Linear Regression With Biological Data

Linearization on Michaelis Menten equation

Michaelis menten equation , is a mathematical model on which we can investigate the reaction kinetics of an enzyme when it binds to its substrate , it gives us full understanding on the underlying rate of reactions and the effects on the substrate by the catalysis of the enzyme.

Importance of this mathematical model lies in revealing the catalytic mechanism of this enzyme, its role in metabolism, how its activity is controlled and how a drug or an agonist might inhibit the enzyme.

General mathematical mode of michaelis menten is as follows

The plot of v versus [S] is not linear, this nonlinearity could make it difficult to estimate Km and Vmax accurately , therefore, several researchers developed some linearization of the above mathematical model such as lineweaver-burk plot , Eadie-Hofstee diagram and the hanes-Woolf plot. In this assignment, I am going to explain the lineweaver-Burk plot In more details.

LineWeaver-Burk Plot

LineWeaver-Burk plot or double reciprocal plot is a common way of illustrating kinetic data. It is produced by taking the reciprocal of both sides of the michaelis menten equation. As follows 

The above double reciprocal method yields a linear equation on the rate of catalytic reaction of an enzyme which can be plotted against the observed data .

According to the general formulae for a straight line of the form

Y = M * X + b

M è Which is the slope of the regression line, in LineWeaver-Burk plot represents Km / Vmax

B è The intercept , Y-Axis intercept which represents the reciprocal of Vmax . 1 / Vmax

And another intercept with X-Axis which represents the reciprocal of michaelis constant which is 1 / Km

The following plot explains it in more details

Example of Michaelis menten Linearization

In R , I have followed the following steps to produce LineWeaver-Burk Plot

Ø substrate <- c(20,40,80,120,160,220,260)

Ø reaction <- c(41.0,77.0,113,144,162,185,197)

Ø dataset <- data.frame(substrate,reaction)

Ø attach(dataset)

Ø plot(reaction~substrate,main=”Michaelis Menten Kinetics Plot”)

Producing a lineWeaver Burk plot implies taking the double reciprocal on both sides of michaelis menten equation , so taking the reciprocal of both substrate and reaction variables as follows

Ø recipSubstrate <- 1 / substrate

Ø recipReaction <- 1 / reaction

Ø plot(recipReaction~recipSubstrate,main=”Reciprocal of Michaelis menten variables”)

Now to generate a linear model from the following data , we are going to issue the lm Command in R.

Ø  linearmodel <- lm(recipReaction~recipSubstrate)

Ø Call:

Ø lm(formula = recipReaction ~ recipSubstrate)

Ø  

Ø Coefficients:

Ø    (Intercept)  recipSubstrate 

Ø       0.003468        0.412298 

Ø Summary(linearmodel)

Ø Call:

Ø lm(formula = recipReaction ~ recipSubstrate)

Ø  

Ø Residuals:

Ø          1          2          3          4          5          6          7

Ø  3.072e-04 -7.886e-04  2.277e-04  4.048e-05  1.278e-04  6.317e-05  2.223e-05

Ø  

Ø Coefficients:

Ø                 Estimate Std. Error t value Pr(>|t|)   

Ø (Intercept)    0.0034682  0.0002145   16.17 1.65e-05 ***

Ø recipSubstrate 0.4122980  0.0096963   42.52 1.36e-07 ***

Ø ---

Ø Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Ø  

Ø Residual standard error: 0.0003976 on 5 degrees of freedom

Ø Multiple R-squared:  0.9972,   Adjusted R-squared:  0.9967

Ø F-statistic:  1808 on 1 and 5 DF,  p-value: 1.357e-07

To generate the regression line on the following data

Ø Abline(linearmodel)

Interpretation

From the above data , we can predict both Vmax and Km as
follows

According to line-Weaver Burk plot , the Y-axis intercept is
1 / Vmax

b = 1 / Vmax  =
0.0034682

Vmax = 288.334

Where the X-Axis intercept with the Reciprocal of the
substrate on the above plot is 

1 / Km = 0.4122980

Km = 2.42543

Now we have predicted both Vmax and km values, Therefore, we
can calculate the slope of the line which is represented by 

M = Km / Vmax = 2.42543 / 288.334 = 0.008411

Coefficient of Determination R-Squared is  = 0.9972 , that means about 99 % of the data
will be predicted by the regression line and 1 % will be left for unknown
reasons.

Generally , the regression line is considered a good fit for
predicting the relationship between the two variables.