Intro to Pharmacology and Toxicology Topics   

Enzyme Systems

Enzyme systems may be simple, cooperative or exhibit substrate inhibition.

A simple enzyme system in which the reaction depends upon substrate concentration under constant conditions is described by the Michaelis-Menton equation:

                     k1                      kp
[E] + [S]     [ES]     [E] + [P]
                     k-1

v   =  Vmax [S]         (derivation)
        Km + [S]

When Km and Vmax are known, the rate of reaction can be predicted for any known substrate concentration. Whether the system under study is in vivo or in vitro, knowledge of Km and Vmax is crucial to understand the system.

The full progress curves for enzymatic reactions may be followed using specialized data capture systems. This allows for direct determination of Km and Vmax by integrating the Michaelis-Menton equation. More commonly, initial velocities are measured with varying substrate concentrations under steady-state (not equilibrium) conditions. The initial velocity data can then be analyzed by one of three methods: direct fit, double reciprocal or single reciprocal.

A graphing computer program is needed to directly fit the velocity data to the Michaelis-Menton equation. It will yield a plot of v versus [S] in the form of a rectangular hyperbola, and calculate Km and Vmax with standard errors.

A double reciprocal or Lineweaver-Burk plot yields a straight line, based on a rearrangement of the Michaelis-Menton equation:

v   =  Vmax [S]       Þ     1    =    Km + [S]   =    [S] (Km/[S] +1)
        Km + [S]                 v           Vmax [S]                Vmax [S]

Þ     1    =    Km/[S] + 1   
          v              Vmax         
               =       Km     +     1   
                     Vmax[S]      Vmax

Þ     1    =      Km       1    +     1   
          v            Vmax   [S]        Vmax

         y     =        m     x     +     b

         If   y = 0   Þ   x = -1/Km

         If   x = 0   Þ   y = 1/Vmax

Note that in a Lineweaver-Burk plot the x-intercept is the negative reciprocal of Km and the y-intercept is the reciprocal of Vmax.

Although the double reciprocal transformation is commonly used, it has several limitations. Errors are magnified at low substrate concentrations (upper end of line), as shown in the graph above. Line fit b in the graph is a better fit obtained by ignoring the lowest substrate concentration point.

The single reciprocal or Eadie-Hoffstee plot is another rearrangement of the Michaelis-Menton equation to fit a straight line:

v   =  Vmax [S]     =      Vmax [S]     
        Km + [S]          [S] (Km/[S]+1)

Þ     v (Km/[S] + 1)   =   Vmax

         vKm/[S]   =   Vmax - v

         v/[S]   =   Vmax/Km - v/Km

or      v/[S]   =   -1/Km v  +  Vmax/Km

            y      =      m    x   +     b

         If   y = 0   Þ   x = Vmax

         If   x = 0   Þ   y = Vmax/Km

Note that in a Eadie-Hoffstee plot the x-intercept is Vmax and the y-intercept is Vmax/Km.

The Eadie-Hoffstee transformation also distorts errors, especially at the extremes of the data (see line fit b in the graph above, done by ignoring the outliner point). The dependent variable (v) is in both axes, thus significant errors will reflect in both the x and y axes.

Although many enzyme systems follow or closely approximate Michaelis-Menton kinetics, some differ from that model. Some more complex enzyme systems are cooperative enzymes and enzyme inhibition systems. Cooperativity has been observed in many enzymes at important regulatory points in metabolic pathways.

Cooperativity may occur in enzymes with multiple monomers and several sites for substrate binding. The sites interact to influence catalytic activity. If the binding of the first molecule of substrate increases the activity of the enzyme towards the second molecule of substrate, the system is positively cooperative. If the binding of the first substrate molecule lowers the activity toward subsequent molecules of substrate, the system is negatively cooperative. Cooperative enzymes are described by the Hill equation:

where n is the number of catalytically equivalent substrate binding sites per enzyme molecule and K• is a constant including many interaction factors (a, b, c, etc.) and the intrinsic dissociation constant Ks. If n = 1 the equation reverts to Th. Michaelis-Menton equation. For enzymes exhibiting positive cooperativity n>1, while those exhibiting negative cooperativity have n<1.

Depending on the number of interacting subunits and the strength of interactions, the form of the v vs. [S] curve may be different from a rectangular hyperbola. The form of the curve for a positively cooperative enzyme is strongly sigmoidal. The shape of a curve may be described by its X0.9/X0.1 ratio, in this case a [S]0.9/[S]0.1 ratio. For a hyperbolic curve the ratio is 81, while for a sigmoidal curve the ratio equals 9. Therefore, a positively cooperative enzyme moves from nearly inactive to almost fully active within a narrow range of substrate concentrations.

Negatively cooperative enzymes show a more rapid rise but a depressed maximum relative to the original rectangular hyperbola. The [S]0.9/[S]0.1 ratio for a negatively cooperative enzyme is over 6000.

When enzymes have the ability to bind more than one molecule of substrate or bind substrate in more than one way, the possibility exists for one molecule of substrate to interfere with the binding and catalytic conversion of another molecule of substrate. In one model for such an interaction, a substrate molecule binds two sites on the enzyme, but when substrate concentrations are high may result in unproductive enzyme binding.

                    k1                          kp
[E] + [S]       [ES]     [E] + [P]
                    k-1             +    
                                    [S]
                                 k-2k2
                                [ESwrong]

v   =           Vmax [S]              
        Km + [S] + [S]^2 /Kis

Kis is the dissociation constant governing the wrong binding of substrate. For many enzymes Kis is 100 times the value of Km. Note that when [S] is small, the term [S]^2 /Kis becomes very small and the system closely approximates Michaelis-Menton kinetics.

The v vs. [S] plot for an enzyme showing substrate inhibition may be subtly or markedly different from a rectangular hyperbola, but will always show a depressed maximum. Working from this plot is very difficult to estimate Vmax and therefore Km cannot be determined. Curve fitting to the actual equation may be the only way to study this system.

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