Team:UESTC-China/modeling
Overview
Our system is a three-enzyme chain reaction composition，We hope that the three enzymes (DhA31, HheC, EchA) will be used to degrade TCP into glycerol. In our experiments, we hope that according to the related research of enzyme kinetic engineering, building a suitable mathematical model to characterize TCP and glycerol the time-varying theoretical curve evaluates our actual measured values and finds the interaction of the three foreign genes at the expression level. And we want to analyze the parameters of the model, find a better optimization strategy to increase our conversion efficiency
The picture shows the degradation pathway of TCP
Model Foundation
Basic Model
Because the kinetics of the multi-enzyme system, no only each enzymatic reaction is reversible, the overall reaction could reach the equilibria, but also the inhibition of the various products on the enzyme, and the affinity of the enzyme for different substrates, it is difficult to study at the same time.
So we first look at the whole reaction as independent kinetic kinetics, we suppose that the reaction of the enzyme is consistent with the Michaelis-Menten equation, so chain reaction in the first step of the Dha31 catalytic TCP into DCP reaction(Eq.(1)), obeying the Michaelis-Menten model (where complex for the enzyme and substrate intermediates). Its reaction rate equations for the changes of substrate is described in Eq. (2).
We also obtain the equation for the reaction rate of all the intermediates in terms of the Michaelis-Menten model. which both similar to the Eq.(2). Because our reactions are single-substrate reaction to get a single product, and their material is approximately equal to 1: 1, so we can get a relationship based on this in Eqs. (3) - (7).
We integrate the reponse rate to each step Can get the TCP and the various intermediates and glycerol concentration with respect to the time expression. Then we can throw these Eqs. (2)-(7) can get the relationship between the amount of glycerol (GLY) produced and the other parameters, where t is the reaction time (h).
Improved Model
Although we obtained the GDL expression by the relationship between the Michaelis-Menten equation and the chain reaction, the model only considers the case of single enzyme reaction, and the affinity of the enzyme to different substrates is too complicated to analyze. But the reaction process has two enzymes were used twice, we hope that through this aspect to improve the model.
As the Eq. (8) shows, we can use HheC and EchA as the same reaction process to establish the model, the two substrates together to share the same enzyme can be approximated as competitive inhibition of the kinetic equation. (E is the enzyme, S, I is the two substrates, ES, EI are the intermediate complex, P, Q are the product)
In Eqs. (9), E is Total enzyme concentration, after the reaction began, divided into [ES], [EI], and the free enzyme [E_f].
Eq. (11) is constructed based on the 'steady state' theory and the law of quality, combine with Eqs. (10), (11) we can get the Eqs. (12) – (15), then the changes in the concentrations of substrates (i.e DCP), product TCP, and intermediates (i.e ECH, CPD, GDL), can be calculated by using Eqs. (3) - (7) and Eqs. (12) - (15).
Table.1 Kinetic parameters of three enzymes used for simulation modeling.
Model fitting with experimental data
( pH8.5,37℃，200mM Tris-SO4,mM/L TCP ) , The timedependent profiles of substrates (i.e DCP), product TCP, and intermediates (i.e ECH, CPD, GDL) are shown in Fig.1. Three parameter settings for the model are listed in Table 1. The first setting was based on kinetic parameters of the three enzymes under the same experimental reaction conditions. But there were some differences in terms of reaction time required for reaching the equilibrium. Kinetic parameters of an individual enzyme may not be the same as its apparent catalytic activity in the mixture of multiple enzymes. The first set of parameters need be adjusted for the best fitting. Among a number of parameters tested, it was found out that adjusting one kinetic parameters. Simulation curves based on the secend set of data exhibited the best fitting (Fig. 1).
Fig.1 (A) Simulation modeling and experimental results of the synthesis of GLY from TCP. Experimental conditions: 50 mM HEPES buffer (pH 7.5) containing 100 mM sucrose, 10 mM phosphate, 1 U/mL SP, 1 U/mL GI, and 1 U/mL CBP at 45 ℃，GLY, glycerinum ; (B) Under the best fitting, the curves of TCP and various intermediates, glycerol concentration over time.
Modeling prediction and experimental validation
Optimum catalytic constant of the enzyme
Smulation modeling of the synthesis of glycerinum is related to thirteen parameters, of which 10 are catalytic constant of the enzyme, and three are the concentrations of the enzyme. At present, there are many studies related to DhA, and DhA31 is already available the highest activity of the mutant, so the value of
and
is difficult to change. The activity of the EchA, which exists in the plant, is high enough. And its amount of change has little effect on the rate of glycerol production. So we tried to find the best k
cat value for EchA. After a lot of tests we found that
had the greatest impact on Smulation modeling of the synthesis of glycerinum. Fig.3 shows the change in the concentration of glcerol over time at different
values. We find that as the value of K is increasing, the effect on the concentration curve is getting smaller and smaller, and when approaching the critical
, changing it loading only slightly promotes the productivity. We define the kcat when the reaction completion time is not greater than 110% of the minimum time to completion of the reaction for finding the optimal mutant. So we construct the mutant when the reference to the th.eoretical critical value of the model, do not need to blindly pursue the maximum kcat value. According to the Smulation modeling we determined the kcat value, and we constructed the x mutants, as shown in the figure we used the model to predict all the mutant under the glycerol concentration curve, found that although all the kcat although the critical value, but which HheC-W249P achieve the best results, so We finally used the mutant to construct the plasmid.
Fig. 3 (A) In the case of different K value, simulation modeling of the synthesis of glycerinum.
(B) simulation modeling of the synthesis of glycerinum in all of the
Optimal enzyme loading ratio
Another important function of modeling is the prediction of the optimal reaction conditions (e.g., enzyme ratio and catalytic constant of the enzyme) Predictive simulation of the kinetic model was conducted based on the second set of parameters under conditions: 100mM TCP. Because each enzymatic reaction is reversible, the overall reaction could reach the equilibria when the reaction time is long enough. We defined the time when glycerinum concentration reached 90% of the equilibrium concentration for calculating the volumetric productivity. The simulation curve of GLY loading (Fig. 2A) indicates that the critical DhA31 loading is approximately 0.50 U/mL.
Below this concentration, decreasing DhA31 loading greatly decreases the volumetric productivity; beyond this concentration, increasing DhA31 loading only slightly promotes the productivity. Similarly, the critical loading of HheC and EchA were estimated to be 1.00 U/mL and 1.50 U/mL, respectively (Fig. 2A). These results suggest that the optimal enzyme loading ratio of Dha31:HheC:EchA is 0.5:1.0:1.5. Experiment at this enzyme ratio was conducted to validate the model prediction. As shown in Fig. 2B, the simulation curves of TCP, intermediates and the glycerinum fit well with experimental data, indicating the validity of the model. At this enzyme ratio, reaction time to reach 90% of the glycerinum equilibrium was decreased from 30 to 15 h.
Fig. 2 (A) Predictive simulation for the effects of enzyme loading on the synthesis of glycerinum in ….，(B) Smulation modeling of the synthesis of glycerinum in …..，
Summary
We developed the mathematical model of the three-chain reaction by the kinetic study of the enzyme. We predict the changes in the concentration of substrate TCP, each intermediate product, and glycerol over time. Then we get the catalytic constant of the enzyme through a lot of experiments. We use these constant to predict the ideal concentration curve, and compared with the experimental data, we found that we can get a high degree of fitting. We decided to improve the HheC's catalytic constant of the enzyme by analyzing and testing the final decision. We found that the greater the k value, the smaller the effect on the glycerol production curve. We predicted the correlation curves of various mutants, and finally chose the HheC-W249P mutant to construct the vector. Then we predict the optimal enzyme ratio of 1: 2: 4, so although EchA is expressed in the plant, we still construct the gene, hoping to let the plant overexpress the enzyme to a certain extent, Compared to the rate at which TCP degrades to glycerol.
