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