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ComiXology Thousands of Digital Comics. East Dane Designer Men's Fashion. Shopbop Designer Fashion Brands. Withoutabox Submit to Film Festivals. When coupling is strong and all four events are concomitant, the only possible way of capturing them is within the framework of QM-MD approaches, such as Car-Parinello molecular dynamics. Free—energy calculations in either biological or abiological assays can be arbitrarily distinguished in terms of alchemical and geometrical transformations.

Alchemical free—energy calculations exploit the malleability of the potential energy function and the virtually unlimited possibilities of computer simulations to transform between chemically distinct states.

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Alchemical transformations require the definition of a progress variable, often referred to as coupling parameter, 62 , 63 which connects the end states of the chemical reaction and generally delineates a nonphysical path. Traditionally, variation of the free energy upon reorganization of bonded and nonbonded interactions is determined using a perturbative approach, i. However, system size of the biological motors in particular limits the feasibility of these calculations.

In stark contrast, geometrical free—energy calculations embrace positional, orientational, and conformational transitions described by a suitably chosen reaction coordinate model, often consisting of a collective variable or combinations thereof. Measure of the free—energy change that underlies these geometric transitions can be achieved employing a host of methods, which can be roughly classified into four distinct categories, namely i histogram—based approaches, 62 , 63 e.

It ought to be emphasized at this stage that while these methods have different merits and drawbacks, they also suffer from similar shortcomings, notably in connection with the upstream modeling of the reaction coordinate. It would, therefore, be misleading to assert that one method — or one class of methods, is clearly superior to the others. A number of weaknesses endogenous to geometric free—energy calculations, notably non—ergodicity behaviors, can, however, be addressed cogently by means of a seamless combination of the free—energy method at hand and generalized—ensemble techniques, 75 , 76 e.

In the BEUS scheme, 58 , 77 biasing potentials are exchanged between neighboring windows according to an exchange rule to specifically accelerate sampling of those degrees of freedom most relevant to the transition of interest. In contrast, in the MW—ABF algorithm, 57 , 78 a series of walkers explore the free—energy landscape in an independent fashion, exchanging regularly information about the gradient being measured locally.

The algorithm can be refined by means of Darwinian selection rules, whereby the most efficient walkers are promoted and spawned at the expense of the least efficient ones. The concerted movements of motor components that underlie the conversion of chemical energy into mechanical work is captured employing transition path—optimization methods. SMST 34 is one such method that refines iteratively a trial transition pathway, referred in this context to as a string, 79 until the pathway is converged.

The string is defined in a high—dimensional space of collective variables by a number of conformations or images, the position of which is updated at each iteration. The average drift applied to each image is estimated from a swarm of short unbiased MD trajectories, shot at the instantaneous position of the image. Each set of unbiased trajectories is followed by a biased equilibration stage, moving the replicas to their updated position. The iterative process continues until the string converges to an MFEP characterizing the conformational transition. As has been outlined previously, geometric free—energy calculations require the definition of a reaction coordinate model, 30 , 62 which can be provided by such approaches as SMST.

Once the converged string is obtained, the free—energy change can be determined using one the methods enumerated in the last subsection. The missing link between the optimized pathway and the free—energy method is the progress variable or general—extent parameter that tracks conversion between the end—states of the geometric transformation. This progress variable can either be implicit or explicit. In the former case, use is made of geometric variables, i.

Conversely, in the latter case, a path—collective variable 80 is defined as a function of the Cartesian coordinates and continuously changes between 0 and 1 as the image morphs between the end—states. So far, we have carefully utilized the vocabulary reaction coordinate model to describe the variable along which the free—energy change is determined.

This conservative nomenclature stems from the fact that, in general, our knowledge of the actual reaction coordinate — a high—dimensional mathematical object that defines the MFEP between the end—states of the transformation, is limited. While this consequence has no bearing on the measured free—energy change by virtue of its path independence, the kinetic properties of interest are prone to be erroneous. It should be clearly understood that a string optimization with swarms of trajectories 34 can approach the true reaction coordinate only under the assumption that it encompasses all relevant collective variables.

In practice, it is generally recommended to verify a posteriori that the string coincides with the MFEP by means of a committor analysis. Free—energy changes measured from the aforementioned alchemical and geometrical transformations represent, respectively, the energy input into a motor from a fuel source and its subsequent consumption by the motor components to perform mechanical work during the conformational transition. The ratio of these two energy contributions yields the energy conversion efficiency of a molecular motor.

Furthermore, from the rate constants determined along the MFEP, one can infer the turnover rate of the motor. Results from the theoretical strategy presented here are, therefore, readily comparable with experiment. More importantly, the simulations from whence they are obtained provide a detailed picture of the chemomechanical coupling that underlies motor action, the mechanism of which remains only partially understood by experiments.

In this section, we put forward a computational workflow for capturing the action steps of biological and abiological motors, summarized in Figure 2. Towards this end, we will dissect the different contributions of the chemomechanical coupling that eventually leads to spontaneous motor action, and outline how these contributions can be quantified. The reliability of the transition pathway determined using the aforementioned string algorithm, 34 and, hence, that of the thermodynamic and kinetic properties inferred from geometric free—energy calculations, 30 depend acutely on the quality of the trial pathway — i.

Optimization of an unphysical trial pathway often results in convergence to functionally irrelevant transitions, which can severely depart from the true reaction coordinate. Thus, construction of a physically meaningful trial pathway is key to the accuracy of the string calculation. An array of techniques have been developed that supply the guess initial pathway. Targeted MD 83 constitutes one such technique, which allows a path to be constructed by minimizing the RMSD between its two end—states. Alternatively, elastic network models ENM are designed, 84 , 85 employing harmonic spring approximations to provide the lowest potential energy pathway connecting the end—states of the geometric transformation, which represent basins of the potential energy hyperplane.

In certain cases, only one end—state of the transition pathway is known. Under these premises, the other end—state can be constructed employing SMD simulations, 33 steering the relevant degrees of freedom away from the energy minima characterizing the known end—state to an alternate one that represents the other end—state of the pathway. It ought to be understood that this naive approach does not offer any guarantee that i another basin will be found, and, ii whatever is found corresponds, indeed, to a functionally relevant state.

Monitoring the height of potential—energy barriers in ENM trial pathways, or the magnitude of non—equilibrium work in SMD trial pathways provide a qualitative idea on the functional relevance of the conformations that have been sampled, as well as the sequence of events that delineate the initial pathway. Generally, the lower the height of the potential—energy barrier, or the lesser the magnitude of the work required to induce a transition, the more reliable the initial pathway.

Notwithstanding proper initiation through a convincingly designed trial pathway, the accuracy of an SMST motor—action pathway is dependent on the choice of the set of collective variables that underlies the string and determines its dimension. As already commented on, SMST with an incomplete set of collective variables converges to unphysical pathways, yielding erroneous kinetic properties. In practice, collective variables are introduced to specifically capture those degrees of freedom that are most relevant to the transition of interest.

These collective variables are often constructed by analyzing the structural changes in the trial transition pathway. For example, Cartesian positions of a set of atoms can be chosen as collective variables if, within the initial transition pathway, the root mean-square deviation, or the change in the interaction energy characterizing this reduced set of atoms is comparable to that of the entire molecular structure.

Various combinations of atomic positions in the form of generalized coordinates, e.

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Identification of the relevant collective variables allow images equally spaced along the transition path to be defined. Iterations of swarms of biased and unbiased MD simulations are carried out at every image, prefacing reparametrization of the string until it converges to the sought MFEP. Shorter MD trajectories and larger swarms are found to provide richer ensembles than longer MD trajectories with smaller swarms.

Invoking thousands of MD trajectories simultaneously, the method is amenable to both serial and parallel implementation on supercomputers. Once the string has converged and coincides with the putative MFEP, the free—energy change along the latter can be determined using methods germane to geometric transformations. Such methods have been introduced in the Theoretical Underpinnings section. Here, w t represents the unnormalized weight of configuration x t , and can be estimated via: The mixing of the replicas in BEUS guarantees continuity in the sampling of a conformational tube that embraces the string, 89 thereby yielding a more reliable free—energy profile for the process than ordinary US.

With appropriate reweighting, the potential of mean force PMF can be reconstructed in the collective—variable space wherein the string is defined, under the assumption of adequate sampling. One of the techniques used for this purpose is nonparametric reweighting, which assigns a weight to each individual conformation sampled. As has been mentioned previously, a possible alternative approach to BEUS is MW—ABF, 57 , 78 wherein a number of walkers explore the free—energy hyperplane concomitantly in the course of an unconstrained MD simulation, exchanging periodically information on the gradient of the free energy.

In stark contrast with probability—based methods like US and its variants, a purely local estimate of the gradient is utilized, allowing the biasing force to be updated continuously.

One attractive feature of motor modeling lies in the possibility to compute turnover rates, which can be readily compared with experiment. Access to this quantity surmises knowledge of the position—dependent diffusivity, D s , along the MFEP, together with the PMF, the determination of which has been outlined above. Although there are a number of avenues for the estimation of D s , 92 — 94 one promising approach relies on Bayesian inferences and may be viewed as the inverse solution of the Smoluchowski equation, 95 which supplies consistent estimates of F s and D s , based on the trajectory obtained from a biased simulation.

Under the stringent assumption of a diffusive regime, the motion is propagated using a discretized Brownian integrator, whereby the trajectory can be seen as a series of discrete hops of finite time. The probability of the complete trajectory given the parameter of interest, namely D s , is the product of the probabilities at each step. Using Bayes theorem, one can then infer the probability of D s given the trajectory. Put together, finding the optimal position—dependent diffusivity becomes a maximum—likelihood problem of finding a parameter that maximizes the posterior probability, which can be handled by generating a Markov chain of states, employing a Metropolis—Hastings algorithm.

From the knowledge of F s and D s , one can then estimate the mean—first—passage time 96 — or the inverse of the rate constant, between the end—states of the string. So far, we have addressed how the conformational transition of motor action, that is its mechanical component, ought to be handled computationally. To complete the chemomechanical coupling, we now outline the treatment of the chemical component, the fuel source of the motor. The latter by and large consists of a chemical reaction involving the reorganization on bonded and nonbonded interactions, e.

Strictly speaking, rearrangement of electronic structures would require specific quantum—mechanical calculations to capture bond creations and ruptures. First, the timescale separation of the conformational transition and the chemical reaction allows the two to be handled independently. Second, whereas the choice of the reaction path is crucial in the modeling of the conformational transition, it is clearly less so for the chemical reaction, which is a corollary of the first consideration.

ATP hydrolysis: Gibbs free energy

Besides, our interest lies chiefly in the end states of the chemical reaction — not its actual pathway. Modeling of the chemical reaction can be performed using alchemical free—energy calculations, 30 in which the reference state, i. One common theoretical tool towards this end is perturbation theory, 62 , 63 whereby the target state can be seen as a perturbation of the reference state.

In practice, as a way to reduce the systematic error in FEP calculations, 64 , 65 the path connecting the two end states is broken into nonphysical intermediates separated by incremental perturbations. Furthermore, in order to minimize the variance of the calculation, the transformation is carried out bidirectionally between the reactant and the product, 97 and the statistical data accrued in the forward and the backward directions is combined to yield a maximum—likelihood estimator of the free energy, referred to as Bennett acceptance ratio. The reorganization of bonded and nonbonded interactions that fuels motor action is often accompanied by significant variations of configurational entropy, which are not easily captured in MD simulations — e.

A rigorous theoretical framework has been proposed to address this limitation of MD and relies upon the introduction of a series of geometric restraints enforced as the substrate is being coupled or decoupled reversibly from its environment. To summarize, we have put forth a practical workflow for the modeling of motor action, embodied in Figure 2 , which yields the chemical and the mechanical contributions to the chemomechanical coupling. The ratio of these two contributions quantifies the energy—conversion efficiency of the molecular motor. Figuratively speaking, the aryl moiety of the motor can be viewed as the piston, whereas the cyclodextrin would correspond to the cylinder.

The compression stroke arises from the binding of the aryl moiety by the cavity of the macrocycle cyclodextrin and is triggered by the extraction of the substrate 1—adamantanol in the proper solvent condition — a competition between intra— and intermolecular complexation. It is shown to occur with the Z—isomer of the amide group to which the aryl moiety, i. Conversely, the decompression stroke results from the addition of the substrate, and occurs with the E—isomer of the amide group. The upper binding process corresponds to the Z—isomer of the piston amide group and the lower binding process to its E—isomer.

K eq Z and K eq E are the respective equilibrium constants for the latter binding processes. In the case of the Z—isomer, the aryl moiety is initially ensconced inside the macrocycle, whereas for the E—isomer, it stretches outside of the host. The molecular assembly consisted of the motor and the substrate in a bath of about 4, water molecules. The MD simulations were performed in the isobaric—isothermal at 1 atm and Details of the simulations can be found in reference The simulation system size is about 0.

Maintaining the same harmonic potentials, a 1 ns equilibration was carried out in the isobaric—isothermal ensemble 1 atm at K , followed by a 4 ns canonical—ensemble simulation, gradually decreasing the spring constant to zero during the latter stage. In this first illustration of a rudimentary, cyclodextrin—based abiological motor, the transition coordinate is well identified, thereby obviating the need for tedious search of the most probable MFEP.

This nearly ideal scenario is, however, not representative of simple, abiological motors, for which the conformational transition may consist of more entangled movements. For instance, in certain rotaxanes, e. Post—hoc assessment of the model reaction coordinate, for instance, by means of a committor analysis, 37 , 81 is recommended, in particular if the kinetic properties of the abiological motor are sought.

From the onset, it can be seen that, irrespective of the isomer, complexation of the substrate by the macrocycle corresponds to a metastability on the free—energy landscape. The minimum of the one—dimensional free—energy profile for the E—isomer is, however, somewhat deeper than that for the Z—isomer. The difference of approximately 2. Under the assumption of proper averaging of all other relevant degrees of freedom and considering that association follows a cylindrical symmetry, the binding constant of the substrate to the cavity of the cyclodextrin can be expressed as, The hemomechanical work, i.

A second example showcasing the simulation protocol presented here is provided by the pathway of ADP inhibition in the hexameric ring of the molecular motor V 1 —ATPase. Boyer developed a binding—change model, 4 , 6 , 7 which postulates that ATP synthesis is coupled with a conformational change in the ATP synthase generated by rotation of its central stalk.

The occupancy of these sites is strongly correlated to the conformations of the protein, denoted respectively A e B e , A b B b , and A t B t , according to the binding interface they form during the ATP hydrolysis cycle. This rotation, coined conformational rotation, carries a near zero moment of inertia. Conformational rotation of the A 3 B 3 ring is examined employing anisotropic network models ANM to build a trial pathway, which will be subsequently refined by means of string calculations.

All three protein dimer states are allowed to change simultaneously in the first step, i. Next, two protein dimer states change simultaneously, i. Last, only one dimer state changes at a time, i. The product of the pathway endowed with the lowest energetic barrier among the seven distinct ANM pathways, i.

The former pathway was found to exhibit a lower barrier. The lowest potential energy pathway was resolved at full—atomic detail via TMD simulations. This ANM—based TMD pathway is discretized into equally distant images and optimized subsequently via string calculations, 34 as described in the next section, in order to obtain the MFEP for the conformational rotation of the A 3 B 3 ring. These collective variables are identified to be the positions of key residues pertaining to the six subunit—subunit interfaces of the A 3 B 3 trimer. These residues are found to be major contributors to changes in the interface energy, as well as to changes in the overall RMSD observed in the ANM pathways.

Construction of the reduced space for the string simulations of the V 1 —ring complex as well as the V 1 —rotor. An additional 47 residues are derived from the non—nucleotide binding interface. The A subunit residues included in the subspace are number 7 8 9 10 11 12 54 55 56 57 58 59 60 61 62 83 85 91 92 94 95 from the chains A, B and C in the PDB file of V 1 —rotor, and similarly for the B subunit are residue number 23 24 25 26 27 46 47 48 49 76 from chains D, E and F. Ten replicas of 0. The pathway was, thus, optimized iteratively until it converged to the most probable transition path, which is an approximation of the MFEP.

A stable solution of the transition pathway for the V 1 —ring was obtained within 50 iterations. The converged string trajectory reveals a distinct series of events that follows ATP hydrolysis. This transition prefaces binding of an ATP into the neighboring e site. Finally, the third site, previously in a b conformation, i. The BEUS simulations 58 , 77 for the putative MFEP were performed employing the aforementioned 1, Cartesian coordinates restrained to the position of the images along the optimized string.

To ensure sufficient window overlap in the biased simulations, the number of images was increased to , i.

An exchange of biases was attempted every 1 ps between adjacent and circumjacent images — i. Ten replicas per image were employed in 8-ns long BEUS simulations. The statistical data accrued in the course of this simulation was employed to construct the relevant histograms and subsequently infer the underlying free—energy change, using the non—parametric reweighting scheme outlined above. The primary free—energy barrier, about 4.

This barrier reflects a reorganization of the gatekeeper, R, residue lining the B subunit, which allows ingress of ATP into its binding pocket in the A subunit.

The positive net free—energy change highlights an endothermic reaction and implies that rotation is thermodynamically unfavorable in the presence of products bound to the empty site. The initial, the final and the two intermediate structures are labeled by their image number, i. Both thermodynamic quantities, i.

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It has been suggested by crystallographic studies that opening of the tight A t B t interface after ATP hydrolysis into an empty A e B e interface promotes opening of the neighboring A e B e interface even further, allowing it to bind an incoming ATP by facilitating isomerization of the gatekeeper R residue. Consequently, the incoming ATP associates only partially to its binding pocket at the A e B e interface. Furthermore, presence of bound ATP in all three binding pockets, as was observed in the present trajectory, morphs the A 3 B 3 ring into a more symmetric state than when only two of the pockets are occupied.

Such symmetric states are expected to engender deleterious elastic stress, making the pathway overall unfavorable. We, therefore, conclude that release of the reaction product from the t site allows thermodynamically favorable binding of ATP in the neighboring e site.

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