Candidate states of Helicobacter pylori's genome-scale metabolic network upon application of "loop law" thermodynamic constraints

Constraint-based modeling has proven to be a useful tool in the analysis of biochemical networks. To date, most studies in this field have focused on the use of linear constraints, resulting from mass balance and capacity constraints, which lead to the definition of convex solution spaces. One additional constraint arising out of thermodynamics is known as the "loop law" for reaction fluxes, which states that the net flux around a closed biochemical loop must be zero because no net thermodynamic driving force exists. The imposition of the loop-law can lead to nonconvex solution spaces making the analysis of the consequences of its imposition challenging. A four-step approach is developed here to apply the loop-law to study metabolic network properties: 1), determine linear equality constraints that are necessary (but not necessarily sufficient) for thermodynamic feasibility; 2), tighten V(max) and V(min) constraints to enclose the remaining nonconvex space; 3), uniformly sample the convex space that encloses the nonconvex space using standard Monte Carlo techniques; and 4), eliminate from the resulting set all solutions that violate the loop-law, leaving a subset of steady-state solutions. This subset of solutions represents a uniform random sample of the space that is defined by the additional imposition of the loop-law. This approach is used to evaluate the effect of imposing the loop-law on predicted candidate states of the genome-scale metabolic network of Helicobacter pylori.     

MapMaker and PathTracer for tracking carbon in genome‐scale metabolic models

Constraint‐based reconstruction and analysis (COBRA) modeling results can be difficult to interpret given the large numbers of reactions in genome‐scale models. While paths in metabolic networks can be found, existing methods are not easily combined with constraint‐based approaches. To address this limitation, two tools (MapMaker and PathTracer) were developed to find paths (including cycles) between metabolites, where each step transfers carbon from reactant to product. MapMaker predicts carbon transfer maps (CTMs) between metabolites using only information on molecular formulae and reaction stoichiometry, effectively determining which reactants and products share carbon atoms. MapMaker correctly assigned CTMs for over 97% of the 2,251 reactions in an Escherichia coli metabolic model (iJO1366). Using CTMs as inputs, PathTracer finds paths between two metabolites. PathTracer was applied to iJO1366 to investigate the importance of using CTMs and COBRA constraints when enumerating paths, to find active and high flux paths in flux balance analysis (FBA) solutions, to identify paths for putrescine utilization, and to elucidate a potential CO₂ fixation pathway in E. coli. These results illustrate how MapMaker and PathTracer can be used in combination with constraint‐based models to identify feasible, active, and high flux paths between metabolites.     

Monte Carlo sampling can be used to determine the size and shape of the steady-state flux space

Constraint-based modeling results in a convex polytope that defines a solution space containing all possible steady-state flux distributions. The properties of this polytope have been studied extensively using linear programming to find the optimal flux distribution under various optimality conditions and convex analysis to define its extreme pathways (edges) and elementary modes. The work presented herein further studies the steady-state flux space by defining its hyper-volume. In low dimensions (i.e. for small sample networks), exact volume calculation algorithms were used. However, due to the #P-hard nature of the vertex enumeration and volume calculation problem in high dimensions, random Monte Carlo sampling was used to characterize the relative size of the solution space of the human red blood cell metabolic network. Distributions of the steady-state flux levels for each reaction in the metabolic network were generated to show the range of flux values for each reaction in the polytope. These results give insight into the shape of the high-dimensional solution space. The value of measuring uptake and secretion rates in shrinking the steady-state flux solution space is illustrated through singular value decomposition of the randomly sampled points. The V(max) of various reactions in the network are varied to determine the sensitivity of the solution space to the maximum capacity constraints. The methods developed in this study are suitable for testing the implication of additional constraints on a metabolic network system and can be used to explore the effects of single nucleotide polymorphisms (SNPs) on network capabilities.     

Genome-scale models of microbial cells: evaluating the consequences of constraints

Microbial cells operate under governing constraints that limit their range of possible functions. With the availability of annotated genome sequences, it has become possible to reconstruct genome-scale biochemical reaction networks for microorganisms. The imposition of governing constraints on a reconstructed biochemical network leads to the definition of achievable cellular functions. In recent years, a substantial and growing toolbox of computational analysis methods has been developed to study the characteristics and capabilities of microorganisms using a constraint-based reconstruction and analysis (COBRA) approach. This approach provides a biochemically and genetically consistent framework for the generation of hypotheses and the testing of functions of microbial cells.     

Stoichiometric network theory for nonequilibrium biochemical systems.

We introduce the basic concepts and develop a theory for nonequilibrium steady-state biochemical systems applicable to analyzing large-scale complex isothermal reaction networks. In terms of the stoichiometric matrix, we demonstrate both Kirchhoff's flux law sigma(l)J(l)=0 over a biochemical species, and potential law sigma(l) mu(l)=0 over a reaction loop. They reflect mass and energy conservation, respectively. For each reaction, its steady-state flux J can be decomposed into forward and backward one-way fluxes J = J+ - J-, with chemical potential difference deltamu = RT ln(J-/J+). The product -Jdeltamu gives the isothermal heat dissipation rate, which is necessarily non-negative according to the second law of thermodynamics. The stoichiometric network theory (SNT) embodies all of the relevant fundamental physics. Knowing J and deltamu of a biochemical reaction, a conductance can be computed which directly reflects the level of gene expression for the particular enzyme. For sufficiently small flux a linear relationship between J and deltamu can be established as the linear flux-force relation in irreversible thermodynamics, analogous to Ohm's law in electrical circuits.     

Exploring the gap between dynamic and constraint-based models of metabolism

Systems biology provides new approaches for metabolic engineering through the development of models and methods for simulation and optimization of microbial metabolism. Here we explore the relationship between two modeling frameworks in common use namely, dynamic models with kinetic rate laws and constraint-based flux models. We compare and analyze dynamic and constraint-based formulations of the same model of the central carbon metabolism of Escherichia coli. Our results show that, if unconstrained, the space of steady states described by both formulations is the same. However, the imposition of parameter-range constraints can be mapped into kinetically feasible regions of the solution space for the dynamic formulation that is not readily transferable to the constraint-based formulation. Therefore, with partial kinetic parameter knowledge, dynamic models can be used to generate constraints that reduce the solution space below that identified by constraint-based models, eliminating infeasible solutions and increasing the accuracy of simulation and optimization methods.     

Toward predicting metabolic fluxes in metabolically engineered strains

Predicting metabolic fluxes of a genetically engineered organism is an important step toward rational pathway design. However, because of various regulatory mechanisms, which are complex, often ill-characterized, and sometimes undiscovered, predicting metabolic fluxes using kinetic simulation is difficult. We propose to incorporate regulatory constraints in flux calculation to allow prediction of the steady-state fluxes without complete kinetics. The regulatory constraint, in its linear form, is derived from the dynamic metabolic control theory and involves the flux control coefficients. It is shown that with these constraints, the responses to metabolic perturbation can be predicted. Conversely, the regulatory constraints and the control coefficients can be determined by comparing the experimental data with the prediction. Therefore, this approach may offer a practical direction toward prediction of fluxes for metabolically engineered organisms.     

Estimating Metabolic Fluxes Using a Maximum Network Flexibility Paradigm

MotivationGenome-scale metabolic networks can be modeled in a constraint-based fashion. Reaction stoichiometry combined with flux capacity constraints determine the space of allowable reaction rates. This space is often large and a central challenge in metabolic modeling is finding the biologically most relevant flux distributions. A widely used method is flux balance analysis (FBA), which optimizes a biologically relevant objective such as growth or ATP production. Although FBA has proven to be highly useful for predicting growth and byproduct secretion, it cannot predict the intracellular fluxes under all environmental conditions. Therefore, alternative strategies have been developed to select flux distributions that are in agreement with experimental “omics” data, or by incorporating experimental flux measurements. The latter, unfortunately can only be applied to a limited set of reactions and is currently not feasible at the genome-scale. On the other hand, it has been observed that micro-organisms favor a suboptimal growth rate, possibly in exchange for a more “flexible” metabolic network. Instead of dedicating the internal network state to an optimal growth rate in one condition, a suboptimal growth rate is used, that allows for an easier switch to other nutrient sources. A small decrease in growth rate is exchanged for a relatively large gain in metabolic capability to adapt to changing environmental conditions.ResultsHere, we propose Maximum Metabolic Flexibility (MMF) a computational method that utilizes this observation to find the most probable intracellular flux distributions. By mapping measured flux data from central metabolism to the genome-scale models of Escherichia coli and Saccharomyces cerevisiae we show that i) indeed, most of the measured fluxes agree with a high adaptability of the network, ii) this result can be used to further reduce the space of feasible solutions iii) this reduced space improves the quantitative predictions made by FBA and contains a significantly larger fraction of the measured fluxes compared to the flux space that was reduced by a uniform sampling approach and iv) MMF can be used to select reactions in the network that contribute most to the steady-state flux space. Constraining the selected reactions improves the quantitative predictions of FBA considerably more than adding an equal amount of flux constraints, selected using a more naïve approach. Our method can be applied to any cell type without requiring prior information.AvailabilityMMF is freely available as a MATLAB plugin at: http://cs.ru.nl/~wmegchel/mmf.     

A thermodynamic view of networks

Networks can be described by the frequency distribution of the number of links associated with each node (the degree of the node). Of particular interest are the power law distributions, which give rise to the so-called scale-free networks, and the distributions of the form of the simplified canonical law (SCL) introduced by Mandelbrot, which give what we shall call the Mandelbrot networks. Many dynamical methods have been obtained for the construction of scale-free networks, but no dynamical construction of Mandelbrot networks has been demonstrated. Here we develop a systematic technique to obtain networks with any given distribution of the degrees of the nodes. This is done using a thermodynamic approach in which we maximise the entropy associated with degree distribution of the nodes of the network subject to certain constraints. These constraints can be chosen systematically to produce the desired network architecture. For large networks we therefore replace a dynamical approach to the stationary state by a thermodynamical viewpoint. We use the method to generate scale-free and Mandelbrot networks with arbitrarily chosen parameters. We emphasise that this approach opens the possibility of insights into a thermodynamics of networks by suggesting thermodynamic relations between macroscopic variables for networks.     

Optimization in integrated biochemical systems

As of yet, steady-state optimization in biochemical systems has been limited to a few studies in which networks of fluxes were optimized. These networks of fluxes are represented by linear (stoichiometric) equations that are used as constraints in a linear program, and a flux or a sum of weighted fluxes is used as the objective function. In contrast to networks of fluxes, systems of metabolic processes have not been optimized in a comparable manner. The primary reason is that these types of integrated biochemical systems are full of synergisms, antagonisms, and regulatory mechanisms that can only be captured appropriately with nonlinear models. These models are mathematically complex and difficult to analyze. In most cases it is not even possible to compute, let alone optimize, steady-state solutions analytically. Rare exceptions are S-system representations. These are nonlinear and able to represent virtually all types of dynamic behaviors, but their steady states are characterized by linear equations that can be evaluated both analytically and numerically. The steady-state equations are expressed in terms of the logarithms of the original variables, and because a function and its logarithms of the original variables, and because a function and its logarithm assume their maxima for the same argument, yields or fluxes can be optimized with linear programs expressed in terms of the logarithms of the original variables. (c) 1992 John Wiley & Sons, Inc.     

Quantification of intracellular metabolic fluxes from fractional enrichment and 13C-13C coupling constraints on the isotopomer distribution in labeled biomass components

A method for the quantification of intracellular metabolic flux distributions from steady-state mass balance constraints and from the constraints posed by the measured 13C labeling state of biomass components is presented. Two-dimensional NMR spectroscopy is used to analyze the labeling state of cell protein hydrolysate and cell wall components. No separation of the biomass hydrolysate is required to measure the degree of 13C-13C coupling and the fractional 13C enrichment in various carbon atom positions. A mixture of [1-13C]glucose and uniformly labeled [13C6]glucose is applied to make fractional 13C enrichment data and measurements of the degree of 13C-13C coupling informative with respect to the intracellular flux distribution. Simulation models that calculate the complete isotopomer distribution in biomass components on the basis of isotopomer mapping matrices are used for the estimation of intracellular fluxes by least-squares minimization. The statistical quality of the estimated intracellular flux distributions is assessed by Monte Carlo methods. Principal component analysis is performed on the outcome of the Monte Carlo procedure to identify groups of fluxes that contribute major parts to the total variance in the multiple flux estimations. The methods described are applied to a steady-state culture of a glucoamylase-producing recombinant Aspergillus niger strain.     

基于酶约束的代谢网络模型研究进展及其应用

基因组尺度代谢网络模型已经成功地应用于指导代谢工程改造,但由于传统通量平衡分析法仅考虑化学计量学和反应方向约束,模拟得到的是理论最优结果,对一些现象如代谢溢流、底物层级利用等无法准确描述。近年来人们通过在代谢网络模型中引入新的蛋白量、热力学等约束发展了新的约束优化计算方法,可以更准确真实地模拟细胞在不同条件下的代谢行为。文中主要对近年来提出的多种酶约束模型进行评述,对酶约束引入的基本思路、酶约束的数学方程表示及优化目标设定、引入酶约束后对代谢通量计算结果的影响及酶约束模型在代谢工程菌种改造中的应用等进行了全面深入的介绍,并提出了已有各种方法存在的主要问题,展望了相关方法的未来发展方向。通过引入新的约束,代谢网络模型能够更精确模拟和预测细胞在环境和基因扰动下的代谢行为,为代谢工程菌种改造提供更准确可靠的指导。     

From annotated genomes to metabolic flux models and kinetic parameter fitting

Significant advances in system-level modeling of cellular behavior can be achieved based on constraints derived from genomic information and on optimality hypotheses. For steady-state models of metabolic networks, mass conservation and reaction stoichiometry impose linear constraints on metabolic fluxes. Different objectives, such as maximization of growth rate or minimization of flux distance from a reference state, can be tested in different organisms and conditions. In particular, we have suggested that the metabolic properties of mutant bacterial strains are best described by an algorithm that performs a minimization of metabolic adjustment (MOMA) upon gene deletion. The increasing availability of many annotated genomes paves the way for a systematic application of these flux balance methods to a large variety of organisms. However, such a high throughput goal crucially depends on our capacity to build metabolic flux models in a fully automated fashion. Here we describe a pipeline for generating models from annotated genomes and discuss the current obstacles to full automation. In addition, we propose a framework for the integration of flux modeling results and high throughput proteomic data, which can potentially help in the inference of whole-cell kinetic parameters.     

Combining flux and energy balance analysis to model large-scale biochemical networks

Stoichiometric Network Theory is a constraints-based, optimization approach for quantitative analysis of the phenotypes of large-scale biochemical networks that avoids the use of detailed kinetics. This approach uses the reaction stoichiometric matrix in conjunction with constraints provided by flux balance and energy balance to guarantee mass conserved and thermodynamically allowable predictions. However, the flux and energy balance constraints have not been effectively applied simultaneously on the genome scale because optimization under the combined constraints is non-linear. In this paper, a sequential quadratic programming algorithm that solves the non-linear optimization problem is introduced. A simple example and the system of fermentation in Saccharomyces cerevisiae are used to illustrate the new method. The algorithm allows the use of non-linear objective functions. As a result, we suggest a novel optimization with respect to the heat dissipation rate of a system. We also emphasize the importance of incorporating interactions between a model network and its surroundings.     

Investigation of metabolic objectives in cultured hepatocytes

Using optimization based methods to predict fluxes in metabolic flux balance models has been a successful approach for some microorganisms, enabling construction of in silico models and even inference of some regulatory motifs. However, this success has not been translated to mammalian cells. The lack of knowledge about metabolic objectives in mammalian cells is a major obstacle that prevents utilization of various metabolic engineering tools and methods for tissue engineering and biomedical purposes. In this work, we investigate and identify possible metabolic objectives for hepatocytes cultured in vitro. To achieve this goal, we present a special data-mining procedure for identifying metabolic objective functions in mammalian cells. This multi-level optimization based algorithm enables identifying the major fluxes in the metabolic objective from MFA data in the absence of information about critical active constraints of the system. Further, once the objective is determined, active flux constraints can also be identified and analyzed. This information can be potentially used in a predictive manner to improve cell culture results or clinical metabolic outcomes. As a result of the application of this method, it was found that in vitro cultured hepatocytes maximize oxygen uptake, coupling of urea and TCA cycles, and synthesis of serine and urea. Selection of these fluxes as the metabolic objective enables accurate prediction of the flux distribution in the system given a limited amount of flux data; thus presenting a workable in silico model for cultured hepatocytes. It is observed that an overall homeostasis picture is also emergent in the findings.     

Comparison of metabolite production capability indices generated by network analysis methods

A framework of constraint-based reconstruction and analysis (COBRA) is used for modeling large-scale metabolic networks. In COBRA, extreme pathway and optimization analyses are commonly used to study the properties of networks. While the results of both methods are completely consistent, extreme pathway analysis is considered to be better because of its wider representational ability. In this study, we assessed these two methods by computational knockout experiments. We examined a simple pathway model and found that the extreme pathway method led to misguided conclusions in specific cases, while optimization analysis calculated the correct knockout effects. We also investigated the Escherichia coli metabolic pathway model, and found that these methods result in inconsistent interpretations of the network properties. IN CONCLUSION: it has been claimed that these two methods result in the same producible metabolites, but we found a difference in individual results for a biological pathway. Our results could provide helpful guidance for when to use the methods, particularly extreme pathway analysis.     

Improving the accuracy of flux balance analysis through the implementation of carbon availability constraints for intracellular reactions

Constraint-based modeling methods, such as Flux Balance Analysis (FBA), have been extensively used to decipher complex, information rich -omics datasets to elicit system-wide behavioral patterns of cellular metabolism. FBA has been successfully used to gain insight in a wide range of applications, such as range of substrate utilization, product yields and to design metabolic engineering strategies to improve bioprocess performance. A well-known challenge associated with large genome-scale metabolic networks is that they result in underdetermined problem formulations. Consequently, rather than unique solutions, FBA and related methods examine ranges of reaction flux values that are consistent with the studied physiological conditions. The wider the reported flux ranges, the higher the uncertainty in the determination of basic reaction properties, limiting interpretability of and confidence in the results. Herein, we propose a new, computationally efficient approach that refines flux range predictions by constraining reaction fluxes on the basis of the elemental balance of carbon. We compared carbon constraint FBA (ccFBA) against experimentally-measured intracellular fluxes using the latest CHO GEM (iCHO1766) and were able to substantially improve the accuracy of predicted flux values compared with FBA. ccFBA can be used as a stand-alone method but is also compatible with and complimentary to other constraint-based approaches.     

Conformational states of the switch I region of Ha-ras-p21 in hinge residue mutants studied by fluorescence lifetime and fluorescence anisotropy measurements

The hinge residues (Val29 and Ile36) of the switch I region (also known as the effector loop) of the Ha-ras-p21 protein have been mutated to glycines to accelerate the conformational changes typical for the effector loop. In this work, we have studied the influence of the combined mutations on the steady-state structure of the switch I region of the protein in both the inactive GDP-bound conformation as in the active GTP-bound conformation. Here, we use the fluorescence properties of the single tryptophan residue in the Y32W mutant of Ha-ras-p21. This mutant has already been used extensively as a reference form of the protein. Reducing the size of the side chains of the hinge residues not only accelerates the conformational changes but also affects the steady-state structures of the effector loop as indicated by the changes in the fluorescence properties. A thorough analysis of the fluorescence changes (quantum yield, lifetimes, etc.) proves that these changes are from a reshuffling between the rotamer populations of Trp. The population reshuffling is caused by the overall structural rearrangement along the switch I region. The effects are clearly more pronounced in the inactive GDP-bound conformation than in the active GTP-bound conformation. The effect of both mutations seems to be additive in the GDP-bound state, but cooperative in the GTP-bound state.