In allosteric proteins, the association of ligands to enzymes, a bimolecular step, may precede or follow the unimolecular conformational change or, if binding occurs at states far from equilibrium, the bimolecular step may occur during the unimolecular change. In the first two cases, the unimolecular transition described here will be independent of the bimolecular components of the reaction. The free energy difference between the two surfaces in Fig. 3 can be determined between the ligand-bound conformations in the first case and for the ligand-free conformations in the second one. If ligand binding, however, occurs concurrently with the structural transformation, then the situation becomes substantially more complex. For example, if ligand binding is fast compared with protein structural changes, the free energy drive on the reaction would be ligand-concentration-dependent, with the caveat that the existence of ligand preequilibrium states might destroy this effect. A similarity to the nonadiabatic and adiabatic patterns of electron transfer is immediately apparent (37).
When unfolding or cracking is allowed, the rate dependence on the stability difference of open and closed forms differs from the purely elastic models. Without unfolding, the energy surfaces are quadratic, so the activation barrier varies quadratically with the free energy difference of the two forms, much as in the Marcus theory for electron transfer (37). On the other hand, when unfolding occurs, the free energy surfaces are locally linear, giving a linear dependence of the activation energy on the reaction free energy change. Using protein engineering to differentially affect the stability of the two forms is the natural way to test this model.
Recent observations of the Kern group (D. Kern, personal communication) suggest that the structural transition may be the rate-determining step for catalysis in adenylate kinase. This result encourages us to make specific quantitative predictions for this system, which we now describe. Fig. 4 shows how the transition-state barrier depends on the reaction driving force. Although a clear curvature characteristic of intersecting parabolas is observed for a fully elastic model, inclusion of cracking makes the dependence linear for a very large range of driving forces. Although the calculations have been performed for adenylate kinase, we note such a large linear range is observed in many systems. In particular, linearity with load has been observed by Bustamante et al. (34) in motor proteins where such linearity has been argued to be essential to the motor's efficiency. More importantly the cracking model allows residues to become unfolded in the transition state that are folded in both stable states. Thus an unexpected dependence on folding stability change should be seen: simultaneously lowering the stability of both conformations without changing their relative stability will speed the reaction. Using site-directed mutagenesis, one can determine the probability of individual residues to remain structured or to crack during the transition as in folding Φ value analysis (38). Evidence for cracking can also be gleaned by using the global effects of denaturants, such as urea, on conformational change kinetics. Denaturant will enhance the local unfolding and therefore should lower the barrier height. Such anomalous activation of enzymes by adding low concentrations of denaturant has been observed in this and other systems (39, 40). Under the assumption that the conformational change is the rate-determining step, we have calculated the reaction rate dependence on denaturant. This relationship is not monotonic. At higher concentration, as anyone would expect, denaturation reduces the activity. But at low levels of denaturant the activity does indeed increase in accord with our model. The activity increase observed experimentally by adding denaturant (39), corresponds to a ¶DG*/¶[urea] in the range of 0.2 and 0.7 kcal/mol per M. Here M is the molar concentration of urea. We have computed the analogous slope for our theoretical model of the conformational change by itself, ¶DG*/¶[urea]. Details are presented in Fig. 5 for a reasonable driving force DGeq = –3 kcal/mol and in the range between 0.15 and 0.1 kcal/mol, the model yields results consistent with the experimental speedup, ¶DG*/¶[urea] of between 0.22 and 0.65 kcal/mol per M. More careful direct measurement of the conformational change kinetics by itself, not as a composite with other chemical steps, as a function of denaturant, however, is needed to test our model more completely. Although the conformational change step has been kinetically isolated in the related systems of carboxyl-terminal Src kinase and cAMP-dependent protein kinase (41, 42), the denaturant dependence has yet to be studied. Although the present model can be refined, this initial agreement between theory and experiment shows the potential of the present approach for quantitatively understanding the underlying mechanisms governing allosteric motions.