Medicinal Chemistry Applet

Lineweaver-Burk plots

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Introduction

Enzymes catalyze reactions in physiological systems.  In an equilibrium, an enzyme (E) binds a substrate (S) to form an enzyme-substrate complex (E-S).  The E-S complex can dissociate or irreversibly convert the substrate to a product (P) (Scheme 1).  The Michaelis-Menten equation describes the relationship between the rate of substrate conversion by an enzyme to the concentration of the substrate (Equation 1).  In this equation, V is the rate of conversion, V_max is the maximum rate of conversion, [S] is the substrate concentration, and K_m is the Michaelis constant, the substrate concentration at which the rate of conversion is half of V_max.  A more illustrative version of the Michaelis-Menten equation is the Lineweaver-Burk equation (Equation 2).  The Lineweaver-Burk equation affords a line with a slope of K_m/V_max and y-intercept of 1/V_max.  The x-intercept, a theoretical point since 1/[S] cannot be negative, is -1/K_m.

Scheme 1 - enzyme-substrate complex and product formation model

     (1)

     (2)

Enzyme inhibition is a common goal for the pharmaceutical industry.  All inhibitors cause the substrate to react at a lower rate than without the inhibitor.  Reversible enzyme inhibitors fall into three categories - competitive, non-competitive, and uncompetitive.  Furthermore, non-competitive inhibitors can be divided into two additional categories - pure and mixed.  The Lineweaver-Burk equation can be used to categorize different inhibitors.  Understanding the type of inhibitor will give clues on how its structure might be modified to increase its potency.

Competitive inhibitors bind at the active site of the enzymes to form an E-I complex (Scheme 2).  The inhibitor blocks the active site, and the substrate cannot bind until the inhibitor dissociates.  Since the inhibitor and substrate compete for the same site, raising the substrate concentration can eventually overcome the inhibitor, and V_max can be achieved.  Although V_max can be reached, a competitive inhibitor raises K_m, indicating that the affinity of the enzyme for the substrate is lower in the presence of the inhibitor.  The effect of a competitive inhibitor in a Lineweaver-Burk plot is both to move the x-intercept and increase the slope.  Plots made with varying amounts of a competitive inhibitor will all cross at the same y-intercept.

Scheme 2 - competitive inhibitor model

Non-competitive inhibitors bind at an allosteric site on the enzyme and leave the active site unblocked.  In a pure non-competitive system, the substrate has an identical affinity for both the E-I complex and enzyme.  Unlike the E-S complex, the E-I-S complex cannot convert the substrate to product (Scheme 3).  With a pure non-competitive inhibitor, the K_m value is unchanged while V_max is lowered.  So, the x-intercept will be constant, and the slope will increase with more inhibitor.  Note that pure non-competitive inhibitors are virtually unknown.  With a mixed non-competitive inhibitor, the affinity of the E-I complex for the substrate is not the same as the unbound enzyme.  In this case, not only is V_max lowered, but K_m is also raised.  The Lineweaver-Burk plot will show changes in the x-intercept and increasing slope.

Scheme 3 - non-competitive inhibitor model

Uncompetitive inhibitors are thought to bind the E-S complex and not the enzyme.  As with non-competitive inhibitors, the E-S-I complex cannot form the product.  The product can only be formed from the E-S complex (Scheme 4).  The effect of an uncompetitive inhibitor is to decrease both V_max and K_m.  The drop in K_m deserves some comment.  K_m is a measure of substrate affinity for the enzyme.  A lower K_m corresponds to a higher affinity.  The presence of an uncompetitive inhibitor actually increases the affinity of the enzyme for the substrate.  This surprising fact can be understood through the binding equilibrium.  Since the inhibitor binds the E-S complex, the inhibitor decreases the concentration of the E-S complex.  By Le Chatlier's principle, equilibrium of the enzyme and substrate will shift to form more E-S complex.  Therefore, the enzyme demonstrates a higher affinity for the substrate eventhough this higher affinity does not lead to a higher V_max.  In a Lineweaver-Burk plot, uncompetitive inhibitors shift the line higher with a raised y-intercept.

Scheme 4 - uncompetitive inhibitor model

Applet

This applet plots 1/V vs 1/[S] data for up to three compounds/situations.  The data points are plotted in a solid line to 1/[S] = 0.05.  The remainder of the lines are dotted back to the x-intercept of -1/K_m.  For each compound, K_m and V_max must be specified.  The units of K_m and [S] are concentration, e.g. mM or µM.  The units of V_max and V are amount of product over time, typically µmol/min or similar.

Problem information

Fumarase catalyzes the hydration of fumarate (1) to L-maleate (2).  This hydration reaction has a V_max of 5.1x10^-3 M min^-1 and a K_m of 2.0x10^-3.  Kinetic data for the inhibitor citrate (3) are shown below in the table.

Problems

1. Using the applet, generate Lineweaver-Burk plots for fumarase with three concentrations of citrate - 0.0, 3.5, and 7.0x10^-3 M. 
2. What type of inhibitor is citrate for fumarase? Based on the structure of citrate, is this reasonable?
3. Assume that compound 4 was found to also be an inhibitor of fumarase?  Without knowing the kinetic data, what type of inhibitor might you expect 4 to be (or not be)?

   

Reference

Teipel, J. W.; Hill, R. L. J. Biol. Chem. 1968, 243, 5679.

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