Multiple pathways in conformational transitions of the alanine dipeptide: An application of dynamic importance sampling

Analyzing Molecular Dynamics Trajectories Thermodynamically through Artificial Intelligence

Molecular dynamics simulations produce trajectories that correspond to vast amounts of structure when exploring biochemical processes. Extracting valuable information, e.g., important intermediate states and collective variables (CVs) that describe the major movement modes, from molecular trajectories to understand the underlying mechanisms of biological processes presents a significant challenge. To achieve this goal, we introduce a deep learning approach, coined DIKI (deep identification of key intermediates), to determine low-dimensional CVs distinguishing key intermediate conformations without a-priori assumptions. DIKI dynamically plans the distribution of latent space and groups together similar conformations within the same cluster. Moreover, by incorporating two user-defined parameters, namely, coarse focus knob and fine focus knob, to help identify conformations with low free energy and differentiate the subtle distinctions among these conformations, resolution-tunable clustering was achieved. Furthermore, the integration of DIKI with a path-finding algorithm contributes to the identification of crucial intermediates along the lowest free-energy pathway. We postulate that DIKI is a robust and flexible tool that can find widespread applications in the analysis of complex biochemical processes.

Accelerated Simulations of Molecular Systems through Learning of Effective Dynamics

Simulations are vital for understanding and predicting the evolution of complex molecular systems. However, despite advances in algorithms and special purpose hardware, accessing the time scales necessary to capture the structural evolution of biomolecules remains a daunting task. In this work, we present a novel framework to advance simulation time scales by up to 3 orders of magnitude by learning the effective dynamics (LED) of molecular systems. LED augments the equation-free methodology by employing a probabilistic mapping between coarse and fine scales using mixture density network (MDN) autoencoders and evolves the non-Markovian latent dynamics using long short-term memory MDNs. We demonstrate the effectiveness of LED in the Müller-Brown potential, the Trp cage protein, and the alanine dipeptide. LED identifies explainable reduced-order representations, i.e., collective variables, and can generate, at any instant, all-atom molecular trajectories consistent with the collective variables. We believe that the proposed framework provides a dramatic increase to simulation capabilities and opens new horizons for the effective modeling of complex molecular systems.

Dynamics Sampling in Transition Pathway Space

The minimum energy pathway contains important information describing the transition between two states on a potential energy surface (PES). Chain-of-states methods were developed to efficiently calculate minimum energy pathways connecting two stable states. In the chain-of-states framework, a series of structures are generated and optimized to represent the minimum energy pathway connecting two states. However, multiple pathways may exist connecting two existing states and should be identified to obtain a full view of the transitions. Therefore, we developed an enhanced sampling method, named as the direct pathway dynamics sampling (DPDS) method, to facilitate exploration of a PES for multiple pathways connecting two stable states as well as addition minima and their associated transition pathways. In the DPDS method, molecular dynamics simulations are carried out on the targeting PES within a chain-of-states framework to directly sample the transition pathway space. The simulations of DPDS could be regulated by two parameters controlling distance among states along the pathway and smoothness of the pathway. One advantage of the chain-of-states framework is that no specific reaction coordinates are necessary to generate the reaction pathway, because such information is implicitly represented by the structures along the pathway. The chain-of-states setup in a DPDS method greatly enhances the sufficient sampling in high-energy space between two end states, such as transition states. By removing the constraint on the end states of the pathway, DPDS will also sample pathways connecting minima on a PES in addition to the end points of the starting pathway. This feature makes DPDS an ideal method to directly explore transition pathway space. Three examples demonstrate the efficiency of DPDS methods in sampling the high-energy area important for reactions on the PES.

Recovering hidden dynamical modes from the generalized Langevin equation

In studying large molecular systems, insights can better be extracted by selecting a limited number of physical quantities for analysis rather than treating every atomic coordinate in detail. Some information may, however, be lost by projecting the total system onto a small number of coordinates. For such problems, the generalized Langevin equation (GLE) is shown to provide a useful framework to examine the interaction between the observed variables and their environment. Starting with the GLE obtained from the time series of the observed quantity, we perform a transformation to introduce a set of variables that describe dynamical modes existing in the environment. The introduced variables are shown to effectively recover the essential information of the total system that appeared to be lost by the projection.

Computing ensembles of transitions with molecular dynamics simulations

A molecular understanding of conformational change is important for connecting structure and function. Without the ability to sample on the meaningful large-scale conformational changes, the ability to infer biological function and to understand the effect of mutations and changes in environment is not possible. Our Dynamic Importance Sampling method (DIMS), part of the CHARMM simulation package, is a method that enables sampling over ensembles of transition intermediates. This chapter outlines the context for the method and the usage within the program.

Recovering position-dependent diffusion from biased molecular dynamics simulations

All atom molecular dynamics (MD) models provide valuable insight into the dynamics of biophysical systems, but are limited in size or length by the high computational demands. The latter can be reduced by simulating long term diffusive dynamics (also known as Langevin dynamics or Brownian motion) of the most interesting and important user-defined parts of the studied system, termed collective variables (colvars). A few hundred nanosecond-long biased MD trajectory can therefore be extended to millisecond lengths in the colvars subspace at a very small additional computational cost. In this work, we develop a method for determining multidimensional anisotropic position- and timescale-dependent diffusion coefficients (D) by analysing the changes of colvars in an existing MD trajectory. As a test case, we obtained D for dihedral angles of the alanine dipeptide. An open source Mathematica(®) package, capable of determining and visualizing D in one or two dimensions, is available at https://github.com/lbf-ijs/DiffusiveDynamics. Given known free energy and D, the package can also generate diffusive trajectories.

Computational Recipe for Efficient Description of Large-Scale Conformational Changes in Biomolecular Systems

Characterizing large-scale structural transitions in biomolecular systems poses major technical challenges to both experimental and computational approaches. On the computational side, efficient sampling of the configuration space along the transition pathway remains the most daunting challenge. Recognizing this issue, we introduce a knowledge-based computational approach toward describing large-scale conformational transitions using (i) nonequilibrium, driven simulations combined with work measurements and (ii) free energy calculations using empirically optimized biasing protocols. The first part is based on designing mechanistically relevant, system-specific reaction coordinates whose usefulness and applicability in inducing the transition of interest are examined using knowledge-based, qualitative assessments along with nonequilirbrium work measurements which provide an empirical framework for optimizing the biasing protocol. The second part employs the optimized biasing protocol resulting from the first part to initiate free energy calculations and characterize the transition quantitatively. Using a biasing protocol fine-tuned to a particular transition not only improves the accuracy of the resulting free energies but also speeds up the convergence. The efficiency of the sampling will be assessed by employing dimensionality reduction techniques to help detect possible flaws and provide potential improvements in the design of the biasing protocol. Structural transition of a membrane transporter will be used as an example to illustrate the workings of the proposed approach.

Microscopically computing free-energy profiles and transition path time of rare macromolecular transitions

We introduce a rigorous method to microscopically compute the observables which characterize the thermodynamics and kinetics of rare macromolecular transitions for which it is possible to identify a priori a slow reaction coordinate. In order to sample the ensemble of statistically significant reaction pathways, we define a biased molecular dynamics (MD) in which barrier-crossing transitions are accelerated without introducing any unphysical external force. In contrast to other biased MD methods, in the present approach the systematic errors which are generated in order to accelerate the transition can be analytically calculated and therefore can be corrected for. This allows for a computationally efficient reconstruction of the free-energy profile as a function of the reaction coordinate and for the calculation of the corresponding diffusion coefficient. The transition path time can then be readily evaluated within the dominant reaction pathways approach. We illustrate and test this method by characterizing a thermally activated transition on a two-dimensional energy surface and the folding of a small protein fragment within a coarse-grained model.

Coarse-grained simulations of transitions in the E2-to-E1 conformations for Ca ATPase (SERCA) show entropy-enthalpy compensation

SERCA is a membrane transport protein that has been extensively studied. There are a large number of highly resolved X-ray structures and several hundred mutations that have been characterized functionally. Despite this, the molecular details of the catalytic cycle, a cycle that includes large conformational changes, is not fully understood. In this computational study, we provide molecular dynamics descriptions of conformational changes during the E2→E1 transitions. The motivating point for these calculations was a series of insertion mutants in the A-M3 linker region that led to significant shifts in measured rates between the E2 and E1 states, as shown by experimental characterization. Using coarse-grained dynamic importance sampling within the context of a population shift framework, we sample on the intermediates along the transition pathway to address the mechanism for the conformational changes and the effects of the insertion mutations on the kinetics of the transition. The calculations define an approximation for the relative changes in entropy and enthalpy along the transition. These are found to be important for understanding the experimentally observed differences in rates. In particular, the interactions between cytoplasmic domains, water interactions, and the shifts in protein degrees of freedom with the insertion mutations show mutual compensation for the E2→E1 transitions in wild-type and mutant systems.

The role of domain: domain interactions versus domain: water interactions in the coarse-grained simulations of the E1P to E2P transitions in Ca-ATPase (SERCA)

SERCA is an important model system for understanding the molecular details of conformational change in membrane transport systems. This reflects the large number of solved X-ray structures and the equally large database of mutations that have been assayed. In this computational study, we provide a molecular dynamics description of the conformational changes during the E1P → E2P transitions. This set of states further changes with insertion mutants in the A-M3 linker region. These mutants were experimentally shown to lead to significant shifts in rates between the E1P → E2P states. Using the population shift framework and dynamic importance sampling method along with coarse-grained representations of the protein, lipid, and water, we suggest why these changes are found. The calculations sample on intermediates and suggest that changes in interactions, individual helix interactions, and water behavior are key elements in the molecular compositions that underlie shifts in kinetics. In particular, as the insertion length grows, it attracts more water and disrupts domain interactions, creating changes as well at the sites of key helix interactions between the A-Domain and the P-Domain. This provides a conceptual picture that aids understanding of the experimental results.

Towards the prediction of order parameters from molecular dynamics simulations in proteins

A molecular understanding of how protein function is related to protein structure requires an ability to understand large conformational changes between multiple states. Unfortunately these states are often separated by high free energy barriers and within a complex energy landscape. This makes it very difficult to reliably connect, for example by all-atom molecular dynamics calculations, the states, their energies, and the pathways between them. A major issue needed to improve sampling on the intermediate states is an order parameter--a reduced descriptor for the major subset of degrees of freedom--that can be used to aid sampling for the large conformational change. We present a method to combine information from molecular dynamics using non-linear time series and dimensionality reduction, in order to quantitatively determine an order parameter connecting two large-scale conformationally distinct protein states. This new method suggests an implementation for molecular dynamics calculations that may be used to enhance sampling of intermediate states.

Fluctuations in the ensemble of reaction pathways

The dominant reaction pathway is a rigorous framework to microscopically compute the most probable trajectories, in nonequilibrium transitions. In the low-temperature regime, such dominant pathways encode the information about the reaction mechanism and can be used to estimate nonequilibrium averages of arbitrary observables. On the other hand, at sufficiently high temperatures, the stochastic fluctuations around the dominant paths become important and have to be taken into account. In this work, we develop a technique to systematically include the effects of such stochastic fluctuations, to order k(B)T. This method is used to compute the probability for a transition to take place through a specific reaction channel and to evaluate the reaction rate.

Beyond microscopic reversibility: Are observable non-equilibrium processes precisely reversible?

Although the principle of microscopic reversibility has been studied for many decades, there remain ambiguities in its application to non-equilibrium processes of importance to chemistry, physics and biology. Examples include whether protein unfolding should follow the same pathways and in the same proportions as folding, and whether unbinding should likewise mirror binding. Using continuum-space calculations which extend previous kinetic analyses, we demonstrate formally that the precise symmetry of forward and reverse processes is expected only under certain special conditions. Approximate symmetry will be exhibited under a separate set of conditions. Exact, approximate, and broken symmetry scenarios are verified in several ways: using numerical calculations on toy and molecular systems; using exact calculations on kinetic models of induced fit in protein-ligand binding; and based on reported experimental results. The analysis highlights intrinsic challenges and ambiguities in the design and analysis of both experiments and simulations.

Cy3-DNA stacking interactions strongly depend on the identity of the terminal basepair

We characterized the effect of the first basepair on the conformational dynamics of the fluorescent dye Cy3 attached to the 5' end of double-stranded DNA using gaussian-mixture adaptive umbrella sampling simulations. In the simulations, the sampling of all five dihedral angles along the linker was enhanced, so that both stacked and unstacked states were sampled. The affinity of Cy3 for a T·A basepair (with the dye attached to T) was found to be significantly less than for the other basepairs. This was verified experimentally by measuring the activation energies for cis-trans isomerization of the dye. The simulation and experimental results indicate the existence of partially unstacked conformations amenable to photoisomerization. The simulations also showed that stacking of Cy3 straightens the DNA while stabilizing the first basepair. Our findings indicate that fluorescence is modulated by Cy3-DNA interactions in a sequence-dependent manner.

Computing ensembles of transitions from stable states: Dynamic importance sampling

There is an increasing dataset of solved biomolecular structures in more than one conformation and increasing evidence that large-scale conformational change is critical for biomolecular function. In this article, we present our implementation of a dynamic importance sampling (DIMS) algorithm that is directed toward improving our understanding of important intermediate states between experimentally defined starting and ending points. This complements traditional molecular dynamics methods where most of the sampling time is spent in the stable free energy wells defined by these initial and final points. As such, the algorithm creates a candidate set of transitions that provide insights for the much slower and probably most important, functionally relevant degrees of freedom. The method is implemented in the program CHARMM and is tested on six systems of growing size and complexity. These systems, the folding of Protein A and of Protein G, the conformational changes in the calcium sensor S100A6, the glucose-galactose-binding protein, maltodextrin, and lactoferrin, are also compared against other approaches that have been suggested in the literature. The results suggest good sampling on a diverse set of intermediates for all six systems with an ability to control the bias and thus to sample distributions of trajectories for the analysis of intermediate states.

β-Barrel topology of Alzheimer's β-amyloid ion channels

Emerging evidence supports the ion channel mechanism for Alzheimer's disease pathophysiology wherein small β-amyloid (Aβ) oligomers insert into the cell membrane, forming toxic ion channels and destabilizing the cellular ionic homeostasis. Solid-state NMR-based data of amyloid oligomers in solution indicate that they consist of a double-layered β-sheets where each monomer folds into β-strand-turn-β-strand and the monomers are stacked atop each other. In the membrane, Aβ peptides are proposed to be β-type structures. Experimental structural data available from atomic force microscopy (AFM) imaging of Aβ oligomers in membranes reveal heterogeneous channel morphologies. Previously, we modeled the channels in a non-tilted organization, parallel with the cross-membrane normal. Here, we modeled a β-barrel-like organization. β-Barrels are common in transmembrane toxin pores, typically consisting of a monomeric chain forming a pore, organized in a single-layered β-sheet with antiparallel β-strands and a right-handed twist. Our explicit solvent molecular dynamics simulations of a range of channel sizes and polymorphic turns and comparisons of these with AFM image dimensions support a β-barrel channel organization. Different from the transmembrane β-barrels where the monomers are folded into a circular β-sheet with antiparallel β-strands stabilized by the connecting loops, these Aβ barrels consist of multimeric chains forming double β-sheets with parallel β-strands, where the strands of each monomer are connected by a turn. Although the Aβ barrels adopt the right-handed β-sheet twist, the barrels still break into heterogeneous, loosely attached subunits, in good agreement with AFM images and previous modeling. The subunits appear mobile, allowing unregulated, hence toxic, ion flux.

Steady-state simulations using weighted ensemble path sampling

We extend the weighted ensemble (WE) path sampling method to perform rigorous statistical sampling for systems at steady state. A straightforward steady-state implementation of WE is directly practical for simple landscapes, but not when significant metastable intermediates states are present. We therefore develop an enhanced WE scheme, building on existing ideas, which accelerates attainment of steady state in complex systems. We apply both WE approaches to several model systems, confirming their correctness and efficiency by comparison with brute-force results. The enhanced version is significantly faster than the brute force and straightforward WE for systems with WE bins that accurately reflect the reaction coordinate(s). The new WE methods can also be applied to equilibrium sampling, since equilibrium is a steady state.

Exploring Multidimensional Free Energy Landscapes Using Time-Dependent Biases on Collective Variables

A new implementation of the adaptive biasing force (ABF) method is described. This implementation supports a wide range of collective variables and can be applied to the computation of multidimensional energy profiles. It is provided to the community as part of a code that implements several analogous methods, including metadynamics. ABF and metadynamics have not previously been tested side by side on identical systems. Here, numerical tests are carried out on processes including conformational changes in model peptides and translocation of a halide ion across a lipid membrane through a peptide nanotube. On the basis of these examples, we discuss similarities and differences between the ABF and metadynamics schemes. Both approaches provide enhanced sampling and free energy profiles in quantitative agreement with each other in different applications. The method of choice depends on the dimension of the reaction coordinate space, the height of the barriers, and the relaxation times of degrees of freedom in the orthogonal space, which are not explicitly described by the chosen collective variables.

Zipping and unzipping of adenylate kinase: atomistic insights into the ensemble of open<-->closed transitions

Adenylate kinase (AdK), a phosphotransferase enzyme, plays an important role in cellular energy homeostasis. It undergoes a large conformational change between an open and a closed state, even in the absence of substrate. We investigate the apo-AdK transition at the atomic level both with free-energy calculations and with our new dynamic importance sampling (DIMS) molecular dynamics method. DIMS is shown to sample biologically relevant conformations as verified by comparing an ensemble of hundreds of DIMS transitions to AdK crystal structure intermediates. The simulations reveal in atomic detail how hinge regions partially and intermittently unfold during the transition. Conserved salt bridges are seen to have important structural and dynamic roles; in particular, four ionic bonds that open in a sequential, zipper-like fashion and, thus, dominate the free-energy landscape of the transition are identified. Transitions between the closed and open conformations only have to overcome moderate free-energy barriers. Unexpectedly, the closed state and the open state encompass broad free-energy basins that contain conformations differing in domain hinge motions by up to 40 degrees . The significance of these extended states is discussed in relation to recent experimental Förster resonance energy transfer measurements. Taken together, these results demonstrate how a small number of cooperative key interactions can shape the overall dynamics of an enzyme and suggest an "all-or-nothing" mechanism for the opening and closing of AdK. Our efficient DIMS molecular dynamics computer simulation approach can provide a detailed picture of a functionally important macromolecular transition and thus help to interpret and suggest experiments to probe the conformational landscape of dynamic proteins such as AdK.

How to lose a kink and gain a helix: pH independent conformational changes of the fusion domains from influenza hemagglutinin in heterogeneous lipid bilayers

We have simulated two conformations of the fusion domain of influenza hemagglutinin (HA) within explicit water, salt, and heterogeneous lipid bilayers composed of POPC:POPG (4:1). Each conformation has seven different starting points in which the initial peptide structure is the same for each conformation, but the location across the membrane normal and lipid arrangement around the peptide are varied, giving a combined total simulation time of 140 ns. For the HA5 conformation (primary structure from recent NMR spectroscopy at pH = 5), the peptide exhibits a stable and less kinked structure in the lipid bilayer compared to that from the NMR studies. The relative fusogenic behavior of the different conformations has been investigated by calculation of the relative free energy of insertion into the hydrophobic region of lipid bilayer as a function of the depth of immersion. For the HA7 conformations (primary structure from recent NMR spectroscopy at pH = 7.4), while the N-terminal helix preserves its initial structure, the flexible C-terminal chain produces a transient helical motif inside the lipid bilayer. This conformational change is pH-independent, and is closely related to the peptide insertion into the lipid bilayer.

Improved sampling methods for molecular simulation

Molecular simulation has broad application in the biological sciences. One of the greatest challenges in molecular simulation is the limited conformational sampling due to slow barrier crossing on the rugged energy landscape of complex biomolecules and to the relatively short simulation time. Many enhanced sampling techniques have been developed over the years to alleviate this problem. Significant progress has been made in the past couple of years, with emerging methods targeting specific aspects of the potential energy surface and new variants of the replica exchange method.