Metabolic Control Analysis and Enzyme Kinetics

1. Drawbacks of rate-limiting step concept

At a steady state, the flux through each pathway in a biochemical network is a function of the individual enzyme kinetic properties. The activities of the enzyme affect the concentration of its reactants and products and influence the flux through pathways. Metabolic control analysis (MCA) provides a mathematical framework to study the distribution of metabolic fluxes and concentrations among the pathways that comprise the model. It replaces the principle of the rate-limiting step, which proved to be ineffective in practice. The control of the system as a whole is much more distributed than it was appreciated, making rate-limited step not very useful.

2. Purpose of MCA

The purpose of the MCA is to identify the steps which have the strongest effect on the levels of metabolites and fluxes. Its basis is the overall steady state flux with respect to the individual enzyme activities.

3. MCA coefficients

The challenge in analysing a metabolic network is determination of flux control coefficients (FCC). The FCC is a measure of how the flux changes in response to small perturbations in the activity or concentration of the enzyme. The value of the FCC is a measure of how important a particular enzyme is in the determination of the steady state flux. Another set of variables are elasticity coefficients. They quantify the influence of the pool levels on the individual pathway reactions.

4. MCA theorems

MCA uses two theorems. First is the summation theorem, which states that the sum of all FCC related to a particular pathway equal to 1. A more important theorem is the connectivity theorem; as it provides understanding of the way enzyme kinetics affect the values of FCC. It states that the sum of the products of the FCC of all steps that are affected by X and their elasticity coefficients towards X, is zero

5. Estimating FCC

There are several ways of estimating FCC, which can be roughly divided into experimental estimation and modelling.

5.1 Experimental estimation

• Changes can be introduced into enzyme activities and changes in flux measured.
• Elasticity coefficients can be calculated if the kinetics of each step of the pathway are known, then FCC can be calculated from elasticity coefficients
• In-vitro titration of enzyme activities

5.2 Estimation through modelling

• From their definition by small change in reaction rate and calculation of the resulting change in flux or concentration
• From matrix methods that use summation and connectivity theorems. The first approach is based on two matrices, one containing elasticity coefficients and another containing FCC. This approach works but is hard to implement in software. Alternative approach, developed by Reder, requires only knowledge of stoichiometry matrix and elasticity coefficients. This method is best for software calculation of FCC from elasticity coefficients.

by Evgeny. Also posted on my website

Estimating Michaelis-Menten Parameters

1. Introduction

The asparatic proteinase from HIV is a target for antiviral drug design. Kuzmic et al (1996) studied the influence of mechanical stirring on the process of asparatic proteinase catalysed hydrolysis of a fluorogenic peptide. The kinetics were measured for the two reactions types, one with mechanical stirring and one without, under otherwise identical conditions. Initial velocities from both sets of reactions were plotted at initial substrate concentrations of 6, 4, 3, 1.5, 1 and 0.66 micromoles. Kuzmic et al (1996) did not include the raw data into the article, but the data was presented in Figure 1.

Figure 1. Raw data used in the exercise

For the purpose of this exercise, the raw data was visually translated from the graph into the numeric values. The data then was plotted using the following R script

conc<-c(0.66,1,1.5,3,4,6)
rate1<-c(0.0611,0.0870,0.1,0.1407,0.1685,0.1796)
rate2<-c(0.0685,0.0944,0.1148,0.1592,0.1796,0.1907)

plot(conc,rate1,ylim=range(c(rate1,rate2)),lwd=2,col="red",xlab="[S], micromoles",ylab="V0")
par(new=TRUE)
plot(conc,rate2,ylim=range(c(rate1,rate2)),lwd=2,col="green",axes=FALSE,xlab="",ylab="")

Figure 2. Raw data plotted with R

2. Using the Lineweaver-Burk plot

Next, the reciprocal values were produced and the Lineweaver-Burk plot was constructed and plotted using the following R script

concLB<-1/conc
rate1LB<-1/rate1
rate2LB<-1/rate2
plot(concLB,rate1LB,ylim=range(c(rate1LB,rate2LB)),lwd=2,col="red",xlab="1/[S]",ylab="1/V0")
par(new=TRUE)
plot(concLB,rate2LB,ylim=range(c(rate1LB,rate2LB)),lwd=2,col="green",xlab="",ylab="")

Figure 3. Lineweaver-Burk plot with R

Next, the least squares regression was applied to the plot of reciprocals to establish the values of K_m and V_max

fit<-lm(rate1LB~concLB)
fit

Call:
lm(formula = rate1LB ~ concLB)

Coefficients:
(Intercept)       concLB  
      4.050        7.626

From the results of the script, the regression line for the data gathered in the absence of stirring is described by the following equation
1/ V_max = 4.050 + 7.626*(1/ K_m)
V_max = 1/4.050 = 0.2469
-7.626*(1/ K_m) = 4.050
-(1/ K_m)=0.5311
K_m = 1.8829

Similarly, the following script was applied to the second set of data

fit<-lm(rate2LB~concLB)
fit

Call:
lm(formula = rate2LB ~ concLB)

Coefficients:
(Intercept)       concLB  
      3.957        6.932  
Call:
lm(formula = rate2LB ~ concLB)

Coefficients:
(Intercept)       concLB  
      4.166        7.170 

The regression line was described by the following equation.
1/ V_max = 4.166 + 7.170*(1/ K_m), from which
V_max = 0.24, K_m = 1.7212

3. Using the Hanes-Woolf plot

The Lineweaver-Burk plot is prone to errors as small errors in reaction rate measurement increase the reciprocal. Among alternative linear forms is the Hanes-Woolf plot where the ratio of the substrate concentration to the reaction velocity is plotted against substrate concentration. The data was used to plot the Hanes-Woolf plot using the following R script

rate1HW<-conc/rate1
rate2HW<-conc/rate2
plot(conc,rate1HW,ylim=range(c(rate1HW,rate2HW)),lwd=2,col="red",xlab="[S]",ylab="[S]/V0")
par(new=TRUE)
plot(conc,rate2HW,ylim=range(c(rate1HW,rate2HW)),lwd=2,col="green",xlab="",ylab="")

The least squares regression was applied

fit<-lm(rate1HW~conc)
fit

Call:
lm(formula = rate1HW ~ conc)

Coefficients:
(Intercept)         conc  
      8.005        4.191  

The regression line is described by the following equation
(S/v) = 8.005 + 4.191*S
K_m = 8.005/4.191 = 1.91
K_m / V_max = 8.005
V_max = 1.91/8.005 = 0.2386

Similar script was used to process the data taken in the presence of stirring.

fit<-lm(rate2HW~conc)
fit

Call:
lm(formula = rate2HW ~ conc)

Coefficients:
(Intercept)         conc  
      6.726        4.054

The regression line is described by the following equation
(S/v) = 6.726 + 4.054*S
K_m = 1.659, V_max = 0.2467

4. Using the R language package

Finally, the non-linear self-starting Michaelis-Menten model, which is part of the R language, was used to estimate the values of V_max and K_m in the absence of stirring using the following R script

model<nls(rate1~SSmicmen(conc,a,b))
summary(model)

and providing the following output

Formula: rate1 ~ SSmicmen(conc, a, b)

Parameters:
  Estimate Std. Error t value Pr(>|t|)    
a  0.23953    0.01188  20.166 3.57e-05 ***
b  1.92898    0.23804   8.104  0.00126 ** 
---
Signif. codes:  0 *** 0.001 ** 0.01 * 0.05 . 0.1   1 

Residual standard error: 0.005621 on 4 degrees of freedom

Number of iterations to convergence: 0 
Achieved convergence tolerance: 1.244e-06 

Which estimated V_max = 0.23953 and K_m =1.92898

Similar results for the reaction with stirring were

model<-nls(rate2~SSmicmen(conc,a,b))
summary(model)

Formula: rate2 ~ SSmicmen(conc, a, b)

Parameters:
  Estimate Std. Error t value Pr(>|t|)    
a 0.248409   0.006281   39.55 2.44e-06 ***
b 1.682884   0.111784   15.05 0.000113 ***
---
Signif. codes:  0 *** 0.001 ** 0.01 * 0.05 . 0.1   1 

Residual standard error: 0.003319 on 4 degrees of freedom

Number of iterations to convergence: 0 
Achieved convergence tolerance: 2.261e-06 

Which estimated V_max = 0.248409 and K_m =1.682884

5. Conclusion

For reference, Kuzmic et al (1996) estimated the best-fit values of kinetic constants as K_m (1.71 ± 0.28) μM, and V_max (0.235 ± 0.015) abstract units/s in the absence of stirring, and K_m (2.31 ± 0.24) μM, and V_max (0.279 ± 0.012) with stirring. The results obtained by different methods were summarised in Table 1.

	L-B Plot	H-W Plot	SSmicmen	Kuzmic et al
No Stirring, Km, μM	1.8829	1.91	1.92898	1.71 ± 0.28
No Stirring, Vmax	0.2469	0.2386	0.23953	0.235 ± 0.015
Stirring, Km, μM	1.7212	1.659	1.682884	2.31 ± 0.24
Stirring, Vmax	0.24	0.2467	0.248409	0.279 ± 0.012

Table 1. Michaelis-Menten parameters estimated by various methods

References:

Crawley, M, The R Book, Wiley, 2007

Kuzmic P, Peranteau A, Garcia-Echeverria C, Rich D, Mechanical Effects on the Kinetics of the HIV Proteinase Deactivation, Biochemical and Biophysical Research Communications 221, 313-317 (1996)

by Evgeny. Also posted on my website