Re: how Lineweaver-Burk plots(1/V vs. 1/[S]]determine kind of enzyme inhibition

Before going into Lineweaver-Burk plots (sorry, this is going to be long), let's look at the standard diagram of enzyme kinetics:

       Ks      kcat

E + S  ´  ES  Æ E + P

Or, Enzyme (E) and Substrate (S) bind to form a complex (ES) which allows the substrate to be converted into Product (P). The binding constant for Substrate to enzyme is K_s, and the Catalytic constant for conversion of substrate to product is k_cat. Working from this, Michaelis and Menton derived their equation for enzyme kinetics, defining V_max as the maximum rate of the reaction if all of the substrate were in ES. They also noticed that at half this rate, K _s was equal to the concentration of Substrate [S]. They called this the Michaelis constant (K_M), so K_M = [S] at V_max /2. Continuing on, Lineweaver and Burk rearranged the Michaelis - Menton equation and graphed it to make the Lineweaver - Burk plot pictured here. The key features of the Lineweaver - Burk plot are: the Y- intercept = -1/K_M; the X-intercept = 1/V _max; and the slope = K_M /V_max. Now we're ready to look at inhibitors. There are three classical types of inhibitors, Competitive, Non-competitive, and Uncompetitive.

Competitive inhibitors work by competing with the substrate for binding with the free enzyme. This reduces the supply of available enzyme, and thus slows the rate of reaction.

       Ks      kcat

E + S  ´  ES  Æ E + P
Ø+I
EI

Where inhibitor (I) binds free enzyme (E) to form an unreactive complex (EI). The result of this is an apparent increase in K_M, since it takes more substrate [S] to reach V _max/2. Since the inhibition can be overcome by adding huge amounts of substrate, the value of V_max remains unchanged. On the Lineweaver - Burk plot, Competitive Inhibition looks like this, notice the X-intercept (1/V _max) doesn't move but the Y-intercept (-1/K _M) does.

Non-competitive inhibitors work by binding to the enzyme in a way that doesn't affect its ability to bind substrate, but does prevent it from converting the substrate into product. Because the binding of the non- competitive inhibitor is independent of substrate binding, we get:

       Ks      kcat

E + S  ´  ES  Æ E + P
Ø+I        Ø+I
EI         ESI

Where inhibitor (I) can bind either free enzyme (E) or bound enzyme (ES) to form an unreactive complex (EI or ESI, respectively). The result of this is a decrease in V_max, but no effect on K _M, since the binding of enzyme to substrate is unaffected. On the Lineweaver - Burk plot, Non-competitive Inhibition looks like this, notice the X-intercept (1/V_max ) moves but the Y-intercept (-1/K_M ) doesn't.

Uncompetitive inhibitors only bind to the enzyme when it is complexed with substrate (ES). This both prevents product from being formed, and pulls more enzyme into the ES complex, affecting the apparent rate of substrate binding.

       Ks      kcat

E + S  ´  ES  Æ E + P
           Ø+I
           ESI

Where inhibitor (I) binds the enzyme/substrate complex (ES) to form an unreactive complex (ESI). The result is a decrease in V _max and an equivalent decrease in K_M , such that K_M/V _max is the same. The Lineweaver - Burk plot would look something like this, if the lines were parallel (i.e. the slope remains unchanged).

Most non-competitive inhibitors show some preference for either the free enzyme or the ES complex, such that the K_M is not unaffected by the addition of inhibitor. In these cases, the non- competitive inhibition is called mixed, since it is somewhere between Competitive and Uncompetitive without being truly Non-competitive.

So, where the lines intersect on the Lineweaver - Burk plot is dependent on the changes in K_M and V _max with the addition of your specific inhibitor. This can be related directly to the mechanism of inhibition and thus the type inhibitor you are using.