Hybrid metabolic flux analysis: combining stoichiometric and statistical constraints to model the formation of complex recombinant products

Background  

Stoichiometric models constitute the basic framework for fluxome quantification in the realm of metabolic engineering. A recurrent bottleneck, however, is the establishment of consistent stoichiometric models for the synthesis of recombinant proteins or viruses. Although optimization algorithms for in silico metabolic redesign have been developed in the context of genome-scale stoichiometric models for small molecule production, still rudimentary knowledge of how different cellular levels are regulated and phenotypically expressed prevents their full applicability for complex product optimization.     

Advantages and disadvantages of aggregating fluxes into synthetic and degradative fluxes when modelling metabolic pathways.

It is now widely accepted that mathematical models are needed to predict the behaviour of complex metabolic networks in the cell, in order to have a rational basis for planning metabolic engineering with biotechnological or therapeutical purposes. The great complexity of metabolic networks makes it crucial to simplify them for analysis, but without violating key principles of stoichiometry or thermodynamics. We show here, however, that models for branched complex systems are sometimes obtained that violate the stoichiometry of fluxes at branch points and as a result give unrealistic metabolite concentrations at the steady state. This problem is especially important when models are constructed with the S-system form of biochemical systems theory. However, the same violation of stoichiometry can occur in metabolic control analysis if control coefficients are assumed to be constant when trying to predict the effects of large changes. We derive the appropriate matrix equations to analyse this type of problem systematically and to assess its extent in any given model.     

Optimization of regulatory architectures in metabolic reaction networks

Successful biotechnological applications, such as amino acid production, have demonstrated significant improvement in bioprocess performance by genetic modifications of metabolic control architectures and enzyme expression levels. However, the stoichiometric complexity of metabolic pathways, along with their strongly nonlinear nature and regulatory coupling, necessitates the use of structured kinetic models to direct experimental applications and aid in quantitative understanding of cellular bioprocesses. A novel optimization problem is introduced here, the objective of which is to identify changes in the regulatory characteristics of pertinent enzymes and in their cellular content which should be implemented to optimize a particular metabolic process. The mathematical representation of the metabolic reaction networks used is the S-system representation, which at steady state is characterized by linear equations. Exploiting the linearity of the representation, we formulated the optimization problem as a mixed-integer linear programming (MILP) problem. This formulation allows the consideration of a regulatory superstructure that contains all alternative regulatory structures that can be considered for a given pathway. The proposed approach is developed and illustrated using a simple linear pathway. Application of the framework on a complicated pathway-namely, the xanthine monophosphate (XMP) and guanosine monophosphate (GMP) synthesis pathway-identified the modification of the regulatory architecture that, along with changes in enzyme expression levels, can increase the XMP and GMP concentration by over 114 times the reference value, which is 50 times more than could be achieved by changes in enzyme expression levels only. (c) 1996 John Wiley & Sons, Inc.     

Experimental and mathematical approaches to modeling plant metabolic networks

To support their sessile and autotrophic lifestyle higher plants have evolved elaborate networks of metabolic pathways. Dynamic changes in these metabolic networks are among the developmental forces underlying the functional differentiation of organs, tissues and specialized cell types. They are also important in the various interactions of a plant with its environment. Further complexity is added by the extensive compartmentation of the various interconnected metabolic pathways in plants. Thus, although being used widely for assessing the control of metabolic flux in microbes, mathematical modeling approaches that require steady-state approximations are of limited utility for understanding complex plant metabolic networks. However, considerable progress has been made when manageable metabolic subsystems were studied. In this article, we will explain in general terms and using simple examples the concepts underlying stoichiometric modeling (metabolic flux analysis and metabolic pathway analysis) and kinetic approaches to modeling (including metabolic control analysis as a special case). Selected studies demonstrating the prospects of these approaches, or combinations of them, for understanding the control of flux through particular plant pathways are discussed. We argue that iterative cycles of (dry) mathematical modeling and (wet) laboratory testing will become increasingly important for simulating the distribution of flux in plant metabolic networks and deriving rational experimental designs for metabolic engineering efforts.     

Competition for enzymes in metabolic pathways: implications for optimal distributions of enzyme concentrations and for the distribution of flux control

The structures of biochemical pathways are assumed to be determined by evolutionary optimization processes. In the framework of mathematical models, these structures should be explained by the formulation of optimization principles. In the present work, the principle of minimal total enzyme concentration at fixed steady state fluxes is applied to metabolic networks. According to this principle there exists a competition of the reactions for the available amount of enzymes such that all biological functions are maintained. In states which fulfil these optimization criteria the enzyme concentrations are distributed in a non-uniform manner among the reactions. This result has consequences for the distribution of flux control. It is shown that the flux control matrix c, the elasticity matrix epsilon, and the vector e of enzyme concentrations fulfil in optimal states the relations c(T)e = e and epsilon(T)e = 0. Starting from a well-balanced distribution of enzymes the minimization of total enzyme concentration leads to a lowering of the SD of the flux control coefficients.     

Mathematical simulation and analysis of cellular metabolism and regulation.

MOTIVATION: A better understanding of the biological phenomena observed in cells requires the creation and analysis of mathematical models of cellular metabolism and physiology. The formulation and study of such models must also be simplified as far as possible to cope with the increasing complexity demanded and exponential accumulation of the metabolic reconstructions computed from sequenced genomes. RESULTS: A mathematical simulation workbench, DBsolve, has been developed to simplify the derivation and analysis of mathematical models. It combines: (i) derivation of large-scale mathematical models from metabolic reconstructions and other data sources; (ii) solving and parameter continuation of non-linear algebraic equations (NAEs), including metabolic control analysis; (iii) solving the non-linear stiff systems of ordinary differential equations (ODEs); (iv) bifurcation analysis of ODEs; (v) parameter fitting to experimental data or functional criteria based on constrained optimization. The workbench has been successfully used for dynamic metabolic modeling of some typical biochemical networks (Dolgacheva et al., Biochemistry (Moscow), 6, 1063-1068, 1996; Goldstein and Goryanin, Mol. Biol. (Moscow), 30, 976-983, 1996), including microbial glycolytic pathways, signal transduction pathways and receptor-ligand interactions. AVAILABILITY: DBsolve 5. 00 is freely available from http://websites.ntl.com/ approximately igor.goryanin. CONTACT: gzz78923@ggr.co.uk     

A survey on methods for modeling and analyzing integrated biological networks

Understanding how cellular systems build up integrated responses to their dynamically changing environment is one of the open questions in Systems Biology. Despite their intertwinement, signaling networks, gene regulation and metabolism have been frequently modeled independently in the context of well-defined subsystems. For this purpose, several mathematical formalisms have been developed according to the features of each particular network under study. Nonetheless, a deeper understanding of cellular behavior requires the integration of these various systems into a model capable of capturing how they operate as an ensemble. With the recent advances in the "omics" technologies, more data is becoming available and, thus, recent efforts have been driven toward this integrated modeling approach. We herein review and discuss methodological frameworks currently available for modeling and analyzing integrated biological networks, in particular metabolic, gene regulatory and signaling networks. These include network-based methods and Chemical Organization Theory, Flux-Balance Analysis and its extensions, logical discrete modeling, Petri Nets, traditional kinetic modeling, Hybrid Systems and stochastic models. Comparisons are also established regarding data requirements, scalability with network size and computational burden. The methods are illustrated with successful case studies in large-scale genome models and in particular subsystems of various organisms.     

From genomes to in silico cells via metabolic networks

Genome-scale metabolic models are the focal point of systems biology as they allow the collection of various data types in a form suitable for mathematical analysis. High-quality metabolic networks and metabolic networks with incorporated regulation have been successfully used for the analysis of phenotypes from phenotypic arrays and in gene-deletion studies. They have also been used for gene expression analysis guided by metabolic network structure, leading to the identification of commonly regulated genes. Thus, genome-scale metabolic modeling currently stands out as one of the most promising approaches to obtain an in silico prediction of cellular function based on the interaction of all of the cellular components.     

Simplified modelling of metabolic pathways for flux prediction and optimization: lessons from an in vitro reconstruction of the upper part of glycolysis

Explicit modelling of metabolic networks relies on well-known mathematical tools and specialized computer programs. However, identifying and estimating the values of the very numerous enzyme parameters inherent to the models remain a tedious and difficult task, and the rate equations of the reactions are usually not known in sufficient detail. A way to circumvent this problem is to use 'non-mechanistic' models, which may account for the behaviour of the systems with a limited number of parameters. Working on the first part of glycolysis reconstituted in vitro, we showed how to derive, from titration experiments, values of effective enzyme activity parameters that do not include explicitly any of the classical kinetic constants. With a maximum of only two parameters per enzyme, this approach produced very good estimates for the flux values, and enabled us to determine the optimization conditions of the system, i.e. to calculate the set of enzyme concentrations that maximizes the flux. This fast and easy method should be valuable in the context of integrative biology or for metabolic engineering, where the challenge is to deal with the dramatic increase in the number of parameters when the systems become complex.     

Applications of metabolic modeling to drive bioprocess development for the production of value-added chemicals

Increasing numbers of value added chemicals are being produced using microbial fermentation strategies. Computational modeling and simulation of microbial metabolism is rapidly becoming an enabling technology that is driving a new paradigm to accelerate the bioprocess development cycle. In particular, constraint-based modeling and the development of genome-scale models of industrial microbes are finding increasing utility across many phases of the bioprocess development workflow. Herein, we review and discuss the requirements and trends in the industrial application of this technology as we build toward integrated computational/experimental platforms for bioprocess engineering. Specifically we cover the following topics: (1) genome-scale models as genetically and biochemically consistent representations of metabolic networks; (2) the ability of these models to predict, assess, and interpret metabolic physiology and flux states of metabolism; (3) the model-guided integrative analysis of high throughput ‘omics’ data; (4) the reconciliation and analysis of on- and off-line fermentation data as well as flux tracing data; (5) model-aided strain design strategies and the integration of calculated biotransformation routes; and (6) control and optimization of the fermentation processes. Collectively, constraint-based modeling strategies are impacting the iterative characterization of metabolic flux states throughout the bioprocess development cycle, while also driving metabolic engineering strategies and fermentation optimization.     

基于代谢网络预测菌种基因改造靶点方法的研究进展

高产特定产品的人工细胞工厂的构建需要对野生菌株进行大量的基因工程改造,近年来随着大量基因组尺度代谢网络模型的构建,人们提出了多种基于代谢网络分析预测基因改造靶点以使某一目标化合物合成最优的方法。这些方法利用基因组尺度代谢网络模型中的反应计量关系约束和反应不可逆性约束等,通过约束优化的方法预测可使产物合成最大化的改造靶点,避免了传统的通过相关途径的直观分析确定靶点的方法的局限性和主观性,为细胞工厂的理性设计提供了新的思路。以下结合作者的实际研究经验,对这些菌种优化方法的原理、优缺点及适用性等进行详细介绍,并讨论了目前存在的主要问题和未来的研究方向,为人们针对不同目标产品选择合适的方法及预测结果的可靠性评估提供了指导。     

Optimization strategies for metabolic networks

Background  

The increasing availability of models and data for metabolic networks poses new challenges in what concerns optimization for biological systems. Due to the high level of complexity and uncertainty associated to these networks the suggested models often lack detail and liability, required to determine the proper optimization strategies. A possible approach to overcome this limitation is the combination of both kinetic and stoichiometric models. In this paper three control optimization methods, with different levels of complexity and assuming various degrees of process information, are presented and their results compared using a prototype network.     

Automated Bayesian model development for frequency detection in biological time series

Background

The ability to perform quantitative studies using isotope tracers and metabolic flux analysis (MFA) is critical for detecting pathway bottlenecks and elucidating network regulation in biological systems, especially those that have been engineered to alter their native metabolic capacities. Mathematically, MFA models are traditionally formulated using separate state variables for reaction fluxes and isotopomer abundances. Analysis of isotope labeling experiments using this set of variables results in a non-convex optimization problem that suffers from both implementation complexity and convergence problems.

Results

This article addresses the mathematical and computational formulation of ¹³C MFA models using a new set of variables referred to as fluxomers. These composite variables combine both fluxes and isotopomer abundances, which results in a simply-posed formulation and an improved error model that is insensitive to isotopomer measurement normalization. A powerful fluxomer iterative algorithm (FIA) is developed and applied to solve the MFA optimization problem. For moderate-sized networks, the algorithm is shown to outperform the commonly used 13CFLUX cumomer-based algorithm and the more recently introduced OpenFLUX software that relies upon an elementary metabolite unit (EMU) network decomposition, both in terms of convergence time and output variability.

Conclusions

Substantial improvements in convergence time and statistical quality of results can be achieved by applying fluxomer variables and the FIA algorithm to compute best-fit solutions to MFA models. We expect that the fluxomer formulation will provide a more suitable basis for future algorithms that analyze very large scale networks and design optimal isotope labeling experiments.     

Inferring general relations between network characteristics from specific network ensembles

Different network models have been suggested for the topology underlying complex interactions in natural systems. These models are aimed at replicating specific statistical features encountered in real-world networks. However, it is rarely considered to which degree the results obtained for one particular network class can be extrapolated to real-world networks. We address this issue by comparing different classical and more recently developed network models with respect to their ability to generate networks with large structural variability. In particular, we consider the statistical constraints which the respective construction scheme imposes on the generated networks. After having identified the most variable networks, we address the issue of which constraints are common to all network classes and are thus suitable candidates for being generic statistical laws of complex networks. In fact, we find that generic, not model-related dependencies between different network characteristics do exist. This makes it possible to infer global features from local ones using regression models trained on networks with high generalization power. Our results confirm and extend previous findings regarding the synchronization properties of neural networks. Our method seems especially relevant for large networks, which are difficult to map completely, like the neural networks in the brain. The structure of such large networks cannot be fully sampled with the present technology. Our approach provides a method to estimate global properties of under-sampled networks in good approximation. Finally, we demonstrate on three different data sets (C. elegans neuronal network, R. prowazekii metabolic network, and a network of synonyms extracted from Roget's Thesaurus) that real-world networks have statistical relations compatible with those obtained using regression models.     

An optimization framework of biological dynamical systems

Different biological dynamics are often described by different mathematical equations. On the other hand, some mathematical models describe many biological dynamics universally. Here, we focus on three biological dynamics: the Lotka-Volterra equation, the Hopfield neural networks, and the replicator equation. We describe these three dynamical models using a single optimization framework, which is constructed with employing the Riemannian geometry. Then, we show that the optimization structures of these dynamics are identical, and the differences among the three dynamics are only in the constraints of the optimization. From this perspective, we discuss the unified view for biological dynamics. We also discuss the plausible categorizations, the fundamental nature, and the efficient modeling of the biological dynamics, which arise from the optimization perspective of the dynamical systems.     

Simultaneous modelling of metabolic, genetic and product-interaction networks

The creation of cell models from annotated genome information, as well as additional data from other databases, requires both a format and medium for its distribution. Standards are described for the representation of the data in the form of Document Type Definitions (DTDs) for XML files. Separate DTDs are detailed for genetic, metabolic and gene product-interaction networks, which can be used to hold information on individual subsystems, or which may be combined to create a whole cell DTD. In the execution of this work, a fifth DTD was also created for a metabolite thesaurus, which allows incorporation of metabolite synonyms and generic nomenclature data into the models. A gene-regulation classification scheme was also created, to facilitate incorporation of gene regulatory information in an efficient manner. The work is described with particular reference to the metabolic network of Escherichia coli, which contains 808 individual enzymes. The assignment of confidence levels to these data, through the use of Gene Ontology evidence codes, is highlighted. In silico investigations may now be performed using the mathematical simulation workbench, DBsolve, which incorporates the facility to introduce data directly from XML.     

FluxAnalyzer: exploring structure,pathways, and flux distributions in metabolic networks on interactive flux maps

MOTIVATION: The analysis of structure, pathways and flux distributions in metabolic networks has become an important approach for understanding the functionality of metabolic systems. The need of a user-friendly platform for stoichiometric modeling of metabolic networks in silico is evident. RESULTS: The FluxAnalyzer is a package for MATLAB and facilitates integrated pathway and flux analysis for metabolic networks within a graphical user interface. Arbitrary metabolic network models can be composed by instances of four types of network elements. The abstract network model is linked with network graphics leading to interactive flux maps which allow for user input and display of calculation results within a network visualization. Therein, a large and powerful collection of tools and algorithms can be applied interactively including metabolic flux analysis, flux optimization, detection of topological features and pathway analysis by elementary flux modes or extreme pathways. The FluxAnalyzer has been applied and tested for complex networks with more than 500,000 elementary modes. Some aspects of the combinatorial complexity of pathway analysis in metabolic networks are discussed. AVAILABILITY: Upon request from the corresponding author. Free for academic users (license agreement). Special contracts are available for industrial corporations. SUPPLEMENTARY INFORMATION: http://www.mpi-magdeburg.mpg.de/projects/fluxanalyzer.     

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.