Topological Methods for Polymeric Materials: Characterizing the Relationship Between Polymer Entanglement and Viscoelasticity

Topology-Based Detection and Tracking of Deadlocks Reveal Aging of Active Ring Melts

Connecting the viscoelastic behavior of stressed ring melts to the various forms of entanglement that can emerge in such systems is still an open challenge. Here, we consider active ring melts, where stress is generated internally, and introduce a topology-based method to detect and track consequential forms of ring entanglements, namely, deadlocks. We demonstrate that, as stress accumulates, more and more rings are co-opted in a growing web of deadlocks that entrap many other rings by threading, bringing the system to a standstill. The method ought to help the study of topological aging in more general polymer contexts.

Force transmission during repose of flexible granular chains

We study the mechanics of standing columns formed during the repose of flexible granular chains. It is one of the many intriguing behaviours exhibited by granular materials when links capable of transmitting tension exist between particles. We develop and calibrate a discrete element method contact model to simulate the mechanics of the macroscopic flexible granular chains and conduct simulations of the angle of repose experiments of these chains by extracting a chain-filled cylinder and allowing the material to flow out under gravity and repose. We evaluate various micro-mechanical, topological and macroscopic parameters to elucidate the mechanics of the repose behaviour of chain ensembles. It is the ability of the links connecting the individual particles to transmit tensile forces along the chain backbone that provides lateral stability to the column, enabling them to stand. In particular, the contact force rearrangement inside the columns generates a self-confining radial stress near the base of the columns, which provides an important stabilizing stress.

Topological damping in an ultrafast giant cell

Cellular systems are known to exhibit some of the fastest movements in biology, but little is known as to how single cells can dissipate this energy rapidly and adapt to such large accelerations without disrupting internal architecture. To address this, we investigate Spirostomum ambiguum-a giant cell (1-4 mm in length) well-known to exhibit ultrafast contractions (50% of body length) within 5 ms with a peak acceleration of 15[Formula: see text]. Utilizing transmitted electron microscopy and confocal imaging, we identify an association of rough endoplasmic reticulum (RER) and vacuoles throughout the cell-forming a contiguous fenestrated membrane architecture that topologically entangles these two organelles. A nearly uniform interorganelle spacing of 60 nm is observed between RER and vacuoles, closely packing the entire cell. Inspired by the entangled organelle structure, we study the mechanical properties of entangled deformable particles using a vertex-based model, with all simulation parameters matching 10 dimensionless numbers to ensure dynamic similarity. We demonstrate how entangled deformable particles respond to external loads by an increased viscosity against squeezing and help preserve spatial relationships. Because this enhanced damping arises from the entanglement of two networks incurring a strain-induced jamming transition at subcritical volume fractions, which is demonstrated through the spatial correlation of velocity direction, we term this phenomenon "topological damping." Our findings suggest a mechanical role of RER-vacuolar meshwork as a metamaterial capable of damping an ultrafast contraction event.

Ultrafast reversible self-assembly of living tangled matter

Tangled active filaments are ubiquitous in nature, from chromosomal DNA and cilia carpets to root networks and worm collectives. How activity and elasticity facilitate collective topological transformations in living tangled matter is not well understood. We studied California blackworms (Lumbriculus variegatus), which slowly form tangles in minutes but can untangle in milliseconds. Combining ultrasound imaging, theoretical analysis, and simulations, we developed and validated a mechanistic model that explains how the kinematics of individual active filaments determines their emergent collective topological dynamics. The model reveals that resonantly alternating helical waves enable both tangle formation and ultrafast untangling. By identifying generic dynamical principles of topological self-transformations, our results can provide guidance for designing classes of topologically tunable active materials.

The Jones polynomial of collections of open curves in 3-space

Measuring the entanglement complexity of collections of open curves in 3-space has been an intractable, yet pressing mathematical problem, relevant to a plethora of physical systems, such as in polymers and biopolymers. In this manuscript, we give a novel definition of the Jones polynomial that generalizes the classic Jones polynomial to collections of open curves in 3-space. More precisely, first we provide a novel definition of the Jones polynomial of linkoids (open link diagrams) and show that this is a well-defined single variable polynomial that is a topological invariant, which, for link-type linkoids, coincides with that of the corresponding link. Using the framework introduced in (Panagiotou E, Kauffman L. 2020 Proc. R. Soc. A 476 , 20200124. (( doi:10.1098/rspa.2020.0124 )), this enables us to define the Jones polynomial of collections of open and closed curves in 3-space. For collections of open curves in 3-space, the Jones polynomial has real coefficients and it is a continuous function of the curves’ coordinates. As the endpoints of the curves tend to coincide, the Jones polynomial tends to that of the resultant link. We demonstrate with numerical examples that the novel Jones polynomial enables us to characterize the topological/geometrical complexity of collections of open curves in 3-space for the first time.

The Local Topological Free Energy of the SARS-CoV-2 Spike Protein

The novel coronavirus SARS-CoV-2 infects human cells using a mechanism that involves binding and structural rearrangement of its Spike protein. Understanding protein rearrangement and identifying specific amino acids where mutations affect protein rearrangement has attracted much attention for drug development. In this manuscript, we use a mathematical method to characterize the local topology/geometry of the SARS-CoV-2 Spike protein backbone. Our results show that local conformational changes in the FP, HR1, and CH domains are associated with global conformational changes in the RBD domain. The SARS-CoV-2 variants analyzed in this manuscript (alpha, beta, gamma, delta Mink, G614, N501) show differences in the local conformations of the FP, HR1, and CH domains as well. Finally, most mutations of concern are either in or in the vicinity of high local topological free energy conformations, suggesting that high local topological free energy conformations could be targets for mutations with significant impact of protein function. Namely, the residues 484, 570, 614, 796, and 969, which are present in variants of concern and are targeted as important in protein function, are predicted as such from our model.

Microwave-Assisted Physically Cross-Linked Chitosan-Sodium Alginate Hydrogel Membrane Doped with Curcumin as a Novel Wound Healing Platform

This project purposes to develop chitosan and sodium alginate-based hydrogel membranes loaded with curcumin through microwave-based physical cross-linking technique and its evaluation for wound healing potential. For the purpose, curcumin-loaded chitosan and sodium alginate membranes were developed using microwave at fixed frequency of 2450 MHz, power 350 W for 60 s, and tested for their physicochemical attributes like swelling, erosion, surface morphology, drug content, and in vitro drug release. The membranes were also subjected to tensile strength and vibrational and thermal analysis followed by testing in vivo on animals. The results indicated that microwave treatment significantly enhanced the swelling ability, reduced the erosion, and ensured smooth surface texture with optimal drug content. The drug was released in a slow fashion releasing total of 41 ± 4.2% within 24-h period with a higher tensile strength of 16.4 ± 5.3 Mpa. The vibrational analysis results revealed significant fluidization of hydrophilic domains and defluidization of hydrophobic domains which translated into a significant rise in the melting temperature and corresponding enthalpy which were found to be 285.2 ± 3.2 °C and 4.89 ± 1.4 J/g. The in vivo testing revealed higher percent re-epithelialization (75 ± 2.3%) within 14 days of the treatment application in comparison to only gauze and other treatments applied, with higher extent of collagen deposition having well-defined epidermis and stratum corneum formation. The microwave-treated chitosan-sodium alginate hydrogel membranes loaded with curcumin may prove to be another alternative to treat skin injuries. Graphical Abstract.

Efficient compressed database of equilibrated configurations of ring-linear polymer blends for MD simulations

To effectively archive configuration data during molecular dynamics (MD) simulations of polymer systems, we present an efficient compression method with good numerical accuracy that preserves the topology of ring-linear polymer blends. To compress the fraction of floating-point data, we used the Jointed Hierarchical Precision Compression Number - Data Format (JHPCN-DF) method to apply zero padding for the tailing fraction bits, which did not affect the numerical accuracy, then compressed the data with Huffman coding. We also provided a dataset of well-equilibrated configurations of MD simulations for ring-linear polymer blends with various lengths of linear and ring polymers, including ring complexes composed of multiple rings such as polycatenane. We executed 10⁹ MD steps to obtain 150 equilibrated configurations. The combination of JHPCN-DF and SZ compression achieved the best compression ratio for all cases. Therefore, the proposed method enables efficient archiving of MD trajectories. Moreover, the publicly available dataset of ring-linear polymer blends can be employed for studies of mathematical methods, including topology analysis and data compression, as well as MD simulations.

Measurement(s)	equilibrated configurations of ring-linear polymer blends
Technology Type(s)	molecular dynamics simulation
Factor Type(s)	length of linear and ring polymer

Machine-accessible metadata file describing the reported data: 10.6084/m9.figshare.18742097

Alkali Concentration and Diluent Effects on Properties of Grape Cane Fiber-Reinforced Polymer Composites

The main objective of this study was to investigate the properties of polymer composites reinforced with grape cane fibers. The fibers were subjected to a sodium hydroxide (NaOH) treatment at two treatment concentrations to extract the fibers as well as fiber surface treatment. Panels were fabricated by hand lay-up and compression molding according to different fiber types, namely outer bark (OB) and whole (W) fibers. The whole fiber was a mixture of OB and inner bark (IB) fibers. Grape cane fibers were used as the reinforcement material for unsaturated polyester (UPE) resin panels. Acrylated epoxidized soybean oil (AESO) was used as a reactive diluent material with the UPE resin, and the results were compared with panels prepared with commercial styrene-UPE. There were inconsistent alkali treatment concentration effects on the mechanical properties and water absorption. However, panels fabricated with the whole bark fibers that have been treated with 1 wt % NaOH and had AESO-UPE resin resulted in the best tensile and flexural strength.

Topological and physical links in soft matter systems

Linking, or multicomponent topological entanglement, is ubiquitous in soft matter systems, from mixtures of polymers and DNA filaments packedin vivoto interlocked line defects in liquid crystals and intertwined synthetic molecules. Yet, it is only relatively recently that theoretical and experimental advancements have made it possible to probe such entanglements and elucidate their impact on the physical properties of the systems. Here, we review the state-of-the-art of this rapidly expanding subject and organize it as follows. First, we present the main concepts and notions, from topological linking to physical linking and then consider the salient manifestations of molecular linking, from synthetic to biological ones. We next cover the main physical models addressing mutual entanglements in mixtures of polymers, both linear and circular. Finally, we consider liquid crystals, fluids and other non-filamentous systems where topological or physical entanglements are observed in defect or flux lines. We conclude with a perspective on open challenges.

Vassiliev measures of complexity of open and closed curves in 3-space

In this article, we define Vassiliev measures of complexity for open curves in 3-space. These are related to the coefficients of the enhanced Jones polynomial of open curves in 3-space. These Vassiliev measures are continuous functions of the curve coordinates; as the ends of the curve tend to coincide, they converge to the corresponding Vassiliev invariants of the resulting knot. We focus on the second Vassiliev measure from the enhanced Jones polynomial for closed and open curves in 3-space. For closed curves, this second Vassiliev measure can be computed by a Gauss code diagram and it has an integral formulation, the double alternating self-linking integral. The double alternating self-linking integral is a topological invariant of closed curves and a continuous function of the curve coordinates for open curves in 3-space. For polygonal curves, the double alternating self-linking integral obtains a simpler expression in terms of geometric probabilities.

Liquid-liquid phase separation of Tau by self and complex coacervation

The liquid-liquid phase separation (LLPS) of Tau has been postulated to play a role in modulating the aggregation property of Tau, a process known to be critically associated with the pathology of a broad range of neurodegenerative diseases including Alzheimer's Disease. Tau can undergo LLPS by homotypic interaction through self-coacervation (SC) or by heterotypic association through complex-coacervation (CC) between Tau and binding partners such as RNA. What is unclear is in what way the formation mechanisms for self and complex coacervation of Tau are similar or different, and the addition of a binding partner to Tau alters the properties of LLPS and Tau. A combination of in vitro experimental and computational study reveals that the primary driving force for both Tau CC and SC is electrostatic interactions between Tau-RNA or Tau-Tau macromolecules. The liquid condensates formed by the complex coacervation of Tau and RNA have distinctly higher micro-viscosity and greater thermal stability than that formed by the SC of Tau. Our study shows that subtle changes in solution conditions, including molecular crowding and the presence of binding partners, can lead to the formation of different types of Tau condensates with distinct micro-viscosity that can coexist as persistent and immiscible entities in solution. We speculate that the formation, rheological properties and stability of Tau droplets can be readily tuned by cellular factors, and that liquid condensation of Tau can alter the conformational equilibrium of Tau.

Knot polynomials of open and closed curves

In this manuscript, we introduce a method to measure entanglement of curves in 3-space that extends the notion of knot and link polynomials to open curves. We define the bracket polynomial of curves in 3-space and show that it has real coefficients and is a continuous function of the curve coordinates. This is used to define the Jones polynomial in a way that it is applicable to both open and closed curves in 3-space. For open curves, the Jones polynomial has real coefficients and it is a continuous function of the curve coordinates and as the endpoints of the curve tend to coincide, the Jones polynomial of the open curve tends to that of the resulting knot. For closed curves, it is a topological invariant, as the classical Jones polynomial. We show how these measures attain a simpler expression for polygonal curves and provide a finite form for their computation in the case of polygonal curves of 3 and 4 edges.

GISA: using Gauss Integrals to identify rare conformations in protein structures

The native structure of a protein is important for its function, and therefore methods for exploring protein structures have attracted much research. However, rather few methods are sensitive to topologic-geometric features, the examples being knots, slipknots, lassos, links, and pokes, and with each method aimed only for a specific set of such configurations. We here propose a general method which transforms a structure into a ”fingerprint of topological-geometric values” consisting in a series of real-valued descriptors from mathematical Knot Theory. The extent to which a structure contains unusual configurations can then be judged from this fingerprint. The method is not confined to a particular pre-defined topology or geometry (like a knot or a poke), and so, unlike existing methods, it is general. To achieve this our new algorithm, GISA, as a key novelty produces the descriptors, so called Gauss integrals, not only for the full chains of a protein but for all its sub-chains. This allows fingerprinting on any scale from local to global. The Gauss integrals are known to be effective descriptors of global protein folds. Applying GISA to sets of several thousand high resolution structures, we first show how the most basic Gauss integral, the writhe, enables swift identification of pre-defined geometries such as pokes and links. We then apply GISA with no restrictions on geometry, to show how it allows identifying rare conformations by finding rare invariant values only. In this unrestricted search, pokes and links are still found, but also knotted conformations, as well as more highly entangled configurations not previously described. Thus, an application of the basic scan method in GISA’s tool-box revealed 10 known cases of knots as the top positive writhe cases, while placing at the top of the negative writhe 14 cases in cis-trans isomerases sharing a spatial motif of little secondary structure content, which possibly has gone unnoticed. Possible general applications of GISA are fold classification and structural alignment based on local Gauss integrals. Others include finding errors in protein models and identifying unusual conformations that might be important for protein folding and function. By its broad potential, we believe that GISA will be of general benefit to the structural bioinformatics community. GISA is coded in C and comes as a command line tool. Source and compiled code for GISA plus read-me and examples are publicly available at GitHub (https://github.com).

Characterization of entanglements in glassy polymeric ensembles using the Gaussian linking number

We propose a method for enumerating entanglements between long chained, linear polymers that is based on the Gaussian linking number. The linking number is calculated between closely approaching segments of the macromolecular chains. Topological features of an entanglement, i.e., the extent to which one open segment winds around another, are reflected by the linking number. We show that using this measure, we can track disentanglement events through a deformation history and gain insights into how large scale disentanglements lead to failure. Incorporating an additional step where the topological entanglements identified along each chain are optimally clustered using standard clustering algorithms, we can also obtain a measure of the average number of rheological constraints that exist along each chain in an ensemble. Comparisons with other methods of enumerating entanglements, especially the primitive path analysis, are also made. Our results indicate that the linking number between two entangled segments in the undeformed state is a good indicator of the strength of the entanglement. Also, disentanglements occurring overwhelmingly around chain ends are an important cause of failure when a triaxial stress state exists in the polymer.