For an action of a finite group on a finite EI quiver, we construct its ‘orbifold’ quotient EI quiver. The free EI category associated to the quotient EI quiver is equivalent to the skew group category with respect to the given group action. Specializing the result to a finite group action on a finite acyclic quiver, we prove that, under reasonable conditions, the skew group category of the path category is equivalent to a finite EI category of Cartan type. If the ground field is of characteristic $p$ and the acting group is a cyclic $p$-group, we prove that the skew group algebra of the path algebra is Morita equivalent to the algebra associated to a Cartan matrix, defined in [C. Geiss, B. Leclerc, and J. Schröer, Quivers with relations for symmetrizable Cartan matrices I: Foundations, Invent. Math. 209 (2017), 61–158]. We apply the Morita equivalence to construct a categorification of the folding projection between the root lattices with respect to a graph automorphism. In the Dynkin cases, the restriction of the categorification to indecomposable modules corresponds to the folding of positive roots.

Solving the mechanism of a chemical reaction requires determining the structures of all the ground states on the pathway and the elusive transition states linking them. 2024 is the centenary of Brønsted’s landmark paper that introduced the β-value and structure-activity studies as the only experimental means to infer the structures of transition states. It involves making systematic small changes in the covalent structure of the reactants and analysing changes in activation and equilibrium-free energies. Protein engineering was introduced for an analogous procedure, Φ-value analysis, to analyse the noncovalent interactions in proteins central to biological chemistry. The methodology was developed first by analysing noncovalent interactions in transition states in enzyme catalysis. The mature procedure was then applied to study transition states in the pathway of protein folding – ‘part (b) of the protein folding problem’. This review describes the development of Φ-value analysis of transition states and compares and contrasts the interpretation of β- and Φ-values and their limitations. Φ-analysis afforded the first description of transition states in protein folding at the level of individual residues. It revealed the nucleation-condensation folding mechanism of protein domains with the transition state as an expanded, distorted native structure, containing little fully formed secondary structure but many weak tertiary interactions. A spectrum of transition states with various degrees of structural polarisation was then uncovered that spanned from nucleation-condensation to the framework mechanism of fully formed secondary structure. Φ-analysis revealed how movement of the expanded transition state on an energy landscape accommodates the transition from framework to nucleation-condensation mechanisms with a malleability of structure as a unifying feature of folding mechanisms. Such movement follows the rubric of analysis of classical covalent chemical mechanisms that began with Brønsted. Φ-values are used to benchmark computer simulation, and Φ and simulation combine to describe folding pathways at atomic resolution.

In this article we rethink the chronotope approach by examining what happened to religious space-times in a Chinese urban development project that completely transformed what had once been five relatively rural townships. What happens to chronotopes when a place is so completely transformed? We focus on multiple chronotopic dimensions in the religious experience of those villagers whose families had long occupied this land, but who now live separated from their old neighbors, without their old livelihoods, having lost their old temples, and surrounded by new migrants who are generally wealthier and better educated. Building on recent anthropological work on chronotopes, coupled with insights taken from Maurice Merleau-Ponty and Gilles Deleuze, this article explores the complex interrelationships and workings of chronotopes through the idea of the fold. This approach reconsiders what the boundaries between chronotopes might look like—not necessarily straight lines that are difficult to cross, but more like the infinite inflections of curves as those curves intersect and interact with each other. Rather than thinking of chronotopes as structured wholes separated by clear boundaries—much as we also tend to think about “states,” “cultures,” or “ontologies”—folding allows us to reconceptualize the kinds of interactions that take place when one space-time touches another. We examine in particular three ways in which folding elucidates how chronotopic boundaries can work: they can make the distant near, separate inside from outside, and complicate the boundary by interdigitating.

The Sierras Pampeanas consist of basement uplifts that are mainly controlled by W-verging crustal thrusts. One of these, the Sierra Pie de Palo, is controlled by a blind thrust whose kinematics is debated. We analysed the reverse faults located along the eastern side of the range and show that these are characterized by small displacements (few hundreds of metres) of the Neogene and Quaternary deposits controlled by the inherited basement metamorphic foliation and shear zones. At the sierra scale, the inherited foliation pattern appears already folded, before the Cenozoic shortening, forming a series of antiforms and synforms with various wavelengths. We propose that this structure is amplified during the Plio-Quaternary shortening, resulting in a large asymmetrical basement-cored anticline where foliation-parallel slip can act as folding mechanism. The overall kinematics is consistent with an E-verging fault-propagation fold; this proposed kinematics is an alternative to the frequently suggested W-verging fault-bend fold model. Finally, we propose a new, alternative and speculative crustal and lithospheric structure for this region consisting of two E-verging basement thrusts coeval to W-verging Pampean thrusts, both rooting below the Precordillera, above an eclogitized lower crust.

Quantified balanced and restored crustal cross-sections across the NW Zagros Mountains are presented in this work integrating geological and geophysical local and global datasets. The balanced crustal cross-section reproduces the surficial folding and thrusting of the thick cover succession, including the near top of the Sarvak Formation (~90 Ma) that forms the top of the restored crustal cross-section. The base of the Arabian crust in the balanced cross-section is constrained by recently published seismic receiver function results showing a deepening of the Moho from 42 ± 2 km in the undeformed foreland basin to 56 ± 2 km beneath the High Zagros. The internal parts of the deformed crustal cross-section are constrained by new seismic tomographic sections imaging a ~50° NE-dipping sharp contact between the Arabian and Iranian crusts. These surfaces bound an area of 10800 km2 that should be kept constant during the Zagros orogeny. The Arabian crustal cross-section is restored using six different tectonosedimentary domains according to their sedimentary facies and palaeobathymetries, and assuming Airy isostasy and area conservation. While the two southwestern domains were directly determined from well-constrained surface data, the reconstruction of the distal domains to the NE was made using the recent margin model of Wrobel-Daveau et al. (2010) and fitting the total area calculated in the balanced cross-section. The Arabian continental–oceanic boundary, at the time corresponding to the near top of the Sarvak Formation, is located 169 km to the NE of the trace of the Main Recent Fault. Shortening is estimated at ~180 km for the cover rocks and ~149 km for the Arabian basement, including all compressional events from Late Cretaceous to Recent time, with an average shortening rate of ~2 mm yr−1 for the last 90 Ma.

We describe the geomorphology of a large (~10000 km2) internally draining region within the Zagros fold-and-thrust belt of Fars province, Iran. A series of wind gaps through fold crests and a continuous line of low-slope pixels in digital elevation models indicate the presence of an older, and now abandoned, through-going river system. We suggest, from the presence of the wind gaps, that the original through-going river system was abandoned as a direct result of fold growth. At present, through-going drainage in Fars is restricted to only two major rivers, the Kul and the Mand, which bound the margins of the internally drained region. The low gradients of the Kul and the Mand rivers are similar to those in topographic profiles drawn along the course of the abandoned drainage. The Mand and Kul rivers may be defeated in the future, causing an expansion of the internally drained region, and resulting in a profound change in the distribution of sediment and surface elevations within the Zagros. The internally draining part of the Zagros resembles the Central Iranian Plateau both in its geomorphology and in the apparently slow rates of deformation within it. We speculate that the development of internally drained basins and distribution of shortening within the range may be causally linked. The geomorphology that we describe might, therefore, record a stage in the southward expansion of the non-deforming and topographically high Central Iranian Plateau.

Cryptosporidium parvum is an intracellular protozoan parasite that causes cryptosporidiosis in mammals including humans. In the current study, the gene encoding the cysteine protease of C. parvum (cryptopain-1) was identified and the biochemical properties of the recombinant enzyme were characterized. Cryptopain-1 shared common structural properties with cathepsin L-like papain family enzymes, but lacked a typical signal peptide sequence and contained a possible transmembrane domain near the amino terminus and a unique insert in the front of the mature domain. The recombinant cryptopain-1 expressed in Escherichia coli and refolded to the active form showed typical biochemical properties of cathepsin L-like enzymes. The folding determinant of cryptopain-1 was characterized through multiple constructs with or without different lengths of the pro-domain of the enzyme expressed in E. coli and assessment of their refolding abilities. All constructs, except one that did not contain the full-length mature domain, successfully refolded into the active enzymes, suggesting that cryptopain-1 did not require the pro-domain for folding. Western blot analysis showed that cryptopain-1 was expressed in the sporozoites and the enzyme preferentially degraded proteins, including collagen and fibronectin, but not globular proteins. This suggested a probable role for cryptopain-1 in host cell invasion and/or egression by the parasite.

This paper provides a detailed documentation of microstructures developed in the banded iron formation (BIF) of Gua mine, located in the Bonai Synclinorium (eastern India), where the rocks have been subjected to three deformations (D1 to D3). Folded iron ores, quartz strain fringes around rigid core objects and folded iron ore layers, and refracted quartz veins are described from samples taken from D2 folds in the banded iron formation. Orientations of microstructures are compared with mesoscopic structures to interpret the generations of ore minerals, planar structures and the time relationship between deformation and development of different microstructures. The mechanism of D2 folding is worked out and its bearing on microstructure development is discussed. The D2 folds are inferred to have developed by a combination of tangential longitudinal strain in the competent layer, flexural flow in the incompetent layers and flexural slip at the interface between layers of differing competence. Homogeneous flattening strain superposed the earlier strain, which led to modification of the folds in the competent layer from class 1B to 1C. This strain is quantified and is found to be higher in the limb than the hinge of a fold. Diffusive mass transfer by solution and bulging dynamic recrystallization in quartz are inferred as the dominant deformation processes during folding. Moreover, based on comparison with published deformation microstructure maps, the microstructures of the present study are estimated to have developed between 300 and 350 °C temperatures at a strain rate of 10−14–10−12 s−1, which are geologically realistic conditions for naturally deformed rocks.

Thermoplastic matrix composites with a low glass transition temperature (Tg) have been developed for the Space Solar Power Truss (SSP). In this application, the truss is folded and packaged for launch, then expanded and deployed in space using a heat source. The present paper describes a resin film infusion process (RFI) used to fabricate laboratory-scale laminate tubes consisting of polyurethane and plain weave carbon fabrics. Subscale (1:5) sample tubes were formed to approximate the real tubes. The performance of the folded and unfolded tubes was measured under compression loading and compared with as-fabricated tubes at 25 and -75°C. Results show that elastic modulus was restored and even increased after bending. Stitched samples were also examined to evaluate the potential for reducing delamination at folds.

Structures deforming Llandovery turbidites of the Gala Group in the Southern Uplands terrane are spectacularly exposed in the Berwickshire coastal section, southeastern Scotland. The upward-facing, upright to NW-vergent folds and associated structures appear to record a single regional phase of subhorizontal NW–SE contractional deformation, with a steeply dipping direction of bulk finite extension. These structures are markedly different from those developed in rocks correlated with the Upper Llandovery Hawick Group exposed some 5 km to the south in the Eyemouth–Burnmouth coastal section. Here a highly domainal system of sinistral transpressional strain occurs, with zones of steeply plunging curvilinear folds, clockwise cleavage transection and bedding-parallel sinistral detachment faults. The markedly different bulk strain patterns in the Berwickshire coastal sections are thought to reflect the regionally diachronous nature of transpressional deformation in the Southern Uplands terrane. There are striking similarities in the structures recognized in the Berwickshire coastal sections and those developed in stratigraphically equivalent units along strike in southwestern Scotland and Northern Ireland. This confirms the lateral structural continuity and correlation of tracts and tract boundaries along the entire length of the Southern Uplands terrane. The regional structure suggests that a phase of top-to-the-NW backthrusting and backfolding associated with the southern margin of the Gala Group outcrop marks the transition from orthogonal contraction to sinistral transpression in the Southern Upland thrust wedge during late Llandovery times.

The folding of ribonuclease A (RNase A) has been extensively studied by characterizing the disulfide containing intermediates using different experimental conditions and analytical techniques. So far, some aspects still remain unclear such as the role of the loop 65–72 in the folding pathway. We have studied the oxidative folding of a RNase A derivative containing at position 67 the substitution Asn → isoAsp where the local structure of the loop 65–72 has been modified keeping intact the C65–C72 disulfide bond. By comparing the folding behavior of this mutant to that of the wild-type protein, we found that the deamidation significantly decreases the folding rate and alters the folding pathway of RNase A. Results presented here shed light on the role of the 65–72 region in the folding process of RNase A and also clarifies the effect of the deamidation on the folding/unfolding processes. On a more general ground, this study represents the first characterization of the intermediates produced along the folding of a deamidated protein.

The DNA-binding domain of the yeast heat shock transcription factor (HSF) contains a strictly conserved proline that is at the center of a kink. To define the role of this conserved proline-centered kink, we replaced the proline with a number of other residues. These substitutions did not diminish the ability of the full-length protein to support growth of yeast or to activate transcription, suggesting that the proline at the center of the kink is not conserved for function. The stability of the isolated mutant DNA-binding domains was unaltered from the wild-type, so the proline is not conserved to maintain the stability of the protein. The crystal structures of two of the mutant DNA-binding domains revealed that the helices in the mutant proteins were still kinked after substitution of the proline, suggesting that the proline does not cause the α-helical kink. So why are prolines conserved in this and the majority of other kinked α-helices if not for structure, function, or stability? The mutant DNA-binding domains are less soluble than wild-type when overexpressed. In addition, the folding kinetics, as measured by stopped-flow fluorescence, is faster for the mutant proteins. These two results support the premise that the presence of the proline is critical for the folding pathway of HSF's DNA-binding domain. The finding may also be more general and explain why kinked helices maintain their prolines.

A direct conflict between the stabilization free energy parameters of cytochrome c determined by optical methods and by hydrogen exchange (HX) is quantitatively explained when the partially folded intermediates seen by HX are taken into account. The results support the previous HX measurements of intermediate populations, show how intermediates can elude the standard melting analysis, and illustrate how they confuse the analysis when they are significantly populated within the melting transition region.

We previously reported the de novo design of an amphiphilic peptide [YGG(IEKKIEA)4] that forms a native-like, parallel triple-stranded coiled coil. Starting from this peptide, we sought to regulate the assembly of the peptide by a metal ion. The replacement of the Ile18 and Ile22 residues with Ala and Cys residues, respectively, in the hydrophobic positions disrupted of the triple-stranded α-helix structure. The addition of Cd(II), however, resulted in the reconstitution of the triple-stranded α-helix bundle, as revealed by circular dichroism (CD) spectroscopy and sedimentation equilibrium analysis. By titration with metal ions and monitoring the change in the intensity of the CD spectra at 222 nm, the dissociation constant Kd was determined to be 1.5 ± 0.8 μM for Cd(II). The triple-stranded complex formed by the 113Cd(II) ion showed a single 113Cd NMR resonance at 572 ppm whose chemical shift was not affected by the presence of Cl− ions. The 113Cd NMR resonance was connected with the βH protons of the cysteine residue by 1H–113Cd heteronuclear multiple quantum correlation spectroscopy. These NMR results indicate that the three cysteine residues are coordinated to the cadmium ion in a trigonal-planar complex. Hg(II) also induced the assembly of the peptide into a triple-stranded α-helical bundle below the Hg(II)/peptide ratio of 1/3. With excess Hg(II), however, the α-helicity of the peptide was decreased, with the change of the Hg(II) coordination state from three to two. Combining this construct with other functional domains should facilitate the production of artificial proteins with functions controlled by metal ions.

The dimeric β-barrel is a characteristic topology initially found in the transcriptional regulatory domain of the E2 DNA binding domain from papillomaviruses. We have previously described the kinetic folding mechanism of the human HPV-16 domain, and, as part of these studies, we present a structural characterization of the urea-denatured state of the protein. We have obtained a set of chemical shift assignments for the C-terminal domain in urea using heteronuclear NMR methods and found regions with persistent residual structure. Based on chemical shift deviations from random coil values, 3JNHNα coupling constants, heteronuclear single quantum coherence peak intensities, and nuclear Overhauser effect data, we have determined clusters of residual structure in regions corresponding to the DNA binding helix and the second β-strand in the folded conformation. Most of the structures found are of nonnative nature, including turn-like conformations. Urea denaturation at equilibrium displayed a loss in protein concentration dependence, in absolute parallel to a similar deviation observed in the folding rate constant from kinetic experiments. These results strongly suggest an alternative folding pathway in which a dimeric intermediate is formed and the rate-limiting step becomes first order at high protein concentrations. The structural elements found in the denatured state would collide to yield productive interactions, establishing an intermolecular folding nucleus at high protein concentrations. We discuss our results in terms of the folding mechanism of this particular topology in an attempt to contribute to a better understanding of the folding of dimers in general and intertwined dimeric proteins such as transcription factors in particular.

Whereas previously we have successfully utilized the folding funnels concept to rationalize binding mechanisms (Ma B, Kumar S, Tsai CJ, Nussinov R, 1999, Protein Eng 12:713–720) and to describe binding (Tsai CJ, Kumar S, Ma B, Nussinov R, 1999, Protein Sci 8:1181–1190), here we further extend the concept of folding funnels, illustrating its utility in explaining enzyme pathways, multimolecular associations, and allostery. This extension is based on the recognition that funnels are not stationary; rather, they are dynamic, depending on the physical or binding conditions (Tsai CJ, Ma B, Nussinov R, 1999, Proc Natl Acad Sci USA 96:9970–9972). Different binding states change the surrounding environment of proteins. The changed environment is in turn expressed in shifted energy landscapes, with different shapes and distributions of populations of conformers. Hence, the function of a protein and its properties are not only decided by the static folded three-dimensional structure; they are determined by the distribution of its conformational substates, and in particular, by the redistributions of the populations under different environments. That is, protein function derives from its dynamic energy landscape, caused by changes in its surroundings.

To investigate the role of some tertiary interactions, the disulfide bonds, in the early stages of refolding of hen lysozyme, we report the kinetics of reoxidation of denatured and reduced lysozyme under the same refolding conditions as those previously used to investigate the kinetics of regain of its circular dichroism (CD), fluorescence, and activity. At different stages of the refolding, the oxidation of the protein was blocked by alkylation of the free cysteines with iodoacetamide and the various oxidation states present in the samples were identified by electrospray-mass spectrometry. Thus, it was possible to monitor the appearance and/or disappearance of the species with 0 to 4 disulfide bonds. Using a simulation program, these kinetics were compared with those of regain of far-UV CD, fluorescence, and enzymatic activity and were discussed in terms of a refined model for the refolding of reduced hen egg white lysozyme.

The effect of pressure on the unfolding of bovine α-lactalbumin was investigated by ultraviolet absorption methods. The change of molar volume associated with unfolding, ΔV, was measured in the presence or absence of guanidine hydrochloride at pH 7. The ΔV was estimated to be −63 cm3/mol in the absence of a chemical denaturant. While in the presence of guanidine hydrochloride (GuHCl), it was found that ΔV was −66 cm3/mol at 25 °C and was independent of the concentration of GuHCl, despite the fact that the molten globule fraction in the total unfolding product decreased with the increase of GuHCl concentration. The results indicate that the volume of α-lactalbumin only changes at the transition from a native to a molten globule state, and almost no volume change has been found during the transition from a molten globule to the unfolded state.