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RNA cotranscriptional folding is the phenomenon in which an RNA transcript folds upon itself while being synthesized out of a gene. The oritatami system is a computation model of this phenomenon, which lets its sequence (transcript) of beads (abstract molecules) fold cotranscriptionally by the interactions between beads according to the binding ruleset. In such models based on self-assembly, one of the key questions is the ability to construct fractal structures. We focus on the problem of generating an infinite fractal curves using a cyclic oritatami system, which has an infinite periodic transcript. We first establish a formal definition of drawing a curve using an oritatami system, proposing conditions and restrictions with reference to prior oritatami designs for possibly infinite conformations. Under such definition, we prove that it is impossible to draw a Koch curve or a Minkowski curve infinitely. We then establish sufficient conditions of infinite aperiodic curves that a cyclic oritatami system cannot fold.

In this paper, introduce algorithm on complete graph K4, when the graph weighted, and discusses the folding of algorithm graph of weighted complete graph K4, the folding at some cases such as folding of the edges as all cases, and folding of the vertices, some theorems related to these result are obtained and prove of this theorems are obtained, also some life applications are introduced.

Because of their potential utility in predicting conformational changes and assessing folding dynamics, coarse-grained (CG) RNA folding models are appealing for rapid characterization of RNA molecules. Previously, we reported the iterative simulated RNA reference state (IsRNA) method for parameterizing a CG force field for RNA folding, which consecutively updates the simulation force field to reflect marginal distributions of folding coordinates in the structure database and extract various energy terms. While the IsRNA model was validated by showing close agreement between the IsRNA-simulated and experimentally observed distributions, here, we expand our theoretical understanding of the model and, in doing so, improve the parameterization process to optimize the subset of included folding coordinates, which leads to accelerated simulations. Using statistical mechanical theory, we analyze the underlying, Bayesian concept that drives parameterization of the energy function, providing a general method for developing predictive, knowledge-based, polymer force fields on the basis of limited data. Furthermore, we propose an optimal parameterization procedure, based on the principal of maximum entropy.

The snake cube puzzle made of a linear array of 27 cubes and its modified and extended versions are used as theoretical models to study the mechanism of folding of proteins into their sequence-specific native three-dimensional structures. Each of the three versions is characterized by the respective set of characteristics attributed to each of its constituent cubes and an array is characterized by its specific sequence of the cube characteristics. The aim of the puzzles is to fold the cube array into a compact 3×3×3 cubic structure. In all three versions, out of all possible sequences, only a limited fraction of sequences are found foldable into the compact cube. Even among foldable sequences, the structures folded into the compact 3×3×3 cube are found often not uniquely determined from the sequence. By comparing the results obtained for the three versions of models, we conclude that the power of the hydrophobic interactions to make the folded structure unique to the sequence is much weaker than the geometrical varieties of constituent cubes as modelled in the original snake cube puzzle. However, when this weak cube attribute is compounded to that of the original snake cube puzzle, the power is enhanced very effectively. This is a strong manifestation of the consistency principle: The sequence-specific native structure of protein is realized as a result of consistency of various types of interactions working in protein.

Computational coarse-grained models play a fundamental role as a research tool in protein folding, and they are important in bridging theory and experiments. Folding mechanisms are generally discussed using the energy landscape framework, which is well mapped within a class of simplified structure-based models. In this chapter, simplified computer models are discussed with special focus on structure-based ones.

Cryo-electron microscopy (EM) requires molecular modeling to refine structural details from data. Ensemble models arrive at low free-energy molecular structures, but are computationally expensive and limited to resolving only small proteins that cannot be resolved by cryo-EM. Here, we introduce CryoFold - a pipeline of molecular dynamics simulations that determines ensembles of protein structures directly from sequence by integrating density data of varying sparsity at 3-5 Å resolution with coarse-grained topological knowledge of the protein folds. We present six examples showing its broad applicability for folding proteins between 72 to 2000 residues, including large membrane and multi-domain systems, and results from two EMDB competitions. Driven by data from a single state, CryoFold discovers ensembles of common low-energy models together with rare low-probability structures that capture the equilibrium distribution of proteins constrained by the density maps. Many of these conformations, unseen by traditional methods, are experimentally validated and functionally relevant. We arrive at a set of best practices for data-guided protein folding that are controlled using a Python GUI.

The computer-aided investigation of protein folding has greatly benefited from coarse-grained models, that is, simplified representations at a resolution level lower than atomistic, providing access to qualitative and quantitative details of the folding process that would be hardly attainable, via all-atom descriptions, for medium to long molecules. Nonetheless, the effectiveness of low-resolution models is itself hampered by the presence, in a small but significant number of proteins, of nontrivial topological self-entanglements. Features such as native state knots or slipknots introduce conformational bottlenecks, affecting the probability to fold into the correct conformation; this limitation is particularly severe in the context of coarse-grained models. In this work, we tackle the relationship between folding probability, protein folding pathway, and protein topology in a set of proteins with a nontrivial degree of topological complexity. To avoid or mitigate the risk of incurring in kinetic traps, we make use of the elastic folder model, a coarse-grained model based on angular potentials optimized toward successful folding via a genetic procedure. This light-weight representation allows us to estimate in silico folding probabilities, which we find to anti-correlate with a measure of topological complexity as well as to correlate remarkably well with experimental measurements of the folding rate. These results strengthen the hypothesis that the topological complexity of the native state decreases the folding probability and that the force-field optimization mimics the evolutionary process these proteins have undergone to avoid kinetic traps.

Conformational sampling of biomolecules using molecular dynamics simulations often produces large amount of high dimensional data that makes it difficult to interpret using conventional analysis techniques. Dimensionality reduction methods are thus required to extract useful and relevant information. Here we devise a machine learning method, Gaussian mixture variational autoencoder (GMVAE) that can simultaneously perform dimensionality reduction and clustering of biomolecular conformations in an unsupervised way. We show that GMVAE can learn a reduced representation of the free energy landscape of protein folding with highly separated clusters that correspond to the metastable states during folding. Since GMVAE uses a mixture of Gaussians as the prior, it can directly acknowledge the multi-basin nature of protein folding free-energy landscape. To make the model end-to-end differentialble, we use a Gumbel-softmax distribution. We test the model on three long-timescale protein folding trajectories and show that GMVAE embedding resembles the folding funnel with folded states down the funnel and unfolded states outer in the funnel path. Additionally, we show that the latent space of GMVAE can be used for kinetic analysis and Markov state models built on this embedding produce folding and unfolding timescales that are in close agreement with other rigorous dynamical embeddings such as time independent component analysis (TICA).

Significance Protein-folding kinetics is often described as Markovian (i.e., memoryless) diffusion in a one-dimensional free energy landscape, governed by an instantaneous friction coefficient that is fitted to reproduce experimental or simulated folding times. For the α-helix forming polypeptide alanine9 and a specific reaction coordinate that consists of the summed native hydrogen-bond lengths, we demonstrate that the friction extracted from molecular dynamics simulations exhibits significant memory with a decay time that is in the nanosecond range and thus, of the same order as the folding and unfolding times. Our non-Markovian modeling not only reproduces the molecular dynamics simulations accurately but also demonstrates that memory friction effects lead to anomalous and drastically accelerated protein kinetics., We extract the folding free energy landscape and the time-dependent friction function, the two ingredients of the generalized Langevin equation (GLE), from explicit-water molecular dynamics (MD) simulations of the α-helix forming polypeptide alanine9 for a one-dimensional reaction coordinate based on the sum of the native H-bond distances. Folding and unfolding times from numerical integration of the GLE agree accurately with MD results, which demonstrate the robustness of our GLE-based non-Markovian model. In contrast, Markovian models do not accurately describe the peptide kinetics and in particular, cannot reproduce the folding and unfolding kinetics simultaneously, even if a spatially dependent friction profile is used. Analysis of the GLE demonstrates that memory effects in the friction significantly speed up peptide folding and unfolding kinetics, as predicted by the Grote–Hynes theory, and are the cause of anomalous diffusion in configuration space. Our methods are applicable to any reaction coordinate and in principle, also to experimental trajectories from single-molecule experiments. Our results demonstrate that a consistent description of protein-folding dynamics must account for memory friction effects.

Designing a robot or structure that can fold into a target shape is a process that involves challenges originated from multiple sources. For example, the designer of self-folding robots must consider foldability from geometric and kinematic aspects to avoid self-collisions and undesired deformations. Recent works have shown success in estimating foldability of a design using robot motion planners. However, many foldable structures are actuated using physically coupled reactions, e.g., folding originated from thermal, chemical, or electromagnetic loads. Therefore, a reliable folding process must consider additional constraints that result from these critical and coupled phenomena. This work investigates the idea of efficiently incorporating computationally intensive physics simulations within the folding motion planner to affect the physical folding results. We will use the manufacturing process of laser-forming origami as an example to demonstrate the benefits of considering the physical properties of the foldable structure beyond its kinematics. We show that designs produced by the proposed method can be fabricated more efficiently.

Standard low resolution coarse-grained modeling techniques have difficulty capturing multiple configurations of protein systems. Here, we present a method for creating accurate coarse-grained (CG) models with multiple configurations using a linear combination of functions or “states”. Individual CG models are created to capture the individual states, and the approximate coupling between the two states is determined from an all-atom potential of mean force. We show that the resulting multiconfiguration coarse-graining (MCCG) method accurately captures the transition state as well as the free energy between the two states. We have tested this method on the folding of dodecaalanine, as well as the amphipathic helix of endophilin.

For a given polygon P and a polyhedron Q, the folding problem asks if Q can be obtained from P by folding it. This simple problem is quite complicated, and there is no known efficient algorithm that solves this problem in general. In this paper, we focus on the case that Q is a box, and the size of Q is not given. That is, input of the box folding problem is a polygon P, and it asks if P can fold to boxes of certain sizes. We note that there exist an infinite number of polygons P that can fold into three boxes of different sizes. In this paper, we give a pseudo polynomial time algorithm that computes all possible ways of folding of P to boxes., WALCOM: Algorithms and Computation. WALCOM 2019.

The simplest class of structures formed by \(N\ge 2\) interacting RNAs consists of all crossing-free base pairs formed over linear arrangements of the constituent RNA sequences. For each permutation of the N strands the structure prediction problem is algorithmically very similar – but not identical – to folding of a single, contiguous RNA. The differences arise from two sources: First, “nicks”, i.e., the transitions from one to the next piece of RNA, need to be treated with special care. Second, the connectedness of the structures needs to guaranteed. For the forward recursions, i.e., the computation of folding energies or partition functions, these modifications are rather straightforward and retain the cubic time complexity of the well-known folding algorithms. This is not the case for a straightforward implementation of the corresponding outside recursion, which becomes quartic. Cubic running times, however, can be restored by introducing linear-size auxiliary arrays. Asymptotically, the extra effort over the corresponding algorithms for a single RNA sequence of the same length is negligible in both time and space. An implementation within the framework of the ViennaRNA package conforms to the theoretical performance bounds and provides access to several algorithmic variants, include the handling of user-defined hard and soft constraints.

In this paper, we study dimension folding for matrix/array structured predictors with categorical variables. The categorical variable information is incorporated into dimension folding for regression and classification. The concepts of marginal, conditional, and partial folding subspaces are introduced, and their connections to central folding subspace are investigated. Three estimation methods are proposed to estimate the desired partial folding subspace. An empirical maximal eigenvalue ratio criterion is used to determine the structural dimensions of the associated partial folding subspace. Effectiveness of the proposed methods is evaluated through simulation studies and an application to a longitudinal data.

An airbag automatic inflator was designed, which consists of a column pressure vessel, a bidirectional solenoid valve and a spherical airbag, etc. By introducing a thermodynamic model, its key parameters were analyzed and calculated according to the mass flow formula equation system. The numerical analysis models of the airbag with folding modes such as planar direct folding, symmetrical folding and roll folding were established to study the dynamic characteristics of the airbag inflating and unfolding, and the effects of the folding modes, the inflating rate and the inflating amount on the change of the curves of the volume and internal pressure of the airbag in the unfolding process were discussed respectively.

The RNA molecule is capable of folding into different shapes, with some being more stable than others. The structural space of the RNA can be described by Stochastic Context-free Grammars (SCFG), offering a probabilisitic distribution of structural scenarios. In a more accurate folding model, the probability associated with a structural scenario is more informative of its stability. Here, we offer two different ways of calculating the Shannon Entropy of the SCFG-modeled RNA structural space; Grammar Space (GS) Entropy and SCFG Entropy. The former is the Shannon Entropy of the infinite number of structures produced by the model and the latter is the Shannon Entropy of a limited subset of structures all of which belong to the same RNA sequence. After a brief introduction on the two measures, we explore the relationship between these measures on a given set of RNA folding models and biologically functional RNA sequences. We show that these two measures of entropy are indeed correlated. While more theoretical work is needed in understanding the convergence behavior between the two, this observation suggests that GS Entropy is a promising characteristic in future model training approaches.

International audience; The predictive capabilities of deep neural networks (DNNs) continue to evolve to increasingly impressive levels. However, it is still unclear how training procedures for DNNs succeed in finding parameters that produce good results for such high-dimensional and nonconvex loss functions. In particular, we wish to understand why simple optimization schemes, such as stochastic gradient descent, do not end up trapped in local minima with high loss values that would not yield useful predictions. We explain the optimizability of DNNs by characterizing the local minima and transition states of the loss-function landscape (LFL) along with their connectivity. We show that the LFL of a DNN in the shallow network or data-abundant limit is funneled, and thus easy to optimize. Crucially, in the opposite low-data/deep limit, although the number of minima increases, the landscape is characterized by many minima with similar loss values separated by low barriers. This organization is different from the hierarchical landscapes of structural glass formers and explains why minimization procedures commonly employed by the machine-learning community can navigate the LFL successfully and reach low-lying solutions.

Geometry and topology are the main factors that determine the functional properties of proteins. In this work, we show how to use the Gauss linking integral (GLN) in the form of a matrix diagram - for a pair of a loop and a tail - to study both the geometry and topology of proteins with closed loops e.g. lassos. We show that the GLN method is a significantly faster technique to detect entanglement in lasso proteins in comparison with other methods. Based on the GLN technique, we conduct comprehensive analysis of all proteins deposited in the PDB and compare it to the statistical properties of the polymers. We found that there are significantly more lassos with negative crossings than those with positive ones in proteins, the average value of maxGLN (maximal GLN between loop and pieces of tail) depends logarithmically on the length of a tail similarly as in the polymers. Next, we show the how high and low GLN values correlate with the internal exibility of proteins, and how the GLN in the form of a matrix diagram can be used to study folding and unfolding routes. Finally, we discuss how the GLN method can be applied to study entanglement between two structures none of which are closed loops. Since this approach is much faster than other linking invariants, the next step will be evaluation of lassos in much longer molecules such as RNA or loops in a single chromosome., 25 pages, 8 figures, submitted to the journal and waiting for reviews, having Supplementary Material

We used the tube model with Go-like potential for native contacts to study the folding transition of a designed three-helix bundle and a designed protein G-like structure. It is shown that both proteins in this model are two-state folders with a cooperative folding transition coincided with the collapse transition. We defined the transition states as protein conformations in a small region around the saddle point on a free energy surface with the energy and the conformational root-mean-square deviation (RMSD) from the native state as the coordinates. The transition state region on the free energy surface then was sampled by using the umbrella sampling technique. We show that the transition state ensemble is broad consisting of different conformations that have different folded and unfolded elements.

Molecular simulation trajectories represent high-dimensional data. Such data can be visualized by methods of dimensionality reduction. Non-linear dimensionality reduction methods are likely to be more efficient than linear ones due to the fact that motions of atoms are non-linear. Here we test a popular non-linear t-distributed stochastic neighbor embedding (t-SNE) method on analysis of trajectories of alanine dipeptide dynamics and Trp-cage folding and unfolding. Furthermore, we introduced a time-lagged variant of t-SNE in order to focus on slow motions in the molecular system. This time-lagged t-SNE efficiently visualizes slow dynamics of molecular systems., Comment: 7 pages, 3 figures, supplementary material (4 pages, 3 figures)

At the intersection of mathematics and biology, we find mathematical models built to address biological questions as well as new mathematical theories inspired by biological structures. In this chapter, we explore a combinatorial model for folding words around plane trees which is inspired by the bonds that form between nucleotides in a single-stranded RNA molecule. This chapter walks the reader through the construction of valid plane trees, structures formed by folding a word in a complementary alphabet around a plane tree, and enumerates the class of words with exactly one such folding. Valid plane trees are relatively unexplored combinatorial objects, and while we present several potential research projects, a careful reader can come up with many additional directions for further study.

In this chapter, we consider a special set of polygons and convex polyhedra folded from it. After giving the (counterintuitive) answers to the puzzle given in this book, we consider the folding problem of (bumpy) pyramids folded from a special set of polygons called “petal polygons”.

Inspired by the NP-hardness of string folding problems modeling the natural process of protein folding, we discuss the idea of solving instances of NP-hard problems (e.g., string folding problems) of moderate size by letting artificially assembled proteins to fold. The accuracy with which one can combinatorially model the protein folding process, e.g., by string folding, as well as the precision with which one could experimentally estimate the energy of folded artificial proteins are crucial issues.

International audience; The RNA Inverse Folding problem comes from computational biology. The goal is to find a molecule that has a given folding. It is important for scientific fields such as bioengineering, pharmaceutical research, biochemistry, synthetic biology and RNA nanostructures. Nested Monte Carlo Search has given excellent results for this problem. We propose to adapt and evaluate different Monte Carlo Search algorithms for the RNA Inverse Folding problem.

Over the past two decades, one of the standard models of protein folding has been the "two-state" model, in which a protein only resides in the folded or fully unfolded states with a single pathway between them. Recent advances in spatial and temporal resolution of biophysical measurements have revealed "beyond-two-state" complexity in protein folding, even for small, single-domain proteins. In this work, we used high-resolution optical tweezers to investigate the folding/unfolding kinetics of the B1 domain of immunoglobulin-binding protein G (GB1), a well-studied model system. Experiments were performed for GB1 both in and out of equilibrium using force spectroscopy. When the force was gradually ramped, simple single-peak folding force distributions were observed, while multiple rupture peaks were seen in the unfolding force distributions, consistent with multiple force-dependent parallel unfolding pathways. Force-dependent folding and unfolding rate constants were directly determined by both force-jump and fixed-trap measurements. Monte Carlo modeling using these rate constants was in good agreement with the force ramp data. The unfolding rate constants exhibited two different behaviors at low vs high force. At high force, the unfolding rate constant increased with increasing force, as previously reported by high force, high pulling speed force ramp measurements. However, at low force, the situation reversed and the unfolding rate constant decreased with increasing force. Taken together, these data indicate that this small protein has multiple distinct pathways to the native state on the free energy landscape.

In this paper, the induced sequence of folding and unfolding on the fundamental group will be obtained from a sequence of folding and unfolding on a manifold. The limit of folding and unfolding on the fundamental group is deduced. The sequences of the commutative diagram of fundamental groups will be achieved from the sequences of the commutative diagram of manifolds. The connection between a manifold and a fundamental group is assigned.

In this paper, the definition of a jumping folding of a graph has been introduced. The relations betweenthis type of jumping folding of the graph and its chromatic number will be obtained. Theorems governing theserelations have been introduced. Mobile radio frequency assignment has been given as an application.

Geometric folding processes are ubiquitous in natural systems ranging from protein biochemistry to patterns of insect wings and leaves. In a previous study, a folding operation between strings of formal languages was introduced as a model of such processes. The operation was then used to define a folding system (F-system) as a construct consisting of a core language, containing the strings to be folded, and a folding procedure language, which defines how the folding is done. This paper reviews main definitions associated with F-systems and next it determines necessary conditions for a language to belong to classes generated by such systems. The conditions are stated in the form of pumping lemmas and four classes are considered, in which the core and folding procedure languages are both regular, one of them is regular and the other context-free, or both are context-free. Full demonstrations of the lemmas are provided, and the analysis is illustrated with examples., 12 pages, 6 figures. This is a preprint (pre-refereeing) version of a manuscript accepted for publication in Natural Computing

Made available in DSpace on 2020-12-10T19:41:27Z (GMT). No. of bitstreams: 0 Previous issue date: 2019-11-01 Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) NSF CAREER award Center for Theoretical Biological Physics - NSF Protein folding occurs in a high dimensional phase space, and the representation of the associated energy landscape is nontrivial. A widely applied approach to studying folding landscapes is to describe the dynamics along a small number of reaction coordinates. However, other strategies involve more elaborate analysis of the complex phase space. There have been many attempts to obtain a more detailed representation of all available conformations for a given system. In this work, we address this problem using a metric based on internal distances between amino acids to describe the differences between any two conformations. Using an effective projection method, we are able to go beyond the typical one-dimensional representation and provide intuitive two dimensional visualizations of the landscape. We refer to this method as the energy landscape visualization method (ELViM). We have applied this methodology using a C-alpha structure-based model to study the folding of two well-known proteins: SH3 domain and protein-A. Our visualization method yields a detailed description of the folding process, making possible the identification of transition state regions, and establishing the paths that lead to the native state. For SH3, we have analyzed structural differences in the distribution of folding routes. The competition between the native and mirror structures in protein A is also discussed. Finally, the method is applied to study conformational changes in the protein elongation factor thermally unstable. Distinct features of ELViM are that it does not require or assume a reaction coordinate, and it does not require analysis of kinetic aspects of the system. Univ Estadual Paulista, Inst Biociencias Letras & Ciencias Exatas, Dept Fis, BR-15054000 Sao Jose Do Rio Preto, SP, Brazil Northeastern Univ, Dept Phys, Boston, MA 02115 USA Rice Univ, Ctr Theoret Biol Phys, Houston, TX 77005 USA Univ Estadual Paulista, Inst Biociencias Letras & Ciencias Exatas, Dept Fis, BR-15054000 Sao Jose Do Rio Preto, SP, Brazil FAPESP: 2018/18668-1 FAPESP: 2016/19766-1 FAPESP: 2014/50739-5 NSF CAREER award: MCB-1350312 Center for Theoretical Biological Physics - NSF: PHY-1427654

Origami-inspired devices have been widely applied in engineering for converting two-dimensional materials into innovative and functional three-dimensional objects. This paper proposed a novel sensing method for identifying folding angles, which was able to be employed to foldable materials like a sheet of paper with no constraint. A fundamental examination was conducted in which a proposed fiber wiring pattern enabled to convert folding angle into moderate curvature of an optical fiber, and therefore the optical fiber detected the folding as an optical loss changes. Furthermore, the sensitivity to folding angle could be enhanced by introducing a hetero-core fiber optic macro-bending sensor.

Coarse-grained models are an attractive tool for studying the long time scale dynamics of large macromolecules at a level that cannot be studied directly by experiment and is still out of reach for atomistic simulation. However, coarse models involve approximations that may affect their predictive power. We propose a modeling framework that allows us to design simplified models to accurately reproduce experimental observables. We demonstrate the approach on the folding mechanism of a WW domain. We show that when the correct coarsening resolution is used not only do the optimized models match the Reference model simulated experimental data accurately but additional observables not directly targeted during the optimization procedure are also reproduced. Additionally, the analysis of the results shows that localized frustration plays an important role in the folding mechanism of this protein and suggests that nontrivial aspects of the protein dynamics are evolutionary conserved.

Structural analysis and reconstruction of folding of the Ukrainian (flysh) Carpathians have been executed according to materials (cross-sections) of medium-scale geological survey. Methodic development has been used within the limits of bending and out-of-flow theories. It has been shown with the help of asymmetric folding simulation that the recapture of folds into intermediate deformed state by removal of out-of-line flow and the statistic search of the most probable orientation of compression axis is the effective method of diagnostics. Conditions of appearance of Carparhians structures have been identified which can be a result of both longitudinal-transverse bending and horizontal out-of-line flow superimposed upon initially symmetric or asymmetrical bending folds. Within the limits of models of out-of-line deformation inclinations of compression axis and intensity of deformation are being determined which finally resulted in observed folding forms, specification of folds from different structural-facial zones of Carpathians according to the grade of their compactness and asymmetry has been shown, their genetic interpretation has been given.

We show how multi-ensemble Markov state models can be combined with constant-force equilibrium simulations. Besides obtaining the unfolding/folding rates, Markov state models allow gaining detailed insights into the folding dynamics and pathways through identifying folding intermediates and misfolded structures. For two specific peptides, we demonstrate that the end-to-end distance is an insufficient reaction coordinate. This problem is alleviated through constructing models with multiple collective variables, for which we employ the time-lagged independent component analysis requiring only minimal prior knowledge. Our results show that combining Markov state models with constant-force simulations is a promising strategy to bridge the gap between simulation and experiments even for medium-sized biomolecules.

Protein folding is a central challenge in computational biology, with important applications in molecular biology, drug discovery and catalyst design. As a hard combinatorial optimisation problem, it has been studied as a potential target problem for quantum annealing. Although several experimental implementations have been discussed in the literature, the computational scaling of these approaches has not been elucidated. In this article, we present a numerical study of quantum annealing applied to a large number of small peptide folding problems, aiming to infer useful insights for near-term applications. We present two conclusions: that even naive quantum annealing, when applied to protein lattice folding, has the potential to outperform classical approaches, and that careful engineering of the Hamiltonians and schedules involved can deliver notable relative improvements for this problem. Overall, our results suggest that quantum algorithms may well offer improvements for problems in the protein folding and structure prediction realm., Comment: 45 pages, 18 figures

Macromolecular systems are composed of a very large number of atomic degrees of freedom. There is strong evidence suggesting that structural changes occurring in large biomolecular systems at long time scale dynamics may be captured by models coarser than atomistic, although a suitable or optimal coarse-graining is a priori unknown. Here we propose a systematic approach to learning a coarse representation of a macromolecule from microscopic simulation data. In particular, the definition of effective coarse variables is achieved by partitioning the degrees of freedom both in the structural (physical) space and in the conformational space. The identification of groups of microscopic particles forming dynamical coherent states in different metastable states leads to a multiscale description of the system, in space and time. The application of this approach to the folding dynamics of two proteins provides a revised view of the classical idea of prestructured regions (foldons) that combine during a protein-folding process and suggests a hierarchical characterization of the assembly process of folded structures.

Origami structures, such as those with a Miura-Ori fold pattern, have one principle direction of kinematic freedom where rigid foldability holds true. However, motions such as twisting and curling, and bending require that origami panels bend, thus adding complexity to the system. The goal of this work is to study the actuation of Miura-Ori structures for damage detection. The combination of form-finding for large displacements and finite element model for dynamic characterization is successful to capture the kinematic properties of the Miura-Ori origami structure movements of folding and curling. Simulations and experimental tests are useful as a starting point for determining the behavior of mid-scale actuated origami structures. Stiffness values for folding of creases and bending of panels must be considered when modeling shape change of Miura-Ori origami structures using dynamic relaxation. Energy produced by the folding and bending stiffness values activated in the structure through actuation is a useful measure to rank elements in terms of their potential for damage.

Origami robots self-reconfigure from a quasi two-dimensional manufactured state to three-dimensional mobile robots. By folding, they excel in transforming their initial spatial configuration to expand their functionalities. However, unlike paper-based origamis, where the materials can remain homogeneous, origami robots require varying payloads and controllability of their reconfigurations. Therefore, the mechanisms to achieve automated folding adapt flat thin panels and folding hinges that are often of different materials to achieve the folding. While the fundamental working principle of an origami hinge remains simple, these multi-component, multi-material origami joints can no longer be modeled by beam theory without considering the semi-rigid connections at the material interfaces. Currently, there is no comprehensive model to analyze physical behavior of an actuated folding hinge accurately. In this work, we propose a model based on the plate theory to predict the origami folding joint: we adapt a torsional spring to capture this semi-rigid connection, predict the folding stiffness and bending of origami joints. Herein, the semi-rigid connection is calibrated by quasi-static folding tests on a series of physical origami folding joints, and the accuracy of our model is compared to finite element simulations. With this analytical model, we can accurately simulate the mechanics of physical origami folding joints. (C) 2021 Published by Elsevier Ltd.

We address the accurate single arm robotic garment folding. The folding capability is influenced mostly by the folding path which is performed by the robotic arm. This paper presents a new method for the folding path generation based on the static equilibrium of forces. The existing approach based on a similar principle confirmed to be accurate for one-dimensional strips only. We generalize the method to two-dimensional shapes by modeling the garment as an elastic shell. The path generated by our method prevents the garment from slipping while folding on a low friction surface. We demonstrate the accuracy of this approach by comparing our paths (a) with the existing method when one-dimensional strips of different materials were modeled, and (b) experimentally with real robotic folding.

It has been a long-standing and intriguing issue to develop robust methods to identify metastable states and interstate transitions from simulations or experimental data to understand the functional conformational changes of proteins. It is usually hard to define the complicated boundaries of the states in the conformational space using most of the existing methods, and they often lead to parameter-sensitive results. Here, we present a new approach, visualized Trajectory Map (vTM), to identify the metastable states and the rare interstate transitions, by considering both the conformational similarity and the temporal successiveness of conformations. The vTM is able to give a nonambiguous description of slow dynamics. The case study of a β-hairpin peptide shows that the vTM can reveal the states and transitions from all-atom MD trajectory data even when a single observable (i.e, one-dimensional reaction coordinate) is used. We also use the vTM to refine the folding/unfolding mechanism of HP35 in explicit water by analyzing a 125 μs all-atom MD trajectory and obtain folding/unfolding rates of about 1/μs, which are in good agreement with the experimental values.

The purpose of this short paper is to describe a new proof of the Pythagorean Theorem that involves paper folding.

We propose a new approach for modelling the process of RNA folding as a graph transformation guided by the global value of free energy. Since the folding process evolves towards a configuration in which the free energy is minimal, the global behaviour resembles the one of a self-adaptive system. Each RNA configuration is a graph and the evolution of configurations is constrained by precise rules that can be described by a graph grammar., In Proceedings GaM 2016, arXiv:1612.01053

This article investigates (experimentally and numerically) the time and temperature-dependent folding and unfolding behavior, including the structural nonlinearity, of thin-walled softenable composite structures. The composites are fabricated with woven fabric fibers and a heat softenable material of the shape memory polymer (SMP) resin. A numerical model is developed. The model is based on the elastic properties of the woven fabric fibers, their fabric geometry and the thermo-viscoelastic properties of the SMP resin. A thin-walled cylindrical boom, which can be flexibly folded into an arbitrary configuration and be self-deployed, is fabricated using the heat softenable composite. The viscoelasticity and the shape memory behaviors are investigated during the folding and unfolding of the boom at elevated temperatures.

The stretching of fluid elements combined with folding leads to a fluid dispersive property, which is an indicator of chaotic advection. A numerical calculation method was proposed for a di...

Made available in DSpace on 2020-12-10T19:36:37Z (GMT). No. of bitstreams: 0 Previous issue date: 2019-09-21 Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) National Science Foundation (NSF) National Science Foundation CAREER Award The stochastic drift-diffusion (DrDiff) theory is an approach used to characterize the dynamical properties of simulation data. With new features in transition times analyses, the framework characterized the thermodynamic free-energy profile [F(Q)], the folding time (tau(f)), and transition path time (tau(TP)) by determining the coordinate-dependent drift-velocity [v(Q)] and diffusion [D(Q)] coefficients from trajectory time traces. In order to explore the DrDiff approach and to tune it with two other methods (Bayesian analysis and fep1D algorithm), a numerical integration of the Langevin equation with known D(Q) and F(Q) was performed and the inputted coefficients were recovered with success by the diffusion models. DrDiff was also applied to investigate the prion protein (PrP) kinetics and thermodynamics by analyzing folding/unfolding simulations. The protein structure-based model, the well-known Go over bar -model, was employed in a coarse-grained C-alpha level to generate long constant-temperature time series. PrP was chosen due to recent experimental single-molecule studies in D and tau(TP) that stressed the importance and the difficulty of probing these quantities and the rare transition state events related to prion misfolding and aggregation. The PrP thermodynamic double-well F(Q) profile, the X shape of tau(f)(T), and the linear shape of tau(TP)(T) were predicted with v(Q) and D(Q) obtained by the DrDiff algorithm. With the advance of single-molecule techniques, the DrDiff framework might be a useful ally for determining kinetic and thermodynamic properties by analyzing time observables of biomolecular systems. The code is freely available at . Univ Fed Triangulo Mineiro, Inst Ciencias Exatas Nat & Educ, Dept Fis, Lab Biofis Teor, Uberaba, MG, Brazil Univ Fed ABC, Lab Biol Computac & Bioinformat, Santo Andre, SP, Brazil Rice Univ, Ctr Theoret Biol Phys, Houston, TX 77005 USA Univ Estadual Paulista, Dept Fis, Sao Jose Do Rio Preto, SP, Brazil Brazilian Ctr Res Energy & Mat CNPEM, Brazilian Biorenewables Natl Lab LNBR, Campinas, SP, Brazil Northeastern Univ, Dept Phys, Boston, MA 02115 USA Univ Estadual Paulista, Dept Fis, Sao Jose Do Rio Preto, SP, Brazil FAPEMIG: APQ-00941-14 FAPEMIG: CEX-RED-00010-14 CNPq: 438316/2018-5 FAPESP: 2016/13998-8 FAPESP: 2017/09662-7 National Science Foundation (NSF): PHY-1427654 National Science Foundation CAREER Award: MCB-1350312

We have developed an enhanced conformational sampling method combining replica-exchange umbrella sampling (REUS) with Gaussian accelerated molecular dynamics (GaMD). REUS enhances the sampling along predefined reaction coordinates, while GaMD accelerates the conformational dynamics by adding a boost potential to the system energy. The method, which we call GaREUS (Gaussian accelerated replica-exchange umbrella sampling), enhances the sampling more efficiently than REUS or GaMD, while the computational resource for GaREUS is the same as that required for REUS. The two-step reweighting procedure using the multistate Bennett acceptance ratio method and the cumulant expansion for the exponential average is applied to the simulation trajectories for obtaining the unbiased free-energy landscapes. We apply GaREUS to the calculations of free-energy landscapes for three different cases: conformational equilibria of N-glycan, folding of chignolin, and conformational change of adenyl kinase. We show that GaREUS speeds up the convergences of free-energy calculations using the same amount of computational resources as REUS. The free-energy landscapes reweighted from the trajectories of GaREUS agree with previously reported ones. GaREUS is applicable to free-energy calculations of various biomolecular dynamics and functions with reasonable computational costs.

Although a wide variety of quantum computers are currently being developed, actual computational results have been largely restricted to contrived, artificial tasks. Finding ways to apply quantum computers to useful, real-world computational tasks remains an active research area. Here we describe our mapping of the protein design problem to the D-Wave quantum annealer. We present a system whereby Rosetta, a state-of-the-art protein design software suite, interfaces with the D-Wave quantum processing unit to find amino acid side chain identities and conformations to stabilize a fixed protein backbone. Our approach, which we call the QPacker, uses a large side-chain rotamer library and the full Rosetta energy function, and in no way reduces the design task to a simpler format. We demonstrate that quantum annealer-based design can be applied to complex real-world design tasks, producing designed molecules comparable to those produced by widely adopted classical design approaches. We also show through large-scale classical folding simulations that the results produced on the quantum annealer can inform wet-lab experiments. For design tasks that scale exponentially on classical computers, the QPacker achieves nearly constant runtime performance over the range of problem sizes that could be tested. We anticipate better than classical performance scaling as quantum computers mature.

Computing surface folding maps has numerous applications ranging from computer graphics to material design. In this work we propose a novel way of computing surface folding maps via solving a linear PDE. This framework is a generalization of the existing computational quasi-conformal geometry and allows precise control of the geometry of folding. This property comes from a crucial quantity that occurs as the coefficient of the equation, namely, the alternating Beltrami coefficient. This approach also enables us to solve an inverse problem of parametrizing the folded surface given only partial data with known folding topology. Various interesting applications such as fold sculpting on 3 dimensional models, study of Miura-ori patterns, and self-occlusion reasoning are demonstrated to show the effectiveness of our method.

State-free reversible VAMPnets (SRVs) are a neural network-based framework capable of learning the leading eigenfunctions of the transfer operator of a dynamical system from trajectory data. In molecular dynamics simulations, these data-driven collective variables (CVs) capture the slowest modes of the dynamics and are useful for enhanced sampling and free energy estimation. In this work, we employ SRV coordinates as a feature set for Markov state model (MSM) construction. Compared to the current state of the art, MSMs constructed from SRV coordinates are more robust to the choice of input features, exhibit faster implied timescale convergence, and permit the use of shorter lagtimes to construct higher kinetic resolution models. We apply this methodology to study the folding kinetics and conformational landscape of the Trp-cage miniprotein. Folding and unfolding mean first passage times are in good agreement with prior literature, and a nine macrostate model is presented. The unfolded ensemble comprises a central kinetic hub with interconversions to several metastable unfolded conformations and which serves as the gateway to the folded ensemble. The folded ensemble comprises the native state, a partially unfolded intermediate "loop" state, and a previously unreported short-lived intermediate that we were able to resolve due to the high time-resolution of the SRV-MSM. We propose SRVs as an excellent candidate for integration into modern MSM construction pipelines.

Nearest neighbor parameters for estimating the folding stability of RNA are commonly used in secondary structure prediction, for generating folding ensembles of structures, and for analyzing RNA function. Previously, we demonstrated that we could quantify the uncertainties in each nearest neighbor parameter by perturbing the underlying optical melting data within experimental error and rederiving the parameters, which accounts for the substantial correlations that exist between the parameters. In this contribution, we describe a method to estimate uncertainty in the estimated folding stabilities of RNA structures, accounting for correlations in the nearest neighbor parameters. This method is incorporated in the RNA structure software package.

Single-molecule force spectroscopy has proven extremely beneficial in elucidating folding pathways for membrane proteins. Here, we simulate these measurements, conducting hundreds of unfolding trajectories using our fast Upside algorithm for slow enough speeds to reproduce key experimental features that may be missed using all-atom methods. The speed also enables us to determine the logarithmic dependence of pulling velocities on the rupture levels to better compare to experimental values. For simulations of atomic force microscope measurements in which force is applied vertically to the C-terminus of bacteriorhodopsin, we reproduce the major experimental features including even the back-and-forth unfolding of single helical turns. When pulling laterally on GlpG to mimic the experiment, we observe quite different behavior depending on the stiffness of the spring. With a soft spring, as used in the experimental studies with magnetic tweezers, the force remains nearly constant after the initial unfolding event, and a few pathways and a high degree of cooperativity are observed in both the experiment and simulation. With a stiff spring, however, the force drops to near zero after each major unfolding event, and numerous intermediates are observed along a wide variety of pathways. Hence, the mode of force application significantly alters the perception of the folding landscape, including the number of intermediates and the degree of folding cooperativity, important issues that should be considered when designing experiments and interpreting unfolding data.