Energy landscapes and properties of biomolecules

Energy landscapes of combinatorial optimization in Ising machines

Physics-based Ising machines (IM) have been developed as dedicated processors for solving hard combinatorial optimization problems with higher speed and better energy efficiency. Generally, such systems employ local search heuristics to traverse energy landscapes in searching for optimal solutions. Here, we quantify and address some of the major challenges met by IMs by extending energy-landscape geometry visualization tools known as disconnectivity graphs. Using efficient sampling methods, we visually capture landscapes of problems having diverse structure and hardness manifesting as energetic and entropic barriers for IMs. We investigate energy barriers, local minima, and configuration space clustering effects caused by locality reduction methods when embedding combinatorial problems to the Ising hardware. To this end, we sample disconnectivity graphs of PUBO energy landscapes and their different QUBO mappings accounting for both local minima and saddle regions. We demonstrate that QUBO energy-landscape properties lead to the subpar performance for quadratic IMs and suggest directions for their improvement.

Virtual indentation of the empty capsid of the minute virus of mice using a minimal coarse-grained model

A minimal coarse-grained model for T=1 viral capsids assembled from 20 protein rigid trimers has been designed by extending a previously proposed form of the interaction energy written as a sum of anisotropic pairwise interactions between the trimeric capsomers. The extension of the model has been performed to properly account for the coupling between two internal coordinates: the one that measures the intercapsomer distance and the other that gives the intercapsomer dihedral angle. The model has been able to fit with less than a 10% error the atomic force microscopy (AFM) indentation experimental data for the empty capsid of the minute virus of mice (MVM), providing in this way an admissible picture of the main mechanisms behind the capsid deformations. In this scenario, the bending of the intercapsomer dihedral angle is the angular internal coordinate that can support larger deformations away from its equilibrium values, determining important features of the AFM indentation experiments as the elastic constants along the three symmetry axes of the capsid and the critical indentations. From the value of one of the parameters of our model, we conclude that trimers in the MVM must be quite oblate tops, in excellent agreement with their known structure. The transition from the linear to the nonlinear regimes sampled in the indentation process appears to be an interesting topic for future research in physical virology.

MasitinibL shows promise as a drug-like analog of masitinib that elicits comparable SARS-Cov-2 3CLpro inhibition with low kinase preference

SARS-CoV-2 infection has led to several million deaths worldwide and ravaged the economies of many countries. Hence, developing therapeutics against SARS-CoV-2 remains a core priority in the fight against COVID-19. Most of the drugs that have received emergency use authorization for treating SARS-CoV-2 infection exhibit a number of limitations, including side effects and questionable efficacy. This challenge is further compounded by reinfection after vaccination and the high likelihood of mutations, as well as the emergence of viral escape mutants that render SARS-CoV-2 spike glycoprotein-targeting vaccines ineffective. Employing de novo drug synthesis or repurposing to discover broad-spectrum antivirals that target highly conserved pathways within the viral machinery is a focus of current research. In a recent drug repurposing study, masitinib, a clinically safe drug against the human coronavirus OC43 (HCoV-OC43), was identified as an antiviral agent with effective inhibitory activity against the SARS-CoV-2 3CLpro. Masitinib is currently under clinical trial in combination with isoquercetin in hospitalized patients (NCT04622865). Nevertheless, masitinib has kinase-related side effects; hence, the development of masitinib analogs with lower anti-tyrosine kinase activity becomes necessary. In this study, in an attempt to address this limitation, we executed a comprehensive virtual workflow in silico to discover drug-like compounds matching selected pharmacophore features in the SARS-CoV-2 3CLpro-bound state of masitinib. We identified a novel lead compound, "masitinibL", a drug-like analog of masitinib that demonstrated strong inhibitory properties against the SARS-CoV-2 3CLpro. In addition, masitinibL further displayed low selectivity for tyrosine kinases, which strongly suggests that masitinibL is a highly promising therapeutic that is preferable to masitinib.

Minimal Design Principles for Icosahedral Virus Capsids

The geometrical structures of single- and multiple-shell icosahedral virus capsids are reproduced as the targets that minimize the cost corresponding to relatively simple design functions. Capsid subunits are first identified as building blocks at a given coarse-grained scale and then represented in these functions as point particles located on an appropriate number of concentric spherical surfaces. Minimal design cost is assigned to optimal spherical packings of the particles. The cost functions are inspired by the packings favored for the Thomson problem, which minimize the electrostatic potential energy between identical charged particles. In some cases, icosahedral symmetry constraints are incorporated as external fields acting on the particles. The simplest cost functions can be obtained by separating particles in disjoint nonequivalent sets with distinct interactions, or by introducing interacting holes (the absence of particles). These functions can be adapted to reproduce any capsid structure found in real viruses. Structures absent in Nature require significantly more complex designs. Measures of information content and complexity are assigned to both the cost functions and the capsid geometries. In terms of these measures, icosahedral structures and the corresponding cost functions are the simplest solutions.

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.

A minimal coarse-grained model for the low-frequency normal mode analysis of icosahedral viral capsids

The main goal of this work is the design of a coarse-grained theoretical model of minimal resolution for the study of the physical properties of icosahedral virus capsids within the linear-response regime. In this model the capsid is represented as an interacting many-body system whose composing elements are capsid subunits (capsomers), which are treated as three-dimensional rigid bodies. The total interaction potential energy is written as a sum of pairwise capsomer-capsomer interactions. Based on previous work [Gomez Llorente et al., Soft Matter, 2014, 10, 3560], a minimal and complete anisotropic binary interaction that includes a full Hessian matrix of independent force constants is proposed. In this interaction model, capsomers have rotational symmetry around an axis of order n > 2. The full coarse-grained model is applied to analyse the low-frequency normal-mode spectrum of icosahedral T = 1 capsids. The model performance is evaluated by fitting its predicted spectrum to the full-atom results for the Satellite Tobacco Necrosis Virus (STNV) capsid [Dykeman and Sankey, Phys. Rev. Lett., 2008, 100, 028101]. Two capsomer choices that are compatible with the capsid icosahedral symmetry are checked, namely pentamers (n = 5) and trimers (n = 3). Both subunit types provide fair fits, from which the magnitude of the coarse-grained force constants for a real virus is obtained. The model is able to uncover latent instabilities whose analysis is fully consistent with the current knowledge about the STNV capsid, which does not self-assemble in the absence of RNA and is thermally unstable. The straightforward generalisability of the model beyond the linear regime and its completeness make it a promising tool to theoretically interpret many experimental data such as those provided by the atomic force microscopy or even to better understand processes far from equilibrium such as the capsid self-assembly.

Efficient prediction of reaction paths through molecular graph and reaction network analysis

Despite remarkable advances in computational chemistry, prediction of reaction mechanisms is still challenging, because investigating all possible reaction pathways is computationally prohibitive due to the high complexity of chemical space. A feasible strategy for efficient prediction is to utilize chemical heuristics. Here, we propose a novel approach to rapidly search reaction paths in a fully automated fashion by combining chemical theory and heuristics. A key idea of our method is to extract a minimal reaction network composed of only favorable reaction pathways from the complex chemical space through molecular graph and reaction network analysis. This can be done very efficiently by exploring the routes connecting reactants and products with minimum dissociation and formation of bonds. Finally, the resulting minimal network is subjected to quantum chemical calculations to determine kinetically the most favorable reaction path at the predictable accuracy. As example studies, our method was able to successfully find the accepted mechanisms of Claisen ester condensation and cobalt-catalyzed hydroformylation reactions.

Pathways to self-organization: Crystallization via nucleation and growth

Crystallization, a prototypical self-organization process during which a disordered state spontaneously transforms into a crystal characterized by a regular arrangement of its building blocks, usually proceeds by nucleation and growth. In the initial stages of the transformation, a localized nucleus of the new phase forms in the old one due to a random fluctuation. Most of these nuclei disappear after a short time, but rarely a crystalline embryo may reach a critical size after which further growth becomes thermodynamically favorable and the entire system is converted into the new phase. In this article, we will discuss several theoretical concepts and computational methods to study crystallization. More specifically, we will address the rare event problem arising in the simulation of nucleation processes and explain how to calculate nucleation rates accurately. Particular attention is directed towards discussing statistical tools to analyze crystallization trajectories and identify the transition mechanism.

Viral Evolved Inhibition Mechanism of the RNA Dependent Protein Kinase PKR's Kinase Domain, a Structural Perspective

The protein kinase PKR activated by viral dsRNA, phosphorylates the eIF2α, which inhibit the mechanism of translation initiation. Viral evolved proteins mimicking the eIF2α block its phosphorylation and help in the viral replication. To decipher the molecular basis for the PKR’s substrate and inhibitor interaction mechanisms, we carried the molecular dynamics studies on the catalytic domain of PKR in complex with substrate eIF2α, and inhibitors TAT and K3L. The studies conducted show the altered domain movements of N lobe, which confers open and close state to the substrate-binding cavity. In addition, PKR exhibits variations in the secondary structural transition of the activation loop residues, and inter molecular contacts with the substrate and the inhibitors. Phosphorylation of the P+1 loop at the Thr-451 increases the affinity of the binding proteins exhibiting its role in the phosphorylation events. The implications of structural mechanisms uncovered will help to understand the basis of the evolution of the host-viral and the viral replication mechanisms.

Systematically constructing kinetic transition network in polypeptide from top to down: trajectory mapping

Molecular dynamics (MD) simulation is an important tool for understanding bio-molecules in microscopic temporal/spatial scales. Besides the demand in improving simulation techniques to approach experimental scales, it becomes more and more crucial to develop robust methodology for precisely and objectively interpreting massive MD simulation data. In our previous work [J Phys Chem B 114, 10266 (2010)], the trajectory mapping (TM) method was presented to analyze simulation trajectories then to construct a kinetic transition network of metastable states. In this work, we further present a top-down implementation of TM to systematically detect complicate features of conformational space. We first look at longer MD trajectory pieces to get a coarse picture of transition network at larger time scale, and then we gradually cut the trajectory pieces in shorter for more details. A robust clustering algorithm is designed to more effectively identify the metastable states and transition events. We applied this TM method to detect the hierarchical structure in the conformational space of alanine-dodeca-peptide from microsecond to nanosecond time scales. The results show a downhill folding process of the peptide through multiple pathways. Even in this simple system, we found that single common-used order parameter is not sufficient either in distinguishing the metastable states or predicting the transition kinetics among these states.

Energy landscapes of atomic clusters as black box optimization benchmarks

We present the energy minimization of atomic clusters as a promising problem class for continuous black box optimization benchmarks. Finding the arrangement of atoms that minimizes a given potential energy is a specific instance of the more general class of geometry optimization or packing problems, which are generally NP-complete. Atomic clusters are a well-studied subject in physics and chemistry. From the large set of available cluster optimization problems, we propose two specific instances: Cohn-Kumar clusters and Lennard-Jones clusters. The potential energies of these clusters are governed by distance-dependent pairwise interaction potentials. The resulting collection of landscapes is composed of smooth and rugged single-funnel topologies, as well as tunable double-funnel topologies. In addition, all problems possess a feature that is not covered by the synthetic functions in current black box optimization test suites: isospectral symmetry. This property implies that any atomic arrangement is uniquely defined by the pairwise distance spectrum, rather than the absolute atomic positions. We hence suggest that the presented problem instances should be included in black box optimization benchmark suites.

Improvements of network approach for analysis of the folding free-energy surface of peptides and proteins

Folding network is an effective approach to investigate the high-dimensional free-energy surface of peptide and protein folding, and it can avoid the limitations of the projected free-energy surface based on two-order parameters. In this article, we present improvements of the effectiveness and accuracy of the folding network analysis based on Markov cluster (MCL) algorithm. We used this approach to investigate the folding free-energy surface of the beta-hairpin peptide trpzip2 and found the folding network is able to determine the basins and folding paths of trpzip2 more clearly and accurately than the two-dimensional free-energy surface.

Global view of bionetwork dynamics: adaptive landscape

Based on recent work, I will give a nontechnical brief review of a powerful quantitative concept in biology, adaptive landscape, initially proposed by S. Wright over 70 years ago, reintroduced by one of the founders of molecular biology and by others in different biological contexts, but apparently forgotten by modern biologists for many years. Nevertheless, this concept finds an increasingly important role in the development of systems biology and bionetwork dynamics modeling, from phage lambda genetic switch to endogenous network for cancer genesis and progression. It is an ideal quantification to describe the robustness and stability of bionetworks. Here, I will first introduce five landmark proposals in biology on this concept, to demonstrate an important common thread in theoretical biology. Then I will discuss a few recent results, focusing on the studies showing theoretical consistency of adaptive landscape. From the perspective of a working scientist and of what is needed logically for a dynamical theory when confronting empirical data, the adaptive landscape is useful both metaphorically and quantitatively, and has captured an essential aspect of biological dynamical processes. Though at the theoretical level the adaptive landscape must exist and it can be used across hierarchical boundaries in biology, many associated issues are indeed vague in their initial formulations and their quantitative realizations are not easy, and are good research topics for quantitative biologists. I will discuss three types of open problems associated with the adaptive landscape in a broader perspective.

Chasing funnels on protein-protein energy landscapes at different resolutions

Studies of intermolecular energy landscapes are important for understanding protein association and adequate modeling of protein interactions. Landscape representation at different resolutions can be used for the refinement of docking predictions and detection of macro characteristics, like the binding funnel. A representative set of protein-protein complexes was used to systematically map the intermolecular landscape by grid-based docking. The change of the resolution was achieved by varying the range of the potential, according to the variable resolution GRAMM methodology. A formalism was developed to consistently parameterize the potential and describe essential characteristics of the landscape. The results of gradual landscape smoothing, from high to low resolution, indicate that i), the number of energy basins, the landscape ruggedness, and the slope decrease accordingly; ii), the number of near-native matches, defined as those inside the funnel, increases until the trend breaks down at critical resolution; the rate of the increase and the critical resolution are specific to the type of a complex (enzyme inhibitor, antigen-antibody, and other), reflect known underlying recognition factors, and correlate with earlier determined estimates of the funnel size; iii), the native/nonnative energy gap, a major characteristic of the energy minima hierarchy, remains constant; and iv), the putative funnel (defined as the deepest basin) has the largest average depth-related ruggedness and slope, at all resolutions. The results facilitate better understanding of the binding landscapes and suggest directions for implementation in practical docking protocols.

Calculations of protein-ligand binding entropy of relative and overall molecular motions

In the context of virtual database screening, calculations of protein-ligand binding entropy of relative and overall molecular motions are challenging, owing to the inherent structural complexity of the ligand binding well in the energy landscape of protein-ligand interactions and computing time limitations. We describe a fast statistical thermodynamic method for estimation the binding entropy to address the challenges. The method is based on the integration of the configurational integral over clusters obtained from multiple docked positions. We apply the method in conjunction with 11 popular scoring functions (AutoDock, ChemScore, DrugScore, D-Score, F-Score, G-Score, LigScore, LUDI, PLP, PMF, X-Score) to evaluate the binding entropy of 100 protein-ligand complexes. The averaged values of binding entropy contribution vary from 6.2 to 9.1 kcal/mol, showing good agreement with literature. We calculate positional sizes and the angular volume of the native ligand wells. The averaged geometric mean of positional sizes in principal directions varies from 0.8 to 1.4 A. The calculated range of angular volumes is 3.3-11.8 rad(2). Then we demonstrate that the averaged six-dimensional volume of the native well is larger than the volume of the most populated non-native well in energy landscapes described by all of 11 scoring functions.