An individual-based modeling approach to simulate the effects of cellular nutrient competition on Escherichia coli K-12 MG1655 colony behavior and interactions in aerobic structured food systems.

Traditional kinetic models in predictive microbiology reliably predict macroscopic dynamics of planktonically-growing cell cultures in homogeneous liquid food systems. However, most food products have a semi-solid structure, where microorganisms grow locally in colonies. Individual colony cells exhibit strongly different and non-normally distributed behavior due to local nutrient competition. As a result, traditional models considering average population behavior in a homogeneous system do not describe colony dynamics in full detail. To incorporate local resource competition and individual cell differences, an individual-based modeling approach has been applied to Escherichia coli K-12 MG1655 colonies, considering the microbial cell as modeling unit. The first contribution of this individual-based model is to describe single colony growth under nutrient-deprived conditions. More specifically, the linear and stationary phase in the evolution of the colony radius, the evolution from a disk-like to branching morphology, and the emergence of a starvation zone in the colony center are simulated and compared to available experimental data. These phenomena occur earlier at more severe nutrient depletion conditions, i.e., at lower nutrient diffusivity and initial nutrient concentration in the medium. Furthermore, intercolony interactions have been simulated. Higher inoculum densities lead to stronger intercolony interactions, such as colony merging and smaller colony sizes, due to nutrient competition. This individual-based model contributes to the elucidation of characteristic experimentally observed colony behavior from mechanistic information about cellular physiology and interactions.

Multi-objective optimal control of dynamic bioprocesses using ACADO Toolkit.

The optimal design and operation of dynamic bioprocesses gives in practice often rise to optimisation problems with multiple and conflicting objectives. As a result typically not a single optimal solution but a set of Pareto optimal solutions exist. From this set of Pareto optimal solutions, one has to be chosen by the decision maker. Hence, efficient approaches are required for a fast and accurate generation of the Pareto set such that the decision maker can easily and systematically evaluate optimal alternatives. In the current paper the multi-objective optimisation of several dynamic bioprocess examples is performed using the freely available ACADO Multi-Objective Toolkit ( http://www.acadotoolkit.org ). This toolkit integrates efficient multiple objective scalarisation strategies (e.g., Normal Boundary Intersection and (Enhanced) Normalised Normal Constraint) with fast deterministic approaches for dynamic optimisation (e.g., single and multiple shooting). It has been found that the toolkit is able to efficiently and accurately produce the Pareto sets for all bioprocess examples. The resulting Pareto sets are added as supplementary material to this paper.

Parameter estimations in predictive microbiology: Statistically sound modelling of the microbial growth rate.

When building models to describe the effect of environmental conditions on the microbial growth rate, parameter estimations can be performed either with a one-step method, i.e., directly on the cell density measurements, or in a two-step method, i.e., via the estimated growth rates. The two-step method is often preferred due to its simplicity. The current research demonstrates that the two-step method is, however, only valid if the correct data transformation is applied and a strict experimental protocol is followed for all experiments. Based on a simulation study and a mathematical derivation, it was demonstrated that the logarithm of the growth rate should be used as a variance stabilizing transformation. Moreover, the one-step method leads to a more accurate estimation of the model parameters and a better approximation of the confidence intervals on the estimated parameters. Therefore, the one-step method is preferred and the two-step method should be avoided.

An interaction model for the combined effect of temperature, pH and water activity on the growth rate of E. coli K12.

Previous research has indicated that more complex model structures than the commonly used gamma model are needed to obtain an accurate prediction of the effect of multiple environmental conditions on the microbial growth rate. Due to the complexity associated with the development of such model structures, it is recommended that the model structure is compatible with a modular model building method. In this research, a gamma-interaction model was built to describe the combined effect of temperature, pH and water activity on the microbial growth rate of E. coli K12 based on a dataset of 68 bioreactor experiments. This novel interaction model was compared with the standard gamma model. The model structures were tested separately for the combined effects of (i) temperature and pH, (ii) pH and water activity, (iii) temperature and water activity and (iv) temperature, pH and water activity. Based on the results of this research, it was concluded that models for the combined effect of environmental conditions need to allow for sufficient flexibility for the description of combined effects of environmental conditions to obtain accurate model predictions. In the current study, this flexibility was successfully introduced by using the gamma-interaction model. A cross-validation study also demonstrated that the predictions of the interaction model are more robust with respect to the specific data used than the gamma model. As such, the gamma-interaction model provides food producers and food safety authorities with a more accurate and reliable tool for the prediction of the microbial growth rate as a function of multiple environmental conditions.

Including experimental uncertainty on the independent variables when modelling microbial dynamics: The combined effect of pH and acetic acid on the growth rate of E. coli K12.

Modelling methods applied in predictive microbiology generally neglect the importance of uncertainty on the measurement of the independent variables. The Ordinary Least Squares (OLS) method that is commonly applied in predictive microbiology is only applicable if the experimental error on the inputs of the model are insignificant. However, this does not apply for many types of experimental measurements of the independent variables. Therefore, a parameter estimation method was adapted in this research for the estimation of the parameters of secondary models, taking into account uncertainty on the measurement of the influencing food characteristics. This parameter estimation method was based on the work of Stortelder (1996) and is referred to as the Weighted Total Least Squares method (WTLS). The method is formulised as an extension of the commonly used OLS method. Consequently the current WTLS method (i) is easily implemented using similar numerical methods, (ii) reduces to an OLS method when the measurement error on the model inputs is negligible and (iii) enables the evaluation of the accuracy of the model parameter estimates based on the same approximations.

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.

Introducing a novel interaction model structure for the combined effect of temperature and pH on the microbial growth rate.

Efficient modelling of the microbial growth rate can be performed by combining the effects of individual conditions in a multiplicative way, known as the gamma concept. However, several studies have illustrated that interactions between different effects should be taken into account at stressing environmental conditions to achieve a more accurate description of the growth rate. In this research, a novel approach for modeling the interactions between the effects of environmental conditions on the microbial growth rate is introduced. As a case study, the effect of temperature and pH on the growth rate of Escherichia coli K12 is modeled, based on a set of computer controlled bioreactor experiments performed under static environmental conditions. The models compared in this case study are the gamma model, the model of Augustin and Carlier (2000), the model of Le Marc et al. (2002) and the novel multiplicative interaction model, developed in this paper. This novel model enables the separate identification of interactions between the effects of two (or more) environmental conditions. The comparison of these models focuses on the accuracy, interpretability and compatibility with efficient modeling approaches. Moreover, for the separate effects of temperature and pH, new cardinal parameter model structures are proposed. The novel interaction model contributes to a generic modeling approach, resulting in predictive models that are (i) accurate, (ii) easily identifiable with a limited work load, (iii) modular, and (iv) biologically interpretable.

Identifiability of large-scale non-linear dynamic network models applied to the ADM1-case study.

In this work, both the structural and practical identifiability of the Anaerobic Digestion Model no. 1 (ADM1) is investigated, which serves as a relevant case study of large non-linear dynamic network models. The structural identifiability is investigated using the probabilistic algorithm, adapted to deal with the specifics of the case study (i.e., a large-scale non-linear dynamic system of differential and algebraic equations). The practical identifiability is analyzed using a Monte Carlo parameter estimation procedure for a 'non-informative' and 'informative' experiment, which are heuristically designed. The model structure of ADM1 has been modified by replacing parameters by parameter combinations, to provide a generally locally structurally identifiable version of ADM1. This means that in an idealized theoretical situation, the parameters can be estimated accurately. Furthermore, the generally positive structural identifiability results can be explained from the large number of interconnections between the states in the network structure. This interconnectivity, however, is also observed in the parameter estimates, making uncorrelated parameter estimations in practice difficult.

Dynamic optimization of biological networks under parametric uncertainty.

BACKGROUND: Micro-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. RESULTS: Three 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. CONCLUSIONS: The 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.

Dynamic estimation of specific fluxes in metabolic networks using non-linear dynamic optimization.

BACKGROUND: Metabolic network models describing the biochemical reaction network and material fluxes inside microorganisms open interesting routes for the model-based optimization of bioprocesses. Dynamic metabolic flux analysis (dMFA) has lately been studied as an extension of regular metabolic flux analysis (MFA), rendering a dynamic view of the fluxes, also in non-stationary conditions. Recent dMFA implementations suffer from some drawbacks, though. More specifically, the fluxes are not estimated as specific fluxes, which are more biologically relevant. Also, the flux profiles are not smooth, and additional constraints like, e.g., irreversibility constraints on the fluxes, cannot be taken into account. Finally, in all previous methods, a basis for the null space of the stoichiometric matrix, i.e., which set of free fluxes is used, needs to be chosen. This choice is not trivial, and has a large influence on the resulting estimates. RESULTS: In this work, a new methodology based on a B-spline parameterization of the fluxes is presented. Because of the high degree of non-linearity due to this parameterization, an incremental knot insertion strategy has been devised, resulting in a sequence of non-linear dynamic optimization problems. These are solved using state-of-the-art dynamic optimization methods and tools, i.e., orthogonal collocation, an interior-point optimizer and automatic differentiation. Also, a procedure to choose an optimal basis for the null space of the stoichiometric matrix is described, discarding the need to make a choice beforehand. The proposed methodology is validated on two simulated case studies: (i) a small-scale network with 7 fluxes, to illustrate the operation of the algorithm, and (ii) a medium-scale network with 68 fluxes, to show the algorithm's capabilities for a realistic network. The results show an accurate correspondence to the reference fluxes used to simulate the measurements, both in a theoretically ideal setting with no experimental noise, and in a realistic noise setting. CONCLUSIONS: Because, apart from a metabolic reaction network and the measurements, no extra input needs to be given, the resulting algorithm is a systematic, integrated and accurate methodology for dynamic metabolic flux analysis that can be run online in real-time if necessary.