A circuit topology approach to categorizing changes in biomolecular structure

The biological world is composed of folded linear molecules of bewildering topological complexity and diversity. The topology of folded biomolecules such as proteins and ribonucleic acids is often subject to change during biological processes. Despite intense research, we lack a solid mathematical framework that summarizes these operations in a principled manner. Circuit topology, which formalizes the arrangements of intramolecular contacts, serves as a general mathematical framework to analyze the topological characteristics of folded linear molecules. In this work, we translate familiar molecular operations in biology, such as duplication, permutation, and elimination of contacts, into the language of circuit topology. We show that for such operations there are corresponding matrix representations as well as basic rules that serve as a foundation for understanding these operations within the context of a coherent algebraic framework. We present several biological examples and provide a simple computational framework for creating and analyzing the circuit diagrams of proteins and nucleic acids. We expect our study and future developments in this direction to facilitate a deeper understanding of natural molecular processes and to provide guidance to engineers for generating complex polymeric materials

Biology helps to construct weighted scale free networks

In this work we study a simple evolutionary model of bipartite networks which its evolution is based on the duplication of nodes. Using analytical results along with numerical simulation of the model, we show that the above evolutionary model results in weighted scale free networks. Indeed we find that in the one mode picture we have weighted networks with scale free distributions for interesting quantities like the weights, the degrees and the weighted degrees of the nodes and the weights of the edges.Comment: 15 pages, 7 figures, Revte

Mapping a single-molecule folding process onto a topological space

Physics of protein folding has been dominated by conceptual frameworks including nucleation-propagation mechanism and the diffusion-collision model, none address topological properties of a chain during a folding process. Single-molecule interrogation of folded biomolecules has enabled real-time monitoring of the folding processes at an unprecedented resolution. Despite these advances, the topology landscape has not been fully mapped for any chain. Using a novel circuit topology approach, we map the topology landscape of a model polymeric chain. Inspired by single-molecule mechanical interrogation studies, we restrained the ends of a chain and followed fold nucleation dynamics. We find that, before the nucleation, transient local entropic loops dominate. Although the nucleation length of globules is dependent on the cohesive interaction, the ultimate topological states of the collapsed polymer are largely independent of the interaction but depend on the speed of the folding process. After the nucleation, transient topological rearrangements are observed that converge to a steady-state, where the fold grows in a self-similar manner

Droplet microfluidics: a tool for biology, chemistry and nanotechnology

The ability to perform laboratory operations on small scales using miniaturized devices provides numerous benefits, including reduced quantities of reagents and waste as well as increased portability and controllability of assays. These operations can involve reaction components in the solution phase and as a result, their miniaturization can be accomplished through microfluidic approaches. One such approach, droplet microfluidics, provides a high-throughput platform for a wide range of assays and approaches in chemistry, biology and nanotechnology. We highlight recent advances in the application of droplet microfluidics in chip-based technologies, such as single-cell analysis tools, small-scale cell cultures, in-droplet chemical synthesis, high-throughput drug screening, and nanodevice fabrication

Circuit Topology Analysis of Polymer Folding Reactions

Analytical BioScience

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.Pharmacolog

A tile model of circuit topology for self-entangled biopolymers

Building on the theory of circuit topology for intra-chain contacts in entangled proteins, we introduce tiles as a way to rigorously model local entanglements which are held in place by molecular forces. We develop operations that combine tiles so that entangled chains can be represented by algebraic expressions. Then we use our model to show that the only knot types that such entangled chains can have are 3(1), 4(1), 5(1), 5(2), 6(1), 6(2), 6(3), 7(7), 8(12) and connected sums of these knots. This includes all proteins knots that have thus far been identified.Pharmacolog

Investigation of a Protein Complex Network

The budding yeast {\it Saccharomyces cerevisiae} is the first eukaryote whose genome has been completely sequenced. It is also the first eukaryotic cell whose proteome (the set of all proteins) and interactome (the network of all mutual interactions between proteins) has been analyzed. In this paper we study the structure of the yeast protein complex network in which weighted edges between complexes represent the number of shared proteins. It is found that the network of protein complexes is a small world network with scale free behavior for many of its distributions. However we find that there are no strong correlations between the weights and degrees of neighboring complexes. To reveal non-random features of the network we also compare it with a null model in which the complexes randomly select their proteins. Finally we propose a simple evolutionary model based on duplication and divergence of proteins.Comment: 19 pages, 9 figures, 1 table, to appear in Euro. Phys. J.

Atorvastatin treatment softens human red blood cells: An optical tweezers study

Optical tweezers are proven indispensable single-cell micro-manipulation and mechanical phenotyping tools. In this study, we have used optical tweezers for measuring the viscoelastic properties of human red blood cells (RBCs). Comparison of the viscoelastic features of the healthy fresh and atorvastatin treated cells revealed that the drug softens the cells. Using a simple modeling approach, we proposed a molecular model that explains the drug-induced softening of the RBC membrane. Our results suggest that direct interactions between the drug and cytoskeletal components underlie the drug-induced softening of the cells. © 2018 Optical Society of America

Lassa hemorrhagic shock syndrome‐on‐a‐chip

Analytical BioScience