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Model

Abstract

      We have two models this year. The first model was designed to predict the lipid production of algae after we co-cultured C.minutissima and E.coli It is worth mentioning that in the model we also consider the natural growth of C.minutissima and E.coli, using the logistic and ordinary differential models as a basis. The second model is an expression model to reflect the expression of our synthesized IAA, and we also want to combine Michaelis-Menten equation to better represent the role of the two enzymes in the IAA pathway.

Co-culture Model

Introduction

      Under the co-culture system of this project, Chlorella will use IAA produced by E. coli to promote growth and synthesize lipids from carbon sources in the wastewater. Our preliminary study concluded that Chlorella utilizes the carbon source in the wastewater mainly as carbohydrates, so in this model we represent it as glucose. The self-growth of Chlorella, its lipid production and glucose consumption then form a certain mathematical relationship. Our aim is to develop a mathematical model that reflects the growth, lipid production and glucose consumption of Chlorella. We want to apply the model to determine the optimal values of each control parameter of the wastewater treatment process.

Model Design

      Most microbial growth processes can be explained by Logistic equations. In the present model we treat algal growth as consistent with microorganisms and use Logistic equations to calculate biologically and geometrically meaningful parameters simply by means of s-shaped curves independent of substrate concentration. ^[1]

       A. The growth model of Chlorella was then constructed as follows:

      Where  is the growth rate of C.minutissima; μ_max is the biggest growth rate for C.minutissima; X is the concentration of C.minutissima in the reaction vessel and X_max is the maximum concentration of C.minutissima.                                   

      When t equals to zero, the concentration of C.minutissima is the initial concentration value (X = X₀). After integration, the equation becomes:                                 

     We can find out μ_max and X_max through the experiment. So as long as we determine X₀, the equation as a function of Chlorella concentration X and time t is then obtained.                           

     B. We describe the lipid formation by the Luedeking-Piret equation^[2]. The rate of lipid formation is linearly related to the instantaneous biomass concentration X and the growth rate :       

     Where,  is the rate of lipid formation; α is the coefficient of lipid formation; ß is the non-growth related coefficient.    

     Gaden^[3] classified the modes of product formation according to the relationship between product formation and microbial growth into: category 1, where product formation is related to microbial growth; category 2, where product formation is partially related to microbial growth; and category 3, where product formation is not related to microbial growth.

      For the above equation, when α=0 and ß≠0, the relationship between product formation and microbial growth is of category 3. For α≠0 and ß≠0, the relationship between product formation and microbial growth is partial and therefore belongs to category 2. When α≠0 and ß=0, the relationship between product formation and microbial growth is linear and fits into category 1.

      We concluded that lipid formation was initially linearly related to Chlorella growth, so we grouped the experimental cases into the first category (α≠0 and ß=0). Thus, integrating the above equation:

      C. Finally we found that the kinetics of glucose consumption can be expressed as the conversion of substrate to product plus the consumption of substrate.^[4] The equation is:

      C. Finally we found that the kinetics of glucose consumption can be expressed as the conversion of substrate to product plus the consumption of substrate.^[4] The equation is:

      Where  is the total consumption rate of glucose, Y_x/s is the maximum growth coefficient of Chlorella; Y_p/s is the maximum lipid production coefficient; S is the glucose concentration; and m is the maximum consumption rate (maintenance coefficient) to sustain the life of Chlorella.

      Lipids are intracellular metabolites, and in our scenario, glucose is used in three main parts: growth of Chlorella, accumulation of lipids and maintenance of Chlorella life. Under this assumption, the above equation can be simplified as :

       Substituting Equation 1 and integrating it, we get:

      S₀ is the initial concentration of glucose. As a result of the Y_p/s by formula  are obtained. The unknowns are Y_x/s and m. From this we obtained the glucose concentration as a function of time.

      We can use this to predict how much glucose we can add to culture Chlorella to obtain more lipid production and how glucose will change over time.

Conclusion

      However, due to the epidemic, the experimental supplies were not delivered on time, so we could not carry out the experiment to get the data, so the model could not be continued.

IAA expression pathway

Introduction

      Indole-3-acetic acid (IAA) is a plant growth hormone, which is an important hormone synthesized by plants for regulating growth and physiological activities. In addition, IAA controls plant metabolism and senescence, landward and polar transport, and responses to drought, alkali salts, pathogenic bacteria, and heavy metal stresses. Many plant-related microorganisms, such as Agrobacterium, Pseudomonas aeruginosa, and Streptomyces, can convert L-tryptophan to IAA. in recent years, the biosynthetic pathway for converting L-tryptophan to IAA in microorganisms has been elucidated.^[1] We decided to introduce the synthetic pathway of IAM of indoleacetic acid into E.coli for better interaction between E.coli and Chlorella, so that E. coli secretes IAA to promote the growth of Chlorella. To measure the yield of IAA, we introduced the expression model and the Rice equation enzyme reaction model for yield prediction.11

Model Design

      Because the intermediate product in this model is IAM, which is reacted by ami1 enzyme to generate the end product IAA, and IAM is generated from L-Trp by iaaM enzyme reaction, the expression models of iaaM and IAM enzymes and the Rice equation enzyme reaction models of L-Trp to generate IAM and IAM to IAA are involved here. And the combination of both is needed to achieve the best prediction.

      We plan to use the expression model to obtain the concentration of the two enzymes during the stabilization period and use this concentration as the initial enzyme concentration in the Michaelis-Menten equation to perform the product-related calculations.

   

Expression model

Assumption

     (1) At first,we assume that the overall transcription rate is only determined by the copies of the plasmids carrying the target gene fragment in E.coli and the strength of the promoter.

     (2) Once the degradation process starts, the translation of the mRNA will stop and the protein will become ineffective. With similar molecular weight sizes, we assume that proteins’ degradation rates are the same, as are the mRNAs.

     (3) The concentration of mRNAs and proteins studied in the model are zero at the beginning.

Variables & Parameters

Table 1

      The expression requires several important constant parameters such as the copy number of psb1c3 in E.coli is 100-300, while the size of psb1c3 is 1872bp, the size of psb1c3 is 7705bp, the size of iaaM is 623aa, and the size of ami1 is 425aa.We also have the following parameters in table.2.（It is not clear what transcriptases and ribosomes are actually used in the project, and it is not possible to estimate the specific transcription and translation rates.）

Table 1

Cellular Equations

Results

      We import the equation into matlab2021 and solve it to obtain the following figure 1.    

Figure 1

      Because the reaction pathway intermediates as well as the final product require enzyme catalysis and the amount of enzyme can be derived from the expression model,
the Mie equation is established to determine the yield of the product.

Variables & Parameters

      We look for the variables in the model as shown in Table 2:

Table 2

Cellular Equations

    

Conclusion

      In our expectation, the reaction rate of the first step reaction should first increase and then decrease to a stable value with the influence of the second step reaction, and the rate of the second step reaction should gradually increase and then level off to a stable value. Due to various force majeure factors this year, such as the failure delivery of the transportation company, the experiment was not conducted as scheduled, resulting in missing data needed to construct the model, which could not be constructed.