Irving M. Klotz Northwestern University Evanston, Illinois 6 0 2 0 1
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Free Energy Diagrams and Concentration Profiles for Enzyme-Catalyzed Reactions
Catalytic effects of enzymes have been interpreted in the terminology of different conceptual models. One of the most universally used adopts the terms of transition-state theory and constructs therefrom a diagram presenting the relative free energies of species in ground state and in activated states. A typical example of such a diagram is illustrated in Figure 1 for a reaction involving a single species s. The central concept of transition-state theory is that reactants are in equilibrium with the transition-state species and that the rate of reaction is proportional to the concentration of that species. On this basis one can introduce the thermodynamic concept of an equilibrium constant Kt for the formation of the transition state species from reactants and consequently one can calculate a standard free energy change, AGOf,for the activation process. These AGOt values provide the information necessary for the construction of a free energy diagram, such as that illustrated in Fieure 1. T o an inexperienced reader the diagram implies that the catalvtic effect of the enzvme. . . E,. on substrate, s, is due to the &ct that the energy level Got for the enzGe:activated complex (E.s)f is below that for the corresponding nunenzymatic species st. The very use of the letter E predisposes one to presume a catalytic effect on a reaction rate. In fact if E' were a molecule that combined strongly with s but slowed 'down the chemical transformation (i.e. if E' were an inhibitor) the free energy level of E'.s would still he below that of s and hence that of (E'.s)f could also lie below that of st. Indeed, as has been pointed out in numerno acceleration in rate occurs (in ous critical dis~ussions,'-~ for E.s the simplest, single-species reaction) unless AGO~E.~ (E.s)t is smaller than AG,Of for s st. Thus a free energy diagram can actually he misleading. In any event, it does not reveal the nature of enzyme catalytic effects in the terminolow of a thermodvnamic exoression of transitionstate tLory. On the other hand if one looks hack a t the central concept of transition-state theory, it is clear that what we need to see is the relative concentrations of participating
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FREE ENERGY DIAGRAM
Species A l m
With Catolyot
With inhibitor
CONCENTRATION PROFILE Species Alone
With Cotolvtic Macromolecule
Figure 2. Concennation pmfile presenting concentrations of substrate s in ground states and in transition states under various circumstances. This p r p file is for a reaction involving only a single solute species.
species. In terms of a chart what we need is a concentration profile. Let us consider first a simple single-substrate reaction (Fig. 2). In the nun-enzymatic reaction (left side of Fig. 2) the concentration of substrate in the ground state is represented by the height of the har so; that of the transition state species st, which is in equilibrium4 with s, is shown by the too of the bar snt. The rate of reaction is . oronortional . to the concentration of sot. In the presence of an enzyme or catalvtic macromolecule E. the substrate mav be hound (in the ground state) to produce the species E.s. The concentration of the latter will depend on the magnitude of the binding constant K E ~If. this constant is appreciable, a significant concentration of E.s will be produced at equilibrium and that of free s will drop perceptibly. Insofar as the focal question, the rate of chemical change of s, is cuncerned, the crucial issue is, what is the concentration of (E.s)t? If the equilibrium constant for the formation of (E.s)f from E.s is the same5 as that for the formation of sot from so in the absence of macromolecule, then no acceleration in rate will he effected by the macromolecule. For under these circumstances
and hence the ratio of concentration of transition state species to that of ground state species is given hy
' Pauling, L., Amer. Sei., 36,58 (1948).
Figure 1. Relative (standard) free energies of ground state and transition state species in the absence and in the presence of a catalytic macromolecule such as an enzyme. Substrate denoted by s, enzyme by E, inhibitor by E'.
Wolfenden, R., Aeets. Chem. Res., 5.10 (1972). Lienhard, G., Science, 180,149 (1973). The standard free energy AGOt for the formation of sf from s is ~rooortionalto the difference in heights - of the bars in the concen&ion profile chart. .. In other words, the AGot is the same for s bound by the macromolecule as it is for s free in solution. Volume 53, Nornber3, March 1976 / 159
Thus if the total concentration of s is the same in the oresence of macromolecule as in its absence, then the fraction of activated species is the same in the presence as in the absence of enzyme and hence ((E.s)X)+(sf) = ,f
CONCENTRATION PROFILE Spscies Alone
With Cotaiytic Mocrorndecuie
(3)
that is, the concentration of activated species is the same in the presence or absence of macromolecule. The presence of combining macromolecule just does not affect the total concentration of transition state species. Consequently the macromolecule would not affect the velocity of the reaction. I t follows, therefore, that in a single-suhstrate reaction, a catalytically effective enzyme or macromolecule must have a K E that ~ is larger than that, Kt, in the absence of macrom ~ l e c u l e Only . ~ then does it follow (from algebraic manipulations analogous to those in eqns. (1)-(3)) that (4) i(E-)') + ist) > isof) In a single-substrate reaction, a macromolecular catalyst must provide an environment conducive to the formation of activated species6 if the macromolecule is to effect an acceleration in rate of reaction. If the macromolecule does favor formation of the activated soecies. then the concentration of the latter deoends not only on K E but ~ also on the association constant, K E ~for , the formation of E.s (E.4 E + s = E.s; K E =~ -
(EM
(5)
Algebraic analysis leads to the following expression for the fraction, at, of activated molecules
Thus the fraction of.molecules activated, and hence the rate of reaction, depends on the concentration of macromolecule (E) and on its affinity for substrate, as expressed in KR-. If K -d is neater7 than Kt and is independent of K E ~then , the stronger the binding of s, the more accelerated would he its rate of reaction. I t is also evident from eon. (61,or Figure 2, that a t very high (E) or as KE, becomes increasingly large, uf tends toward a limit and the velocity of the reaction will attain a plateau value. Extending our concentration profile analysis to a twosubstrate reaction, we must distinguish between different modes of participation of the macromolecule. Figure 3 illustrates concentration relations in a reaction in which the primary substrate, s, must interact with a second (e.g. a nucleophile), n. In the absence of macromolecule, so and no form collision pairs no.so, whose concentration will usually be verv small since net attractive forces between these species in solution are usually negligible. The collision pair will then establish an eouilihrium concentration of activated complex (no.so)f. If the macromolecule has the second participant covalently linked to it, as E-n denotes, and if the macromolecule has a significant affinity for s, then an appreciable concentration of E-n.s will appear. In particular A
(E-n.s) > ( w s o )
(7)
the actual magnitude of (E-n.s) depending on the magnitude of the binding constant KE.",.. Consequently, even if the equilibrium constant KE;"~for the formation of (E-n.s)t is the same as that, Kt, in the absence of the mac-
160 / Journal of Chemical Education
Figure 3. Concentration profile presenting cancenbations of reacting species under various circumstances. The primary reactant is represented by s;the second participating small molecule species (or resaue if enached to enzyme) is denoted by n. This profile is for a reaction with two solute species. one of which, n, is not consumed.
romolecule, the concentration of activated species is greater in the presence of the macromolecule (8) ((E-n& > ( ( n o 4 ) Thus the velocity of the transformation is increased in the nresence of a bindine macromolecule even if AGOt is unchanged. If the macromolecule in addition provides an environment conducive to the existence of activated soecies. then the velocity will be increased even further. I t is oossihle also to have a two-substrate reaction in which neither subrrate is covalently linked to the mncromolecule. In that case the concentration orofile would have an appearance similar to Figure 3, hut wkh a bond hetween E and n absent. The predecessor of the activated complex (E.n.s)t would be E.n.s. Thus the comments made with respect to E-n.s are valid for E.n.s, but one must recognize one complicating feature in the equilibrium association generating the latter. Since E.n and E.s can also he formed, in addition to E.n.s, increasing concentrations of E relative to n and s will favor formation of the suhstrate-separated complexes Em and E.s and decrease the concentration of the two-participants complex E.n.s. Thus with increasing polymer concentration the acceleration in velocity may reach a maximum and thereafter actually decline. Summine uo. .. we find that to increase the concentration of transition-state species a catalytic macromolecule must as a minimum bind substrates. Thereafter, if the polymer provides an environment facilitating activation, further accelerations in rates of reaction will occur. A concentration profile illustrates the interplay between interrelated species and reveals exolicitlv the concentrations of activated molecules in the &ansition state which determine the reaction rate.8
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In other wards, AGof of activation must be lower in the presence of polymer than in its absence. If K E is ~ smaller than K > then the macromolecule decreases the rate of reaction, the degree of attenuation depending on the strength af binding and the concentration of (El. This investigation was supported in part by a grant (No.GM09280) from the National Institute of General Medical Sciences, US. Public Health Service.
