Many enzymes mold their structures to enclose substrates in their active sites such that conformational remodeling may be required during each catalytic cycle. In adenylate kinase (AK), this involves a large-amplitude rearrangement of the enzyme's lid domain. Using our method of high-resolution single-molecule FRET, we directly followed AK's domain movements on its catalytic time scale. To quantitatively measure the enzyme's entire conformational distribution, we have applied maximum entropy-based methods to remove photon-counting noise from single-molecule data. This analysis shows unambiguously that AK is capable of dynamically sampling two distinct states, which correlate well with those observed by x-ray crystallography. Unexpectedly, the equilibrium favors the closed, active-site-forming configurations even in the absence of substrates. Our experiments further showed that interaction with substrates, rather than locking the enzyme into a compact state, restricts the spatial extent of conformational fluctuations and shifts the enzyme's conformational equilibrium toward the closed form by increasing the closing rate of the lid. Integrating these microscopic dynamics into macroscopic kinetics allows us to model lid opening-coupled product release as the enzyme's rate-limiting step.conformational equilibrium ͉ rate-limiting step ͉ single-molecule FRET ͉ adenylate kinase P roteins such as enzymes are flexible with a range of motions spanning from picoseconds for localized vibrations to seconds for concerted global conformational rearrangements (1). Despite their randomly fluctuating environment, in which stochastic collisions with solvent molecules drive changes in tertiary structure, enzymes have evolved to catalyze reactions efficiently and specifically. Indeed, conformational transitions have been postulated to play a central role in enzyme functions in a wide variety of ways, including direct contribution to catalysis (2), allosteric regulation (3), and large-scale conformational changes in response to ligand binding (4). Most of our current understanding of structural motions in solution comes from NMR experiments (5) as well as from molecular dynamics simulations (6), approaches that are best suited to study dynamics in the picoto millisecond time scales. Because catalysis in enzymes frequently occurs in the submillisecond to minute time regime, our current understanding of the relationship between enzyme function and conformational dynamics comes from NMR experiments involving relatively localized motions of active site forming loops on the submillisecond time scale (7-10). However, many enzymes contain active sites located in between domains in which large-amplitude, low-frequency domain motions are required to complete their Michaelis-Menten enzyme-substrate complexes. Even simple questions regarding these transitions remain generally unanswered: What is the number and range of conformational states accessible to enzymes during their catalytic cycle? How does the enzyme's conformation respond to interac...

We characterized the conformational change of adenylate kinase (AK) between open and closed forms by conducting five all-atom molecular-dynamics simulations, each of 100 ns duration. Different initial structures and substrate binding configurations were used to probe the pathways of AK conformational change in explicit solvent, and no bias potential was applied. A complete closed-to-open and a partial open-to-closed transition were observed, demonstrating the direct impact of substrate-mediated interactions on shifting protein conformation. The sampled configurations suggest two possible pathways for connecting the open and closed structures of AK, affirming the prediction made based on available x-ray structures and earlier works of coarse-grained modeling. The trajectories of the all-atom molecular-dynamics simulations revealed the complexity of protein dynamics and the coupling between different domains during conformational change. Calculations of solvent density and density fluctuations surrounding AK did not show prominent variation during the transition between closed and open forms. Finally, we characterized the effects of local unfolding of an important hinge near Pro(177) on the closed-to-open transition of AK and identified a novel mechanism by which hinge unfolding modulates protein conformational change. The local unfolding of Pro(177) hinge induces alternative tertiary contacts that stabilize the closed structure and prevent the opening transition.

Two methods are developed to enhance the stability, efficiency, and robustness of reaction path optimization using a chain of replicas. First, distances between replicas are kept equal during path optimization via holonomic constraints. Finding a reaction path is, thus, transformed into a constrained optimization problem. This approach avoids force projections for finding minimum energy paths (MEPs), and fast-converging schemes such as quasi-Newton methods can be readily applied. Second, we define a new objective function - the total Hamiltonian - for reaction path optimization, by combining the kinetic energy potential of each replica with its potential energy function. Minimizing the total Hamiltonian of a chain determines a minimum Hamiltonian path (MHP). If the distances between replicas are kept equal and a consistent force constant is used, then the kinetic energy potentials of all replicas have the same value. The MHP in this case is the most probable isokinetic path. Our results indicate that low-temperature kinetic energy potentials (<5 K) can be used to prevent the development of kinks during path optimization and can significantly reduce the required steps of minimization by 2-3 times without causing noticeable differences between a MHP and MEP. These methods are applied to three test cases, the C7eq-to-Cax isomerization of an alanine dipeptide, the (4)C1-to-(1)C4 transition of an α-d-glucopyranose, and the helix-to-sheet transition of a GNNQQNY heptapeptide. By applying the methods developed in this work, convergence of reaction path optimization can be achieved for these complex transitions, involving full atomic details and a large number of replicas (>100). For the case of helix-to-sheet transition, we identify pathways whose energy barriers are consistent with experimental measurements. Further, we develop a method based on the work energy theorem to quantify the accuracy of reaction paths and to determine whether the atoms used to define a path are enough to provide quantitative estimation of energy barriers.

A mechanical view provides an attractive alternative for predicting the behavior of complex systems since it circumvents the resource-intensive requirements of atomistic models; however, it remains extremely challenging to characterize the mechanical responses of a system at the molecular level. Here, the structural distribution is proposed to be an effective means to extracting the molecular mechanical properties. End-to-end distance distributions for a series of short poly-L-proline peptides with the sequence PnCG3K-biotin (n = 8, 12, 15 and 24) were used to experimentally illustrate this new approach. High-resolution single-molecule Förster-type resonance energy transfer (FRET) experiments were carried out and the conformation-resolving power was characterized and discussed in the context of the conventional constant-time binning procedure for FRET data analysis. It was shown that the commonly adopted theoretical polymer models—including the worm-like chain, the freely jointed chain, and the self-avoiding chain—could not be distinguished by the averaged end-to-end distances, but could be ruled out using the molecular details gained by conformational distribution analysis because similar polymers of different sizes could respond to external forces differently. Specifically, by fitting the molecular conformational distribution to a semi-flexible polymer model, the effective persistence lengths for the series of short poly-L-proline peptides were found to be size-dependent with values of ~190 Å, ~67 Å, ~51 Å, and ~76 Å for n = 8, 12, 15, and 24, respectively. A comprehensive computational modeling was carried out to gain further insights for this surprising discovery. It was found that P8 exists as the extended all-trans isomaer whereas P12 and P15 predominantly contained one proline residue in the cis conformation. P24 exists as a mixture of one-cis (75%) and two-cis (25%) isomers where each isomer contributes to an experimentally resolvable conformational mode. This work demonstrates the resolving power of the distribution-based approach, and the capacity of integrating high-resolution single-molecule FRET experiments with molecular modeling to reveal detailed structural information about the conformation of molecules on the length scales relevant to the study of biological molecules.

of the activation energy barrier, and Dx U, the distance to the transition state. For all protecting osmolytes we measure DG U increases, demonstrating that the I27 protein is stabilized. More striking is the measurement of Dx U . Unfolding the I27 protein in water gives a Dx U ¼ 2.5 Å , a distance similar to the size of a water molecule. Water molecules have been identified as integral components of the unfolding transition state of the I27 protein, forming a solvent bridge between two b-strands. By varying osmolyte molecule size we rigorous test this solvent bridging hypothesis. We find that Dx U correlates with osmolyte size for molecules ranging in size from 2.5 Å to 5.6 Å . However, for larger molecules (> 5.6 Å ) Dx U remains unchanged relative to the value measured in water, suggesting these osmolytes do not participate in solvent bridging in the transition state. These studies uniquely probe the length scales over which solvent molecules can modify the molecular architecture of the unfolding transition of a protein, an area which remains beyond the reach of other experimental techniques.