The Kinetoplast DNA (kDNA) is a two-dimensional Olympic-ring-like network of mutually linked 2.5 kb-long DNA minicircles found in certain parasites called Trypanosomes. Understanding the self-assembly and replication of this structure are not only major open questions in biology but can also inform the design of synthetic topological materials. Here we report the first high-resolution, single-molecule study of kDNA network topology using AFM and steered molecular dynamics simulations. We map out the DNA density within the network and the distribution of linking number and valence of the minicircles. We also characterise the DNA hubs that surround the network and show that they cause a buckling transition akin to that of a 2D elastic thermal sheet in the bulk. Intriguingly, we observe a broad distribution of density and valence of the minicircles, indicating heterogeneous network structure and individualism of different kDNA structures. Our findings explain outstanding questions in the field and offer single-molecule insights into the properties of a unique topological material.
In this work we present a computational study of the Kinetoplast genome, modelled as a large number of semiflexible unknotted loops, which are allowed to link with each other. As the DNA density increases, the systems shows a percolation transition between a gas of unlinked rings and a network of linked loops which spans the whole system. Close to the percolation transition, we find that the mean valency of the network, i.e. the average number of loops which are linked to any one loop, is around 3 as found experimentally for the Kinetoplast DNA. Even more importantly, by simulating the digesti…
We study theoretically the replication of Kinetoplast DNA consisting of several thousands separate mini-circles found in organisms of the class Kinetoplastida. When the cell is not actively dividing these are topologically connected in a marginally linked network of rings with only one connected component. During cell division each mini-circle is removed from the network, duplicated and then re-attached, along with its progeny. We study this process under the hypothesis that there is a coupling between the topological state of the mini-circles and the expression of genetic information encoded …
Recent experiments have elucidated the physical properties of kinetoplasts, which are chain-mail-like structures found in the mitochondria of trypanosome parasites formed from catenated DNA rings. Inspired by these studies, we use Monte Carlo simulations to examine the behavior of two-dimensional networks ("membranes") of linked rings. For simplicity, we consider only identical rings that are circular and rigid and that form networks with a regular linking structure. We find that the scaling of the eigenvalues of the shape tensor with membrane size are consistent with the behavior of the flat …
Molecular topology of polymers plays a key role in determining their physical properties. We studied herein the topological effects on the static and dynamic properties of a 2D catenated network of DNA rings called a kinetoplast. Restriction enzymes, that cleave DNA at sequence-specific sites, are used to selectively cut and remove rings from the network and hence tune the molecular topology while maintaining overall structural integrity. We find that topology has minimal effects over the spatial extension of the 2D network, however, it significantly affects the relaxation behavior. The shape …
Understanding and controlling the rheology of polymeric complex fluids that are pushed out-of-equilibrium is a fundamental problem in both industry and biology. For example, to package, repair, and replicate DNA, cells use enzymes to constantly manipulate DNA topology, length, and structure. Inspired by this, here we engineer and study DNA-based complex fluids that undergo enzymatically-driven topological and architectural alterations via restriction endonuclease (RE) reactions. We show that these systems display time-dependent rheological properties that depend on the concentrations and prope…
Polymer composites are ubiquitous in biology and industry alike, owing to their emergent desirable mechanical properties not attainable in single-species systems. At the same time, polymer topology has been shown to play a key role in tuning the rheology of polymeric fluids. However, how topology impacts the rheology of composites remains poorly understood. Here, we create composites of rigid rods (microtubules) polymerized within entangled solutions of flexible linear and ring polymers (DNA). We couple linear and nonlinear optical tweezers microrheology with confocal microscopy and scaled par…
We use a recently developed coarse-grained model for DNA to study kissing complexes formed by hybridization of complementary hairpin loops. The binding of the loops is topologically constrained because their linking number must remain constant. By studying systems with linking numbers -1, 0 or 1 we show that the average number of interstrand base pairs is larger when the topology is more favourable for the right-handed wrapping of strands around each other. The thermodynamic stability of the kissing complex also decreases when the linking number changes from -1 to 0 to 1. The structures of the…
Cytoskeletal crowding plays a key role in the diffusion of DNA molecules through the cell, acting as a barrier to effective intracellular transport and conformational stability required for such processes as transfection, viral infection, and gene therapy. Here we elucidate the transport properties and conformational dynamics of linear and ring DNA molecules diffusing through entangled and crosslinked composite networks of actin and microtubules. We couple single-molecule conformational tracking with differential dynamic microscopy to reveal that ring and linear DNA exhibit surprisingly distin…
Atomic Force Microscopy analysis is employed to study the geometrical and topological properties of $3000$kbp DNA molecules fixed in mica substrates with $MgCl_{2}$. We found that the aggregates on the substrate surface for certain salt concentrations form a honeycomb stable structure with the addition of salt. The honeycomb, and the transition to other structures, were characterized by the \emph{Betti numbers}, which is a topological invariant property and by \emph{Hausdorff-Besicovitch fractal dimension}.
Gel electrophoresis is a powerful experimental method to probe the topology of DNA and other biopolymers. While there is a large body of experimental work which allows us to accurately separate different topoisomers of a molecule, a full theoretical understanding of these experiments has not yet been achieved. Here we show that the mobility of DNA knots depends crucially and subtly on the physical properties of the gel, and in particular on the presence of dangling ends. The topological interactions between these and DNA molecules can be described in terms of an "entanglement number", and yiel…
We present a theoretical treatment of DNA stretching and twisting experiments, in which we discuss global topological subtleties of self avoiding ribbons and provide an underlying justification for the worm like rod chain (WLRC) model proposed by Bouchiat and Mezard. Some theoretical points regarding the WLRC model are clarified: the writhe of open curves and the use of an adjustable cutoff parameter to ``regularise'' the model. Our treatment brings out the precise relation between the worm like chain (WLC), the paraxial worm like chain (PWLC) and the WLRC models. We describe the phenomenon of…
There is a long-standing experimental observation that the melting of topologically constrained DNA, such as circular-closed plasmids, is less abrupt than that of linear molecules. This finding points to an intriguing role of topology in the physics of DNA denaturation, which is however poorly understood. Here, we shed light on this issue by combining large-scale Brownian Dynamics simulations with an analytically solvable phenomenological Landau mean field theory. We find that the competition between melting and supercoiling leads to phase coexistence of denatured and intact phases at the sing…
Overstretching of B-DNA is currently understood as force-induced melting. Depending on the geometry of the stretching experiment, the force threshold for the overstretching transition is around 65 or 110 pN. Although the mechanisms behind force-induced melting have been correctly described by Rouzina and Bloomfield \cite{RouzinaBloomfield2001a}, neither force threshold has been exactly calculated by theory. In this work, a detailed analysis of the force-extension curve is presented, based on a description of single-stranded DNA in terms of the discrete Kratky-Porod model, consistent with (i) t…
We investigate quantum transport characteristics of a ladder model, which effectively mimics the topology of a double-stranded DNA molecule. We consider the interaction of tunneling charges with a selected internal vibrational degree of freedom and discuss its influence on the structure of the current-voltage characteristics. Further, molecule-electrode contact effects are shown to dramatically affect the orders of magnitude of the current. Recent electrical transport measurements on suspended DNA oligomers with a complex base-pair sequence, revealing strikingly high currents, are also present…
Topological entanglements severely interfere with important biological processes. For this reason, genomes must be kept unknotted and unlinked during most of a cell cycle. Type II Topoisomerase (TopoII) enzymes play an important role in this process but the precise mechanisms yielding systematic disentanglement of DNA in vivo are not clear. Here we report computational evidence that Structural Maintenance of Chromosomes (SMC) proteins -- such as cohesins and condensins -- can cooperate with TopoII to establish a synergistic mechanism to resolve topological entanglements. SMC-driven loop extrus…
The elasticity of DNA catenanes, i.e. multiply linked DNA rings, is investigated using the Gauss invariant as a minimal model for topology conservation. An effective elastic free energy as a function of the distance $R$ between segments located on different rings is obtained. An anharmonic part at large distances, growing as $R^{4}$, if $R\gg R_{G}$ ($R_{G}$ being the radius of gyration of a random walk ring) is found, while for $R\ll R_{G}$ the interaction is strongly repulsive. Treating the attractive interaction as the dominant one, distribution functions for the distance between segments l…
We discuss the entropy of a circular polymer under a topological constraint. We call it the {\it topological entropy} of the polymer, in short. A ring polymer does not change its topology (knot type) under any thermal fluctuations. Through numerical simulations using some knot invariants, we show that the topological entropy of a stiff ring polymer with a fixed knot is described by a scaling formula as a function of the thickness and length of the circular chain. The result is consistent with the viewpoint that for stiff polymers such as DNAs, the length and diameter of the chains should play …
Abundant in nature, colloids also find increasingly important applications in science and technology, ranging from direct probing of kinetics in crystals and glasses to fabrication of third-generation quantum-dot solar cells. Because naturally occurring colloids have a shape that is typically determined by minimization of interfacial tension (for example, during phase separation) or faceted crystal growth, their surfaces tend to have minimum-area spherical or topologically equivalent shapes such as prisms and irregular grains (all continuously deformable - homeomorphic - to spheres). Although …
Time-resolved single-molecule biophysical experiments yield data that contain a wealth of dynamic information, in addition to the equilibrium distributions derived from histograms of the time series. In typical force spectroscopic setups the molecule is connected via linkers to a read-out device, forming a mechanically coupled dynamic network. Deconvolution of equilibrium distributions, filtering out the influence of the linkers, is a straightforward and common practice. We have developed an analogous dynamic deconvolution theory for the more challenging task of extracting kinetic properties o…
Ring polymers exhibit unique flow properties due to their closed chain topology. Despite recent progress, we have not yet achieved a full understanding of the nonequilibrium flow behavior of rings in nondilute solutions where intermolecular interactions greatly influence chain dynamics. In this work, we directly observe the dynamics of DNA rings in semidilute ring-linear polymer blends using single molecule techniques. We systematically investigate ring polymer relaxation dynamics from high extension and transient and steady-state stretching dynamics in planar extensional flow for a series of …
