Comparison and analysis of objective functions in flux balance analysis

Flux balance analysis (FBA) is currently one of the most important and used techniques for estimation of metabolic reaction rates (fluxes). This mathematical approach utilizes an optimization criterion in order to select a distribution of fluxes from the feasible space delimited by the metabolic reactions and some restrictions imposed over them, assuming that cellular metabolism is in steady state. Therefore, the obtained flux distribution depends on the specific objective function used. Multiple studies have been aimed to compare distinct objective functions at given conditions, in order to determine which of those functions produces values of fluxes closer to real data when used as objective in the FBA; in other words, what is the best objective function for modeling cell metabolism at a determined environmental condition. However, these comparative studies have been designed in very dissimilar ways, and in general, several factors that can change the ideal objective function in a cellular condition have not been adequately considered. Additionally, most of them have used only one dataset for representing one condition of cell growth, and different measuring techniques have been used. For these reasons, a rigorous study on the effect of factors such as the quantity of used data, the number and type of fluxes utilized as input data, and the selected classification of growth conditions, are required in order to obtain useful conclusions for these comparative studies, allowing limiting clearly the application range on any of those results. © 2014 American Institute of Chemical Engineers Biotechnol. Prog., 30:985–991, 2014     

A reliable simulator for dynamic flux balance analysis

Dynamic flux balance analysis (DFBA) provides a platform for detailed design, control and optimization of biochemical process technologies. It is a promising modeling framework that combines genome‐scale metabolic network analysis with dynamic simulation of the extracellular environment. Dynamic flux balance analysis assumes that the intracellular species concentrations are in equilibrium with the extracellular environment. The resulting underdetermined stoichiometric model is solved under the assumption of a biochemical objective such as growth rate maximization. The model of the metabolism is coupled with the dynamic mass balance equations of the extracellular environment via expressions for the rates of substrate uptake and product excretion, which imposes additional constraints on the linear program (LP) defined by growth rate maximization of the metabolism. The linear program is embedded into the dynamic model of the bioreactor, and together with the additional constraints this provides an accurate model of the substrate consumption, product secretion, and biomass production during operation. A DFBA model consists of a system of ordinary differential equations for which the evaluation of the right‐hand side requires not only function evaluations, but also the solution of one or more linear programs. The numerical tool presented here accurately and efficiently simulates large‐scale dynamic flux balance models. The main advantages that this approach has over existing implementation are that the integration scheme has a variable step size, that the linear program only has to be solved when qualitative changes in the optimal flux distribution of the metabolic network occur, and that it can reliably simulate behavior near the boundary of the domain where the model is defined. This is illustrated through large‐scale examples taken from the literature. Biotechnol. Bioeng. 2013; 110: 792–802. © 2012 Wiley Periodicals, Inc.     

Peroxisomal cholesterol biosynthesis and Smith-Lemli-Opitz syndrome

Smith-Lemli-Opitz syndrome (SLOS), caused by 7-dehydrocholesterol-reductase (DHCR7) deficiency, shows variable severity independent of DHCR7 genotype. To test whether peroxisomes are involved in alternative cholesterol synthesis, we used [1-(14)C]C24:0 for peroxisomal beta-oxidation to generate [1-(14)C]acetyl-CoA as cholesterol precursor inside peroxisomes. The HMG-CoA reductase inhibitor lovastatin suppressed cholesterol synthesis from [2-(14)C]acetate and [1-(14)C]C8:0 but not from [1-(14)C]C24:0, implicating a peroxisomal, lovastatin-resistant HMG-CoA reductase. In SLOS fibroblasts lacking DHCR7 activity, no cholesterol was formed from [1-(14)C]C24:0-derived [1-(14)C]acetyl-CoA, indicating that the alternative peroxisomal pathway also requires this enzyme. Our results implicate peroxisomes in cholesterol biosynthesis but provide no link to phenotypic variation in SLOS.     

DFBAlab: a fast and reliable MATLAB code for dynamic flux balance analysis

Background

Dynamic Flux Balance Analysis (DFBA) is a dynamic simulation framework for biochemical processes. DFBA can be performed using different approaches such as static optimization (SOA), dynamic optimization (DOA), and direct approaches (DA). Few existing simulators address the theoretical and practical challenges of nonunique exchange fluxes or infeasible linear programs (LPs). Both are common sources of failure and inefficiencies for these simulators.

Results

DFBAlab, a MATLAB-based simulator that uses the LP feasibility problem to obtain an extended system and lexicographic optimization to yield unique exchange fluxes, is presented. DFBAlab is able to simulate complex dynamic cultures with multiple species rapidly and reliably, including differential-algebraic equation (DAE) systems. In addition, DFBAlab’s running time scales linearly with the number of species models. Three examples are presented where the performance of COBRA, DyMMM and DFBAlab are compared.

Conclusions

Lexicographic optimization is used to determine unique exchange fluxes which are necessary for a well-defined dynamic system. DFBAlab does not fail during numerical integration due to infeasible LPs. The extended system obtained through the LP feasibility problem in DFBAlab provides a penalty function that can be used in optimization algorithms.

Electronic supplementary material

The online version of this article (doi:10.1186/s12859-014-0409-8) contains supplementary material, which is available to authorized users.     

Identification of active constraints in dynamic flux balance analysis

This study deals with the calibration of dynamic metabolic flux models that are formulated as the maximization of an objective subject to constraints. Two approaches were applied for identifying the constraints from data. In the first approach a minimal active number of limiting constraints is found based on data that are assumed to be bounded within sets whereas, in the second approach, the limiting constraints are found based on parametric sensitivity analysis. The ability of these approaches to finding the active limiting constraints was verified through their application to two case studies: an in‐silico (simulated) data‐based study describing the growth of E. coli and an experimental data‐based study for Bordetella pertussis (B. pertussis). © 2016 American Institute of Chemical Engineers Biotechnol. Prog., 33:26–36, 2017     

Insights into pH‐induced metabolic switch by flux balance analysis

Lactate accumulation in mammalian cell culture is known to impede cellular growth and productivity. The control of lactate formation and consumption in a hybridoma cell line was achieved by pH alteration during the early exponential growth phase. In particular, lactate consumption was induced even at high glucose concentrations at pH 6.8, whereas highly increased production of lactate was obtained at pH 7.8. Consequently, constraint‐based metabolic flux analysis was used to examine pH‐induced metabolic states in the same growth state. We demonstrated that lactate influx at pH 6.8 led cells to maintain high fluxes in the TCA cycle and malate‐aspartate shuttle resulting in a high ATP production rate. In contrast, under increased pH conditions, less ATP was generated and different ATP sources were utilized. Gene expression analysis led to the conclusion that lactate formation at high pH was enabled by gluconeogenic pathways in addition to facilitated glucose uptake. The obtained results provide new insights into the influence of pH on cellular metabolism, and are of importance when considering pH heterogeneities typically present in large scale industrial bioreactors. © 2015 American Institute of Chemical Engineers Biotechnol. Prog., 31:347–357, 2015     

Chemical synthesis of 7- and 8-dehydro derivatives of pregnane-3,17alpha,20-triols,potential steroid metabolites in Smith-Lemli-Opitz syndrome

Pregnane-3,17 alpha,20-triols bearing unsaturation at delta(7), delta(8), delta(5,7), or delta(5,8) have been tentatively identified as steroid metabolites in Smith-Lemli-Opitz syndrome (SLOS). Starting with 17 alpha-hydroxypregnenolone diacetate, we have synthesized 13 unsaturated C(21) triols by four different routes in one to four steps. These multifunctional steroids were prepared by a series of regio- and stereoselective transformations chosen to minimize facile olefin isomerization and 17-deoxygenation. The results include a study of stereoselectivity in the reduction of 17 alpha-hydroxy-20-ketosteroids, an alternative method for reducing diethyl azodicarboxylate adducts of delta(5,7) steroids, and an efficient oxidation-isomerization of a delta(5,7) steroid using cholesterol oxidase. The 13 triols and their synthetic precursors were fully characterized by 1H and 13C nuclear magnetic resonance (NMR) spectroscopy. The NMR data, together with molecular modeling, indicated unanticipated conformational heterogeneity for two synthetic intermediates, 17 alpha-hydroxypregna-4,7-diene-3,20-dione and 17 alpha-hydroxy-5 beta-pregn-7-ene-3,20-dione. The unsaturated C(21) triols are useful as reference standards to study adrenal steroid production in SLOS and to develop methods for pre- and postnatal diagnosis of this congenital disorder.     

Revealing metabolic mechanisms of interaction in the anaerobic digestion microbiome by flux balance analysis

Anaerobic digestion is a key biological process for renewable energy, yet the mechanistic knowledge on its hidden microbial dynamics is still limited. The present work charted the interaction network in the anaerobic digestion microbiome via the full characterization of pairwise interactions and the associated metabolite exchanges. To this goal, a novel collection of 836 genome-scale metabolic models was built to represent the functional capabilities of bacteria and archaea species derived from genome-centric metagenomics. Dominant microbes were shown to prefer mutualistic, parasitic and commensalistic interactions over neutralism, amensalism and competition, and are more likely to behave as metabolite importers and profiteers of the coexistence. Additionally, external hydrogen injection positively influences microbiome dynamics by promoting commensalism over amensalism. Finally, exchanges of glucogenic amino acids were shown to overcome auxotrophies caused by an incomplete tricarboxylic acid cycle. Our novel strategy predicted the most favourable growth conditions for the microbes, overall suggesting strategies to increasing the biogas production efficiency. In principle, this approach could also be applied to microbial populations of biomedical importance, such as the gut microbiome, to allow a broad inspection of the microbial interplays.     

The implications of 7-dehydrosterol-7-reductase deficiency (Smith-Lemli-Opitz syndrome) to neurosteroid production

Smith-Lemli-Opitz syndrome (SLOS) is an autosomal recessive, multiple malformation/mental retardation syndrome with an estimated incidence among individuals of European ancestry of 1 in 20000 to 1 in 30000. It is caused by inactivity of the enzyme 7-dehydrosterol-delta(7)-reductase, which catalyses the terminal transformation in cholesterol synthesis. Patients show not only an increased level of 7-dehydrocholesterol in blood and tissues, but also increased 8-dehydrocholesterol because of the presence of an active delta(8)-delta(7) isomerase. A major consequence of these biochemical abnormalities is the alteration of normal embryonic and fetal somatic development causing postnatal abnormalities of growth, learning, language and behavior. While deficient cholesterol during early development is the primary cause of central nervous system (CNS) abnormalities and retardation, we questioned whether neurosteroids could also be involved since they can have a profound influence on behavioral characteristics. Disordered neurosteroidogenesis would be expected in SLOS and could be caused by a deficiency in classical neurosteroid synthesis secondary to cholesterol deficiency, or by synthesis from 7- and 8-dehydrocholesterol of novel neurosteroids with delta(7) or delta(8) unsaturation which may have altered activity compared with conventional neurosteroids. In particular we sought analogues of dehydroepiandrosterone sulfate, pregnenolone sulfate and the pregnanolone epimers. We targeted urine from post-pubertal females, as this type of sample would be most likely to yield identifiable amounts of the pregnanolone metabolites of progesterone. Analysis by GC/MS of urinary steroids excreted by post-pubertal females confirmed the presence of neurosteroid-like compounds in SLOS patient's urine. Even though the new neuroactive steroids identified were unlikely to have been formed in the brain, it is likely that mechanisms for their synthesis are operable in this organ.     

亚热带毛竹林生态系统能量通量及平衡分析

利用开路涡度相关系统和常规气象观测仪器,对亚热带(浙江省)毛竹林生态系统2011年的净辐射、显热通量、潜热通量、土壤热通量以及气温、地温、降雨量等气象要素进行了连续观测,定量分析了毛竹林生态系统能量通量的变化和各能量分量的分配特征,并计算了能量闭合度以及波文比。结果表明:毛竹林全年净辐射为2628.00 MJ/m2,显热通量为576.80 MJ/m2,潜热通量为1666.77 MJ/m2,土壤热通量为-7.52 MJ/m2,土壤为热源,各能量分量季节变化明显,日变化基本呈单峰型曲线变化。显热通量占净辐射的22.0%,潜热通量占63.4%,毛竹林生态系统潜热通量为能量散失的主要形式。波文比逐月变化规律不明显,波动较大,在0.07—1.77之间变化,能量平衡比率法得出毛竹林年能量闭合度为0.85,月平均闭合度为0.84,能量闭合度高于线性回归法计算结果,但仍有15%的能量不闭合。     

Stoichiometric model and flux balance analysis for a mixed culture of Leptospirillum ferriphilum and Ferroplasma acidiphilum

The oxidation process of sulfide minerals in natural environments is achieved by microbial communities from the Archaea and Bacteria domains. A metabolic reconstruction of two dominant species, Leptospirillum ferriphilum and Ferroplasma acidiphilum, which are always found together as a mixed culture in this natural environments, was made. The metabolic model, composed of 152 internal reactions and 29 transport reactions, describes the main interactions between these species, assuming that both use ferrous iron as energy source, and F. acidiphilum takes advantage of the organic compounds secreted by L. ferriphilum for chemomixotrophic growth. A first metabolic model for a mixed culture used in bacterial leaching is proposed in this article, which pretends to represent the characteristics of the mixed culture in a simplified manner. It was evaluated with experimental data through flux balance analysis (FBA) using as objective function the maximization of biomass. The growth yields on ferrous iron obtained for each microorganism are consistent with experimental data, and the flux distribution obtained allows understanding of the metabolic capabilities of both microorganisms growing together in a bioleaching process. The model was used to simulate the growth of F. acidiphilum on different substrates, to determine in silico which compounds maximize cell growth, and which are essential. Knockout simulations were carried out for L. ferriphilum and F. acidiphilum metabolic models, predicting key enzymes of central metabolism. The results of this analysis are consistent with experimental data from literature, showing a robust behavior of the metabolic model. © 2014 American Institute of Chemical Engineers Biotechnol. Prog., 31:307–315, 2015     

Development of thermodynamic optimum searching (TOS) to improve the prediction accuracy of flux balance analysis

Flux balance analysis (FBA) has been widely used in calculating steady‐state flux distributions that provide important information for metabolic engineering. Several thermodynamics‐based methods, for example, quantitative assignment of reaction directionality and energy balance analysis have been developed to improve the prediction accuracy of FBA. However, these methods can only generate a thermodynamically feasible range, rather than the most thermodynamically favorable solution. We therefore developed a novel optimization method termed as thermodynamic optimum searching (TOS) to calculate the thermodynamically optimal solution, based on the second law of thermodynamics, the minimum magnitude of the Gibbs free energy change and the maximum entropy production principle (MEPP). Then, TOS was applied to five physiological conditions of Escherichia coli to evaluate its effectiveness. The resulting prediction accuracy was found significantly improved (10.7–48.5%) by comparing with the ¹³C‐fluxome data, indicating that TOS can be considered an advanced calculation and prediction tool in metabolic engineering. Biotechnol. Bioeng. 2013; 110: 914–923. © 2012 Wiley Periodicals, Inc.     

Metabolic flux analysis of lactic acid fermentation: effects of pH and lactate ion concentration

A detailed metabolic flux analysis for lactic acid production by Streptococcus lactis has been carried out. A metabolic reaction set was constructed for the metabolism of S. lactis. Fluxes through these reactions were estimated by using accumulation rates of biomass, product and consumption rates of the substrate, which were obtained through experiments. The changes in the flux movement are shown for different pHs and initial lactate concentrations of the medium. The analysis indicated that pH only affected the uptake rates of lactose, whereas lactate ion concentration influenced the movement of flux through the network.     

Quantifying the propagation of parametric uncertainty on flux balance analysis

Flux balance analysis (FBA) and associated techniques operating on stoichiometric genome-scale metabolic models play a central role in quantifying metabolic flows and constraining feasible phenotypes. At the heart of these methods lie two important assumptions: (i) the biomass precursors and energy requirements neither change in response to growth conditions nor environmental/genetic perturbations, and (ii) metabolite production and consumption rates are equal at all times (i.e., steady-state). Despite the stringency of these two assumptions, FBA has been shown to be surprisingly robust at predicting cellular phenotypes. In this paper, we formally assess the impact of these two assumptions on FBA results by quantifying how uncertainty in biomass reaction coefficients, and departures from steady-state due to temporal fluctuations could propagate to FBA results. In the first case, conditional sampling of parameter space is required to re-weigh the biomass reaction so as the molecular weight remains equal to 1 g mmol⁻¹, and in the second case, metabolite (and elemental) pool conservation must be imposed under temporally varying conditions. Results confirm the importance of enforcing the aforementioned constraints and explain the robustness of FBA biomass yield predictions.     

Dynamic flux balance analysis of biomass and lipid production by Antarctic thraustochytrid Oblongichytrium sp. RT2316-13

Production of biomass and lipids in batch cultures of the Antarctic thraustochytrid Oblongichytrium sp. RT2316-13, is reported. The microorganism proved capable of producing nearly 67% docosahexaenoic acid (DHA) and 15% eicosapentaenoic acid (EPA) in its total lipid fraction. Biomass with a maximum total lipid content of 33.5% (wt/wt) could be produced at 15°C in batch culture using a medium containing glucose (20 g/L), yeast extract (10.5 g/L), and other minor components. A lower culture temperature (5°C) reduced biomass and lipid productivities compared to culture at 15°C, but enhanced the DHA and EPA content of the lipids by 6.4- and 3.3-fold, respectively. Both a simple minimally structured mathematical model and a more complex genome-scale metabolic model (GEM) allowed the fermentation profiles in batch cultures to be satisfactorily simulated, but the GEM provided much greater insight in the biochemical and physiological phenomena underlying the observed behavior. Unlike the simpler model, the GEM could be interrogated for the possible effects of various external factors such as oxygen supply, on the expected outcomes. In silico predictions of oxygen effects were consistent with literature observations for DHA producing thraustochytrids.     

Bayesian flux balance analysis applied to a skeletal muscle metabolic model

In this article, the steady state condition for the multi-compartment models for cellular metabolism is considered. The problem is to estimate the reaction and transport fluxes, as well as the concentrations in venous blood when the stoichiometry and bound constraints for the fluxes and the concentrations are given. The problem has been addressed previously by a number of authors, and optimization-based approaches as well as extreme pathway analysis have been proposed. These approaches are briefly discussed here. The main emphasis of this work is a Bayesian statistical approach to the flux balance analysis (FBA). We show how the bound constraints and optimality conditions such as maximizing the oxidative phosphorylation flux can be incorporated into the model in the Bayesian framework by proper construction of the prior densities. We propose an effective Markov chain Monte Carlo (MCMC) scheme to explore the posterior densities, and compare the results with those obtained via the previously studied linear programming (LP) approach. The proposed methodology, which is applied here to a two-compartment model for skeletal muscle metabolism, can be extended to more complex models.     

Applying heat and mass balance theory to determine the flow rate for the measurement of benthic material flux in a flow-through system

Movement of water overlying the sediments has not been taken into consideration in most of the experiments conducted to estimate the dissolved material flux from the sediment. Even in recent experiments that incorporated the stirring motion, interpretation of the data is difficult, because the mixing rate used may be different from actual mixing rate in the field. We propose a method to estimate the in situ mixing rate that should be used to set the flow rate in a flow-through core incubation system. The flow rate is calculated from the vertical mixing rate of the water that is deduced from the heat diffusivity. Release rates of NO₃+NO₂–N obtained from our flow-through incubation system were higher by 1–3 orders of magnitude than those from the conventional diffusion calculation method that estimates the flux from the gradient of nutrient concentration across the sediment-water interface. Increase in NO₃+NO₂–N flux is considered to be due to intensification of the nitrification process as a result of an increase in dissolved oxygen (DO) supply with the motion of water. DO supply is also considered to be an important factor controlling macrofaunal abundance and consequently their excretory contributions to the fluxes of dissolved organic nitrogen as well as NH₄–N. From this point of view, we strongly recommend the application of heat and mass balance theory to estimate nitrogen flux using a flow-through experimental system.