Cell metabolic model of an interpretable modular structured network

Abstract

The application discloses a cell metabolism model of an interpretable modular structure network. The modular network comprises an input layer, two hidden layers and an output layer. Firstly, the functions of each module are determined, the input layer is corresponded to the enzyme concentration in cells, the enzyme abundance data is used for calculation instead, the two hidden layers are corresponded to the upper bound of reaction flux and the size of reaction flux respectively, and finally, the output layer is defined as the concentration change of metabolites in cells. Four modules are integrated into a completed network model, and the connection between layers has practical significance. The connection parameters between layers are pretreated, the logical relationship between the processed parameters is determined, and the parameters and structure of the model are determined. The mathematical formula of the network is analyzed, the formula is deduced, and a solving method based on a differential evolution algorithm and a matrix null space search is developed. Finally, iteration parameters are set, and the metabolic model is simulated and calculated. The application can be applied to cell modeling in the fields of pharmaceutical, food and other industrial biology.

Description

Technical Field

[0001] This invention belongs to the interdisciplinary field of biotechnology and information technology, and involves the construction of interpretable modular genome-scale metabolic network models of microorganisms. Background Technology

[0002] Metabolism refers to a series of chemical reactions that occur within living organisms, and metabolic systems play a crucial role in cellular activities within microorganisms. Genome-scale metabolic network models (GSMMs) have long been considered capable of expressing the relationships between metabolites, metabolic reactions, and proteins during cellular metabolism. As a bottom-up systems biology tool, GSMMs enable the prediction of cellular metabolism and phenotypes based on specific constraints. Integrating other omics data into GSMMs to enhance their performance is one of the development directions for improving GSMM performance. Among these, enzyme-constrained metabolic network models are considered to be effective in improving model performance by integrating protease usage data for flux constraints.

[0003] When using genome-scale metabolic network models constrained by microbial enzymes for phenotypic and metabolic prediction, a commonly used method is flux balance analysis. Flux balance analysis is a mathematical method that analyzes the fluxes of metabolite-related reactions through cellular metabolic networks. This method defines a linear programming problem based on a single objective, where the optimization objective is to maximize the specific growth rate of the microorganism. The optimization result obtained through flux balance analysis is a unique set of flux distributions, which can serve as a tool for analyzing and designing microbial metabolic processes.

[0004] With the introduction of the concept of whole-cell digital models, the importance of metabolic networks has been further enhanced, and the requirements for the results of metabolic network models are becoming increasingly stringent. When the solution results of metabolic network models reach a certain level of accuracy, they can be used to guide research in metabolic studies or genetic engineering. Therefore, this paper proposes an interpretable modular network structure cellular metabolic model based on the homeostasis assumption, constructing a two-hidden-layer network model integrated from four modules. Simultaneously, through analysis and derivation of the network's calculation formulas, a new solution method is developed, introducing a difference evolutionary algorithm with stochasticity to make the calculation results less singular and consistent with the stochastic laws of biological growth. This presents a new approach to the construction of metabolic network models.

[0005] Differential Evolutionary Algorithm (DEA) is a simple and efficient heuristic parallel search algorithm for solving problems. It boasts advantages such as fast convergence, few and simple control parameters, and robust optimization results. It is a population-based global search algorithm. DEA uses the difference information of a certain proportion of multiple individuals as the perturbation of each individual, making the algorithm adaptive in the search direction. Because the initial iteration individuals are randomly given during initialization, and random parameters are introduced in each iteration, DEA possesses excellent randomness and the ability to learn from the population, giving it unparalleled performance compared to other evolutionary algorithms. Summary of the Invention

[0006] The purpose of this invention is to propose an interpretable modular network structure cellular metabolism model based on the homeostasis assumption, constructing a two-hidden-layer network model integrated from four modules. After deriving the network's computational formula, a differential evolution algorithm is introduced to solve the model.

[0007] To achieve the above-mentioned objectives, the specific technical solution of this invention is as follows:

[0008] A cell metabolism model with an interpretable modular network structure is constructed from a two-hidden-layer network model integrating four modules. After deriving the network's computational formula, a differential evolution algorithm is introduced to solve the model, resulting in a cell metabolism model with an interpretable modular network structure. The specific steps include:

[0009] Step 1: Identify the four modules of the metabolic network;

[0010] Step 2: Establish connections between the four modules, integrate the four modules into a complete network model, and give practical meaning to the connections between each layer;

[0011] Step 3: Preprocess the parameters of each layer and the connections between them to determine the model parameters and structure;

[0012] Step 4: Analyze the relationship between the network's input and output, and establish mathematical formulas. Derive these formulas and construct new calculation methods.

[0013] Step 5: Set the iteration parameters and perform simulation calculations on the metabolic model.

[0014] Specifically, for the network modules described in step 1, the interpretable modular network structure of the cell metabolism model can be divided into four modules: enzyme concentration module, flux upper bound module, flux data module, and metabolite concentration module. Considering the non-negligible catalytic effect of proteases on the reaction, this effect will cause changes in the metabolic reaction flux. The input part of the entire network model is defined as the enzyme concentration module, and the input layer is defined as the enzyme concentration. The first hidden layer and the second hidden layer are the flux upper bound module and the flux magnitude module, respectively. The metabolite concentration module serves as the network output layer, representing the concentration changes of each metabolite during the metabolic process.

[0015] Specifically, regarding the module connections described in step 2, based on the definitions of each sub-module, the four modules can be integrated into a complete, interpretable network model. The modules are connected via weights. Specifically, the input layer and the first hidden layer are connected via the enzyme's turnover number k. cat By connecting these layers and calculating the upper bound of the reaction flux using enzyme concentration and turnover rate, a protease constraint is formed. The first and second hidden layers are connected by a weighted one-to-one correspondence, meaning the flux of each reaction is determined by the different proportions of their flux upper bounds. The second hidden layer, together with the output layer, constitutes the steady-state equilibrium condition, and all the weights within it also have practical significance. If the metabolic reaction represented by a node in the second hidden layer includes a metabolite represented by a node in the output layer, then a stoichiometric coefficient exists between them to represent their relationship. Therefore, the relationship between the nodes in the second hidden layer and the output layer is expressed by the stoichiometric coefficient. All four modules are connected by interpretable weights, constructing a complete and highly interpretable modular network structure cellular metabolism model.

[0016] Specifically, regarding the parameter preprocessing described in step 3, some key parameters need to be given in advance before solving the model, including the input enzyme concentration, enzyme turnover, and the model's stoichiometry matrix. Since the input to the interpretable modular network cell structure-metabolic model is the concentration of each protease, the flux upper bound of metabolic reactions is constrained by specifying the concentration of each protein. However, the number of proteins within a cell is enormous, making it extremely difficult to measure the concentration of each protease in real time during cell growth. Therefore, we can choose to obtain protein abundance data from a database, which, after calculation, represents the intracellular protease concentration, thereby constraining metabolic reactions. Under different enzyme catalytic conditions, proteases catalyze metabolic reactions with certain logical relationships, such as "or" and "and". Therefore, proteases that simultaneously catalyze the same reaction need to be analyzed and processed before the network model is calculated. The model's stoichiometry matrix is ​​a very sparse matrix, where each non-zero element represents the stoichiometric relationship between metabolites and metabolic reactions. All of these parameters need to be processed before solving the model.

[0017] Specifically, regarding the formula derivation in step 4, the structure of the interpretable modular network cell metabolism model shows that since most parameters in the network need to be pre-defined, the key to solving the model lies in finding the weights connecting the first and second hidden layers. Furthermore, most parameters in this modular network model are fixed, and as the model's dimensionality increases, using general gradient-based calculation methods may not be effective in finding the optimal solution. Therefore, to introduce a degree of randomness into the solution process to better reflect the laws of cell growth, we derived the model's formulas and developed a new solution method based on differential evolution and matrix null space search.

[0018] Specifically, for the metabolic model in step 5, the metabolic reactions and enzyme catalytic relationships in the model are compared with those in the Kegg and Uniprot databases to determine the specific data of the original model, based on the stoichiometric coefficients between each metabolic reaction and metabolite. Finally, the enzyme turnover of the protease in the central metabolism at each metabolic reaction is obtained from the BRENDA database.

[0019] Compared with existing technologies, the method of this invention has the following advantages and beneficial effects: (1) It proposes a new structure for the metabolic network model, enabling the model to match more complex processes; (2) Each node and connection in the model contains practical meaning, making the model highly interpretable; (3) It replaces enzyme concentration with calculated enzyme abundance data, solving the limitation that enzyme concentration cannot be measured; (4) It develops a model solving method based on differential evolution algorithm through formula derivation, introducing randomness to make the results more reasonable; (5) It provides more accurate simulation results for microbial specific growth rates, which can improve the accuracy of the model. Attached Figure Description

[0020] Figure 1 This is a structural diagram of the cell metabolism model that can explain the modular network structure in this invention.

[0021] Figure 2 This is the iterative curve of the overall objective function of the solution algorithm proposed in this invention on the central metabolism of Saccharomyces cerevisiae.

[0022] Figure 3 This is an iterative curve showing the degree of violation of upper and lower bounds in the central metabolism of Saccharomyces cerevisiae by the solution algorithm proposed in this invention.

[0023] Figure 4 This is the specific growth rate iteration curve of the solution algorithm proposed in this invention on the central metabolism of Saccharomyces cerevisiae. Detailed Implementation

[0024] The present invention will be further illustrated by the following examples:

[0025] In this example, the central metabolic model of *Saccharomyces cerevisiae* is used in the interpretable modular network structure cellular metabolic model. This model contains 113 metabolic reactions, 81 metabolites, and 151 proteases. This embodiment utilizes this model to construct an interpretable modular network metabolic model of *Saccharomyces cerevisiae* and solves it using a novel solution method. Figure 1 This is a diagram illustrating the overall structure of the modular network cell metabolism model that can be explained in the invention.

[0026] 1. Design and integration of each module

[0027] The interpretable modular network structure cell metabolism model can be divided into four modules (see...). Figure 1The network consists of four modules: enzyme concentration, flux upper bound, flux data, and metabolite concentration. Enzyme concentration serves as the input to the entire network; different enzyme concentrations yield the corresponding reaction flux under specific conditions. The flux upper bound module, as the first hidden layer, represents the allowable upper bound of flux for each metabolic reaction, defining the solution range for each reaction during network computation. The second hidden layer, the flux data module, characterizes the specific flux of each metabolic reaction. It is the core of the interpretable modular network structure cellular metabolism model, as the goal of solving the metabolic network is to obtain a set of flux data that meets the conditions; therefore, the data in this module is crucial for evaluating the network model's computational results. The metabolite concentration module, as the network output layer, represents the concentration changes of each metabolite during metabolism. This module plays a vital role in solving the model and forms the foundation of the interpretable modular network structure cellular metabolism model.

[0028] This network is built upon a model of cellular homeostasis. Based on the definitions of each sub-module, the four modules can be integrated into a complete and interpretable network model. The proposed modular network model is a two-hidden-layer neural network model. The input layer represents the concentration of the protease, the first hidden layer represents the upper bound of the flux for each metabolic reaction, the second hidden layer represents the flux magnitude of each metabolic reaction, and the output layer represents the concentration change of the metabolite. The modules are connected by weights. Specifically, the input layer and the first hidden layer are connected through enzyme turnover; the upper bound of the reaction flux is calculated from the enzyme concentration and turnover number, forming the protease constraint. The first and second hidden layers are connected by a one-to-one correspondence of weights, meaning the flux magnitude of each reaction is determined by the different proportions of their flux upper bounds. The second hidden layer, together with the output layer, constitutes the homeostatic equilibrium condition, and all the weights within it also have practical significance. If the metabolic reaction represented by a node in the second hidden layer includes a metabolite represented by the output layer, then a stoichiometric coefficient exists between them to represent their relationship. Therefore, the relationship between the nodes in the second hidden layer and the output layer is expressed by the stoichiometric coefficient. The four modules are connected together by interpretable weights, forming a complete and highly interpretable modular network structure cell metabolism model.

[0029] 2. Interpretability of Modular Networks

[0030] Unlike typical neural networks that only define inputs and outputs, each node and connection in this modular network model contains practical meaning, meaning the model is highly interpretable.

[0031] from Figure 1As can be seen, the interpretable modular network structure cellular metabolism model integrates its modules in the form of a neural network. For the enzyme constraint module, each input node represents a protease in the metabolic process. Therefore, the connection weights between the input layer and the first hidden layer should be the enzyme turnover rate, thereby establishing a protease constraint to limit the reaction rate.

[0032] The first hidden layer is the upper bound of the metabolic reaction rate. According to formula (1), the product between enzyme turnover and enzyme concentration is defined as the flux upper bound for the existence of a catalytic relationship [9]. If there is no catalytic relationship between the protease and the metabolic reaction, the connection will be set to zero. It can be seen that the R6 node has no preceding input node, which means that the reaction is not catalyzed by the protease, so an upper bound value needs to be given manually.

[0033] R = k cat ·E (1)

[0034] Depend on Figure 1 The connection between the input and output layers also shows that enzyme catalysis can be divided into three types. The first type is a single enzyme catalyzing a single reaction, i.e., a one-to-one connection. Therefore, it is only necessary to obtain the corresponding enzyme concentration and turnover rate and then calculate the upper bound according to formula (1). The second type is a single enzyme catalyzing multiple different reactions, i.e., a one-to-many connection. This indicates that a certain protease can simultaneously produce catalytic effects on multiple reactions, and their catalytic abilities for these reactions are different, so their enzyme turnover rates for that reaction are also different. Therefore, after obtaining the enzyme concentration data, it is necessary to obtain the corresponding k based on the catalytic reaction. cat The parameters are then calculated according to formula (1). The third case is that multiple enzymes catalyze the same reaction, i.e., many to one. This case is the most complex because when multiple enzymes catalyze the same reaction, there will be certain relationships between these enzymes, which will affect the final catalytic effect. Therefore, after obtaining the corresponding enzyme concentration and turnover parameters, it is necessary to further analyze the relationship between them. The analysis method is described in detail in the parameter section.

[0035] The first and second hidden layers form the reaction flux module. The second hidden layer represents the flux of each metabolic reaction, calculated by multiplying a weight parameter W by the rate upper bound R, as shown in formula (2). Therefore, the weight parameter W should be a real number between 0 and 1. Thus, the first and second hidden layers are connected one-to-one. The flux range of each metabolic reaction is determined by the flux upper bound output by the preceding enzyme constraint module.

[0036] V = W·R (2)

[0037] The final steady-state equilibrium module comprises a second hidden layer and an output layer. The output layer represents the changes in the concentration of each metabolite, where S is the stoichiometric coefficient of the corresponding metabolic reaction and metabolite. This model is still based on steady-state modeling, therefore it is still assumed that the metabolite concentration does not change over time. The calculation formula for this layer is shown in formula (3). Similarly, connections between unrelated metabolites and metabolic reactions will be set to zero.

[0038] S·V=M=0 (3)

[0039] As can be seen from the formula, the key to solving the model lies in finding a suitable set of weight parameters W. After inputting a set of protease concentrations, the upper bound of each reaction rate is first calculated using formula (1), and then a suitable set of W is found so that the flux of each reaction meets the steady-state condition of formula (3). Among them, the enzyme turnover number k cat The stoichiometric coefficient S needs to be used as a pre-input for the network. After inputting the enzyme concentration E, the network solves to obtain a set of W, and the specific reaction flux simulation results are obtained according to formula (2).

[0040] Therefore, this modular network model demonstrates strong interpretability, with all components containing practical meaning. The model's construction can be fully represented by intuitive formulas, and the data flow from the input layer to the output layer can also be directly expressed using formulas. The mechanisms of each module and their integration can also be clearly explained.

[0041] 3. Model parameter preprocessing

[0042] according to Figure 1 As can be seen from the model structure, some key parameters need to be given in advance before solving the model, including the input enzyme concentration E and the enzyme turnover k. cat And the stoichiometric matrix S of the model.

[0043] Since the input to the interpretable modular network structure cell metabolism model is the concentration of each protease, the upper bound of the metabolic flux is constrained by giving the concentration of each protein. However, the number of proteins in a cell is enormous, making it very difficult to measure the concentration of each protease in real time during cell growth. Therefore, we can choose to obtain protein abundance data from a database, calculate it to represent the concentration of proteases in the cell, and thus constrain the metabolic response. Therefore, we need to process and calculate the abundance data. The specific method for calculating enzyme concentration is shown in formula (4).

[0044]

[0045] Here, abundance is the protein abundance data obtained from the database, and MW is the molecular weight of the protein. The molecular weight of each protein can be obtained from the UniProt database. Therefore, an estimated enzyme concentration based on protein abundance can be calculated.

[0046] Before performing calculations in the network model, it is necessary to analyze and process the proteases that simultaneously catalyze the same reaction. The logical relationship "or" indicates that several enzymes have comparable effects on catalyzing the same reaction; when the catalytic activity of one enzyme reaches its maximum, the catalytic activity of other enzymes can be ignored. Therefore, for such protease catalytic logical relationships, and to avoid constraining the flux upper bound to be too small, we choose the enzyme that maximizes the upper bound of the reaction flux and its corresponding k. cat The next step of the calculation is to use E, where the enzyme turnover number can be obtained from the BRENDA database. For example, for two enzymes with an "or" logical relationship and catalyzing the same metabolic reaction, their concentrations and turnover numbers in that reaction are E1 and E2, respectively, and k. cat1 and k cat2 The formula for calculating the upper bound of the rate of this metabolic reaction is given in formula (5).

[0047] E1or E2:R=max{k cat1 ·E1,k cat2 ·E2} (5)

[0048] Similarly, when several enzymes with an "and" relationship simultaneously catalyze the same reaction, it means that all of these enzymes are indispensable in the metabolic reaction. Each enzyme participates in the catalytic process, and the absence of any one enzyme will prevent catalysis. Therefore, the magnitude of the catalytic effect is determined by the enzyme with the least catalytic effect. In the model, this means finding the minimum value among the calculated values ​​of these enzymes for this reaction using formula (1). Similarly, the case where two enzymes with an "and" relationship simultaneously catalyze the same reaction can be represented by formula (6).

[0049] E1and E2:R=min{k cat1 ·E1,k cat2 ·E2} (6)

[0050] Before solving for W in the proposed model, the structure and parameters from the input layer to the first hidden layer need to be determined. This is because this part of the processing and computation can be directly obtained after determining the enzyme turnover parameter in the model and inputting the enzyme concentration. However, adding extra logical operations during network computation will significantly reduce its computational efficiency.

[0051] Finally, the stoichiometric matrix S of the model can be extracted from the metabolic model database BIGG. It is a very sparse matrix, where each non-zero element represents the stoichiometric relationship between metabolites and metabolic reactions. Before solving the IMNMM, the parameters S between the second hidden layer and the output layer need to be given in advance. After completing the above parameter settings, the most important W weight vector needs to be solved.

[0052] 4. Solution method for the model

[0053] As can be seen from the structure of the interpretable modular network cell metabolism model, since the parameters from the input layer to the first hidden layer and from the second hidden layer to the output layer need to be given in advance, the key to solving the model lies in finding the weights W between the first and second hidden layers. Furthermore, most parameters in this modular network model are fixed, and as the model's dimensionality increases, using general gradient-based calculation methods may no longer be effective in finding the optimal solution. Therefore, to introduce a certain degree of randomness into the solution process to better reflect the laws of cell growth, we derived the model's formulas in an attempt to discover some new calculation methods.

[0054] Let net(X) be the output of the interpretable modular network structure cell metabolism model. b is the bias from the input layer to the first hidden layer, where b can be understood as the upper bound of the flux of a given metabolic reaction not constrained by proteases. Then, when there are n proteases, m metabolic reactions, and k metabolites in the model, according to the calculation formula of the neural network model, we can express the calculation process of the metabolic model in the form of formula (7).

[0055]

[0056] The dimensions of each matrix are shown in Table 1. It should be noted that since the first and second hidden layers have a one-to-one correspondence, W is a diagonal matrix.

[0057] Table 1. Matrix and its dimensions

[0058]

[0059] Therefore, we can simply write formula (7) in the form of formula (8).

[0060]

[0061] For equation (8), by transposing both sides of the equation, we can obtain the following formula.

[0062] S·W·(k cat ·XT +b T )=0 (9)

[0063] Let k cat ·X T +b T =D, then D is an m×1 vector. Therefore, formula (9) can be rewritten as formula (10).

[0064] S·W·D=0 (10)

[0065] According to the distributive law of matrix multiplication, we can first calculate formula (11).

[0066] S·(W·D)=0 (11)

[0067] Since W is a diagonal matrix and D is a vector, we can transform the calculation of (W·D) using formula (12).

[0068]

[0069] That is, if we swap the elements of the diagonal matrix W and the vector D, their product will not change. We will name the transformed matrix W... and Then at this time For an m×1 vector, Let m be a diagonal matrix. Then formula (11) can be rewritten as formula (13).

[0070]

[0071] make H is a k×m matrix. Then... When considered as parameters to be solved, they can form a system of linear homogeneous equations.

[0072]

[0073] Generally, the number of metabolites in a metabolic network is less than the number of metabolic reactions, i.e., k < m. Furthermore, metabolic networks have a large degree of freedom, so this system of linear homogeneous equations should have infinitely many solutions. Therefore, we attempt to find the general solution to this system of equations.

[0074] Let rank(H) = z < m, then we can obtain a fundamental solution set of the system of equations consisting of mz vectors by finding the null space of the H matrix.

[0075] P = null(H) (15)

[0076] Then P should be an m×(mz) dimensional matrix, which can be expressed in the form of formula (16).

[0077] P = (p1p2p3…p) m-z (16)

[0078] Where p is an m×1 vector. Then, the general solution of the linear homogeneous system of equations (14) is... It can be expressed in the form of formula (17).

[0079]

[0080] Where q is a real number. The general solution of the linear homogeneous system of equations. It can be represented as a linear combination of mz fundamental solution sets. Rewritten in matrix form, we get formula (18).

[0081]

[0082] Then we obtain... This satisfies the requirements of formula (7). The calculation method for the flux of the metabolic reaction at this time is shown in formula (19).

[0083]

[0084] At this point, solving the model becomes an optimization problem for the Q vector. Since Q consists of mz real numbers, and to introduce a degree of randomness, we consider using a differential evolution algorithm to optimize the Q vector.

[0085] Differential Evolution (DE) is a simple and effective heuristic parallel search optimization algorithm. It is one of the most powerful random search algorithms and has been widely used in many fields. In the interpretable modular network structure cell metabolism model, we use the standard DE algorithm to optimize the Q vector. The standard DE algorithm process includes iterative processes such as initialization, mutation, crossover, and selection. In the solution process of the interpretable modular network structure cell metabolism model, a random initialization method, a DE / rand / 1 mutation strategy, and a binomial crossover strategy are adopted. We hope to perform a global search in the search space to find the optimal solution as much as possible. At the same time, according to the derivation of the above formula, it can be guaranteed that all individuals in the iteration process can theoretically make the model output net(X) = M = 0. Then there are two objectives to be optimized in the optimization process: (1) the specific growth rate reaches the maximum; (2) the reaction flux V is within the feasible upper and lower bounds. Since these two objectives are not contradictory, they can reach the optimal at the same time under certain conditions. Therefore, the multi-objective selection method cannot be used for optimization. So, the optimization problem can be combined into a single objective, which is described in the form of formula (20).

[0086] minf = -M·(P·Q)obj +∑max(-(ub-P·Q)(P·Q-lb)·correction 2 ,0) (20)

[0087] The unconstrained single-objective problem constructed using formula (20) is used to optimize Q. In the formula, the first term represents the specific growth rate, and the second term represents the degree to which the flux exceeds the upper and lower bounds. Here, ub and lb represent the upper and lower limits of the flux for each reaction, both within the range [0,1]. obj represents the specific growth rate, M is a variable parameter, and correction is a correction parameter. The M parameter in the formula is an amplification factor when integrating multiple optimization objectives into a single objective, to better match the two objectives. Since the fluxes of different metabolic reactions are of different magnitudes, if a correction term is not introduced when calculating the degree to which the flux violates the upper and lower bounds, the impact of different reactions exceeding the upper and lower bounds on the iterative calculation will vary greatly. Therefore, the correction parameter is introduced to balance this difference.

[0088] 5. Construct an interpretable modular network cellular metabolic model of the central metabolism of Saccharomyces cerevisiae.

[0089] Protein abundance data for *Saccharomyces cerevisiae* can be obtained from the PAXdb database. The PAXdb database contains 18 sets of abundance data, with coverage ranging from a minimum of 16% to a maximum of 96%. Due to differences in measurement conditions and the total amount of protein measured, there are discrepancies in the measurement results for the same protein between each set. Similarly, to avoid overly limiting the upper bound of metabolic flux, we compared the 18 sets of protein abundance data obtained from PAXdb. For the same protein, we selected the highest abundance value from the 18 sets as a reference and calculated their estimated protein concentration and corresponding upper bound of metabolic flux using formulas (1) and (4).

[0090] Considering that the growth of Saccharomyces cerevisiae is inhibited in the presence of ethanol, especially when the ethanol is produced by the Saccharomyces cerevisiae itself during metabolism, and that the intracellular glucose content will greatly affect the growth rate of microorganisms, M is defined as the form of Equation (21).

[0091]

[0092] Similarly, we define the correction parameter as in the form of formula (22).

[0093]

[0094] Among them, if the metabolic reaction ub i =0, then a very small value is given as the upper bound ub.i =10 -8 This is to ensure that formula (22) can be calculated normally.

[0095] The objective function for optimization can be divided into two parts: maximizing the specific growth rate and minimizing the degree of violation of the upper and lower bounds, as shown in formula (23).

[0096] min f=f1+f2 (23)

[0097] Where f1 = -M·(P·Q) obj Representing the specific growth rate, f2=∑max(-(ub-P·Q)(P·Q-lb)·correction 2 ,0) represents the degree of violation of the upper and lower bounds, and Q is the parameter to be optimized.

[0098] We used experimental data on glucose consumption, oxygen consumption, ethanol production, acetate production, and carbon dioxide production as constraints. We set the maximum number of generations to 50,000, the crossover probability CR = 0.5, and the population size to 1000. We performed 20 iterations of the growth calculations. Three iterative graphs were plotted: the overall objective function f, the degree of violation of upper and lower bounds f², and the specific growth rate. (See attached graphs.) Figure 2 , Figure 3 and Figure 4 .

Claims

1. A method for constructing a cellular metabolic model with an interpretable modular network structure, characterized in that, It is constructed from a dual-hidden-layer network model integrating four modules; after deriving the network's calculation formula, the differential evolution algorithm is introduced to solve the model; thus, an interpretable modular network-based cellular metabolism model is obtained; specifically, the following steps are included: Step 1: Determine the four modules of the metabolic network. These four modules are: enzyme concentration module, flux upper bound module, flux data module, and metabolite concentration module. The input layer of the entire network model is defined as the enzyme concentration module. The first hidden layer and the second hidden layer are the flux upper bound module and flux size module, respectively. The metabolite concentration module serves as the network output layer, representing the concentration changes of various metabolites during metabolism. Step 2 establishes connections between the four modules, integrating them into a complete network model and assigning practical meaning to the connections between each layer. The connections between the network modules in Step 2 integrate the four modules from Step 1 into a complete, interpretable network model. This network model is a modular, double-hidden-layer neural network model. The input layer represents the concentration of the protease, the first hidden layer represents the upper bound of the flux for each metabolic reaction, the second hidden layer represents the flux magnitude for each metabolic reaction, and the output layer represents the change in metabolite concentration. The modules in this network model are connected by weights. In this model, the input layer and the first hidden layer are connected by the enzyme turnover rate. The upper limit of the reaction flux is calculated by combining the enzyme concentration and the turnover rate, forming a protease constraint. The first hidden layer and the second hidden layer are connected by a one-to-one correspondence of weights, meaning that the flux of each reaction is determined by the different proportions of their flux upper limits. The second hidden layer and the output layer together constitute the steady-state equilibrium condition, and all the weights in it also have practical significance. If the metabolic reaction represented by the node of the second hidden layer includes the metabolite represented by the output layer, then there is a stoichiometric coefficient between them to represent their relationship. The relationship between the nodes of the second hidden layer and the output layer is expressed by the stoichiometric coefficient. All four modules are connected together by interpretable weights, constructing a complete and highly interpretable modular network structure cell metabolism model. Step 3: Preprocess the parameters of each layer and the connections between them to determine the model parameters and structure; Step 4: Analyze the relationship between the network's input and output, and establish mathematical formulas; derive the formulas and construct new calculation methods; the establishment of mathematical formulas and the derivation of the formulas in Step 4 involve finding the weights of the connection between the first hidden layer and the second hidden layer; by transforming the original problem into a linear homogeneous equation system formula, a new solution calculation method based on the differential evolution algorithm and matrix null space search is obtained. Step 5: Set the iteration parameters and perform simulation calculations on the metabolic model to obtain a cell metabolic model with an interpretable modular structure network.

2. The method for constructing a cell metabolism model with an interpretable modular network structure according to claim 1, characterized in that, The network parameter preprocessing in step 3 includes the input enzyme concentration, enzyme turnover, and the stoichiometry matrix of the model.

3. The method for constructing a cell metabolism model with an interpretable modular network structure according to claim 1, characterized in that, The metabolic model described in step 5 is determined by comparing the model with the metabolic reactions and enzyme catalysis relationships in the database, determining the specific data of the original model, and obtaining the enzyme turnover of the protease in the central metabolism based on the stoichiometric coefficients between each metabolic reaction and metabolite.

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