A Tale of Two Chains: Geometries of a Chain Model and Protein Native State Structures.

Linear chain molecules play a central role in polymer physics with innumerable industrial applications. They are also ubiquitous constituents of living cells. Here, we highlight the similarities and differences between two distinct ways of viewing a linear chain. We do this, on the one hand, through the lens of simulations for a standard polymer chain of tethered spheres at low and high temperatures and, on the other hand, through published experimental data on an important class of biopolymers, proteins. We present detailed analyses of their local and non-local structures as well as the maps of their closest contacts. We seek to reconcile the startlingly different behaviors of the two types of chains based on symmetry considerations.

III. Geometrical framework for thinking about globular proteins: Turns in proteins.

We have shown recently that the notion of poking pairwise interactions along a chain provides a unifying framework for understanding the formation of both secondary and the tertiary protein structure based on symmetry and geometry. α-helices and ß-sheets are found to be special geometries that have systematic poking contacts in a repetitive manner with the contacts being local along the α-helix and non-local along a pair of adjacent strands within a ß-sheet. Pairwise poking interactions also govern tertiary structure formation, but they are weaker and there are no special geometrical constraints as in secondary structure formation. Here we demonstrate that protein turns, the most prevalent non-repetitive structural element in proteins, are instances of local (as in α-helices) and isolated (non-repetitive) poking pairwise contacts for which the geometrical constraints are partially relaxed. This simple and purely geometrical definition of protein turns (also sometimes known as reverse turns, ß-turns, ß-bends, hairpin bends, 310 bends, kinks, widgets, etc.) provides a simple framework for unifying them. We present the results of a systematic analysis and identify their structural classes as well as their respective amino acid preferences.

II. Geometrical framework for thinking about globular proteins: The power of poking.

Recently, we presented a framework for understanding protein structure based on the idea that simple constructs of holding hands or touching of objects can be used to rationalize the common characteristics of globular proteins. We developed a consistent approach for understanding the formation of the two key common building blocks of helices and sheets as well as the compatible assembly of secondary structures into the tertiary structure through the notion of poking pairwise interactions. Here we benchmark our predictions with a detailed analysis of structural data of over 4000 proteins from the Protein Data Bank. We also present the results of detailed computer simulations of a simplified model demonstrating a pre-sculpted free energy landscape, determined by geometry and symmetry, comprising numerous minima corresponding to putative native state structures. We explore the consequences of our model. Our results suggest that symmetry and geometry are a powerful guide to capture the simplicity underlying protein complexity.

A geometrical framework for thinking about proteins.

We present a model, based on symmetry and geometry, for proteins. Using elementary ideas from mathematics and physics, we derive the geometries of discrete helices and sheets. We postulate a compatible solvent-mediated emergent pairwise attraction that assembles these building blocks, while respecting their individual symmetries. Instead of seeking to mimic the complexity of proteins, we look for a simple abstraction of reality that yet captures the essence of proteins. We employ analytic calculations and detailed Monte Carlo simulations to explore some consequences of our theory. The predictions of our approach are in accord with experimental data. Our framework provides a rationalization for understanding the common characteristics of proteins. Our results show that the free energy landscape of a globular protein is pre-sculpted at the backbone level, sequences and functionalities evolve in the fixed backdrop of the folds determined by geometry and symmetry, and that protein structures are unique in being simultaneously characterized by stability, diversity, and sensitivity.

Energetic preference and topological constraint effects on the formation of DNA twisted toroidal bundles.

DNA toroids are compact torus-shaped bundles formed by one or multiple DNA molecules being condensed from the solution due to various condensing agents. It has been shown that the DNA toroidal bundles are twisted. However, the global conformations of DNA inside these bundles are still not well understood. In this study, we investigate this issue by solving different models for the toroidal bundles and performing replica-exchange molecular dynamics (REMD) simulations for self-attractive stiff polymers of various chain lengths. We find that a moderate degree of twisting is energetically favorable for toroidal bundles, yielding optimal configurations of lower energies than for other bundles corresponding to spool-like and constant radius of curvature arrangements. The REMD simulations show that the ground states of the stiff polymers are twisted toroidal bundles with the average twist degrees close to those predicted by the theoretical model. Constant-temperature simulations show that twisted toroidal bundles can be formed through successive processes of nucleation, growth, quick tightening, and slow tightening of the toroid, with the two last processes facilitating the polymer threading through the toroid's hole. A relatively long chain of 512 beads has an increased dynamical difficulty to access the twisted bundle states due to the polymer's topological constraint. Interestingly, we also observed significantly twisted toroidal bundles with a sharp U-shaped region in the polymer conformation. It is suggested that this U-shaped region makes the formation of twisted bundles easier by effectively reducing the polymer length. This effect can be equivalent to having multiple chains in the toroid.

Effects of inter-organism interactions in biofouling on microtopographic surfaces.

An extended model of the surface energetic attachment (SEA) model is introduced to study the fouling of marine organisms on microtopographic surfaces, taking into account the excluded volume interaction and the attraction between the organisms. It is shown that the excluded volume interaction leads to changes in the site-typed attachment probabilities which increase with the average spore density on the surface. As a result of these changes, the spore density map is flattened under very high density fouling. The attractive interaction on the other hand leads to aggregation of spores and the average aggregate size increased with the strength of attraction. The model can be mapped to a specific experiment to determine the attachment energy parameters. In contrast to various prior empirical approaches, the extended SEA model is rigorous from the statistical mechanics viewpoint, thus it provides a reliable tool for studying complex attachment behaviors of microorganisms on topographic surfaces.

Theoretical Model for the Structural Relaxation Time in Coamorphous Drugs.

We propose a simple approach to investigate the structural relaxation time and glass transition of amorphous drugs. Amorphous materials are modeled as a set of equal sized hard spheres. The structural relaxation time over many decades in hard-sphere fluids is theoretically calculated using the elastically collective nonlinear Langevin equation theory associated with Kramer's theory. Then, new thermal mapping from a real material to an effective hard-sphere fluid provides temperature-dependent relaxation time, which can be compared to experiments. Numerical results quantitatively agree with previous experiments for pharmaceutical binary mixtures having different weight ratios. We carry out experiments to test our calculations for an ezetimibe-simvastatin-Kollidon VA64 mixture. Our approach would provide a simple but comprehensive description of glassy dynamics in amorphous composites.

The elixir phase of chain molecules.

A phase of matter is a familiar notion for inanimate physical matter. The nature of a phase of matter transcends the microscopic material properties. For example, materials in the liquid phase have certain common properties independent of the chemistry of the constituents: liquids take the shape of the container; they flow; and they can be poured-alcohol, oil, and water as well as a Lennard-Jones computer model exhibit similar behavior when poised in the liquid phase. Here, we identify a hitherto unstudied "phase" of matter, the elixir phase, in a simple model of a polymeric chain whose backbone has the correct local cylindrical symmetry induced by the tangent to the chain. The elixir phase appears on breaking the cylindrical symmetry by adding side spheres along the negative normal direction, as in proteins. This phase, nestled between other phases, has multiple ground states made up of building blocks of helices and almost planar sheets akin to protein native folds. We discuss the similarities of this "phase" of a finite size system to the liquid crystal and spin glass phases. Our findings are relevant for understanding proteins; the creation of novel bioinspired nanomachines; and also may have implications for life elsewhere in the cosmos.

Sequence dependent aggregation of peptides and fibril formation.

Deciphering the links between amino acid sequence and amyloid fibril formation is key for understanding protein misfolding diseases. Here we use Monte Carlo simulations to study the aggregation of short peptides in a coarse-grained model with hydrophobic-polar (HP) amino acid sequences and correlated side chain orientations for hydrophobic contacts. A significant heterogeneity is observed in the aggregate structures and in the thermodynamics of aggregation for systems of different HP sequences and different numbers of peptides. Fibril-like ordered aggregates are found for several sequences that contain the common HPH pattern, while other sequences may form helix bundles or disordered aggregates. A wide variation of the aggregation transition temperatures among sequences, even among those of the same hydrophobic fraction, indicates that not all sequences undergo aggregation at a presumable physiological temperature. The transition is found to be the most cooperative for sequences forming fibril-like structures. For a fibril-prone sequence, it is shown that fibril formation follows the nucleation and growth mechanism. Interestingly, a binary mixture of peptides of an aggregation-prone and a non-aggregation-prone sequence shows the association and conversion of the latter to the fibrillar structure. Our study highlights the role of a sequence in selecting fibril-like aggregates and also the impact of a structural template on fibril formation by peptides of unrelated sequences.

Effective stiffness and formation of secondary structures in a protein-like model.

We use Wang-Landau and replica exchange techniques to study the effect of an increasing stiffness on the formation of secondary structures in protein-like systems. Two possible models are considered. In both models, a polymer chain is formed by tethered beads where non-consecutive backbone beads attract each other via a square-well potential representing the tendency of the chain to fold. In addition, smaller hard spheres are attached to each non-terminal backbone bead along the direction normal to the chain to mimic the steric hindrance of side chains in real proteins. The two models, however, differ in the way bending rigidity is enforced. In the first model, partial overlap between consecutive beads is allowed. This reduces the possible bending angle between consecutive bonds thus producing an effective entropic stiffness that competes with a short-range attraction, and leads to the formation of secondary structures characteristic of proteins. We discuss the low-temperature phase diagram as a function of increasing interpenetration and find a transition from a planar, beta-like structure, to helical shape. In the second model, an energetic stiffness is explicitly introduced by imposing an infinitely large energy penalty for bending above a critical angle between consecutive bonds, and no penalty below it. The low-temperature phase of this model does not show any sign of protein-like secondary structures. At intermediate temperatures, however, where the chain is still in the coil conformation but stiffness is significant, we find the two models to predict a quite similar dependence of the persistence length as a function of the stiffness. This behaviour is rationalized in terms of a simple geometrical mapping between the two models. Finally, we discuss the effect of shrinking side chains to zero and find the above mapping to still hold true.

Transport properties through graphene grain boundaries: strain effects versus lattice symmetry.

As most materials available at the macroscopic scale, graphene samples usually appear in a polycrystalline form and thus contain grain boundaries. In the present work, the effect of uniaxial strain on the electronic transport properties through graphene grain boundaries is investigated using atomistic simulations. A systematic picture of transport properties with respect to the strain and lattice symmetry of graphene domains on both sides of the boundary is provided. In particular, it is shown that strain engineering can be used to open a finite transport gap in all graphene systems where the two domains are arranged in different orientations. This gap value is found to depend on the strain magnitude, on the strain direction and on the lattice symmetry of graphene domains. By choosing appropriately the strain direction, a large transport gap of a few hundred meV can be achieved when applying a small strain of only a few percents. For a specific class of graphene grain boundary systems, strain engineering can also be used to reduce the scattering on defects and thus to significantly enhance the conductance. With a large strain-induced gap, these graphene heterostructures are proposed to be promising candidates for highly sensitive strain sensors, flexible electronic devices and p-n junctions with non-linear I-V characteristics.

Phase diagram of the ground states of DNA condensates.

The phase diagram of the ground states of DNA in a bad solvent is studied for a semiflexible polymer model with a generalized local elastic bending potential characterized by a nonlinearity parameter x and effective self-attraction promoting compaction. x=1 corresponds to the wormlike chain model. Surprisingly, the phase diagram as well as the transition lines between the ground states are found to be a function of x. The model provides a simple explanation for the results of prior experimental and computational studies and makes predictions for the specific geometries of the ground states. The results underscore the impact of the form of the microscopic bending energy at macroscopic observable scales.

Repulsive interactions of a lipid membrane with graphene in composite materials.

The van der Waals interaction between a lipid membrane and a substrate covered by a graphene sheet is investigated using the Lifshitz theory. The reflection coefficients are obtained for a layered planar system submerged in water. The dielectric response properties of the involved materials are also specified and discussed. Our calculations show that a graphene covered substrate can repel the biological membrane in water. This is attributed to the significant changes in the response properties of the system due to the monolayer graphene. It is also found that the van der Waals interaction is mostly dominated by the presence of graphene, while the role of the particular substrate is secondary.

Sequence repeats and protein structure.

Repeats are frequently found in known protein sequences. The level of sequence conservation in tandem repeats correlates with their propensities to be intrinsically disordered. We employ a coarse-grained model of a protein with a two-letter amino acid alphabet, hydrophobic (H) and polar (P), to examine the sequence-structure relationship in the realm of repeated sequences. A fraction of repeated sequences comprises a distinct class of bad folders, whose folding temperatures are much lower than those of random sequences. Imperfection in sequence repetition improves the folding properties of the bad folders while deteriorating those of the good folders. Our results may explain why nature has utilized repeated sequences for their versatility and especially to design functional proteins that are intrinsically unstructured at physiological temperatures.

Inference of the solvation energy parameters of amino acids using maximum entropy approach.

We present a novel technique, based on the principle of maximum entropy, for deriving the solvation energy parameters of amino acids from the knowledge of the solvent accessible areas in experimentally determined native state structures as well as high quality decoys of proteins. We present the results of detailed studies and analyze the correlations of the solvation energy parameters with the standard hydrophobic scale. We study the ability of the inferred parameters to discriminate between the native state structures of proteins and their decoy conformations.

Geometry of proteins: hydrogen bonding, sterics, and marginally compact tubes.

The functionality of proteins is governed by their structure in the native state. Protein structures are made up of emergent building blocks of helices and almost planar sheets. A simple coarse-grained geometrical model of a flexible tube barely subject to compaction provides a unified framework for understanding the common character of globular proteins. We argue that a recent critique of the tube idea is not well founded.

Common attributes of native-state structures of proteins, disordered proteins, and amyloid.

We show that a framework derived from the common character of globular proteins can be used to understand the design of protein sequences, the behavior of intrinsically unstructured proteins, and the formation of amyloid fibrils in a unified manner. Our studies provide compelling support for the idea that protein native-state structures, the structures adopted by intrinsically unstructured proteins on binding as well as those of amyloid aggregates, all reside in a physical state of matter in which the free energy landscape is sculpted not by the specific sequence of amino acids, but rather by considerations of geometry and symmetry. We elucidate the key role played by sequence design in selecting the structure of choice from the predetermined menu of putative native-state structures.

Unified perspective on proteins: a physics approach.

We study a physical system which, while devoid of the complexity one usually associates with proteins, nevertheless displays a remarkable array of proteinlike properties. The constructive hypothesis that this striking resemblance is not accidental not only leads to a unified framework for understanding protein folding, amyloid formation, and protein interactions but also has implications for natural selection.