Ultrafast electron diffraction: dynamical structures on complex energy landscapes.

In this contribution, we report studies in ultrafast electron diffraction (UED), with the aim of exploring new directions. The main focus is on the determination of complex structures and their dynamics with spatial and temporal resolutions sufficient to give an atomic-scale picture for the evolution in chemical or biological change. We also provide the theoretical framework for UED, and compare the experimental findings of UED to those predicted by density functional and charge density calculations. Selected applications are given in order to highlight phenomena related to concepts such as bifurcation of trajectories in dynamics, far-from-equilibrium coherent structures, and conformational robustness in biological structures. For the former two cases, we consider chemical systems, and, for the latter, we examine proteins of 200 atoms (angiotensin I) or more.     

Potential energy and free energy landscapes

Familiar concepts for small molecules may require reinterpretation for larger systems. For example, rearrangements between geometrical isomers are usually considered in terms of transitions between the corresponding local minima on the underlying potential energy surface, V. However, transitions between bulk phases such as solid and liquid, or between the denatured and native states of a protein, are normally addressed in terms of free energy minima. To reestablish a connection with the potential energy surface we must think in terms of representative samples of local minima of V, from which a free energy surface is projected by averaging over most of the coordinates. The present contribution outlines how this connection can be developed into a tool for quantitative calculations. In particular, stepping between the local minima of V provides powerful methods for locating the global potential energy minimum, and for calculating global thermodynamic properties. When the transition states that link local minima are also sampled we can exploit statistical rate theory to obtain insight into global dynamics and rare events. Visualizing the potential energy landscape helps to explain how the network of local minima and transition states determines properties such as heat capacity features, which signify transitions between free energy minima. The organization of the landscape also reveals how certain systems can reliably locate particular structures on the experimental time scale from among an exponentially large number of local minima. Such directed searches not only enable proteins to overcome Levinthal's paradox but may also underlie the formation of "magic numbers" in molecular beams, the self-assembly of macromolecular structures, and crystallization.     

Conformational ensembles and sampled energy landscapes: Analysis and comparison

We present novel algorithms and software addressing four core problems in computational structural biology, namely analyzing a conformational ensemble, comparing two conformational ensembles, analyzing a sampled energy landscape, and comparing two sampled energy landscapes. Using recent developments in computational topology, graph theory, and combinatorial optimization, we make two notable contributions. First, we present a generic algorithm analyzing height fields. We then use this algorithm to perform density‐based clustering of conformations, and to analyze a sampled energy landscape in terms of basins and transitions between them. In both cases, topological persistence is used to manage (geometric) frustration. Second, we introduce two algorithms to compare transition graphs. The first is the classical earth mover distance metric which depends only on local minimum energy configurations along with their statistical weights, while the second incorporates topological constraints inherent to conformational transitions. Illustrations are provided on a simplified protein model (BLN69), whose frustrated potential energy landscape has been thoroughly studied. The software implementing our tools is also made available, and should prove valuable wherever conformational ensembles and energy landscapes are used. © 2015 Wiley Periodicals, Inc.     

Exploring protein energy landscapes with hierarchical clustering

In this work we present a new method for investigating local energy minima on a protein energy landscape. The CABS (CAlpha, CBeta and the center of mass of the Side chain) method was employed for generating protein models, but any other method could be used instead. Cα traces from an ensemble of models are hierarchical clustered with the HCPM (Hierarchical Clustering of Protein Models) method. The efficiency of this method for sampling and analyzing energy landscapes is shown.     

Intrinsic energy landscapes of amino acid side-chains

Amino acid side-chain conformational properties influence the overall structural and dynamic properties of proteins and, therefore, their biological functions. In this study, quantum mechanical (QM) potential energy surfaces for the rotation of side-chain χ(1) and χ(2) torsions in dipeptides in the alphaR, beta, and alphaL backbone conformations were calculated. The QM energy surfaces provide a broad view of the intrinsic conformational properties of each amino acid side-chain. The extent to which intrinsic energetics dictates side-chain orientation was studied through comparisons of the QM energy surfaces with χ(1) and χ(2) free energy surfaces from probability distributions obtained from a survey of high resolution crystal structures. In general, the survey probability maxima are centered in minima of the QM surfaces as expected for sp(3) (or sp(2) for χ(2) of Asn, Phe, Trp, and Tyr) atom centers with strong variations between amino acids occurring in the energies of the minima indicating intrinsic differences in rotamer preferences. High correlations between the QM and survey data were found for hydrophobic side-chains except Met, suggesting minimal influence of the protein and solution environments on their conformational distributions. Conversely, low correlations for polar or charged side-chains indicate a dominant role of the environment in stabilizing conformations that are not intrinsically favored. Data also link the presence of off-rotamers in His and Trp to favorable interactions with the backbone. Results also suggest that the intrinsic energetics of the side-chains of Phe and Tyr may play important roles in protein folding and stability. Analyses on whether intrinsic side-chain energetics can influence backbone preference identified a strong correlation for residues in the alphaL backbone conformation. It is suggested that this correlation reflects the intrinsic instability of the alphaL backbone such that assumption of this backbone conformation is facilitated by intrinsically favorable side-chain conformations. Together our results offer a broad overview of the conformational properties of amino acid side-chains and the QM data may be used as target data for force field optimization.     

Analyzing energy landscapes for folding model proteins

A new benchmark 20-bead HP model protein sequence (on a square lattice), which has 17 distinct but degenerate global minimum (GM) energy structures, has been studied using a genetic algorithm (GA). The relative probabilities of finding particular GM conformations are determined and related to the theoretical probability of generating these structures using a recoil growth constructor operator. It is found that for longer successful GA runs, the GM probability distribution is generally very different from the constructor probability, as other GA operators have had time to overcome any initial bias in the originally generated population of structures. Structural and metric relationships (e.g., Hamming distances) between the 17 distinct GM are investigated and used, in conjunction with data on the connectivities of the GM and the pathways that link them, to explain the GM probability distributions obtained by the GA. A comparison is made of searches where the sequence is defined in the normal (forward) and reverse directions. The ease of finding mirror image solutions are also compared. Finally, this approach is applied to rationalize the ease or difficulty of finding the GM for a number of standard benchmark HP sequences on the square lattice. It is shown that the relative probabilities of finding particular members of a set of degenerate global minima depend critically on the topography of the energy landscape in the vicinity of the GM, the connections and distances between the GM, and the nature of the operators used in the chosen search method.     

Conformational energy landscapes of liquid crystal molecules

The conformational energy landscape of the prototypical nematic liquid crystal 4-pentyl-4cyanobiphenyl (5CB) is studied using first principles computer modelling. It is found that the most favourable conformation occurs when the two constituent phenyl rings are inclined at an angle of 31 with respect to each other. Also, the orientation of the alkyl chain is found to have an important influence on the ring-ring torsional potential. We fit the energy surface of these coupled torsions to yield an accurate intramolecular potential for use in empirical modelling. To test the strength of the coupling between the alkyl tail and the phenyl rings and the cyano group, we also calculate potentials for the relative orientation of the phenyl rings in biphenyl and cyanobiphenyl (0CB). Our calculations are performed using density functional theory using pseudo-potentials and the generalized gradient approximation to exchange and correlation. The molecular electronic wavefunction is expanded in terms of a plane wave basis set. We compare our results with recent NMR and Gaussian-based quantum chemistry calculations where available.     

Visualizing energy landscapes with metric disconnectivity graphs

The visualization of multidimensional energy landscapes is important, providing insight into the kinetics and thermodynamics of a system, as well the range of structures a system can adopt. It is, however, highly nontrivial, with the number of dimensions required for a faithful reproduction of the landscape far higher than can be represented in two or three dimensions. Metric disconnectivity graphs provide a possible solution, incorporating the landscape connectivity information present in disconnectivity graphs with structural information in the form of a metric. In this study, we present a new software package, PyConnect, which is capable of producing both disconnectivity graphs and metric disconnectivity graphs in two or three dimensions. We present as a test case the analysis of the 69‐bead BLN coarse‐grained model protein and show that, by choosing appropriate order parameters, metric disconnectivity graphs can resolve correlations between structural features on the energy landscape with the landscapes energetic and kinetic properties. © 2014 The Authors Journal of Computational Chemistry Published by Wiley Periodicals, Inc.     

Calculation of free energy landscapes: A histogram reweighted metadynamics approach

We present an efficient method for the calculation of free energy landscapes. Our approach involves a history‐dependent bias potential, which is evaluated on a grid. The corresponding free energy landscape is constructed via a histogram reweighting procedure a posteriori. Because of the presence of the bias potential, it can be also used to accelerate rare events. In addition, the calculated free energy landscape is not restricted to the actual choice of collective variables and can in principle be extended to auxiliary variables of interest without further numerical effort. The applicability is shown for several examples. We present numerical results for the alanine dipeptide and the Met‐Enkephalin in explicit solution to illustrate our approach. Furthermore, we derive an empirical formula that allows the prediction of the computational cost for the ordinary metadynamics variant in comparison with our approach, which is validated by a dimensionless representation. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011     

Kinetics of model energy landscapes: an approach to complex systems

An idealized potential energy surface (PES), simply a PES-like network of stationary points, is demonstrated to be a useful tool to study kinetic relaxation of complex energy landscapes. Combined with a master equation, we show that if constructed with proper regularity, the kinetics of the PES is easy to predict and understand by carefully examining the eigenmodes of the master equation. By modifying the idealized PES model to make it more and more complicated, we demonstrate a systematic method to study the complex kinetics on large PES. The idealized PES model is used to explore the feasibility and the robustness of statistical sampling of large PES. We develop several sampling strategies, such as the "rough topography sampling" and the "low barrier saddle sampling" in the idealized PES model and find they are also applicable to a realistic PES of the 13-atom Morse cluster with range parameter rho= 6. To measure the robustness of the sampling methods, we compare the eigenvalue spectra, the eigenvector similarity and the relaxation times of the total energy of the full and sample PESs.     

Randomized tree construction algorithm to explore energy landscapes

In this work, a new method for exploring conformational energy landscapes is described. The method, called transition-rapidly exploring random tree (T-RRT), combines ideas from statistical physics and robot path planning algorithms. A search tree is constructed on the conformational space starting from a given state. The tree expansion is driven by a double strategy: on the one hand, it is naturally biased toward yet unexplored regions of the space; on the other, a Monte Carlo-like transition test guides the expansion toward energetically favorable regions. The balance between these two strategies is automatically achieved due to a self-tuning mechanism. The method is able to efficiently find both energy minima and transition paths between them. As a proof of concept, the method is applied to two academic benchmarks and the alanine dipeptide.     

Imaging energy landscapes with concentrated diffusing colloidal probes

The ability to locally interrogate interactions between particles and energetically patterned surfaces provides essential information to design, control, and optimize template directed self-assembly processes. Although numerous techniques are capable of characterizing local physicochemical surface properties, no current method resolves interactions between colloids and patterned surfaces on the order of the thermal energy kT, which is the inherent energy scale of equilibrium self-assembly processes. Here, the authors describe video microscopy measurements and an inverse Monte Carlo analysis of diffusing colloidal probes as a means to image three dimensional free energy and potential energy landscapes due to physically patterned surfaces. In addition, they also develop a consistent analysis of self-diffusion in inhomogeneous fluids of concentrated diffusing probes on energy landscapes, which is important to the temporal imaging process and to self-assembly kinetics. Extension of the concepts developed in this work suggests a general strategy to image multidimensional and multiscale physical, chemical, and biological surfaces using a variety of diffusing probes (i.e., molecules, macromolecules, nanoparticles, and colloids).     

Global optimization and the energy landscapes of Dzugutov clusters

The global minima of clusters bound by a Dzugutov potential form non-compact polytetrahedral clusters mainly composed of interpenetrating and face-sharing 13-atom icosahedra. As the size increases, these icosahedral units first form linear arrays, then two-dimensional rings, then three-dimensional networks. Characterization of the energy landscapes of these clusters shows that they are particularly rough and generally exhibit a multiple-funnel topography. These results provide new insights into the structure and dynamics of bulk supercooled Dzugutov liquids and the form of the bulk phase diagram.     

Mapping patterned potential energy landscapes with diffusing colloidal probes

We report a new method for mapping patterned surfaces based on monitoring the interactions of freely diffusing colloidal probes with pattern features to generate measured potential energy landscapes. Evanescent wave scattering and video microscopy are used to track 3D center positions of nominal 2 microm silica colloids as they diffuse over 5-20-nm-thick patterned gold films. An analysis of ensemble-averaged particle height histograms on different pattern features using Boltzmann's equation produces local electrostatic and van der Waals potentials in good agreement with independent measurements and predictions. Absolute separation is obtained from theoretical fits to measured potential-energy profiles and direct measurement by depositing silica colloids onto gold surfaces via electrophoretic deposition. As colloidal probe and pattern feature dimensions become comparable, potential energy profiles suffer some distortion due to the increased probability of probes sampling pattern feature edges. An analysis of interfacial colloidal probe diffusion in conjunction with potential energy measurements demonstrates a consistent interpretation of dissipative and conservative forces in these measurements. Future extensions of this work should produce similar approaches for interrogating physical, chemical, and biomolecular heterogeneous/patterned surfaces and structures with diffusing colloidal probes.     

Statistical theory for protein ensembles with designed energy landscapes

Combinatorial protein libraries provide a promising route to investigate the determinants and features of protein folding and to identify novel folding amino acid sequences. A library of sequences based on a pool of different monomer types are screened for folding molecules, consistent with a particular foldability criterion. The number of sequences grows exponentially with the length of the polymer, making both experimental and computational tabulations of sequences infeasible. Herein a statistical theory is extended to specify the properties of sequences having particular values of global energetic quantities that specify their energy landscape. The theory yields the site-specific monomer probabilities. A foldability criterion is derived that characterizes the properties of sequences by quantifying the energetic separation of the target state from low-energy states in the unfolded ensemble and the fluctuations of the energies in the unfolded state ensemble. For a simple lattice model of proteins, excellent agreement is observed between the theory and the results of exact enumeration. The theory may be used to provide a quantitative framework for the design and interpretation of combinatorial experiments.     

Smart resolution replica exchange: an efficient algorithm for exploring complex energy landscapes

A coarse-grained representation of a condensed phase system can significantly reduce the number of system degrees of freedom, making coarse-grained simulations very computationally efficient. Moreover, coarse graining can smoothen the free energy landscape of the system. Thus coarse-grained dynamics is usually faster than its fully atomistic counterpart. In this work, the smart resolution replica exchange method is introduced that incorporates the information from coarse-grained simulations into atomistic simulations in order to accelerate the sampling of rough, complex atomistic energy landscapes. Within this methodology, interactions between particles are defined by a potential energy that interpolates between a fully atomistic potential and a fully coarse-grained effective potential according to a parameter lambda. Instead of exchanging the configurations from neighboring resolutions directly, as has been done in the resolution replica exchange methods [E. Lyman et al., Phys. Rev. Lett. 96, 028105 (2006); M. Christen and W. F. v. Gunsteren, J. Chem. Phys. 124, 154106 (2006)], the configuration described at the coarser resolution is first relaxed before an exchange is attempted, similar to the smart walking method [R. Zhou and B. J. Berne, J. Chem. Phys. 107, 9185 (1997)]. This approach greatly increases the acceptance ratio of exchange and only two replicas, one at the atomistic level and one at the coarse-grained level, are usually required (although more can be implemented if desired). This new method can approximately obtain the correct canonical sampling if the exchange interval is sufficiently large to allow the system to explore the local energy landscape. The method is demonstrated for a two-dimensional model system, where the ideal population distribution can be recovered, and also for an alanine polypeptide (Ala(15)) model with explicit water, where its native structure, an alpha helix, is obtained from the extended structure within 1 ns.     

Comparing search strategies for finding global optima on energy landscapes

We provide some tests of the convex global underestimator (CGU) algorithm, which aims to find global minima on funnel-shaped energy landscapes. We use two different potential functions—the reduced Lennard–Jones cluster potential, and the modified Sun protein folding potential, to compare the CGU algorithm with the simplest versions of the traditional trajectory-based search methods, simulated annealing (SA), and Monte Carlo (MC). For both potentials, the CGU reaches energies lower on the landscapes than both SA and MC, even when SA and MC are given the same number of starting points as in a full CGU run or when all methods are given the same amount of computer time. The CGU consistently finds the global minima of the Lennard–Jones potential for all cases with up to at least n=30 degrees of freedom. Finding the global or near-global minimum in the CGU method requires polynomial time [scaling between O(n³) and O(n⁴)], on average. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 1527–1532, 1999     

Manipulating energy landscapes to tune ordering in biotemplated nanoparticle arrays

Two-dimensional non-close-packed crystals of the protein streptavidin, grown on phospholipid membranes, can serve as nanoscale templates capable of directing the formation of ordered nanoparticle arrays through site-specific electrostatic adsorption. Here we examine the effects of both interparticle and nanoparticle/lipid membrane electrostatic interactions on the degree of structural order exhibited by the templated nanoparticle array. Interparticle electrostatic repulsion is shown to have only marginal influence on nanoparticle ordering. In contrast, the degree of order exhibited by the templated array can be tuned by controlling the charge on the lipid membrane. Analysis of the local and global structure of arrays generated with negatively charged gold nanoparticles (～6 nm) indicate improved long-range order when the lipid membrane supporting the protein crystal is derived from cationic lipid molecules as opposed to zwitterionic phospholipids. Furthermore, as nanoparticle size is reduced (～3 nm), the presence of a charged lipid membrane is found to be essential, as smaller particles do not adhere to streptavidin crystals grown on zwitterionic membranes. These findings demonstrate that the composition of the lipid support can influence the efficacy of directed-assembly processes which utilize protein templates and are important results toward enhancing control over bottom-up nanofabrication applications.     

Systematic coarse-graining of potential energy landscapes and dynamics in liquids

Recent efforts have shown that the dynamic properties of a wide class of liquids can be mapped onto semi-universal scaling laws and constitutive relations that are motivated by thermodynamic analyses of much simpler models. In particular, it has been found that many systems exhibit dynamics whose behavior in state space closely follows that of soft-sphere particles interacting through an inverse power repulsion. In the present work, we show that a recently developed coarse-graining theory provides a natural way to understand how arbitrary liquids can be mapped onto effective soft-sphere models and hence how one might potentially be able to extract underlying dynamical scaling laws. The theory is based on the relative entropy, an information metric that quantifies how well a soft-sphere approximation to a liquid's multidimensional potential energy landscape performs. We show that optimization of the relative entropy not only enables one to extract effective soft-sphere potentials that suggest an inherent scaling of thermodynamic and dynamic properties in temperature-density space, but that also has rather interesting connections to excess entropy based theories of liquid dynamics. We apply the approach to a binary mixture of Lennard-Jones particles, and show that it gives effective soft-sphere scaling laws that well-describe the behavior of the diffusion constants. Our results suggest that the relative entropy formalism may be useful for "perturbative" type theories of dynamics, offering a general strategy for systematically connecting complex energy landscapes to simpler reference ones with better understood dynamic behavior.