Schedule for: 21w5232 - Novel Mathematical Methods in Material Science: Applications to Biomaterials (Online)

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Abstract I will report on a series of studies where we looked at how the static and dynamics of entangled polymers is affected by confinement. Specifically, I will first by consider the knotting of semi-flexible chains inside channels of different size and discuss how the size and complexity evolves during the free or externally-driven dynamics of the chain[1,2]. Next, I will turn to the case of linked rings inside channels and slits and discuss how the size and dynamics of their linked portion responds to different types of confinement[3,4]. References [1] C. Micheletti and E. Orlandini, ”Knotting and unknotting dynamics of DNA strands in nanochannels”, ACS Macro Letters, 3 , 876-880 (2014) [2] D. Michieletto, E. Orlandini, M.S. Turner and C. Micheletti, ”Separation of Geometrical and Topological entangle- ment in Confined polymers Driven out of Equilibrium”, ACS Macro Letters, 9 , 1081-1085 (2020) [3] G. D’Adamo, E. Orlandini and C. Micheletti, ”Linking of ring polymers in slit-like confinement”, Macromolecules,, 50 , 1713-1718 (2017) [4] G. Amici, M. Caraglio, E. Orlandini and C. Micheletti, ”Topologically Linked Chains in Confinement”, ACS Macro Lett., 8 , 442-446 (2019)

The mechanics of cells and tissues are largely governed by scaffolds of filamentous proteins that make up the cytoskeleton, as well as extracellular matrices. Evidence is emerging that such networks can exhibit rich mechanical phase behavior. A classic example of a mechanical phase transition was identified by Maxwell for macroscopic engineering structures: networks of struts or springs exhibit a continuous, second-order phase transition at the isostatic point, where the number of constraints imposed by connectivity just equals the number of mechanical degrees of freedom. We will present recent theoretical predictions and experimental evidence for a strain-controlled mechanical phase transition in biopolymer networks below Maxwell’s isostatic point. We will outline a theoretical framework to understand and quantify the critical phenomena associated with this transition. As we show, this transition also governs elastic anomalies, including an anomalously large Poisson ratio and inverse Poynting effect.

While nucleosomes are dynamic entities that must undergo structural deformations to perform their functions, the general view from available high-resolution structures is a largely static one. Even though numerous examples of twist defects have been documented, the DNA wrapped around the histone core is generally thought to be overtwisted. Analysis of available high-resolution structures reveals a heterogeneous distribution of twist along the nucleosomal DNA, with clear patterns that are consistent with the literature, and a significant fraction of structures that are undertwisted. The subtle differences in nucleosomal DNA folding, which extend beyond twist, have implications for nucleosome disassembly and modeled higher-order structures. Simulations of oligonucleosome arrays built with undertwisted models behave very differently from those constructed from overtwisted models, in terms of compaction and inter-nucleosome contacts, introducing configurational changes equivalent to those associated with 2-3 base-pair changes in nucleosome spacing. Differences in the nucleosomal DNA pathway, which underlie the way that DNA enters and exits the nucleosome, give rise to different nucleosome-decorated minicircles and affect the topological mix of configurational states.

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A knotoid is a generalization of a 1-1 tangle in classical knot theory to a diagram with ends so that the ends can be in distinct regions. Such diagrams are taken up to Reidemeister moves that do not allow passage of strands across the ends of the diagram. In this way one obtains a concept of an open ended diagram that can be classified topologically just as are the closed diagrams of classical knot theory. By constructions due to Vladimir Turaev (for diagrams on the two-sphere) and the author and Neslihan Gugumcu (for diagrams in the plane) one can interpret knotoids as projections from open-ended curves in three dimsensional space. By natural restrictions of the isotopies of such space curves (in relation to the projection) one then has a way to handle the topology of open-ended curves in three dimensional space. This talk will discuss the relationship between open-ended curves in three dimensional space and their corresponding knotoid classes. We will discuss basic invariants such as the Jones polynomial, relationships of knotoids with viritual knot theory and aspects of our joint work with Nesilhan Gugumcu, Sofia Lambropoulou,Manos Manouras and with Eleni Panagiotou.

Entanglement in polymer melts is an effect of the uncrossability constraint of polymer chains which dramatically affects their mechanical properties. Is topological entanglement, as measured using tools from knot theory, relevant to polymer entanglement? In this talk we use the Gauss linking integral, as well as new measures of topological complexity of both open and closed curves in 3-space - the Jones polynomial of open curves and Vassiliev measures of open curves in 3-space- and apply them to polymer melts in several contexts though Molecular Dynamics simulations. Our results show that topological entanglement indeed correlates with polymer viscoelastic properties and can provide further insight in polymer mechanics.

In confined active anisotropic soft mater, the interplay of ordering, elasticity, chirality, confinement, surface anchoring, external fields, flows, and activity leads to numerous complex static and dynamic structures. Their orientational ordering fields include singular topological defects and nonsingular solitonic deformations. Increasing interest in active systems stimulated us to model topology of three-dimensional extensile activity driven nematodynamics in a spherical confinement providing a topological constrain [1,2]. We used a simple mesoscopic modelling of active nematic fluids [3] that enables numerical simulations of active nematodynamics. It reasonably well describes experiments in thin layers and shells with active complex fluids that are mostly biological systems driven by internal conversion of stored chemical energy into motion [3,4]. We demonstrated that at low activity stationary dynamic structures occur that with increasing activity undergo transitions from stationary to chaotic 3D motions - active nematic turbulence. In this talk I will present how in a such regime the time evolution can be for a specific confinement characterized by a series of elementary topological events where nematic disclinations divide, merge, annihilate, and crossover. I will focus to homeotropic anchoring, no-slip surface, and for selected activities illustrate our findings by simulated dynamics of nematic disclinations & flows accompanied by simulated optical microscopy. Our simple confined system could be a nice test ground for recently introduced machine learning approach to active nematics [5]. The research was done in collaboration with S. Čopar, J. Aplinc, Ž. Kos, and M. Ravnik. [1] S. Čopar, J. Aplinc, Ž. Kos, S. Žumer, and M. Ravnik, Topology of three-dimensional active nematic turbulence confined to droplets, Physical Review X 9, 031051 (2019), [2] J. Binysh, Z. Kos, S. Čopar, M. Ravnik, and G. P. Alexander, Three-dimensional active defect loops, Physical Review Letters 124, 088001 (2020). [3] A. Doostmohammadi, J. Ignés-Mullol, and J. M. Yeomans, F. Sagúes, Active nematics, Nature Communications 9: 3246, 1 (2018). [4] G. Duclos, R. Adkins, D. Banerjee, M. S. Peterson, M. Varghese, I. Kolvin, A. Baskaran, R. A. Pelcovits, T. R. Powers, A. Baskaran, F. Toschi, M. F. Hagan, S.J. Streichan, V. Vitelli, D. A. Beller, and Z. Dogic, Topological structure and dynamics of three dimensional active nematics, Science 367, 1120 (2020). [5] J. Colen, M.Han, R. Zhang, S. A. Redford, L. M. Lemma, L. Morgan, P. V Ruijgrok, R.Adkins, Z. Bryant, Z. Dogic, M. L. Gardel, J. J de Pablo, V. Vitelli, Machine learning active-nematic hydrodynamics, Proc. Natl. Acad. Sci. USA 118, e2016708118 (2021).

In this talk, I will present our on-going work related to understanding topological effects in polymer melts and solutions. In particular, issue of Gauge invariance in the field theory of polymers will be discussed and it will be shown that Gauge fixing can be used to discover topological invariants. A specific example using the Coulomb gauge will be used to demonstrate that the helicity is one of the topological invariants for both, linear and ring polymers. Furthermore, a numerical recipe to generate knotted vector fields will be presented for studying topological configurations near equilibrium using the self-consistent field theory of polymers.

Cell differentiation, the process by which stem cells become specialized cells, is associated with chromatin reorganization inside the cell nucleus. Here, we measure the chromatin distribution and dynamics in embryonic stem cells in vivo before and after differentiation. We find that undifferentiated chromatin is less compact, more homogeneous and more dynamic than differentiated chromatin. Further, we present a noninvasive rheological analysis using intrinsic chromatin dynamics, which reveals that undifferentiated chromatin behaves like a Maxwell fluid, while differentiated chromatin shows a coexistence of fluid-like (sol) and solid-like (gel) phases. Our data suggest that chromatin undergoes a local sol-gel transition upon cell differentiation, corresponding to the formation of the more dense and transcriptionally inactive heterochromatin (Eshghi I, Eaton JA and Zidovska A, Phys. Rev. Lett., 2021).

Presented will be two methods that allow the study of the stochastic and dynamical behavior of knotted and confined DNA molecules. One method is based on exact statistical sampling of closed configurations, the other on dynamical simulations performed using on generalized immersed boundary method. The equations of motion of the rod include the fluid–structure interaction, sequence-dependent elasticity and a combination of two interactions that prevent self-contact, namely the electrostatic interaction and hard-core repulsion. I will discuss the dynamics of DNA trefoils and configurations of DNA Hopf links with relevance to kinetoplast DNA.

The three dimensional organization of genomes is a key player in multiple biological processes including the genome packaging and release in viruses. The genome of some viruses, such as bacteriophages or human herpes, is a double stranded DNA (dsDNA) molecule that is stored inside a viral protein capsid at a concentration of 200 mg/ml-800mg/ml and an osmotic pressure of 70 atmospheres. The organization of the viral genome under these extreme physical conditions is believed to be liquid crystalline but remains to be properly understood. A general picture of this organization has been recently given by cryoelectron microscopy (cryoEM) studies that show a series of concentric layers near the surface of the viral capsid followed by a disordered arrangement of DNA fibers near the center of the capsid. In this talk I will present computational and experimental results modeling the structure and packing of DNA in bacteriophage P4. P4 is characterized for producing DNA knots and for being one of the smallest bacteriophages with only 45nm in diameter. I will discuss experimental results concerning the structure of P4 and how liquid crystal models can help predict the properties of DNA in P4 and the formation of knots.

The Gaussian network plays a central role in the study on the fundamental elastic behavior of various polymer networks such as rubbers and gels [1, 2]. Here we remark that many bio-materials are made of gels. Recently, a new method has been introduced for generating an ensemble of random conformations of graph-shaped polymers in terms of topologically constrained Gaussian random walks (TCRW) or Gaussian random graph embeddings [3]. It is one of the key properties of TCRW that the probability distribution function of the bond vectors in polymer conformations of TCRW is composed of the normal distributions with unit variance. In this talk we critically study Flory’s approximate expression for the mean square fluctuation of the end-to-end vector r around its average value \(r\) with functionality \(f\) [4] $$⟨2 ⟨(r − ⟨r⟩) ⟩2 2Nb f$$ Here N is the number of the Kuhn segments in the network subchain connecting a crosslinking point to another one. We express the fluctuation ⟨(Δr)2⟩ in terms of resistance distances, and evaluate it rigorously. We argue that Flory’ s expression should be valid if the functionality f is very large, based on the numerical experiments of large random graphs with functionality f, i.e., regular graphs with functionality f. We also discuss the results of Ref. [5]. The results of the present talk should be important not only in materials science but also in applications of biomaterials.

The faithful segregation of chromosomes during mitosis is a fundamental and important process. Errors in mitosis have severe implications and are often detrimental to development, health and survival of the organism. We know that microtubules, in particular kinetochore microtubules, exert forces on chromosomes to initially position them on the metaphase plate and consequently divide them to the two daughter cells. The forces generated by microtubules are in balance during metaphase resulting in a mechanical steady-state and a stable long-lived spindle shape and length. Previous studies have identified the proteins involved in metaphase spindle assembly. Yet, we do not understand how those proteins lead to force generation through interactions of microtubules, motor proteins and chromosomes in submicron scale, and the collective effect of these forces on spindle shape function at larger scales. One major barrier in answering this question is the limitation of light microscopy in visualizing details of spindle microstructure in submicron resolutions. We have developed a novel approach of visualizing entire spindles in 3D by electron tomography and automatic microtubule segmentation. Using this approach, we can resolve single microtubules, which provides a unique perspective and offers a plethora of completely new information about the microstructure of spindles. Specifically, we can resolve chromosome surfaces, identify microtubules that are in contact with chromosomes (kinetochore microtubules), determine microtubules’ nucleation profile, length distribution and local curvature. We combine electron tomography, light microcopy, biophysical modeling and large-scale simulations to develop a detailed and unprecedented understanding of force generation inside the spindle from individual microtubules to the mitotic spindle composed of thousands of microtubules.

Topological modifications play a central role in morphogenesis of biological systems, in processes such as tissue formation, embryogenesis, wound healing, cancer proliferation, to name a few. More than a century ago, D’Arcy Thomson suggested that one of the guiding principles in shaping epithelian tissues is a balance of minimum surface tension and close packing of cells. This balance of surface and bulk properties is evident at many different length scales, from multicellular organisms to subcellular entities. For example, double strands of DNA condense into nanometer scale structures of various topologies in presence of condensing agents such as multivalent salts and polymers. The shapes of these DNA condensates vary from rod-like to spheres and tori. Another prominent geometry met in biological systems is a spindle-like organization of cytoskeletal structures, composed of microtubules and proteins and responsible for segregation of chromosomes during cell divisions; these spindle-like structures are known in materials science as nematic tactoids. The complex mechanisms by which living matter changes its topology in response to various cues are far from being understood. The field would certainly benefit from simple model systems in which the topological change can be triggered by well-controlled factors such as temperature or concentration and observed in situ by optical microscopy. In this work, we describe such a system, representing droplets of a water-based nematic lyotropic chromonic liquid crystal (LCLC) that coexists with its isotropic melt in presence of a condensing agent, a polymer polyethylene glycole (PEG). LCLCs form phases such as the nematic (N) and, at yet higher concentrations or lower temperatures, the columnar (Col) phase. We demonstrate that a simple change in temperature or concentration dramatically alters the topology of the nematic droplets from that of a sphere to that of a torus. The transition is reversible and is driven by the strong temperature and concentration dependence of the elastic and surface properties of the system.

Motivated in part by experimental and molecular dynamics studies of the entanglement characteristics of DNA in nanonchannels, we have been studying the statistics of knotting and linking for equilibrium lattice models of polymers confined to lattice tubes. In this talk I will present our theorems and transfer-matrix-based numerical results for the link statistics for self-avoiding polygon models in small tubes. The main focus will be on the special case of pairs of polygons which span a lattice tube. In this case, it is known that all but exponentially few of the configurations will be linked as the span of the polygons goes to infinity. However there are many interesting open questions about configurational statistics for pairs of polygons with fixed link type and I will introduce some of those.

Starting from the Oseen-Frank theory, we derive a simple model for the dynamics of a nematic liquid crystal director field under the influence of an electric field. The resulting nonlinear system of partial differential equations consists of the electrostatics equations for the electric field coupled with the damped wave map equation for the evolution of the liquid crystal director field, which is a normal vector pointing in the direction of the main orientation of the liquid crystal molecules. The liquid crystal director field enters the electrostatics equations in the constitutiverelations while the electric field enters the wave map equation in the form of a nonlinear source term. Since it is a normal vector, the variable for the liquid crystal director field has to satisfy the constraint that it takes values in the unit sphere. We derive an energy-stable and constraint preserving numerical method for this system and prove convergence of a subsequence of approximations to a weak solution of the system of partial differential equations. In particular, this implies the existence of weak solutions for this model.

Polycontinuous patterns appear as microphase separation of block copolymers. In this talk, we discuss handlebody decompositions of the 3-torus and their application to the study of polycontinuous patterns.

Using periodic surfaces as a scaffold is a convenient route to making periodic entanglements. I will present a systematic way of building new tangled periodic structures, using low-dimensional topology and combinatorics, posing the question of how to characterise the structures more completely.

A multitude of knot invariants, including quantum invariants and their categorifications, have been introduced to aid with characterizing and classifying knots and their topological properties. Relations between knot invariants and their relative strengths at distinguishing knots are still mostly elusive. We propose adapting TDA tools such as Ball Mapper and Mapper to analyzing these infinite data sets where representative sampling is impossible or impractical. We focus on Jones and HOMFLYPT polynomials, and Khovanov homology and along the way introduce enhancements that deal with assumed or perceived symmetries in data, as well as a combination of Mapper and Ball Mapper approaches to enhance their strengths and provide a way to visualize maps between high dimensional Euclidean spaces. This is joint work with P. Dlotko and D. Gurnari.

Polymeric structures are ubiquitous in biology, and perform diverse functions at multiple length-scales. DNA carries the genetic code through the chemistry of the constituent bases at an atomic level, but also plays an active role in its own regulation through its ability to store and transmit mechanical stress over genomic length-scales. Long polymeric coiled-coils are a common protein structural motif, and as well as forming the basis of robust super-macromolecular hierarchical structures such as collagen, also have an active role in regulating the chemo-mechanical cycle of molecular motors such as dynein and myosin. Intrinsically disordered proteins present a particular enigma; some undergo disorder to order transitions on encountering their binding partner and so participate in highly specific molecular recognition in spite of their apparent lack of structure, whereas others appear to generate vital emergent behaviour over far longer length-scales than their own structure, such as the self-assembly of membraneless organelles. Here I will compare and contrast multi-scale representations of polymeric biomacromolecules from the fully atomistic up to the continuum level. I will discuss open challenges to development and biological questions that would benefit from robust mathematical and computational models of biological polymers.

Drop-Carrier Particles (DCPs) are solid microparticles designed to capture uniform microscale drops of a target solution without using costly microfluidic equipment and techniques. DCPs are useful for automated and high-throughput biological assays and reactions, as well as single cell analyses. The ability of the DCPs to enable templated uniform-sized droplets can be understood theoretically using surface energy minimization for multiple droplet interactions. We compare the theoretical prediction for the volume distribution to macroscale experiments of pairwise droplet splitting, with good agreement. This leads to a theory for the number of pairwise interactions of DCPs needed to reach a uniform volume distribution. We develop a probabilistic pairwise interaction model for a system of such DCPs exchanging fluid volume to minimize surface energy. Heterogeneous mixtures of DCPs with different sized particles require fewer interactions to reach a minimum energy distribution for the system. We present ideas for the optimization of the DCP geometry for minimal required target solution and uniformity in droplet volume.

Many knots in nature are open knots, not the closed knots from knot theory. There are several definitions of knotting in open curves, each of which have their own advantages and disadvantages. The speaker's favorite open knot definition involves extending rays to infinity in a common direction from the endpoints to create a closed knot for each such direction. In such a case, the knotting in an open chain is classified as the distribution of knot types seen over the different directions of closure. In most cases, there is a knot type that appears in over 50% of the closure directions, in which case we might all be able to agree that the open knot has the essence of that closed knot type. However, there are many cases where there is no knot type that appears in over 50% of the closure directions, especially near transitions between different knot types. We present the accumulated knot probability as a way of making sense of these more ambiguous situations. The short story is that, for a given knot type K, we compute the probability that the closures are a knot type which "includes" K in some sense. In this talk, we use the partial ordering on knots developed by Diao, Ernst, and Stasiak based on crossing changes in minimal knot diagrams, which creates a sort of family tree of knots. However, any sort of family tree could be substituted here depending on what one is trying to model. We show how some of the knotting classifications change for some proteins and tight knot configurations.

Bacteriophages densely pack their long dsDNA genome inside a protein capsid. The conformation of the viral genome inside the capsid is consistent with a hexagonal liquid crystalline structure. Experiments have confirmed that the details of the hexagonal packing depend on the electrochemistry of the capsid and its environment. In this work, we propose a biophysical model that quantifies the relationship between DNA configurations inside bacteriophage capsids and the types and concentrations of ions present in a biological system. We introduce an expression for the free energy which combines the electrostatic energy with contributions from bending of individual segments of DNA and Lennard--Jones-type interactions between these segments. The equilibrium points of this energy solve a partial differential equation that defines the distributions of DNA and the ions inside the capsid. We develop a computational approach that allows us to simulate much larger systems than what is possible using the existing molecular-level methods. In particular, we are able to estimate bending and repulsion between the DNA segments as well as the full electrochemistry of the solution, both inside and outside of the capsid. The numerical results show good agreement with existing experiments and with molecular dynamics simulations.

The superposition of HOMFLY-PT polynomials of collections of open chains provides an "average" of the polynomials associated to individual closures and, consequently, a HOMFLY-PT polynomial for the open link. Following a brief description of this procedure, I will present an overview of data from a study of open two component links of varying lengths confined to balls of varying radii. This is joint work with Eleni Panagiotou.

Monte Carlo simulations are a big part of understanding the statistical properties of knots. Unfortunately, if one wishes to study curves of fixed knot types then there are very few methods available. This work, with Nick Beaton and Nathan Clisby, is an attempt to adapt existing algorithms to polygons in R3 of fixed topology. It is very much a work in progress, but I will report on our work adapting BFACF to polygons in R3, and also our attempts at trying to coerce the (very fast) pivot algorithm to respect topology.