Combining multi-mutant and modular thermodynamic cycles to measure energetic coupling networks in enzyme catalysis

One of our oldest and most inclusive insights about proteins is that protein folding coordinates the behavior of the amino acids linked together in the polypeptide chain. Proposals implicating the resulting cooperative protein motions, or “dynamics,” in enzyme catalysis are commonplace.^1–11 A central notion underlying such proposals is that low frequency motions—loop closing, domain motion, etc.—transmit the kinetic energy along directions aligned with chemical changes, thereby contributing to the overall rate acceleration.

Such proposals have been challenged, however, for three main reasons.¹² (i) Frequencies of domain motions are many orders of magnitude slower than those of the chemical reactions. (ii) Forces that can be imposed with such motions are substantially weaker than those necessary to stabilize the transition states. (iii) In order to definitively implicate motions in catalysis, it is necessary to map the accurate free energy landscapes of chemical reactions and how they integrate with the protein-dynamical changes, and cases where these have been done do not reveal a significant catalytic contribution.

Nonetheless, many enzymes, especially NTP (nucleotide triphosphate)-dependent motors, synthetases, polymerases, and signaling GTPases, exhibit multi-state, vectorial behavior^13–19 in which biological function depends critically on coupling catalysis to conformational changes. Purposes served by such coupling include many of the most interesting aspects of biology. For this reason, the creation of, and transitions between, these multiple states are biologically relevant. It is therefore important to determine the mechanisms by which conformational transitions can be coupled to active-site chemistry.^13,14,20

Our work on Bacillus stearothermophilus tryptophanyl-tRNA synthetase (TrpRS) occupies a specialized niche amidst these questions. We try to answer such questions as:

• What remote parts of a protein exert a substantive influence on catalysis?

• What techniques are necessary to verify these effects?

• What is the magnitude of these influences?

• How cooperative are they?

• What is their functional significance?

TrpRS furnishes an attractive laboratory model for exploring long-range coupling comprehensively.

Our interest in these questions was initially stimulated by communications with Arieh Warshel, who has argued forcefully that the essence of catalysis is electrostatic stabilization of the transition state^21–23 and cannot arise from dynamical effects. Reviewing an earlier paper of ours,²⁴ he challenged our contention that long-range forces might be involved in catalysis, and argued that the TrpRS rate acceleration was due primarily to electrostatic stabilization by the single Mg²⁺ ion.

Testing that idea experimentally, we found that only about a third of the overall transition-state stabilization could be attributed to the Mg²⁺ ion.²⁵ This posed two questions:

• What accounts for the metal-independent two-thirds of the transition-state stabilization?

• How does metal-protein coupling work if Mg²⁺ does not interact with the protein, has very little effect on the rate in water, but accelerates the rate 10⁵-fold in the presence of TrpRS?

The first question remains to be answered, although we have taken some steps^26,27 in that direction. We tentatively answer the second question here. We conclude that Warshel is correct that domain movements themselves cannot directly stabilize transition states. However, what he calls the “active site pre-organization”²⁸ appears to occur only during the domain movement, which has the effect of synchronizing catalysis with domain movement. The resulting conditional dependence of catalysis is what couples catalysis to domain movements,^20,29,30 enabling the vectorial behavior. Readers may wish to refer to Figs. 10 and 11, which are discussed in detail later on, and provide a useful visualization of the proposed reaction path.

TrpRS uses Adenosine 5′ triphosphate (ATP) to activate the carboxyl group of the amino acid tryptophan to form Trp-5′ adenylate plus pyrophosphate (PPi). This reaction is the sine qua non of protein synthesis, and therefore of fundamental biological significance. Domain movements along the TrpRS structural reaction profile were elucidated by solving the crystal structures of TrpRS without ligands³¹ and complexed to substrates—tryptophan, ATP;³² non-reactive substrate analog combinations—tryptophanamide plus ATP;³² tryptophan plus Adenosine-5′ monophosphate (AMP) plus PPi;³³ a transition-state analog—adenosine-5′tetraphosphate;³⁴ and products—trp-5′AMP,^35,36 trp-5′sulfoamyl-AMP, trp-methenyl-AMP;³⁷ representing different states along the reaction path. These crystal structures group into three distinct, narrowly defined conformations that correspond to nodes connected by the three canonical stages of enzyme catalysis: induced-fit active-site assembly, catalysis, and product release.³⁸ In this respect, the three resulting “states”—open, pre-transition state (PreTS), and Products—correspond functionally and structurally to the three conformations observed in the successive stages of the chemical reactions of the ATP by the catalytic F1-ATPase β-subunit.³⁹ Specifically, an open conformation is associated with the unliganded state, a closed, twisted conformation is associated with the ATP bound state, and a closed, untwisted conformation is associated with the aminoacyl-AMP-bound state. We previously commented on the possible relationship between the three-state behavior and free energy transduction.³⁸ 

The three TrpRS conformations are produced by the relative rigid-body movement of the anticodon-binding domain (ABD) relative to the catalytic domain.³³ The induced-fit and catalytic transitions are rate-limited by high-energy conformational transition states^24,40 along the conformational free energy surface. We identified the structures responsible for these high-energy conformations using searches for the most probable paths that connect the respective ground-state conformations.^30,41 An important barrier to domain movement associated with induced-fit is a change in aromatic side chain packing within the D1 switch, a motif located in the N-terminal crossover connection of the Rossmann fold.^41,42

Seeking to understand the evolutionary sources of TrpRS catalysis, we have experimentally deconstructed the monomer (Fig. 1) into three nested, catalytically active components—Protozyme (46 residues), Urzyme (130 residues), and Catalytic Domain (204 residues)—which, together with the anticodon-binding domain (ABD) form the full-length TrpRS monomer.²⁶ The Protozyme and Urzyme retain a considerable catalytic activity and hence likely represent ancestral forms of contemporary aminoacyl-tRNA synthetases. Full-length synthetases combine the Urzyme with two additional modules added later in evolution: connecting peptide 1 (CP1), which completes the catalytic domain and provides the context for editing domains in the larger synthetases specific for aliphatic side chains; and the anticodon-binding domain, which enables recognition of the tRNA anticodon (Fig. 1(a)). The CP1 module does not actually move as a rigid body with the Urzyme, but moves relative to it. We, and others, have used computational methods to identify the correlated motions of the ABD domain and CP1 in several of the Class I synthetases.^40,43

CP1 and the ABD together form a “cone” that is fitted around the Urzyme (Fig. 1(b)). The active-site opening is accessible only through this cone, which therefore appears to physically constrain the relative positions of the Urzyme's modular ATP and amino acid binding sites. The active site is thus sensitive to the configuration of the CP1 and ABD modules in full-length TrpRS. We have measured the impact of these interactions on catalysis^20,25,29 and specificity.^30,44

A key concept underlying this work is that of “energetic coupling.” Coupling implies the existence of sources of free energy that enforce joint behaviors; such joint behaviors enable new, higher levels of functionality. It is generally true that coupling energies across a very broad spectrum of circumstances can be assessed from the non-additivity of expected contributions. A useful example is that the intrinsic proton angular momentum, or spin, cannot be explained using only the intrinsic properties of its three constituent quarks, but instead requires an energetic coupling between them and the gluons that hold them together.⁴⁵ In nuclear physics, the data come from an inelastic electron and proton scattering experiments.⁴⁶ In structural biology, non-additivities are evaluated quantitatively for combinatorial perturbations using a device that in a biophysical context is called a thermodynamic cycle (Fig. 2).

A thermodynamic cycle is a 2^N factorial experiment that tests all possible combinations of specific perturbations for their impact on kinetic or thermodynamic behavior. Perturbations that have non-additive effects in contexts of other perturbations imply coupled interactions. Use of multi-mutant cycles was introduced in biochemistry by Jencks^47,48 and exploited by Fersht soon after the development of directed mutagenesis enhanced their feasibility.^49–51 High-order thermodynamic cycles have demonstrated three-dimensional coupling interactions capable of changing the direction of conformational equilibria by up to 10⁵-fold.⁵² 

The ³²PPi exchange reaction used to assay the aminoacyl-tRNA synthetases is done under equilibrium conditions, so both K_M and k_cat are related to the thermodynamic values, ΔGK_M and ΔGk_cat. Free energies are additive, so linear regression methods can solve the simultaneous equations necessary to estimate and assess the statistical significance of coefficients for main and higher-order interaction effects.²⁰ The latter represent the accessible coupling energies for a given experimental design. Experimental perturbations according to a full-factorial design can thus be used to estimate both the strength and significance of energetic coupling between parts of a protein during enzyme-catalyzed reactions.

In the course of this work, we were led to investigate a series of apparently unrelated combinatorial perturbations. To our surprise, three qualitatively quite different thermodynamic cycles give the same quantitative energetic coupling during TrpRS catalysis.

Prompted by the suggestion that the TrpRS catalysis should be essentially eliminated in the absence of Mg²⁺ ion, we prepared extensively depleted stocks of enzyme, assay buffer, and reagents.²⁵ The resulting assays exhibited reduced, but still substantial rate accelerations over the uncatalyzed rate, which we initially attributed to residual remaining Mg²⁺ ion (Fig. 3(b)). Titration of the assays with increasing amounts of ethylene diamine tetraacetate (EDTA) caused an additional rate reduction, but that reached a plateau beyond which there was, surprisingly, no further decrease.²⁵ 

Iterated solution of the simultaneous binding equilibria using known dissociation constants for the Mg²⁺·ATP and Mg²⁺·EDTA equilibria allowed us to fit the activity data at two TrpRS concentrations to a nonlinear model with two parameters: the proportional decrease in rate in the absence, relative to that in the presence of Mg²⁺ ion (1.1 × 10⁻⁵) and the dissociation constant of Mg²⁺ from the transition-state complex (120 nM)²⁵ (Fig. 3(c)). The former result implies that the metal accounts for only about 33% of the transition-state stabilization free energy, and is unexpected for several reasons: (i) It implies that the TrpRS active site alone is capable of ∼−12 kcal/mole of transition-state stabilization free energy, even in the absence of the metal; (ii) Mg²⁺ ion accelerates ATP-dependent reactions in water by only ∼5-fold.⁵³ The two-factor thermodynamic cycle mapping the effects of TrpRS and Mg²⁺ ion thus exhibits a non-additivity of −6.4 kcal/mole, implying that the metal and protein are coupled by that energy in the transition state for ATP-dependent tryptophan activation.²⁵ 

Mg²⁺ has been observed in crystal structures only of the PreTS state. In those structures (1MAU, 1M83), it interacts directly only with three phosphate oxygen atoms and two water molecules, and not with any TrpRS functional groups. The absence of protein-metal contacts necessarily implied that the energy coupling them together arises from as yet unidentified sources that, in turn, interact with the Mg²⁺ ion indirectly, via the ATP phosphate oxygens. We first sought to identify these unknown coupling interactions within the active site, via combinatorial mutagenesis of the active-site lysine residues that are physically coupled to the metal because they compete with the metal for electrostatic interactions with phosphate oxygen atoms (Fig. 4).²⁹ Those experiments showed that the Mg²⁺·protein interactions within the active-site actually retard the reaction,²⁹ forcing us to look further afield for the sources of catalytic assist by Mg²⁺.

To locate these distant interactions, we used a combination of two bioinformatic filters—one geometric, the other energetic—to identify side chains likely to influence conformational equilibria (Fig. 5).²⁴ First, we quantitatively identified the local packing interactions that were invariant throughout the conformational cycle using Delaunay tessellation.^54–56 Eliminating from consideration all invariant tetrahedra of nearest neighbors—those within rigid bodies—greatly reduced the number of such simplices that are “dynamic” and change during catalysis. Then, we used Rosetta^57–59 to identify point mutants that might stabilize the high-energy PreTS state, relative to the OPEN and PRODUCTS ground states. The intersection of Venn diagrams identified by both Delaunay tessellation and by the multi-state design algorithm was surprisingly small,²⁴ consisting of one large, centrally located dynamic packing motif and three smaller peripheral motifs. We called these dynamic motifs “Switches” D1–D4, from “Delaunay.”

These bioinformatic filters are discussed further in the Methods section. Their combined effects reduced the number of candidate residues to a small enough number that experimental perturbation was feasible. The majority of the residues that survived this two-stage filter were all part of the D1 switch, a packing motif at the C-terminus of the first alpha helix of the Rossmann fold. We subsequently identified this packing motif as one of the most widely distributed motifs in the proteome.⁴² The widespread phylogenetic distribution and key location of this motif in the first crossover connection of the Rossmann fold reinforced our surmise that the residues engaged in dynamic switching in this region might be responsible for the long-range coupling to the active-site Mg²⁺ ion. Although the D1 Switch motif consists of seven residues, we decided to mutate only four, because we sought a full factorial design, and 15 variants was an accessible number to construct and assay. Moreover, because the Rosetta multi-state design algorithm had given us the specific mutations for each site—I4V, F26L, Y33F, F37I—and because these mutations all consistently reduced differences between the Rosetta scores of the PreTS and Product structures, we used these specific residues, rather than alanine mutations.

Alan Fersht used the term “multi-mutant cycle analysis” to describe the factorial directed point mutagenesis designs.⁵¹ Such experiments measure the coupling energy between individual residues and/or active site ligands. Extensive combinatorial multi-mutant cycle analysis of active-site functional groups subsequently showed that none of these coupling energies between protein and metal resides in the active-site residues,²⁹ creating a strong argument for functionally relevant effects arising from parts of the protein distant from the active-site.

In order to detect coupling to the active-site metal, we first verified that the catalytic assist by Mn²⁺ ion was a suitable perturbation by showing that the metal substitution reduced activity to ∼15% of that with Mg²⁺ and that the concentration dependencies of catalysis by the two metals were very similar over a 10⁻⁵-fold range of metal concentrations, both exhibiting a maximal effect at approximately the same concentrations.²⁹ All possible combinations of the four D1 mutations were constructed, purified, and their active fractions were determined by active-site titration.⁶⁰ 

The resulting experimental design involved 32 separate combinations of point mutant and metal (Mg²⁺ vs. Mn²⁺) and was thus perhaps the most ambitious of its kind yet attempted.^19,61 All variants were assayed by ATP-dependent Michaelis-Menten steady-state kinetic analysis. Approximately, three-fold replication of the experimental data afforded a fit in which essentially all the main and higher-order interaction free energies of transition-state stabilization could be estimated with high statistical significance.²⁰ Overall, the regression model for ΔGk_cat has coefficients with only modest magnitudes, of both signs, for the main effects and intermediate-order interactions. The 5-way interaction between the four permuted sites and the metal ion, however, contributes ∼−5 kcal/mole. The 5-dimensional multi-mutant cycle showed that ∼85% of the catalytic effect of Mg²⁺ ion can be attributed to the 5-way interaction between the metal and four residues that mediate the shear during domain movement (Fig. 6).

The regression model is a variant of principle component analysis, and its coefficients furnish a detailed histogram of contributions to transition-state stabilization (Fig. 6(b)). The dominant five-way interaction term has a definitive interpretation: at the moment of transition state stabilization, motions of all four D1 switch side chains and the active-site Mg²⁺ ion are tightly coupled, such that their simultaneous movement can reduce the free energy of the transition state.²⁰ This interpretation implies that the change in side-chain packing that occurs during domain motion occurs essentially simultaneously with some, as yet undocumented, change in the configuration of the Mg²⁺ ion in Mg²⁺·ATP and that this configurational change allows the metal ion to interact more tightly with the transition state. Thus, this multi-mutant cycle is in substantial quantitative agreement with the conclusions drawn from the two-way thermodynamic cycle involving the intact TrpRS enzyme and the Mg²⁺ ion described in the section on Identification of the D1 switch.

If the interpretation of high-order coupling described in the section on Combinatorial mutagenesis is correct, then it should also be the case that the two domains that rotate relative to the active site should also be coupled together in the transition state by similar free energies.⁴⁴ We recently confirmed the prediction quantitatively, by constructing and carrying out a modular thermodynamic cycle.⁴⁴ Modular cycles are enabled for TrpRS by the characterization of Urzymes^62–64 (and potentially of protozymes²⁶). As the CP1 and ABD domains are definitively deleted, the TrpRS Urzyme itself plays the role of a molecular knockout. Its remaining catalytic activity thus measures the impact of simultaneously losing both modules. We therefore constructed each modular perturbation necessary to complete the cycle by adding back either CP1 or the ABD to measure the effects of each modular deletion separately (refer to Fig. 1).

Combinatorial domain perturbation (Fig. 7) reveals that neither CP1 nor the ABD contribute materially to transition state stabilization by the Urzyme alone, yet the presence of both modules together in full length TrpRS boosts the catalytic effect by 10⁵-fold! Non-additivity in the resulting modular thermodynamic cycle measured the free energy coupling between the two mobile domains to be −5 ∼kcal/mole in aminoacylation of tRNA^Trp. This coupling energy closely matched that obtained in the five-way multi-mutant cycle.^20,44

Multi-mutant and modular models validate one another in specificity as well as in catalysis. The 4-way interaction between I4, F26, Y33, and F37 also makes a similar contribution to the specific recognition of Tryptophan vs. Tyrosine (k_cat/K_M)_Trp/(k_cat/K_M)_Tyr in tryptophan activation.³⁰ Simultaneous motions of CP1 and the ABD are necessary in order to achieve a fully functional TrpRS catalysis and specific tryptophan recognition: in the absence of either domain, the overall rate acceleration falls by ∼10⁻⁵-fold, almost exactly to that of the Urzyme alone, and the free energy of discriminating between tryptophan and tyrosine falls from ∼6 kcal/mole to 1 kcal/mole (Fig. 8).^30,44

The coupling evident in Fig. 6(b) is notable for several reasons. First, it is contrary to that expected for higher order factorial experiments⁶⁵ in which intrinsic effects normally dominate, and higher order interactions diminish in magnitude. That expected behavior has been verified in a comparable five-dimensional full factorial analysis of the catalytic roles of residues in the active site of alkaline phosphatase⁶¹ (Fig. 9) whose catalytic side chains link only into small clusters that are themselves dominated by the intrinsic effects of individual side chains. Here (Fig. 6(b)), the dominant effect, by a substantial margin, is the highest order interaction term. Second, the highly cooperative nature of protein tertiary structures suggests the possibility that any sufficiently high-dimensional combinatorial mutagenesis would result in high-order coupling. The pattern in alkaline phosphatase serves as an important “negative control” by showing that high-order interactions do not necessarily result from any high-dimensional mutagenesis experiment. Finally, as discussed by Sunden et al., an evolutionary argument holds that high-order coupling is indeed unexpected, and hence represents the product of a complex series of selected properties.⁶¹ We also have advanced similar arguments regarding the conundrum evident in Fig. 7(b), which suggests that the full-length enzyme could not have evolved by a simple process of accumulating first one, then other of the two domains missing in the Urzyme.⁴⁴ 

The stunning quantitative agreement between the coupling energies estimated from multi-mutant and modular combinatorial experiments with those anticipated from the characterization of metal-free catalysis²⁵ reinforces our earlier suggestions^32,34,38,66 that domain motions are necessary to form the TrpRS·transition state complex, and hence that differences observed between TrpRS crystal structures are relevant to the catalytic cycle. Thus, the different TrpRS crystal structures probably approximate those states connected by motions necessary for transition-state stabilization. Because the coupling energies contribute to ΔGk_cat, and hence to transition-state stabilization, they represent the transient interactions that occur only in the transition state. Hence, movement of parts of TrpRS—the D1 switch residues, the CP1 and ABD domains, and the catalytic Mg²⁺ ion—relative to one another is required to generate them, as Fersht noted when interpreting his early directed mutagenesis of the TyrRS active site^50,67–69 which had to be done without reference to multiple crystal structures.

In view of the recent critique of the functional relevance of domain motions¹² it is worth discussing these results cautiously. The concluding sections address separate aspects of this question.

In order for a protein to exhibit the multistate behavior, its conformation must have multiple distinct potential energy wells, separated by passes of high conformational free energy. We have used a minimum action path algorithm (PATH)⁷⁰ to characterize how the three TrpRS conformational states connect to one another via such conformational transition states (Fig. 10(a) (Multimedia view)).^30,33,41 Replica exchange discrete molecular dynamics⁷¹ showed that the conformational transition state for TrpRS catalytic transition is narrowly defined structurally, and that it coincides with a saddle point in the respective conformational free energy landscape⁴¹ (Fig. 10(b)). Importantly, that landscape refutes the contention⁷² that the saddle point in the free energy surface between multiple states of a protein is relatively flat in the direction normal to the path, and hence helps assure that the most probable path does result from minimizing the Onsager-Machlup action functional expressing the energy dissipation for a stochastic, overdamped system.⁷³ 

In the same study, we also compared the conformational transition states of the TrpRS Induced-Fit transition to those associated with the Pre-power stroke to rigor transition of the myosin VI converter domain⁷⁴ and calcium release by calmodulin. These three transition states are remarkably similar: each entails re-packing of multiple aromatic side chains, which is much slower than other types of repacking. These results, together with our observation that the volume of the tryptophan binding site assumes a minimum just prior to the conformational transition state for the catalytic transition, and hence is likely the moment at which a decision is made regarding whether or not the correct amino acid is bound³⁰ all reinforce the conformational transition state as a real, functionally relevant conformation.

Although TrpRS dynamic behavior appears to be closely linked to catalysis by long-range energetic coupling influences on k_cat, the domain movement itself probably does not actually lower the energy of the transition state. As noted,¹² the time frame of the motions is too slow, and the molecular forces generated are too weak to couple directly to transition-state stabilization. Several apparently unrelated observations suggest a novel, perhaps counterintuitive interpretation of the relationship between domain motion and catalysis.

Rather than contributing directly to transition state stabilization, TrpRS domain motion may contribute instead to dynamic active-site preorganization²⁸ (Fig. 11). By this hypothesis, motions allow the active site to non-randomly populate configurations complementary to the chemical transition states. The active site is properly configured only transiently, because it coincides with the maximum rate of conformational change and hence exists only for a brief portion of the domain motion. Because preorganization occurs transiently and with a low frequency, the overall chemical reaction may effectively proceed more slowly than it would if the active-site remained pre-organized throughout the substrate residence time.

If this hypothesis is correct, then it would account for three observations that previously seemed unrelated:

• Active-site titration experiments show that the chemical step occurs in about the same time frame as the domain motions, i.e., milliseconds (Fig. 12(a)). Thus, the chemical reaction itself may actually be controlled by, i.e., retarded, by reduced time during which the active-site retains the catalytically active configuration.

• The free energy of activation for the chemical step measured in single-turnover active site titration experiments, ΔGk_chem, is closely correlated to the D1 Switch mutations and their higher-order interactions in almost identical fashion to that observed for the ΔGk_cat values (Figs. 12(b) and 12(c)). Turnover itself, estimated from the steady-state rate from such an experiment is, however, uncorrelated with the pattern of mutations. This result is unexpected in light of the evident correlation between mutant sites and ΔGk_cat values from steady-state experiments. It can be understood, however, by recognizing that differences in a small fraction of the total reaction time in the latter context are amplified many fold by turnover. The significance of this correlation, evidenced by the very small t-test P-values (P < 0.0001), is that structural perturbations influence primarily the chemical step of the reaction, supporting the conclusion that acceleration of the chemical step itself is tightly coupled to the domain motion.

• Studies of the mechanism by which indolmycin inhibits bacterial TrpRSs^75,76 show that indolmycin affinity for TrpRS is enhanced by the presence of Mg²⁺·ATP, but not by ATP alone. We have argued elsewhere^75,76 that this unusual effect can be attributed to a small shift in the crystallographically observed metal position, which moves it from a high-energy (and catalytically competent) position to a low-energy (and catalytically inactive) conformation.

Enzyme mechanisms, especially those that couple NTP hydrolysis to work and/or information, use sophisticated dynamic networks to transduce the active-site chemistry into domain motions that change the macromolecular binding affinities. The dual energetic coupling of catalysis to domain motion and D1 switch repacking with Mg²⁺ ion movement renders catalytic utilization of ATP conditional, or contingent upon the domain motion. TrpRS domain motions apparently constitute a timing mechanism.

The variety of different TrpRS X-ray structures has identified each conformation with a distinctive spectrum of ligand affinities,^32–34 including those ligand configurations associated with the transition state. Decelerating the chemical step from what it might be in a more static active site, synchronizing chemical and mechanical events and thereby optimizing the efficiency of free energy transfer. Controlling the pre-organization of the catalytic metal ion can thus be seen as part of an orchestrated sequence that couples these differential binding changes associated with ATP binding, hydrolysis, and product release, to the domain configuration, ensuring that ATP hydrolysis occurs if and only if the conformation changes.²⁰ 

Our recent demonstration that the equilibrium for this catalytic domain movement is driven only by the release of pyrophosphate⁴¹ appears to be an additional mechanism ensuring that catalysis is coupled to (subsequent) domain movement, presumably regulated by tRNA binding. Using PPi release to change the overall conformational equilibrium resembles the behavior observed for myosin, for which the power stroke is initiated by orthophosphate release, whereas the release of adenosine-diphosphate happens only as the power stroke proceeds toward the rigor state.^77–79 As both orthophosphate and pyrophosphate are much more strongly solvated by water than is ATP,⁸⁰ an early release of the phosphoryl group may be more widely used to orchestrate the vectorial domain motion in transducing enzymes.

Serial conditionality furnishes qualitatively new insights into the mechanism of chemical free energy transduction and vectorial processes. Perhaps the clearest discussion of these underlying issues is that by Jencks,^13,14 who outlined how sequential changes in ligand-binding affinity lead to profound changes in the overall free energy of ATP hydrolysis, and recognized the necessity for understanding what he called the “coupling rules.” As the structural landscape of transducing enzymes was as yet unknown, Jencks was mute about what mechanisms actually linked conformational changes to catalysis, and this issue remains unresolved.⁸¹ Identifying^24,40 and mutating²⁹ key residues associated with the conformational transition state(s), together with modular thermodynamic cycles⁴⁴ to measure energetic coupling between domains, now enhance the understanding of how idiosyncratic structural and mechanistic details implement the coupling rules. These methods should also be useful in studying a large number of other transducing NTPases.

The accompanying paper⁸² describes an improved analysis of TrpRS conformational transition states whose computationally derived thermodynamic and kinetic parameters correlate strongly with structures and kinetic data for all sixteen D1 switch variants, unifying and further validating the results summarized here.

A full-length enzyme has almost innumerable separate interactions. Consider only 2^N experiments obtained by constructing all possible combinations of mutations for one type of substitution per residue in TrpRS (328 residues). This amounts to 2³²⁸= e¹⁹⁰⁰ different variants. Clearly, the number of variants must be massively reduced in order even to begin to evaluate long-range energetic coupling in an enzyme. Fortunately, several aspects of domain motion simplify the problem if sufficient structural information is available for the conformational variations along the reaction path.

Many conformational changes consist of rigid-body motions. As these rigid bodies preserve their internal structure, amino acid residues within them can be ignored to a first approximation in designing the combinatorial cycles. Delaunay tessellation discriminates the invariant tertiary interactions from those that change during the structural reaction profile. The remaining, dynamic, interactions constitute a far smaller reservoir of candidates for permutation studies. Rigid body changes therefore entail either hinge-bending or shear, both of which impact small numbers of residues, reducing the number to be considered for combinatorial point mutagenesis.

Algorithms developed for multistate protein design^57–59 furnish a second, complementary source of useful, residue-specific estimates of free energies associated with the domain motion when crystal structures are available for multiple states. Multi-state design identifies the side chain mutations that alter the relative conformational energies. Dynamic residues identified as potential sites of mutation to alter conformational equilibria form an overlapping, but distinct set of residues.

The Venn diagram of sidechains identified by the Delaunay SNAPP (Simplicial Network Analysis of Protein Packing)^54–56 and Rosetta energy function tests (Fig. 5) reduces the number of mutants to a manageable number.

Finally, the Rosetta energy function identifies the specific point mutants that have the greatest impact. This contribution cannot be ignored, because it greatly reduces the number of possible mutations at each site to a single amino acid. The process described here thus identifies specific mutations to a sufficiently small number of sites, to compose a sensible set of combinatorial experiments.

Regression analysis of experimental results recorded in a factorial design is implemented in most statistics programs. We use JMP statistical discovery software (JMP),⁸³ which combines a wide range of state-of-the-art statistical and presentation modules from the SAS system with a very user-friendly interface. Linear models require the use of free energies, which are additive. Thus, all rates and equilibrium constants must be converted first into Gibbs energies. Then, the Gibbs energies contributing either to kcat, K_M, or k_cat/K_M can be expressed as the linear combination (1)⁸⁴ 

where i−l are either 1 or −1 depending on whether the residue at position i is wild-type or mutant, and M is 1 or −1 depending on whether the catalytic metal is Mg²⁺ or Mn²⁺.

A decisive advantage of this approach to the analysis is that coefficients for main effects and interaction terms furnish intrinsic energetic contributions and coupling free energies from the cross terms in units of kcal/mole. A detailed example, clarifying various questions about the method as an alternative to manual calculation of all quantities was described previously⁵¹ is discussed in the supplementary material for Ref. 84.

We are grateful to B. Kuhlman, J. Hermans, J. Das, R. Wolfenden, and N. Dokholyan for advice and assistance. This work was supported by NIGMS (78227 and 40906).