Knotted proteins embed a physical (i.e., open) knot within their native structures. For decades, significant effort has been devoted to elucidating the functional role of knots in proteins, yet no consensus has been reached. Here, using extensive Monte Carlo off-lattice simulations of a simple structure-based model, we isolate the effect of topology by comparing simulations that preserve the linear topology of the chain with simulations that allow chain crossings. This controlled framework enables us to isolate topological effects from sequence, structure and energetic contributions. We show that protein kinetic stability, defined as resistance to unfolding at a fixed temperature, is higher in knotted proteins. Additionally, kinetic stability increases significantly with knot depth, whereas foldability (or folding efficiency) is comparatively less affected. By considering a simple model of protein evolution in which amino-acid alphabet size is used as a proxy for evolutionary time, we find that increasing primary-sequence complexity through the addition of biotic amino acids predominantly enhances kinetic stability. Taken together, these results indicate that kinetic stability is a functional advantage conferred by protein knots and suggest that evolutionary pressure for kinetic stability could contribute to the persistence of knotted proteins.

Abstract UCH-L1 is a monomeric deubiquitinating enzyme whose native structure embeds a shallow 5 2 knot located near the N-terminus, placing the knotted topology in direct proximity to both the substrate-binding pocket and the catalytic site. While our previous work established that N-terminal integrity is critical for catalytic activity, the energetic cost of unknotting and its structural consequences remained unquantified. Here, we combine steered molecular dynamics with an umbrella sampling scheme to generate topologically modified variants of UCH-L1 and, for the first time, reconstruct the free-energy profile of UCH-L1 unknotting. The potential of mean force reveals a steep energetic barrier to knot disruption, consistent with knotting being a late, ratelimiting folding step that is effectively locked in once the native structure is established. Long unbiased MD simulations of fully unknotted variants in both apo and holo states show that knot removal increases local flexibility at the N-terminus without inducing significant global structural destabilization. Binding energy calculations indicate that the unknotted variant binds to ubiquitin less tightly than the wild-type ( ∼ -62 vs ∼ 76 kcal/mol), suggesting that topological integrity contributes to substrate affinity. Together, these results show that the 5 2 knot in UCH-L1 is not a passive structural feature but a functional element that fine-tunes folding kinetics and contributes to substrate binding efficiency.

For several decades, experimental and computational studies have been used to investigate the potential functional role of knots in protein structures. A property that has attracted considerable attention is thermal stability, i.e., the extent to which a protein retains its native conformation and biological activity at high temperatures, without undergoing denaturation or aggregation. Thermal stability is quantified by the melting temperature Tm, an equilibrium property that corresponds to the peak of heat capacity in differential scanning calorimetry (DSC) experiments. Experimental and computational studies report conflicting effects of knotting on protein thermal stability [M. A. Soler and P. F. N. Faisca, PLoS One 8, e74755 (2013);Y.-C. Chuang, I.-C. Hu, P.-C. Lyu, and S.-T. D. Hsu, J. Mol. Biol. 431, 857 (2019)]. Here, we use extensive Monte Carlo simulations of a simple C-alpha model of protein YibK, with energetics modeled by the Go potential, to show that Tm does not depend on the topological state of the protein. Our simulations further support the view that the discrepancy between the experimental and computational results stems from a pronounced separation of timescales for unknotting and unfolding that is inherent to deeply knotted proteins like YibK. In particular, the timescale separation implies that the complete unfolding-untying transition may not be accessible within the duration of a DSC experiment, whose apparent Tm measurements likely reflect a nonequilibrium distribution lacking unfolded states that are also unknotted.

We developed a simple reverse evolution method to explore the protein folding transition and knotting process in globular proteins throughout evolution using as a proxy for evolutionary time the length of the amino acid alphabet. Three small proteins were considered. An unknotted one featuring a β-sandwich fold (FN3), a protein embedding a shallow trefoil knot (MJ0366), and a deeply knotted trefoil protein (YibK). Results from Monte Carlo simulations of a native-centric Cα model show that thermal stability increases throughout evolution for all considered proteins suggesting it provided a selective pressure for the introduction of biosynthetic amino acids in the alphabet repertoire. Additionally, the thermodynamic cooperativity, the two-state nature of the folding transition, and the kinetic efficiency of folding (and knotting) displayed by the unknotted (and shallow knotted) protein have been roughly conserved throughout evolution, indicating that early alphabets, and, in particular the putative ancestral one composed of 10 amino acids, is competent to self-assemble relatively complex native structures. However, while the ancestral alphabet is clearly more efficient in collapsing the polypeptide chain to nonspecific (i.e., non-native) knots, it significantly compromises the thermodynamic cooperativity and the folding ability of the deeply knotted protein YibK. The findings presented here collectively suggest that the alphabet of amino acids evolved to improve folding, but early alphabets might not have favored the folding of native structures with deep knots. This could help to explain why deep knots in proteins are statistically uncommon.

Knotted proteins are present in nature, but there is still an open issue regarding the existence of a universal role for these remarkable structures. To address this question, we used classical molecular dynamics (MD) simulations combined with in vitro experiments to investigate the role of the Gordian knot in the catalytic activity of UCH-L1. To create an unknotted form of UCH-L1, we modified its amino acid sequence by truncating several residues from its N-terminus. Remarkably, we find that deleting the first two N-terminal residues leads to a partial loss of enzyme activity with conservation of secondary structural content and knotted topological state. This happens because the integrity of the N-terminus is critical to ensure the correct alignment of the catalytic triad. However, the removal of five residues from the N-terminus, which significantly disrupts the native structure and the topological state, leads to a complete loss of enzymatic activity. Overall, our findings indicate that UCH-L1's catalytic activity depends critically on the integrity of the N-terminus and the secondary structure content, with the latter being strongly coupled with the knotted topological state.

The last years have witnessed remarkable advances in our understanding of the emergence and consequences of topological constraints in biological and soft matter. Examples are abundant in relation to (bio)polymeric systems and range from the characterization of knots in single polymers and proteins to that of whole chromosomes and polymer melts. At the same time, considerable advances have been made in the description of the interplay between topological and physical properties in complex fluids, with the development of techniques that now allow researchers to control the formation of and interaction between defects in diverse classes of liquid crystals. Thanks to technological progress and the integration of experiments with increasingly sophisticated numerical simulations, topological biological and soft matter is a vibrant area of research attracting scientists from a broad range of disciplines. However, owing to the high degree of specialization of modern science, many results have remained confined to their own particular fields, with different jargon making it difficult for researchers to share ideas and work together towards a comprehensive view of the diverse phenomena at play. Compelled by these motivations, here we present a comprehensive overview of topological effects in systems ranging from DNA and genome organization to entangled proteins, polymeric materials, liquid crystals, and theoretical physics, with the intention of reducing the barriers between different fields of soft matter and biophysics. Particular care has been taken in providing a coherent formal introduction to the topological properties of polymers and of continuum materials and in highlighting the underlying common aspects concerning the emergence, characterization, and effects of topological objects in different systems. The second half of the review is dedicated to the presentation of the latest results in selected problems, specifically, the effects of topological constraints on the viscoelastic properties of polymeric materials; their relation with genome organization; a discussion on the emergence and possible effects of knots and other entanglements in proteins; the emergence and effects of topological defects and solitons in complex fluids.This review is dedicated to the memory of Marek Cieplak.

Atomic force microscopy (AFM) imaging enables the visualization of protein molecules with high resolution, providing insights into their shape, size, and surface topography. Here, we use AFM to study the aggregation process of protein S100A9 in physiological conditions, in the presence of calcium at a molar ratio 4Ca2+:S100A9. We find that S100A9 readily assembles into a worm-like fibril, with a period dimension along the fibril axis of 11.5 nm. The fibril’s chain length extends up to 136 periods after an incubation time of 144 h. At room temperature, the fibril’s bending stiffness was found to be 2.95×10−28 Nm2, indicating that the fibrils are relatively flexible. Additionally, the values obtained for the Young’s modulus (Ex=6.96×105 Pa and Ey=3.37×105 Pa) are four orders of magnitude lower than those typically reported for canonical amyloid fibrils. Our findings suggest that, under the investigated conditions, a distinct aggregation mechanism may be in place in the presence of calcium. Therefore, the findings reported here could have implications for the field of biomedicine, particularly with regard to Alzheimer’s disease.

The field of protein science is witnessing a revolution in the wake of recent breakthroughs such as the artificial intelligence (AI) system AlphaFold which can predict the three-dimensional structure of almost every globular protein from its protein sequence alone. Inevitably, there is some expectation that structure-prediction modelling approaches based on AI may also learn the physics of protein folding to the extent of predicting folding pathways. Motivated by these advances, we briefly review the thermodynamic, kinetic, and structural aspects governing the folding transition from an unfolded protein chain into its final native structure and discuss directions for future research.

Determining the role of non-native interactions in folding dynamics, kinetics, and mechanisms is a classic problem in protein folding. More recently, this question has witnessed a renewed interest in light of the hypothesis that knotted proteins require the assistance of non-native interactions to fold efficiently. Here, we conduct extensive equilibrium and kinetic Monte Carlo simulations of a simple off-lattice C-alpha model to explore the role of non-native interactions in the thermodynamics and kinetics of three proteins embedding a trefoil knot in their native structure. We find that equilibrium knotted conformations are stabilized by non-native interactions that are non-local, and proximal to native ones, thus enhancing them. Additionally, non-native interactions increase the knotting frequency at high temperatures, and in partially folded conformations below the transition temperatures. Although non-native interactions clearly enhance the efficiency of transition from an unfolded conformation to a partially folded knotted one, they are not required to efficiently fold a knotted protein. Indeed, a native-centric interaction potential drives the most efficient folding transition, provided that the simulation temperature is well below the transition temperature of the considered model system.

Protein aggregation is a complex process, strongly dependent on environmental conditions and highly structurally heterogeneous, both at the final level of fibril structure and intermediate level of oligomerization. Since the first step in aggregation is the formation of a dimer, it is important to clarify how certain properties of the latter (e.g., stability or interface geometry) may play a role in self-association. Here, we report a simple model that represents the dimer's interfacial region by two angles and combine it with a simple computational method to investigate how modulations of the interfacial region occurring on the ns-μs time scale change the dimer's growth mode. To illustrate the proposed methodology, we consider 15 different dimer configurations of the β2m D76N mutant protein equilibrated with long Molecular Dynamics simulations and identify which interfaces lead to limited and unlimited growth modes, having, therefore, different aggregation profiles. We found that despite the highly dynamic nature of the starting configurations, most polymeric growth modes tend to be conserved within the studied time scale. The proposed methodology performs remarkably well taking into consideration the nonspherical morphology of the β2m dimers, which exhibit unstructured termini detached from the protein's core, and the relatively weak binding affinities of their interfaces, which are stabilized by nonspecific apolar interactions. The proposed methodology is general and can be applied to any protein for which a dimer structure has been experimentally determined or computationally predicted.

Monte Carlo simulations are a powerful technique and are widely used in different fields. When applied to complex molecular systems with long chains, such as those in synthetic polymers and proteins, they have the advantage of providing a fast and computationally efficient way to sample equilibrium ensembles and calculate thermodynamic and structural properties under desired conditions. Conformational Monte Carlo techniques employ a move set to perform the transitions in the simulation Markov chain. While accepted conformations must preserve the sequential bonding of the protein chain model and excluded volume among its units, the moves themselves may take the chain across itself. We call this a break in linear topology preservation. In this manuscript, we show, using simple protein models, that there is no difference in equilibrium properties calculated with a move set that preserves linear topology and one that does not. However, for complex structures, such as those of deeply knotted proteins, the preservation of linear topology provides correct equilibrium results but only after long relaxation. In any case, to analyze folding pathways, knotting mechanisms and folding kinetics, the preservation of linear topology may be an unavoidable requirement.

Protein beta-2-microglobulin (β2m) is classically considered the causative agent of dialysis related amyloidosis, a conformational disorder that affects patients undergoing long-term hemodialysis. The wild type (WT) form, the ΔN6 structural variant, and the D76N mutant have been extensively used as model systems ofβ2m aggregation. In all of them, the native structure is stabilized by a disulfide bridge between the sulphur atoms of the cysteine residues 25 (at B strand) and 80 (at F strand), which has been considered fundamental inβ2m fibrillogenesis. Here, we use extensive discrete molecular dynamics simulations of a full atomistic structure-based model to explore the role of this disulfide bridge as a modulator of the folding space ofβ2m. In particular, by considering different models for the disulfide bridge, we explore the thermodynamics of the folding transition, and the formation of intermediate states that may have the potential to trigger the aggregation cascade. Our results show that the dissulfide bridge affects folding transition and folding thermodynamics of the considered model systems, although to different extents. In particular, when the interaction between the sulphur atoms is stabilized relative to the other intramolecular interactions, or even locked (i.e. permanently established), the WT form populates an intermediate state featuring a well preserved core and two unstructured termini, which was previously detected only for the D76N mutant. The formation of this intermediate state may have important implications in our understanding ofβ2m fibrillogenesis.

The D76N mutant of the β2m protein is a biologically motivated model system to study protein aggregation. There is strong experimental evidence, supported by molecular simulations, that D76N populates a highly dynamic conformation (which we originally named I2) that exposes aggregation-prone patches as a result of the detachment of the two terminal regions. Here, we use Molecular Dynamics simulations to study the stability of an ensemble of dimers of I2 generated via protein–protein docking. MM-PBSA calculations indicate that within the ensemble of investigated dimers the major contribution to interface stabilization at physiological pH comes from hydrophobic interactions between apolar residues. Our structural analysis also reveals that the interfacial region associated with the most stable binding modes are particularly rich in residues pertaining to both the N- and C-terminus, as well residues from the BC- and DE-loops. On the other hand, the less stable interfaces are stabilized by intermolecular interactions involving residues from the CD- and EF-loops. By focusing on the most stable binding modes, we used a simple geometric rule to propagate the corresponding dimer interfaces. We found that, in the absence of any kind of structural rearrangement occurring at an early stage of the oligomerization pathway, some interfaces drive a self-limited growth process, while others can be propagated indefinitely allowing the formation of long, polymerized chains. In particular, the interfacial region of the most stable binding mode reported here falls in the class of self-limited growth.

Native interactions are crucial for folding, and non-native interactions appear to be critical for efficiently knotting proteins. Therefore, it is important to understand both their roles in the folding of knotted proteins. It has been proposed that non-native interactions drive the correct order of contact formation, which is essential to avoid backtracking and efficiently self-tie. In this study, we ask if non-native interactions are strictly necessary to tangle a protein or if the correct order of contact formation can be assured by a specific set of native, but otherwise heterogeneous (i.e., having distinct energies), interactions. In order to address this problem, we conducted extensive Monte Carlo simulations of lattice models of protein-like sequences designed to fold into a preselected knotted conformation embedding a trefoil knot. We were able to identify a specific set of heterogeneous native interactions that drives efficient knotting and is able to fold the protein when combined with the remaining native interactions modeled as homogeneous. This specific set of heterogeneous native interactions is strictly enough to efficiently self-tie. A distinctive feature of these native interactions is that they do not backtrack because their energies ensure the correct order of contact formation. Furthermore, they stabilize a knotted intermediate state, which is en route to the native structure. Our results thus show that-at least in the context of the adopted model-non-native interactions are not necessary to knot a protein. However, when they are taken into account in protein energetics, it is possible to find specific, nonlocal non-native interactions that operate as a scaffold that assists the knotting step.

A small fraction of all protein structures characterized so far are entangled. The challenge of understanding the properties of these knotted proteins, and the why and the how of their natural folding process, has been taken up in the past decade with different approaches, such as structural characterization, in vitro experiments, and simulations of protein models with varying levels of complexity. The simplest among these are the lattice G\=o models, which belong to the class of structure-based models, i.e., models that are biased to the native structure by explicitly including structural data. In this review we highlight the contributions to the field made in the scope of lattice G\=o models, putting them into perspective in the context of the main experimental and theoretical results and of other, more realistic, computational approaches.

Protein β2-microglobulin is the causing agent of two amyloidosis, dialysis related amyloidosis (DRA), affecting the bones and cartilages of individuals with chronic renal failure undergoing long-term hemodialysis, and a systemic amyloidosis, found in one French family, which impairs visceral organs. The protein’s small size and its biomedical significance attracted the attention of theoretical scientists, and there are now several studies addressing its aggregation mechanism in the context of molecular simulations. Here, we review the early phase of β2-microglobulin aggregation, by focusing on the identification and structural characterization of monomers with the ability to trigger aggregation, and initial small oligomers (dimers, tetramers, hexamers etc.) formed in the so-called nucleation phase. We focus our analysis on results from molecular simulations and integrate our views with those coming from in vitro experiments to provide a broader perspective of this interesting field of research. We also outline directions for future computer simulation studies.

We have come a long way since coining of the term protein and the early findings that proteins are charged macromolecules composed of strings of amino acids linked by peptide bonds. Today, structural biologists have technologies that allow in many cases to achieve an atomic-level understanding of protein structure, dynamics and folding; protein physics approaches have made substantial contributions to understanding the intricacies of folding mechanisms and its energetics; biochemists have developed conceptual frameworks to relate protein structure with biological functions. Yet, despite the efforts of a vibrant community of protein scientists, a lot of questions remain to be answered in the field of protein structure and folding.

In the summer of 2018, Professor Michael Kosterlitz visited Portugal as a plenary speaker of the FÍSICA2018 conference organised by the Portuguese Physical Society (SPF) and by the University of Beira Interior. FÍSICA 2018 comprised two meetings: the 28th Iberian Meeting for Physics Teaching, and the 21st National Conference of Physics, a biannual event that brings together researchers from all areas of Physics working in Portugal.

This Brief provides a snapshot of the field of protein folding, being a unique resource for newcomers to the field. It elucidates how protein structure is acquired and maintained, explores folding kinetics and mechanisms, and also covers protein misfolding and presents methods for protein folding.