving dynamic model equations of reaction systems. State variables of this approach are thermokinetic potentials that describe the ability of compounds to drive a reaction. A compound has a parameter called capacity, which is the ratio of its concentration and thermokinetic potential. A reaction is described by its resistance which is the ratio of the thermokinetic driving force and flux. In these aspects, the formalism is similar to the modeling formalism for electrical networks and an analogous graphical representation is possible. The thermodynamic‐kinetic modeling formalism is equivalent to the traditional kinetic modeling formalism with the exception that it is not possible to build thermodynamically infeasible models. Here, the thermodynamic‐kinetic modeling formalism is reviewed, compared to other approaches, and some of its advantages are worked out. In contrast to other approaches, thermodynamic‐kinetic modeling does not rely on an explicit enumeration of stoichiometric cycles. It is capable of describing rate laws far from equilibrium. Further, the parameterization by capacities and resistances is particularly intuitive and powerful.     

Sequential modelling of a full-scale wastewater treatment plant using an artificial neural network

This work proposes a sequential modelling approach using an artificial neural network (ANN) to develop four independent multivariate models that are able to predict the dynamics of biochemical oxygen demand (BOD), chemical oxygen demand (COD), suspended solid (SS), and total nitrogen (TN) removal in a wastewater treatment plant (WWTP). Suitable structures of ANN models were automatically and conveniently optimized by a genetic algorithm rather than the conventional trial and error method. The sequential modelling approach, which is composed of two parts, a process disturbance estimator and a process behaviour predictor, was also presented to develop multivariate dynamic models. In particular, the process disturbance estimator was first employed to estimate the influent quality. The process behaviour predictor then sequentially predicted the effluent quality based on the estimated influent quality from the process disturbance estimator with other process variables. The efficiencies of the developed ANN models with a sequential modelling approach were demonstrated with a practical application using a data set collected from a full-scale WWTP during 2 years. The results show that the ANN with the sequential modelling approach successfully developed multivariate dynamic models of BOD, COD, SS, and TN removal with satisfactory estimation and prediction capability. Thus, the proposed method could be used as a powerful tool for the prediction of complex and nonlinear WWTP performance.     

Translating biochemical network models between different kinetic formats

Mechanistic biochemical network models describe the dynamics of intracellular metabolite pools in terms of substance concentrations, stoichiometry and reaction kinetics. Data from stimulus response experiments are currently the most informative source for in-vivo parameter estimation in such models. However, only a part of the parameters of classical enzyme kinetic models can usually be estimated from typical stimulus response data. For this reason, several alternative kinetic formats using different “languages” (e.g. linear, power laws, linlog, generic and convenience) have been proposed to reduce the model complexity. The present contribution takes a rigorous “multi-lingual” approach to data evaluation by translating biochemical network models from one kinetic format into another. For this purpose, a new high-performance algorithm has been developed and tested. Starting with a given model, it replaces as many kinetic terms as possible by alternative expressions while still reproducing the experimental data. Application of the algorithm to a published model for Escherichia coli's sugar metabolism demonstrates the power of the new method. It is shown that model translation is a powerful tool to investigate the information content of stimulus response data and the predictive power of models. Moreover, the local and global approximation capabilities of the models are elucidated and some pitfalls of traditional single model approaches to data evaluation are revealed.     

Efficient and accurate experimental design for enzyme kinetics: Bayesian studies reveal a systematic approach

In areas such as drug development, clinical diagnosis and biotechnology research, acquiring details about the kinetic parameters of enzymes is crucial. The correct design of an experiment is critical to collecting data suitable for analysis, modelling and deriving the correct information. As classical design methods are not targeted to the more complex kinetics being frequently studied, attention is needed to estimate parameters of such models with low variance. We demonstrate that a Bayesian approach (the use of prior knowledge) can produce major gains quantifiable in terms of information, productivity and accuracy of each experiment. Developing the use of Bayesian Utility functions, we have used a systematic method to identify the optimum experimental designs for a number of kinetic model data sets. This has enabled the identification of trends between kinetic model types, sets of design rules and the key conclusion that such designs should be based on some prior knowledge of K(M) and/or the kinetic model. We suggest an optimal and iterative method for selecting features of the design such as the substrate range, number of measurements and choice of intermediate points. The final design collects data suitable for accurate modelling and analysis and minimises the error in the parameters estimated.     

A generative spiking neural-network model of goal-directed behaviour and one-step planning

In mammals, goal-directed and planning processes support flexible behaviour used to face new situations that cannot be tackled through more efficient but rigid habitual behaviours. Within the Bayesian modelling approach of brain and behaviour, models have been proposed to perform planning as probabilistic inference but this approach encounters a crucial problem: explaining how such inference might be implemented in brain spiking networks. Recently, the literature has proposed some models that face this problem through recurrent spiking neural networks able to internally simulate state trajectories, the core function at the basis of planning. However, the proposed models have relevant limitations that make them biologically implausible, namely their world model is trained ‘off-line’ before solving the target tasks, and they are trained with supervised learning procedures that are biologically and ecologically not plausible. Here we propose two novel hypotheses on how brain might overcome these problems, and operationalise them in a novel architecture pivoting on a spiking recurrent neural network. The first hypothesis allows the architecture to learn the world model in parallel with its use for planning: to this purpose, a new arbitration mechanism decides when to explore, for learning the world model, or when to exploit it, for planning, based on the entropy of the world model itself. The second hypothesis allows the architecture to use an unsupervised learning process to learn the world model by observing the effects of actions. The architecture is validated by reproducing and accounting for the learning profiles and reaction times of human participants learning to solve a visuomotor learning task that is new for them. Overall, the architecture represents the first instance of a model bridging probabilistic planning and spiking-processes that has a degree of autonomy analogous to the one of real organisms.     

Kinetic modelling of plant metabolic pathways

This paper provides a review of kinetic modelling of plant metabolic pathways as a tool for analysing their control and regulation. An overview of different modelling strategies is presented, starting with those approaches that only require a knowledge of the network stoichiometry; these are referred to as structural. Flux-balance analysis, metabolic flux analysis using isotope labelling, and elementary mode analysis are briefly mentioned as three representative examples. The main focus of this paper, however, is a discussion of kinetic modelling, which requires, in addition to the stoichiometry, a knowledge of the kinetic properties of the constituent pathway enzymes. The different types of kinetic modelling analysis, namely time-course simulation, steady-state analysis, and metabolic control analysis, are explained in some detail. An overview is presented of strategies for obtaining model parameters, as well as software tools available for simulation of such models. The kinetic modelling approach is exemplified with discussion of three models from the general plant physiology literature. With the aid of kinetic modelling it is possible to perform a control analysis of a plant metabolic system, to identify potential targets for biotechnological manipulation, as well as to ascertain the regulatory importance of different enzymes (including isoforms of the same enzyme) in a pathway. Finally, a framework is presented for extending metabolic models to the whole-plant scale by linking biochemical reactions with diffusion and advective flow through the phloem. Future challenges include explicit modelling of subcellular compartments, as well as the integration of kinetic models on the different levels of the cellular and organizational hierarchy.     

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.     

Integration of Boolean models exemplified on hepatocyte signal transduction

The number of mathematical models for biological pathways is rapidly growing. In particular, Boolean modelling proved to be suited to describe large cellular signalling networks. Systems biology is at the threshold to holistic understanding of comprehensive networks. In order to reach this goal, connection and integration of existing models of parts of cellular networks into more comprehensive network models is necessary. We discuss model combination approaches for Boolean models. Boolean modelling is qualitative rather than quantitative and does not require detailed kinetic information. We show that these models are useful precursors for large-scale quantitative models and that they are comparatively easy to combine. We propose modelling standards for Boolean models as a prerequisite for smooth model integration. Using these standards, we demonstrate the coupling of two logical models on two different examples concerning cellular interactions in the liver. In the first example, we show the integration of two Boolean models of two cell types in order to describe their interaction. In the second example, we demonstrate the combination of two models describing different parts of the network of a single cell type. Combination of partial models into comprehensive network models will take systems biology to the next level of understanding. The combination of logical models facilitated by modelling standards is a valuable example for the next step towards this goal.     

Neural networks for perceptual processing: from simulation tools to theories

Neural networks are modelling tools that are, in principle, able to capture the input-output behaviour of arbitrary systems that may include the dynamics of animal populations or brain circuits. While a neural network model is useful if it captures phenomenologically the behaviour of the target system in this way, its utility is amplified if key mechanisms of the model can be discovered, and identified with those of the underlying system. In this review, we first describe, at a fairly high level with minimal mathematics, some of the tools used in constructing neural network models. We then go on to discuss the implications of network models for our understanding of the system they are supposed to describe, paying special attention to those models that deal with neural circuits and brain systems. We propose that neural nets are useful for brain modelling if they are viewed in a wider computational framework originally devised by Marr. Here, neural networks are viewed as an intermediate mechanistic abstraction between 'algorithm' and 'implementation', which can provide insights into biological neural representations and their putative supporting architectures.     

The complexity of comparing reaction systems

MOTIVATION: As more genomic data becomes available there is increased attention on understanding the mechanisms encoded in the genome. New XML dialects like CellML and Systems Biology Markup Language (SBML) are being developed to describe biological networks of all types. In the absence of detailed kinetic information for these networks, stoichiometric data is an especially valuable source of information. Network databases are the next logical step beyond storing purely genomic information. Just as comparison of entries in genomic databases has been a vital algorithmic problem through the course of the sequencing project, comparison of networks in network databases will be a crucial problem as we seek to integrate higher-order network knowledge. RESULTS: We show that comparing the stoichiometric structure of two reactions systems is equivalent to the graph isomorphism problem. This is encouraging because graph isomorphism is, in practice, a tractable problem using heuristics. The analogous problem of searching for a subsystem of a reaction system is NP-complete. We also discuss heuristic issues in implementations for practical comparison of stoichiometric matrices.     

A Hypothesis Test for Equality of Bayesian Network Models

Bayesian network models are commonly used to model gene expression data. Some applications require a comparison of the network structure of a set of genes between varying phenotypes. In principle, separately fit models can be directly compared, but it is difficult to assign statistical significance to any observed differences. There would therefore be an advantage to the development of a rigorous hypothesis test for homogeneity of network structure. In this paper, a generalized likelihood ratio test based on Bayesian network models is developed, with significance level estimated using permutation replications. In order to be computationally feasible, a number of algorithms are introduced. First, a method for approximating multivariate distributions due to Chow and Liu (1968) is adapted, permitting the polynomial-time calculation of a maximum likelihood Bayesian network with maximum indegree of one. Second, sequential testing principles are applied to the permutation test, allowing significant reduction of computation time while preserving reported error rates used in multiple testing. The method is applied to gene-set analysis, using two sets of experimental data, and some advantage to a pathway modelling approach to this problem is reported.     

Linear systems approach to analysis of complex dynamic behaviours in biochemical networks

Central functions in the cell are often linked to complex dynamic behaviours, such as sustained oscillations and multistability, in a biochemical reaction network. Determination of the specific mechanisms underlying such behaviours is important, e.g. to determine sensitivity, robustness, and modelling requirements of given cell functions. In this work we adopt a systems approach to the analysis of complex behaviours in intracellular reaction networks, described by ordinary differential equations with known kinetic parameters. We propose to decompose the overall system into a number of low complexity subsystems, and consider the importance of interactions between these in generating specific behaviours. Rather than analysing the network in a state corresponding to the complex non-linear behaviour, we move the system to the underlying unstable steady state, and focus on the mechanisms causing destabilisation of this steady state. This is motivated by the fact that all complex behaviours in unforced systems can be traced to destabilisation (bifurcation) of some steady state, and hence enables us to use tools from linear system theory to qualitatively analyse the sources of given network behaviours. One important objective of the present study is to see how far one can come with a relatively simple approach to the analysis of highly complex biochemical networks. The proposed method is demonstrated by application to a model of mitotic control in Xenopus frog eggs, and to a model of circadian oscillations in Drosophila. In both examples we are able to identify the subsystems, and the related interactions, which are instrumental in generating the observed complex non-linear behaviours.     

Combining model structure identification and hybrid modelling for photo-production process predictive simulation and optimisation

Integrating physical knowledge and machine learning is a critical aspect of developing industrially focused digital twins for monitoring, optimisation, and design of microalgal and cyanobacterial photo-production processes. However, identifying the correct model structure to quantify the complex biological mechanism poses a severe challenge for the construction of kinetic models, while the lack of data due to the time-consuming experiments greatly impedes applications of most data-driven models. This study proposes the use of an innovative hybrid modelling approach that consists of a simple kinetic model to govern the overall process dynamic trajectory and a data-driven model to estimate mismatch between the kinetic equations and the real process. An advanced automatic model structure identification strategy is adopted to simultaneously identify the most physically probable kinetic model structure and minimum number of data-driven model parameters that can accurately represent multiple data sets over a broad spectrum of process operating conditions. Through this hybrid modelling and automatic structure identification framework, a highly accurate mathematical model was constructed to simulate and optimise an algal lutein production process. Performance of this hybrid model for long-term predictive modelling, optimisation, and online self-calibration is demonstrated and thoroughly discussed, indicating its significant potential for future industrial application.