Discrete path sampling

Bioinformatics and Biosimulations as Toolbox for Peptides and Peptidomimetics Design: Where Are We?

Peptides and peptidomimetics are strongly re-emerging as amenable candidates in the development of therapeutic strategies against a plethora of pathologies. In particular, these molecules are extremely suitable to treat diseases in which a major role is played by protein-protein interactions (PPIs). Unlike small organic compounds, peptides display both a high degree of specificity avoiding secondary off-targets effects and a relatively low degree of toxicity. Further advantages are provided by the possibility to easily conjugate peptides to functionalized nanoparticles, so improving their delivery and cellular uptake. In many cases, such molecules need to assume a specific three-dimensional conformation that resembles the bioactive one of the endogenous ligand. To this end, chemical modifications are introduced in the polypeptide chain to constrain it in a well-defined conformation, and to improve the drug-like properties. In this context, a successful strategy for peptide/peptidomimetics design and optimization is to combine different computational approaches ranging from structural bioinformatics to atomistic simulations. Here, we review the computational tools for peptide design, highlighting their main features and differences, and discuss selected protocols, among the large number of methods available, used to assess and improve the stability of the functional folding of the peptides. Finally, we introduce the simulation techniques employed to predict the binding affinity of the designed peptides for their targets.

Defining, Calculating, and Converging Observables of a Kinetic Transition Network

Kinetic transition networks (KTNs) of local minima and transition states are able to capture the dynamics of numerous systems in chemistry, biology, and materials science. However, extracting observables is numerically challenging for large networks and generally will be sensitive to additional computational discovery. To have any measure of convergence for observables, these sensitivities must be regularly calculated. We present a matrix formulation of the discrete path sampling framework for KTNs, deriving expressions for branching probabilities, transition rates, and waiting times. Using the concept of the quasi-stationary distribution, a clear hierarchy of expressions for network observables is established, from exact results to steady-state approximations. We use these results in combination with the graph transformation method to derive the sensitivity, with respect to perturbations of the known KTN, giving explicit terms for the pairwise sensitivity and discussing the pathwise sensitivity. These results provide guidelines for converging the network, with respect to additional sampling, focusing on the estimates obtained for the overall rate coefficients between product and reactant states. We demonstrate this procedure for transitions in the double-funnel landscape of the 38-atom Lennard-Jones cluster.

Structural transitions in the RNA 7SK 5' hairpin and their effect on HEXIM binding

7SK RNA, as part of the 7SK ribonucleoprotein complex, is crucial to the regulation of transcription by RNA-polymerase II, via its interaction with the positive transcription elongation factor P-TEFb. The interaction is induced by binding of the protein HEXIM to the 5′ hairpin (HP1) of 7SK RNA. Four distinct structural models have been obtained experimentally for HP1. Here, we employ computational methods to investigate the relative stability of these structures, transitions between them, and the effects of mutations on the observed structural ensembles. We further analyse the results with respect to mutational binding assays, and hypothesize a mechanism for HEXIM binding. Our results indicate that the dominant structure in the wild type exhibits a triplet involving the unpaired nucleotide U40 and the base pair A43-U66 in the GAUC/GAUC repeat. This conformation leads to an open major groove with enough potential binding sites for peptide recognition. Sequence mutations of the RNA change the relative stability of the different structural ensembles. Binding affinity is consequently lost if these changes alter the dominant structure.

Energy Landscapes of a Pair of Adsorbed Peptides

The wide relevance of peptide adsorption in natural and synthetic contexts means it has attracted much attention. Molecular dynamics (MD) simulation has been widely used in these endeavors. Much of this has focused on single peptides due to the computational effort required to capture the rare events that characterize their adsorption. This focus is, however, of limited practical relevance as in reality, most systems of interest operate in the nondilute regime where peptides will interact with other adsorbed peptides. As an alternative to MD simulation, we have used energy landscape mapping (ELM) to investigate two met-enkephalin molecules adsorbed at a gas/graphite interface. Major conformations of the adsorbed peptides and the connecting transition states are elucidated along with the associated energy barriers and rates of exchange. The last of these makes clear that MD simulations are currently of limited use in probing the co-adsorption of two peptides, let alone more. The constant volume heat capacity as a function of temperature is also presented. Overall, this study represents a significant step toward characterizing peptide adsorption beyond the dilute limit.

Energy landscape of Au13: a global view of structure transformation

It has long been a challenge in physics and chemistry to acquire a global picture of the energy landscape of a specific material, as well as the kinetic transformation process between configurations of interest. Here we have presented a comprehensive approach to deal with the structure transformation problem, along with the illustration of the energy landscape, as exemplified with the case of Au13. A configuration space based on interatomic distances was proposed and demonstrated to have a strong correlation between structure and energy, with application in structure analysis to screen for trial transition pathways. As several representative configurations and their transition pathways ascertained and by projecting on a plane, a visual two-dimensional contour map was sketched revealing the unique energy landscape of Au13. It shows that the 2D and 3D clusters form two funnels in the high-dimensional configuration space, with a transition pathway with a 0.976 eV barrier bridging them.

Thermal fracture kinetics of heterogeneous semiflexible polymers

The fracture and severing of polymer chains plays a critical role in the failure of fibrous materials and the regulated turnover of intracellular filaments. Using continuum wormlike chain models, we investigate the fracture of semiflexible polymers via thermal bending fluctuations, focusing on the role of filament flexibility and dynamics. Our results highlight a previously unappreciated consequence of mechanical heterogeneity in the filament, which enhances the rate of thermal fragmentation particularly in cases where constraints hinder the movement of the chain ends. Although generally applicable to semiflexible chains with regions of different bending stiffness, the model is motivated by a specific biophysical system: the enhanced severing of actin filaments at the boundary between stiff bare regions and mechanically softened regions that are coated with cofilin regulatory proteins. The results presented here point to a potential mechanism for disassembly of polymeric materials in general and cytoskeletal actin networks in particular by the introduction of locally softened chain regions, as occurs with cofilin binding.

A multifunnel energy landscape encodes the competing α-helix and β-hairpin conformations for a designed peptide

Depending on the amino acid sequence, as well as the local environment, some peptides have the capability to fold into multiple secondary structures. Conformational switching between such structures is a key element of protein folding and aggregation. Specifically, understanding the molecular mechanism underlying the transition from an α-helix to a β-hairpin is critical because it is thought to be a harbinger of amyloid assembly. In this study, we explore the energy landscape for an 18-residue peptide (DP5), designed by Araki and Tamura to exhibit equal propensities for the α-helical and β-hairpin forms. We find that the degeneracy is encoded in the multifunnel nature of the underlying free energy landscape. In agreement with experiment, we also observe that mutation of tyrosine at position 12 to serine shifts the equilibrium in favor of the α-helix conformation, by altering the landscape topography. The transition from the α-helix to the β-hairpin is a complex stepwise process, and occurs via collapsed coil-like intermediates. Our findings suggest that even a single mutation can tune the emergent features of the landscape, providing an efficient route to protein design. Interestingly, the transition pathways for the conformational switch seem to be minimally perturbed upon mutation, suggesting that there could be universal microscopic features that are conserved among different switch-competent protein sequences.

Energy Landscapes and Hybridization Pathways for DNA Hexamer Duplexes

Strand hybridization is not only a fundamental molecular mechanism underlying the biological functions of nucleic acids but is also a key step in the design of efficient nanodevices. Despite recent efforts, the microscopic rules governing the hybridization mechanisms remain largely unknown. In this study, we exploit the energy landscape framework to assess how sequence-specificity modulates the hybridization mechanisms in DNA. We find that GG-tracts hybridize much more rapidly compared to GC-tracts, via either zippering or slithering pathways. For the hybridization of GG-tracts, both zippering and slithering mechanisms appear to be kinetically relevant. In contrast, for the GC-tracts, the zippering mechanism is dominant. Our work reveals that even for the relatively small systems considered, the energy landscapes feature multiple metastable states and kinetic traps, which is at odds with the conventional "all-or-nothing" model of DNA hybridization formulated on the basis of thermodynamic arguments alone. Interestingly, entropic effects are found to play an important role in determining the thermal stability of competing conformational ensembles and in determining the preferred hybridization pathways.

Computational methods in the study of self-entangled proteins: a critical appraisal

The existence of self-entangled proteins, the native structure of which features a complex topology, unveils puzzling, and thus fascinating, aspects of protein biology and evolution. The discovery that a polypeptide chain can encode the capability to self-entangle in an efficient and reproducible way during folding, has raised many questions, regarding the possible function of these knots, their conservation along evolution, and their role in the folding paradigm. Understanding the function and origin of these entanglements would lead to deep implications in protein science, and this has stimulated the scientific community to investigate self-entangled proteins for decades by now. In this endeavour, advanced experimental techniques are more and more supported by computational approaches, that can provide theoretical guidelines for the interpretation of experimental results, and for the effective design of new experiments. In this review we provide an introduction to the computational study of self-entangled proteins, focusing in particular on the methodological developments related to this research field. A comprehensive collection of techniques is gathered, ranging from knot theory algorithms, that allow detection and classification of protein topology, to Monte Carlo or molecular dynamics strategies, that constitute crucial instruments for investigating thermodynamics and kinetics of this class of proteins.

Mean first passage times reconstruct the slowest relaxations in potential energy landscapes of nanoclusters

Relaxation modes are the collective modes in which all probability deviations from equilibrium states decay with the same relaxation rates. In contrast, a first passage time is the required time for arriving for the first time from one state to another. In this paper, we discuss how and why the slowest relaxation rates of relaxation modes are reconstructed from the first passage times. As an illustrative model, we use a continuous-time Markov state model of vacancy diffusion in KCl nanoclusters. Using this model, we reveal that all characteristics of the relaxations in KCl nanoclusters come from the fact that they are hybrids of two kinetically different regions of the fast surface and slow bulk diffusions. The origin of the different diffusivities turns out to come from the heterogeneity of the activation energies on the potential energy landscapes. We also develop a stationary population method to compute the mean first passage times as mean times required for pair annihilations of particle-hole pairs, which enables us to obtain the symmetric results of relaxation rates under the exchange of the sinks and the sources. With this symmetric method, we finally show why the slowest relaxation times can be reconstructed from the mean first passage times.

Go-Kit: A Tool To Enable Energy Landscape Exploration of Proteins

Coarse-grained Go̅-like models, based on the principle of minimal frustration, provide valuable insight into fundamental questions in the field of protein folding and dynamics. In conjunction with commonly used molecular dynamics (MD) simulations, energy landscape exploration methods like discrete path sampling (DPS) with Go̅-like models can provide quantitative details of the thermodynamics and kinetics of proteins. Here we present Go-kit, a software that facilitates the setup of MD and DPS simulations of several flavors of Go̅-like models. Go-kit is designed for use with MD (GROMACS) and DPS (PATHSAMPLE) simulation engines that are open source. The Go-kit code is written in python2.7 and is also open source. A case study for the ribosomal protein S6 is discussed to illustrate the utility of the software, which is available at https://github.com/gokit1/gokit .

Structural properties of sub-nanometer metallic clusters

At the nanoscale, the investigation of structural features becomes fundamental as we can establish relationships between cluster geometries and their physicochemical properties. The peculiarity lies in the variety of shapes often unusual and far from any geometrical and crystallographic intuition clusters can assume. In this respect, we should treat and consider nanoparticles as a new form of matter. Nanoparticle structures depend on their size, chemical composition, ordering, as well as external conditions e.g. synthesis method, pressure, temperature, support. On top of that, at finite temperatures nanoparticles can fluctuate among different structures, opening new and exciting horizons for the design of optimal nanoparticles for advanced applications. This article aims to overview geometrical features of transition metal clusters and of their various rearrangements.

Computational Investigation of RNA A-Bulges Related to the Microtubule-Associated Protein Tau Causing Frontotemporal Dementia and Parkinsonism

Mutations in the human tau gene result in alternative splicing of the tau protein, which causes frontotemporal dementia and Parkinsonism. One disease mechanism is linked to the stability of a hairpin within the microtubule-associated protein tau (MAPT) mRNA, which contains an A-bulge. Here we employ computational methods to investigate the structural and thermodynamic properties of several A-bulge RNAs with different closing base-pairs. We find that the current amber RNA force field has a preference to overstabilize base-triple over stacked states, even though some of the A-bulges are known to prefer stacked states according to NMR studies. We further determined that if the neighboring base-pairs of A-bulges are AU, this situation can lead to base slippage. However, when the 3'-side of the A-bulge has an UA base-pair, the stacked state is stabilized by an extra interaction that is not observed in the other sequences. We suggest that these A-bulge RNA systems could be used as benchmarks to improve the current RNA force fields.

Using Data-Reduction Techniques to Analyze Biomolecular Trajectories

This chapter discusses the way in which dimensionality reduction algorithms such as diffusion maps and sketch-map can be used to analyze molecular dynamics trajectories. The first part discusses how these various algorithms function as well as practical issues such as landmark selection and how these algorithms can be used when the data to be analyzed comes from enhanced sampling trajectories. In the later part a comparison between the results obtained by applying various algorithms to two sets of sample data is performed and discussed. This section is then followed by a summary of how one algorithm in particular, sketch-map, has been applied to a range of problems. The chapter concludes with a discussion on the directions that we believe this field is currently moving.

Improvement of RNA Simulations with Torsional Revisions of the AMBER Force Field

Our current knowledge on the unique roles of RNA in cells makes it vital to investigate the properties of RNA systems using computational methods because of the potential pharmaceutical applications. With the continuous advancement of computer technology, it is now possible to study RNA folding. Molecular mechanics calculations are useful in discovering the structural and thermodynamic properties of RNA systems. Yet, the predictions depend on the quality of the RNA force field, which is a set of parameters describing the potential energy of the system. Torsional parameters are one of the terms in a force field that can be revised using physics-based approaches. This chapter focuses on improvements provided by revisions of torsional parameters of the AMBER (Assisted Model Building with Energy Refinement) RNA force field. The theory behind torsional revisions and re-parameterization of several RNA torsions is briefly described. Applications of the revised torsional parameters to study RNA nucleosides, single-stranded RNA tetramers, and RNA repeat expansions are described in detail. It is concluded that RNA force fields require constant revisions and should be benchmarked against diverse RNA systems such as single strands and internal loops in order to test their qualities.

Defining an Optimal Metric for the Path Collective Variables

Path Collective Variables (PCVs) are a set of path-like variables that have been successfully used to investigate complex chemical and biological processes and compute their associated free energy surfaces and kinetics. Their current implementation relies on general, but at times inefficient, metrics (such as RMSD or DRMSD) to evaluate the distance between the instantaneous conformational state during the simulation and the reference coordinates defining the path. In this work, we present a new algorithm to construct optimal PCVs metrics as linear combinations of different CVs weighted through a spectral gap optimization procedure. The method was tested first on a simple model, trialanine peptide, in vacuo and then on a more complex path of an anticancer inhibitor binding to its pharmacological target. We also compared the results to those obtained with other path-based algorithms. We find that not only our proposed approach is able to automatically select relevant CVs for the PCVs metric but also that the resulting PCVs allow for reconstructing the associated free energy very efficiently. What is more, at difference with other path-based methods, our algorithm is able to explore nonlocally the reaction path space.

Analysis of the Ub to Ub-CR Transition in Ubiquitin

A new conformation has recently been reported for ubiquitin (Ub). This invisible conformation (Ub-CR), where the C-terminal tail is retracted, has a key functional role in phosphorylation of the Ser65 residue, a trigger for PINK1 and Parkin dependent mitophagy. Here we calculate the transition mechanism and associated rates for the Ub to Ub-CR pathway in the wild-type protein and a selection of mutants. We predict a cooperative one-step process with a transition state that resembles the Ub configuration, characterized by a loss of all interactions of the C-terminal tail with surrounding residues, and an open ubiquitin scaffold. The calculated observables agree well with reported values, and we make a range of predictions to guide future experiments. In particular, the effect of mutations on the pathway and the corresponding structural ensembles should have observable consequences. This feedback between theory and experiment promises new insight into key cellular processes.

Multimodal transport and dispersion of organelles in narrow tubular cells

Intracellular components explore the cytoplasm via active motor-driven transport in conjunction with passive diffusion. We model the motion of organelles in narrow tubular cells using analytical techniques and numerical simulations to study the efficiency of different transport modes in achieving various cellular objectives. Our model describes length and time scales over which each transport mode dominates organelle motion, along with various metrics to quantify exploration of intracellular space. For organelles that search for a specific target, we obtain the average capture time for given transport parameters and show that diffusion and active motion contribute to target capture in the biologically relevant regime. Because many organelles have been found to tether to microtubules when not engaged in active motion, we study the interplay between immobilization due to tethering and increased probability of active transport. We derive parameter-dependent conditions under which tethering enhances long-range transport and improves the target capture time. These results shed light on the optimization of intracellular transport machinery and provide experimentally testable predictions for the effects of transport regulation mechanisms such as tethering.

Single-root networks for describing the potential energy surface of Lennard-Jones clusters

Potential energy surface (PES) holds the key in understanding a number of atomic clusters or molecular phenomena. However, due to the high dimension and incredible complexity of PES, only indirect methods can be used to characterize a PES of a given system in general. In this paper, a branched dynamic lattice searching method was developed to travel the PES, which was described in detail by a single-root network (SRN). The advantage of SRN is that it reflects the topological relation between different conformations and highlights the size of each structure energy trap. On the basis of SRN, to demonstrate how to transform one conformation to another, the transition path that connects two local minima in the PES was constructed. Herein, we take Lennard-Jones (LJ) clusters at the sizes of 38, 55, and 75 as examples. It is found that the PES of these three clusters have many local funnels and each local funnel represents one morphology. If a morphology is located more frequently, it will lie in a larger local funnel. Besides, certain steps of the transition path were generated successfully, such as changing from icosahedral to truncated octahedral of the LJ₃₈-cluster. Though we do not exhibit all the parts of the PES or all transition paths, this method indeed works well in the local area and can be used more widely.

A spin-1 representation for dual-funnel energy landscapes

The interconversion between the left- and right-handed helical folds of a polypeptide defines a dual-funneled free energy landscape. In this context, the funnel minima are connected through a continuum of unfolded conformations, evocative of the classical helix-coil transition. Physical intuition and recent conjectures suggest that this landscape can be mapped by assigning a left- or right-handed helical state to each residue. We explore this possibility using all-atom replica exchange molecular dynamics and an Ising-like model, demonstrating that the energy landscape architecture is at odds with a two-state picture. A three-state model-left, right, and unstructured-can account for most key intermediates during chiral interconversion. Competing folds and excited conformational states still impose limitations on the scope of this approach. However, the improvement is stark: Moving from a two-state to a three-state model decreases the fit error from 1.6 kBT to 0.3 kBT along the left-to-right interconversion pathway.

Exploring Energy Landscapes

Recent advances in the potential energy landscapes approach are highlighted, including both theoretical and computational contributions. Treating the high dimensionality of molecular and condensed matter systems of contemporary interest is important for understanding how emergent properties are encoded in the landscape and for calculating these properties while faithfully representing barriers between different morphologies. The pathways characterized in full dimensionality, which are used to construct kinetic transition networks, may prove useful in guiding such calculations. The energy landscape perspective has also produced new procedures for structure prediction and analysis of thermodynamic properties. Basin-hopping global optimization, with alternative acceptance criteria and generalizations to multiple metric spaces, has been used to treat systems ranging from biomolecules to nanoalloy clusters and condensed matter. This review also illustrates how all this methodology, developed in the context of chemical physics, can be transferred to landscapes defined by cost functions associated with machine learning.

Nested Transition Path Sampling

We introduce a novel transition path (TPS) sampling scheme employing nested sampling. Analogous to how nested sampling explores the entire configurational phase space for atomistic systems, nested TPS samples the entire available trajectory space in one simulation. Thermodynamic and path observables can be constructed a posteriori for all temperatures simultaneously. We exploit this to compute the rate of rare processes at arbitrarily low temperature through the coupling to easily accessible rates at high temperature. We illustrate the method on several model systems.

Loss surface of XOR artificial neural networks

Training an artificial neural network involves an optimization process over the landscape defined by the cost (loss) as a function of the network parameters. We explore these landscapes using optimization tools developed for potential energy landscapes in molecular science. The number of local minima and transition states (saddle points of index one), as well as the ratio of transition states to minima, grow rapidly with the number of nodes in the network. There is also a strong dependence on the regularization parameter, with the landscape becoming more convex (fewer minima) as the regularization term increases. We demonstrate that in our formulation, stationary points for networks with N_{h} hidden nodes, including the minimal network required to fit the XOR data, are also stationary points for networks with N_{h}+1 hidden nodes when all the weights involving the additional node are zero. Hence, smaller networks trained on XOR data are embedded in the landscapes of larger networks. Our results clarify certain aspects of the classification and sensitivity (to perturbations in the input data) of minima and saddle points for this system, and may provide insight into dropout and network compression.

Evolved Minimal Frustration in Multifunctional Biomolecules

Protein folding is often viewed in terms of a funneled potential or free energy landscape. A variety of experiments now indicate the existence of multifunnel landscapes, associated with multifunctional biomolecules. Here, we present evidence that these systems have evolved to exhibit the minimal number of funnels required to fulfill their cellular functions, suggesting an extension to the principle of minimum frustration. We find that minimal disruptive mutations result in additional funnels, and the associated structural ensembles become more diverse. The same trends are observed in an atomic cluster. These observations suggest guidelines for rational design of engineered multifunctional biomolecules.

Energy Landscapes of Mini-Dumbbell DNA Octanucleotides

Single-stranded DNA structures play a significant role in biological systems, in particular during replication, translation, and DNA repair. Tracts of simple repetitive DNA are associated with slipped-strand mispairing, which can lead to genetic diseases. Recent NMR studies of TTTA and CCTG repeats have shown that these sequences form mini-dumbbells (MDBs), leading to frameshift mutations. Here we explore the energy landscapes of (CCTG)₂ and (TTTA)₂, which are currently the smallest known molecules to form MDBs. While (CCTG)₂ MDBs are stable, (TTTA)₂ exhibits numerous other structures with lower energies. A key factor identified in the stabilization of MDB structures is the bonding strength between residues 1 and 4, and 5 and 8.

Atomic clusters with addressable complexity

A general formulation for constructing addressable atomic clusters is introduced, based on one or more reference structures. By modifying the well depths in a given interatomic potential in favour of nearest-neighbour interactions that are defined in the reference(s), the potential energy landscape can be biased to make a particular permutational isomer the global minimum. The magnitude of the bias changes the resulting potential energy landscape systematically, providing a framework to produce clusters that should self-organise efficiently into the target structure. These features are illustrated for small systems, where all the relevant local minima and transition states can be identified, and for the low-energy regions of the landscape for larger clusters. For a 55-particle cluster, it is possible to design a target structure from a transition state of the original potential and to retain this structure in a doubly addressable landscape. Disconnectivity graphs based on local minima that have no direct connections to a lower minimum provide a helpful way to visualise the larger databases. These minima correspond to the termini of monotonic sequences, which always proceed downhill in terms of potential energy, and we identify them as a class of biminimum. Multiple copies of the target cluster are treated by adding a repulsive term between particles with the same address to maintain distinguishable targets upon aggregation. By tuning the magnitude of this term, it is possible to create assemblies of the target cluster corresponding to a variety of structures, including rings and chains.

RNA Structural Dynamics As Captured by Molecular Simulations: A Comprehensive Overview

With both catalytic and genetic functions, ribonucleic acid (RNA) is perhaps the most pluripotent chemical species in molecular biology, and its functions are intimately linked to its structure and dynamics. Computer simulations, and in particular atomistic molecular dynamics (MD), allow structural dynamics of biomolecular systems to be investigated with unprecedented temporal and spatial resolution. We here provide a comprehensive overview of the fast-developing field of MD simulations of RNA molecules. We begin with an in-depth, evaluatory coverage of the most fundamental methodological challenges that set the basis for the future development of the field, in particular, the current developments and inherent physical limitations of the atomistic force fields and the recent advances in a broad spectrum of enhanced sampling methods. We also survey the closely related field of coarse-grained modeling of RNA systems. After dealing with the methodological aspects, we provide an exhaustive overview of the available RNA simulation literature, ranging from studies of the smallest RNA oligonucleotides to investigations of the entire ribosome. Our review encompasses tetranucleotides, tetraloops, a number of small RNA motifs, A-helix RNA, kissing-loop complexes, the TAR RNA element, the decoding center and other important regions of the ribosome, as well as assorted others systems. Extended sections are devoted to RNA-ion interactions, ribozymes, riboswitches, and protein/RNA complexes. Our overview is written for as broad of an audience as possible, aiming to provide a much-needed interdisciplinary bridge between computation and experiment, together with a perspective on the future of the field.

Slowest kinetic modes revealed by metabasin renormalization

Understanding the slowest relaxations of complex systems, such as relaxation of glass-forming materials, diffusion in nanoclusters, and folding of biomolecules, is important for physics, chemistry, and biology. For a kinetic system, the relaxation modes are determined by diagonalizing its transition rate matrix. However, for realistic systems of interest, numerical diagonalization, as well as extracting physical understanding from the diagonalization results, is difficult due to the high dimensionality. Here, we develop an alternative and generally applicable method of extracting the long-time scale relaxation dynamics by combining the metabasin analysis of Okushima et al. [Phys. Rev. E 80, 036112 (2009)PLEEE81539-375510.1103/PhysRevE.80.036112] and a Jacobi method. We test the method on an illustrative model of a four-funnel model, for which we obtain a renormalized kinematic equation of much lower dimension sufficient for determining slow relaxation modes precisely. The method is successfully applied to the vacancy transport problem in ionic nanoparticles [Niiyama et al., Chem. Phys. Lett. 654, 52 (2016)CHPLBC0009-261410.1016/j.cplett.2016.04.088], allowing a clear physical interpretation that the final relaxation consists of two successive, characteristic processes.

Exploring biomolecular energy landscapes

The potential energy landscape perspective provides both a conceptual and a computational framework for predicting, understanding and designing molecular properties. In this Feature Article, we highlight some recent advances that greatly facilitate structure prediction and analysis of global thermodynamics and kinetics in proteins and nucleic acids. The geometry optimisation procedures, on which these calculations are based, can be accelerated significantly using local rigidification of selected degrees of freedom, and through implementations on graphics processing units. Results of progressive local rigidification are first summarised for trpzip1, including a systematic analysis of the heat capacity and rearrangement rates. Benchmarks for all the essential optimisation procedures are then provided for a variety of proteins. Applications are then illustrated from a study of how mutation affects the energy landscape for a coiled-coil protein, and for transitions in helix morphology for a DNA duplex. Both systems exhibit an intrinsically multifunnel landscape, with the potential to act as biomolecular switches.

Energy Landscapes for the Aggregation of Aβ17-42

The aggregation of the Aβ peptide (Aβ_1-42) to form fibrils is a key feature of Alzheimer's disease. The mechanism is thought to be a nucleation stage followed by an elongation process. The elongation stage involves the consecutive addition of monomers to one end of the growing fibril. The aggregation process proceeds in a stop-and-go fashion and may involve off-pathway aggregates, complicating experimental and computational studies. Here we present exploration of a well-defined region in the free and potential energy landscapes for the Aβ_17-42 pentamer. We find that the ideal aggregation process agrees with the previously reported dock-lock mechanism. We also analyze a large number of additional stable structures located on the multifunnel energy landscape, which constitute kinetic traps. The key contributors to the formation of such traps are misaligned strong interactions, for example the stacking of F19 and F20, as well as entropic contributions. Our results suggest that folding templates for aggregation are a necessity and that aggregation studies could employ such species to obtain a more detailed description of the process.

Probing helical transitions in a DNA duplex

The complex conformational change from B-DNA to Z-DNA requires inversion of helix-handedness. Multiple degrees of freedom are intricately coupled during this transition, and formulating an appropriate reaction coordinate that captures the underlying complexity would be problematic. In this contribution, we adopt an alternative approach, based on the potential energy landscape perspective, to construct a kinetic transition network. Microscopic insight into the B → Z transition is provided in terms of geometrically defined discrete paths consisting of local minima and the transition states that connect them. We find that the inversion of handedness can occur via two competing mechanisms, either involving stretched intermediates, or a B-Z junction, in agreement with previous predictions. The organisation of the free energy landscape further suggests that this process is likely to be slow under physiological conditions. Our results represent a key step towards decoding the more intriguing features of the B → Z transition, such as the role of ionic strength and negative supercoiling in reshaping the landscape.

Energy landscapes for machine learning

Machine learning techniques are being increasingly used as flexible non-linear fitting and prediction tools in the physical sciences. Fitting functions that exhibit multiple solutions as local minima can be analysed in terms of the corresponding machine learning landscape. Methods to explore and visualise molecular potential energy landscapes can be applied to these machine learning landscapes to gain new insight into the solution space involved in training and the nature of the corresponding predictions. In particular, we can define quantities analogous to molecular structure, thermodynamics, and kinetics, and relate these emergent properties to the structure of the underlying landscape. This Perspective aims to describe these analogies with examples from recent applications, and suggest avenues for new interdisciplinary research.