Molecular dynamics studies within a coarse-grained structure based model were used on two similar proteins belonging to the transcarbamylase family to probe the effects in the native structure of a knot. The first protein, N-acetylornithine transcarbamylase, contains no knot whereas human ormithine transcarbamylase contains a trefoil knot located deep within the sequence. In addition, we also analyzed a modified transferase with the knot removed by the appropriate change of a knot-making crossing of the protein chain. The studies of thermally- and mechanically-induced unfolding processes suggest a larger intrinsic stability of the protein with the knot.
An increasing number of proteins are being discovered with a remarkable and somewhat surprising feature, a knot in their native structures. How the polypeptide chain is able to knot itself during the folding process to form these highly intricate protein topologies is not known. Here, we perform a computational study on the 160-amino acid homodimeric protein YibK which, like other proteins in the SpoU family of MTases, contains a deep trefoil knot in its C-terminal region. In this study, we use a coarse-grained C-alpha-chain representation and Langevin dynamics to study folding kinetics. We fi…
Using a structure-based coarse-grained model of proteins, we study the mechanism of unfolding of knotted proteins through heating. We find that the dominant mechanisms of unfolding depend on the temperature applied and are generally distinct from those identified for folding at its optimal temperature. In particular, for shallowly knotted proteins, folding usually involves formation of two loops whereas unfolding through high-temperature heating is dominated by untying of single loops. Untying the knots is found to generally precede unfolding unless the protein is deeply knotted and the heatin…
Polymers can be modeled as open polygonal paths and their closure generates knots. Knotted proteins detection is currently achieved via high-throughput methods based on a common framework insensitive to the handedness of knots. Here we propose a topological framework for the computation of the HOMFLY polynomial, an handedness-sensitive invariant. Our approach couples a multi-component reduction scheme with the polynomial computation. After validation on tabulated knots and links the framework was applied to the entire Protein Data Bank along with a set of selected topological checks that allow…
Knots in proteins have been proposed to resist proteasomal degradation. Ample evidence associates proteasomal degradation with neurodegeneration. One interesting possibility is that indeed knotted conformers stall this machinery leading to toxicity. However, although the proteasome is known to unfold mechanically its substrates, at present there are no experimental methods to emulate this particular traction geometry. Here, we consider several dynamical models of the proteasome in which the complex is represented by an effective potential with an added pulling force. This force is meant to ind…
Proteins can sometimes be knotted, and for many reasons the study of knotted proteins is rapidly becoming very important. For example, it has been proposed that a knot increases the stability of a protein. Knots may also alter enzymatic activities and enhance binding. Moreover, knotted proteins may even have some substantial biomedical significance in relation to illnesses such as Parkinson's disease. But to a large extent the biological role of knots remains a conundrum. In particular, there is no explanation why knotted proteins are so scarce. Here we argue that knots are relatively rare bec…
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 Gō 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.…
Although rare, an increasing number of proteins have been observed to contain entanglements in their native structures. To gain more insight into the significance of protein knotting, researchers have been investigating protein knot formation using both experimental and theoretical methods. Motivated by the hypothesized folding pathway of $α$-haloacid dehalogenase (DehI) protein, Flapan, He, and Wong proposed a theory of how protein knots form, which includes existing folding pathways described by Taylor and Bölinger et al. as special cases. In their topological descriptions, two loops in an u…
Understanding the biological function of knots in proteins and their folding process is an open and challenging question in biology. Recent studies classify the topology and geometry of knotted proteins by analysing the distribution of a protein's planar projections using topological objects called knotoids. We approach the analysis of proteins with the same topology by introducing a topologically inspired statistical metric between their knotoid distributions. We detect geometric differences between trefoil proteins by characterising their entanglement and we recover a clustering by sequence …
The computer artificial intelligence system AlphaFold has recently predicted previously unknown three-dimensional structures of thousands of proteins. Focusing on the subset with high-confidence scores, we algorithmically analyze these predictions for cases where the protein backbone exhibits rare topological complexity, i.e. knotting. Amongst others, we discovered a $7_1$-knot, the most topologically complex knot ever found in a protein, as well several 6-crossing composite knots comprised of two methyltransferase or carbonic anhydrase domains, each containing a simple trefoil knot. These dee…
The backbone of most proteins forms an open curve. To study their entanglement, a common strategy consists in searching for the presence of knots in their backbones using topological invariants. However, this approach requires to close the curve into a loop, which alters the geometry of curve. Knoto-ID allows evaluating the entanglement of open curves without the need to close them, using the recent concept of knotoids which is a generalization of the classical knot theory to open curves. Knoto-ID can analyse the global topology of the full chain as well as the local topology by exhaustively s…
We perform theoretical studies of stretching of 20 proteins with knots within a coarse grained model. The knot's ends are found to jump to well defined sequential locations that are associated with sharp turns whereas in homopolymers they diffuse around and eventually slide off. The waiting times of the jumps are increasingly stochastic as the temperature is raised. Larger knots do not return to their native locations when a protein is released after stretching.
Knots are abundant in globular homopolymers but rare in globular proteins. To shed new light on this long-standing conundrum, we study the influence of sequence on the formation of knots in proteins under native conditions within the framework of the hydrophobic-polar (HP) lattice protein model. By employing large scale Wang-Landau simulations combined with suitable Monte Carlo trial moves we show that, even though knots are still abundant on average, sequence introduces large variability in the degree of self-entanglements. Moreover, we are able to design sequences which are either almost alw…
We study here global and local entanglements of open protein chains by implementing the concept of knotoids. Knotoids have been introduced in 2012 by Vladimir Turaev as a generalization of knots in 3-dimensional space. More precisely, knotoids are diagrams representing projections of open curves in 3D space, in contrast to knot diagrams which represent projections of closed curves in 3D space. The intrinsic difference with classical knot theory is that the generalization provided by knotoids admits non-trivial topological entanglement of the open curves provided that their geometry is frozen a…
A number of recently discovered protein structures incorporate a rather unexpected structural feature: a knot in the polypeptide backbone. These knots are extremely rare, but their occurrence is likely connected to protein function in as yet unexplored fashion. Our analysis of the complete Protein Data Bank reveals several new knots which, along with previously discovered ones, can shed light on such connections. In particular, we identify the most complex knot discovered to date in human ubiquitin hydrolase, and suggest that its entangled topology protects it against unfolding and degradation…
Recent studies classify the topology of proteins by analysing the distribution of their projections using knotoids. The approximation of this distribution depends on the number of projection directions that are sampled. Here we investigate the relation between knotoids differing only by small perturbations of the direction of projection. Since such knotoids are connected by at most a single forbidden move, we characterise forbidden moves in terms of equivariant band attachment between strongly invertible knots and of strand passages between $θ$-curves. This allows for the determination of the …
Proper folding of deeply knotted proteins has a very low success rate even in structure-based models which favor formation of the native contacts but have no topological bias. By employing a structure-based model, we demonstrate that cotranslational folding on a model ribosome may enhance the odds to form trefoil knots for protein YibK without any need to introduce any non-native contacts. The ribosome is represented by a repulsive wall that keeps elongating the protein. On-ribosome folding proceeds through a a slipknot conformation. We elucidate the mechanics and energetics of its formation. …
Quantification and classification of protein structures, such as knotted proteins, often requires noise-free and complete data. Here we develop a mathematical pipeline that systematically analyzes protein structures. We showcase this geometric framework on proteins forming open-ended trefoil knots, and we demonstrate that the mathematical tool, persistent homology, faithfully represents their structural homology. This topological pipeline identifies important geometric features of protein entanglement and clusters the space of trefoil proteins according to their depth. Persistence landscapes q…
We study the folding process in the shallowly knotted protein MJ0366 within two variants of a structure-based model. We observe that the resulting topological pathways are much richer than identified in previous studies. In addition to the single knot-loop events, we find novel, and dominant, two-loop mechanisms. We demonstrate that folding takes place in a range of temperatures and the conditions of most successful folding are at temperatures which are higher than those required for the fastest folding. We also demonstrate that nascent conditions are more favorable to knotting than off-riboso…
Stochastic simulations of coarse-grained protein models are used to investigate the propensity to form knots in early stages of protein folding. The study is carried out comparatively for two homologous carbamoyltransferases, a natively-knotted N-acetylornithine carbamoyltransferase (AOTCase) and an unknotted ornithine carbamoyltransferase (OTCase). In addition, two different sets of pairwise amino acid interactions are considered: one promoting exclusively native interactions, and the other additionally including non-native quasi-chemical and electrostatic interactions. With the former model …
Long, flexible physical filaments are naturally tangled and knotted, from macroscopic string down to long-chain molecules. The existence of knotting in a filament naturally affects its configuration and properties, and may be very stable or disappear rapidly under manipulation and interaction. Knotting has been previously identified in protein backbone chains, for which these mechanical constraints are of fundamental importance to their molecular functionality, despite their being open curves in which the knots are not mathematically well defined; knotting can only be identified by closing the…
Proteins with nontrivial topology, containing knots and slipknots, have the ability to fold to their native states without any additional external forces invoked. A mechanism is suggested for folding of these proteins, such as YibK and YbeA, which involves an intermediate configuration with a slipknot. It elucidates the role of topological barriers and backtracking during the folding event. It also illustrates that native contacts are sufficient to guarantee folding in around 1-2% of the simulations, and how slipknot intermediates are needed to reduce the topological bottlenecks. As expected, …
Over the years, advances in experimental and computational methods have helped us to understand the role of thermodynamic, kinetic and active (chaperone-aided) effects in coordinating the folding steps required to achieving a knotted native state. Here, we review such developments by paying particular attention to the complementarity of experimental and computational studies. Key open issues that could be tackled with either or both approaches are finally pointed out.
The folding pathway and rate coefficients of the folding of a knotted protein are calculated for a potential energy function with minimal energetic frustration. A kinetic transition network is constructed using the discrete path sampling approach, and the resulting potential energy surface is visualized by constructing disconnectivity graphs. Owing to topological constraints, the low-lying portion of the landscape consists of three distinct regions, corresponding to the native knotted state and to configurations where either the N- or C-terminus is not yet folded into the knot. The fastest fol…
Apart from the knots formed by the main-chain, the proteins can form numerous topological structures, when included the covalent and ion-mediated interactions. In this work, we define the protein non-trivial $θ$-curves and identify 7 different topologies in all structures known up to date. We study the correlation of the motif with the function and organism of origin, and pointing the similarity with main-chain knots, we show that some motifs may indeed be functional. We also analyze the folding and bridge-induced stability of an exemplary protein with $θ$-curve motif and provide a catalogue o…
The ongoing effort to detect and characterize physical entanglement in biopolymers has so far established that knots are present in many globular proteins and also abound in viral DNA packaged inside bacteriophages. RNA molecules, on the other hand, have not yet been systematically screened for the occurrence of physical knots. We have accordingly undertaken the systematic profiling of the ~6,000 RNA structures present in the protein data bank. The search identified no more than three deeply-knotted RNA molecules. These are ribosomal RNAs solved by cryo-em and consist of about 3,000 nucleotide…
We incorporate hydrodynamic interactions in a structure-based model of ubiquitin and demonstrate that the hydrodynamic coupling may reduce the peak force when stretching the protein at constant speed, especially at larger speeds. Hydrodynamic interactions are also shown to facilitate unfolding at constant force and inhibit stretching by fluid flows.
We consider mechanical stability of dimeric and monomeric proteins with the cystine knot motif. A structure based dynamical model is used to demonstrate that all dimeric and some monomeric proteins of this kind should have considerable resistance to stretching that is significantly larger than that of titin. The mechanisms of the large mechanostability are elucidated. In most cases, it originates from the induced formation of one or two cystine slipknots. Since there are four termini in a dimer, there are several ways of selecting two of them to pull by. We show that in the cystine knot system…
We study local conformational biases in the dynamics of α-synuclein by using all-atom simulations with explicit and implicit solvents. The biases are related to the frequency of the specific contact formation. In both approaches, the protein is intrinsically disordered, and its strongest bias is to make bend and turn local structures. The explicit-solvent conformations can be substantially more extended which allows for formation of transient trefoil knots, both deep and shallow, that may last for up to 5 μs. The two-chain self-association events, both short- and long-lived, are dominated by f…
We fit the Fourier transforms of solvent accessibility and hydrophobicity profiles of a representative set of proteins to a joint multi-variable Gaussian. This allows us to separate the intrinsic tendencies of sequence and structure profiles from the interactions that correlate them; for example, the $α$-helix periodicity in sequence hydrophobicity is dictated by the solvent accessibility of structures. The distinct intrinsic tendencies of sequence and structure profiles are most pronounced at long periods, where sequence hydrophobicity fluctuates more, while solvent accessibility fluctuations…
The electrochemical behaviour of the biomedical and metallic alloys, especially in the orthopaedic implants fields, raises many questions. This study is dedicated for studying the Ti-6Al-4V alloy, by electrochemical impedance spectroscopy, EIS, in various physiological media,: Ringer solution, phosphate buffered solution (PBS), PBS solution and albumin, PBS solution with calf serum and PBS solution with calf serum and an antioxidant (sodium azide). Moreover, the desionised water was considered as the reference solution. The tests reproducibility was investigated. The time-frequency-Module grap…
A recent survey of 17 134 proteins has identified a new class of proteins which are expected to yield stretching induced force-peaks in the range of 1 nN. Such high force peaks should be due to forcing of a slip-loop through a cystine ring, i.e. by generating a cystine slipknot. The survey has been performed in a simple coarse grained model. Here, we perform all-atom steered molecular dynamics simulations on 15 cystine knot proteins and determine their resistance to stretching. In agreement with previous studies within a coarse grained structure based model, the level of resistance is found to…
Cotranslational folding depends on the folding speed and stability of the nascent protein. It remains difficult, however, to predict which proteins cotranslationally fold. Here, we simulate evolution of model proteins to investigate how native structure influences evolution of cotranslational folding. We developed a model that connects protein folding during and after translation to cellular fitness. Model proteins evolved improved folding speed and stability, with proteins adopting one of two strategies for folding quickly. Low contact order proteins evolve to fold cotranslationally. Such pro…
Understanding the influence of macromolecular crowding and nanoparticles on the formation of in-register $β$-sheets, the primary structural component of amyloid fibrils, is a first step towards describing \emph{in vivo} protein aggregation and interactions between synthetic materials and proteins. Using all atom molecular simulations in implicit solvent we illustrate the effects of nanoparticle size, shape, and volume fraction on oligomer formation of an amyloidogenic peptide from the transthyretin protein. Surprisingly, we find that inert spherical crowding particles destabilize in-register $…
The quantum-mechanical mechanisms by which the enzymes catalyze the hydrogen transfer in biochemical reactions are considered. Up to date it was established both experimentally and theoretically that in many cases the proton tunnelling through the intermolecular potential barrier is essential. We argue that in this case the enzyme excitation and internal motion facilitate proton transfer between reactants by squeezing the potential barrier which otherwise is practically impenetrable. In the similar fashion, the enzymes can facilitate the formation of hydrogen (H) bonds between the molecules. B…
We investigate the influences of the excluded volume of molecules on biochemical reaction processes on 2-dimensional surfaces using a model of signal transduction processes on biomembranes. We perform simulations of the 2-dimensional cell-based model, which describes the reactions and diffusion of the receptors, signaling proteins, target proteins, and crowders on the cell membrane. The signaling proteins are activated by receptors, and these activated signaling proteins activate target proteins that bind autonomously from the cytoplasm to the membrane, and unbind from the membrane if activate…
We present a theoretical study of the folding of small proteins inside confining potentials. Proteins are described in the framework of an effective potential model which contains the Ramachandran angles as degrees of freedom and does not need any {\it a priori} information about the native state. Hydrogen bonds, dipole-dipole- and hydrophobic interactions are taken explicitly into account. An interesting feature displayed by this potential is the presence of some intermediates between the unfolded and native states. We consider different types of confining potentials in order to study the str…
Proteins must fold quickly to acquire their biologically functional three-dimensional native structures. Hence, these are mainly stabilized by local contacts, while intricate topologies such as knots are rare. Here, we reveal the existence of specific patterns adopted by protein sequences and structures to deal with backbone self-entanglement. A large scale analysis of the Protein Data Bank shows that loops significantly intertwined with another chain portion are typically closed by weakly bound amino acids. Why is this energetic frustration maintained? A possible picture is that entangled loo…
We propose a new deterministic methodology to predict RNA sequence and protein folding. Is stem enough for structure prediction? The main idea is to consider all possible stem formation in the given sequence. With the stem loop energy and the strength of stem, we explore how to deterministically utilize stem information for RNA sequence and protein folding structure prediction. We use graph notation, where all possible stems are represented as vertices, and co-existence as edges. This full Stem-graph presents all possible folding structure, and we pick sub-graph(s) which give the best matching…
Breast cancer's complexity and variability pose significant challenges in understanding its progression and guiding effective treatment. This study aims to integrate protein sequence data with expression levels to improve the molecular characterization of breast cancer subtypes and predict clinical outcomes. Using ProtGPT2, a language model specifically designed for protein sequences, we generated embeddings that capture the functional and structural properties of proteins. These embeddings were integrated with protein expression levels to form enriched biological representations, which were a…
Many native structures of proteins accomodate complex topological motifs such as knots, lassos, and other geometrical entanglements. How proteins can fold quickly even in the presence of such topological obstacles is a debated question in structural biology. Recently, the hypothesis that energetic frustration might be a mechanism to avoid topological frustration has been put forward based on the empirical observation that loops involved in entanglements are stabilized by weak interactions between amino-acids at their extrema. To verify this idea, we use a toy lattice model for the folding of p…
The influence of three well-known disaccharides, namely trehalose, maltose and sucrose, on some structural and dynamical properties of lysozyme has been investigated by means of molecular dynamics computer simulations in the 37-60 wt % concentration range. The effects of sugars on the protein conformation are found relatively weak, in agreement with the preferential hydration of lysozyme. Conversely, sugars seem to increase significantly the relaxation times of the protein. These effects are shown to be correlated to the fractional solvent accessibilities of lysozyme residues and further suppo…
Understanding the observed variability in the number of homologs of a gene is a very important, unsolved problem that has broad implications for research into co-evolution of structure and function, gene duplication, pseudogene formation and possibly for emerging diseases. Here we attempt to define and elucidate the reasons behind this observed unevenness in sequence space. We present evidence that sequence variability and functional diversity of a gene or fold family is influenced by certain quantitative characteristics of the protein structure that reflect potential for sequence plasticity i…
The zinc finger structure where a Zn2+ ion binds to 4 cysteine or histidine amino acids in a tetrahedral structure is very common motif of nucleic acid binding proteins. The corresponding interaction model is present in 3% of the genes of human genome. As a result, zinc finger has been shown to be extremely useful in various therapeutic and research capacities, as well as in biotechnology. In stable configuration, the cysteine amino acids are deprotonated and become negatively charged. This means the Zn2+ ion is overscreened by 4 cysteine charges (overcharged). It is question of whether this o…
The conformational complexity of linear polymers far exceeds that of point-like atoms and molecules. Polymers can bend, twist, even become knotted. Thus they may also display a much richer phase structure than point particles. But it is not very easy to characterize the phase of a polymer. Essentially, the only attribute is the radius of gyration. The way how it changes when the degree of polymerization becomes different, and how it evolves when the ambient temperature and solvent properties change, discloses the phase of the polymer. Moreover, in any finite length chain there are corrections …
We review the status of protein-based molecular electronics. First we discuss fundamental concepts of electron transfer and transport in and across proteins and proposed mechanisms for these processes. We then describe the immobilization of proteins to solid-state surfaces in both nanoscale and macroscopic approaches, and highlight how different methodologies can alter protein electronic properties. Because immobilizing proteins while retaining biological activity is crucial to the successful development of bioelectronic devices, we discuss this process at length. We briefly discuss computatio…
A complex network approach to protein folding is proposed. The graph object is the network of shortcut edges present in a native-state protein (SCN0). Although SCN0s are found via an intuitive message passing algorithm (S. Milgram, Psychology Today, May 1967 pp. 61-67), they are meaningful enough that the logarithm form of their contact order (SCN0_lnCO) correlates significantly with protein kinetic rates, regardless of protein size. Further, the clustering coefficient of a SCN0 (CSCN0) can be used to combine protein segments iteratively within the Restricted Binary Collision model to form the…
Though the problem of sequence-reversed protein folding is largely unexplored, one might speculate that reversed native protein sequences should be significantly more foldable than purely random heteropolymer sequences. In this article, we investigate how the reverse-sequences of native proteins might fold by examining a series of small proteins of increasing structural complexity (α-helix, \b{eta}-hairpin, α-helix bundle, and α/\b{eta}-protein). Employing a tandem protein structure prediction algorithmic and molecular dynamics simulation approach, we find that the ability of reverse sequences…
Protein complexes are macromolecules essential to the functioning and well-being of all living organisms. As the structure of a protein complex, in particular its region of interaction between multiple protein subunits (i.e., chains), has a notable influence on the biological function of the complex, computational methods that can quickly and effectively be used to refine and assess the quality of a protein complex's 3D structure can directly be used within a drug discovery pipeline to accelerate the development of new therapeutics and improve the efficacy of future vaccines. In this work, we …
