Resolving the complex role of enzyme conformational dynamics in catalytic function

Abstract

Despite growing evidence suggesting the importance of enzyme conformational dynamics (ECD) in catalysis, a consensus on how precisely ECD influences the chemical step and reaction rates is yet to be reached. Here, we characterize ECD in Cyclophilin A, a well-studied peptidyl-prolyl cis-trans isomerase, using normal and accelerated, atomistic molecular dynamics simulations. Kinetics and free energy landscape of the isomerization reaction in solution and enzyme are explored in unconstrained simulations by allowing significantly lower torsional barriers, but in no way compromising the atomistic description of the system or the explicit solvent. We reveal that the reaction dynamics is intricately coupled to enzymatic motions that span multiple timescales and the enzyme modes are selected based on the energy barrier of the chemical step. We show that Kramers’ rate theory can be used to present a clear rationale of how ECD affects the reaction dynamics and catalytic rates. The effects of ECD can be incorporated into the effective diffusion coefficient, which we estimate to be about ten times slower in enzyme than in solution. ECD thereby alters the preexponential factor, effectively impeding the rate enhancement. From our analyses, the trend observed for lower torsional barriers can be extrapolated to actual isomerization barriers, allowing successful prediction of the speedup in rates in the presence of CypA, which is in notable agreement with experimental estimates. Our results further reaffirm transition state stabilization as the main effect in enhancing chemical rates and provide a unified view of ECD’s role in catalysis from an atomistic perspective.

Enzymes accelerate reaction rates by several orders of magnitude, allowing them to occur at timescales relevant for cellular functions (1). One of the long-standing issues in biochemistry is how enzymes achieve this remarkable speedup. It is commonly accepted that the most dominating effect arises from significant reduction in the free energy barrier compared to the corresponding noncatalyzed reaction in solution. It is also well established that this predominant effect is mainly electrostatic in nature (2, 3), which is more favorable for the transition state than the reactant or the product (1). However, to what degree and how other factors such as desolvation, steric strain, and enzyme dynamics contribute to catalysis remains disputable. Of particular interest is the role of enzyme dynamics in catalysis that has stirred considerable debate (4–11) partly because it has not been clearly defined, leading to a semantic issue. Also, the link between enzyme dynamics and catalysis is difficult to address both experimentally and theoretically. Currently, the implications of enzyme dynamics are from ensemble- and time-averaged experiments, as the temporal behavior of every atom cannot be observed directly. Although standard molecular dynamics (MD) simulations can provide an atomistic picture of enzyme dynamics, they are still not amenable to study catalytic reactions that usually occur in milliseconds. Computational approaches that have investigated the effects of millisecond-timescale enzyme dynamics on the chemical reaction have been possible only with the use of coarse-grained models (4, 10). NMR relaxation dispersion experiments that can probe microsecond-millisecond timescale motions have detected backbone and side-chain motions in and around the active site that occur on the same millisecond-timescale as the chemical step (12). It has been further shown that such slow motions are already present in the free enzyme (13). Furthermore, loss of conformational fluctuations occurring in milliseconds in the active site of mutant enzyme has been observed with concomitant reduction in activity (11). Single molecule studies on enzymes have also revealed that catalytic rates can fluctuate over five orders of magnitude—from milliseconds to hundreds of seconds, similar to the range of timescale for conformational fluctuations (14). These observations are not surprising given that protein dynamics comprise motions that span multiple timescales and occur in either a more localized or collective manner (15, 16). Nevertheless, protein dynamics has been suggested to directly contribute to catalytic function and rate enhancement. The exact nature of this dynamical contribution cannot be understood, unless specific questions regarding whether dynamical motions of enzymes help in lowering the activation barrier (i.e., barrier effects) or aid the substrate to surmount the barrier (i.e., prefactor effects) are addressed.

The energy landscape of proteins is characterized by several energy minima that represent conformational substates separated by barriers of varying heights (17, 18). Simultaneous motions of many degrees of freedom constitute protein dynamics and bring about equilibrium interconversions (16, 19). We sought to understand the role of enzyme conformational dynamics (ECD) in catalytic functions by employing a combination of normal MD (nMD) and accelerated MD (aMD) (20) approach that provides atomistic detail with extended timescale. We chose to study Cyclophilin A (CypA) (Fig. 1A), an extensively studied peptidyl-prolyl cis-trans isomerase, that catalyzes isomerization of the peptide (ω) bond preceding proline residues in proteins. Such system is ideal to study using classical molecular mechanics because no bond breaking or formation is involved in the catalytic process.

Fig. 1.

Structure of CypA and the influence of its dynamics on the kinetics of prolyl isomerization in its substrate. (A) CypA (gray) with its substrate Ace-Ala-Ala-Pro-Phe-Nme in the binding pocket. The binding cavity (VDW surface) is defined by ten residues (nonpolar (white): Phe60, Met61, Ala101, Ala103, Phe113, Leu122 and polar (cyan): Arg55, Gln63, Asn102, and His126) that are within 4 Å of the substrate’s Ala-Pro motif. Decay of probability of survival in the trans well as a function of time obtained from nMD simulations of (B) R_O and (C) R_C with V₂ set to 11.0 (cyan), 9.0 (red), 7.0 (blue), 5.0 (yellow), 4.0 (green), and 0.0 (magenta) kcal/mol. Continuous lines are fits to single exponential in (B) and multiexponential functions in (C). (Inset to B) Plots of survival probability functions on an extended timescale when V₂ = 7.0, 9.0 and 11.0 kcal/mol. Parameters from exponential fits of isomerization kinetics, i.e. the amplitudes and the time constants, τ, are plotted for (D) R_O and (E) R_C with the same color code as in (B) and (C).

Probing the influence of CypA dynamics on the chemical step

The uncatalyzed isomerization reaction (R_O) is an extremely slow process with an activation barrier of approximately 20 kcal/mol and occurs readily in hundreds of seconds in solution (21, 22). Cyclophilins are known to accelerate prolyl isomerization by 10⁵–10⁶ times, reducing the timescale to around milliseconds (23, 24). It is not feasible to simulate even the catalyzed reaction (R_C) with nMD, since it is currently limited to only hundreds of nanoseconds. Therefore, to probe the effects of ECD in catalysis, we used several lowered torsional energy barriers around the -Ala-Pro- ω bond of a well-studied substrate analogue, Ace-Ala-Ala-Pro-Phe-Nme (Fig. 1A). We then took advantage of the linearity of the behavior to extrapolate to the desired barriers. This approach is similar to carrying out experiments in an optimum temperature range and extrapolating to a temperature outside that range. The lower barriers allowed us to track the dwell times in the trans well before going over the barrier with sufficiently good statistics. Using nMD we investigated the kinetics of prolyl isomerization in the free and the enzyme-bound substrate with the same value of the AMBER force field parameter V₂ (see Methods and SI Text, section 1.4), which is the main determinant of torsional barrier. We found that the decay of the survival probability function, S(t), of dwell times for the reference R_O in the free substrate unambiguously exhibited single exponential behavior (Fig. 1B). Progressively slower kinetics resulted as V₂ was systematically increased from 0 to 11 kcal/mol. Interestingly, for R_C, the resulting kinetics exhibited multiexponential decays (Fig. 1C).

In view of the notion that protein dynamics is not independent of its environment (19), we observed that peptide isomerization dynamics was influenced by the environment, whether it be the fluctuations in the solvent or the active site of the environment. From Fig. 1 B and D, it became evident that in terms of timescale, the dynamics of the aqueous environment was relatively uniform with the solvent motions occurring on a single or a very narrow timescale. In contrast, enzymatic motions were dispersed over an extensive timescale and were coupled to substrate dynamics. Consequently, the different enzyme dynamic modes became apparent in the isomerization kinetics, yielding multiexponential behavior (Fig. 1 C and E). As the torsional barrier became progressively greater (with the increase in V₂), the distributions of timescales showed the following trend: the relative amplitudes of the faster phases diminished and those of the slower phases showed a gradual increase. These results suggested that, depending on the barrier, and hence characteristic timescale, of the chemical step, the reaction dynamics got coupled to the slightly faster and slower enzyme motions, resulting in multiexponential decays. Overall, the time constant for R_O in solution was always larger than the average lifetime of the chemical step in the enzyme-bound substrate, R_C (Table I). It was evident from our results that the chemical step was coupled to, and would be affected by, the enzyme motions.

Table 1.

Free energy barriers, diffusion and time constants for the uncatalyzed and catalyzed prolyl isomerization^*

	V2 (kcal/mol)	Ro			Rc
		ΔG# (kcal/mol)	(ns)	Deff(deg2/s)	ΔG#(kcal/mol)	(ns)	Deff(deg2/s)
nMD	0.0	0.81	0.05	18.1 × 1014	-	0.01	1.37 × 1014
nMD	4.0	2.3	0.28	18.1 × 1014	-	0.07	1.37 × 1014
nMD	5.0	3.16	0.54	18.1 × 1014	-	0.16	1.37 × 1014
nMD	7.0	3.93	2.48	18.1 × 1014	0.99	0.50	1.37 × 1014
nMD	9.0	5.57	16.11	18.1 × 1014	2.31	1.35	1.37 × 1014
nMD	11.0	7.42	61.65	18.1 × 1014	3.49	6.24	1.37 × 1014
aMD	28.0	20.0a	1.22 × 1011b		10.74a	2.75 × 105b
aMD I	7.0				2.09	0.26	5.443 × 1014
aMD II	7.0				2.37	0.14	13.362 × 1014
aMD III	7.0				2.7	0.09	31.229 × 1014

^*ΔG^#'s were calculated from potentials of mean force obtained from umbrella sampling except for (a) where they were obtained after reweighting free energy profiles resulting from aMD. (b) Time constants extrapolated from linear fits (red and green lines) in figure 3c. AMD levels I, II and III are the lowest, intermediate and the highest extents of acceleration (same as in Fig. 2D) subjected only on CypA. D_eff = 2πk_BT. exp(ln(k/w_ow_b)_ΔG^#=0).

Accelerating CypA Dynamics and Its Effects on the Chemical Step

Another question of interest is whether the contribution from ECD can significantly enhance the rate of the chemical step as compared to that in solution. In answering this question, we subjected only CypA to increasing levels of aMD with the substrate still stimulated with nMD (SI Text, section 1.3). In order to confirm that aMD indeed resulted in faster ECD in CypA, we characterized the fluctuations in free CypA(C_o) from independent nMD and aMD (SI Text, section 1.2 and 1.3). Accelerated MD brought about an increase in not only conformational plasticity; i.e., greater amplitudes of fluctuations as depicted from the shift of backbone (Fig. 2 A and B) and side chain (Fig. S1) order parameters (S²) to lower values, but also conformational heterogeneity (Fig. S2) at the active site of C_o. Similar to our results, aMD has been shown to successfully increase the rate of conformational sampling, thereby characterizing millisecond-timescale protein/peptide dynamical motions and achieving notable agreement with experimental data (25–29). Our simulations further confirmed recent experimental observations (13, 14) that ECD in CypA takes place over a broad range of timescales even in the substrate-free state (Fig. S1). Accelerating CypA dynamics clearly affected the kinetics of prolyl isomerization in the bound substrate, resulting in faster decay of the survival probability (Fig. 2C). The decays fitted to multiexponential functions with only three phases as opposed to five phases in the nMD of CypA (Fig. 2D). Since the enzyme modes sped up, the relative contribution of the faster phases increased as slower phases became faster (Fig. 2D). The net result was a gradual speed up in the average lifetimes as the extent of acceleration of CypA motions was increased (Table I).

Fig. 2.

Effects of accelerating CypA dynamics on prolyl isomerization in the substrate. Distribution of order parameters (S²) (SI Text, section 1.8) obtained from (A) nMD and (B) aMD of free CypA using the highest level of acceleration. CypA structure is color-coded based on the S² values of each backbone N-H bond vector (see color scale). The active site residues are shown with stick representation. Fluctuational motions with the largest amplitudes are indicated by the smallest S² (red) while those with the smallest amplitudes are depicted with the largest S² (blue). (C) Decay of probability of survival in the trans well as a function of time when V₂ = 7 kcal/mol. (D) Parameters of exponential fits in C; i.e., amplitudes and time constants, τ, are plotted with the same color code as in (C). Shown are the isomerization kinetics in the free substrate (cyan, S) when subjected to nMD and the enzyme-bound substrate when CypA was subjected to nMD (blue, ES) as well as aMD at the lowest (orange, A1), intermediate (dark red, A2), and the highest (dark green, A3) level of acceleration. Continuous black lines are mono- or multiexponential fits.

Using Kramers’ Rate Theory to Explain the Effects of CypA Dynamics

Although the usage of traditional rate theories to explain enzyme kinetics has been a contentious matter (8, 30, 31), our results could be rationalized within the framework of Kramers’ theory (32, 33) in the high friction regime:, where k is the rate of escape from the trans well with curvature ω_o over the free energy barrier ΔG^# with curvature ω_b, k_B is the Boltzmann constant and T is the temperature. D_eff is the effective diffusion coefficient on a one-dimensional free energy profile and assumed to be independent of the reaction coordinate. D_eff incorporates the effects of the environment, as well as those inherent in proteins, for example, frictional, dynamical effects and energetic roughness. Frictional effects arise from solvent viscosity and internal friction that impede protein motions. Dynamical effects originate from enzyme with inhomogeneous diffusivity, arising from ECD occurring on a wide continuum of timescales, or an aqueous medium that offers a more homogeneous environment with essentially single (or very narrowly distributed) diffusion coefficient. The substrate undergoes desolvation while moving into the active site of the enzyme from aqueous solvent, as a result the energetic roughness may reduce, leading to a less hindered substrate (34). The above Kramers’ relation can be rearranged in the log form; i.e., . From the plots of vs.ΔG^#,v₂ (where rate constants, curvatures, and free energy barrier heights correspond to various values of V₂) with a well-defined slope of 1/k_BT, the relative contributions from the preexponential factor and the barrier effects were estimated for R_O and R_C. (Fig. 3). Clearly, there was speedup in the isomerization rate in each case of the enzyme-bound substrate as compared to the corresponding R_O (Fig. 3A). The speedup was the consequence of two opposing effects: Increase in rate as a result of the reduction in free energy barrier, for example, 237 times from a barrier reduction of 3.26 kcal/mol in case of V₂ = 9.0 kcal/mol, which was offset by approximately 2.6 times due to the modification in the curvatures of the free energy profiles and by approximately 13 times due to the differences in the D_eff in solution and in enzyme-bound substrate, bringing the net rate enhancement to only about seven times. It can be seen from Fig. 3A that the y-intercept, from which the effective diffusion coefficient can be estimated, is smaller for R_C than R_O (Table I). Therefore, ECD does not enhance but rather hinders the rate enhancement. When the dynamics of CypA was accelerated, we clearly noticed an increase in the isomerization rates (Table I). Enzymatic CD does not directly modify the properties of the free energy landscape, as that scenario would violate Boltzmann statistics. However, for each level of acceleration subjected on CypA, the solvated CypA-substrate complex should be considered a distinct system associated with its Hamiltonian and characteristic free energy profile. And indeed, the curvatures and barrier heights were modified (Table I) when ECD was accelerated. As the levels of acceleration were raised, both ΔG^# and D_eff showed an increase (Fig. 3B) relative to the case in which ECD was not sped up (i.e., R_C). Noticeably, even with the highest level of acceleration, D_eff was not significantly faster (i.e., by only approximately 1.7 times) than the one in aqueous solution, given the errors associated with the calculation of quantities from logarithmic scale and MD simulations. Further acceleration of CypA dynamics could reach a limiting case where the integrity of the active site might be lost as a result of the unfolding of the enzyme brought about by very fast dynamical motions. Our analysis therefore suggested that altering motions associated with ECD made the enzyme’s active site environment more aqueous-like and directly affected the preexponential factor. At the same time, the favorable interactions between CypA and its substrate were possibly perturbed such that the free energy height of isomerization is increased relative to R_C but still remains lower than that for R_O.

Fig. 3.

Comparison of prolyl isomerization kinetics in the free and the enzyme-bound substrate. Kramers’ plots are shown in the form of ln (k/ω_oω_b) vs. ΔG^#. (A) nMD data points for R_O (open blue circles) and R_C (filled blue circles) when V₂ = 0,4, 5, 7, 9, and 11 kcal/mol. (B) Same plot as in (A). For clarity, only data points from nMD corresponding to V₂ = 7.0 kcal/mol are shown. Also plotted are the data points for R_C from aMD when the lowest (violet), intermediate (magenta), and the highest (red) level of acceleration are applied on CypA. All continuous lines are linear fits with slope = 1/k_BT. (C) Same plot as in (A) with linear fits that are extrapolated to higher free energy barriers. Data points for R_O (open orange circle) and R_C (filled orange circle) are shown when the reoptimized V₂ = 28 kcal/mol was used in aMD and assumed to follow the corresponding linear trends (i.e., red and green lines, respectively). Gray dashed lines above the red and below the green lines (with the same slope = 1/k_BT) represent illustrative kinetic trends for R_C in CypA mutants with faster and slower dynamics, respectively than the wild-type enzyme. Horizontal and vertical dashed lines with arrows depict reduction in free energy barriers and speedup in isomerization rates (along with sharper curvatures), respectively.

In an independent aMD study, setting V₂ to the reoptimized value of 28 kcal/mol (35), we calculated the free energy profiles with the expected actual barriers for R_O and R_C (SI Text, section 1.3, and Fig. S4). We estimated the rate constants (Table 1) corresponding to the actual R_O and R_C from their respective linear fits in Kramers’ plots (Fig. 3C). R_C showed a speed up of approximately 4.5 × 10⁵ times over R_O, which was strikingly similar to experimental estimates (23). The above result therefore validated Kramers’ theory in the analysis of prolyl isomerization kinetics and its catalysis. We would like to note from Fig. 3C. that in the case of CypA, as one goes to the higher barrier regime, the relative and dominant contribution from the reduction in barrier heights to the rate enhancement will continue to increase while the difference in D_eff between R_C and R_C will remain at approximately 13 times.

Recently, ambient temperature X-ray and relaxation NMR studies on CypA have shown that impediment of motions that help in the interconversion of conformational substates in a mutant enzyme is accompanied with the reduction in catalytic rate (36). It was therefore concluded that protein dynamic motions contribute directly to the catalytic power of the enzyme. Such results can now be explained with Kramers’ theory that allows us to understand the nature of the dynamical contribution. The CypA-mutant with slower dynamics implies that the isomerization reaction would take place on a distinct free-energy profile with barrier heights and curvatures that are different from the wild-type CypA and with an effective diffusion coefficient that is perhaps slower than the wild-type enzyme (Fig. 3C). As we show above, it is equally important to investigate the dynamical effects as well as the free-energy barrier effects of the mutant, which, in most cases, is missing from experimental analyses. The recent studies linking enzyme dynamics to catalysis have focused on mutants with either slower or total absence of fluctuations in the active site as compared to the wild-type enzyme (11, 36). It would be interesting to investigate mutants with faster dynamics (Fig. 3C) and observe whether the catalytic rates are enhanced or not and how they compare with wild-type- and nonenzymatic rates.

Characterizing ECD in the Free and Bound CypA

We compared the dynamic motions in the active site of CypA in the absence and the presence of the substrate. Three separate nMD were carried out for the CypA-bound complex (Fig. 1A), when the -Ala-Pro- peptidyl-prolyl ω-bond of the substrate, Ace-Ala-Ala-Pro-Phe-Nme, was allowed to fluctuate in the trans (C_trans), transition (C_TS) and cis (C_cis) states (SI Text, section 1.2). No significant differences were observed in either the distribution of backbone dihedral angles (Fig. S5) or amplitudes of backbone amide bonds in the C_o versus bound-CypA or between C_trans, C_TS, and C_cis (Fig. S7). However, the distribution of side-chain dihedrals of the active site residues (Fig. S6) and the fluctuations of selected side-chain bonds (Fig. S8) showed notable differences in the absence and the presence of the substrate. Prominently, we found that the side-chain rotamers that were preferred in the bound state are already sampled in C_o and there is simply a redistribution of rotameric population in the bound-CypA as compared to C_o. Conforming to earlier studies (37, 38), the catalytically important Arg55, which have been shown to form stabilizing electrostatic interactions with the substrate in the transition state, exhibited side-chain motions that were of smaller amplitudes and much more restricted in C_TS than the end states (Figs. S6 and S8). The trajectories of C_o, C_trans, C_TS, and C_cis were analyzed using principal component analysis (PCA) that decomposes the fluctuations in the atomic coordinates into modes ranked according to their relative contribution to the overall protein motion (SI Text, section 1.6). Projection of the trajectories onto the first three modes that accounted for 90% of the total fluctuations resulted in two-dimensional representation of the multidimensional phase space (Fig. 4 A to C). The conformational space sampled by the active site residues in C_o was not only much larger than that in C_trans, C_TS, and C_cis, but also showed considerable overlap with them, indicating that certain fluctuations observed in the enzyme in the presence of the substrate are already preserved in the free enzyme. C_TS occupied a much more restricted conformational space as compared to C_trans and C_cis, in agreement with the reduction in conformational heterogeneity observed in side chains of active site residues in C_TS; i.e., mainly a single side-chain rotamer is populated for all active site residues in C_TS (Fig. S6). Also, as compared to C_o, C_trans, or C_cis, the side-chain fluctuations in C_TS showed decreased amplitudes; i.e., higher S² values (Fig. S8). As seen in Fig. 4, the smaller ensemble of C_TS consisted of conformations that were a subset of both C_trans and C_cis ensembles, each of which also had its own unique set of conformations. Such connectivity between the conformations of the ensembles is important for multiple numbers of possible catalytic pathways. The above results clearly affirmed the picture of conformational selection (39) wherein the substrate prefers to bind a subset of enzyme conformations and upon catalysis there is a shift toward the population of enzyme conformations bound to the product.

Fig. 4.

Conformational selection and transition state stabilization by CypA. (A, B, C) Projection of the multidimensional Cartesian space onto the first three principal components. Shown are the configurations of the binding site residues (see SI Text, section 1.6) resulting from the snapshots of a 300-ns long trajectory of free CypA (black) and 50-ns long trajectory each of C_trans (blue), C_TS (green), C_cis, (red) complexes. (D) Probability distribution of binding free energies, ΔG_bind, of C_trans, C_TS, and C_cis with average values of -20.8, -34.1, and -24.3 kcal/mol, respectively. The color code is the same as in A–C (Inset) Binding site of CypA (gray VDW) in C_TS where the substrate makes hydrogen bonds (dashed green lines) with active site residues.

Estimating Binding Free Energies of the CypA-substrate Complexes

We estimated the binding free energies from each snapshot of C_trans, C_TS, and C_cis (Fig. 4D and SI Text, section 1.7). In support of earlier studies, Fig. 4D revealed that CypA has maximum affinity for the substrate in the transition state and binds the cis isomer more favorably than the trans isomer (23, 38). The ability of CypA to bind transition states better than the ground states resulted in transition state stabilization by approximately 10-13 kcal/mol greater than the cis and the trans isomers, respectively (Fig. 4D). The estimates of binding free energy were, however, approximate and did not include contributions from the changes in configurational entropy (SI Text, section 1.7). As seen in Fig. S4 and Table 1, we indeed found that the trans to cis free-energy barrier height for the catalyzed isomerization of -Ala-Pro- ω bond was reduced by approximately 9.3 kcal/mol. Thus, the transition state was stabilized considerably greater as compared to that of the cis or the trans isomers and this alone accounted for a speed up of approximately 6 × 10⁶ in isomerization rate if the same preexponential factor was assumed for R_O and R_C. However, we showed D_eff for R_C to be approximately 13 times smaller than that of R_O. Our findings therefore also reinforce the idea of selective binding and transition state stabilization as the major barrier effect in enhancing the rate of catalysis.

Conclusions

Although a growing body of experimental data suggests that enzyme motions play an important role in its catalytic function, the exact nature of this dynamical contribution has never been explained earlier. Here, we show that substrate dynamics involved in the chemical step, is coupled to the dynamics of the surrounding medium which can either be the solvent or the active site of the enzyme. If the environment relaxes much faster than the chemical step such that there is a clear separation of timescales, then the motions are not coupled; e.g., second-timescale cis-trans isomerization in aqueous solution. Provided the timescale of the chemical step falls within that of the fluctuations in the environment, not only the motions on the same timescale but also those that are slightly faster and slower than the chemical step get coupled to substrate dynamics and result in multiexponential kinetic decays. Since our studies were carried out in the regime of lower free-energy barriers in which the timescale of substrate dynamics was shifted to nanoseconds, we could observe the coupling with nanosecond-motions of the enzyme. But for the actual cis-trans isomerization that involves larger barriers and takes place in milliseconds in CypA, increasingly slower modes of the enzyme clustered in the millisecond-timescale will be selected for coupling with substrate dynamics. We further show that one-dimensional Kramers’ theory in the high friction regime, which has proved sufficiently valid in analyzing and explaining the kinetics of various problems of biological interest—which includes protein folding (40–43) and cis-trans isomerization in peptides (25, 29), can be equally useful to interpret the kinetics of enzyme-catalyzed chemical step and understand the role of enzyme dynamics. The enzymatic motions do not and cannot modify the free energy landscape of the enzyme-substrate complex, but rather reduce the effective diffusion coefficient as compared to the reference nonenzymatic reaction in solution. These effects that are incorporated in the preexponential factor of the chemical reaction reduce the speedup possible just from barrier effects and, as we show, modify the diffusion coefficient by more than an order of magnitude. Therefore, as often implied, enzyme dynamics do not accelerate the chemical step. Enzymatic motions are important for reorganization of the active site so that the transition state is better stabilized. However, enzyme dynamics does not have to enhance the catalytic rates to be important to the chemical step. Perturbation of conformational motions caused due to mutations or acceleration affects the catalytic rates resulting from modification of both—the free energy profile and the effective diffusion coefficient. In any case, it is the alteration of free-energy barriers that remains the dominant effect in either impeding or enhancing catalytic rates. Our results provide the missing link between assertions made from theoretical studies and observations from experimental studies, thereby unifying apparently disparate views about the role of enzyme dynamics in catalysis.

Methods

We carried out an extensive characterization of CypA dynamics (including backbone of all residues and side chains of active site residues) in the absence and presence of its substrate using all-atom nMD (SI Text). In order to access long-timescale motions, we further subjected CypA to aMD. For details on aMD methodology, see SI Text. We then characterized the isomerization reaction in solution and in enzyme both kinetically and thermodynamically. The parameter (V₂) in the AMBER force field is the force constant in the dihedral energy functional where n, ω, and γ are the periodicity, dihedral angle, and phase angle, respectively. V₂ (for which n = 2 and γ = 180°) predominantly controls the rotational barrier only around the peptidyl-prolyl bonds with almost isoenergetic cis and trans states. Changing V₂ modifies the total potential and hence the free-energy barriers of isomerization. The default or reoptimized (35) value of V₂ would yield such high barriers that sufficient number of trans-cis transitions will not be observed in a standard MD trajectory of even several hundred nanoseconds. Therefore, to make the simulation of kinetics of cis-trans isomerization feasible and obtain reasonable statistics, we shifted the isomerization timescale from seconds to nanoseconds by reducing V₂ to lower values of 11.0, 9.0, 7.0, 5.0, 4.0, and 0 kcal/mol. It should be noted that V₂ = 0 kcal/mol does not imply zero torsional barrier. With each value of V₂, we performed individual nMD simulations on the free solvated substrate to model the uncatalyzed reaction in solution as well as on the CypA-substrate complex to model the catalyzed isomerization in the active site of the enzyme. In a different set of simulations, the CypA-bound substrate was simulated with nMD with V₂ set to 7.0 kcal/mol while the enzyme was subjected to three increasing levels of acceleration (SI Text, section 1.3). For each case, free energy profiles projected onto the ω dihedral were calculated from umbrella sampling (SI Text, section 1.5). ΔG^#, ω_o, and ω_b were estimated from such one-dimensional free energy profiles (Fig. S3). The probability of survival in the trans well for time t or longer was then calculated from the distribution of dwell times, p(τ), as follows: . We analyzed our results using the DISCRETE (44, 45) software program that provides nonlinear least square solution of multiexponential decays without any a priori guesses for the number of exponentials or initial parameters. The survival probability function, S(t), was fitted to a sum of discrete multiexponential decays; i.e.,. The average rate constant where the average lifetime . A_i and τ_i are amplitudes and time constants, respectively, of phase i in the above function. Using the MM/PBSA (SI Text, section 1.7) method, we further calculated the binding free energies of C_trans, C_TS, and C_cis to estimate the contribution of transition state stabilization to the speedup in the isomerization rate.