# Multi-objective optimization of enzyme manipulations in metabolic networks considering resilience effects

- Wu-Hsiung Wu
^{1}, - Feng-Sheng Wang
^{2}Email author and - Maw-Shang Chang
^{1}

**5**:145

https://doi.org/10.1186/1752-0509-5-145

© Wu et al; licensee BioMed Central Ltd. 2011

**Received: **15 May 2011

**Accepted: **19 September 2011

**Published: **19 September 2011

## Abstract

### Background

Improving the synthesis rate of desired metabolites in metabolic systems is one of the main tasks in metabolic engineering. In the last decade, metabolic engineering approaches based on the mathematical optimization have been used extensively for the analysis and manipulation of metabolic networks. Experimental evidence shows that mutants reflect resilience phenomena against gene alterations. Although researchers have published many studies on the design of metabolic systems based on kinetic models and optimization strategies, almost no studies discuss the multi-objective optimization problem for enzyme manipulations in metabolic networks considering resilience phenomenon.

### Results

This study proposes a generalized fuzzy multi-objective optimization approach to formulate the enzyme intervention problem for metabolic networks considering resilience phenomena and cell viability. This approach is a general framework that can be applied to any metabolic networks to investigate the influence of resilience phenomena on gene intervention strategies and maximum target synthesis rates. This study evaluates the performance of the proposed approach by applying it to two metabolic systems: *S. cerevisiae* and *E. coli*. Results show that the maximum synthesis rates of target products by genetic interventions are always over-estimated in metabolic networks that do not consider the resilience effects.

### Conclusions

Considering the resilience phenomena in metabolic networks can improve the predictions of gene intervention and maximum synthesis rates in metabolic engineering. The proposed generalized fuzzy multi-objective optimization approach has the potential to be a good and practical framework in the design of metabolic networks.

## Keywords

## Background

Improving the synthesis rate of desired metabolites in metabolic systems is one of the main tasks in metabolic engineering. Two recent advancements in this area are promising to increase the performance of metabolic systems. The first factor is a significantly better understanding of the structure of metabolic networks and the kinetics and thermodynamics of biochemical reactions that take place in living cells. In many cases, this understanding is not merely qualitative but quantitative, and can be expressed in terms of kinetics equations. The second factor is the current advance in molecular biological techniques and the development of numerous useful vectors. This has enabled microbiologists to change the protein content in a given organism and alter its enzymatic profile, enhancing the synthesis of specific end-products or intermediates. The combination of these two factors permits the modification of the metabolic structure and the improvement of the synthesis rate of some desired metabolites in an organism.

In the last decade, many researchers have used model-based optimization strategies to analyze and manipulate metabolic networks [1–12]. The mathematical models used in these model-based optimization problems can be classified as stoichiometric and kinetic models. Stoichiometric models can be obtained through the reaction topology of a metabolic network. Though stoichiometric models do not require kinetic data and are easy to construct, there is a shortage of handling regulatory dynamics of metabolic networks in them. On the other hand, kinetic models, e.g., generalized mass action (GMA) and Michaelis-Menten formulations, require more information to describe system characteristics. However, kinetic models are in general expressed as nonlinear models that are more complex than linear models and require more computational time for analysis and optimization. Logarithmic transformation can convert a non-linear model represented by the S-system formalism used widely in biochemical systems theory (BST) to a linear model if we consider systems at steady state only [7, 12]. Indirect optimization methods (IOMs) convert a nonlinear kinetic model into an S-system model, and then solve the optimization problem at steady state using a linear programming method [10–13]. On the contrary, stochastic optimization methods and deterministic branch-and-reduce methods are directly applied to nonlinear models to obtain a global optimum [4, 15]. Optimization problems for metabolic network systems can be categorized as single-objective and multi-objective formulations, depending on the design purpose. Most studies on microbial metabolic engineering focus on only a single objective to maximize the synthesis rate of the desired metabolite [4, 16]. In contrast, a multi-objective optimization approach attempts to find the solutions that are optimal for many objectives simultaneously. The multi-objective indirect optimization method (MOIOM) has been applied to maximize ethanol productivity and to minimize intermediate concentrations simultaneously [10].

Selecting a proper genetic manipulation strategy for metabolic network optimization problem is a tedious task. The regulatory structure of metabolic networks can be determined by model-based optimization strategies [4, 16]. Researchers have used mixed-integer linear programming to determine an optimal regulatory structure and the synthesis rate of metabolic systems described by linear models [4, 5]. However, the minimum set of enzymes (or corresponding genes) in a metabolic system that should be manipulated to obtain a viable strain under the situation of producing the maximum possible flux or yield of a desired final product remains unclear. This study introduces a multi-objective optimization formulation to find an optimal regulatory structure to cope with these problems. Experimental results show that a strain may reflect resilience phenomenon after stressful environmental changes and genetic perturbations [17, 18]. This resilience phenomenon means that the mutant strain may respond with rapid and dramatic alterations to global genetic perturbations. However, after genetic perturbations, the mutant tries to evolve to a new steady state that may be only slightly different from its previous steady state. This new steady state indicates that the mutant strain tries to recover from its "wild-type" characteristics and maintain relative stability on metabolism. Accurately predicting the steady state of a microbial strain after gene manipulations is not a trivial job, since the adaption of metabolic systems against gene alterations is complex. Segrèt et al. introduced the minimization of metabolic adjustment (MOMA) method to calculate the minimum distance solution relative to the original "wild-type" solution for the mutant strain [17]. Shlomi et al. applied regulatory on/off minimization (ROOM) to determine minimum number of changes between the mutant strain and the original strain [18]. However, both of these models are based on the stoichiometric model derived from the flux balance analysis (FBA). Almost no studies discuss the resilience phenomena for metabolic systems described by kinetic models.

This study introduces a generalized fuzzy multi-objective optimization problem (GFMOOP) to determine the optimal enzymatic manipulations for metabolic network systems considering resilience effects. This study first formulates a multi-objective optimization problem that simultaneously considers the resilience effects and minimum set of manipulated enzymes by combining the concepts of MOMA and ROOM into an optimization framework. Since nonlinear kinetic models offer a more detailed description of metabolic networks than stoichiometric models and the gene manipulations, including gene repressions and over-expressions, in metabolic networks can directly correspond to the changes of maximum flux parameters and reaction rates in the kinetic models, this study uses a nonlinear kinetic model in the optimization formulation. Integer variables are also introduced to model gene over-expression and repression. Thus, the optimization formulation must be solved using mixed-integer nonlinear programming (MINLP) methods. The metabolic networks for ethanol production by *Saccharomyces cerevisiae* and amino acid synthesis rates in *Escherichia coli* were employed to evaluate the applicability of the GFMOOP. Suitable membership functions are used to quantify the resilience effects and cell viability constraints. Results show that the maximum synthesis rates of target products by genetic interventions are always over-estimated in metabolic networks that do not consider the resilience effects.

## Results and discussion

Each following example solves two optimization problems and compares their results. The first problem is a primal optimization problem for determining the optimal enzyme manipulations, corresponding gene over-expression or repression, in metabolic networks without considering cell viability and metabolic adjustment. The second problem is the fuzzy optimization problem, which is similar to the primal optimization problem, but considers cell viability and metabolic adjustment.

### Maximization of the ethanol production by *S. cerevisiae*

*S. cerevisiae*is still the most important microorganism for ethanol production to date. Researchers have developed many strategies to enhance ethanol productivity using yeast, and its metabolic network is well studied. Figure 1 shows a scheme of the simple metabolic network of

*S. cerevisiae*for anaerobic ethanol production. Curto et al. developed a GMA model for analyzing the anaerobic ethanol fermentation of

*S. cerevisiae*at steady state [19]. This model consists of five nonlinear ordinal differential equations and eight nonlinear rate equations. Detailed information about this model can be found in the Additional File 1.

*S. cerevisiae*. The feasible region for each metabolite and enzyme can be estimated through biological understanding or global optimization techniques [22, 23]. This study sets the feasible region for each metabolite and enzyme to expand/shrink 5-fold based on its basal value. The primal optimization problem for maximizing the ethanol productivity in

*S. cerevisiae*was first solved by MIHDE to obtain the Pareto front, shown as the red curve in Figure 2. The larger the allowable number of the manipulated enzymes in the metabolic network, the higher the improved ethanol flux ratio, ${v}_{PYK}\u2215{v}_{PYK}^{basal}$. The highest improvement (about 5.2) was achieved when the allowable number of the manipulated enzymes was greater than six. Figure 2 shows all feasible solutions (red data points) for the primal optimization problem. Many improvements in the ethanol flux ratio are close to the highest value; for example, if at most two enzymes can be modulated, seven out of 28 feasible solutions with the improved ethanol flux ratio greater than 2.0 are obtained. The highest improved ratio is 2.452 and the corresponding modulated enzymes are HXT and PFK in this case.

The optimal solution for maximizing ethanol productivity by S. cerevisiae

| ${v}_{PYK}^{*}\u2215{v}_{PYK}^{basal}$ | Modulated enzymes |
---|---|---|

1 | 2.092 | HXT |

2 | 2.452 | HXT, PFK |

2.434 (SBB) | HXT, ATPase | |

3 | 3.152 | HXT, PFK, PYK |

4 | 3.592 | HXT, PFK, PYK, TDH |

3.326 (LINDOGlobal, BARON) | HXT, PFK, PYK, ATPase | |

5 | 4.428 | HXT, PFK, PYK, TDH, GLK |

6 | 5.191 | HXT, PFK, PYK, TDH, GLK, ATPase |

4.458 (DICOPT) | HXT, PFK, PYK, TDH, GLK, GOL | |

7 | 5.231 | HXT, PFK, PYK, TDH, GLK, ATPase, GOL |

3.651 (DICOPT) | HXT, PFK, PYK, TDH, ATPase, GOL, TPS | |

8 | 5.231 | HXT, PFK, PYK, TDH, GLK, ATPase, GOL, TPS |

*v*

_{ PYK }, was activated by the concentrations of metabolites, [f6p] and [pep], and inhibited by [atp] so that the improved ratio of

*v*

_{ PYK }to its basal value can be expressed as

The optimal solution for maximizing ethanol productivity by S. cerevisiae considering resilience effects

| ${v}_{PYK}^{*}\u2215{v}_{PYK}^{basal}$ | Modulated enzymes |
---|---|---|

1 | 1.482 | HXT |

2 | 1.710 | HXT, TDH |

1.618 | HXT, PFK | |

1.519 | HXT, ATPase | |

3 | 1.991 | HXT, TDH, ATPase |

1.877 | HXT, TDH, PFK | |

1.663 | HXT, PFK, PYK | |

4 | 2.340 | HXT, TDH, PFK, PYK |

5 | 2.741 | HXT, TDH, PFK, PYK, GLK |

6 | 3.080 | HXT, TDH, PFK, PYK, GLK, ATPase |

7 | 3.106 | HXT, TDH, PFK, PYK, GLK, ATPase, GOL |

8 | 3.105 | HXT, TDH, PFK, PYK, GLK, ATPase, GOL, TPS |

The exponents in equation (1) indicate that the ethanol synthesis flux, *v*_{
PYK
} , increases when the concentrations of metabolites [f6p] or [pep] increase and the concentration of metabolite [atp] decreases. Figure 1 shows the optimal flux ratios for the modulated enzymes HXT and TDH (red numbers) and modulated enzymes HXT and PFK (blue numbers), respectively. Figure 1 also shows the changed concentration ratios of the metabolites for the corresponding modulated enzymes. For the allowable modulated enzyme set {HXT, TDH}, the optimal concentration ratios of [f6p] and [atp] are smaller than those obtained by modulating enzymes HXT and PFK. This indicates that lower [atp] and higher [pep] increase the ethanol flux rate *v*_{
PYK
} . This result makes sense from a biological viewpoint, since a lower [atp] level slows down cell growth and allows yeast to carry out the anaerobic fermentation required to produce ethanol. Although the exponent of [f6p] is positive, the small value causes an insignificant effect on the improved ethanol flux ratio. As a result, the maximum value of *v*_{
PYK
} obtained by modulating enzymes HXT and TDH exceeds that by modulating enzymes HXT and PFK. Following the similar procedures, the best selected enzymes to be modulated for the resilience problem when at most three enzymes can be modulated are HXT, TDH, and ATPase. These results differ from those obtained from the primal optimization problem, as Tables 1 and 2 indicate. The ATPase enzyme does not appear in the suggested modulated enzyme set obtained from the resilience problem when four and five manipulated enzymes are allowed. As a result, we cannot prioritize the selection of enzymatic modulations in the optimization problems when at most three enzymes can be modulated. However, both primal optimization and resilience problems have the identical selected enzymes when the allowable number of manipulated enzymes is greater than three.

### Multi-objective maximization of amino acid synthesis rates in *Escherichia coli*

*E. coli*. The kinetic model for the network is complex and highly nonlinear. Chassagnole et al. developed a nonlinear dynamic model for part of central carbon metabolism of

*E. coli*[24]. Their model has the ability to describe the experimentally observed dynamic behavior of metabolites in metabolic networks and is also capable of describing the intracellular metabolite oscillations observed in experiments [25]. This model links the kinetics of sugar transporter PTS (phosphor-transferase system) with glycolysis and pentose-phosphate pathways, and is used to support the exploration of the central carbon metabolism of

*E. coli*. Figure 4 presents a schematic diagram of central carbon metabolism of

*E. coli*. It depicts 30 enzymatic reactions, 18 metabolites or precursors, and seven co-metabolites (amp, adp, atp, nadp, nadph, nad, and nadh). These co-metabolite concentrations are assumed to be constant in the mathematical model. The reaction rates can be accessed from the model database of JWS Online Cellular Systems Modeling (http://jjj.biochem.sun.ac.za/). The detail information of the kinetic model for the central carbon metabolism in

*E. coli*can be found in the Additional File 2.

Many researchers have applied optimization methods to enhance synthesis capabilities of microbial strains [16, 26, 27]. Most of the works on optimization of microbial strains focused on single-objective optimization. For example, Vital-Lopez et al. used a kinetic model of the central carbon metabolism of *E. coli* to identify optimal intervention strategies under the maximization of serine synthesis [16]. Lee et al. applied bi-objective optimization methods to investigate the influences of gene interventions on the amino acid synthesis using *E. coli*[26]. Lee et al. were interested in maximizing DAHPS, PEPC, and SERS enzymatic flux ratios that correspond to the enhancement of synthesis of aromatic amino acids, serine, and oxaloacetate, respectively. Their study solves two bi-objective optimization problems for maximizing DAHPS and PEPC flux ratios and maximizing DAHPS and SERS flux ratios, respectively, to determine the optimal gene manipulation strategies. However, this current study determines the optimal enzyme manipulation strategies to maximize the flux ratios of DAHPS, PEPC, and SERS simultaneously through genetic manipulations.

The optimal solution for multi-synthesis maximization by E. coli

| $\frac{{v}_{PEPC}^{*}}{{v}_{PEPC}^{basal}}$ | $\frac{{v}_{SERS}^{*}}{{v}_{SERS}^{basal}}$ | $\frac{{v}_{DAHPS}^{*}}{{v}_{DAHPS}^{basal}}$ | Modulated enzymes | Optimal objective value$({\eta}_{D}^{*})$ |
---|---|---|---|---|---|

1 | 1.271 | 1.081 | 1.560 | PK | 0.970 |

1.342 | 1.068 | 1.780 | G6PDH | 0.975 | |

2 | 1.248 | 1.518 | 1.652 | G6PDH, SERS | 0.846 |

1.211 | 1.409 | 1.456 | PK, SERS | 0.878 | |

1.778 | 1.106 | 2.185 | PK, G6PDH | 0.969 | |

3 | 1.578 | 1.860 | 2.027 | G6PDH, PK, SERS | 0.782 |

1.388 | 1.730 | 1.872 | G6PDH, SERS, RPPK | 0.814 | |

4 | 1.801 | 1.973 | 2.225 | G6PDH, PK, SERS, RPPK | 0.778 |

1.492 | 1.934 | 2.175 | G6PDH, PK, SERS, DAHPS | 0.787 | |

5 | 1.597 | 2.258 | 2.467 | G6PDH, PK, SERS, RPPK, DAHPS | 0.763 |

1.958 | 2.134 | 2.322 | G6PDH, PK, SERS, RPPK, SYN1 | 0.786 |

The optimal solution for multi-synthesis maximization by E.coli considering resilience effects

| $\frac{{v}_{PEPC}^{*}}{{v}_{PEPC}^{basal}}$ | $\frac{{v}_{SERS}^{*}}{{v}_{SERS}^{basal}}$ | $\frac{{v}_{DAHPS}^{*}}{{v}_{DAHPS}^{basal}}$ | Modulated enzymes | Optimal objective value$({\eta}_{D}^{*})$ |
---|---|---|---|---|---|

1 | 1.262 | 1.079 | 1.545 | PK | 0.971 |

1.342 | 1.068 | 1.780 | G6PDH | 0.975 | |

2 | 1.214 | 1.447 | 1.586 | G6PDH, SERS | 0.867 |

1.186 | 1.365 | 1.407 | PK, SERS | 0.891 | |

1.763 | 1.105 | 2.174 | PK, G6PDH | 0.968 | |

3 | 1.443 | 1.740 | 1.884 | G6PDH, PK, SERS | 0.811 |

1.314 | 1.591 | 1.764 | G6PDH, SERS, RPPK | 0.849 | |

4 | 1.582 | 1.829 | 2.044 | G6PDH, PK, SERS, RPPK | 0.810 |

1.412 | 1.782 | 1.985 | G6PDH, PK, SERS, DAHPS | 0.821 | |

5 | 1.479 | 2.010 | 2.177 | G6PDH, PK, SERS, RPPK, DAHPS | 0.809 |

1.704 | 1.980 | 2.143 | G6PDH, PK, SERS, RPPK, SYN1 | 0.815 |

The optimal enzyme PK is selected from the 30-enzyme network if only one enzyme can be manipulated. The improved flux ratios of PEPC, SERS, and DAHPS for the primal optimal problem are nearly identical to those obtained from the resilience problem. This indicates that the resilience phenomenon has little effect on the cell response when only one enzyme alteration is allowed. In contrast, the optimal selected enzymes are G6PDH and SERS if two enzyme manipulations are allowed. Both flux ratios of SERS and DAHPS are enhanced even though the PEPC flux ratio is smaller than that obtained by modulating PK only. As a result, the pair of the selected enzymes (G6PDH and SERS) is a Pareto optimal solution. To confirm this result, solve the primal optimization problem and resilience problem using the fixed modulation pairs of (PK, SERS) and (PK, G6PDH). The three maximum flux ratios obtained by modulating PK and SERS are less than those obtained by manipulating G6PDH and SERS. This indicates that the modulation pair of (PK, SERS) is dominated by (G6PDH, SERS). The maximum flux ratios of PEPC and DAHPS increase and the maximum flux ratio of SERS decreases in comparison with those obtained by using the modulation pair (G6PDH, SERS). Thus, the result for the modulation pair (PK, G6PDH) is also a Pareto solution. However, the optimal objective value for (PK, G6PDH) exceeds that of manipulating G6PDH and SERS, so we obtain a single Pareto solution. The Pareto solution for manipulating G6PDH is similar to those show in Tables 3 and 4.

Similar procedures were also used to obtain the optimal modulations for the primal optimization problem using different allowable numbers of manipulated enzymes ranging from three to five, respectively. Tables 3 and 4 show the convergent solutions obtained by seven solvers in GAMS. For the case of the allowable numbers of three and four, we obtained two convergent solutions, but a Pareto optimal solution only. Tables 3 and 4 also show that the optimal solution for *n* allowable modulated enzymes includes the suggested enzyme set for *n* - 1 allowable modulated enzymes. This trend is same as that obtained by Voit and Signore resulting from the effect of experimental imprecision [28]. Finally, we could qualitatively speculate that the suggested enzymes to be modulated for both primal and resilience optimization problems are {G6PDH, PK, SERS, RPPK, DAHPS, SYN1} in decreasing order of priority.

*E. coli*cells and the ratios of [

*nad*]/[

*nadh*] and [

*nad*]/[

*nadh*] in the redox metabolic network are equal to the ratios of their basal levels and can be considered as constants [29]. Their relationships are expressed as

where *c*_{
i
} are constants. This study also evaluated the effects of the assumption of constant co-metabolite concentrations by applying equations (2)-(4) to the optimization problems. Similar procedures were also used to obtain the optimal modulations for the primal and resilience optimization problems. The computational results can be found in Tables S1 and S2 of Additional File 3. We could conclude that the improved flux ratios are a little different from those obtained from the optimization problems with constant co-metabolite concentrations. The order of priority of the suggested modulated enzymes is nearly identical to that obtained based on the assumption of constant co-metabolite concentrations except when the number of allowable modulated enzymes is five.

## Conclusions

The optimization of biological systems, which is a branch of metabolic engineering, has generated a lot of industrial and academic interest for a long time. The ultimate goal of this optimization is to find the optimal mutation strategy for improving productivity. Model-based optimization strategies have been applied to analyze and design metabolic networks during the last decade. The accuracy of optimization results depends heavily on the development of essential kinetic models of metabolic networks. Kinetic models can quantitatively capture the experimentally observed regulation data of metabolic systems and are often used to find the optimal manipulation of external inputs. To address the issues of optimizing the regulatory structure of metabolic networks, it is necessary to consider qualitative effects, e.g., the resilience phenomena and cell viability constraints. Combining the qualitative and quantitative descriptions for metabolic networks makes it possible to design a viable strain and accurately predict the maximum possible flux rates of desired products.

This study introduces a generalized fuzzy multi-objective optimization approach to determine the optimal enzymatic manipulations for metabolic network systems to obtain the maximal flux ratios of desired metabolites of interest. The goal of this optimization problem is to find the maximal synthesis rates of desired metabolites and the minimum set of manipulated enzymes simultaneously based on kinetic models. The kinetic model was directly used for the optimization problem and MINLP solvers were applied to determine which enzymes to manipulate and how their corresponding activities changed. This study applies fuzzy equal and fuzzy inequality operations to the optimization problem to deal with the resilience phenomena and cell viability constraints. This study tests the practical utility of the proposed approach by applying it to two metabolic networks of *S. cerevisiae* and *E. coli*. The resulting optimal enzymatic manipulations for metabolic networks and the maximum flux ratios for desired metabolites are more justifiable based on biological knowledge. We could qualitatively speculate the priority of modulated enzymes obtained by iteratively solving the optimization problems using various allowable numbers of manipulated enzymes. These results can help microbiologists make a proper decision when genetically modifying a microorganism.

## Methods

### Kinetic model

where **x** ∈ ℝ ^{
n
} is a vector of concentrations of metabolites or pools, **e** ∈ ℝ ^{
m
} is a vector of enzyme levels corresponding to the enzyme activities, *θ* ∈ ℝ ^{
p
} is a vector of parameters, **v** ∈ ℝ ^{
m
} is a vector of reaction rates, and *S* ∈ ℝ ^{
n×m
} is the stoichiometric matrix describing the interconnecting fluxes. The stoichiometry of biochemical reactions is constant. Kinetic aspects are used to capture the dynamics of a system and may change rather quickly as they are driven by the state of the system. The stoichiometry of a biochemical pathway determines the wiring diagram of the network, describes which fluxes enter or leave which pool, and ensures that mass is conserved in the process. The reaction rate can be expressed by the power-law functions or Michaelis-Menten-based rate laws in the field of biological systems.

### Primal optimization problem

*i*

^{ th }flux

*v*

_{ i }∈

**v**, Σ

_{ O }∈ ℕ

^{ r }is the set of indices of production rates to be maximized,

*r*is the number of target fluxes to be maximized, and the binary variable

*y*

_{ j }∈

**y**indicates whether the

*j*

^{ th }enzyme should be modulated and is defined as

Equation (6) is a general formula for simultaneously maximizing a set of metabolite synthesis rates. Several researchers have introduced genetic manipulations to redistribute various metabolic fluxes in a metabolic network to enhance the desired synthesis rates [16, 27]. Equation (7) obtains the minimum set of modulated enzymes in the metabolic network.

*i*is not modulated, then its activity can have a small variance around its basal value ${e}_{i}^{basal}$. The lower and upper perturbation bounds, ${b}_{i}^{LB}$ and ${b}_{i}^{UB}$, restrict the activity variance of

*i*

^{ th }un-modulated enzyme due to other enzyme alterations. A similar non-significant variance in enzyme activity discussed in the assumption of ROOM [18]. Constraint (9) indicates that at least one enzyme/gene should be manipulated. The concentration for each metabolite is restricted by its lower and upper bounds,

where ${\gamma}_{{x}_{i}}^{LB}$ and ${\gamma}_{{x}_{i}}^{UB}$ are the lower and upper bounded factors for each metabolite, respectively, and ${x}_{i}^{basal}$ is the basal value of the *i*^{th} metabolite *x*_{
i
} ∈ **x**.

where ζ _{
x
} and ζ _{
e
} are the restriction factors for the constraints on total metabolite concentration and total enzyme concentration, respectively.

The primal multi-objective optimization problem formulated by equations (5) to (12) is a multi-objective mixed-integer nonlinear programming problem. Many methods are capable of solving multi-objective optimization problems (MOOPs) to obtain the Pareto front [30–32] and generally fall into one of two categories: generating methods and preference-based methods. Each method has its advantages and disadvantages, as discussed in several articles [30–32]. Generating methods can apply a scalarization approach to convert an MOOP into a single-objective optimization problem (SOOP) with different factors to find one Pareto optimal solution. A series of the SOOP with various factors must be solved to find a Pareto front of the MOOP. Evolutionary algorithms can be directly applied to the MOOP to find the Pareto front, but they are time-consuming. A decision maker (DM) then selects a desired solution from the Pareto front. In contrast, the preference-based methods require preferences in advance from the DM and then find a satisfactory solution. However, preferences are generally difficult to specify with limited knowledge of the values of objective functions. Therefore, an interactive algorithm must be carried out to find a compromised solution.

*ε*-constraint, one of the generating methods, retains only one of the objective functions as the criterion and converts the others into inequality constraints. This approach is suitable for the MOOP with objective functions, which can easily assign the

*ε*-values. The weighted infinite norm method is a reference-goal method that can conveniently determine a trade-off solution if the lower and upper bounds of each objective value are known in advance. This study combines the

*ε*-constraint method and weighted infinite norm method to solve the primal multi-objective optimization problem. The objective function in equation (7) can be straightforwardly converted into an

*ε*-constraint because the number of enzymes is an integer value. The primal multi-objective optimization problem can be transformed into a weighted infinite norm problem defined as follows:

where the user provides the allowable number *ε* of the manipulated enzymes in advance, the lower bound ${v}_{i}^{LB}$ is equal to its basal flux ${v}_{i}^{basal}$, the upper bound ${v}_{i}^{UB}$ can be estimated by SOOP that maximizes *v*_{
i
} only, and the feasible set Ω consists of all feasible solutions that satisfy the material balance equations in the steady state and the constraints in equations (8)-(12).

### Resilience phenomena

where Σ _{
X
} ∈ ℕ ^{
n
} is the set of metabolite indices and Σ _{
E
} ∈ ℕ ^{
m
} is the set of enzyme indices. Here, the symbols, ≿ and ≈ denote a relaxed or fuzzy version of the ordinary inequality "≥" and equality "=", respectively. The fuzzy maximization, "$\stackrel{\u0303}{max}$", in equation (15) means that the enzyme manipulation is completely acceptable if the *i*^{
th
} flux ratio exceeds its upper bound ${f}_{i}^{UB}$, which can be estimated from the previous primal optimization problem. Conversely, the design is completely unacceptable if the *i*^{
th
} flux ratio is less than the lower bound ${f}_{i}^{LB}$. The lower bound is generally equal to one, meaning that the modified flux should exceed its basal value. Equations (16) and (17) are "fuzzy equal $\left(\stackrel{\u0303}{equal}\right)$" objective functions that represent the fuzzy goals. For example, the metabolite concentration *x*_{
j
} and enzyme activity *e*_{
k
} should be restored to a state that is as close to the wild-type as possible. Equation (18) is the crisp objective function as same as Equation (7).

where the symbol "≾" denotes a fuzzy version of the ordinary inequality "≤". Here, ${\zeta}_{x\u2215e}^{LB}$ and ${\zeta}_{x\u2215e}^{UB}$ are the lower and upper restriction factors for the fuzzy constraints on total metabolite/enzyme concentrations, respectively. The interval bound $\left[{\zeta}_{x\u2215e}^{LB},{\zeta}_{x\u2215e}^{UB}\right]$ indicates that the microbes have some degree of satisfaction if each total metabolite/enzyme concentration is within its boundary. The lower bounds of the fuzzy inequality constraints mean that the microbes are completely survival if both total metabolite/enzyme concentration constraints in equations (19) and (20) are less than their lower limits. Conversely, the microbes completely die if one of the total metabolite/enzyme concentration constraints exceeds its upper limit. This situation indicates that the solution is infeasible.

### Goal-attainment problem

*ε*-constraints, abbreviated as

*ε*-FMOOP, by transforming the crisp objective function to an

*ε*-constraint. To solve the

*ε*-FMOOP, each fuzzy objective functions for maximizing synthesis rate can be quantified by eliciting the following membership function:

where *d*_{
i
} is a strictly monotonically increasing function for evaluating the degree of satisfaction. The maximum synthesis rate becomes somewhat acceptable if its objective value lies between the lower and upper bounds. The membership function value is zero if the synthesis rate is less than the desired lower bound. Conversely, the grade of membership function is one when the synthesis rate exceeds its upper bound.

where ${x}_{j}^{LB}$ and ${x}_{j}^{UB}$ are the lower and upper bounds for the concentration of the *j*^{
th
} metabolite, and the user-provided functions ${d}_{j}^{\prime}$ and ${d}_{j}^{\u2033}$ are strictly monotonically increasing and decreasing functions, respectively.

where ${d}_{k}^{\prime \prime \prime}$ is a strictly monotonically decreasing function for evaluating the degree of satisfaction. The value of the membership function is one if the total amount of the metabolite concentrations is less than the desired lower bound. Conversely, the grade of membership function is zero when the total amount of the metabolite concentrations exceeds its upper bound. The cell viability becomes somewhat acceptable if the total amount lies between the bounded interval.

*ε*-FMOOP can be expressed as the goal attainment problem:

where ${\stackrel{\u0304}{\eta}}_{i}$ is the ideal preferred goal, Σ = Σ _{
O
} ∪ Σ _{
X
} ∪ Σ*E*, and *η*_{
D
} denotes an aggregation function defined on the crisp domain Ω, which consists of the feasible solutions satisfied equation (5), the crisp bounds in equations (8)-(9), and the *ε*-constraint in equation (14). Sakawa introduced several aggregation functions in which the value of the aggregation function can be interpreted as an overall degree of satisfaction with user's fuzzy goals [31]. This study uses the first term of the aggregation function in the brace of equation (24) to identify the optimal trade-off solution that is nearest to the ideal preferred goal, ${\stackrel{\u0304}{\eta}}_{i}$, which indicates 100% satisfaction. The second term avoids testing the uniqueness for optimality of the solution and the constant *δ* is a small positive value within 10^{-3} - 10^{-5}. The fuzzy goal attainment approach can directly find a satisfactory solution in the Pareto set without yielding the Pareto frontier of the problem.

## Declarations

### Acknowledgements

The financial support from the National Science Council, Taiwan, ROC (Grant NSC100-2221-E-194-028-MY3 and NSC100-2627-B-194-001), is highly appreciated.

## Authors’ Affiliations

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