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We present a braided circuit topology framework for investigating topology and structural phase transitions in aggregates of semiflexible polymers. In the conventional approach to circuit topology, which specifically applies to single isolated folded linear chains, the number and arrangement of contacts within the circuitry of a folded chain give rise to increasingly complex fold topologies. Another avenue for achieving complexity is through the interaction and entanglement of two or more folded linear chains. The braided circuit topology approach describes the topology of such multiple-chain systems and offers topological measures such as writhe, complexity, braid length, and isotopy class. This extension of circuit topology to multichains reveals the interplay between collapse, aggregation, and entanglement. In this work, we show that circuit topological motif fractions are ideally suited order parameters to characterise structural phase transitions in entangled systems that can detect structural re-ordering other measures cannot., Comment: v2: accepted version in iScience

Circuit topology employs fundamental units of entanglement, known as soft contacts, for constructing knots from the bottom up, utilising circuit topology relations, namely parallel, series, cross, and concerted relations. In this article, we further develop this approach to facilitate the analysis of chirality, which is a significant quantity in polymer chemistry. To achieve this, we translate the circuit topology approach to knot engineering into a braid-theoretic framework. This enables us to calculate the Jones polynomial for all possible binary combinations of contacts in cross or concerted relations and to show that, for series and parallel relations, the polynomial factorises. Our results demonstrate that the Jones polynomial provides a powerful tool for analysing the chirality of molecular knots constructed using circuit topology. The framework presented here can be used to design and engineer a wide range of entangled chain with desired chiral properties, with potential applications in fields such as materials science and nanotechnology., Comment: 8 pages, 5 figures, 3 tables

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.

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 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 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 4 fundamental structural units subjected to 3 (or 5) 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. Prime knots, which are viewed by knot theory as undecomposable, are also made of these structural units connected in some specific way. In turn, this kind of connection shows the fundamental reason why prime knots cannot be decomposed in the rigorous sense of knot theory. We believe that such a detailed, bottom-up understanding of the structure of molecular knots should be beneficial for molecular engineering.

Folded linear molecular chains are ubiquitous in biology. Folding is mediated by intra-chain interactions that "glue" two or more regions of a chain. The resulting fold topology is widely believed to be a determinant of biomolecular properties and function. Recently, knot theory has been extended to describe the topology of folded linear chains such as proteins and nucleic acids. To classify and distinguish chain topologies, algebraic structure of quandles has been adapted and applied. However, the approach is limited as apparently distinct topologies may end up having the same number of colorings. Here, we enhance the resolving power of the quandle coloring approach by introducing Boltzmann weights. We demonstrate that the enhanced coloring invariants can distinguish fold topologies with an improved resolution., Comment: 22 pages. 23 figures. Comments are welcome

Proteins are linear molecular chains that often fold to function. The topology of folding is widely believed to define its properties and function, and knot theory has been applied to study protein structure and its implications. More that 97% of proteins are, however, classified as unknots when intra-chain interactions are ignored. This raises the question as to whether knot theory can be extended to include intra-chain interactions and thus be able to categorize topology of the proteins that are otherwise classified as unknotted. Here, we develop knot theory for folded linear molecular chains and apply it to proteins. For this purpose, proteins will be thought of as an embedding of a linear segment into three dimensions, with additional structure coming from self-bonding. We then project to a two-dimensional diagram and consider the basic rules of equivalence between two diagrams. We further consider the representation of projections of proteins using Gauss codes, or strings of numbers and letters, and how we can equate these codes with changes allowed in the diagrams. Finally, we explore the possibility of applying the algebraic structure of quandles to distinguish the topologies of proteins. Because of the presence of bonds, we extend the theory to define bondles, a type of quandle particularly adapted to distinguishing the topological types of proteins., Comment: This is the version that will appear in the Journal of Mathematical Chemistry

We study a diagnostic strategy which is based on the anticipation of the diagnostic process by simulation of the dynamical process starting from the initial findings. We show that such a strategy could result in more accurate diagnoses compared to a strategy that is solely based on the direct implications of the initial observations. We demonstrate this by employing the mean-field approximation of statistical physics to compute the posterior disease probabilities for a given subset of observed signs (symptoms) in a probabilistic model of signs and diseases. A Monte Carlo optimization algorithm is then used to maximize an objective function of the sequence of observations, which favors the more decisive observations resulting in more polarized disease probabilities. We see how the observed signs change the nature of the macroscopic (Gibbs) states of the sign and disease probability distributions. The structure of these macroscopic states in the configuration space of the variables affects the quality of any approximate inference algorithm (so the diagnostic performance) which tries to estimate the sign/disease marginal probabilities. In particular, we find that the simulation (or extrapolation) of the diagnostic process is helpful when the disease landscape is not trivial and the system undergoes a phase transition to an ordered phase.

Efficient search acts as a strong selective force in biological systems ranging from cellular populations to predator-prey systems. The search processes commonly involve finding a stationary or mobile target within a heterogeneously structured environment where obstacles limit migration. An open generic question is whether random or directionally biased motions or a combination of both provide an optimal search efficiency and how that depends on the motility and density of targets and obstacles. To address this question, we develop a simple model that involves a random walker searching for its targets in a heterogeneous medium of bond percolation square lattice and used mean first passage time (MFPT, $\langle T \rangle$) as an indication of average search time. Our analysis reveals a dual effect of directional bias on the minimum value of $\langle T \rangle$. For a homogeneous medium, directionality always decreases $\langle T \rangle$ and a pure directional migration (a ballistic motion) serves as the optimized strategy; while for a heterogeneous environment, we find that the optimized strategy involves a combination of directed and random migrations. The relative contribution of these modes is determined by the density of obstacles and motility of targets. Existence of randomness and motility of targets add to the efficiency of search. Our study reveals generic and simple rules that govern search efficiency. Our findings might find application in a number of areas including immunology, cell biology, and ecology.

A fundamental problem in medicine and biology is to assign states, e.g. healthy or diseased, to cells, organs or individuals. State assignment or making a diagnosis is often a nontrivial and challenging process and, with the advent of omics technologies, the diagnostic challenge is becoming more and more serious. The challenge lies not only in the increasing number of measured properties and dynamics of the system (e.g. cell or human body) but also in the co-evolution of multiple states and overlapping properties, and degeneracy of states. We develop, from first principles, a generic rational framework for state assignment in cell biology and medicine, and demonstrate its applicability with a few simple theoretical case studies from medical diagnostics. We show how disease-related statistical information can be used to build a comprehensive model that includes the relevant dependencies between clinical and laboratory findings (signs) and diseases. In particular, we include disease-disease and sign-sign interactions. We then study how one can infer the probability of a disease in a patient with given signs. We perform comparative analysis with simple benchmark models to check the performances of our models. This first principles approach, as we show, enables the construction of consensus diagnostic flow charts and facilitates the early diagnosis of disease. Additionally, we envision that our approach will find applications in systems biology, and in particular, in characterizing the phenome via the metabolome, the proteome, the transcriptome, and the genome.

Here we report on the translocation of folded polymers through nano-pores using molecular dynamic simulations. Two cases are studied; one in which a folded molecule unfolds upon passage and one in which the folding remains intact as the molecule passes through the nano-pore. The topology of a folded polymer chain is defined as the arrangement of the intramolecular contacts, known as circuit topology. In the case where intramolecular contacts remain intact, we show that the dynamics of passage through a nano-pore varies for molecules with differing topologies: a phenomenon that can be exploited to enrich certain topologies in mixtures. We find that the nano-pore allows reading of topology for short chains. Moreover, when the passage is coupled with unfolding, the nano-pore enables discrimination between pure states, i.e., states for which the majority of contacts are arranged identically. In this case, as we show here, it is also possible to read the positions of the contact sites along a chain. Our results demonstrate the applicability of nano-pore technology to characterize and sort molecules based on their topology., Comment: 7 pages, 5 figures

Conformational transitions are ubiquitous in biomolecular systems, have significant functional roles and are subject to evolutionary pressures. Here we provide a first theoretical framework for topological transition, i.e. conformational transitions that are associated with changes in molecular topology. For folded linear biomolecules, arrangement of intramolecular contacts is identified as a key topological property, termed as circuit topology. Distance measures are proposed as reaction coordinates to represent progress along a pathway from initial topology to final topology. Certain topological classes are shown to be more accessible from a random topology. We study dynamic stability and pathway degeneracy associated with a topological reaction and found that off-pathways might seriously hamper evolution to desired topologies. Finally we present an algorithm for estimating the number of intermediate topologies visited during a topological reaction. The results of this study are relevant to, among others, structural studies of RNA and proteins, analysis of topologically associated domains in chromosomes, and molecular evolution., Comment: 21 pages, 7 figures, 4 appendixes

Circuit topology refers to the arrangement of interactions between objects belonging to a linearly ordered object set. Linearly ordered set of objects are common in nature and occur in a wide range of applications in economics, computer science, social science and chemical synthesis. Examples include linear bio-polymers, linear signaling pathways in cells as well as topological sorts appearing in project management. Using a statistical mechanical treatment, we study circuit topology landscapes of linear polymer chains with intra-chain contacts as a prototype of linearly sorted objects with interactions. We find generic features of the topological space and study the statistical properties of the space under the most basic constraints on the occupancy of arrangements and topological interactions. We observe that a set of correlated contact sites (a sector) could nontrivially influence the entropy of circuits as the number of involved sites increases. Finally, we discuss how constraints can be inferred from the information provided by local contact distributions in presence of a sector., Comment: 17 pages, 6 figures, 3 appendixes

Nucleic acids, like DNA and RNA, are molecules that are present in any life form. Their most notable function is to encode biological information. Why then would a physicist be interested in these molecules? As we will see, DNA is an interesting molecular tool for physicists to test and explore physical laws and theories, like the ergodic theorem, the theory of elasticity and information theory. DNA also has unique material properties, which attract material scientists, nanotechnologists and engineers. Among interesting developments in this field are DNA-based hybrid materials and DNA origami., Comment: 6 pages, 3 figures