Dynamic Flux Balance Analysis Models Research Articles

Nonlinear programming reformulation of dynamic flux balance analysis models

Dynamic flux balance analysis (dFBA) models are very popular in systems biology because of the many possible applications. dFBA consists of dynamic mass balance equations that describe the concentration of external metabolites and an optimization model to compute the internal flux distribution of the cell. Here, we applied a nonlinear programming formulation for solving dFBA models. The ODE system is discretized using the orthogonal collocation technique and the optimization problem is replaced by its first-order optimality conditions. In addition, an adaptive mesh scheme is applied to improve performance by efficiently handling changes in the active set of constraints. This formulation ensures total differentiability of the model, which can be solved by large-scale solvers and take advantage of automatic differentiation packages. The new approach presents results equivalent to state-of-the-art methods and outperforms them in dynamic optimization problems by avoiding calling an external solver to handle the embedded optimization model.

Fast uncertainty quantification for dynamic flux balance analysis using non-smooth polynomial chaos expansions.

We present a novel surrogate modeling method that can be used to accelerate the solution of uncertainty quantification (UQ) problems arising in nonlinear and non-smooth models of biological systems. In particular, we focus on dynamic flux balance analysis (DFBA) models that couple intracellular fluxes, found from the solution of a constrained metabolic network model of the cellular metabolism, to the time-varying nature of the extracellular substrate and product concentrations. DFBA models are generally computationally expensive and present unique challenges to UQ, as they entail dynamic simulations with discrete events that correspond to switches in the active set of the solution of the constrained intracellular model. The proposed non-smooth polynomial chaos expansion (nsPCE) method is an extension of traditional PCE that can effectively capture singularities in the DFBA model response due to the occurrence of these discrete events. The key idea in nsPCE is to use a model of the singularity time to partition the parameter space into two elements on which the model response behaves smoothly. Separate PCE models are then fit in both elements using a basis-adaptive sparse regression approach that is known to scale well with respect to the number of uncertain parameters. We demonstrate the effectiveness of nsPCE on a DFBA model of an E. coli monoculture that consists of 1075 reactions and 761 metabolites. We first illustrate how traditional PCE is unable to handle problems of this level of complexity. We demonstrate that over 800-fold savings in computational cost of uncertainty propagation and Bayesian estimation of parameters in the substrate uptake kinetics can be achieved by using the nsPCE surrogates in place of the full DFBA model simulations. We then investigate the scalability of the nsPCE method by utilizing it for global sensitivity analysis and maximum a posteriori estimation in a synthetic metabolic network problem with a larger number of parameters related to both intracellular and extracellular quantities.

Simulation and optimization of dynamic flux balance analysis models using an interior point method reformulation

This work presents a novel, differentiable, way of solving dynamic Flux Balance Analysis (dFBA) problems by embedding flux balance analysis of metabolic network models within lumped bulk kinetics for biochemical processes. The proposed methodology utilizes transformation of the bounds of the embedded linear programming problem of flux balance analysis via a logarithmic barrier (interior point) approach. By exploiting the first-order optimality conditions of the interior-point problem, and with further transformations, the approach results in a system of implicit ordinary differential equations. Results from four case studies, show that the CPU and wall-times obtained using the proposed method are competitive with existing state-of-the art approaches for solving dFBA simulations, for problem sizes up to genome-scale. The differentiability of the proposed approach allows, using existing commercial packages, its application to the optimal control of dFBA problems at a genome-scale size, thus outperforming existing formulations as shown by two dynamic optimization case studies.

Dynamic flux balance analysis with nonlinear objective function.

Dynamic flux balance analysis (DFBA) extends flux balance analysis and enables the combined simulation of both intracellular and extracellular environments of microbial cultivation processes. A DFBA model contains two coupled parts, a dynamic part at the upper level (extracellular environment) and an optimization part at the lower level (intracellular environment). Both parts are coupled through substrate uptake and product secretion rates. This work proposes a Karush-Kuhn-Tucker condition based solution approach for DFBA models, which have a nonlinear objective function in the lower-level part. To solve this class of DFBA models an extreme-ray-based reformulation is proposed to ensure certain regularity of the lower-level optimization problem. The method is introduced by utilizing two simple example networks and then applied to a realistic model of central carbon metabolism of wild-type Corynebacterium glutamicum.

Xylose-induced dynamic effects on metabolism and gene expression in engineered Saccharomyces cerevisiae in anaerobic glucose-xylose cultures.

Xylose is present with glucose in lignocellulosic streams available for valorisation to biochemicals. Saccharomyces cerevisiae has excellent characteristics as a host for the bioconversion, except that it strongly prefers glucose to xylose, and the co-consumption remains a challenge. Further, since xylose is not a natural substrate of S. cerevisiae, the regulatory response it induces in an engineered strain cannot be expected to have evolved for its utilisation. Xylose-induced effects on metabolism and gene expression during anaerobic growth of an engineered strain of S. cerevisiae on medium containing both glucose and xylose medium were quantified. The gene expression of S. cerevisiae with an XR-XDH pathway for xylose utilisation was analysed throughout the cultivation: at early cultivation times when mainly glucose was metabolised, at times when xylose was co-consumed in the presence of low glucose concentrations, and when glucose had been depleted and only xylose was being consumed. Cultivations on glucose as a sole carbon source were used as a control. Genome-scale dynamic flux balance analysis models were simulated to analyse the metabolic dynamics of S. cerevisiae. The simulations quantitatively estimated xylose-dependent flux dynamics and challenged the utilisation of the metabolic network. A relative increase in xylose utilisation was predicted to induce the bi-directionality of glycolytic flux and a redox challenge even at low glucose concentrations. Remarkably, xylose was observed to specifically delay the glucose-dependent repression of particular genes in mixed glucose-xylose cultures compared to glucose cultures. The delay occurred at a cultivation time when the metabolic flux activities were similar in the both cultures.

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.

Hybrid neural modeling framework for simulation and optimization of diauxie-involved fed-batch fermentative succinate production

In this theoretical study, a framework for simulation and optimization of diauxie-involved fed-batch fermentative succinate production is proposed. In the proposed method, for the first time, diauxie is employed as a considerable phenomenon to decrease the level of undesired byproduct during the fermentative process. For this aim, an engineered mutant strain of Escherichia coli K-12, termed AB3, is considered. A hybrid-neural modeling framework is constructed based on dynamic flux balance analysis (DFBA) in which aerobic-condition-associated diauxie phenomenon (consumption of produced acetate) is included. The DFBA model is built using an in silico genome-scale metabolic network reconstruction with deletions of pfl, adhE, and ldhA genes. The intracellular description of carbon metabolism is simulated by a multilayer-perceptron (MLP) network to prepare the hybrid-neural model. First, model parameters are calculated by dynamic optimization. Then, a two-step bioprocess is considered in which a fed-batch aerobic step is followed by an anaerobic one with time-varying exponential feeding profiles. Optimal operating policies are determined using differential evolution method. The obtained strategy significantly increases the production of succinate compared to acetate (the major undesired byproduct). The proposed method can be used to study the potential of mutant strains for producing valuable metabolites in time-varying scenarios. In addition, it is shown that diauxie plays a key role in bioprocesses that can be used for preliminary purifications.

A dynamic, genome-scale flux model of Lactococcus lactis to increase specific recombinant protein expression

Recently, lactic acid bacteria (LAB) have attracted a great deal of interest because of their potential to serve as oral delivery vehicles for recombinant protein vaccines. An important limitation to their use is the typically low level of heterologous expression obtained in LAB. To address this, a dynamic flux balance analysis (DFBA) model was used to identify gene targets for increasing specific expression of Green Fluorescent Protein (GFP), a model heterologous protein, in Lactococcus lactis IL1403. Two strains, each targeting one of the top model-identified genes, were constructed and tested in vivo. Data show that both strains, by a conservative estimate, achieved 15% higher GFP per cell than the control strain, a qualitative confirmation of the model predictions. A genome-scale DFBA model for L. lactis growing on M17 medium is presented along with the procedure for screening gene targets and a powerful method for visualizing fluxes in genome-scale metabolic networks.

DFBA-LQR: An Optimal Control Approach to Flux Balance Analysis

Systems biology is becoming an emerging field, as systems science is demonstrated as an essential tool in untangling biocomplexity. While various steady-state-based methods have been developed for study of metabolic networks, the study of the network dynamics is a less explored area, as it is inherently related to the regulatory network. The difficulty of regulatory network inference opens up the possibility of employing control techniques and methods for analysis of metabolic networks. In this work, the potential for application of optimal control theory to metabolic control problems is investigated. Considering that the regulatory network in essence is a cascade control structure, the proposed modeling approach relies on the postulate that evolution is a genetic optimization scheme leading to “optimal” regulatory networks. Thus the approach utilizes a modified linear quadratic regulator formulation to evaluate the optimal regulatory network for a given flux model. Utilizing a homeostasis-type objective, the proposed dynamic flux balance analysis with linear quadratic regulator (DFBA-LQR) approach enables constructing dynamic flux balance analysis models and produces analytical descriptions of the responses of the metabolic fluxes to disturbances. The analytical description can also be used to calculate the flux control coefficients and linearized reaction kinetics for the system. The application potential of the approach is demonstrated by a sample problem of modeling of lipid accumulation in rat hepatocytes cultures.

Dynamic analysis of optimality in myocardial energy metabolism under normal and ischemic conditions

To better understand the dynamic regulation of optimality in metabolic networks under perturbed conditions, we reconstruct the energetic-metabolic network in mammalian myocardia using dynamic flux balance analysis (DFBA). Additionally, we modified the optimal objective from the maximization of ATP production to the minimal fluctuation of the profile of metabolite concentration under ischemic conditions, extending the hypothesis of original minimization of metabolic adjustment to create a composite modeling approach called M-DFBA. The simulation results are more consistent with experimental data than are those of the DFBA model, particularly the retentive predominant contribution of fatty acid to oxidative ATP synthesis, the exact mechanism of which has not been elucidated and seems to be unpredictable by the DFBA model. These results suggest that the systemic states of metabolic networks do not always remain optimal, but may become suboptimal when a transient perturbation occurs. This finding supports the relevance of our hypothesis and could contribute to the further exploration of the underlying mechanism of dynamic regulation in metabolic networks.