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176                                                  Essentials of Physical Chemistry



                                        No inhibitor        V max


                                                            Competitive
                                   V I
                                                             inhibition
                                                          V max
                               1/2 V max
                                                        Noncompetitive
                                                          inhibition
                               1/2 V max





                                                     [S]
                                      K M
                                           K M
            FIGURE 8.9  Types of inhibition in terms of substrate concentration. (From Kimball, J., Enzyme kinetics,
            http:==users.rcn.com=jkimball.ma.ultranet=BiologyPages=E=EnzymeKinetics.html. With permission.)



            so,


                                   1      (1 þ K I [I])K M  1    1
                                                                     :
                                      ¼                     þ
                                   V         V max      [S]     V max
            This is just like the Michaelis–Menten equation with a modified slope but the same y-intercept.

                             1                                        1       1
              Note that when     ¼ 0, we have the same y-intercept as before,  ¼  . However,
                            [S]                                       V      V max
                  1
            when      ¼ 0, the x-intercept changes if K I [I] > 0. Therefore, one can add a suspected inhibitor
                  V
            molecule to the enzyme solution and carry out the rate measurement and do the Michaelis–Menten
            analysis. If the slope of the Lineweaver–Burk line changes, that is proof of competitive inhibition by

                            (1 þ K I [I])K M
            species ‘‘I’’ since          ¼ slope.
                               V max
              An excellent example of the effect of inhibition is given on the Internet [16] at http:==users.rcn.
            com=jkimball.ma.ultranet=BiologyPages=E=EnzymeKinetics.html with data for o-diphenol oxidase
            shown in Figures 8.9–8.11. The enzyme is prepared from the supernatant liquid of homogenized
            apples. This enzyme is responsible for the fact that apples turn brown when the skin is removed.
            A Lineweaver–Burk plot is easily obtained using a spectrometer to measure the reaction as an
            increase in optical density at 540 nm with time. A similar set of data can be obtained with the
            presence of para-hydroxybenzoic acid which is shown to be an inhibitor and a third set of data is
            obtained showing the effect of phenylthiourea as a noncompetitive inhibitor. There are many other
            useful descriptions of enzyme kinetics on the Internet. Our function here is to provide the algebraic
            derivation which is not often given.



            MICHAELIS–MENTEN SUMMARY
            Enzymes are very efficient with high reaction rates when the substrate concentration is low but they
            can become saturated when the substrate concentration is high [16]. Competitive inhibition results
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