Multidimensional Free-energy Landscapes Research Articles

The GTPase hGBP1 converts GTP to GMP in two steps via proton shuttle mechanisms.

GTPases play a crucial role in the regulation of many biological processes by catalyzing the hydrolysis of GTP into GDP. The focus of this work is on the dynamin-related large GTPase human guanine nucleotide binding protein-1 (hGBP1) which is able to hydrolyze GTP even to GMP. Here, we studied the largely unknown mechanisms of both GTP and GDP hydrolysis steps utilizing accelerated ab initio QM/MM metadynamics simulations to compute multi-dimensional free energy landscapes. We find an indirect substrate-assisted catalysis (SAC) mechanism for GTP hydrolysis involving transfer of a proton from the water nucleophile to a nonbridging phosphoryl oxygen via a proton relay pathway where the rate-determining first step is concerted-dissociative nature. A "composite base" consisting of Ser73, Glu99, a bridging water molecule, and GTP was found to activate the nucleophilic water, thus disclosing the complex nature of the general base in hGBP1. A nearly two-fold reduction in the free energy barrier was obtained for GTP hydrolysis in the enzyme in comparison to bulk solvent. The subsequent GDP hydrolysis in hGBP1 was also found to follow a water-mediated proton shuttle mechanism. It is expected that the proton shuttle mechanisms unravelled for hGBP1 apply to many classes of GTPases/ATPases that possess an optimally-arranged hydrogen bonding network, which connects the catalytic water to a proton acceptor.

Free-Energy Landscape and Proton Transfer Pathways in Oxidative Deamination by Methylamine Dehydrogenase.

The rate-determining step in the reductive half-reaction of the bacterial enzyme methylamine dehydrogenase, which is proton abstraction from the native substrate methylamine, is investigated using accelerated QM/MM molecular dynamics simulations at room temperature. Generation of the multidimensional thermal free-energy landscape without restriction of the degrees of freedom beyond a multidimensional reaction subspace maps two rather similar pathways for the underlying proton transfer to one of two aspartate carboxyl oxygen atoms, termed OD1 and OD2, which hydrogen bond with Thr122 and Trp108, respectively. Despite significant large-amplitude motion perpendicular to the one-dimensional proton transfer coordinate, due to fluctuations of the donor-acceptor distance of about 3 Å, it is found that the one-dimensional proton transfer free-energy profiles are essentially identical to the minimum free-energy pathways on the multidimensional free-energy landscapes for both proton transfer channels. Proton transfer to one of the acceptor oxygen atoms-the OD2 site-is slightly favored in methylamine dehydrogenase by approximately 2 kcal mol-1 , both kinetically and thermodynamically. Mechanistic analyses reveal that the hydrogen bond between Thr122β and OD1 is always present in the transition state independently of the proton transfer channel. Population analysis confirms that the electronic charge gained upon oxidation of the substrate is delocalized within the ring systems of the tryptophan tryptophylquinone cofactor.

Global Langevin model of multidimensional biomolecular dynamics.

Molecular dynamics simulations of biomolecular processes are often discussed in terms of diffusive motion on a low-dimensional free energy landscape F(𝒙). To provide a theoretical basis for this interpretation, one may invoke the system-bath ansatz á la Zwanzig. That is, by assuming a time scale separation between the slow motion along the system coordinate x and the fast fluctuations of the bath, a memory-free Langevin equation can be derived that describes the system's motion on the free energy landscape F(𝒙), which is damped by a friction field and driven by a stochastic force that is related to the friction via the fluctuation-dissipation theorem. While the theoretical formulation of Zwanzig typically assumes a highly idealized form of the bath Hamiltonian and the system-bath coupling, one would like to extend the approach to realistic data-based biomolecular systems. Here a practical method is proposed to construct an analytically defined global model of structural dynamics. Given a molecular dynamics simulation and adequate collective coordinates, the approach employs an "empirical valence bond"-type model which is suitable to represent multidimensional free energy landscapes as well as an approximate description of the friction field. Adopting alanine dipeptide and a three-dimensional model of heptaalanine as simple examples, the resulting Langevin model is shown to reproduce the results of the underlying all-atom simulations. Because the Langevin equation can also be shown to satisfy the underlying assumptions of the theory (such as a delta-correlated Gaussian-distributed noise), the global model provides a correct, albeit empirical, realization of Zwanzig's formulation. As an application, the model can be used to investigate the dependence of the system on parameter changes and to predict the effect of site-selective mutations on the dynamics.

The true nature of rotary movements in rotaxanes

Disentangling the different movements observed in rotaxanes is critical to characterize their function as molecular and biological motors. How to achieve unidirectional rotation is an important question for successful construction of a highly efficient molecular motor. The motions within a rotaxane composed of a benzylic amide ring threaded on a fumaramide moiety were investigated employing atomistic molecular dynamics simulations. The free-energy profiles describing the rotational process of the ring about the thread were determined from multi-microsecond simulations. Comparing the theoretical free-energy barriers with their experimental counterpart, the syn-anti isomerization of the amide bond within the ring was ruled out. The free-energy barriers arise in fact from the disruption of hydrogen bonds between the ring and the thread. Transition path analysis reveals that complete description of the reaction coordinate requires another collective variable. The free-energy landscape spanned by the two variables characterizing the coupled rotational and shuttling processes of the ring in the rotaxane was mapped. The calculated free-energy barrier, amounting to 9.3 kcal mol-1, agrees well with experiment. Further analysis shows that shuttling is coupled with the isomerization of the ring, which is not limited to a simplistic chair-to-chair transition. This work provides a cogent example that contrary to chemical intuition, molecular motion can result from complex, entangled movements requiring for their accurate description careful modeling of the underlying reaction coordinate. The methodology described here can be used to evaluate the different components of the multifaceted motion in rotaxanes, and constitutes a robust tool for the rational design of molecular machines.

Efficient Sampling of High-Dimensional Free-Energy Landscapes with Parallel Bias Metadynamics.

Metadynamics accelerates sampling of molecular dynamics while reconstructing thermodynamic properties of selected descriptors of the system. Its main practical difficulty originates from the compromise between keeping the number of descriptors small for efficiently exploring their multidimensional free-energy landscape and biasing all of the slow motions of a process. Here we illustrate on a model system and on the tryptophan-cage miniprotein parallel bias metadynamics, a method that overcomes this issue by simultaneously applying multiple low-dimensional bias potentials.

Rational Modulation of the Induced-Fit Conformational Change for Slow-Onset Inhibition in Mycobacterium tuberculosis InhA.

Slow-onset enzyme inhibitors are the subject of considerable interest as an approach to increasing the potency of pharmaceutical compounds by extending the residence time of the inhibitor on the target (the lifetime of the drug-receptor complex). However, rational modulation of residence time presents significant challenges because it requires additional mechanistic insight, such as the nature of the transition state for postbinding isomerization. Our previous work, based on X-ray crystallography, enzyme kinetics, and molecular dynamics simulation, suggested that the slow step in inhibition of the Mycobacterium tuberculosis enoyl-ACP reductase InhA involves a change in the conformation of the substrate binding loop from an open state in the initial enzyme-inhibitor complex to a closed state in the final enzyme-inhibitor complex. Here, we use multidimensional free energy landscapes for loop isomerization to obtain a computational model for the transition state. The results suggest that slow-onset inhibitors crowd key side chains on helices that slide past each other during isomerization, resulting in a steric clash. The landscapes become significantly flatter when residues involved in the steric clash are replaced with alanine. Importantly, this lower barrier can be increased by rational inhibitor redesign to restore the steric clash. Crystallographic studies and enzyme kinetics confirm the predicted effects on loop structure and flexibility, as well as inhibitor residence time. These loss and regain of function studies validate our mechanistic hypothesis for interactions controlling substrate binding loop isomerization, providing a platform for the future design of inhibitors with longer residence times and better in vivo potency. Similar opportunities for slow-onset inhibition via the same mechanism are identified in other pathogens.

A theoretical treatment of energy transfer in a molecular motor based on Brownian motion on a multidimensional free-energy landscape: The derivation of formal analytic results in the Kramers regime of deep potential wells

A theoretical treatment of energy transfer in a molecular motor based on Brownian motion on a multidimensional free-energy landscape: The derivation of formal analytic results in the Kramers regime of deep potential wells

Energy Transfer in a Molecular Motor in Kramers' Regime

We present a theoretical treatment of energy transfer in a molecular motor described in terms of Brownian motion on a multidimensional free-energy landscape. We implement the classical analog of the tight-binding model of quantum mechanics to transform the continuous diffusion equation to a discrete master equation that is analytically tractable [1]. This treatment applies for multidimensional non-separable periodic potentials enabling energy transfer between degrees of freedom to be described explicitly in both the strong and weak coupling regimes. Our approach provides an opportunity to connect the continuous multidimensional Brownian-motion based theory with experiments, phenomenological models, and established results from non-equilibrium thermodynamics. In Kramers' regime of deep potential wells, we derive a range of formal expressions for molecular motors. We determine physical properties including the drift and diffusion, the rate and efficiency of energy transfer, and the entropy generated, thereby unifying known results within a single theoretical framework [2]. We also consider thermal fluctuations for the motor and show that energy transfer between degrees of freedom creates statistical correlations between fluctuations in those degrees of freedom [3]. The fluctuation statistics provide an opportunity to distinguish the different operating regimes of the motor. [1] K. J. Challis and M. W. Jack, Phys. Rev. E 87, 052102 (2013). [2] K. J. Challis and M. W. Jack, arXiv:1208.5818, to appear in Phys. Rev. E. [3] K. J. Challis and M. W. Jack, arXiv:1308.6354.

Multi-Dimensional Free Energy Landscape of Voltage Sensor Domain Activation

Many modulators, such as toxins, anesthetics or drugs, act on voltage-gated channels by altering the activation and/or deactivation mechanism of their voltage-sensing domains (VSD). So far, we have proposed a static model of the Kv1.2 VSD activation using brute force and modified molecular dynamics simulations and that agrees with a large body of experimental data. This model involves 5 states: α (activated), β, γ, δ (three intermediate) and e (resting) [Delemotte et al. 2011, Proc. Natl. Acad. Sci. USA, 108:6109-6114], and has enabled to gain access to the contribution of transmembrane voltage to the free energy of activation via calculation of the corresponding gating charge. Crucial details, however, are still missing, among which an estimation of the thermodynamic stability of these states or of the minimum energy transition pathway linking them. In order to complete our understanding of VSD function, we produce the multi-dimensional free energy landscape (FES) of the four transitions linking the Kv1.2 VSD conformations, enabling to follow for the first time the pathway of activation of a VSD. We then investigate how the free energy landscape of VSD activation is modified by a change in the lipid environment and by the mutation of key residues.

Highly Enhanced Conformational Sampling of the Transmembrane Domain of Egf Receptor Sheds Light on the Activation Mechanism

The epidermal growth factor (EGF) receptor plays critical roles in regulating proliferation, survival and migration of cells. The activation of this receptor tyrosine kinase involves the dimerization of alpha helices in the transmembrane (TM) domain. Single point mutations in this region can modulate efficiency of helix dimerization and abnormal activation leads to human cancers. Therefore, it is crucial to determine modes and affinities of association in functional receptors and oncogenic forms. Various experimental and computational techniques have been recently used in this respect, which have provided substantial results. The emergence of innovative approaches might strongly contribute to improve our understanding of the activation mechanisms.Here, we develop a new computational method that combines coarse-grained molecular dynamics and metadynamics to investigate the dimerization of TM helices of EGF receptor. We perform simulations above the millisecond timescale to explore multiple events of association and dissociation. Our enhanced sampling protocol enables reconstruction of the multidimensional free energy landscape of the TM domain in its lipid bilayer environment. The results reveal complex free energy surfaces on which pathways of dimerization are clearly located. Then, cluster analyses enable identification of dimer conformations with N-terminal contacts of marginally higher stability than conformations displaying C-terminal contacts. Overall, the results suggest a mechanism in which the TM domain is a soft pivot and the dynamics of both ectodomain and juxta-membrane domain would drive the conformational change from inactive to active states of EGF receptor dimers. Application of the method to the homolog receptor Her2 will be also discussed to illustrate potential differences in the mechanisms of activation.

Graph representation of protein free energy landscape

The thermodynamics and kinetics of protein folding and protein conformational changes are governed by the underlying free energy landscape. However, the multidimensional nature of the free energy landscape makes it difficult to describe. We propose to use a weighted-graph approach to depict the free energy landscape with the nodes on the graph representing the conformational states and the edge weights reflecting the free energy barriers between the states. Our graph is constructed from a molecular dynamics trajectory and does not involve projecting the multi-dimensional free energy landscape onto a low-dimensional space defined by a few order parameters. The calculation of free energy barriers was based on transition-path theory using the MSMBuilder2 package. We compare our graph with the widely used transition disconnectivity graph (TRDG) which is constructed from the same trajectory and show that our approach gives more accurate description of the free energy landscape than the TRDG approach even though the latter can be organized into a simple tree representation. The weighted-graph is a general approach and can be used on any complex system.

Following Easy Slope Paths on a Free Energy Landscape: The Case Study of the Trp-Cage Folding Mechanism

In this work a new method for the automatic exploration and calculation of multidimensional free energy landscapes is proposed. Inspired by metadynamics, it uses several collective variables that are relevant for the investigated process and a bias potential that discourages the sampling of already visited configurations. The latter potential allows escaping a local free energy minimum following the direction of slow motions. This is different from metadynamics in which there is no specific direction of the biasing force and the computational effort increases significantly with the number of collective variables. The method is tested on the Ace-Ala3-Nme peptide, and then it is applied to investigate the Trp-cage folding mechanism. For this protein, within a few hundreds of nanoseconds, a broad range of conformations is explored, including nearly native ones, initiating the simulation from a completely unfolded conformation. Finally, several folding/unfolding trajectories give a systematic description of the Trp-cage folding pathways, leading to a unified view for the folding mechanisms of this protein. The proposed mechanism is consistent with NMR chemical shift data at increasing temperature and recent experimental observations pointing to a pivotal role of secondary structure elements in directing the folding process toward the native state.

Explorative Umbrella Sampling

Computer simulation can contribute significantly to our understanding of biomolecules, particularly by elucidating the free energy landscape in which they exist. A major challenge in this type of application is achieving sufficient sampling. Here we highlight a recent improved sampling method which both limits sampling to energetically relevant regions of phase space and reduces resampling of the same regions. This approach reduces the computational burden of studying complex and multidimensional free energy landscapes, extending the type and size of systems that can be simulated.

On‐the‐fly reconstruction of free‐energy profiles using logarithmic mean‐force dynamics

Mean-force dynamics (MFD), which is a fictitious dynamics for a set of collective variables on a potential of mean-force, is a powerful algorithm to efficiently explore free-energy landscapes. Recently, we have introduced logarithmic MFD (LogMFD) (Morishita et al., Phys. Rev. E 2012, 85, 066702) which overcomes difficulties encounterd in free-energy calculations using standard approaches such as thermodynamic integration. Here, we present a guide to implementing LogMFD calculations paying attention to the practical issues in choosing the parameters in LogMFD. A primary focus is given to the effect of the parameters on the accuracy of the reconstructed free-energy profiles. A recipe for reducing the errors due to energy dissipation is presented. We also demonstrate that multidimensional free-energy landscapes can be reconstructed on-the-fly using LogMFD, which cannot be accomplished using any other free-energy calculation techniques.

Collective degrees of freedom involved in absorption and desorption of surfactant molecules in spherical non-ionic micelles

Dynamics of absorption and desorption of a surfactant monomer into and out of a spherical non-ionic micelle is investigated by coarse-grained molecular dynamics (MD) simulations. It is shown that these processes involve a complex interplay between the micellar structure and the monomer configuration. A quantitative model for collective dynamics of these degrees of freedom is developed. This is accomplished by reconstructing a multi-dimensional free energy landscape of the surfactant-micelle system using constrained MD simulations in which the distance between the micellar and monomer centers of mass is held constant. Results of this analysis are verified by direct (unconstrained) MD simulations of surfactant absorption in the micelle. It is demonstrated that the system dynamics is likely to deviate from the minimum energy path on the energy landscape. These deviations create an energy barrier for the monomer absorption and increase an existing barrier for the monomer desorption. A reduced Fokker-Planck equation is proposed to model these effects.

Force-induced Unbinding Dynamics in a Multidimensional Free Energy Landscape

Department of Chemistry, Chung-Ang University, Seoul 156-756, Korea. E-mail: hyeoncb@kias.re.krReceived November 30, 2011, Accepted January 5, 2012We examined theory for force-induced unbinding on a two-dimensional free energy surface where the internaldynamics of biomolecules is coupled with the rupture process under constant tension f. We show that only ifthe transition state ensemble is narrow and activation barrier is high, the f-dependent rupture rate in the 2Dpotential surface can faithfully be described using an effective 1D energy profile.Key Words : Force dynamics, Kramers equation, Multidimensional surface, Hydrodynamic couplingIntroductionSince the birth of chemical dynamics,

Atomistic Simulation Studies of the Permeation of Organic Molecules Through Lipid Membranes

For drugs with intracellular targets the process of permeation through the membrane is of fundamental importance (A. Malkia, et al., Euro J of Pharm Scien, 23,13-47, 2004). Usually the rate of this process is predicted using QSAR or other knowledge-based predictors (R Gozalbes, et al., Bioorganic & Med Chem,19, 2615-2624, 2011). However, this approach is not always accurate. Moreover, it does not provide the atomistic details of the process, and thus its prediction cannot be directly exploited to rationally design drugs with higher permeation rate. We developed a protocol for studying the permeation of small organic molecules (e.g. drugs) through lipid membranes by atomistic simulations. This protocol allows computing accurately the permeability coefficient, and provides a detailed atomistic picture of the process. The approach is based on an enhanced sampling technique, bias exchange metadynamics (S. Piana and A. Laio, J Phys Chem B, 111, 4553-4559, 2007), that allows deriving from atomistic simulations a multidimensional free energy landscape and an accurate kinetic model describing the transitions between the relevant metastable states of the system (F Marinelli, et al., Plos Comp Biol, 5, e1000452, 2009). As a benchmark, we applied this protocol on the permeation of ethanol through palmitoyloleoylphosphatidylcholine (POPC) membrane. We are applying the same procedure to study the permeation of two anti-HIV drugs where unbiased simulation of the permeation process is not possible.

Extracting the underlying effective free energy landscape from single-molecule time series—local equilibrium states and their network

We present a new self-consistent procedure to construct a multidimensional effective free energy landscape from a scalar single molecule time series, when single molecules experience the landscape within a given timescale of "observation." The theory is based on a framework we recently developed to extract a set of local equilibrium states (LESs) and their network from a scalar time series, such as distance between dye molecules tagged in a biomolecule. We scrutinize the appropriateness of the assumptions of local equilibration and local detailed balance among LESs at the single molecule level within the given timescale, rather than postulating them a priori. The self-consistent procedure in this article incorporates the effect of local correlation of the system dynamics inside potential basins, and the effect of finiteness of the sampled data points in assigning the boundary between different LESs. We propose a new simple scheme to assign the dimensionality of the energy landscape from a single molecule time series. We also address the question of what the molecules actually "feel" from the underlying landscape at the single molecule level.

Enhanced Sampling of Multidimensional Free-Energy Landscapes Using Adaptive Biasing Forces

We propose an adaptive biasing algorithm aimed at enhancing the sampling of multimodal measures by Langevin dynamics. The underlying idea consists in generalizing the standard adaptive biasing force method commonly used in conjunction with molecular dynamics to handle in a more effective fashion multidimensional reaction coordinates. The proposed approach is anticipated to be particularly useful for reaction coordinates, the components of which are weakly coupled, as illuminated in a mathematical analysis of the long-time convergence of the algorithm. The strength as well as the intrinsic limitation of the method are discussed and illustrated in two realistic test cases.

Fluctuation power spectra reveal dynamical heterogeneity of peptides

Characterizing the conformational properties and dynamics of biopolymers and their relation to biological activity and function is an ongoing challenge. Single molecule techniques have provided a rich experimental window on these properties, yet they have often relied on simple one-dimensional projections of a multidimensional free energy landscape for a practical interpretation of the results. Here, we study three short peptides with different structural propensity (alpha helical, beta hairpin, and random coil) in the presence (or absence) of a force applied to their ends using Langevin dynamics simulation and an all-atom model with implicit solvation. Each peptide produces fluctuation power spectra with a characteristic dynamic fingerprint consistent with persistent structural motifs of helices, hairpins, and random coils. The spectra for helix formation shows two well-defined relaxation modes, corresponding to local relaxation and cooperative coil to uncoil interconversion. In contrast, both the hairpin and random coil are polymerlike, showing a broad and continuous range of relaxation modes giving characteristic power laws of omega(-5/4) and omega(-3/2), respectively; the -5/4 power law for hairpins is robust and has not been previously observed. Langevin dynamics simulations of diffusers on a potential of mean force derived from the atomistic simulations fail to reproduce the fingerprints of each peptide motif in the power spectral density, demonstrating explicitly that such information is lacking in such one-dimensional projections. Our results demonstrate the yet unexploited potential of single molecule fluctuation spectroscopy to probe more fine scaled properties of proteins and biological macromolecules and how low dimensional projections may cause the loss of relevant information.