# Genetic Optimization Algorithm for Metabolic Engineering Revisited

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. A Basic Genetic Algorithm for Metabolic Engineering

- A genetic representation of solutions. Here, we employ a binary coding.
- Populations of individuals as evolutionary communities.
- A fitness function for evaluating the goodness of individuals.
- Operators, which generate a new population from an existing one and which can be controlled by parameters that shape the fitness-related or random transformation behavior.

#### 2.1.1. Population of Binary Individuals

#### 2.1.2. The Fitness Function

#### 2.1.3. Selection, Mating, and Crossover

#### 2.1.4. Mutation and Elitism

#### 2.1.5. Parallelism

#### 2.2. Adaptive Probabilities of Mutation

#### 2.3. Additional Features

#### 2.3.1. Gene Deletion Targets

#### 2.3.2. Multi-Objective Optimization

#### 2.3.3. Minimization of Perturbations

#### 2.3.4. Non-Native Network Edge Insertions

#### 2.4. Analysis of the Evolution of Populations

#### A Measure of Population Diversity: The Hamming Distance

#### 2.5. Metabolic Model Preprocessing

#### 2.6. General Conduct for the Application of the Genetic Algorithm

## 3. Results

#### 3.1. GA Parameter Sensitivity Analysis

#### 3.1.1. Mutation Rate

#### 3.1.2. Selection Rate and Population Size

#### 3.1.3. Parallelization: Numbers of Generations, Gene-Flow Events, and Threads

#### 3.2. Target Product Varieties and Minimal Intervention Set Sizes

#### 3.3. Multi-Objective Fitness Function Optimization

#### 3.4. Heterologous Reaction Insertion

^{+}-dependent glyceraldehyde-3-phosphate dehydrogenase with its NADP

^{+}-dependent, phosphorylating counterpart (EC 1.2.1.13) and addition of an ATP-dependent citrate lyase (EC 2.3.3.8) frequently occurred in the best individuals. Simultaneously, formation of acetate and ethanol were inhibited, as well as the malic enzyme knocked out, altogether enforcing metabolic flux through the glyoxylate shunt and the reductive branch of the TCA cycle towards succinate. For the glycolytic product ethanol, switching from the NAD

^{+}-dependent to the NADP

^{+}-dependent alcohol dehydrogenase (EC 1.1.1.2) and glyceraldehyde-3-phosphate dehydrogenase (phosphorylating), as well as simultaneously deleting the NAD

^{+}transhydrogenase, led to the most promising strategies. Congruently, NADH/NADPH metabolism was the preferred target for glutamate overproduction, which was spurred by the addition of the NADP

^{+}-dependent glyceraldehyde-3-phosphate dehydrogenase (EC 1.2.1.9) as well as the knockout of NAD

^{+}transhydrogenase. The identified strain designs also suggested to increase flux through the TCA cycle by heterologous expression of the citrate oxaloacetate-lyase (EC 4.1.3.6) to recycle acetate. Interestingly, insertion of the latter in combination with the expression of the non-native NADP

^{+}-dependent glyceraldehyde-3-phosphate dehydrogenase and the deletion of various NADH/NAD

^{+}-dependent reactions also improved lactate overproduction.

#### 3.5. Increasing the Complexity and Predictive Power of Employing Genome-Scale Models

^{+}-dependent glyceraldehyde-3-phosphate dehydrogenase (EC 1.2.1.13) was inserted. However, insertion of novel functionalities did not significantly improve succinate overproduction as compared to the deletion-only strain designs and in case of the quintuple deletion mutants even showed lowered fitness values (Supplementary Figure S14a). Presumably, novel network edges are not of critical concern for optimizing succinate production in E. coli. Identification of significantly better strain design solutions at elevated generation numbers is also unlikely, since the population diversities reached plateau regions indicating approaching fitness convergence (Supplementary Figure S14b). Only for the octuple deletion and double insertion cases did a drop in the Hamming distance approximately from generation 1600 onward suggest incomplete convergence.

^{+}-dependent malate dehydrogenase (Δmdh) or pyruvate kinase (ΔpykA, ΔpykF) occurred frequently, presumably due to the elevated recapture of carbon dioxide. For the complete strategies we refer to the Supplementary Information.

## 4. Discussion

^{+}-dependent reactions with their NADP

^{+}-dependent counterparts, while simultaneously deleting the NADH dehydrogenase, NAD

^{+}transhydrogenase or other NADH-dependent reactions, as was also previously suggested for succinate overproduction by Kim et al. [11] based on their findings employing SimOptStrain. However, by exploiting the E. coli iJO1366 GEM and the full capacity of the GA’s features, we identified a rather different strain design compared to other theoretical or experimental studies. Whereas it has been frequently suggested to directly suppress byproduct formation, e.g., by knocking out ackA, ldhA, or pfl [50], the GA framework applying an E. coli GEM predicted a redirection of the TCA cycle flux towards the glyoxylate shunt to be most beneficial for succinate production. Moreover, and in line with results from the core metabolic model, reduction of NADH generation in favor of NADPH appeared to be a key design principle. This was pronounced by the suggestion to include the non-native NADP

^{+}-dependent glyceraldehyde-3-phosphate dehydrogenase or quinate dehydrogenase, which significantly improved theoretical maximal growth. Since our design suggestions resulted from the simultaneous application of various engineering objectives, a comprehensive metabolic model, and the consideration of actual gene-protein dependencies, as well as detailed wild-type metabolic flux data, it offers the most reliable basis for experimental transfer. Thus, we are looking forward to further investigating or even practically applying the presented designs, which, however, was out of the scope of this work.

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Symbol | Explanation |

B | A bit in the binary representation of an individual |

$\widehat{F}$ | Intervention size-scaled fitness |

$F$ | Objective fitness |

${F}_{R}$ | Fitness of the best discarded individual |

$\mu $ | Specific growth rate |

${N}_{B}$ | Number of bits per individual |

${N}_{D}$ | Number of interventions per individual |

${N}_{G}$ | Number of subsequent generations |

${N}_{GFE}$ | Number of subsequent gene flow events |

${N}_{P}$ | Population size |

${N}_{Pa}$ | Number of possible pairs of individuals |

${N}_{S}$ | Number of selected individuals |

${N}_{T}$ | Number of target reactions |

$R$ | Mutation rate |

${v}_{P}$ | Production rate |

${v}_{S}$ | Substrate uptake rate |

$X$ | Selection rate |

$y$ | Trade-off factor |

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**Figure 2.**Maximal fitness (

**a**) and hamming distance (

**b**) across the populations of every thread in each generation using mutation rates between 0 and 0.7. Deletion of maximally five reactions were allowed while using succinate BPCY as the engineering objective. Hamming distance progressions for mutation rates 0.5 and 0.7 overlap each other.

**Figure 3.**Number of fitness function evaluations until maximal final fitness was reached. Box plots represent five replicate GA runs applying the respective mutation rate. Succinate BPCY was used as the engineering objective.

**Figure 4.**Maximal fitness progressions of GA runs using selection rates of (

**a**) 0.15, (

**b**) 0.3, (

**c**) 0.5, and (

**d**) 0.75. The color codes denote different population sizes ranging between 10 and 50. Generation numbers are plotted on a logarithmic scale. Deletion of maximally seven reactions were allowed. Succinate BPCY was used as the engineering objective.

**Figure 5.**Number of fitness function evaluations until maximal final fitness was reached for GA runs applying population sizes between 10 and 50. Bars are clustered according to the employed selection rate (colored number). Error bars show the standard deviation among five replicates for each population size—selection rate pair. Asterisks denote parameter pairs with which the globally maximal fitness of $0.48{\mathrm{mol}\text{}\mathrm{mol}}^{-1}{\text{}\mathrm{h}}^{-1}$ was not reached in every replicate GA run after 900 generations. Succinate BPCY was used as the engineering objective. The intervention set size was seven.

**Figure 6.**Number of generations (squares) and computation time (triangles) until maximal fitness was reached. Deletion of maximally seven reactions were allowed. Succinate BPCY was used as the engineering objective. Error bars denote the standard deviation of five replicate GA runs using one, three, five, and seven parallel threads.

**Figure 7.**Hamming distance progressions for GA runs applying 5 to 180 GFEs while keeping the total generation number at 900. Deletion of maximally seven reactions were allowed. Succinate BPCY was used as the engineering objective. Error bars denote the standard deviation of five replicate GA runs.

**Figure 8.**Absolute computation time of 900 generations for several pairs of GFEs and generation sizes. Deletion of maximally seven reactions were allowed. Succinate BPCY was used as the engineering objective. Error bars denote the standard deviation of five replicate GA runs.

**Figure 9.**Maximal fitness progression of GA runs optimizing overproduction of succinate (

**a**,

**e**), ethanol (

**b**,

**f**), lactate (

**c**,

**g**), and glutamate (

**d**,

**h**) applying three, five, seven, and nine maximal reaction (

**a**–

**d**) or gene (

**e**–

**h**) deletions.

**Figure 10.**(

**a**) Progressions of the intervention size of the fittest individual throughout GA runs, in which the transformed fitness function was used (cf. Equation (9)) to consider the minimization of the number of deletion targets (cf. Section 2.3.3). Values between 0 and 0.04 were employed for the trade-off factor y to investigate its influence on trade-off between the objective fitness and the number of simultaneous perturbations. Dots illustrate the mean intervention size over a population at a specific generation. The lines represent the corresponding linear interpolations. For the same simulations, the final objective fitness values are shown in Subfigure (

**b**).

**Figure 11.**Yield spaces of wild-type, as well as mutant, E. coli strains optimized for the overproduction of succinate (

**a**), ethanol (

**b**), lactate (

**c**), and glutamate (

**d**) using a combination of BPCY, growth-coupling, and production rate at maximal growth rate as the engineering objective. All mutant yield spaces are based on the substrate uptake rate predicted by MiMBl. Triangles and attached numbers illustrate the phenotype prediction calculated by MiMBl and the fitness value for a strain design with a given number of reaction deletions, respectively. Refer to the Supplementary Section I.2 for a brief description of yield space calculations.

**Figure 12.**Fitness of the best individual after 1800 generations. BPCY of succinate (

**a**), ethanol (

**b**), lactate (

**c**), and glutamate (

**d**) was used as the engineering objective while applying five reaction deletions, as well as one to four novel reaction insertions. The grey bars illustrate the fitness of the best individual after 900 generations without considering any reaction additions (cf. Figure 8).

**Figure 13.**Fitness progressions of GA runs optimizing succinate overproduction in the E. coli iJO1366 model applying gene (red line) and reaction (blue line) deletions. Standard deviation among three replicate GA runs are illustrated as error bands.

**Figure 14.**Yield spaces of wild-type, as well as mutant, E. coli strains optimized for the overproduction of succinate using a combination of BPCY, GCS, and production rate at a maximal growth rate as the engineering objective. The legend shows the maximal allowable numbers of gene deletions (KO) and reaction insertions (Ins). All mutant yield spaces are based on the substrate uptake rate predicted by MiMBl. Triangles and attached numbers illustrate the phenotype prediction calculated by MiMBl and the corresponding BPCY, respectively. Note that yield spaces of the 5KO and 8KO mutant overlap with each other. Refer to the Supplementary Section I.2 for a brief description of yield space calculations.

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**MDPI and ACS Style**

Alter, T.B.; Blank, L.M.; Ebert, B.E.
Genetic Optimization Algorithm for Metabolic Engineering Revisited. *Metabolites* **2018**, *8*, 33.
https://doi.org/10.3390/metabo8020033

**AMA Style**

Alter TB, Blank LM, Ebert BE.
Genetic Optimization Algorithm for Metabolic Engineering Revisited. *Metabolites*. 2018; 8(2):33.
https://doi.org/10.3390/metabo8020033

**Chicago/Turabian Style**

Alter, Tobias B., Lars M. Blank, and Birgitta E. Ebert.
2018. "Genetic Optimization Algorithm for Metabolic Engineering Revisited" *Metabolites* 8, no. 2: 33.
https://doi.org/10.3390/metabo8020033