Topology of Folded Molecular Chains: From Single Biomolecules to Engineered Origami

Cytosolic Interactome Protects Against Protein Unfolding in a Single Molecule Experiment

Single molecule techniques are particularly well suited for investigating the processes of protein folding and chaperone assistance. However, current assays provide only a limited perspective on the various ways in which the cellular environment can influence the folding pathway of a protein. In this study, a single molecule mechanical interrogation assay is developed and used to monitor protein unfolding and refolding within a cytosolic solution. This allows to test the cumulative topological effect of the cytoplasmic interactome on the folding process. The results reveal a stabilization against forced unfolding for partial folds, which are attributed to the protective effect of the cytoplasmic environment against unfolding and aggregation. This research opens the possibility of conducting single molecule molecular folding experiments in quasi-biological environments.

Crystal Structure of de Novo Designed Coiled-Coil Protein Origami Triangle

Coiled-coil protein origami (CCPO) uses modular coiled-coil building blocks and topological principles to design polyhedral structures distinct from those of natural globular proteins. While the CCPO strategy has proven successful in designing diverse protein topologies, no high-resolution structural information has been available about these novel protein folds. Here we report the crystal structure of a single-chain CCPO in the shape of a triangle. While neither cyclization nor the addition of nanobodies enabled crystallization, it was ultimately facilitated by the inclusion of a GCN2 homodimer. Triangle edges are formed by the orthogonal parallel coiled-coil dimers P1:P2, P3:P4, and GCN2 connected by short linkers. A triangle has a large central cavity and is additionally stabilized by side-chain interactions between neighboring segments at each vertex. The crystal lattice is densely packed and stabilized by a large number of contacts between triangles. Interestingly, the polypeptide chain folds into a trefoil-type protein knot topology, and AlphaFold2 fails to predict the correct fold. The structure validates the modular CC-based protein design strategy, providing molecular insight underlying CCPO stabilization and new opportunities for the design.

Circuit topology predicts pathogenicity of missense mutations

The contact topology of a protein determines important aspects of the folding process. The topological measure of contact order has been shown to be predictive of the rate of folding. Circuit topology is emerging as another fundamental descriptor of biomolecular structure, with predicted effects on the folding rate. We analyze the residue‐based circuit topological environments of 21 K mutations labeled as pathogenic or benign. Multiple statistical lines of reasoning support the conclusion that the number of contacts in two specific circuit topological arrangements, namely inverse parallel and cross relations, with contacts involving the mutated residue have discriminatory value in determining the pathogenicity of human variants. We investigate how results vary with residue type and according to whether the gene is essential. We further explore the relationship to a number of structural features and find that circuit topology provides nonredundant information on protein structures and pathogenicity of mutations. Results may have implications for the polymer physics of protein folding and suggest that “local” topological information, including residue‐based circuit topology and residue contact order, could be useful in improving state‐of‐the‐art machine learning algorithms for pathogenicity prediction.

Femtosecond Laser Direct Writing of Optical Overpass

With the rapid increase in information density, problems such as signal crosstalk and crossover restrict the further expansion of chip integration levels and packaging density. Based on this, a novel waveguide structure—photonic jumper wire—is proposed here to break through the technical restrictions in waveguide crossing and parallel line wrapping, which hinder the integration of photonic chips. Furthermore, we fabricated the optical overpass to realize a more complex on-chip optical cross-connection. Our method and structure promote a series of practical schemes for improving optical chip integration.

Circuit topology analysis of cellular genome reveals signature motifs, conformational heterogeneity, and scaling

Reciprocal regulation of genome topology and function is a fundamental and enduring puzzle in biology. The wealth of data provided by Hi-C libraries offers the opportunity to unravel this relationship. However, there is a need for a comprehensive theoretical framework in order to extract topological information for genome characterization and comparison. Here, we develop a toolbox for topological analysis based on Circuit Topology, allowing for the quantification of inter- and intracellular genomic heterogeneity, at various levels of fold complexity: pairwise contact arrangement, higher-order contact arrangement, and topological fractal dimension. Single-cell Hi-C data were analyzed and characterized based on topological content, revealing not only a strong multiscale heterogeneity but also highly conserved features such as a characteristic topological length scale and topological signature motifs in the genome. We propose that these motifs inform on the topological state of the nucleus and indicate the presence of active loop extrusion.

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Circuit topology quantifies heterogeneity in genomic arrangement

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Scale analysis reveals a characteristic length scale of 10 Mb in genome topology

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We identify highly conserved topological structures related to loop extrusion

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We suggest a topological model of chromatin arrangement for loop extrusion, the L-loop

Circuit Topology for Bottom-Up Engineering of Molecular Knots

The art of tying knots is exploited in nature and occurs in multiple applications ranging from being an essential part of scouting programs to engineering molecular knots. Biomolecular knots, such as knotted proteins, bear various cellular functions, and their entanglement is believed to provide them with thermal and kinetic stability. Yet, little is known about the design principles of naturally evolved molecular knots. Intra-chain contacts and chain entanglement contribute to the folding of knotted proteins. Circuit topology, a theory that describes intra-chain contacts, was recently generalized to account for chain entanglement. This generalization is unique to circuit topology and not motivated by other theories. In this conceptual paper, we systematically analyze the circuit topology approach to a description of linear chain entanglement. We utilize a bottom-up approach, i.e., we express entanglement by a set of four fundamental structural units subjected to three (or five) binary topological operations. All knots found in proteins form a well-defined, distinct group which naturally appears if expressed in terms of these basic structural units. We believe that such a detailed, bottom-up understanding of the structure of molecular knots should be beneficial for molecular engineering.

Topological principles of protein folding

What is the topology of a protein and what governs protein folding to a specific topology? This is a fundamental question in biology. The protein folding reaction is a critically important cellular process, which is failing in many prevalent diseases. Understanding protein folding is also key to the design of new proteins for applications. However, our ability to predict the folding of a protein chain is quite limited and much is still unknown about the topological principles of folding. Current predictors of folding kinetics, including the contact order and size, present a limited predictive power, suggesting that these models are fundamentally incomplete. Here, we use a newly developed mathematical framework to define and extract the topology of a native protein conformation beyond knot theory, and investigate the relationship between native topology and folding kinetics in experimentally characterized proteins. We show that not only the folding rate, but also the mechanistic insight into folding mechanisms can be inferred from topological parameters. We identify basic topological features that speed up or slow down the folding process. The approach enabled the decomposition of protein 3D conformation into topologically independent elementary folding units, called circuits. The number of circuits correlates significantly with the folding rate, offering not only an efficient kinetic predictor, but also a tool for a deeper understanding of theoretical folding models. This study contributes to recent work that reveals the critical relevance of topology to protein folding with a new, contact-based, mathematically rigorous perspective. We show that topology can predict folding kinetics when geometry-based predictors like contact order and size fail.

Polynomial Invariant of Molecular Circuit Topology

The topological framework of circuit topology has recently been introduced to complement knot theory and to help in understanding the physics of molecular folding. Naturally evolved linear molecular chains, such as proteins and nucleic acids, often fold into 3D conformations with critical chain entanglements and local or global structural symmetries stabilised by formation contacts between different parts of the chain. Circuit topology captures the arrangements of intra-chain contacts within a given folded linear chain and allows for the classification and comparison of chains. Contacts keep chain segments in physical proximity and can be either mechanically hard attachments or soft entanglements that constrain a physical chain. Contrary to knot theory, which offers many established knot invariants, circuit invariants are just being developed. Here, we present polynomial invariants that are both efficient and sufficiently powerful to deal with any combination of soft and hard contacts. A computer implementation and table of chains with up to three contacts is also provided.

Protein structural features predict responsiveness to pharmacological chaperone treatment for three lysosomal storage disorders

Three-dimensional structures of proteins can provide important clues into the efficacy of personalized treatment. We perform a structural analysis of variants within three inherited lysosomal storage disorders, comparing variants responsive to pharmacological chaperone treatment to those unresponsive to such treatment. We find that predicted ΔΔG of mutation is higher on average for variants unresponsive to treatment, in the case of datasets for both Fabry disease and Pompe disease, in line with previous findings. Using both a single decision tree and an advanced machine learning approach based on the larger Fabry dataset, we correctly predict responsiveness of three Gaucher disease variants, and we provide predictions for untested variants. Many variants are predicted to be responsive to treatment, suggesting that drug-based treatments may be effective for a number of variants in Gaucher disease. In our analysis, we observe dependence on a topological feature reporting on contact arrangements which is likely connected to the order of folding of protein residues, and we provide a potential justification for this observation based on steady-state cellular kinetics.

Pharmacological chaperones are small molecule drugs that bind to proteins to help stabilize the folded state. One set of diseases for which this treatment has been effective is the lysosomal storage disorders, which are caused by defective lysosomal enzymes. However, not all genotypes are equally responsive to treatment. For instance, missense mutants that are particularly destabilized relative to WT are less likely to respond. The availability of datasets containing responsiveness data for large numbers of mutants, along with crystal structures of the protein involved in each disease, make machine learning methods incorporating sequence-based and structural data feasible. We hypothesize that data from two diseases, Fabry and Pompe disease, may be useful for predicting responsiveness of variants in the related Gaucher disease. Results suggest that many rare variants in Gaucher disease could be amenable to existing drugs. Results also suggest that drug responsiveness depends on protein topology in such a way that mutations in early-to-fold residues are more likely to be non-responsive to pharmacological chaperone treatment, which is consistent with a simple kinetic model of stability rescue. This study provides an example of how machine learning can be used to inform further studies towards personalized treatment in medicine.

Coloring Invariant for Topological Circuits in Folded Linear Chains

Circuit topology is a mathematical approach that categorizes the arrangement of contacts within a folded linear chain, such as a protein molecule or the genome. Theses linear biomolecular chains often fold into complex 3D architectures with critical entanglements and local or global structural symmetries stabilised by formation of intrachain contacts. Here, we adapt and apply the algebraic structure of quandles to classify and distinguish chain topologies within the framework of circuit topology. We systematically study the basic circuit topology motifs and define quandle/bondle coloring for them. Next, we explore the implications of circuit topology operations that enable building complex topologies from basic motifs for the quandle coloring approach.

Generalized Circuit Topology of Folded Linear Chains

A wide range of physical systems can be formally mapped to a linear chain of sorted objects. Upon introduction of intrachain interactions, such a chain can "fold" to elaborate topological structures, analogous to folded linear polymer systems. Two distinct chain-topology theories, knot theory and circuit topology, have separately provided insight into the structure, dynamics, and evolution of folded linear polymers such as proteins and genomic DNA. Knot theory, however, ignores intrachain interactions (contacts), whereas chain crossings are ignored in circuit topology. Thus, there is a need for a universal approach that can provide topological description of any folded linear chain. Here, we generalize circuit topology in order to grasp particularities typically addressed by knot theory. We develop a generic approach that is simple, mathematically rigorous, and practically useful for structural classification, analysis of structural dynamics, and engineering applications.