Kai Höffner

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.

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

BackgroundDynamic 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.ResultsDFBAlab, 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.ConclusionsLexicographic 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.

Spatiotemporal modeling of microbial metabolism

BackgroundMicrobial systems in which the extracellular environment varies both spatially and temporally are very common in nature and in engineering applications. While the use of genome-scale metabolic reconstructions for steady-state flux balance analysis (FBA) and extensions for dynamic FBA are common, the development of spatiotemporal metabolic models has received little attention.ResultsWe present a general methodology for spatiotemporal metabolic modeling based on combining genome-scale reconstructions with fundamental transport equations that govern the relevant convective and/or diffusional processes in time and spatially varying environments. Our solution procedure involves spatial discretization of the partial differential equation model followed by numerical integration of the resulting system of ordinary differential equations with embedded linear programs using DFBAlab, a MATLAB code that performs reliable and efficient dynamic FBA simulations. We demonstrate our methodology by solving spatiotemporal metabolic models for two systems of considerable practical interest: (1) a bubble column reactor with the syngas fermenting bacterium Clostridium ljungdahlii; and (2) a chronic wound biofilm with the human pathogen Pseudomonas aeruginosa. Despite the complexity of the discretized models which consist of 900 ODEs/600 LPs and 250 ODEs/250 LPs, respectively, we show that the proposed computational framework allows efficient and robust model solution.ConclusionsOur study establishes a new paradigm for formulating and solving genome-scale metabolic models with both time and spatial variations and has wide applicability to natural and engineered microbial systems.

Metabolic modeling of synthesis gas fermentation in bubble column reactors

BackgroundA promising route to renewable liquid fuels and chemicals is the fermentation of synthesis gas (syngas) streams to synthesize desired products such as ethanol and 2,3-butanediol. While commercial development of syngas fermentation technology is underway, an unmet need is the development of integrated metabolic and transport models for industrially relevant syngas bubble column reactors.ResultsWe developed and evaluated a spatiotemporal metabolic model for bubble column reactors with the syngas fermenting bacterium Clostridium ljungdahlii as the microbial catalyst. Our modeling approach involved combining a genome-scale reconstruction of C. ljungdahlii metabolism with multiphase transport equations that govern convective and dispersive processes within the spatially varying column. The reactor model was spatially discretized to yield a large set of ordinary differential equations (ODEs) in time with embedded linear programs (LPs) and solved using the MATLAB based code DFBAlab. Simulations were performed to analyze the effects of important process and cellular parameters on key measures of reactor performance including ethanol titer, ethanol-to-acetate ratio, and CO and H2 conversions.ConclusionsOur computational study demonstrated that mathematical modeling provides a complementary tool to experimentation for understanding, predicting, and optimizing syngas fermentation reactors. These model predictions could guide future cellular and process engineering efforts aimed at alleviating bottlenecks to biochemical production in syngas bubble column reactors.

From sugars to biodiesel using microalgae and yeast

The economic production of algal biofuels requires novel strategies, such as microbial consortia and synthetic ecologies, to boost the productivity of open pond systems. These strategies have not been fully explored partly due to the lack of reliable and predictive process models. This study uses genome-based metabolic networks to build a process model of a raceway pond. This process model is used as a discovery tool for novel process strategies. First, an algal monoculture with flue gas sparging is modeled. Then, an oleaginous yeast monoculture is modeled. The yeast monoculture is O2 limited and the presence of algae in the culture would result in better resource utilization. Next, an algal/fungal raceway pond with a feed of cellulosic glucose is explored. Finally, an oleaginous yeast that can consume a glucose/xylose mix, resulting from the hydrolysis of lignocellulosic waste, is modeled. This model predicts biomass and lipids productivities comparable to those reported in the literature. Assuming 50% yield loss due to contamination and invasion, a simple economic analysis shows that an algae/yeast coculture can produce biodiesel at competitive prices,

Efficient solution of ordinary differential equations with a parametric lexicographic linear program embedded

2.01 per liter for pure glucose and

Generalized derivatives of dynamic systems with a linear program embedded

1.44 per liter for the sugar mix, whereas the algae monoculture can do so only at very short distances from a flue gas source. This modeling framework will enable the use of optimization algorithms in the design of open pond systems in the near future and will allow the exploration of novel strategies in bioprocesses employing microbial communities.

Representation and Control of Brayton—Moser Systems using a Geometric Decomposition

This work analyzes the initial value problem in ordinary differential equations with a parametric lexicographic linear program (LP) embedded. The LP is said to be embedded since the dynamics depend on the solution of the LP, which is in turn parameterized by the dynamic states. This problem formulation finds application in dynamic flux balance analysis, which serves as a modeling framework for industrial fermentation reactions. It is shown that the problem formulation can be intractable numerically, which arises from the fact that the LP induces an effective domain that may not be open. A numerical method is developed which reformulates the system so that it is defined on an open set. The result is a system of semi-explicit index-one differential algebraic equations, which can be solved with efficient and accurate methods. It is shown that this method addresses many of the issues stemming from the original problem’s intractability. The application of the method to examples of industrial fermentation processes demonstrates its effectiveness and efficiency.

Generalized Derivatives of Lexicographic Linear Programs

Dynamic systems with a linear program (LP) embedded can be found in control and optimization of bioreactor models based on dynamic flux balance analysis (DFBA). Derivatives of the dynamic states with respect to a parameter vector are essential for open and closed-loop dynamic optimization and parameter estimation of such systems. These derivatives, given by a forward sensitivity system, may not exist because the optimal value of a linear program as a function of the right-hand side of the constraints is not continuously differentiable. Therefore, nonsmooth analysis must be applied which provides optimality conditions in terms of subgradients, for convex functions, or Clarkes generalized gradient, for nonconvex functions. This work presents an approach to compute the necessary information for nonsmooth optimization, i.e.,?an element of the generalized gradient. Moreover, a numerical implementation of the results is introduced. The approach is illustrated through a large-scale dynamic flux balance analysis example.

Geometric decomposition and potential-based representation of nonlinear systems

Abstract This paper considers the problem of representing a sufficiently smooth control affine system as a structured Brayton–Moser system and to use the obtained structure to stabilize a desired equilibrium of the system. The present note proposes a geometric decomposition technique to express a given vector field as a Brayton–Moser system with desired structure. The proposed method is based on a decomposition of a differential one-form that encodes the divergence of a given vector field into its exact and anti-exact components, by using a homotopy operator, and into its co-exact and anti-coexact components, by introducing a dual homotopy operator. This enables one to compute, via integration, the potential and the structure generating the drift vector field. By identification of the obtained structure and a desired structure, it is therefore possible to study the feedback realization problem of the Brayton–Moser structure and feedback stabilization of control affine systems. Application of the proposed constructive approach to the control of the three-dimensional rigid-body problem is presented to illustrate the propose approach.