Contact Statistics Highlight Distinct Organizing Principles of Proteins and RNA

Amyloid-β Tetramers and Divalent Cations at the Membrane/Water Interface: Simple Models Support a Functional Role

Charge polarization at the membrane interface is a fundamental process in biology. Despite the lower concentration compared to the abundant monovalent ions, the relative abundance of divalent cations (Ca2+, Mg2+, Zn2+, Fe2+, Cu2+) in particular spaces, such as the neuron synapse, raised many questions on the possible effects of free multivalent ions and of the required protection of membranes by the eventual defects caused by the free forms of the cations. In this work, we first applied a recent realistic model of divalent cations to a well-investigated model of a polar lipid bilayer, di-myristoyl phosphatidyl choline (DMPC). The full atomistic model allows a fairly good description of changes in the hydration of charged and polar groups upon the association of cations to lipid atoms. The lipid-bound configurations were analyzed in detail. In parallel, amyloid-β 1-42 (Aβ42) peptides assembled into tetramers were modeled at the surface of the same bilayer. Two of the protein tetramers' models were loaded with four Cu2+ ions, the latter bound as in DMPC-free Aβ42 oligomers. The two Cu-bound models differ in the binding topology: one with each Cu ion binding each of the monomers in the tetramer; one with pairs of Cu ions linking two monomers into dimers, forming tetramers as dimers of dimers. The models here described provide hints on the possible role of Cu ions in synaptic plasticity and of Aβ42 oligomers in storing the same ions away from lipids. The release of structurally disordered peptides in the synapse can be a mechanism to recover ion homeostasis and lipid membranes from changes in the divalent cation concentration.

Polymer Physics-Based Classification of Neurons

Recognizing that diverse morphologies of neurons are reminiscent of structures of branched polymers, we put forward a principled and systematic way of classifying neurons that employs the ideas of polymer physics. In particular, we use 3D coordinates of individual neurons, which are accessible in recent neuron reconstruction datasets from electron microscope images. We numerically calculate the form factor, F(q), a Fourier transform of the distance distribution of particles comprising an object of interest, which is routinely measured in scattering experiments to quantitatively characterize the structure of materials. For a polymer-like object consisting of n monomers spanning over a length scale of r, F(q) scales with the wavenumber [Formula: see text] as [Formula: see text] at an intermediate range of q, where [Formula: see text] is the fractal dimension or the inverse scaling exponent ([Formula: see text]) characterizing the geometrical feature ([Formula: see text]) of the object. F(q) can be used to describe a neuron morphology in terms of its size ([Formula: see text]) and the extent of branching quantified by [Formula: see text]. By defining the distance between F(q)s as a measure of similarity between two neuronal morphologies, we tackle the neuron classification problem. In comparison with other existing classification methods for neuronal morphologies, our F(q)-based classification rests solely on 3D coordinates of neurons with no prior knowledge of morphological features. When applied to publicly available neuron datasets from three different organisms, our method not only complements other methods but also offers a physical picture of how the dendritic and axonal branches of an individual neuron fill the space of dense neural networks inside the brain.

Dissecting the cosegregation probability from genome architecture mapping

Genome architecture mapping (GAM) is a recently developed methodology that offers the cosegregation probability of two genomic segments from an ensemble of thinly sliced nuclear profiles, enabling us to probe and decipher three-dimensional chromatin organization. The cosegregation probability from GAM binned at 1 Mb, which thus probes the length scale associated with the genomic separation greater than 1 Mb, is, however, not identical to the contact probability obtained from Hi-C, and its correlation with interlocus distance measured with fluorescence in situ hybridization is not so good as the contact probability. In this study, by using a polymer-based model of chromatins, we derive a theoretical expression of the cosegregation probability as well as that of the contact probability and carry out quantitative analyses of how they differ from each other. The results from our study, validated with in silico GAM analysis on three-dimensional genome structures from fluorescence in situ hybridization, suggest that to attain strong correlation with the interlocus distance, a properly normalized version of cosegregation probability needs to be calculated based on a large number of nuclear slices (n>103).

Extracting multi-way chromatin contacts from Hi-C data

There is a growing realization that multi-way chromatin contacts formed in chromosome structures are fundamental units of gene regulation. However, due to the paucity and complexity of such contacts, it is challenging to detect and identify them using experiments. Based on an assumption that chromosome structures can be mapped onto a network of Gaussian polymer, here we derive analytic expressions for n-body contact probabilities (n > 2) among chromatin loci based on pairwise genomic contact frequencies available in Hi-C, and show that multi-way contact probability maps can in principle be extracted from Hi-C. The three-body (triplet) contact probabilities, calculated from our theory, are in good correlation with those from measurements including Tri-C, MC-4C and SPRITE. Maps of multi-way chromatin contacts calculated from our analytic expressions can not only complement experimental measurements, but also can offer better understanding of the related issues, such as cell-line dependent assemblies of multiple genes and enhancers to chromatin hubs, competition between long-range and short-range multi-way contacts, and condensates of multiple CTCF anchors.

Monte Carlo simulation of a lattice model for the dynamics of randomly branching double-folded ring polymers

Supercoiled DNA, crumpled interphase chromosomes, and topologically constrained ring polymers often adopt treelike, double-folded, randomly branching configurations. Here we study an elastic lattice model for tightly double-folded ring polymers, which allows for the spontaneous creation and deletion of side branches coupled to a diffusive mass transport, which is local both in space and on the connectivity graph of the tree. We use Monte Carlo simulations to study systems falling into three different universality classes: ideal double-folded rings without excluded volume interactions, self-avoiding double-folded rings, and double-folded rings in the melt state. The observed static properties are in good agreement with exact results, simulations, and predictions of Flory theory for randomly branching polymers. For example, in the melt state rings adopt compact configurations and exhibit territorial behavior. In particular, we show that the emergent dynamics is in excellent agreement with a recent scaling theory and illustrate the qualitative differences with the familiar reptation dynamics of linear chains.

Revisiting the organization of Polycomb-repressed domains: 3D chromatin models from Hi-C compared with super-resolution imaging

The accessibility of target gene, a factor critical for gene regulation, is controlled by epigenetic fine-tuning of chromatin organization. While there are multiple experimental techniques to study change of chromatin architecture with its epigenetic state, measurements from them are not always complementary. A qualitative discrepancy is noted between recent super-resolution imaging studies, particularly on Polycomb-group protein repressed domains in Drosophila cell. One of the studies shows that Polycomb-repressed domains are more compact than inactive domains and are segregated from neighboring active domains, whereas Hi-C and chromatin accessibility assay as well as the other super-resolution imaging studies paint a different picture. To examine this issue in detail, we analyzed Hi-C libraries of Drosophila chromosomes as well as distance constraints from one of the imaging studies, and modeled different epigenetic domains by employing a polymer-based approach. According to our chromosome models, both Polycomb-repressed and inactive domains are featured with a similar degree of intra-domain packaging and significant intermixing with adjacent active domains. The epigenetic domains explicitly visualized by our polymer model call for extra attention to the discrepancy of the super-resolution imaging with other measurements, although its precise physicochemical origin still remains to be elucidated.

Compressing Θ-Chain in Slit Geometry

When compressed in a slit of width D, a Θ-chain that displays the scaling of size R₀ (diameter) with respect to the number of monomers N, R₀ ∼ aN^1/2, expands in the lateral direction as R_|| ∼ aN^ν(a/D)^2ν-1. Provided that the Θ condition is strictly maintained throughout the compression, the well-known scaling exponent of Θ-chain in two dimensions, ν = 4/7, is anticipated in a perfect confinement. However, numerics shows that upon increasing compression from R₀/D < 1 to R₀/D ≫ 1, ν gradually deviates from ν = 1/2 and plateaus at ν = 3/4, the exponent associated with the self-avoiding walk in two dimensions. Using both theoretical considerations and numerics, we argue that it is highly nontrivial to maintain the Θ condition under confinement because of two major effects. First, as the dimension is reduced from three to two dimensions, the contributions of higher order virial terms, which can be ignored in three dimensions at large N, become significant, making the perturbative expansion used in Flory-type approach inherently problematic. Second and more importantly, the geometrical confinement, which is regarded as an applied external field, alters the second virial coefficient (B₂) changes from B₂ = 0 (Θ condition) in free space to B₂ > 0 (good-solvent condition) in confinement. Our study provides practical insight into how confinement affects the conformation of a single polymer chain.

Heterogeneous Loop Model to Infer 3D Chromosome Structures from Hi-C

Adapting a well-established formalism in polymer physics, we develop a minimalist approach to infer three-dimensional folding of chromatin from Hi-C data. The three-dimensional chromosome structures generated from our heterogeneous loop model (HLM) are used to visualize chromosome organizations that can substantiate the measurements from fluorescence in situ hybridization, chromatin interaction analysis by paired-end tag sequencing, and RNA-seq signals. We demonstrate the utility of the HLM with several case studies. Specifically, the HLM-generated chromosome structures, which reproduce the spatial distribution of topologically associated domains from fluorescence in situ hybridization measurement, show the phase segregation between two types of topologically associated domains explicitly. We discuss the origin of cell-type-dependent gene-expression level by modeling the chromatin globules of α-globin and SOX2 gene loci for two different cell lines. We also use the HLM to discuss how the chromatin folding and gene-expression level of Pax6 loci, associated with mouse neural development, are modulated by interactions with two enhancers. Finally, HLM-generated structures of chromosome 19 of mouse embryonic stem cells, based on single-cell Hi-C data collected over each cell-cycle phase, visualize changes in chromosome conformation along the cell-cycle. Given a contact frequency map between chromatic loci supplied from Hi-C, HLM is a computationally efficient and versatile modeling tool to generate chromosome structures that can complement interpreting other experimental data.

Chain organization of human interphase chromosome determines the spatiotemporal dynamics of chromatin loci

We investigate spatiotemporal dynamics of human interphase chromosomes by employing a heteropolymer model that incorporates the information of human chromosomes inferred from Hi-C data. Despite considerable heterogeneities in the chromosome structures generated from our model, chromatins are organized into crumpled globules with space-filling (SF) statistics characterized by a single universal scaling exponent (ν = 1/3), and this exponent alone can offer a quantitative account of experimentally observed, many different features of chromosome dynamics. The local chromosome structures, whose scale corresponds to that of topologically associated domains (∼ 0.1 − 1 Mb), display dynamics with a fast relaxation time (≲ 1 − 10 sec); in contrast, the long-range spatial reorganization of the entire chromatin (≳O(102) Mb) occurs on a much slower time scale (≳ hour), providing the dynamic basis of cell-to-cell variability and glass-like behavior of chromosomes. Biological activities, modeled using stronger isotropic white noises added to active loci, accelerate the relaxation dynamics of chromatin domains associated with the low frequency modes and induce phase segregation between the active and inactive loci. Surprisingly, however, they do not significantly change the dynamics at local scales from those obtained under passive conditions. Our study underscores the role of chain organization of chromosome in determining the spatiotemporal dynamics of chromatin loci.

Chromosomes are giant chain molecules made of hundreds of megabase-long DNA intercalated with proteins. Structure and dynamics of interphase chromatin in space and time hold the key to understanding the cell type-dependent gene regulation. In this study, we establish that the crumpled and space-filling (SF) organization of chromatin fiber in the chromosome territory, characterized by a single scaling exponent, is sufficient to explain the complex spatiotemporal hierarchy in chromatin dynamics as well as the subdiffusive motion of the chromatin loci. While seemingly a daunting problem at a first glance, our study shows that relatively simple principles, rooted in polymer physics, can be used to grasp the essence of dynamical properties of the interphase chromatin.

DNA Internal Motion Likely Accelerates Protein Target Search in a Packed Nucleoid

Transcription factors must diffuse through densely packed and coiled DNA to find their binding sites. Using a coarse-grained model of DNA and lac repressor (LacI) in the Escherichia coli nucleoid, simulations were performed to examine how LacI diffuses in such a space. Despite the canonical picture of LacI diffusing rather freely, in reality the DNA is densely packed, is not rigid but highly mobile, and the dynamics of DNA dictates to a great extent the LacI motion. A possibly better picture of unbound LacI motion is that of gated diffusion, where DNA confines LacI in a cage, but LacI can move between cages when hindering DNA strands move out of the way. Three-dimensional diffusion constants for unbound LacI computed from simulations closely match those for unbound LacI in vivo reported in the literature. The internal motions of DNA appear to be governed by strong internal forces arising from being crowded into the small space of the nucleoid. A consequence of the DNA internal motion is that protein target search may be accelerated.

Flory theory of randomly branched polymers

Randomly branched polymer chains (or trees) are a classical subject of polymer physics with connections to the theory of magnetic systems, percolation and critical phenomena. More recently, the model has been reconsidered for RNA, supercoiled DNA and the crumpling of topologically-constrained polymers. While solvable in the ideal case, little is known exactly about randomly branched polymers with volume interactions. Flory theory provides a simple, unifying description for a wide range of branched systems, including isolated trees in good and θ-solvent, and tree melts. In particular, the approach provides a common framework for the description of randomly branched polymers with quenched connectivity and for randomly branching polymers with annealed connectivity. Here we review the Flory theory for interacting trees in the asymptotic limit of very large polymerization degree for good solvent, θ-solutions and melts, and report its predictions for annealed connectivity in θ-solvents. We compare the predictions of Flory theory for randomly branched polymers to a wide range of available analytical and numerical results and conclude that they are qualitatively excellent and quantitatively good in most cases.

Genome-scale characterization of RNA tertiary structures and their functional impact by RNA solvent accessibility prediction

As most RNA structures are elusive to structure determination, obtaining solvent accessible surface areas (ASAs) of nucleotides in an RNA structure is an important first step to characterize potential functional sites and core structural regions. Here, we developed RNAsnap, the first machine-learning method trained on protein-bound RNA structures for solvent accessibility prediction. Built on sequence profiles from multiple sequence alignment (RNAsnap-prof), the method provided robust prediction in fivefold cross-validation and an independent test (Pearson correlation coefficients, r, between predicted and actual ASA values are 0.66 and 0.63, respectively). Application of the method to 6178 mRNAs revealed its positive correlation to mRNA accessibility by dimethyl sulphate (DMS) experimentally measured in vivo (r = 0.37) but not in vitro (r = 0.07), despite the lack of training on mRNAs and the fact that DMS accessibility is only an approximation to solvent accessibility. We further found strong association across coding and noncoding regions between predicted solvent accessibility of the mutation site of a single nucleotide variant (SNV) and the frequency of that variant in the population for 2.2 million SNVs obtained in the 1000 Genomes Project. Moreover, mapping solvent accessibility of RNAs to the human genome indicated that introns, 5' cap of 5' and 3' cap of 3' untranslated regions, are more solvent accessible, consistent with their respective functional roles. These results support conformational selections as the mechanism for the formation of RNA-protein complexes and highlight the utility of genome-scale characterization of RNA tertiary structures by RNAsnap. The server and its stand-alone downloadable version are available at http://sparks-lab.org.

Protein function machinery: from basic structural units to modulation of activity

Contemporary protein structure is a result of the trade off between the laws of physics and the evolutionary selection. The polymer nature of proteins played a decisive role in establishing the basic structural and functional units of soluble proteins. We discuss how these elementary building blocks work in the hierarchy of protein domain structure, co-translational folding, as well as in enzymatic activity and molecular interactions. Next, we consider modulators of the protein function, such as intermolecular interactions, disorder-to-order transitions, and allosteric signaling, acting via interference with the protein's structural dynamics. We also discuss the post-translational modifications, which is a complementary intricate mechanism evolved for regulation of protein functions and interactions. In conclusion, we assess an anticipated contribution of discussed topics to the future advancements in the field.

How to fold intricately: using theory and experiments to unravel the properties of knotted proteins

Over the years, advances in experimental and computational methods have helped us to understand the role of thermodynamic, kinetic and active (chaperone-aided) effects in coordinating the folding steps required to achieving a knotted native state. Here, we review such developments by paying particular attention to the complementarity of experimental and computational studies. Key open issues that could be tackled with either or both approaches are finally pointed out.

Basic units of protein structure, folding, and function

Study of the hierarchy of domain structure with alternative sets of domains and analysis of discontinuous domains, consisting of remote segments of the polypeptide chain, raised a question about the minimal structural unit of the protein domain. The hypothesis on the decisive role of the polypeptide backbone in determining the elementary units of globular proteins have led to the discovery of closed loops. It is reviewed here how closed loops form the loop-n-lock structure of proteins, providing the foundation for stability and designability of protein folds/domain and underlying their co-translational folding. Simplified protein sequences are considered here with the aim to explore the basic principles that presumably dominated the folding and stability of proteins in the early stages of structural evolution. Elementary functional loops (EFLs), closed loops with one or few catalytic residues, are, in turn, units of the protein function. They are apparent descendants of the prebiotic ring-like peptides, which gave rise to the first functional folds/domains being fused in the beginning of the evolution of protein structure. It is also shown how evolutionary relations between protein functional superfamilies and folds delineated with the help of EFLs can contribute to establishing the rules for design of desired enzymatic functions. Generalized descriptors of the elementary functions are proposed to be used as basic units in the future computational design.

Looping probability of random heteropolymers helps to understand the scaling properties of biopolymers

Random heteropolymers are a minimal description of biopolymers and can provide a theoretical framework to the investigate the formation of loops in biophysical experiments. The looping probability as a function of polymer length was observed to display in some biopolymers, like chromosomes in cell nuclei or long RNA chains, anomalous scaling exponents. Combining a two-state model with self-adjusting simulated-tempering calculations, we calculate numerically the looping properties of several realizations of the random interactions within the chain. We find a continuous set of exponents upon varying the temperature, which arises from finite-size effects and is amplified by the disorder of the interactions. We suggest that this could provide a simple explanation for the anomalous scaling exponents found in experiments. In addition, our results have important implications notably for the study of chromosome folding as they show that scaling exponents cannot be the sole criteria for testing hypothesis-driven models of chromosome architecture.