# Complex Nonlinear Behavior in Metabolic Processes: Global Bifurcation Analysis of Escherichia coli Growth on Multiple Substrates

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Metabolic Model

#### 2.1. The HCM Framework

_{IN}are the vectors of n

_{x}concentrations of extracellular components in the reactor and feed, respectively, including substrates, products and biomass, r is the vector of n

_{x}fluxes, S

_{x}is the (n

_{x}× n

_{r}) stoichiometric matrix and D is the dilution rate.

**r**=

**Zr**

_{M}_{r}× n

_{z}) matrix composed of EMs as its columns and r

_{M}is the vector of n

_{z}fluxes through EMs. EMs may be viewed as metabolic pathways composed of a minimal set of reactions that can operate alone in steady state. Nonnegative combinations of EMs can represent any feasible metabolic state (i.e., flux distribution) in a network.

_{M,j}is the cybernetic variable controlling enzyme activity, is the kinetic term, and is the relative enzyme level to its theoretical maximum, i.e., e

_{M,j}/ .

_{M,j}is governed by the following dynamic equation, i.e.:

_{M,j}is the cybernetic variable regulating the induction of enzyme synthesis, is the kinetic part of the inducible enzyme synthesis rate, β

_{M,j}is the degradation rate and µ is the specific growth rate. The four terms of the right-hand side denote constitutive and inducible rates of enzyme synthesis and the decrease of enzyme levels by degradation and dilution, respectively. The cybernetic control variables, u

_{M,j}and v

_{M,j}, are computed from the Matching and Proportional laws [21,22], respectively:

_{j}, denotes the carbon uptake flux through the jth EM, i.e., , and f

_{C,j}denotes the factor converting EM flux to the carbon uptake rate. Dynamic shifts among different pathways are realized by two controlling variables, u

_{M,j}and v

_{M,j}.

#### 2.2. HCM for Anaerobic E. coli Growth

## 3. Methods

_{M}-variables. Among many possibilities, we discuss two ideas of handling this issue, i.e., the combinatoric approach used by Namjoshi and Ramkrishna [10] and the smooth approximation to the max function.

#### 3.1. Rigorous Combinatoric Analysis

_{M}variables to be 1, while the others are less than or equal to 1 (Table 2).

Variables or parameters | Equations or parameter values |
---|---|

Extracellular metabolites and biomass | Glucose: Pyruvate: Acetate: Ethanol: Formate: Biomass: |

Enzymes | |

Cybernetic variables | |

Kinetics | |

Parameters and stoichiometric coefficients | |

Notations | c: biomass concentration, g/L D: dilution rate, 1/h e _{M,j}, : level of enzyme that catalyzes the jth EM flux and its maximal level f _{C,j}: factor converting the EM flux (i.e., growth rate) to the carbon uptake rate, C-mmol/gDW (DW = dry weight)k _{F}: rate constant for formate decomposition : maximal rate constant for the jth EM flux, 1/h K _{F}: Michaelis constant for formate decomposition, mM K _{G,j}, K_{P,j}: Michaelis constants for the jth EM flux, mM r _{F}: specific rate of formate decomposition into CO_{2} and H_{2}, mmol/(gDW℘h) r _{M,j}, : regulated and unregulated fluxes through the jth EM, mmol/(gDW h) : kinetic part of inducible enzyme synthesis rate, 1/h s _{A,j}, s_{E,j}, s_{F,j}, s_{G,j}, s_{P,j}: stoichiometric coefficients, mmol/gDW t: time, h u _{M,j}: cybernetic variable regulating the enzyme induction v _{M,j}: cybernetic variable regulating the enzyme activityx _{A}, x_{E}, x_{F}, x_{G}, x_{P}: concentrations of acetate, ethanol, formate, glucose and pyruvate, mM x _{IN,G}, x_{IN,P}: feed concentration of glucose and pyruvate, mM Greek letters α _{M,j}: constitutive enzyme synthesis rate, 1/h β _{M,j}: rate of enzyme degradation, 1/h µ: growth rate, 1/h |

Case | v_{M}_{,1} | v_{M}_{,2} | v_{M}_{,3} | v_{M}_{,4} |
---|---|---|---|---|

I | 1 | ≤1 | ≤1 | ≤1 |

II | ≤1 | 1 | ≤1 | ≤1 |

III | ≤1 | ≤1 | 1 | ≤1 |

IV | ≤1 | ≤1 | ≤1 | 1 |

_{M,j}to be 1 by replacing the denominator of v

_{M,j}, i.e., , with , leading to four independent sets of model equations. Figure 1 shows the resulting four hysteresis curves in the D − c space with a fixed value of γ (i.e., 0.2), obtained from the analysis of Cases I to IV, respectively. Segments highlighted in color represent feasible branches satisfying the constraint, i.e., v

_{M,j}= 1 (j = 1 − 4), i.e., green (b), cyan (c) and magenta (d), respectively. Note that no such colored branch is found in Figure 1a, indicating that there exists no feasible solution satisfying v

_{M}

_{,1}= 1 along the whole profile.

**Figure 1.**Hysteresis curves obtained from four cases considered in Table 2 (x

_{IN,total}= 50 mM and γ = 0.2): (

**a**) Case I (v

_{M}

_{,1}= 1), (

**b**) Case II (v

_{M}

_{,2}= 1), (

**c**) Case III (v

_{M}

_{,3}= 1) and (

**d**) Case IV (v

_{M}

_{,4}= 1). Solid and dotted lines indicate stable and unstable branches, while colored and uncolored lines, feasible and infeasible branches, respectively.

**Figure 2.**Overall hysteresis curve generated by integrating individual pieces of feasible branches: (

**a**) γ = 0.2, (

**b**) γ = 0.4.

#### 3.2. Smooth Approximation to the Max Function

_{p}-norm is considered as an accurate approximation to the max function when p is sufficiently large. That is, we may approximate with .

_{p}-norm approximation with different p-values. No appreciable errors are found when p ≥ 30, while some deviations are observed when p-values are lower than that.

_{p}-norm approximation with a p value of 70.

**Figure 3.**Reproduction of the hysteresis curve of Figure 2a using the L

_{p}-norm approximation with different p-values.

**Figure 4.**Magnified views of two red windows around the catch-up points in the lower-right panel of Figure 3: (

**a**) left upper window, (

**b**) right lower window.

#### 3.3. Integration of Two Methods

## 4. Results and Discussion

_{p}-norm approximation (Section 3.2) to explore the nonlinear behavior of the HCM model by Kim et al. presented in Table 1. The smooth approximation is conveniently implementable with no appreciable errors in our case. The main parameters subject to variation include dilution rate (D) and the fractional molar concentration of glucose in the feed (γ), i.e.,

_{IN,G}and x

_{IN,P}are concentrations of glucose and pyruvate in the feed, respectively. The total sugar concentration (x

_{IN,total}) is the sum of x

_{IN,G}and x

_{IN,P}.

#### 4.1. Hysteresis Behaviors and Bifurcation Diagram

_{IN,total}= 50). The implication of different colors and solid and dashed lines is the same as before. This parameter set yields up to five steady states in a range of D between 0.325 and 0.335. A catch-up point is observed between EM3 and EM4.

_{IN,total}> is fixed to 50 mM. It shows two closed curves in black and gold (left) and four pairs of lines highlighted in the same color, respectively (right). The gold curve represents the neutral saddles, equilibrium points characterized by two real eigenvalues with the opposite sign. Neutral saddles are, however, not bifurcation points of interest and have nothing to do with steady-state multiplicity. Solid lines (other than neutral saddle lines) represent typical limit points, while thick dotted lines, catch-up points. Therefore, inside each envelop, there exist three multiple steady states (i.e., domains I, II, III, IV and V), at least. In the region where two envelops overlap (i.e., domains VI, VII and VIII), five steady states exist. In the remaining region, a unique solution exists.

#### 4.2. Bifurcation Diagram at a Higher Sugar Concentration in the Feed

_{IN,total}) on nonlinear behavior of the E. coli model is examined. When lowering the total sugar concentration from 50 to 25 mM, no qualitative change is observed in bifurcation behavior. Increasing x

_{IN,total}to 100 mL, on the other hand, leads to an additional domain not observed previously.

_{IN,total}of 100 mL. The implication of lines and colors is the same as before. Unlike the previous case, this condition leads to multiplicity regimes with up to seven steady states. That is, seven steady states emerge in the domain (IX) where three different envelops (i.e., red, orange and purple ones) are overlapped. We have highlighted this domain in the figure.

#### 4.3. Experimental Validation

_{IN,total}= 50, yielding a total of three steady states and five steady states, γ = 0.4 and x

_{IN,total}= 25, with a total of five steady states.

## 5. Conclusions

_{p}-norm approximation of the max function tested in this work is a practically useful idea, as it is applicable to general cases considering a large number of metabolic pathway options (i.e., EMs). Replacement of the max function with the L

_{p}-norm representation allows for accurate computation of bifurcation points. While slight errors around non-smooth folds (or catch-up points) are unavoidable, they are negligibly small in our case. When these errors are appreciable in certain cases, however, we can redo rigorous computation only for the non-smooth folds based on the combinatoric idea of Namjoshi and Ramkrishna [10]. Such a combination of these two methods guarantees rigorous results at a minimal level of inconvenience, thus serving as a promising strategy.

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Song, H.-S.; Ramkrishna, D.
Complex Nonlinear Behavior in Metabolic Processes: Global Bifurcation Analysis of *Escherichia coli* Growth on Multiple Substrates. *Processes* **2013**, *1*, 263-278.
https://doi.org/10.3390/pr1030263

**AMA Style**

Song H-S, Ramkrishna D.
Complex Nonlinear Behavior in Metabolic Processes: Global Bifurcation Analysis of *Escherichia coli* Growth on Multiple Substrates. *Processes*. 2013; 1(3):263-278.
https://doi.org/10.3390/pr1030263

**Chicago/Turabian Style**

Song, Hyun-Seob, and Doraiswami Ramkrishna.
2013. "Complex Nonlinear Behavior in Metabolic Processes: Global Bifurcation Analysis of *Escherichia coli* Growth on Multiple Substrates" *Processes* 1, no. 3: 263-278.
https://doi.org/10.3390/pr1030263