Philippe Nimmegeers

Dynamic optimization of biological networks under parametric uncertainty

BackgroundMicro-organisms play an important role in various industrial sectors (including biochemical, food and pharmaceutical industries). A profound insight in the biochemical reactions inside micro-organisms enables an improved biochemical process control. Biological networks are an important tool in systems biology for incorporating microscopic level knowledge. Biochemical processes are typically dynamic and the cells have often more than one objective which are typically conflicting, e.g., minimizing the energy consumption while maximizing the production of a specific metabolite. Therefore multi-objective optimization is needed to compute trade-offs between those conflicting objectives. In model-based optimization, one of the inherent problems is the presence of uncertainty. In biological processes, this uncertainty can be present due to, e.g., inherent biological variability. Not taking this uncertainty into account, possibly leads to the violation of constraints and erroneous estimates of the actual objective function(s). To account for the variance in model predictions and compute a prediction interval, this uncertainty should be taken into account during process optimization. This leads to a challenging optimization problem under uncertainty, which requires a robustified solution.ResultsThree techniques for uncertainty propagation: linearization, sigma points and polynomial chaos expansion, are compared for the dynamic optimization of biological networks under parametric uncertainty. These approaches are compared in two case studies: (i) a three-step linear pathway model in which the accumulation of intermediate metabolites has to be minimized and (ii) a glycolysis inspired network model in which a multi-objective optimization problem is considered, being the minimization of the enzymatic cost and the minimization of the end time before reaching a minimum extracellular metabolite concentration. A Monte Carlo simulation procedure has been applied for the assessment of the constraint violations. For the multi-objective case study one Pareto point has been considered for the assessment of the constraint violations. However, this analysis can be performed for any Pareto point.ConclusionsThe different uncertainty propagation strategies each offer a robustified solution under parametric uncertainty. When making the trade-off between computation time and the robustness of the obtained profiles, the sigma points and polynomial chaos expansion strategies score better in reducing the percentage of constraint violations. This has been investigated for a normal and a uniform parametric uncertainty distribution. The polynomial chaos expansion approach allows to directly take prior knowledge of the parametric uncertainty distribution into account.

A process simulator interface for multiobjective optimization of chemical processes

Abstract The (bio)chemical process industry is under an increasing pressure due to smaller margins and increasing societal and legislative demands for a sustainable future. In this context model-based optimization contributes to the solution because it serves to improve the processes’ performance. Furthermore, multiobjective optimization techniques provide the decision maker with a deeper insight in the tradeoffs when choosing an operating condition. However, an accurate process model is needed to apply these techniques efficiently. In this paper, a novel interface is developed between state-of-the-art gradient-based optimization techniques and the widely used process simulator Aspen Plus. Furthermore, specific challenges and solutions for overcoming the gap between process simulators and optimization tools are highlighted. The resulting interface allows gradient-based techniques to be exploited for optimization of complex industrial processes modeled in the advanced Aspen Plus environment. The interface ensures constraints satisfaction, and a higher computational performance than gradient free methods.

Parametric uncertainty propagation for robust dynamic optimization of biological networks

An important trend in biochemical industry is the search for a more sustainable operation. This requires an improved design and operation of processes with respect to economic, environmental and safety aspects. For an accurate macroscopic process control, multi-scale models based on microscopic systems biology insights are increasingly used. However, model uncertainty as well as external disturbances are inevitably present. Hence, there is a need for approaches which can guarantee process safety despite this presence of uncertainty in the model parameters. Therefore, techniques for the propagation of the parametric uncertainties to the states are vital for a robust process control. In this work, the potential of the polynomial chaos expansion approach is investigated for optimal control of a four step linear pathway with Michaelis-Menten kinetics by manipulating the enzyme expression rates.

Simulation of Escherichia coli Dynamics in Biofilms and Submerged Colonies with an Individual-Based Model Including Metabolic Network Information

Clustered microbial communities are omnipresent in the food industry, e.g., as colonies of microbial pathogens in/on food media or as biofilms on food processing surfaces. These clustered communities are often characterized by metabolic differentiation among their constituting cells as a result of heterogeneous environmental conditions in the cellular surroundings. This paper focuses on the role of metabolic differentiation due to oxygen gradients in the development of Escherichia coli cell communities, whereby low local oxygen concentrations lead to cellular secretion of weak acid products. For this reason, a metabolic model has been developed for the facultative anaerobe E. coli covering the range of aerobic, microaerobic, and anaerobic environmental conditions. This metabolic model is expressed as a multiparametric programming problem, in which the influence of low extracellular pH values and the presence of undissociated acid cell products in the environment has been taken into account. Furthermore, the developed metabolic model is incorporated in MICRODIMS, an in-house developed individual-based modeling framework to simulate microbial colony and biofilm dynamics. Two case studies have been elaborated using the MICRODIMS simulator: (i) biofilm growth on a substratum surface and (ii) submerged colony growth in a semi-solid mixed food product. In the first case study, the acidification of the biofilm environment and the emergence of typical biofilm morphologies have been observed, such as the mushroom-shaped structure of mature biofilms and the formation of cellular chains at the exterior surface of the biofilm. The simulations show that these morphological phenomena are respectively dependent on the initial affinity of pioneer cells for the substratum surface and the cell detachment process at the outer surface of the biofilm. In the second case study, a no-growth zone emerges in the colony center due to a local decline of the environmental pH. As a result, cellular growth in the submerged colony is limited to the colony periphery, implying a linear increase of the colony radius over time. MICRODIMS has been successfully used to reproduce complex dynamics of clustered microbial communities.

A novel algorithm for fast representation of a Pareto front with adaptive resolution: Application to multi-objective optimization of a chemical reactor

Abstract Solving a multi-objective optimization problem yields an infinite set of points in which no objective can be improved without worsening at least another objective. This set is called the Pareto front. A Pareto front with adaptive resolution is a representation where the number of points at any segment of the Pareto front is directly proportional to the curvature of this segment. Such representations are attractive since steep segments, i.e., knees, are more significant to the decision maker as they have high trade-off level compared to the more flat segments of the solution curve. A simple way to obtain such representation is the a posteriori analysis of a dense Pareto front by a smart filter to keep only the points with significant trade-offs among them. However, this method suffers from the production of a large overhead of insignificant points as well as the absence of a clear criterion for determining the required density of the initial dense representation of the Pareto front. This papers contribution is a novel algorithm for obtaining a Pareto front with adaptive resolution. The algorithm overcomes the pitfalls of the smart filter strategy by obtaining the Pareto points recursively while calculating the trade-off level between the obtained points before moving to a deeper recursive call. By using this approach, once a segment of trade-offs insignificant to the decision makers needs is identified, the algorithm stops exploring it further. The improved speed of the proposed algorithm along with its intuitively simple solution process make it a more attractive route to solve multi-objective optimization problems in a way that better suits the decision makers needs.

A low-complexity metabolic network model for the respiratory and fermentative metabolism of Escherichia coli

Over the last decades, predictive microbiology has made significant advances in the mathematical description of microbial spoiler and pathogen dynamics in or on food products. Recently, the focus of predictive microbiology has shifted from a (semi-)empirical population-level approach towards mechanistic models including information about the intracellular metabolism in order to increase model accuracy and genericness. However, incorporation of this subpopulation-level information increases model complexity and, consequently, the required run time to simulate microbial cell and population dynamics. In this paper, results of metabolic flux balance analyses (FBA) with a genome-scale model are used to calibrate a low-complexity linear model describing the microbial growth and metabolite secretion rates of Escherichia coli as a function of the nutrient and oxygen uptake rate. Hence, the required information about the cellular metabolism (i.e., biomass growth and secretion of cell products) is selected and included in the linear model without incorporating the complete intracellular reaction network. However, the applied FBAs are only representative for microbial dynamics under specific extracellular conditions, viz., a neutral medium without weak acids at a temperature of 37℃. Deviations from these reference conditions lead to metabolic shifts and adjustments of the cellular nutrient uptake or maintenance requirements. This metabolic dependency on extracellular conditions has been taken into account in our low-complex metabolic model. In this way, a novel approach is developed to take the synergistic effects of temperature, pH, and undissociated acids on the cell metabolism into account. Consequently, the developed model is deployable as a tool to describe, predict and control E. coli dynamics in and on food products under various combinations of environmental conditions. To emphasize this point,three specific scenarios are elaborated: (i) aerobic respiration without production of weak acid extracellular metabolites, (ii) anaerobic fermentation with secretion of mixed acid fermentation products into the food environment, and (iii) respiro-fermentative metabolic regimes in between the behaviors at aerobic and anaerobic conditions.

A tutorial on uncertainty propagation techniques for predictive microbiology models: A critical analysis of state-of-the-art techniques

Building mathematical models in predictive microbiology is a data driven science. As such, the experimental data (and its uncertainty) has an influence on the final predictions and even on the calculation of the model prediction uncertainty. Therefore, the current research studies the influence of both the parameter estimation and uncertainty propagation method on the calculation of the model prediction uncertainty. The study is intended as well as a tutorial to uncertainty propagation techniques for researchers in (predictive) microbiology. To this end, an in silico case study was applied in which the effect of temperature on the microbial growth rate was modelled and used to make simulations for a temperature profile that is characterised by variability. The comparison of the parameter estimation methods demonstrated that the one-step method yields more accurate and precise calculations of the model prediction uncertainty than the two-step method. Four uncertainty propagation methods were assessed. The current work assesses the applicability of these techniques by considering the effect of experimental uncertainty and model input uncertainty. The linear approximation was demonstrated not always to provide reliable results. The Monte Carlo method was computationally very intensive, compared to its competitors. Polynomial chaos expansion was computationally efficient and accurate but is relatively complex to implement. Finally, the sigma point method was preferred as it is (i) computationally efficient, (ii) robust with respect to experimental uncertainty and (iii) easily implemented.

Comparing design of experiments and optimal experimental design techniques for modelling the microbial growth rate under static environmental conditions

Building secondary models that describe the growth rate as a function of multiple environmental conditions is often very labour intensive and costly. As such, the current research aims to assist in decreasing the required experimental effort by studying the efficacy of both design of experiments (DOE) and optimal experimental designs (OED) techniques. This is the first research in predictive microbiology (i) to make a comparison of these techniques based on the (relative) model prediction uncertainty of the obtained models and (ii) to compare OED criteria for the design of experiments with static (instead of dynamic) environmental conditions. A comparison of the DOE techniques demonstrated that the inscribed central composite design and full factorial design were most suitable. Five conventional and two tailor made OED criteria were tested. The commonly used D-criterion performed best out of the conventional designs and almost equally well as the best of the dedicated criteria. Moreover, the modelling results of the D-criterion were less dependent on the experimental variability and differences in the microbial response than the two selected DOE techniques. Finally, it was proven that solving the optimisation of the D-criterion can be made more efficient by considering the sensitivities of the growth rate relative to its value as Jacobian matrix instead of the sensitivities of the cell density measurements.

The Silent Cooperator: An Epigenetic Model for Emergence of Altruistic Traits in Biological Systems

Spatial evolutionary game theory explains how cooperative traits can survive the intense competition in biological systems. If the spatial distribution allows cooperators to interact with each other frequently, the benefits of cooperation will outweigh the losses due to exploitation by selfish organisms. However, for a cooperative behavior to get established in a system, it needs to be found initially in a sufficiently large cluster to allow a high frequency of intracooperator interactions. Since mutations are rare events, this poses the question of how cooperation can arise in a biological system in the first place. We present a simple model which captures two concepts from genetics that can explain how evolution overcomes the emergence problem. The first concept is, often in nature, a gene may not express its phenotype except under specific environmental conditions, rendering it to be a “silent” gene. The second key idea is that a neutral gene, one that does not harm or improve an organism’s survival chances, can still spread through a population if it is physically near to another gene that is positively selected. Through these two ideas, our model offers a possible solution to the fundamental problem of emergence of cooperation in biological systems.

Investigating practical aspects of the exergy based multi-objective optimization of chemical processes

Abstract When optimizing chemical processes, conflicting objectives are often present (Logist et al, 2011). In industry, the trend towards sustainable development highlights this challenge because it encompasses three pillars: economy, society, and environment. A common example is the search for higher profitability while improving the safety of operation and reducing the energy consumption. In this framework, the exergy concept is highly relevant since it is understood as the confluence of energy, environment, and sustainable development (Rosen and Dincer, 2001). This contribution presents a novel approach to exploit the synergies of the exergy analysis and the gradient-based multi-objective optimization (MOO) of processes. An interface with Aspen Plus has been developed to exploit the added value of this approach, aiming to minimize the lost work in the context of conflicting objectives. The interface links the process simulator to Matlab, which is used as platform to run CasADi, a symbolic framework for algorithmic differentiation and numerical optimization (Andersson, 2013). The traditional butyl acetate production process is chosen as case study. It represents a common set-up of the chemical industry, where the conditions in the separation section are critical regarding quality but also yield and process intensity. Tri-objective constrained problems were formulated to address sustainability goals. The traditional approach based on energy consumption is compared to the case considering lost work minimization. The developed approach reveals how the minimum energy and lost work result from the partial increase in the duty for one of the distillation columns. This reflects the enhanced process understanding obtained, for which the exergy based optimization proves to be more informative, with slightly better results, and highly favorable trade-offs between objectives.