CHAPTER III ENZYME KINETICS
3.3 Kinetics Model in Metabolic Pathway
Depending on the concentration of the inhibitor I, the rate of the reaction can be expressed as the following form:
3.3 Kinetics Model in Metabolic Pathway
Based on the theories proposed in the above two sections, we can build up the mathematical model for some reaction. But it is not applicable to all biochemical reactions.
Owing to the order of addition of reactants to and release of products from the enzyme
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active site, kinetics mechanisms fall into two broad categories [17].
Sequential mechanisms are characterized by the fact that all reactants must be bound
to an enzyme before any reaction occurs.
Ping-Pong mechanics denotes the situation that a product is released between the
consecutive additions of the two substrates with the enzyme.
Sequential mechanisms can be termed either order, when there is a compulsory order of substrate addition and product released, or random, when there is not. The rapid equilibrium indicates a special case in which the equilibrium between an enzyme-reactant and a free-reactant is established.
In ping pong mechanisms, the reaction order for a given enzyme is obtained from the number of reactants in the both reaction directions. The terms Uni, Bi, Ter, and Quad are used for the enzyme-catalyzed reactions with one, two, three, and four reactants in a given reaction direction. For example, a reaction having a reactant A and two products P and Q, are thus called Uni-Bi reaction. We list the classification of reactions in Table 3.1 to help for determining the characteristic of reaction and for choosing a proper model for it.
Table 3.1 Classification of the kinetic mechanism.
Mechanism Type Property
Sequential order Obligatory order
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Random Distinct binding
Rapid equilibrium Special case in equilibrium
Ping pong
Uni One reactant in a direction
Bi Two reactants in a direction
Ter Three reactants in a direction Quad Four reactants in a direction
Besides the categories of reaction, Chassagnole et al. [11] and Kadir et al. [12] built the model to describe E. coli. According to these methods and theories, we modify the equations by combining Michaelis-Menten equations and enzyme inhibition, and consider an example for irreversible Bi-Uni (two substrates and one product) system to derive the equations. Figure 3.5 demonstrates a reaction with competitive inhibition
Figure 3.5 The reaction for a Bi-Uni system with competitive inhibition.
The reaction can be written as
35 principle of mass action, we obtain the equations
1
1
1
1
,We consider the steady-state condition,
1
and introduce the definitions
1 2
Substituting (3.3.3) into the relation of total enzyme concentration, we obtain
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Finally, substituting it into the product equation yields
Next, we proceed to consider the reaction of a Bi-Uni system with an uncompetitive inhibitor as shown in Figure 3.6.
Figure 3.6 The reaction for a Bi-Uni system with uncompetitive inhibition.
Unlike the case of a competitive inhibitor, there are three more inhibition equations
1
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Similarly, we define the equilibrium constants
1 2 3
and substitute (3.3.3) into the equation of total concentration to obtain
So the rate equation of product becomes
The third type of reactions to be considered is the noncompetitive inhibition, which is
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a combination of competitive and uncompetitive inhibitors, as shown in Figure 3.7.
Figure 3.7 The reaction for a Bi-Uni system with a noncompetitive inhibition.
The corresponding inhibition equations are described by
1 2
In terms of the equilibrium constants
1 2 3 4
the total enzyme concentration can be expressed by
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With a further approximation that all the equilibrium constants are equal
1 2 3 4
we obtain the product equation as
obeys the law of conservation of mass for metabolites. Following the same procedures, we will introduce the rate equation of each reaction after deriving the ordinary differential equations of metabolites in E. coli. A mathematical model will be proposed to describe the three main metabolic pathways as shown in Figure 3.8 under the following assumptions:1. TIS is well in equilibrium state so that GAP and DHAP are lumped together.
2. A sequence of enzymatic reactions is in the order of PGK, PGluMu and ENO, which are considered to be in equilibrium so that there is only one reaction from GAP to PEP.
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Figure 3.8 The three metabolic pathways considered in the model of E. coli.
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The general dynamic equation can be described by the mass balance as:
, obtain the ordinary differential equations as follows:
42 Michaelis-Menten equations combined with the three types of enzyme inhibition to the metabolic pathways considered in Figure 3.8. Each rate in Eq. (3.3.9) will be derived according to the number of reactant, inhibition category, reversibility and cooperation, and the detailed derivations are listed in appendix A. In the following, we take the rate equation of enzyme PTS as an example to illustrate the underlying procedures. The reaction to be considered is
𝑃𝐸𝑃 + 𝐺𝐿𝐶𝑃𝑇𝑆→ 𝑃𝑌𝑅 + 𝐺6𝑃,
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and the detailed reaction can be described by
1
2
where A, B, E denote PEP, GLC, PTS, respectively. According to the types of inhibition,
we can list the inhibition equations as:
1
where Q denotes the product and also acts as an inhibitor in this reaction. Meanwhile, on the basis of the conservation of the total enzyme concentration, we can get
ET E EQ EA EB EAQ
EBQ
EABQ
. (3.3.12) Substituting eq. (3.3.11) and (3.3.12) into the noncompetitive inhibition equation, we obtain cooperation, the rate equation has to be modified as44
After the rate equations derived from Appendix A are all substituted into Eq. (3.3.9), we get the complete differential equations that describe the three metabolic pathways in Figure 3.8. These equations will be implemented in the numerical environment of Matlab.
All the constants involved in the equations have to be evaluated numerically, before we can solve the set of differential equations. The numerical values of the constants used in the simulation are listed in Table 4.2, which are quoted from Chassagnole [11] and Kadir [12].
In the next chapter, we will report the simulation results and compare them with the experimental data to verify the proposed model.
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CHAPTER IV
EXPERIMENTAL DATA AND SIMULATION VERIFICATION
After having discussed the main metabolic pathways and the related mathematical model, we will solve the differential equations in this chapter to obtain the changes of metabolites in E. coli O157:H7 and to compare them with the experimental results. Firstly, we introduce the experimental data and find the matching error with the numerical predictions. The matching error is then fed back to a least-square algorithm to adjust the parameters in the model so that the matching error is decreasing as the iteration process goes on.
4.1 Experimental Procedure and Measurement
The experiment was performed by Professor C. S. Chen at Department of Biochemistry and Molecular Biology of National Cheng Kung University. EDL933 is the original culture of E. coli O157:H7, while SDHA is the mutative culture by replacing the enzyme SDH with the antibiotic resistance gene. The difference in the concentration of metabolites between the two cultures is then recorded. The experimental procedures are outlined below.
1. The EDL933 culture is made from the clinical isolations of the E. coli O157:H7 at the
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Bioresource Collection and Research Center.
2. The lambda Red recombination system provides a simple and versatile method for replacing enzyme SDH by the antibiotic resistance gene, Kanamycin.
3. The EDL933 and SDHA mutant are growing in the Luria-Bertani(LB) broth at 37°C and shaking at 220 rpm overnight without adding further antibiotics to the LB broth.
4. Rotate the two cultures in a centrifuge at 8000 rpm for 15min at 4°C.
5. 40ml of bacteria culture broth are collected. Remove the broth and suspend the cell pellet in 5ml deionized water on ice.
6. The pellet is re-suspended in 0.5ml deionized water, and then stored at -80°C to get bacteria powder by freeze drying.
After the above preparation procedures for the two samples, the samples are then analyzed through the following measurement steps.
1. Each sample (powder) is added with 600 μl of 100% MeOH as a ratio of 1 to 3, and metabolites are extracted by vortex for 5 min.
2. After 10 min centrifugation at 14,000 rpm, 200 μl of supernatant is taken for vacuum dry at room temperature.
3. The dry sample is reconstituted in 100 μl of 80% MeOH and then centrifuged at 14,000 rpm for 10 min.
4. The supernatant is collected as a sample and subjected to LC-ESI-MS analysis. The
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acquired data are processed by TargetAnalysis and DataAnalysis software and summarized in an integrated area of signals. Found compounds are selected with the tolerance of LC peaks within 0.3 min and area higher than 1000 counts from established compound identities.
After deleting those measurements with large deviation from the mean value, the remaining measurements are averaged, respectively, for the data of E. coli O157:H7 EDL933 and SDHA. The relative changes of the metabolites in SDHA with respect to those in EDL933 are listed in Table 4.1.
Table 4.1 The relative changes of metabolites between EDL933 and SDHA.
Metabolite EDL933 SDHA Percent increase
G6P 6217.088 5668.813 -8.82%
F6P 6217.088 5668.813 -8.82%
GAP 4698.46 3981.97 -15.25%
PEP 51912 50077 -3.54%
PYR 18415.79 13060.11 -29.08%
AcCoA 2570.83 1052.50 -59.06%
ICIT 37661 24729 -34.34%
SUC 37177.27 79873.87 118.96%
FUM 108541.9 32453.8 -70.10%
MAL 1895.3 2500.1 31.91%
OAA 2969 2733 -7.94%
R5P 1719 1555 -9.54%
E4P 1182.9 1063.6 -10.08%
The relative changes of metabolites between EDL933 and SDHA will be set as the target values, while we adjust the simulation model. In the next section, the model is
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initially simulated with parameters given by [11] and [12]. We then employ the least- square algorithm in Section 4.3 to adjust the parameters so that the simulation results approach the target values given by the measurements in Table 4.1. Finally, in Section 4.4 we will used the modified model to predict the new effects caused by other mutants.
4.2 Initial Simulation Based on Parameters from Literature
The 24 differential equations as expressed by Eq. (3.3.9) involve many free parameters, whose numerical values have to be assigned in advance before we can solve the model for simulation. The initiate our simulation, the parameters given by Chassagnole [11] and Kadir [12], as listed in Table 4.2, will be regarded as the first guess of the correct parameters in order to best fit the measurement data in Table 4.1. With the parameters given by Table 4.2, the set of differential equations (3.3.9) are solved simultaneously to give the time responses of the various metabolites, such as G6P, F6P, GAP, PEP, PYR, AcCoA, ICIT, SUC, FUM, MAL, OAA, R5P and E4P. The results are shown in Figure 4.1 for the metabolites in EDL933 and in Figure 4.2 for the metabolites in SDHA.
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Figure 4.1 Initial simulations of metabolites in E. coli EDL933.
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Figure 4.2 Initial simulations of metabolites in E. coli SDHA.
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Table 4.2 Kinetic parameters[11][12].
Enzyme Parameters Value Parameters Value
PTS vmax
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53
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The simulations for EDL 933 and SDHA will last ten minutes to arrive at their steady state where the concentrations of various metabolites are recorded. The recorded data for EDL933 are treated as the base data, and the relative changes of the simulation data for SFHA with respect to the base data are then computed, as shown in the second column of Table 4.3. By contrast, the third column in Table 4.3 displays the relative changes obtained by the experiments already shown in the last column in Table 4.1.
Table 4.3 The relative changes of metabolites between EDL933 and SDHA obtained, respectively, by simulation and experiment.
metabolites Initial simulation experiment
G6P -1.38% -8.82%
Table 4.3 shows that significant discrepancy exists between the simulation prediction and the experimental results. The matching error arises from the fact that the kinetic parameters of E. coli O157:H7 considered here are not exactly the same as the kinetic parameters of the E. coli listed in Table 4.2. The kinetic parameters thus have to be adjusted to minimize
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the matching error between the simulation and the experiment. The least-square method is very suitable for our need and will be discussed in the next section.
4.3 Refined Simulation Based on Least-Square Algorithm
The least-square method describes a frequently used approach to solving over-determined or inexactly specified systems of equations in an approximate sense.
Instead of solving the equations exactly, we seek only to minimize the sum of the squares of the residuals. A very common source of least-square problems is curve fitting. Assume there are m observations at specific values of t:
, 1,..., . between the observations and the model:
and the sum of the squares of the residuals is defined by
2 2
The purpose of least squares is to find the parameters α’s and β’s that minimize the sum of the squares of the residuals. When applying the least-square method to our system, we
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where K denotes the kinetic parameters we need to modify in the simulation model. The j least-square problem is to determine the parameters K so as to minimize the sum of the j
squares: Table 4.3. The above least-square problem is to be solved by a program called “lsqcurvefit”
in MATLAB. The iteration procedure is illustrated in Figure 4.3.
Figure 4.3 The iteration procedure of modifying the parameters.
While running the program “lsqcurvefit”, we choose the kinetic parameters listed in Table 4.2 as the initial guess K , and the output of the program will give the refined 0 parameters Knew. As the iteration goes on, the sum of the squared residues decreases gradually as shown in Figure 4.4.
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Figure 4.4 The decrease of residuals as the iteration goes on.
When the iteration achieves its steady state, the parameters cease to change and provide us the best least-square solution. After completing the iteration process, the refined parameters are listed in Table 4.4. Also shown in the figure are the initial values of the parameters. Comparing Table 4.4 with Table 4.2, we can see that only some of the parameters in Table 4.2 undergo refinement, while the rest are kept unchanged. The parameters undergoing refinement process are those which are more sensitive to the change of the sum of the squared residues than others.
Table 4.4 Refined parameters generated by the program “lsqcurvefit”.
Parameters of enzyme Initial value Refined value
PPC K6 0.2673 3.26
PCK KOAAm 0.67 0.1146
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PCk KPEPi 0.06 1.1246
PYK KPEP 0.31 0.7947
PGI Keq 0.1725 1.2
G6PDH KG6P 14.4 21
SDH KSUCm 0.1 10.0486
SDH Keq 10 0.0051
PFK Kb,ADP,AMP 1.4514 0.015
ICL KICITm 0.604 0.4278
TAL Keq 1.05 0.0884
FUM Keq 10 1.0428
MDH KMALm 1.33 0.9281
GAPDH KPGP 1.04e-5 1.6107-5
Finally, we substitute the refined parameters into the model and repeat the simulation. The results are shown in Figure 4.5 for the metabolites in EDL933 and in Figure 4.6 for the metabolites in SDHA. The relative changes of the refined simulation data for SDHA with respect to those for EDL933 are shown in the third column of Table 4.5. Also shown in the table are the relative changes of the initial simulation.
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Figure 4.5 Modified simulations of metabolites in E. coli EDL933.
Figure 4.6 Modified simulations of metabolites in E. coli SDHA.
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Table 4.5 The relative changes of metabolites between EDL933 and SDHA obtained, respectively, by initial simulation, refined simulation and experiment.
Metabolites Initial simulation Refined simulation Experiment
G6P -1.38% -8.98% -8.82%
As can be seen in Table 4.5, the most significant reduction in matching error appears at the metabolite SUC, for which the relative change is reduced from 104116.12% to 119.45%, already near to the experimental result 118.96%. Overall speaking, the refined simulation by using the least-square correction of parameters has greatly improved the initial simulation and made the simulation results much closer to the experimental results.
4.4 Prediction on New Mutants
The above simulations have been performed to compare with the existing experiment, which replaced enzyme SDH by the antibiotic resistance gene, Kanamycin. Based on the same model, below we will reveal predictions on new mutants, which have not been confirmed by experiments. Instead of replacing the enzyme SDH, other enzymes in the
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TCA cycle can also be replaced in the same way. In the following simulations, we will replace only one enzyme in each simulation in order to identify the role of the replaced enzyme in the whole cycle. Accordingly, five enzymes, CS, ICDH, αKGDH, FUMe and MDH, will be replaced, respectively. The changes of metabolites caused by the five respective replacements are illustrated in Table 4.6 and Table 4.7. For the purpose of comparison, the corresponding changes caused by the replacement of SDH are also shown in the tables.
Table 4.6 The change of metabolites by replacing CS, ICDH and αKGDH, respectively.
metabolites CS ICDH αKGDH SDH
G6P 0.20% 0.01% 0.20% -8.98%
F6P 0.21% 0.02% 0.21% -8.97%
GAP 0.46% 0.01% 0.46% -15.26%
PEP 0.20% 0.03% 0.20% -12.03%
PYR 0.05% 0.001% 0.05% -1.32%
AcCoA 4.50% 0.0006% 4.50% -0.6%
ICIT -100% 53.2% -100% -14.43%
SUC -7.71% -0.6% -7.71% 119.45%
FUM -7.72% 0.61% -7.72% -70.79%
MAL -7.70% 0.67% -7.70% -70.12%
OAA 3.36% 0.09% 3.36% -9.34%
R5P 0.29% 0.01% 0.29% -10.84%
E4P 0.38% 0.01% 0.38% -13.15%
In Table 4.6, we can see that the effects of replacing CS, ICDH and αKGDH are negligible as compared with the replacement of SDH. In other words, SDH is expected to dominate the biochemical reactions, for example, in the attenuation of toxicity. However,
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not every other enzyme has negligible effects. In Table 4.7, we can see that the changes of metabolites caused by FUMe and MDH mutants are comparable with those caused by the SDH mutant.
Table 4.7 The change of metabolites by replacing FUMe and MDH, respectively.
metabolites FUMe MDH SDH
G6P -10.32% -27.87% -8.98%
F6P -10.33% -27.86% -8.97%
GAP -17.7% -47.22% -15.26%
PEP -13.67% -34.54% -12.03%
PYR -1.57% 0.27% -1.32%
AcCoA 0.42% 4.18% -0.6%
ICIT -18.83% -44.33% -14.43%
SUC 195.92% 411.04% 119.45%
FUM 433.94% 721.93% -70.79%
MAL -70.04% 735.56% -70.12%
OAA -8.49% -9.31% -9.34%
R5P -12.53% -34.01% -10.84%
E4P -15.23% -41.18% -13.15%
The above simulations predict that although the extent of influence may not the same as SDH mutant, FUMe and MDH mutants do possess a similar tendency to attenuate the pathogenicity. Further experiments have to be conducted to confirm the new predictions of the proposed model for E. coli O157:H7.
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CHAPTER V
CONCLUSIONS AND FUTURE WORK
5.1 Conclusions and Discussion
In the framework of enzyme kinetics, the change of the metabolite concentrations, which was originally observed and measured from experiments, now can be reproduced by the mathematical model of the biochemical systems. Experimental studies of biochemical reactions are often time and budget consuming. If they can be simulated in a mathematical model, numerous biochemical reactions under a variety of initial and boundary conditions can be examined in a very short time. Based on the predictions made from the simulations, verification experiments then can focus on the possible new findings revealed by the simulation.
In this thesis, we have discussed the main metabolic pathways of E. coli first, and then employed the enzyme kinetics to build the dynamic equations for biochemical reactions.
After comparing with the existing experimental result, the proposed metabolic model is modified iteratively by a least-square algorithm so as to minimize the matching error between experimental and numerical data. The major contributions of this thesis are summarized as follows.
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(1) Propose a mathematical model for the three metabolic pathways:
A dynamic model encompassing glycolysis, TCA cycle and the pentose phosphate pathway in E. coli has been established and reasonably validated. However, the simulated performance is not good enough to describe the experimental data, because the experimental data were obtained for E. coli O157:H7, while the kinetic parameters used in our model is based on E. coli BW25113, which were the only kinetic parameters available in the open literature.
(2) Refine the model by least-square method
To conquer the aforementioned difficulty, we introduce the least-square method used in the engineering community to modify the proposed metabolic model. An iterative scheme has been proposed to tune the kinetic parameters in the model so as to minimize the matching error between the output of the model and the experimental data. After comparing with the measured metabolite concentrations, the simulation results based on the refined model can fit 80% experimental data. The major error occurs at the MAL metabolite, and further refinement on the kinetic parameters is required to achieve better consistency.
(3) Identify the source of error:
According to the observed discrepancy between the simulation and the experiment, we bring up three possible sources of matching error.
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a. Due to the oversimplified assumptions, the proposed model cannot fully describe the dynamic behavior of the reaction. If necessary, energy viewpoint may be included to reinforce the accuracy of the model.
b. The overall equations of TCA cycle comprise the respiratory chain, while we do not consider it here.
c. The inherent measurement errors appear in the measured concentrations of the
c. The inherent measurement errors appear in the measured concentrations of the