Computational method for inferring objective function of glycerol metabolism in Klebsiella pneumoniae

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1 From the SelectedWorks of Chongyang Liu 2009 Computational method for inferring objective function of glycerol metabolism in Klebsiella pneumoniae Zhaohua Gong, Dalian University of Technology Chongyang Liu Enmin Feng, Dalian University of Technology Qingrui Zhang, Dalian University of Technology Available at:

2 Computational Biology and Chemistry 33 (2009) 1 6 Contents lists available at ScienceDirect Computational Biology and Chemistry journal homepage: Research Article Computational method for inferring objective function of glycerol metabolism in Klebsiella pneumoniae Zhaohua Gong a,b,, Chongyang Liu a,b, Enmin Feng a, Qingrui Zhang c a Department of Applied Mathematics, Dalian University of Technology, Dalian , Liaoning, China b Mathematics and Information Science College, Shandong Institute of Business and Technology, Yantai , Shandong, China c Department of Biotechnology, Dalian University of Technology, Dalian , Liaoning, China article info abstract Article history: Received 18 June 2008 Accepted 22 June 2008 Keywords: Optimization modelling 1,3-Propanediol Metabolism Klebsiella pneumoniae Genetic algorithm Robustness analysis Flux balance analysis (FBA) is an effective tool in the analysis of metabolic network. It can predict the flux distribution of engineered cells, whereas the accurate prediction depends on the reasonable objective function. In this work, we propose two nonlinear bilevel programming models on anaerobic glycerol metabolism in Klebsiella pneumoniae (K. pneumoniae) for 1,3-propanediol (1,3-PD) production. One intends to infer the metabolic objective function, and the other is to analyze the robustness of the objective function. In view of the models characteristic an improved genetic algorithm is constructed to solve them, where some techniques are adopted to guarantee all chromosomes are feasible and move quickly towards the global optimal solution. Numerical results reveal some interesting conclusions, e.g., biomass production is the main force to drive K. pneumoniae metabolism, and the objective functions, which are obtained in term of several different groups of flux distributions, are similar Elsevier Ltd. All rights reserved. 1. Introduction 1,3-Propanediol (1,3-PD) possesses potential applications on a large commercial scale, especially as a monomer of polyesters or polyurethanes, its biosynthesis has attracted worldwide interests (Bibel et al., 1999; Nakamura and Whited, 2003), etc. Among all kinds of microbial production of 1,3-PD, dissimilation of glycerol by Klebsiella pneumoniae (K. pneumoniae) has been widely investigated due to its high productivity since 1980s (Zeng and Biebl, 2002; Menzel et al., 1997). However, compared with the competing chemical process, the microbial production is difficult to obtain a high 1,3-PD concentration in the fermentation broth. Its an area of interest to develop an improved technique to improve the productivity of 1,3-PD. The knowledge of cell physiology and metabolic regulation is helpful to improve productivity of 1,3-PD by a metabolic engineering approach on the strain (Stephanopoulos et al., 1998). Moreover, computational methods for cellular metabolism play an important role in understanding the complex biochemical interactions within the cells. Among those methods, FBA has been proved to be an effectively computational tool for understanding cell phys- Corresponding author at: Department of Applied Mathematics, Dalian University of Technology, Dalian , Liaoning, China. Tel.: ; fax: address: yt gzh@yahoo.com.cn (Z. Gong). iology and regulation of metabolism, and has been successfully applied to different microorganisms (Maczek et al., 2006; Ozkan et al., 2005; Sanchez et al., 2006; Shirai et al., 2005). FBA can predict the metabolic fluxes by using a reduced set of measured fluxes and mass balance equations around intracellular metabolites, and thereby provide a better characterization of cellular phenotypes. However, the accurate prediction depends on a reasonable objective function. Objective functions in practice can take on a linear form, i.e., f = cv, where c denotes the vector defining the coefficients or weights for each flux in v (Beard et al., 2002). The elements of c enable the formulation of a number of diverse objectives. Common objective functions include maximizing biomass or cell growth, maximizing ATP production or maximizing the rate of synthesis of a particular product (Kauffman et al., 2003). Other objective functions include minimizing ATP production in order to determine conditions of optimal metabolic energy efficiency, and minimizing nutrient uptake in order to evaluate the conditions under which a cell will perform its metabolic functions while consuming the minimum amount of nutrients (Lee et al., 2006), etc. Although many hypotheses have been put forward as surrogates for cellular objective functions, substantially less work has been conducted toward systematically validating them with experimentally derived flux distributions of metabolic networks. An optimization-based framework is proved to be effective for inferring the objective function of E. coli metabolism (Burgard and Maranas, 2003). For solving the bilevel problem, the authors transform it into an equivalent nonlinear programming making use of /$ see front matter 2008 Elsevier Ltd. All rights reserved. doi: /j.compbiolchem

3 2 Z. Gong et al. / Computational Biology and Chemistry 33 (2009) 1 6 the duality theory. However, up to date, no attempt has been made to infer the objective function of anaerobic glycerol metabolism in K. pneumoniae for 1,3-PD production. In this work, we introduce a rigorous mathematical model (NBP1) to infer whether a weighted sum of fluxes can explain a set of experimental data observed from anaerobic glycerol metabolism in K. pneumoniae for 1,3-PD production. The model is a nonlinear bilevel programming, whose lower level is a conventional FBA model with undetermined parameters and upper level is a quadratic programming for evaluating the consistency with observed fluxes. Bard (1991) proves that the bilevel linear programming is NP-hard. So it is more difficult to solve NBP1. To obtain the weight coefficients in the model, a fast convergent genetic algorithm is constructed by adopting some techniques, which greatly reduce the searching space and avoid the difficulty to deal with the infeasible chromosomes. In this work, we also propose a model (NBP2) to investigate the effect of deviations between the flux distribution and the experimental ones on the weight coefficients. Some meaningful conclusions are drawn by several groups of data observed at different dilution rates and initial glycerol concentrations. The remainder of this paper is organized as follows. In Section 2, optimization models are formulated to determine the objective function and analyze the robustness of the objective function. In Section 3, we construct an improved genetic algorithm to solve NBP1(2). Section 4 gives the main results. Finally, Section 5 concludes some remarks and future research directions. 2. Optimization Models In this section, we propose a mathematically rigorous model to determine whether the maximization of a weighted combination of fluxes can explain the objective function of anaerobic glycerol metabolism in K. pneumoniae for the production of 1,3-PD Glycerol Metabolism in K. pneumoniae K. pneumoniae German Collection of Microorganisms (DSM) 2026 obtained from the DSM is used in this work. Culture medium composition, culture conditions, and methods for the determination of fermentation products were reported previously (Ahrens et al., 1998; Menzel et al., 1996). It follows from Zeng et al. (1993) and Chen et al. (2003) and the related knowledge that the simplified metabolic network of bio-dissimilation of glycerol to 1,3-PD by K. pneumoniae under anaerobic conditions is constructed by the author Zhang et al. (2006). In the network, the intermediate metabolites without branches are omitted. The network is composed of 22 reactions and 11 intra-metabolites. In this work, we assume each reaction flux associated with a metabolic drain or energy dissipation may be related to the objective function driving cellular metabolism. These fluxes are shown with bold arrows in Fig. 1. We also list all mass balance equations of intracellular metabolites in Table Flux Balance Analysis (FBA) In this paper, we make the following assumptions: (H 1 ). The metabolic system operates at a pseudo-steady state, equivalently, considering the time-averaged behavior for stationary metabolic concentrations are zeros. In other words, there is no intracellular metabolic accumulation in the network; (H 2 ). Weight coefficients are assigned to each reaction flux associated with a metabolic drain or energy dissipation, the index set of these reaction fluxes is denoted by P. Basing on the above assumptions, flux balance analysis model can be formulated as max c T v s.t. Sv = 0 0 v v max. (1) where cost vector c = (c 1,...,c n ) T C R n, and n { C c c Rn c j = 1 and j = 0 ifj/ P c j (0, 1) if j P. j=1 In (1), S is an m n matrix, S ij is the stoichiometric coefficient of metabolite i in reaction j for all i = 1,...,mand j = 1,...,n. v R n denotes the vector of fluxes with elements corresponding to the fluxes in given reactions (column) in S. c j is the weight associated with reaction j. Furthermore, a high coefficient value implies that the corresponding flux is maximized for the cellular metabolism, whereas a low value implies the converse. 0 v v max are box constraints, here v max is the maximal flux vector. Since the number of equations (m) is far less than that of unknown variables (n), the equations Sv = 0 are under-determined and the solution is not unique. Therefore, FBA optimizes the set of fluxes such that the objective is maximized Model for Inferring Objective Function In this study, our purpose is to infer the objective function of the above FBA. That is, we need identify the weights c such that the optimal solution of FBA (denoted by v ) is consistent with the experimental data. Let the experimental fluxes be v e,k E, where k E represents the index set of the fluxes with observed values. Here we measure the consistence between v and v e by the sum of the squared flux deviations, i.e., (v k ve k )2 (2) k E Obviously, the sum is smaller, v is more consistent with the experimental fluxes. Hence the nonlinear bilevel programming model to infer the objective of FBA can be formulated as (NBP1) min J(c, v) = (v k (c) v e c,v k )2 k E s.t. c C where v solves (3) max f (c, v) = c T v v s.t. Sv = 0 0 v v max. In above model, the lower level is a conventional FBA model with undetermined parameters and the upper level is a quadratic programming for evaluating consistency with observed fluxes. J(c, v) and f (c, v) are the objective functions of upper level (termed as leader) and lower level (termed as follower), respectively. c R n and v R n are the decision variables of the upper level and lower level, respectively. v k (c) represents the k th component of the optimal solution of the lower level when c C is given. In NBP1, the leader makes its decision, taking account into the reaction of the follower. In order to facilitate further discussion, some notations and definitions are necessary to introduce following by Bard (2006). Let ={(c, v) R n R n c C,Sv = 0, 0 v v max } denote the constraint region of NBP. Let (c) ={v R n Sv = 0, 0 v v max } be

4 Z. Gong et al. / Computational Biology and Chemistry 33 (2009) Fig. 1. Metabolic network of anaerobic glycerol metabolism in K. pneumoniae for 1,3-PD production. Reaction fluxes assigned with the weight coefficients are shown with bold arrows. the follower s feasible region for fixed c. In fact, (c) is independent of c in our NBP1 model. Let (C) ={c C v s.t.(c, v) } be projection of on the upper decision space, and let M(c) = arg min{f (c, v) v (c)} be the follower s rational reaction set for agivenc. The union of all possible vectors c which the leader may select and the corresponding rational reaction set v M(c) is called the inducible region. Let the inducible region denote by IR ={(c, v) c (C), v M(c)}. Definition 1. A point (c, v) is feasible to NBP1 if (c, v) IR. (c, v ) IR is called to be globally optimal to NBP1 if J(c, v ) J(c, v) holds for any (c, v) IR. Definition 2. A point-to-set mapping : R p 2 Rq is called polyhedral if its graph gr :={(x, y) R p R q y (x)} is equal to the union of a finite number of polyhedral convex sets. It can be shown that NBP1 has the following two properties. Table 1 Mass balance equations of intracellular metabolites for glycerol bioconversion to 1,3-PD by K. pneumoniae in anaerobic continuous culture No. Intra-metabolites Mass balance equations under steady-state condition Property 1. polyhedral. Property 2. The point-to-set mapping M( ) :C 2 Rn is a non-empty NBP1 has a global optimal solution. 1 Glycerol v 1 v 2 v 3 v 4 = 0 2 Phos. v 4 v 5 v 6 = 0 3 Pyruvate v 5 v 8 v 9 v 10 v 16 v 19 = 0 4 Aceytl-CoA v 9 + v 10 v 12 v 13 = 0 5 Acetoin 0.5v 16 v 17 v 7 = 0 6 Formate v 9 v 11 v 18 = 0 7 CO 2 v 10 + v 11 + v 16 v 6 v 20 = 0 8 H 2 v 14 + v 11 v 21 = 0 9 NADH 2 v 3 v 2 + 2v 4 2v 6 v 8 v 7 + v 15 2v 13 = 0 10 ATP 7.5v 3 + v 5 + v 6 + v 12 v 22 = 0 11 FADH 2 v 10 v 15 v 14 = Model for Robustness Analysis Let the minimum value of the upper function of NBP1 be Min and the optimal value of c be c. We assume the range of the sum of the squared deviations between the identified and experimental fluxes is from 0 to r Min due to experimental errors, where r is a positive constant. Now we formulate a model to investigate the effect of deviations on the robustness of the weight coefficient c j in (3), j P, i.e., to determine the largest variation ranges of c j due to

5 4 Z. Gong et al. / Computational Biology and Chemistry 33 (2009) 1 6 experimental errors. The model can be written as (NBP2) max c,v J(c, v) = c j c j s.t. c C where v solves max f (c, v) = c T v v s.t. Sv = 0 (v k v e k )2 Min k E 0 v v max. With respect to NBP2, the corresponding notations and definitions can be introduced. It can also be shown that NBP2 has optimal solutions. 3. Algorithm for Solving NBP1(2) Bilevel programming problems are not easy to solve. In fact they have been proved to be NP hard. In recent years, many approaches have been presented for solving linear bilevel programming, such as branch and bound approach, penalty method, genetic algorithm and many others. While nonlinear bilevel programming is still in its infancy with a hand full of algorithms such as branch and bound, global optimization (Amouzegar, 1999; Fliege and Vicente, 2006; Gümüs and Floudas, 2001; Lan et al., 2007; Wang et al., 2007, etc.). Genetic algorithm (GA) is an iterative random search algorithm. GA maintains a population of candidate solutions where each solution is called a chromosome. A set of chromosomes forms a population which is ranked by a fitness function and evolves to the next generation by randomly selecting parents from the population and reproducing children using genetic operations. A set of selected individuals forms an improved population. In this paper, we construct an improved GA to solve the NBP1(2), which takes advantage of the problem s structural characteristic. In the algorithm, the upper variable is coded as float, and the fitness function is based on the objective function of the upper level. Designing the genetic operators, we adopt some techniques to guarantee the offspring are still in the constraint region of NBP1(2) and they move quickly towards the global optimal solution. Those techniques reduce the searching space and avoid the difficulty to deal with infeasible points. Now we describe the genetic operators in detail. Initial population: Randomly generate N individuals by uniform distribution from C, i.e., c i (0) C,i= 1, 2,...,N. Fitness function: The fitness function is defined as fit(c) = J(c, M(c)), (5) where J(, ) is the upper objective function of NBP1(2), and M(c)is obtained by solving their corresponding lower level optimization problem. Crossover operator: Letp c be the crossover probability. Repeat the following operations from i = 1toi = N: generate a random number r i [0, 1], and if r i <p c, then c i (k) is selected as a parent. Furthermore, randomly mate all parents selected. Let c i (k) and c j (k) be two mated parents, whose offspring are denoted by d i (k) and d j (k). The formula of crossover is as follows: d i (k) = c i (k) + (1 )c j (k), (6) d j (k) = (1 )c i (k) + c j (k), (7) where [0, 1] is a random number. It is easy to find that the generated offspring are still in C. Let the total number of all offspring generated by crossover be N 1. (4) Mutation operator: The best individual of the present individuals is denoted by d (k), so d (k) divide N 1 into two groups, i.e., { N 1+ ={l {1, 2,...,N 1 } d l (k) d (k) ε} N 1 ={l {1, 2,...,N 1 } d l (k) d (k) <ε}. (8) where ε is a small positive number and represents the Euclidean form. For each l N 1+, we mutate d l (k) by the probability p m according to the following formula g m (k) = d l (k) + (1 )d (k), (9) where N(0, 1). For each l N 1, we mutate g l (k) by the probability p m according to the following formula: g m (k) = ˇd l (k) + (1 ˇ)d rand (k), (10) where ˇ N(0, 1) and d rand (k) C is generated randomly. Let the total number of mutated offspring be N 2, then the present population has N + N 1 + N 2 individuals, which are all in C. Selection operator: In order to guarantee the global convergence of new GA, we select the best N individuals from the N + N 1 + N 2 ones to form the next generation. So the steps of the improved GA for solving NBP1(2) can be described as follows: Step 1: Initialize crossover probability p c, mutation probability p m, population size N, the maximal iterations K. Set k = 0, generate the initial population p(k) ={c i (k) i = 1,...,N}. Compute each individual s fitness value fit(c i (k)) by (5). Step 2: Generate the crossover offspring d l (k)(l = 1,...,N 1 )bythe above crossover operator and evaluate their fitness value fit(d l (k)). Step 3: Generate the mutation offspring g m (k)(m = 1,...,N 2 )by the above mutation operator and evaluate their fitness value fit(g m (k)). Step 4: Form the next generation p(k + 1) by selection operator. Step 5: k = k + 1, if either k>kor no progress is made in the last generations, then output the best individual and stop, otherwise go to Step Main Results Three groups of steady-state experimental data (Menzel et al., 1996) at different dilution rates and initial glycerol concentrations are used in this study. These data include the concentrations of glycerol, 1,3-PD, biomass, ethanol, lactate, etc. We may obtain 13 fluxes from each group of experimental data. The three groups of experimental fluxes (v e k ) and computed fluxes (v k) by NBP are all listed in Table 2. InTable 2, both group 1 and group 2 are obtained under substrate-limited conditions, whereas group 3 is got under substrate-excess conditions. The coefficients identified are shown in Fig. 2. The coefficients qualify the contribution of a given flux to the objective function. That is, a high coefficient value implies that the corresponding flux is maximized for the cellular metabolism, whereas a low value implies the converse. The deviations of the weight coefficients are shown in Fig. 3. The parameters chosen in above algorithm is as follows: K = 100, N = 30, p c = 0.8, p m = 0.2. In our study, we draw five conclusions. (i) The three groups of weight coefficients are similar though the flux distributions are quite different. The result is consistent with a conclusion (Burgard and Maranas, 2003), i.e., there exists a single cellular driving force governs the distribution of metabolic fluxes under different conditions. (ii) Biomass production is the main force to drive K. pneumoniae metabolism because its weight coefficients is the largest among all coefficients. (iii) The coefficient of biomass obtained under substrate-excess conditions is a bit smaller than

6 Z. Gong et al. / Computational Biology and Chemistry 33 (2009) Table 2 Three sets of experimentally observed and calculated flux distributions under different D and C s0 conditions (anaerobic case) Flux distributions 1 (0.15,809) a 2 (0.35,443) a 3 (0.35,870) a v e (v1) (25.28) (58.69) (112.50) v e (v2) (10.96) (28.77) (65.18) v e (v3) (1.98) (3.570) (3.469) v e (v6) (0.891) (1.836) (2.590) v e (v7) (0) 0 (0.0479) (8.2) v e (v8) (0.464) (0.615) (3.935) v e (v12) (3.747) (10.089) (10.543) v e (v13) (6.992) (12.459) (9.730) v e (v17) 17 0 (0) (0) (0) v e (v18) (0.541) (2.226) (6.130) v e (v19) (0.004) (1.261) (1.05) v e (v20) (9.452) (18.71) (22.90) v e (v21) (9.306) (18.71) (11.55) Note: D, dilution rate (h 1 ); C s0, initial glycerol concentration (mmol L 1 h 1 ); v e, experimentally observed data; v k k, computated data according to NBP. a (D, C s0). Fig. 2. The values of weight coefficients obtained by three groups of flux distributions. that obtained under substrate-limited conditions. This is consistent with the observed experimental fact that the growth of K. pneumoniae is inhibited by the excess substrate. (iv) For a given flux distribution, the range of allowable value c j is rather narrow, that is, the weight coefficient c j is of good robustness with respect to experimental errors. (v) The errors between experimental fluxes and computed ones are small, so FBA can predict the flux distribution more accurately making use of the obtained objective function. These results can provide theoretical guide for metabolic regulation to improve the 1,3-PD production from glycerol by K. pneumoniae in anaerobic continuous culture. 5. Discussion and Further Studies Undoubtedly, mathematical modelling of cellular metabolism plays an important role to understand biological functions and to provide identification of targets for biotechnological modification. In this paper, we propose two NBP models on anaerobic glycerol metabolism in K. pneumoniae for 1,3-propanediol production. One intends to infer the metabolic objective function, and the other is to analyze the robustness of the objective function. NBP is NP-hard, and is not easy to solve. We construct a fast convergent genetic algorithm to solve the proposed models. At last, we get some interesting results. The results are of significance. On the one hand, it can provide the theoretical guide for metabolic regulation to improve the Fig. 3. Maximal absolute variations of the weight coefficients, i.e., max c j c, j P, j for the (a) group 1, (b) group 2 and (c) group 3 experimental flux distributions when solution constraint is relaxed. Where F(c, v) = (v k v e k E k )2. 1,3-PD production. On the other hand, we can predict the intracellular fluxes more accurately. In addition, the optimization model can also be used to determine the objective functions of other microorganism s metabolism.

7 6 Z. Gong et al. / Computational Biology and Chemistry 33 (2009) 1 6 Acknowledgements This research is supported by National Natural Science Foundation of China (No ) and National Basic Research Program of China (No. 2007AA02Z208). References Ahrens, K., Menzel, K., et al., Kinetic, dynamic, and pathway studies of glycerol metabolism by Klebsiella pneumoniae in anaerobic continuous culture. III. Enzymes and fluxes of glycerol dissimilation and 1,3-propanediol formation. Biotechnol. Bioeng. 59, Amouzegar, M.A., A global optimization method for nonlinear bilevel programming problem. IEEE Trans. Syst. Man, Cybernet. B: Cybernet. 29 (6), Bard, J.F., Some properties of the bilevel programming problem. J. Optim. Theory Appl. 68, Bard, J.F., Practical Bilevel Optimization: Algorithm and Application (Series Nonconvex Optimization and its Application). Springer Verlag. Beard, D.A., Liang, S.D., Qian, H., Energy balance for analysis of complex metabolic networks. Biophys. J. 8 (3), Bibel, H., Memzel, K., Zeng, Z.P., Microbial production of 1,3-propanediol. Appl. Microbiol. Biotechnol. 52, Burgard, A.P., Maranas, C.D., Optimization-based framework for inferring and testing hypothesized metabolic objective functions. Biotechnol. Bioeng. 82 (6), Chen, X., Xiu, Z.L., Wang, J.F., Zhang, D.J., Xu, P., Stoichiometric analysis and experimental investigation of glycerol bioconversion to 1,3-propanediol by Klebsiella pneumoniae under microaerobic conditions. Enzyme Microb. Technol. 33, Fliege, J., Vicente, L.N., Multicriteria approach to bilevel optimization. J. Optim. Theory Appl. 131 (2), Gümüs, Z.H., Floudas, C.A., Global optimization of nonlinear bilevel programming. J. Global Optim. 20, Kauffman, K.J., Prakash, P., Edwards, J.S., Advances in flux balance analysis. Curr. Opin. Biotechnol. 14, Lan, K.W., Wen, U.P., Shin, H.S., Lee, E.S., A hybrid neural network approach to bilevel programming problems. Appl. Mater. Lett. 20 (8), Lee, J.M., Gianchandani, E.P., Papin, J.A., Flux balance analysis in the era of metabolomics. Briefings Bioinform. 7 (2), Maczek, J., Junne, S., et al., Metabolic flux analysis of the sterol pathway in the yeast Saccharomyces cerevisiae. Bioprocess. Biosyst. Eng. 29, Menzel, K., Zeng, A.P., Deckwer, W.D., Kinetic, dynamic, and pathway studies of glycerol metabolism by Klebsiella pneumoniae in anaerobic continuous culture. I. The phenomena and characterization of oscillation and hysteresis. Biotechnol. Bioeng. 52, Menzel, K., Zeng, A.P., Deckwer, W.D., High concentration and productivity of 1,3-propanediol from continuous fermentation of glycerol by Klebsiella pneumoniae. Enzyme Microb. Technol. 20, Nakamura, C.E., Whited, G.M., Metabolic engineering for the microbial production of 1,3-propanediol. Curr. Opin. Biotechnol. 14, Ozkan, P., Sariyar, B., et al., Metabolic flux analysis of recombinant protein overproduction in Escherichia coli. Biochem. Eng. J. 22, Sanchez, A.M., Bennett, G.N., et al., Batch culture characterization and metabolic flux analysis of succinate-producing Escherichia coli strains. Metab. Eng. 8, Shirai, T., Nakato, A., et al., Comparative study of flux redistribution of metabolic pathway in glutamate production by two coryneform bacteria. Metab. Eng. 7, Stephanopoulos, G., Aristidou, A., Nielsen, J., Metabolic Engineering: Principles and Methodologies. Academic Press, San Diego. Wang, G.M., Wang, X.J., Wan, Z.P., Lv, Y.B., A globally convergent algorithm for solving a class of bilevel nonlinear programming problem. Appl. Math. Comput. 188 (1), Zeng, A.P., Biebl, H., Schlieker, H., Deckwer, W.D., Pathway analysis of glycerol fermentation by Klebsiella pneumoniae: regulation of reducing equivalent balance and product formation. Enzyme Microb. Technol. 15, Zeng, A.P., Biebl, H., Bulk-chemicals from biotechnology: the case of 1,3- propanediol production and the new trends. Adv. Biochem. Eng. Biotechnol. 74, Zhang, Q.R., Xiu, Z.L., Zeng, A.P., Optimization of microbial production of 1,3-propanediol by Klebsiella pneumoniae under anaerobic and microaerobic conditions by metabolic flux analysis. J. Chem. Ind. Eng. 57,