Deriving Lineweaver-Burk Reciprocal Plot from Michaelis Menten Equation I

After a recent seminar, a number of students have asked how to derive the Lineweaver-Burk Reciprocal Plot equation from the Michaelis Menten equation... so, here goes.

The enzymatic reaction can be viewed as:

Where E = enzyme; S = substrate; ES = Enzyme substrate complex; and P = product

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The Michaelis-Menton Equation describing the reaction is:

Where v = rate (initial velocity); V_max = maximum velocity (100% of enzyme catalytic sites occupied); K_m = Michaelis constant (concentration of substrate to achieve half V_max); S = substrate concentration

Now, if the equation you are starting with does not look like the one above then have a look at another post (apparently I do it the 'old fashioned way’!)

In the lab, we can change the substrate concentration (S) in a reaction and measure the rate (v). As you know, a plot of substrate concentration (S) against rate (v; initial velocity) gives a curve, which plateaus at V_max, with K_m the concentration at half-V_max:

Plot of Substrate Concentration against Initial Velocity (rate)

The direct measurement for V_max can never be achieved in the lab as the concentration of the substrate needed would never be reached. Also, in the lab, we would use a computer to calculate the values and determine V_max, and K_m. However, you should be able to calculate V_max, and K_m yourself, just so you know the computer is right, and it is possible to calculate these values from experimental data by using a linear plot and an equation derived from the Michaelis-Menton Equation.

As you know, the equation for a straight line is:

Equation for a straight line, where m = the gradient and c = the intercept on the y-axis

So, the problem is, how do we get:


to look like equation 1? The answer is, we rearrange....

Rearranging

We need to 'extract' our substrate concentration S and our rate v so we can plot them on a straight-line graph. Basically, we need to get equation 2 to look like equation 1.

The equation for a straight line is:


And our Michaelis-Menton Equation:


So, the first thing we need to do is invert equation 2 to get:


If you didn’t understand that ‘mathematical move’ consider this:


The above is true. That is, 2 over 1 = 2, 4 over 1 = 4 and 2 times 4 is 8. If I simply invert (flip) all the parts:


It is also true.

Equation 3 is getting closer to what we want. However, the V_max over v is a problem as we don’t know V_max and can only measure v and S in the lab. Therefore, we need to separate out the terms we can measure so we can have x and y as in equation 1.

To ‘remove’ the V_max from the lefthand side we need to divide both sides by V_max:


Note the new V_max term on both sides of the equation - compare to equation 3.

As we now have V_max over v multiplied by V_max we can cancel out both V_max:


to give:

If that bit of maths has lost you, consider this:


if you divide both sides by 2 it is still correct:

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However, on the righthand side the two 2s can be cancelled to give:


which is still correct.

In equation 5 we now have 1/v and this is our y in equation 1. All we need to do now is ‘extract’ x (which is our substrate concentration) from equation 5.

If you consider the following it is true:


we can ‘separate terms’ on this and express it in several other ways:


That is, once we find a ‘common’ element (in the above example 1/2, and in equation 5 1 over [S]V_max) we rearrange, so:


extracting 1 over [S]V_max we get:


multiplying through with 1 over [S]V_max gives:


As you can see in equation 7 we have two terms after the + that can cancel out, and our experimental variable (S) can be separated, so:


Finally, separating out 1/[S] gives:


Which when compare to equation 1:


It can be seen that:

• y = 1/v
• x = 1/[S]
• c, the y-intercept = 1/V_max
• m, the gradient = K_m/V_max

And, the intercept for the x axis, (i.e. when rate (y) = 0) is 1/-K_m.

Hence, the final graph is:

If you struggle with 'Science Maths' then you may like to look at Maths4Biosciences

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Additional Resources

• 📗 - Biochemistry (Stryer) (affiliate link to the book)
• 📗 - Principles of Biochemistry (Lehninger) (affiliate link to the book)
• 📗 - Maths and Chemistry Refresher for Life and Biomedical Scientists
• 📗 - Catchup Chemistry (affiliate link to the book)
• 📗 - Catchup Maths and Stats (affiliate link to the book)