Complex metabolic network of glycerol fermentation by Klebsiella pneumoniae and its system identification via biological robustness

The bioconversion of glycerol to 1,3-propanediol (1,3-PD) by Klebsiella pneumoniae (K. pneumoniae) can be characterized by an intricate metabolic network of interactions among biochemical fluxes, metabolic compounds, key enzymes and genetic regulation. Since there are some uncertain factors in the fermentation, especially the transport mechanisms of substances across cell membrane, the metabolic network contains multiple possible metabolic systems. In this paper, we establish a complex metabolic network and the corresponding nonlinear hybrid dynamical system aiming to determine the most possible metabolic system. The existence, uniqueness and continuity of solutions are discussed. We quantitatively describe biological robustness and present a system identification model on the basis of robustness performance. The identification problem is decomposed into two subproblems and a procedure is constructed to solve them. Numerical results show that it is most possible that both glycerol and 1,3-PD pass the cell membrane by active transport coupling with passive diffusion under substrate-sufficient conditions, whereas, under substrate-limited conditions, glycerol passes cell membrane by active transport coupling with passive diffusion and 1,3-PD by active transport.     

Pathway identification using parallel optimization for a complex metabolic system in microbial continuous culture

The bio-dissimilation of glycerol to 1,3-propanediol (1,3-PD) by Klebsiella pneumoniae (K. pneumoniae) is a complex bioprocess due to the multiple inhibitions of substrate and products onto the cell growth. In consideration of the fact that both the inhibition mechanisms of 3-hydroxypropionaldehyde (3-HPA) onto the cell growth and the transport systems of glycerol and 1,3-PD across the cell membrane are still unclear, we consider 72 possible metabolic pathways, and establish a novel mathematical model which is represented by an eight-dimensional nonlinear dynamical system. The existence, uniqueness, continuous dependence of solutions to the system and the compactness of the solution set are explored. On the basis of biological robustness, we give a quantitative definition of robustness index of the intracellular substances. Taking the robustness index of the intracellular substances together with the relative error between the experimental data and the computational values of the extracellular substances as a performance index, a parameter identification model is proposed for the nonlinear dynamical system, in which 43848 continuous variables and 1152 discrete variables are involved. A parallel particle swarm optimization — pathways identification algorithm (PPSO-PIA) is constructed to find the optimal pathway and parameters under various experiments conditions. Numerical results show that the optimal pathway and the corresponding dynamical system can describe the continuous fermentation reasonably.     

Parameter identification and optimization algorithm in microbial continuous culture

In this study, a novelty mathematical model is established to formulate the continuous culture of glycerol to 1,3-Propanediol (1,3-PD) by Klebsiella pneumoniae, in which the inhibition of 3-hydroxypropionaldehyde (3-HPA) to cells growth and activity of some enzymes (such as glycerol dehydratase (GDHt) and 1,3-PD oxidoreductase (PDOR)), and the passive diffusion and active transport of glycerol and 1,3-PD across cell membrane are all taken into consideration. Taking the mean relative error between the experimental data and calculated values as the performance index, a parameter identification model involving multiple nonlinear dynamic systems is presented. The identifiability of the parameter identification model is also proved. Finally, an improved particle swarm optimization (PSO) algorithm is constructed to find the optimal parameters for the systems under substrate limitation and excess conditions, respectively. Numerical results not only show that the established model can be used to describe the continuous fermentation reasonably, but also the improved PSO algorithm is valid.     

Optimal switching control of a fed-batch fermentation process

Considering the hybrid nature in fed-batch culture of glycerol biconversion to 1,3-propanediol (1,3-PD) by Klebsiella pneumoniae, we propose a state-based switching dynamical system to describe the fermentation process. To maximize the concentration of 1,3-PD at the terminal time, an optimal switching control model subject to our proposed switching system and constraints of continuous state inequality and control function is presented. Because the number of the switchings is not known a priori, we reformulate the above optimal control problem as a two-level optimization problem. An optimization algorithm is developed to seek the optimal solution on the basis of a heuristic approach and control parametrization technique. Numerical results show that, by employing the obtained optimal control strategy, 1,3-PD concentration at the terminal time can be increased considerably.     

Robust parameter identification using parallel global optimization for a batch nonlinear enzyme-catalytic time-delayed process presenting metabolic discontinuities

In this paper, a nonlinear enzyme-catalytic time-delayed switched dynamical system is considered to describe batch culture of glycerol bioconversion to 1,3-propanediol induced by Klebsiella pneumoniae. This system can not only predict the exponential growth phase but also the lag and the stationary growth phases of batch culture since it contains two switching times for representing the starting moment of lag growth phase and the time when the cell specified growth rate reaches the maximum. The biological robustness is expressed in terms of the expectation and variance of the relative deviation. Our aim is to identify the switching times. To this end, a robust parameter identification problem is formulated, where the switching times are decision variables to be chosen such that the biological robustness measure is optimized. This problem, which is governed by the nonlinear system, is subject to a quality constraint and continuous state inequality constraints. Using a hybrid time-scaling transformation to parameterize the switching times into new parameters, an equivalently robust parameter identification problem is investigated. The continuous state inequality constraints are approximated by a conventional inequality constraint, yielding a sequence of approximate robust parameter identification subproblems. The convergence analysis of this approximation is also investigated. Owing to the highly complex nature of these subproblems, a parallel algorithm, based on simulated annealing, is proposed to solve these subproblems. From an extensive simulation study, it is observed that the obtained optimal switching times are satisfactory.     

Optimal switching control for microbial fed-batch culture

In fed-batch culture of glycerol bio-dissimilation to 1,3-propanediol (1,3-PD), the aim of adding glycerol is to obtain as much 1,3-PD as possible. Hence, a proper feed strategy is required during the process. In this paper, we present an optimal switching control model based on our proposed controlled switching system. Some properties of the controlled switching system are obtained. Subsequently, we prove the existence of optimal control. In order to deduce the optimality conditions, we transcribe the optimal switching control model into an equivalent one with fixed switching instants and parameters. Finally, the optimality conditions of the equivalent problem are investigated by calculus of variations.     

Sensitivity analysis and identification of kinetic parameters in batch fermentation of glycerol

A nonlinear dynamical system was established in our preceding work to describe the batch and continuous bioconversions of glycerol to 1,3-propanediol by Klebsiella pneumoniae. The purpose of this article is to analyze the sensitivity of kinetic parameters of the dynamical system and identify their values from experiment. A global sensitivity analysis approach is constructed by combining the local technique with the Monte Carlo method. With only those parameters of higher sensitivity as design variables, we propose a parameter identification model and solve it by a gradient-based simulated annealing algorithm. Numerical results show that our methods are feasible and efficient.     

Optimal control for nonlinear dynamical system of microbial fed-batch culture

In fed-batch culture of glycerol bio-dissimilation to 1, 3-propanediol (1, 3-PD), the aim of adding glycerol is to obtain as much 1, 3-PD as possible. So a proper feeding rate is required during the process. Taking the concentration of 1, 3-PD at the terminal time as the performance index and the feeding rate of glycerol as the control function, we propose an optimal control model subject to a nonlinear dynamical system and constraints of continuous state and non-stationary control. A computational approach is constructed to seek the solution of the above model in two aspects. On the one hand we transcribe the optimal control model into an unconstrained one based on the penalty functions and an extension of the state space; on the other hand, by approximating the control function with simple functions, we transform the unconstrained optimal control problem into a sequence of nonlinear programming problems, which can be solved using gradient-based optimization techniques. The convergence analysis of this approximation is also investigated. Numerical results show that, by employing the optimal control policy, the concentration of 1, 3-PD at the terminal time can be increased considerably.     

Stability of nonlinear microbial bioconversion system concerning glycerol’s active transport and 1,3-PD’s passive transport

In this study, a new nonlinear mathematical model is developed to describe the conversion of glycerol to 1,3- propanediol by Klebsiella pneumoniae and the changes of concentrations of intracellular substances are taken into consideration. We consider the cases that glycerol passes the membrane by active transport and 1,3-propanediol passes the membrane by passive transport. We firstly study the solution’s existence and uniqueness and prove that the solution is bounded in a positive quadrant. Secondly, we discuss the existence of equilibrium solutions and calculate them, dependent on some given steady accuracy by Newton’s Method. Lastly, we analyze the asymptotical behavior and the global stability of the mathematical model.     

An improved model for multistage simulation of glycerol fermentation in batch culture and its parameter identification

The bioconversion of glycerol to 1,3-propanediol is a complex bioprocess. In this study, we aim to explore a novel model for describing the multistage cell growth in batch culture. This is achieved by a modification at the specific rate of cell growth in consideration of its time-dependent changes. The existence, uniqueness and boundedness of solutions to the system and the continuity of solutions with respect to the parameters are discussed. In addition, a parameter identification problem of the system is developed and a feasible optimization algorithm is constructed to solve it. Numerical result shows that the improved model could describe the batch culture well.     

Bilevel parameters identification for the multi-stage nonlinear impulsive system in microorganisms fed-batch cultures

Considering the abrupt jump of the substrate and different characters of the bacterial in the lag phase, the exponential phase and the stationary phase this paper proposes the multi-stage nonlinear impulsive system for the fed-batch fermentation from glycerol to 1,3-propanediol (1,3-PD) and establishes the bilevel identification system for its sensitive parameters. The properties of the solutions for the nonlinear multi-stage dynamical system are investigated and identifiability of the parameters is proved. Finally an optimal algorithm is constructed to obtain the optimal solution of the identification model and the numerical example is then discussed to illustrate the algorithm.