Large amplitude conformational change in proteins explored with a plastic network model: adenylate kinase

How subunit coupling produces the gamma-subunit rotary motion in F1-ATPase

F(o)F(1)-ATP synthase manufactures the energy "currency," ATP, of living cells. The soluble F(1) portion, called F(1)-ATPase, can act as a rotary motor, with ATP binding, hydrolysis, and product release, inducing a torque on the gamma-subunit. A coarse-grained plastic network model is used to show at a residue level of detail how the conformational changes of the catalytic beta-subunits act on the gamma-subunit through repulsive van der Waals interactions to generate a torque that drives unidirectional rotation, as observed experimentally. The simulations suggest that the calculated 85 degrees substep rotation is driven primarily by ATP binding and that the subsequent 35 degrees substep rotation is produced by product release from one beta-subunit and a concomitant binding pocket expansion of another beta-subunit. The results of the simulation agree with single-molecule experiments [see, for example, Adachi K, et al. (2007) Cell 130:309-321] and support a tri-site rotary mechanism for F(1)-ATPase under physiological condition.

An NMA-guided path planning approach for computing large-amplitude conformational changes in proteins

This paper presents a new method for computing macromolecular motions based on the combination of path planning algorithms, originating from robotics research, and elastic network normal mode analysis. The low-frequency normal modes are regarded as the collective degrees of freedom of the molecule. Geometric path planning algorithms are used to explore these collective degrees of freedom in order to find possible large-amplitude conformational changes. To overcome the limits of the harmonic approximation, which is valid in the vicinity of the minimum energy structure, and to get larger conformational changes, normal mode calculations are iterated during the exploration. Initial results show the efficiency of our method, which requires a small number of normal mode calculations to compute large-amplitude conformational transitions in proteins. A detailed analysis is presented for the computed transition between the open and closed structures of adenylate kinase. This transition, important for its biological function, involves large-amplitude domain motions. The obtained motion correlates well with results presented in related works.

Intrinsic motions along an enzymatic reaction trajectory

The mechanisms by which enzymes achieve extraordinary rate acceleration and specificity have long been of key interest in biochemistry. It is generally recognized that substrate binding coupled to conformational changes of the substrate-enzyme complex aligns the reactive groups in an optimal environment for efficient chemistry. Although chemical mechanisms have been elucidated for many enzymes, the question of how enzymes achieve the catalytically competent state has only recently become approachable by experiment and computation. Here we show crystallographic evidence for conformational substates along the trajectory towards the catalytically competent 'closed' state in the ligand-free form of the enzyme adenylate kinase. Molecular dynamics simulations indicate that these partially closed conformations are sampled in nanoseconds, whereas nuclear magnetic resonance and single-molecule fluorescence resonance energy transfer reveal rare sampling of a fully closed conformation occurring on the microsecond-to-millisecond timescale. Thus, the larger-scale motions in substrate-free adenylate kinase are not random, but preferentially follow the pathways that create the configuration capable of proficient chemistry. Such preferred directionality, encoded in the fold, may contribute to catalysis in many enzymes.

Large-scale allosteric conformational transitions of adenylate kinase appear to involve a population-shift mechanism

Large-scale conformational changes in proteins are often associated with the binding of a substrate. Because conformational changes may be related to the function of an enzyme, understanding the kinetics and energetics of these motions is very important. We have delineated the atomically detailed conformational transition pathway of the phosphotransferase enzyme adenylate kinase (AdK) in the absence and presence of an inhibitor. The computed free energy profiles associated with conformational transitions offer detailed mechanistic insights into, as well as kinetic information on, the ligand binding mechanism. Specifically, potential of mean force calculations reveal that in the ligand-free state, there is no significant barrier separating the open and closed conformations of AdK. The enzyme samples near closed conformations, even in the absence of its substrate. The ligand binding event occurs late, toward the closed state, and transforms the free energy landscape. In the ligand-bound state, the closed conformation is energetically most favored with a large barrier to opening. These results emphasize the underlying dynamic nature of the enzyme and indicate that the conformational transitions in AdK are more intricate than a mere two-state jump between the crystal-bound and -unbound states. Based on the existence of the multiple conformations of the enzyme in the open and closed states, a different viewpoint of ligand binding is presented. Our estimated activation energy barrier for the conformational transition is also in reasonable accord with the experimental findings.

Illuminating the mechanistic roles of enzyme conformational dynamics

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.

Protein conformational transitions explored by mixed elastic network models

We develop a mixed elastic network model (MENM) to study large-scale conformational transitions of proteins between two (or more) known structures. Elastic network potentials for the beginning and end states of a transition are combined, in effect, by adding their respective partition functions. The resulting effective MENM energy function smoothly interpolates between the original surfaces, and retains the beginning and end structures as local minima. Saddle points, transition paths, potentials of mean force, and partition functions can be found efficiently by largely analytic methods. To characterize the protein motions during a conformational transition, we follow "transition paths" on the MENM surface that connect the beginning and end structures and are invariant to parameterizations of the model and the mathematical form of the mixing scheme. As illustrations of the general formalism, we study large-scale conformation changes of the motor proteins KIF1A kinesin and myosin II. We generate possible transition paths for these two proteins that reveal details of their conformational motions. The MENM formalism is computationally efficient and generally applicable even for large protein systems that undergo highly collective structural changes.

Second-harmonic generation for studying structural motion of biological molecules in real time and space

SHG and sum-frequency generation (SFG) are surface-selective, nonlinear optical techniques whose ability to measure the average tilt angle of molecules on surfaces is well known in non-biological systems. By labeling molecules with a second-harmonic-active dye probe, SHG detection is extended to any biological molecule. The method has been used in previous work to detect biomolecules at an interface and their ligand-induced conformational changes. Here I demonstrate that SHG can be used to study structural motion quantitatively using a probe placed at a specific site (Cys-77) in adenylate kinase, a protein. The protein is also labeled non-site-specifically via amines. Labeled protein is absorbed to a surface and a baseline SH signal is measured. Upon introducing ATP, AMP or a specific inhibitor, AP(5)A, the baseline signal changes depending on the ligand and the labeling site. In particular, a substantial change in SH intensity is produced upon binding ATP to the amine-labeled protein, consistent with the X-ray crystal structures. In contrast, SHG polarization measurements are used to quantitatively determine that no rotation occurs at site Cys-77, in agreement with the lack of motion observed at this site in the X-ray crystal structures. A method for building a global map of conformational change in real time and space is proposed using a set of probes placed at different sites in a biomolecule. For this purpose, SH-active unnatural amino acids are attractive complements to exogenous labels.

Coarse-grained free energy functions for studying protein conformational changes: a double-well network model

In this work, a double-well network model (DWNM) is presented for generating a coarse-grained free energy function that can be used to study the transition between reference conformational states of a protein molecule. Compared to earlier work that uses a single, multidimensional double-well potential to connect two conformational states, the DWNM uses a set of interconnected double-well potentials for this purpose. The DWNM free energy function has multiple intermediate states and saddle points, and is hence a "rough" free energy landscape. In this implementation of the DWNM, the free energy function is reduced to an elastic-network model representation near the two reference states. The effects of free energy function roughness on the reaction pathways of protein conformational change is demonstrated by applying the DWNM to the conformational changes of two protein systems: the coil-to-helix transition of the DB-loop in G-actin and the open-to-closed transition of adenylate kinase. In both systems, the rough free energy function of the DWNM leads to the identification of distinct minimum free energy paths connecting two conformational states. These results indicate that while the elastic-network model captures the low-frequency vibrational motions of a protein, the roughness in the free energy function introduced by the DWNM can be used to characterize the transition mechanism between protein conformations.

MinActionPath: maximum likelihood trajectory for large-scale structural transitions in a coarse-grained locally harmonic energy landscape

The non-linear problem of simulating the structural transition between two known forms of a macromolecule still remains a challenge in structural biology. The problem is usually addressed in an approximate way using 'morphing' techniques, which are linear interpolations of either the Cartesian or the internal coordinates between the initial and end states, followed by energy minimization. Here we describe a web tool that implements a new method to calculate the most probable trajectory that is exact for harmonic potentials; as an illustration of the method, the classical Calpha-based Elastic Network Model (ENM) is used both for the initial and the final states but other variants of the ENM are also possible. The Langevin equation under this potential is solved analytically using the Onsager and Machlup action minimization formalism on each side of the transition, thus replacing the original non-linear problem by a pair of linear differential equations joined by a non-linear boundary matching condition. The crossover between the two multidimensional energy curves around each state is found numerically using an iterative approach, producing the most probable trajectory and fully characterizing the transition state and its energy. Jobs calculating such trajectories can be submitted on-line at: http://lorentz.dynstr.pasteur.fr/joel/index.php.

Coarse-graining of protein structures for the normal mode studies

The coarse-grained structural model such as Gaussian network has played a vital role in the normal mode studies for understanding protein dynamics related to biological functions. However, for the large proteins, the Gaussian network model is computationally unfavorable for diagonalization of Hessian (stiffness) matrix for the normal mode studies. In this article, we provide the coarse-graining method, referred to as "dynamic model condensation," which enables the further coarse-graining of protein structures consisting of small number of residues. It is shown that the coarser-grained structures reconstructed by dynamic model condensation exhibit the dynamic characteristics, such as low-frequency normal modes, qualitatively comparable to original structures. This sheds light on that dynamic model condensation and may enable one to study the large protein dynamics for gaining insight into biological functions of proteins.

Minimum free energy pathways and free energy profiles for conformational transitions based on atomistic molecular dynamics simulations

An efficient method for the calculation of minimum free energy pathways and free energy profiles for conformational transitions is presented. Short restricted perturbation-targeted molecular dynamics trajectories are used to generate an approximate free energy surface. Approximate reaction pathways for the conformational change are constructed from one-dimensional line segments on this surface using a Monte Carlo optimization. Accurate free energy profiles are then determined along the pathways by means of one-dimensional adaptive umbrella sampling simulations. The method is illustrated by its application to the alanine "dipeptide." Due to the low computational cost and memory demands, the method is expected to be useful for the treatment of large biomolecular systems.

Essential dynamics sampling study of adenylate kinase: Comparison to citrate synthase and implication for the hinge and shear mechanisms of domain motions

Essential dynamics sampling simulations of the domain conformations of unliganded Escherichia coli adenylate kinase have been performed to determine whether the ligand-induced closed-domain conformation is accessible to the open unliganded enzyme. Adenylate kinase is a three- domain protein with a central CORE domain and twoflanking domains, the LID and the NMPbind domains. The sampling simulations were applied to the CORE and NMPbind domain pair and the CORE and LID domain pair separately. One aim is to compare the results to those of a similar study on the enzyme citrate synthase to determine whether a similar domain-locking mechanism operates in adenylate kinase. Although for adenylate kinase the simulations suggest that the closed-domain conformation of the unliganded enzyme is at a slightly higher free energy than the open for both domain pairs, the results are radically different to those found for citrate synthase. In adenylate kinase the targeted domain conformations could always be achieved, whereas this was not the case in citrate synthase due to an apparent free-energy barrier between the open and closed conformations. Adenylate kinase has been classified as a protein that undergoes closure through a hinge mechanism, whereas citrate synthase has been assigned to the shear mechanism. This was quantified here in terms of the change in the number of interdomain contacting atoms upon closure which showed a considerable increase in adenylate kinase. For citrate synthase this number remained largely the same, suggesting that the domain faces slide over each other during closure. This suggests that shear and hinge mechanisms of domain closure may relate to the existence or absence of an appreciable barrier to closure for the unliganded protein, as the latter can hinge comparatively freely, whereas the former must follow a more constrained path. In general though it appears a bias toward keeping the unliganded enzyme in the open-domain conformation may be a common feature of domain enzymes.

Molecular dynamics of apo-adenylate kinase: a distance replica exchange method for the free energy of conformational fluctuations

A large domain motion in adenylate kinase from E. coli (AKE) is studied with molecular dynamics. AKE undergoes a large-scale rearrangement of its lid and AMP-binding domains when the open form closes over its substrates, AMP, and Mg2+-ATP, whereby the AMP-binding and lid domains come closer to the core. The third domain, the core, is relatively stable during this motion. A reaction coordinate that monitors the distance between the AMP-binding and core domains is selected to be able to compare with the results of energy transfer experiments. Sampling along this reaction coordinate is carried out by using a distance replica exchange method (DREM), where systems that differ by a restraint potential enforcing different reaction coordinate values are independently simulated with periodic attempts at exchange of these systems. Several methods are used to study the efficiency and convergence properties of the DREM simulation and compared with an analogous non-DREM simulation. The DREM greatly accelerates the rate and extent of configurational sampling and leads to equilibrium sampling as measured by monitoring collective modes obtained from a principal coordinate analysis. The potential of mean force along the reaction coordinate reveals a rather flat region for distances from the open to a relatively closed AKE conformation. The potential of mean force for smaller distances has a distinct minimum that is quite close to that found in the closed form X-ray structure. In concert with a decrease in the reaction coordinate distance (AMP-binding-to-core distance) the lid-to-core distance of AKE also decreases. Therefore, apo AKE can fluctuate from its open form to conformations that are quite similar to its closed form X-ray structure, even in the absence of its substrates.

Conformational transitions of adenylate kinase: switching by cracking

Conformational heterogeneity in proteins is known to often be the key to their function. We present a coarse grained model to explore the interplay between protein structure, folding and function which is applicable to allosteric or non-allosteric proteins. We employ the model to study the detailed mechanism of the reversible conformational transition of Adenylate Kinase (AKE) between the open to the closed conformation, a reaction that is crucial to the protein's catalytic function. We directly observe high strain energy which appears to be correlated with localized unfolding during the functional transition. This work also demonstrates that competing native interactions from the open and closed form can account for the large conformational transitions in AKE. We further characterize the conformational transitions with a new measure Phi(Func), and demonstrate that local unfolding may be due, in part, to competing intra-protein interactions.

Multiple-basin energy landscapes for large-amplitude conformational motions of proteins: Structure-based molecular dynamics simulations

Biomolecules often undergo large-amplitude motions when they bind or release other molecules. Unlike macroscopic machines, these biomolecular machines can partially disassemble (unfold) and then reassemble (fold) during such transitions. Here we put forward a minimal structure-based model, the "multiple-basin model," that can directly be used for molecular dynamics simulation of even very large biomolecular systems so long as the endpoints of the conformational change are known. We investigate the model by simulating large-scale motions of four proteins: glutamine-binding protein, S100A6, dihydrofolate reductase, and HIV-1 protease. The mechanisms of conformational transition depend on the protein basin topologies and change with temperature near the folding transition. The conformational transition rate varies linearly with driving force over a fairly large range. This linearity appears to be a consequence of partial unfolding during the conformational transition.

Folding-based molecular simulations reveal mechanisms of the rotary motor F1-ATPase

Biomolecular machines fulfill their function through large conformational changes that typically occur on the millisecond time scale or longer. Conventional atomistic simulations can only reach microseconds at the moment. Here, extending the minimalist model developed for protein folding, we propose the "switching Gō model" and use it to simulate the rotary motion of ATP-driven molecular motor F(1)-ATPase. The simulation recovers the unidirectional 120 degrees rotation of the gamma-subunit, the rotor. The rotation was induced solely by steric repulsion from the alpha(3)beta(3) subunits, the stator, which undergoes conformation changes during ATP hydrolysis. In silico alanine mutagenesis further elucidated which residues play specific roles in the rotation. Finally, regarding the mechanochemical coupling scheme, we found that the tri-site model does not lead to successful rotation but that the always-bi-site model produces approximately 30 degrees and approximately 90 degrees substeps, perfectly in accord with experiments. In the always-bi-site model, the number of sites occupied by nucleotides is always two during the hydrolysis cycle. This study opens up an avenue of simulating functional dynamics of huge biomolecules that occur on the millisecond time scales involving large-amplitude conformational change.

Modeling protein conformational changes by iterative fitting of distance constraints using reoriented normal modes

Recently we have developed a normal-modes-based algorithm that predicts the direction of protein conformational changes given the initial state crystal structure together with a small number of pairwise distance constraints for the end state. Here we significantly extend this method to accurately model both the direction and amplitude of protein conformational changes. The new protocol implements a multisteps search in the conformational space that is driven by iteratively minimizing the error of fitting the given distance constraints and simultaneously enforcing the restraint of low elastic energy. At each step, an incremental structural displacement is computed as a linear combination of the lowest 10 normal modes derived from an elastic network model, whose eigenvectors are reorientated to correct for the distortions caused by the structural displacements in the previous steps. We test this method on a list of 16 pairs of protein structures for which relatively large conformational changes are observed (root mean square deviation >3 angstroms), using up to 10 pairwise distance constraints selected by a fluctuation analysis of the initial state structures. This method has achieved a near-optimal performance in almost all cases, and in many cases the final structural models lie within root mean square deviation of 1 approximately 2 angstroms from the native end state structures.