Elementary Folding Research Articles

Energetics of graphene origami and their “spatial resolution”

The extreme thinness of graphene combined with its tensile strength made it a material appealing for discussing and even making complex cut-kirigami or folded-only origami. In the case of origami, its stability is mainly defined by the positive energy of the single- or double-fold curvature deformation counterbalanced by the energy reduction due to favorable van der Waals contacts. These opposite sign contributions also have notably different scaling with the size L of the construction, the contacts contributing in proportion to area ~ L2, single folds as ~ L, and highly strained double-fold corners as only ~ L0 = const. Computational analysis with realistic atomistic-elastic representation of graphene allows one to quantify these energy contributions and to establish the length scale, where a single fold is favored (7 nm 21 nm), defining the size of the smallest possible complex origami designs as L ≫ 21 nm. The flexibility and foldability of graphene are some of its attractive properties inspiring the designs of origami structures with potential use in flexible electronics and electromechanical nanodevices. The aesthetics, precision, and ease of folding and stability, however, have limitations at the nanoscale. Here, by means of large-scale atomistic calculations and continuum models, it is quantified how the dimensions determine the relative robustness of the elementary folds of graphene (a single fold and a double-folded graphene forming a single order-four vertex), thereby mapping the spatial resolution limits and providing important guidance for graphene nano-origami realizations.

Lessons about Protein Folding and Binding from Archetypal Folds.

The function of proteins as biological nanomachines relies on their ability to fold into complex 3D structures, bind selectively to partners, and undergo conformational changes on cue. The native functional structures, and the rates of interconversion between conformational states (folded-unfolded, bound-free), are all encoded in the physical chemistry of their amino acid sequence. However, despite extensive research over decades, this code has proven difficult to fully crack, in terms of both prediction and understanding the molecular mechanisms at play.Earlier work on single-domain proteins reported a commonality of slow rates (10-2-102 s-1) and simple behavior in both kinetic and thermodynamic unfolding experiments, which suggested the process was all-or-none and thereby analogous to a chemical reaction (e.g., A ⇄ B). In the absence of a first-principles pre-exponential factor for protein (un)folding dynamics, the rates could only be interpreted in relative terms, e.g., the changes induced by mutation, and hence, neither the height of nor the entropic contribution to the free energy barriers was known. The rates were also many orders of magnitude too slow for direct atomistic simulations, and the computational focus was on predicting rate changes induced by mutation via coarse grained simulations. However, even the effects of mutation proved to be strikingly homogeneous with all experimental data clustering at ∼1/3 of the free energy perturbation recovered on folding and ∼2/3 on unfolding.The implementation of ultrafast kinetic methods turned the field upside down because they allowed researchers to measure the time scales of elementary (un)folding motions, which set the pre-exponential factor for protein conformational transitions at ∼1 μs. In parallel, we and others set out to investigate the simplest possible protein structures capable of autonomous folding, which we defined as archetypal folds. The rationale was to recapitulate the hierarchical organization of protein structure, starting from the bottom up. The study of fold archetypes ended up opening new research avenues in protein (un)folding, but also making unexpected connections with the folding upon binding of intrinsically disordered proteins and suggesting their functioning as conformational rheostats.This Account describes our work on the kinetic, thermodynamic, mechanistic, and functional analysis of fold archetypes. We first discuss the kinetic studies, emphasizing their impact on our understanding of (un)folding rates, of barrierless (downhill) folding, and as benchmarks for atomistic simulations. We continue with the thermodynamic analysis, introducing the differential scanning calorimetry, multiprobe, and NMR approaches that we developed to dissect their gradual, minimally cooperative (un)folding transitions and to probe the underlying mechanisms with unprecedented detail. The last two sections cover single-molecule analyses and some recent, mostly computational, results on the exploration of possible biological and technological roles for the gradual conformational transitions of fold archetypes.

Automatic labeling of cortical sulci using patch- or CNN-based segmentation techniques combined with bottom-up geometric constraints.

The extreme variability of the folding pattern of the human cortex makes the recognition of cortical sulci, both automatic and manual, particularly challenging. Reliable identification of the human cortical sulci in its entirety, is extremely difficult and is practiced by only a few experts. Moreover, these sulci correspond to more than a hundred different structures, which makes manual labeling long and fastidious and therefore limits access to large labeled databases to train machine learning. Here, we seek to improve the current model proposed in the Morphologist toolbox, a widely used sulcus recognition toolbox included in the BrainVISA package. Two novel approaches are proposed: patch-based multi-atlas segmentation (MAS) techniques and convolutional neural network (CNN)-based approaches. Both are currently applied for anatomical segmentations because they embed much better representations of inter-subject variability than approaches based on a single template atlas. However, these methods typically focus on voxel-wise labeling, disregarding certain geometrical and topological properties of interest for sulcus morphometry. Therefore, we propose to refine these approaches with domain specific bottom-up geometric constraints provided by the Morphologist toolbox. These constraints are utilized to provide a single sulcus label to each topologically elementary fold, the building blocks of the pattern recognition problem. To eliminate the shortcomings associated with the Morphologist's pre-segmentation into elementary folds, we complement this regularization scheme using a top-down perspective which triggers an additional cleavage of the elementary folds when required. All the newly proposed models outperform the current Morphologist model, the most efficient being a CNN U-Net-based approach which carries out sulcus recognition within a few seconds.

A Naturally Occurring Peptide with an Elementary Single Disulfide-Directed β-Hairpin Fold

Certain peptide folds, owing to a combination of intrinsic stability and resilience to amino acid substitutions, are particularly effective for the display of diverse functional groups. Such "privileged scaffolds" are valuable as starting points for the engineering of new bioactive molecules. We have identified a precursor peptide expressed in the venom gland of the marine snail Conus victoriae, which appears to belong to a hitherto undescribed class ofmolluscan neuropeptides. Mass spectrometry matching with the venom confirmed the complete mature peptide sequence as a 31-residue peptide with a single disulfide bond. Solution structure determination revealed a unique peptide fold that we have designated the single disulfide-directed β hairpin (SDH). The SDH fold is highly resistant to thermal denaturation and forms the core of several other multiple disulfide-containing peptide folds, including the inhibitor cystine knot. This elementary fold may offer a valuable starting point for the design and engineering of new bioactive peptides.

A METHOD TO PREDICT EDGE STRANDS IN BETA-SHEETS FROM PROTEIN SEQUENCES

There is a need for rules allowing three-dimensional structure information to be derived from protein sequences. In this work, consideration of an elementary protein folding step allows protein sub-sequences which optimize folding to be derived for any given protein sequence. Classical mechanics applied to this system and the energy conservation law during the elementary folding step yields an equation whose solutions are taken over the field of rational numbers. This formalism is applied to beta-sheets containing two edge strands and at least two central strands. The number of protein sub-sequences optimized for folding per amino acid in beta-strands is shown in particular to predict edge strands from protein sequences. Topological information on beta-strands and loops connecting them is derived for protein sequences with a prediction accuracy of 75%. The statistical significance of the finding is given. Applications in protein structure prediction are envisioned such as for the quality assessment of protein structure models.

Time-resolved circular dichroism: Application to the study of conformal changes in biomolecules

Circular dichroism (CD) is known to be a very sensitive probe of the conformation of molecules and biomolecules. It is therefore tempting to implement CD in a pump-probe experiment in order to measure ultrarapid conformational changes which occur in photochemical processes. We present two technical developments of such time-resolved CD experiments. The first one relies on the modulation of the probe polarization from left to right circular whereas the second one measures the pump-induced ellipticity of the probe with a Babinet-Soleil compensator. Some applications are described and extension of these techniques towards the study of elementary protein folding processes is discussed.

A Universal Type IA Topoisomerase Fold

A class of enzymes, called DNA topoisomerases, is responsible for controlling the topological state of cellular DNA. Among these, type IA topoisomerases form a vast family that is present in all living organisms, including higher eukaryotes, in which they play important roles in genome stability. The known 3D structures of three of these enzymes indicate that they share a common toroidal architecture. We previously showed that the toroidal structure could be split off from the core enzyme of Thermotoga maritima topoisomerase I by limited proteolysis. This structure is produced by the association of two tandemly repeated elementary folds in a head-to-tail orientation. By using a combination of structural and sequence data analysis, we show that the elementary fold of about 150 amino acid residues, referred to as the topofold, is likely to be present in the whole topoisomerase IA family. Within each enzyme, the successive topofolds share two conserved sequence motifs located at the base of the ring, and referred to as the MI and MII motifs. However, the overall sequences of the folds have largely diverged. By contrast, secondary and tertiary structures appear remarkably conserved. We suggest that this twofold repeat has evolved by gene duplication/fusion from an ancestral topofold.

Single-molecule transition-state analysis of RNA folding.

How RNA molecules fold into functional structures is a problem of great significance given the expanding list of essential cellular RNA enzymes and the increasing number of applications of RNA in biotechnology and medicine. A critical step toward solving the RNA folding problem is the characterization of the associated transition states. This is a challenging task in part because the rugged energy landscape of RNA often leads to the coexistence of multiple distinct structural transitions. Here, we exploit single-molecule fluorescence spectroscopy to follow in real time the equilibrium transitions between conformational states of a model RNA enzyme, the hairpin ribozyme. We clearly distinguish structural transitions between effectively noninterchanging sets of unfolded and folded states and characterize key factors defining the transition state of an elementary folding reaction where the hairpin ribozyme's two helical domains dock to make several tertiary contacts. Our single-molecule experiments in conjunction with site-specific mutations and metal ion titrations show that the two RNA domains are in a contact or close-to-contact configuration in the transition state even though the native tertiary contacts are at most partially formed. Such a compact transition state without well formed tertiary contacts may be a general property of elementary RNA folding reactions.

Bifurcation Analysis of the Trans ↔ Cis Rotation of Monorotor Molecules

AbstractThe bifurcation analyses of the internal rotation of three different series of monorotor molecules are investigated in this paper. Our results indicate that the types of catastrophe and topological regions are related to the generic form of potential energy function and the critical points. Namely, when the double‐barrier energy function is represented as the third‐order generic polynomial of the reduced reaction coordinate, the process is isomorphic to the elementary fold catastrophe. Similarly, when the triple‐barrier energy functions are represented as the fourth‐order generic polynomial of the reduced reaction coordinate, the process is isomorphic to the elementary cusp catastrophe.

Characterization of internal rotation of monorotor molecules via bifurcation analysis

The bifurcation analyses of the internal rotation processes of two different series of monorotor molecules along the reduced reaction coordinate are investigated in this Letter. Our topological analyses indicate that the internal rotation processes are characterized by the properties of critical points. Furthermore, when the energy functions are represented as the third-order generic polynomial of the reduced reaction coordinate, the processes are isomorphic to the elementary fold catastrophe.

Back to Units of Protein Folding

In response to the criticism by A. Finkelstein (J. Biomol. Struct. Dyn. 20, 311–314, 2002) of our Communication (J. Biomol. Struct. Dyn. 20, 5–6, 2002) several issues are dealt with. Importance of the notion of elementary folding unit, its size and structure, and the necessity of further characterization of the units for the elucidation of the protein folding in vivo are discussed. The criticism (J. Biomol. Struct. Dyn. 20, 311–314, 2002) on the hierarchical protein folding is also briefly addressed.

ENTROPIES OF BRAIDS

Pseudo-Anosov homeomorphisms are classified by their invariant train tracks. The decomposition of any train track map in elementary folding maps gives a normal form for each train track class. In the case of 4-braids there are three train tracks classes and we give an explicit automaton that generates a normal form for each class. This enables us, for instance, to exhibit the pseudo-Anosov 4-braid with the minimal growth rate. We also show that the growth rate of a pseudo-Anosov braid appears as a root of the Alexander polynomial of a link that shares a common sub-link with the closure of the braid. We finally give a criterion for the faithfulness of the Burau representation for 4-braids.

La structure pliée des espaces de représentation : théorie élémentaire

The folded structure of representational spaces : elementary theory. A conceptual framework is defined to describe the structure of the universe represented by a subject {representational space). This construction is intended to be enough rich to be relevant in numerous fields as psychopathology. A system of representation is developed from an analysis of the relations between analogical and symbolic representations. This system is ruled by a principle of unification of analogical representations, which directly display a content. Representational space is constructed from elementary folds defined as analogical representations that can be actualized in working memory. Elementary folds can be connected by symbolic structures in unifying representations. Representational space is pleated because of some objective constraints and folded because of the finite characteristics of working memory. Several integration degrees of a unifying representation are specified. Some processes of complexification are described. Defences processes are defined in this framework.

The relationship between sequence and structure in elementary folding units

Publisher Summary This chapter describes the relationship between sequence and structure in α-helices, β-hairpins, and β- sheets. First, there is an analysis regarding the individual conformational preferences of 20 amino acids. These tendencies could explain the conformational preferences detected in the random coil state, as well as the secondary structure propensities of the 20 amino acids. This is followed by a description of the extent of the knowledge regarding α-helices. The α-helix is the best known and most easily recognized of the polypeptide regular structures. The chapter focuses on β-hairpins and β-sheets, about which much less is known. Finally, the importance of secondary structure elements in protein folding and stability is discussed. Kinetic analysis of proteins with stabilized α-helices demonstrates that an increase in the contribution of favorable local native interactions can, in some cases, optimize folding rates. Essentially, when a secondary structure element is folded in the transition state, its stabilization will accelerate folding. As in the case of the protein stabilization the increase in folding speed will not be equivalent to the stabilization of the secondary structure element, since the denatured state will also be stabilized.

Identifying the structural boundaries of independent folding domains in the alpha subunit of tryptophan synthase, a beta/alpha barrel protein.

Two equilibrium intermediates have previously been observed in the urea denaturation of the alpha subunit of tryptophan synthase (alphaTS) from Escherichia coli, an eight-stranded beta/alpha barrel protein. In the current study, a series of amino-terminal fragments were characterized to probe the elementary folding units that may be in part responsible for this complex behavior. Stop-codon mutagenesis was used to produce eight fragments ranging in size from 105-214 residues and containing incremental elements of secondary structure. Equilibrium studies by circular dichroism indicate that all of these fragments are capable of adopting secondary structure. All except for the shortest fragment fold cooperatively. The addition of the fourth, sixth, and eighth beta-strands leads to distinct increases in structure, cooperativity, and/or stability, suggesting that folding involves the modular assembly of betaalphabeta supersecondary structural elements. One-dimensional NMR titrations at high concentrations of urea, probing the environment around His92, were also performed to test for the presence of residual structure in the fragments. All fragments that contained the first four betaalpha units of structure exhibited a cooperative unfolding transition at high concentrations of urea with significant but reduced stability relative to the full-length protein. These results suggest that the residual structure in alphaTS requires the participation of hydrophobic residues in multiple beta-strands that span the entire sequence.

Stability of a nematic mesophase with internal degrees of freedom Analysis in terms of catastrophe theory

Abstract A model of a nematic mesophase with internal degrees of freedom is developed using the variational principle by means of catastrophe theory methods. It is shown that the free energy of the model being a function of two order and three control parameters is represented in the form of the superposition of local potentials corresponding to elementary fold and swallow tail catastrophes according to Thom's classification. All of the physical conclusions obtained are a consequence of this result. By means of catastrophe theory a bifurcation set (separatrix) of the model is built which divides the control parameters namely the effective molecular length (∊), their rigidity (γ) and the dimensionless temperature (t) of the system space into six non-intersecting parts each parametrizing qualitatively similar potentials. The lower and upper temperature boundaries of the isotropic liquid and orientationally ordered states, respectively, are determined as sub-sets of the bifurcation set in the control paramet...

The construction of Platonic bodies from constant width continuous strips

In this paper we show how to build the five Platonic bodies, starting from a constant width continuous strip, by means of elementary manual operations, knots and folds, without performing measurements or using geometric instruments. Minimal area developments are obtained by a computational method. It is particularly remarkable that simple knots performed on a continuous strip generate regular pentagons, and that by knotting these knots one may obtain, in a surprising way, a perfect dodecahedron.

A theory of concentric, kink, and sinusoidal folding and of monoclinal flexuring of compressible, elastic multilayers: III. Transition from sinusoidal to concentric-like to chevron folds

A basic, sinusoidal solution to the linearized equations of equilibrium for compressible, elastic materials provides solutions to several problems of folding of multilayers. Theoretical wavelengths are comparable to those predicted by Ramberg, using viscosity theory, and to those predicted by elementary folding theory. The linearized analysis of buckling of a single, stiff, elastic layer, either isolated or within a soft medium, suggests that wavelengths computed with elementary beam theory are remarkably similar to those computed with the linearized theory for wavelength-to-thickness ratios greater than about five. This is half the limit of ten normally assumed for use of the elementary theory. The theory and experiments with deep beams of rubber or gelatin indicate that thick, homogeneous layers folded with short wavelengths assume internal forms strikingly similar to those of the ideal concentric fold. Thus, mechanical layering clearly is not required to produce concentric-like forms. Further, the theory suggests that “arc and cusp” structure, or “pinches”, at edges of deep beams as well as chevron-like forms in single or multiple stiff layers are a result of a peculiar, plastic-like behavior of elastic materials subjected to high normal stresses parallel to layering. In a sense, the elastic material “yields” to form the hinge of the chevron fold, although the strain vanishes if the stresses are released. Accordingly, it may be impossible to distinguish chevron forms produced in elastic-plastic materials, such as cardboard or aluminum and perhaps some rock, from chevron forms produced in purely elastic materials, such as rubber. Analysis of the theory shows that, just as high axial stresses make straight, shortened multilayers the unstable form and sinusoidal waves the stable form, stresses induced by sinusoidal displacements of the multilayer make the sinusoidal waveform unstable and concentric-like waves the stable form. Thus, concentric-like folds appear to be typical of folded multilayers according to our analysis. Further, where the layers have short wavelengths in the cores of the concentric-like folds, the stiff layers “yield” elastically at hinges and straighten in limbs. Thus the concentric-like pattern is replaced by chevron folds as the multilayer is shortened. In this way we can understand the sequence of events from uniform shortening, to sinusoidal folding, to concentric-like folding, to chevron folding in multilayers composed of elastic materials.