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174                                                  Essentials of Physical Chemistry


                               TABLE 8.1
                               Typical Data for ‘‘Enzyme-X’’
                               S,mM=L  Rate, V,mM=s  1=[S], L=mol  (1=V), s=mM
                                0.5       0.0064     2000.0      155.56
                                1.0       0.0136     1000.0       73.33
                                2.0       0.0212      500.0       47.22
                                5.0       0.0391      200.0       25.56
                                6.5 a     0.0450 a    153.85 a    22.22 a
                               10.0       0.0546      100.0       18.33
                               15.0       0.0628       66.7       15.93
                               20.0       0.0679       50.0       14.72
                               25.0       0.0714       40.0       14.00
                               35.0       0.0759       28.57      13.17
                               50.0       0.0797       20.0       12.55
                               100.0 a    0.0900 a     10.0 a     11.11 a
                               a
                                Points added by calculation to show the V max limit and
                                K M ¼ 0.0065 mM.


            so  K M ¼ (0:0722)=(11:11) ¼ 0:00649865 ffi 0:0065 mM  and  V max ¼ 1=11:11 ¼ 0:090009 ffi
            0:090 mM=s. So, with the Lineweaver–Burk plot, we see that when y ¼ 0, we have
            x ¼ 153.85 ¼ 1=K M , which checks the value of K M as 0.0065 mM. Then, when x ¼ 0,
            y ¼ 1=V max and V max ¼ 0.090 mM=s again.
              Thus, we see that enzyme kinetics uses the steady-state approximation and the double-reciprocal
            plot to provide a robust approach to study the reaction and K M is a useful concept that tells us what
            concentration of the substrate will give one half of the maximum rate.


            MICHAELIS–MENTEN WITH COMPETITIVE INHIBITOR
            There are actually several other cases of enzyme reactions, but keeping to our list of ‘‘essential’’
            physical chemistry we will only treat the important case of a competitive inhibitor of the normal
            substrate since this is at the heart of much pharmaceutical research. At the simplest level, one can
            use the ‘‘lock-and-key’’ concept to imagine that there are other molecules slightly different from the
            natural substrate molecule. Suppose the natural substrate has a methyl group exposed in a certain
            place. There could be a similar molecule that is the same but without the methyl group and it can
            probably fit into the same active site cavity but might not do the same reaction. Another molecule
            might be the same as the natural substrate but have an ammonium ion instead of the natural methyl
            group. The ammonium group will probably fit in the same space as the methyl group but the charge
            on the ammonium ion may severely change the chemistry in the active site. Other possibilities exist
            but the point is that there are other molecules, which can compete with the natural substrate but
            which do not do the same chemistry.
              The concept of the active site in a floating, mobile, water-soluble enzyme can be extended to
            biological ‘‘receptors’’ that are fixed in cell membranes and a similar analysis can be applied to
            competitors to natural substrates. This means that it is important to have an analysis procedure for
            competitive inhibition. Consider the Michaelis–Menten equations with an inhibitor ‘‘I’’:

                                             k 1
                                                        k 2
                                      E þ S ! (E  S) ƒƒƒƒ! E þ P

                                            k  1
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