Section II: Enzymes
Topics: introduction to energetics, activation energy and catalysis, interactions between active site and substrate(s), enzyme kinetics, enzymatic mechanisms.
A. Introduction to Energetics ( see pages 35-39)
Enthalpy, H, is defined as H = E + PV, entropy, S, as S = Q / T and free energy, G, as G = H - TS.
B. Activation Energy and Catalysis (see pages 45-47)
In introductory reaction kinetics you learned that a simple reversible reaction such as A<=> B may be decomposed into A => B and B => A and that the velocities of those two reactions may be expressed as v1 = k1 [A] for the first, and v2 = k2 [B] for the second, where the letters k represent are rate constants that are complex functions of temperature. When the two velocities are equal, the process has reached equilibrium and k1[A] = k2[B] which leads to [B] / [A] = k1 / k2. Since the equilibrium constant for this reaction is K = [B] / [A], substitution of the previous result yields K = k1 / k2. If we considered the reverse reaction B A, its equilibrium constant would be equal to k2 / k1.
Chemical reactions proceed through, first, the formation of transition states with higher energy levels than those of the reactants. The energy required for the formation of the transition states is called activation energy. Arrhenius established a quantitative relationship between the rate constant of a reaction and its activation energy based on his collision theory, which assumed that reactant molecules could be considered as hard spheres moving in random directions with velocities proportional to the temperature:
k = A [ exp - (E / RT)], or ln k = A - E / RT
where k is the rate constant, R and T have the usual meanings, E is the activation energy and A, the so called frequency factor, may be interpreted as the fraction of collisions between the molecules with the proper orientation to produce a reaction. The exponential function exp - (E / RT) represents the fraction of molecules of reactant having at least the critical energy E for the reaction to occur. A different approach to reach an expression for the rate constant is followed in the textbook on page 45, where the dependence of the reaction rate constant with respect to the free energy of activation is derived by a different route. Other relationships have been proposed, notably the one based on Eyring's transition state theory, which is a bit more complex.
Figure 1 shows the reaction path for the non-catalyzed reaction with a large activation free energy G*f for the forward reaction, and a still larger activation free energy G*r for the reverse reaction, which means that the forward reaction is faster. Consequently, when equilibrium is reached the concentration of product will be higher than that of reactants. It also shows the delta G for the reaction, which is obviously the same no matter what path the reaction takes. This is a necessary consequence of the fact that all thermodynamic functions (E, H, S, G, etc) are functions of state, i.e., their values depend only on the state of the system, and the difference in their values between any two states is independent of the path followed by the transition from one state to the other. Arrhenius' equation states that, given a certain energy of activation, the higher the temperature the larger the number of molecules capable of reaching the transition state and the larger the value of the rate constant. Similarly, that, for a given temperature, the lower the activation energy the larger the number of molecules capable of reaching the transition state and the higher the value of the rate constant. When the reaction path is complex and shows various energy hills and troughs, the overall rate will be determined by the step with the highest activation energy, which will be rate limiting.
Enzyme action is just a special case of protein binding in which the effect of the binding is to change the ligand's structure in such a way that the activation energy for a chemical transformation of the ligand is decreased. Enzymes are, therefore, biological catalysts, since by reducing the activation energy for a reaction, they increase its rate constant and, thereby, its velocity.
Figure 2
in this page and Figure 2-27 in the
textbook show diagrams that illustrate the
activation energies of the non-catalyzed and the
catalyzed forward reactions, G*f and G,
respectively. Arrhenius' equation states that the
velocity of the reaction increases with increased
temperature, reduced activation energy and with
an increase in the frequency factor. A catalyst as
an enzyme may reduce the activation energy,
increase the frequency factor or, more
commonly, do both. A very large activation
energy results in such a low rate constant that
the reaction is so slow as to be undetectable
even if the change in free energy has a large
negative value. In other words, a reaction may
be thermodynamically quite spontaneous (very far from equilibrium) and still be so slow that for
all practical purposes does not happen.
C. Interactions between the Active Site and Substrate
It was recognized very early that an enzyme and its substrate had to make contact for the catalysis to take place. That this contact was highly specific and that it depended on weak interactions between structurally complementary regions of both enzyme and substrate became progressively clear as more became known about the structure of these proteins. The original view of Emil Fischer of a lock and key relationship, in which the enzyme is presumed to have a rigid structure that exactly matches that of the substrate has been substituted by one proposed by Koshland based on the idea that the protein structure has a degree of flexibility. According to this view referred to as the induced fit hypothesis, the correspondence between the complementary structures is good enough to account for the specificity of the interaction, but, as the binding takes place, both structures undergo slight distortions that considerably improve the fit and make the binding much stronger. At this point, the substrate has achieved the conformation of the transition state (at the top of the activation energy curve) as some of the bonds that need to be broken for the reaction to take place have been weakened. Using the free energy liberated by the binding plus some thermal energy absorbed from the environment.
The meaning of the symbols in Figure 3 are as follows: G is free energy; E is the enzyme; S the
substrate; ES the enzyme substrate complex, which is an imaginary transient state after the
substrate binds, but before it has begun changing towards the activated complex; ES* is the
activated complex in which the substrate
has already reached the transition state; E
+ S* is an imaginary state in which,
although the enzyme is present, it has no
effect on the delta G of activation of the
substrate [dG*(S*)], i.e., the enzyme is
assumed not to play any role; P is the
product; dGb(S) is the delta G of binding
of the substrate to the enzyme and has a
negative value; dG*(ES*) is the delta G
of activation of the enzyme substrate
complex; dGb(S*) is the delta G of
binding of the activated substrate to the
enzyme and has a negative value; and dG
is the delta G of the overall reaction.
Taking into account that the activation
energy of the substrate [dG*(S*)] has a
certain value at a given temperature, it is clear from the figure that the larger the affinity of the
enzyme for the activated substrate [dGb(S*)], the lower the activation energy of the reaction
when the enzyme participates [dG*(ES*)]. This can be shown analytically, but I am not including
it here because it is a little involved. I'll be happy to show it to any of you who so desires.
All these models are just approximations to the real phenomenon and will, therefore, lead to only incomplete representations of it. Still, they are essential to form a reasonable idea of how enzymes work. For example, the transition from E + S to ES to ES* is surely complex and so fast that the process probably goes through many intermediates at such a speed that they may not be detectable with existing techniques. It is now accepted that, as the binding to the enzyme takes place, both enzyme and substrate undergo changes. And, when the distortions suffered are such that the fit improves and more bonds, or stronger ones, can be made, the free energy of binding acquires a larger negative value and becomes capable of compensating for the positive delta G of the distortions that bring the substrate closer to the active state. It has been shown experimentally that molecules that resemble the transition state are frequently powerful inhibitors of the enzyme because they bind to it better than does the natural substrate.
D. Enzymatic Mechanisms.
E. Enzyme Kinetics (see pages 71-74)
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