String Method In Collective Variables Research Articles

Computational Investigation of the Covalent Inhibition Mechanism of Bruton's Tyrosine Kinase by Ibrutinib.

Covalent inhibitors represent a promising class of therapeutic compounds. Nonetheless, rationally designing covalent inhibitors to achieve a right balance between selectivity and reactivity remains extremely challenging. To better understand the covalent binding mechanism, a computational study is carried out using the irreversible covalent inhibitor of Bruton tyrosine kinase (BTK) ibrutinib as an example. A multi-μs classical molecular dynamics trajectory of the unlinked inhibitor is generated to explore the fluctuations of the compound associated with the kinase binding pocket. Then, the reaction pathway leading to the formation of the covalent bond with the cysteine residue at position 481 via a Michael addition is determined using the string method in collective variables on the basis of hybrid quantum mechanical-molecular mechanical (QM/MM) simulations. The reaction pathway shows a strong correlation between the covalent bond formation and the protonation/deprotonation events taking place sequentially in the covalent inhibition reaction, consistent with a 3-step reaction with transient thiolate and enolates intermediate states. Two possible atomistic mechanisms affecting deprotonation/protonation events from the thiolate to the enolate intermediate were observed: a highly correlated direct pathway involving proton transfer to the Cα of the acrylamide warhead from the cysteine involving one or a few water molecules and a more indirect pathway involving a long-lived enolate intermediate state following the escape of the proton to the bulk solution. The results are compared with experiments by simulating the long-time kinetics of the reaction using kinetic modeling.

Experimental and Theoretical Investigation of External Electric-Field-Induced Crystallization of TKX-50 from Solution by Finite-Temperature String with Order Parameters as Collective Variables for Ionic Crystals.

External electric fields are an effective tool to induce phase transformations. The crystallization of ionic crystals from solution is a common phase transformation. However, understanding of mechanisms is poor at the molecular level. In this work, we carried out an experimental and theoretical investigation of the external electric-field-induced crystallization of TKX-50 from saturated formic acid solution by finite-temperature string (FTS) with order parameters (OPs) as collective variables for ionic crystals. The minimum-free-energy path was sketched by the string method in collective variables. The results show that the K-means clustering algorithm based on Euclidean distance and density weights can be used for enhanced sampling of the OPs in external electric-field-induced crystallization of ionic crystal from solution, which improves the conventional FTS. The crystallization from solution is a process of surface-mediated nucleation. The external electric field can accelerate the evolution of the string and decrease the difference in the potential of mean forces between the crystal and the transition state. Due to the significant change in OPs induced by the external electric field in nucleation, the crystalline quality was enhanced, which explains the experimental results that the external electric field enhanced the density, detonation velocity, and detonation pressure of TKX-50. This work provides an effective way to explore the crystallization of ionic crystals from solution at the molecular level, and it is useful for improving the properties of ionic crystal explosives by using external electric fields.

Surface-Accelerated String Method for Locating Minimum Free Energy Paths.

We present a surface-accelerated string method (SASM) to efficiently optimize low-dimensional reaction pathways from the sampling performed with expensive quantum mechanical/molecular mechanical (QM/MM) Hamiltonians. The SASM accelerates the convergence of the path using the aggregate sampling obtained from the current and previous string iterations, whereas approaches like the string method in collective variables (SMCV) or the modified string method in collective variables (MSMCV) update the path only from the sampling obtained from the current iteration. Furthermore, the SASM decouples the number of images used to perform sampling from the number of synthetic images used to represent the path. The path is optimized on the current best estimate of the free energy surface obtained from all available sampling, and the proposed set of new simulations is not restricted to being located along the optimized path. Instead, the umbrella potential placement is chosen to extend the range of the free energy surface and improve the quality of the free energy estimates near the path. In this manner, the SASM is shown to improve the exploration for a minimum free energy pathway in regions where the free energy surface is relatively flat. Furthermore, it improves the quality of the free energy profile when the string is discretized with too few images. We compare the SASM, SMCV, and MSMCV using 3 QM/MM applications: a ribozyme methyltransferase reaction using 2 reaction coordinates, the 2'-O-transphosphorylation reaction of Hammerhead ribozyme using 3 reaction coordinates, and a tautomeric reaction in B-DNA using 5 reaction coordinates. We show that SASM converges the paths using roughly 3 times less sampling than the SMCV and MSMCV methods. All three algorithms have been implemented in the FE-ToolKit package made freely available.

Energetics of Flap Opening in HIV-1 Protease: String Method Calculations.

HIV-1 protease (PR) is the viral protein responsible for virion maturation, and its mechanisms of action remain incompletely understood. PR is dimeric and contains two flexible, symmetry-related flaps, which act as a gate to inhibit access to the binding pocket and hold the polypeptide substrate in the binding pocket once bound. Wide flap opening, a conformational change assumed to be necessary for substrate binding, is a rare event in the closed and bound form. In this study, we use molecular dynamics (MD) simulations and advanced MD techniques including temperature acceleration and string method in collective variables to study the conformational changes associated with substrate unbinding of both wild-type and F99Y mutant PR. The F99Y mutation is shown via MD to decouple the closing of previously unrecognized distal pockets from substrate unbinding. To determine whether or not the F99Y mutation affects the energetic cost of wide flap opening, we use string method in collective variables to determine the minimum free-energy mechanism for wide flap opening in concert with distal pocket closing. The results indicate that the major energetic cost in flap opening is disengagement of the two flap-tip Ile50 residues from each other and is not affected by the F99Y mutation.

Mapping saddles and minima on free energy surfaces using multiple climbing strings.

Locating saddle points on free energy surfaces is key in characterizing multistate transition events in complicated molecular-scale systems. Because these saddle points represent transition states, determining minimum free energy pathways to these saddles and measuring their free energies relative to their connected minima are further necessary, for instance, to estimate transition rates. In this work, we propose a new multistring version of the climbing string method in collective variables to locate all saddles and corresponding pathways on free energy surfaces. The method uses dynamic strings to locate saddles and static strings to keep a history of prior strings converged to saddles. Interaction of the dynamic strings with the static strings is used to avoid the convergence to already-identified saddles. Additionally, because the strings approximate curves in collective-variable space, and we can measure free energy along each curve, identification of any saddle's two connected minima is guaranteed. We demonstrate this method to map the network of stationary points in the 2D and 4D free energy surfaces of alanine dipeptide and alanine tripeptide, respectively.

Mapping Free Energy Pathways for ATP Hydrolysis in the E. coli ABC Transporter HlyB by the String Method.

HlyB functions as an adenosine triphosphate (ATP)-binding cassette (ABC) transporter that enables bacteria to secrete toxins at the expense of ATP hydrolysis. Our previous work, based on potential energy profiles from combined quantum mechanical and molecular mechanical (QM/MM) calculations, has suggested that the highly conserved H-loop His residue H662 in the nucleotide binding domain (NBD) of E. coli HlyB may catalyze the hydrolysis of ATP through proton relay. To further test this hypothesis when entropic contributions are taken into account, we obtained QM/MM minimum free energy paths (MFEPs) for the HlyB reaction, making use of the string method in collective variables. The free energy profiles along the MFEPs confirm the direct participation of H662 in catalysis. The MFEP simulations of HlyB also reveal an intimate coupling between the chemical steps and a local protein conformational change involving the signature-loop residue S607, which may serve a catalytic role similar to an Arg-finger motif in many ATPases and GTPases in stabilizing the phosphoryl-transfer transition state.

Modelling nucleation from solution with the string method in the osmotic ensemble

ABSTRACTDirect molecular simulation of nucleation from solution is a challenging task, requiring a combination of ‘rare events’ techniques and methods to control the chemical potential. Rare event methods usually keep the total number of molecules fixed, resulting in artificial free energy profiles due to the depletion of solute from the solution phase. In order to address this issue, we present a new approach that uses the string method in collective variables in the osmotic ensemble to obtain minimum free energy pathways for nucleation at constant supersaturation. Our method does not require using an explicit reservoir of solute molecules, or making additional assumptions about the activity coefficients in the solution. We apply the new method to the crystallisation of sulfamerazine from acetonitrile and methanol solutions, and compare the resulting potential of mean force profiles to those obtained using analytical corrections previously employed in the literature.

Vapor nucleation paths in lyophobic nanopores

.In recent years, technologies revolving around the use of lyophobic nanopores gained considerable attention in both fundamental and applied research. Owing to the enormous internal surface area, heterogeneous lyophobic systems (HLS), constituted by a nanoporous lyophobic material and a non-wetting liquid, are promising candidates for the efficient storage or dissipation of mechanical energy. These diverse applications both rely on the forced intrusion and extrusion of the non-wetting liquid inside the pores; the behavior of HLS for storage or dissipation depends on the hysteresis between these two processes, which, in turn, are determined by the microscopic details of the system. It is easy to understand that molecular simulations provide an unmatched tool for understanding phenomena at these scales. In this contribution we use advanced atomistic simulation techniques in order to study the nucleation of vapor bubbles inside lyophobic mesopores. The use of the string method in collective variables allows us to overcome the computational challenges associated with the activated nature of the phenomenon, rendering a detailed picture of nucleation in confinement. In particular, this rare event method efficiently searches for the most probable nucleation path(s) in otherwise intractable, high-dimensional free-energy landscapes. Results reveal the existence of several independent nucleation paths associated with different free-energy barriers. In particular, there is a family of asymmetric transition paths, in which a bubble forms at one of the walls; the other family involves the formation of axisymmetric bubbles with an annulus shape. The computed free-energy profiles reveal that the asymmetric path is significantly more probable than the symmetric one, while the exact position where the asymmetric bubble forms is less relevant for the free energetics of the process. A comparison of the atomistic results with continuum models is also presented, showing how, for simple liquids in mesoporous materials of characteristic size of ca. 4nm, the nanoscale effects reported for smaller pores have a minor role. The atomistic estimates for the nucleation free-energy barrier are in qualitative accord with those that can be obtained using a macroscopic, capillary-based nucleation theory.Graphical abstract

Intrusion and extrusion of water in hydrophobic nanopores

Heterogeneous systems composed of hydrophobic nanoporous materials and water are capable, depending on their characteristics, of efficiently dissipating (dampers) or storing ("molecular springs") energy. However, it is difficult to predict their properties based on macroscopic theories-classical capillarity for intrusion and classical nucleation theory (CNT) for extrusion-because of the peculiar behavior of water in extreme confinement. Here we use advanced molecular dynamics techniques to shed light on these nonclassical effects, which are often difficult to investigate directly via experiments, owing to the reduced dimensions of the pores. The string method in collective variables is used to simulate, without artifacts, the microscopic mechanism of water intrusion and extrusion in the pores, which are thermally activated, rare events. Simulations reveal three important nonclassical effects: the nucleation free-energy barriers are reduced eightfold compared with CNT, the intrusion pressure is increased due to nanoscale confinement, and the intrusion/extrusion hysteresis is practically suppressed for pores with diameters below 1.2 nm. The frequency and size dependence of hysteresis exposed by the present simulations explains several experimental results on nanoporous materials. Understanding physical phenomena peculiar to nanoconfined water paves the way for a better design of nanoporous materials for energy applications; for instance, by decreasing the size of the nanopores alone, it is possible to change their behavior from dampers to molecular springs.

Heterogeneous nucleation from a supercooled ionic liquid on a carbon surface.

Classical molecular dynamics simulations were used to study the nucleation of the crystal phase of the ionic liquid [dmim+][Cl-] from its supercooled liquid phase, both in the bulk and in contact with a graphitic surface of D = 3 nm. By combining the string method in collective variables [Maragliano et al., J. Chem. Phys. 125, 024106 (2006)], with Markovian milestoning with Voronoi tessellations [Maragliano et al., J. Chem. Theory Comput. 5, 2589-2594 (2009)] and order parameters for molecular crystals [Santiso and Trout, J. Chem. Phys. 134, 064109 (2011)], we computed minimum free energy paths, the approximate size of the critical nucleus, the free energy barrier, and the rates involved in these nucleation processes. For homogeneous nucleation, the subcooled liquid phase has to overcome a free energy barrier of ∼85 kcal/mol to form a critical nucleus of size ∼3.6 nm, which then grows into the monoclinic crystal phase. This free energy barrier becomes about 42% smaller (∼49 kcal/mol) when the subcooled liquid phase is in contact with a graphitic disk, and the critical nucleus formed is about 17% smaller (∼3.0 nm) than the one observed for homogeneous nucleation. The crystal formed in the heterogeneous nucleation scenario has a structure that is similar to that of the bulk crystal, with the exception of the layers of ions next to the graphene surface, which have larger local density and the cations lie with their imidazolium rings parallel to the graphitic surface. The critical nucleus forms near the graphene surface separated only by these layers of ions. The heterogeneous nucleation rate (∼4.8 × 1011 cm-3 s-1) is about one order of magnitude faster than the homogeneous rate (∼6.6 × 1010 cm-3 s-1). The computed free energy barriers and nucleation rates are in reasonable agreement with experimental and simulation values obtained for the homogeneous and heterogeneous nucleation of other systems (ice, urea, Lennard-Jones spheres, and oxide glasses).

A general method for molecular modeling of nucleation from the melt.

Crystallization is one of the fundamental phase transition processes, and it is also important practically, for example, in the chemical, food, and pharmaceutical industries. Despite its importance, however, our basic understanding of crystallization, and especially crystal nucleation, at the molecular level is still incomplete. In this work, we present a general molecular simulation approach that can be used to investigate the nucleation of crystals from a subcooled liquid. Our method combines a previously proposed general method to construct structure-based order parameters [E. E. Santiso and B. L. Trout, J. Chem. Phys. 134, 064109 (2011)] with the string method in collective variables [L. Maragliano et al., J. Chem. Phys. 125, 024106 (2006)] to obtain a minimum free energy path connecting the liquid and solid basins. We then use Markovian milestoning with Voronoi tessellations [E. Vanden-Eijnden and M. Venturoli, J. Chem. Phys. 130, 194101 (2009); L. Maragliano et al., J. Chem. Theory Comput. 5, 2589-2594 (2009)] to obtain the free energy profile along the path and the nucleation kinetics. We illustrate the application of this method to the nucleation of Benzene-I crystals from the melt, and compare the results to those previously found using transition path sampling [M. Shah et al., J. Phys. Chem. B 115, 10400-10412 (2011)].

Molecular simulation of homogeneous nucleation of crystals of an ionic liquid from the melt.

The homogeneous nucleation of crystals of the ionic liquid [dmim(+)][Cl(-)] from its supercooled liquid phase in the bulk (P = 1 bar, T = 340 K, representing a supercooling of 58 K) was studied using molecular simulations. The string method in collective variables [Maragliano et al., J. Chem. Phys. 125, 024106 (2006)] was used in combination with Markovian milestoning with Voronoi tessellations [Maragliano et al., J. Chem. Theory Comput. 5, 2589-2594 (2009)] and order parameters for molecular crystals [E. E. Santiso and B. L. Trout, J. Chem. Phys. 134, 064109 (2011)] to sketch a minimum free energy path connecting the supercooled liquid and the monoclinic crystal phases, and to determine the free energy and the rates involved in the homogeneous nucleation process. The physical significance of the configurations found along this minimum free energy path is discussed with the help of calculations based on classical nucleation theory and with additional simulation results obtained for a larger system. Our results indicate that, at a supercooling of 58 K, the liquid has to overcome a free energy barrier of the order of 60 kcal/mol and to form a critical nucleus with an average size of about 3.6 nm, before it reaches the thermodynamically stable crystal phase. A simulated homogeneous nucleation rate of 5.0 × 10(10) cm(-3) s(-1) was obtained for our system, which is in reasonable agreement with experimental and simulation rates for homogeneous nucleation of ice at similar degrees of supercooling. This study represents our first step in a series of studies aimed at understanding the nucleation and growth of crystals of organic salts near surfaces and inside nanopores.

Regulation of the Molecular Brake Region in FGFR2 Kinase

Unregulated activity of tyrosine kinases (TKS) is responsible for numerous developmental musculoskeletal diseases and cancers. The regulatory mechanisms of tks and how they are affected by point mutations are still in need of elucidation. Fibroblast growth factor receptor (FGFR) kinase is a TK whose regulatory element, the “molecular brake,” is thought to prevent constitutive activation of the receptor, despite not being near the kinase's active site. germline mutations in this “molecular brake” region in FGFR cause musculoskeletal disorders such as craniosynostosis and dwarfism, while somatic mutations can lead to uninhibited cell growth, causing an array of cancers. furthermore, it is thought that the majority of pathogenic activating mutations in protein kinases are in “molecular brake”-like regions. However, the mechanism through which the “molecular brake” of FGFR kinase is disengaged, allowing for signal transduction, is still unknown. using path-based molecular dynamics (MD) simulations, we have shown that the inward motion of the kinase's activation loop upon tyrosine phosphorylation is correlated with disengagement of the “molecular brake.” results of the string method in collective variables, which finds the minimum free energy path connecting the loop-out and loop-in states, indicate found that inward motion of loop occurs within 1 frame of “molecular brake” disengagement. Furthermore, umbrella sampling shows that in loop-out conformations, the disengaged state of the “molecular brake” is less stable, while in loop-in conformations, the disengaged state becomes more stable. Our results suggest that the “molecular brake” is allosterically controlled by the activation loop motion. An understanding of the “molecular brake” disengagement mechanism will provide avenues for targeted therapies to counteract pathologic over-activation of fgfr kinase leading to uncontrolled signal transduction and uninhibited cell growth.

Energetics and Mechanism of the Normal-to-Amyloidogenic Isomerization of β2-Microglobulin: On-the-Fly String Method Calculations

We use on-the-fly finite temperature string method in collective variables to study the transition from a normal to an amyloidogenic conformation of β2-microglobulin. We show that the protonation state of two histidine residues is of key importance and that under acidic (protonating) conditions, the transition to the amyloidgenic form is facilitated by both displacement of N-terminal residues to disrupt a hydrophobic pocket and by side-chain/side-chain electrostatic attraction, both of which facilitate a cis-trans prolyl isomerization. The free energy barriers for the normal-to-amyloidogenic isomerization are found to be 14.9 and 7.1 kcal/mol for the neutral and protonated cases, respectively, consistent with enhanced amyloidgenesis at low pH observed both in vitro and in hemodialysis-associated amyloidosis, and somewhat lower than experimentally determined barriers for bare prolyl cis-trans isomerization. We suggest specific mutagenesis experiments which could be used to further validate the mechanism observed.

Finding transition pathways using the string method with swarms of trajectories.

An approach to find transition pathways in complex systems is presented. The method, which is related to the string method in collective variables of Maragliano et al. (J. Chem. Phys. 2006, 125, 024106), is conceptually simple and straightforward to implement. It consists of refining a putative transition path in the multidimensional space supported by a set of collective variables using the average dynamic drift of those variables. This drift is estimated on-the-fly via swarms of short unbiased trajectories started at different points along the path. Successive iterations of this algorithm, which can be naturally distributed over many computer nodes with negligible interprocessor communication, refine an initial trial path toward the most probable transition path (MPTP) between two stable basins. The method is first tested by determining the pathway for the C7eq to C7ax transition in an all-atom model of the alanine dipeptide in vacuum, which has been studied previously with the string method in collective variables. A transition path is found with a committor distribution peaked at 1/2 near the free energy maximum, in accord with previous results. Last, the method is applied to the allosteric conformational change in the nitrogen regulatory protein C (NtrC), represented here with a two-state elastic network model. Even though more than 550 collective variables are used to describe the conformational change, the path converges rapidly. Again, the committor distribution is found to be peaked around 1/2 near the free energy maximum between the two stable states, confirming that a genuine transition state has been localized in this complex multidimensional system.

Solvent coarse-graining and the string method applied to the hydrophobic collapse of a hydrated chain.

With computer simulations of >100,000 atoms, the mechanism for the hydrophobic collapse of an idealized hydrated chain was obtained by tiling space with (0.2 nm)(3) cubes and projecting the atomistic water molecule positions onto this grid. With the coarse-grained field thus defined, the string method in collective variables was used to compute a minimum free-energy pathway (MFEP) for the collapsing chain. These calculations provide a proof of principle for a coarse-grained description of water solvent. Furthermore, the calculated MFEP characterizes the mechanism for the collapse of the hydrated chain by providing a path of maximum likelihood for dynamical trajectories. The reliability of the calculated MFEP was confirmed with the use of conventional molecular dynamics trajectories. Analysis of the MFEP provides atomistic confirmation for the mechanism of hydrophobic collapse proposed by ten Wolde and Chandler. In particular, we show that length-scale-dependent hydrophobic dewetting is the rate-limiting step in the hydrophobic collapse of the considered chain.

On-the-fly string method for minimum free energy paths calculation

An improved and simplified version of the string method in collective variables for computing minimum free energy paths is proposed. The string is discretized into a finite number of images and in the new method these images are evolved concurrently with replicas of the original system while keeping their inter-distances equal via reparametrization. There is no need to compute the mean force by time-averaging, nor to project the force perpendicularly to the string. In this Letter, the algorithmic aspects of the on-the-fly string method are presented in detail and the method is tested on the solvated alanine dipeptide molecule.

String method in collective variables: Minimum free energy paths and isocommittor surfaces

A computational technique is proposed which combines the string method with a sampling technique to determine minimum free energy paths. The technique only requires to compute the mean force and another conditional expectation locally along the string, and therefore can be applied even if the number of collective variables kept in the free energy calculation is large. This is in contrast with other free energy sampling techniques which aim at mapping the full free energy landscape and whose cost increases exponentially with the number of collective variables kept in the free energy. Provided that the number of collective variables is large enough, the new technique captures the mechanism of transition in that it allows to determine the committor function for the reaction and, in particular, the transition state region. The new technique is illustrated on the example of alanine dipeptide, in which we compute the minimum free energy path for the isomerization transition using either two or four dihedral angles as collective variables. It is shown that the mechanism of transition can be captured using the four dihedral angles, but it cannot be captured using only two of them.