Rapid knot detection and application to protein structure prediction

Comparison of Fused Deposition Modeling and Color Jet 3D Printing Technologies for the Printing of Mathematical Geometries

Many mathematical geometries act as an optimal structure for functional applications and have always been an area of interest in the research field. Their topology offers properties which are crucial and can be used effectively in various domains. Apart from that, some have a resemblance to naturally occurring compounds which can help us to study their different transformations and behavior. In this paper, we present two such geometries, first, gyroid, which is an iso-minimal surface and second a three-crossing knot, also known as trefoil knot. The structure of gyroid makes it unique and is considered suitable in developing energy-absorbing, structural and lightweight applications. Similarly, some types of knots resemble the DNA structure and have found use in molecular chemistry. This paper discusses different application areas of these geometries. Further, this paper presents modeling and printing by using fused deposition modeling (FDM) and color jet printing (CJP). Comparative analysis has been done by considering various parameters. This paper discusses the potential of these two rapid prototyping technologies and their suitability for specific printing applications.

Computational methods in the study of self-entangled proteins: a critical appraisal

The existence of self-entangled proteins, the native structure of which features a complex topology, unveils puzzling, and thus fascinating, aspects of protein biology and evolution. The discovery that a polypeptide chain can encode the capability to self-entangle in an efficient and reproducible way during folding, has raised many questions, regarding the possible function of these knots, their conservation along evolution, and their role in the folding paradigm. Understanding the function and origin of these entanglements would lead to deep implications in protein science, and this has stimulated the scientific community to investigate self-entangled proteins for decades by now. In this endeavour, advanced experimental techniques are more and more supported by computational approaches, that can provide theoretical guidelines for the interpretation of experimental results, and for the effective design of new experiments. In this review we provide an introduction to the computational study of self-entangled proteins, focusing in particular on the methodological developments related to this research field. A comprehensive collection of techniques is gathered, ranging from knot theory algorithms, that allow detection and classification of protein topology, to Monte Carlo or molecular dynamics strategies, that constitute crucial instruments for investigating thermodynamics and kinetics of this class of proteins.

Topological descriptions of protein folding

How knotted proteins fold has remained controversial since the identification of deeply knotted proteins nearly two decades ago. Both computational and experimental approaches have been used to investigate protein knot formation. Motivated by the computer simulations of Bölinger et al. [Bölinger D, et al. (2010) PLoS Comput Biol 6:e1000731] for the folding of the [Formula: see text]-knotted α-haloacid dehalogenase (DehI) protein, we introduce a topological description of knot folding that could describe pathways for the formation of all currently known protein knot types and predicts knot types that might be identified in the future. We analyze fingerprint data from crystal structures of protein knots as evidence that particular protein knots may fold according to specific pathways from our theory. Our results confirm Taylor's twisted hairpin theory of knot folding for the [Formula: see text]-knotted proteins and the [Formula: see text]-knotted ketol-acid reductoisomerases and present alternative folding mechanisms for the [Formula: see text]-knotted phytochromes and the [Formula: see text]- and [Formula: see text]-knotted proteins.

The energy cost of polypeptide knot formation and its folding consequences

Knots are natural topologies of chains. Yet, little is known about spontaneous knot formation in a polypeptide chain—an event that can potentially impair its folding—and about the effect of a knot on the stability and folding kinetics of a protein. Here we used optical tweezers to show that the free energy cost to form a trefoil knot in the denatured state of a polypeptide chain of 120 residues is 5.8 ± 1 kcal mol⁻¹. Monte Carlo dynamics of random chains predict this value, indicating that the free energy cost of knot formation is of entropic origin. This cost is predicted to remain above 3 kcal mol⁻¹ for denatured proteins as large as 900 residues. Therefore, we conclude that naturally knotted proteins cannot attain their knot randomly in the unfolded state but must pay the cost of knotting through contacts along their folding landscape.

The effect of knots on protein stability and folding kinetics is not well understood. Here the authors combine optical tweezer experiments and calculations to experimentally determine the energy cost for knot formation, which indicates that knotted proteins evolved specific folding pathways because knot formation in unfolded chains is unfavorable.

To Tie or Not to Tie? That Is the Question

In this review, we provide an overview of entangled proteins. Around 6% of protein structures deposited in the PBD are entangled, forming knots, slipknots, lassos and links. We present theoretical methods and tools that enabled discovering and classifying such structures. We discuss the advantages and disadvantages of the non-trivial topology in proteins, based on available data about folding, stability, biological properties and evolutionary conservation. We also formulate intriguing and challenging questions on the border of biophysics, bioinformatics, biology and mathematics, which arise from the discovery of an entanglement in proteins. Finally, we discuss possible applications of entangled proteins in medicine and nanotechnology, such as the chance to design super stable proteins, whose stability could be controlled by chemical potential.

LassoProt: server to analyze biopolymers with lassos

The LassoProt server, http://lassoprot.cent.uw.edu.pl/, enables analysis of biopolymers with entangled configurations called lassos. The server offers various ways of visualizing lasso configurations, as well as their time trajectories, with all the results and plots downloadable. Broad spectrum of applications makes LassoProt a useful tool for biologists, biophysicists, chemists, polymer physicists and mathematicians. The server and our methods have been validated on the whole PDB, and the results constitute the database of proteins with complex lassos, supported with basic biological data. This database can serve as a source of information about protein geometry and entanglement-function correlations, as a reference set in protein modeling, and for many other purposes.

Molecular knots in biology and chemistry

Knots and entanglements are ubiquitous. Beyond their aesthetic appeal, these fascinating topological entities can be either useful or cumbersome. In recent decades, the importance and prevalence of molecular knots have been increasingly recognised by scientists from different disciplines. In this review, we provide an overview on the various molecular knots found in naturally occurring biological systems (DNA, RNA and proteins), and those created by synthetic chemists. We discuss the current knowledge in these fields, including recent developments in experimental and, in some cases, computational studies which are beginning to shed light into the complex interplay between the structure, formation and properties of these topologically intricate molecules.

Comparative analysis of the folding dynamics and kinetics of an engineered knotted protein and its variants derived from HP0242 of Helicobacter pylori

Understanding the mechanism by which a polypeptide chain thread itself spontaneously to attain a knotted conformation has been a major challenge in the field of protein folding. HP0242 is a homodimeric protein from Helicobacter pylori with intertwined helices to form a unique pseudo-knotted folding topology. A tandem HP0242 repeat has been constructed to become the first engineered trefoil-knotted protein. Its small size renders it a model system for computational analyses to examine its folding and knotting pathways. Here we report a multi-parametric study on the folding stability and kinetics of a library of HP0242 variants, including the trefoil-knotted tandem HP0242 repeat, using far-UV circular dichroism and fluorescence spectroscopy. Equilibrium chemical denaturation of HP0242 variants shows the presence of highly populated dimeric and structurally heterogeneous folding intermediates. Such equilibrium folding intermediates retain significant amount of helical structures except those at the N- and C-terminal regions in the native structure. Stopped-flow fluorescence measurements of HP0242 variants show that spontaneous refolding into knotted structures can be achieved within seconds, which is several orders of magnitude faster than previously observed for other knotted proteins. Nevertheless, the complex chevron plots indicate that HP0242 variants are prone to misfold into kinetic traps, leading to severely rolled-over refolding arms. The experimental observations are in general agreement with the previously reported molecular dynamics simulations. Based on our results, kinetic folding pathways are proposed to qualitatively describe the complex folding processes of HP0242 variants.

Two Adhesive Sites Can Enhance the Knotting Probability of DNA

Self-entanglement, or knotting, is entropically favored in long polymers. Relatively short polymers such as proteins can knot as well, but in this case the entanglement is mainly driven by fine-tuned, sequence-specific interactions. The relation between the sequence of a long polymer and its topological state is here investigated by means of a coarse-grained model of DNA. We demonstrate that the introduction of two adhesive regions along the sequence of a self-avoiding chain substantially increases the probability of forming a knot.

A knot in the protein structure - probing the near-infrared fluorescent protein iRFP designed from a bacterial phytochrome

The possibility of engineering near-infrared fluorescent proteins and biosensors from bacterial phytochrome photoreceptors (BphPs) has led to substantial interest in this family of proteins. The near-infrared fluorescent proteins have allowed non-invasive bio-imaging of deep tissues and whole organs in living animals. BphPs and derived near-infrared fluorescent proteins contain a structural element, called a knot, in their polypeptide chains. The formation of knot structures in proteins was refuted for a long time. Here, we studied the denaturation and renaturation processes of the near-infrared fluorescent probe iRFP, engineered from RpBphP2, which utilizes a heme-derived tetrapyrrole compound biliverdin as a chromophore. iRFP contains a unique figure-of-eight knot. The denaturation and renaturation curves of the iRFP apoform coincided well, suggesting efficient refolding. However, the iRFP holoform exhibited irreversible unfolding and aggregation associated with the bound chromophore. The knot structure in the apoform did not prevent subsequent binding of biliverdin, resulting in the functional iRFP holoform. We suggest that the irreversibility of protein unfolding is caused by post-translational protein modifications, such as chromophore binding, rather than the presence of the knot. These results are essential for future design of BphP-based near-infrared probes, and add important features to our knowledge of protein folding.

Knotting pathways in proteins

Most proteins, in order to perform their biological function, have to fold to a compact native state. The increasing number of knotted and slipknotted proteins identified suggests that proteins are able to manoeuvre around topological barriers during folding. In the present article, we review the current progress in elucidating the knotting process in proteins. Although we concentrate on theoretical approaches, where a knotted topology can be unambiguously detected, comparison with experiments is also reviewed. Numerical simulations suggest that the folding process for small knotted proteins is composed of twisted loop formation and then threading by either slipknot geometries or flipping. As the size of the knotted proteins increases, particularly for more deeply threaded termini, the prevalence of traps in the free energy landscape also increases. Thus, in the case of longer knotted and slipknotted proteins, the folding mechanism is probably supported by chaperones. Overall, results imply that knotted proteins can be folded efficiently and survive evolutionary pressure in order to perform their biological functions.

Probing the elasticity of DNA on short length scales by modeling supercoiling under tension

The wormlike-chain (WLC) model is widely used to describe the energetics of DNA bending. Motivated by recent experiments, alternative, so-called subelastic chain models were proposed that predict a lower elastic energy of highly bent DNA conformations. Until now, no unambiguous verification of these models has been obtained because probing the elasticity of DNA on short length scales remains challenging. Here we investigate the limits of the WLC model using coarse-grained Monte Carlo simulations to model the supercoiling of linear DNA molecules under tension. At a critical supercoiling density, the DNA extension decreases abruptly due to the sudden formation of a plectonemic structure. This buckling transition is caused by the large energy required to form the tightly bent end-loop of the plectoneme and should therefore provide a sensitive benchmark for model evaluation. Although simulations based on the WLC energetics could quantitatively reproduce the buckling measured in magnetic tweezers experiments, the buckling almost disappears for the tested linear subelastic chain model. Thus, our data support the validity of a harmonic bending potential even for small bending radii down to 3.5 nm.

PyKnot: a PyMOL tool for the discovery and analysis of knots in proteins

SUMMARY

Understanding the differences between knotted and unknotted protein structures may offer insights into how proteins fold. To characterize the type of knot in a protein, we have developed PyKnot, a plugin that works seamlessly within the PyMOL molecular viewer and gives quick results including the knot's invariants, crossing numbers and simplified knot projections and backbones. PyKnot may be useful to researchers interested in classifying knots in macromolecules and provides tools for students of biology and chemistry with which to learn topology and macromolecular visualization.

AVAILABILITY

PyMOL is available at http://www.pymol.org. The PyKnot module and tutorial videos are available at http://youtu.be/p95aif6xqcM.

CONTACT

rhonald.lua@gmail.com.

Characterizing the existing and potential structural space of proteins by large-scale multiple loop permutations

Worldwide structural genomics projects are increasing structure coverage of sequence space but have not significantly expanded the protein structure space itself (i.e., number of unique structural folds) since 2007. Discovering new structural folds experimentally by directed evolution and random recombination of secondary-structure blocks is also proved rarely successful. Meanwhile, previous computational efforts for large-scale mapping of protein structure space are limited to simple model proteins and led to an inconclusive answer on the completeness of the existing observed protein structure space. Here, we build novel protein structures by extending naturally occurring circular (single-loop) permutation to multiple loop permutations (MLPs). These structures are clustered by structural similarity measure called TM-score. The computational technique allows us to produce different structural clusters on the same naturally occurring, packed, stable core but with alternatively connected secondary-structure segments. A large-scale MLP of 2936 domains from structural classification of protein domains reproduces those existing structural clusters (63%) mostly as hubs for many nonredundant sequences and illustrates newly discovered novel clusters as islands adopted by a few sequences only. Results further show that there exist a significant number of novel potentially stable clusters for medium-size or large-size single-domain proteins, in particular, >100 amino acid residues, that are either not yet adopted by nature or adopted only by a few sequences. This study suggests that MLP provides a simple yet highly effective tool for engineering and design of novel protein structures (including naturally knotted proteins). The implication of recovering new-fold targets from critical assessment of structure prediction techniques (CASP) by MLP on template-based structure prediction is also discussed. Our MLP structures are available for download at the publication page of the Web site http://sparks.informatics.iupui.edu.

Structures and folding pathways of topologically knotted proteins

In the last decade, a new class of proteins has emerged that contain a topological knot in their backbone. Although these structures are rare, they nevertheless challenge our understanding of protein folding. In this review, we provide a short overview of topologically knotted proteins with an emphasis on newly discovered structures. We discuss the current knowledge in the field, including recent developments in both experimental and computational studies that have shed light on how these intricate structures fold.

Pokefind: a novel topological filter for use with protein structure prediction

MOTIVATION

Our focus has been on detecting topological properties that are rare in real proteins, but occur more frequently in models generated by protein structure prediction methods such as Rosetta. We previously created the Knotfind algorithm, successfully decreasing the frequency of knotted Rosetta models during CASP6. We observed an additional class of knot-like loops that appeared to be equally un-protein-like and yet do not contain a mathematical knot. These topological features are commonly referred to as slip-knots and are caused by the same mechanisms that result in knotted models. Slip-knots are undetectable by the original Knotfind algorithm. We have generalized our algorithm to detect them, and analyzed CASP6 models built using the Rosetta loop modeling method.

RESULTS

After analyzing known protein structures in the PDB, we found that slip-knots do occur in certain proteins, but are rare and fall into a small number of specific classes. Our group used this new Pokefind algorithm to distinguish between these rare real slip-knots and the numerous classes of slip-knots that we discovered in Rosetta models and models submitted by the various CASP7 servers. The goal of this work is to improve future models created by protein structure prediction methods. Both algorithms are able to detect un-protein-like features that current metrics such as GDT are unable to identify, so these topological filters can also be used as additional assessment tools.

Tightening the knot in phytochrome by single-molecule atomic force microscopy

A growing number of proteins have been shown to adopt knotted folds. Yet the biological roles and biophysical properties of these knots remain poorly understood. We used protein engineering and atomic force microscopy to explore the single-molecule mechanics of the figure-eight knot in the chromophore-binding domain of the red/far-red photoreceptor, phytochrome. Under load, apo phytochrome unfolds at forces of approximately 47 pN, whereas phytochrome carrying its covalently bound tetrapyrrole chromophore unfolds at approximately 73 pN. These forces are not unusual in mechanical protein unfolding, and thus the presence of the knot does not automatically indicate a superstable protein. Our experiments reveal a stable intermediate along the mechanical unfolding pathway, reflecting the sequential unfolding of two distinct subdomains in phytochrome, potentially the GAF and PAS domains. For the first time (to the best of our knowledge), our experiments allow a direct determination of knot size under load. In the unfolded chain, the tightened knot is reduced to 17 amino acids, resulting in apparent shortening of the polypeptide chain by 6.2 nm. Steered molecular-dynamics simulations corroborate this number. Finally, we find that covalent phytochrome dimers created for these experiments retain characteristic photoreversibility, unexpectedly arguing against a dramatic rearrangement of the native GAF dimer interface upon photoconversion.

How does a knotted protein fold?

The issue of how a newly synthesized polypeptide chain folds to form a protein with a unique three-dimensional structure, otherwise known as the 'protein-folding problem', remains a fundamental question in the life sciences. Over the last few decades, much information has been gathered about the mechanisms by which proteins fold. However, despite the vast topological diversity observed in biological structures, it was thought improbable, if not impossible, that a polypeptide chain could 'knot' itself to form a functional protein. Nevertheless, such knotted structures have since been identified, raising questions about how such complex topologies can arise during folding. Their formation does not fit any current folding models or mechanisms, and therefore represents an important piece of the protein-folding puzzle. This article reviews the progress made towards discovering how nature codes for, and contends with, knots during protein folding, and examines the insights gained from both experimental and computational studies. Mechanisms to account for the formation of knotted structures that were previously thought unfeasible, and their implications for protein folding, are also discussed.

Knotted and topologically complex proteins as models for studying folding and stability

Among proteins of known three-dimensional structure, only a few possess complex topological features such as knotted or interlinked (catenated) protein backbones. Such unusual proteins offer potentially unique insights into folding pathways and stabilization mechanisms. They also present special challenges for both theorists and computational scientists interested in understanding and predicting protein-folding behavior. Here, we review complex topological features in proteins with a focus on recent progress on the identification and characterization of knotted and interlinked protein systems. Also, an approach is described for designing an expanded set of knotted proteins.

Identification of rare slipknots in proteins and their implications for stability and folding

Among the thousands of known three-dimensional protein folds, only a few have been found whose backbones are in knotted configurations. The rarity of knotted proteins has important implications for how natural proteins reach their natively folded states. Proteins with such unusual features offer unique opportunities for studying the relationships between structure, folding, and stability. Here we report the identification of a unique slipknot feature in the fold of a well-known thermostable protein, alkaline phosphatase. A slipknot is created when a knot is formed by part of a protein chain, after which the backbone doubles back so that the entire structure becomes unknotted in a mathematical sense. Slipknots are therefore not detected by computational tests that look for knots in complete protein structures. A computational survey looking specifically for slipknots in the Protein Data Bank reveals a few other instances in addition to alkaline phosphatase. Unexpected similarities are noted among some of the proteins identified. In addition, two transmembrane proteins are found to contain slipknots. Finally, mutagenesis experiments on alkaline phosphatase are used to probe the contribution the slipknot feature makes to thermal stability. The trends and conserved features observed in these proteins provide new insights into mechanisms of protein folding and stability.

A single mutation in the active site swaps the substrate specificity of N-acetyl-L-ornithine transcarbamylase and N-succinyl-L-ornithine transcarbamylase

Transcarbamylases catalyze the transfer of the carbamyl group from carbamyl phosphate (CP) to an amino group of a second substrate such as aspartate, ornithine, or putrescine. Previously, structural determination of a transcarbamylase from Xanthomonas campestris led to the discovery of a novel N-acetylornithine transcarbamylase (AOTCase) that catalyzes the carbamylation of N-acetylornithine. Recently, a novel N-succinylornithine transcarbamylase (SOTCase) from Bacteroides fragilis was identified. Structural comparisons of AOTCase from X. campestris and SOTCase from B. fragilis revealed that residue Glu92 (X. campestris numbering) plays a critical role in distinguishing AOTCase from SOTCase. Enzymatic assays of E92P, E92S, E92V, and E92A mutants of AOTCase demonstrate that each of these mutations converts the AOTCase to an SOTCase. Similarly, the P90E mutation in B. fragilis SOTCase (equivalent to E92 in X. campestris AOTCase) converts the SOTCase to AOTCase. Hence, a single amino acid substitution is sufficient to swap the substrate specificities of AOTCase and SOTCase. X-ray crystal structures of these mutants in complexes with CP and N-acetyl-L-norvaline (an analog of N-acetyl-L-ornithine) or N-succinyl-L-norvaline (an analog of N-succinyl-L-ornithine) substantiate this conversion. In addition to Glu92 (X. campestris numbering), other residues such as Asn185 and Lys30 in AOTCase, which are involved in binding substrates through bridging water molecules, help to define the substrate specificity of AOTCase. These results provide the correct annotation (AOTCase or SOTCase) for a set of the transcarbamylase-like proteins that have been erroneously annotated as ornithine transcarbamylase (OTCase, EC 2.1.3.3).

Protein knots and fold complexity: Some new twists

The current knowledge on topological knots in protein structure is reviewed, considering in turn, knots with three, four and five strand crossings. The latter is the most recent to be identified and has two distinct topological forms. The knot observed in the protein structure is the form that requires the least number of strand crossings to become un-knotted. The position of the chain termini must also correspond to a position that allows (un) knotting in one move. This is postulated as a general property of protein knots and other more complex knots with this property are proposed as the next most likely knots that might be found in a protein. It is also noted that the "Jelly-roll" fold found in some all-beta proteins would provide likely candidates. Alternative measures of knottedness and entanglement are reviewed, including the occurrence of slip-knots. These measures are related to the complexity of the protein fold and may provide useful filters for selecting predicted model structures.