Pairwise Attraction Research Articles

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.

Bonding Analysis of the Ge‐Ge Bonds in the Octagermacubane Ge8(Sit‐butyl2methyl)6

AbstractQuantum chemical calculations have been carried out at the BP86/def2‐SVP level on Ge8(Sit‐butyl2methyl)6(1) and the bonding situation has been analyzed with a variety of methods. The calculated equilibrium geometry of1is in good agreement with the reported x‐ray structure analysis. The D3 correction for dispersion interactions as a sum of pairwise attractions leads to an overestimate of the effect of dispersion forces. Calculations at BP86‐D3(BJ)/def2‐SVP give shorter bonds for Ge(I)−Ge(I) than for Ge(0)−Ge(I), which is in contrast to the experimental values and the BP86/def2‐SVP results. The NBO analysis suggests that the best Lewis structure of1has lone‐pair orbitals at the Ge(0) atoms with occupation numbers of 1.70 e. A lone‐pair character at Ge(0) albeit with less weight is also suggested by the shape of the HOMO, which is an antibonding orbital between the Ge(0) atoms with small contributions from the Ge(I) atoms. The LUMO of1is the corresponding bonding combination of the Ge(0) AOs, which can be explained with the reluctance of the heavier main‐group atoms to s/p hybridization of the valence orbitals. The calculated bond order values suggest significant direct Ge(0)−Ge(0) interactions. This is supported by the shape of the HOMO and by the results of EDA‐NOCV calculations. The deformation densities and the orbitals associated with the pairwise orbital interaction show that there is a direct charge flow between the Ge(0) atoms of the two fragments, but it is not completely separated from the Ge(0)−Ge(I) and Ge(I)−Ge(I) bond formation. The QTAIM calculations suggest that1has a cubic structure with a cage critical point but not a bond critical point for the Ge(0)−Ge(0) interactions. The dispersion interactions of the large substituents in1have a significant influence on the stability of the compound.

Tuning Material States and Functionalities of G-Quadruplex-Modulated RNA-Peptide Condensates

RNA encodes sequence- and structure-dependent interactions to modulate the assembly and properties of biomolecular condensates. RNA G-quadruplexes (rG4s) formed by guanine-rich sequences can trigger the formation of liquid- or solid-like condensates that are involved in many aberrant phase transitions. However, exactly how rG4 motifs modulate different phase transitions and impart distinct material properties to condensates is unclear. Here, using RNA oligonucleotides and cationic peptides as model systems, we show that RNA-peptide condensates exhibit tunability in material properties over a wide spectrum via interactions arising from rG4 folding/unfolding kinetics. rG4-containing oligonucleotides formed strong pairwise attraction with peptides and tended to form solid-like condensates, while their less-structured non-G4 mutants formed liquid-like droplets. We find that the coupling between rG4 dissociation and RNA-peptide complex coacervation triggers solid-to-liquid transition of condensates prior to the complete unfolding of rG4s. This coupling points to a mechanism that material states of rG4-modulated condensates can be finely tuned from solid-like to liquid-like by the addition of less-structured RNA oligonucleotides, which have weak but dominant binding with peptides. We further show that the tunable material states of condensates can enhance RNA aptamer compartmentalization and RNA cleavage reactions. Our results suggest that condensates with complex properties can emerge from subtle changes in RNA oligonucleotides, contributing ways to treat dysfunctional condensates in diseases and insights into prebiotic compartmentalization.

Phase Separation and Multibody Effects in Three-Dimensional Active Brownian Particles.

Simulation studies of the phase diagram of repulsive active Brownian particles in three dimensions reveal that the region of motility-induced phase separation between a high and low density phase is enclosed by a region of gas-crystal phase separation. Near-critical loci and structural crossovers can additionally be identified in analogy with simple fluids. Motivated by the striking similarity to the behavior of equilibrium fluids with short-ranged pairwise attractions, we show that a direct mapping to pair potentials in the dilute limit implies interactions that are insufficiently attractive to engender phase separation. Instead, this is driven by the emergence of multibody effects associated with particle caging that occurs at sufficiently high number density. We quantify these effects via information-theoretical measures of n-body effective interactions extracted from the configurational structure.

Tug of War between Condensate Phases in a Minimal Macromolecular System.

Membraneless organelles, comprising dozens to hundreds of macromolecular components, form heterogeneous phases in space and evolve over time in material properties. Here, using four macromolecules, we demonstrate a range of phase behaviors associated with membraneless organelles and uncover the underlying physicochemical rules. The macromolecules are SH35 (S) and PRM5 (P), two pentameric, oppositely charged protein constructs; heparin (H), an anionic polymer; and lysozyme (L), a cationic single-domain protein. The S:P, S:L, and P:H binaries form droplets, but the H:L binary forms network-like precipitates, therefore setting up a tug of war between different condensate phases within the S:P:H:L quaternary. The H:L exception can partly be attributed to the compactness of L, as supported by ThT binding data. Increasing amounts of P alone or both S and P, but not S alone, can dissolve H:L precipitates into droplets. These differential effects can be explained by the order of the strengths of pairwise attraction: H:L > P:H > S:P > S:L, deduced from the shapes of ternary phase boundaries. When S and P are at subdissolution concentrations, S:P:H:L precipitates change over time to become droplet-like in appearance, although not completely fluidic according to fluorescence recovery after photobleaching. In fact, confocal microscopy reveals separated S:L-rich and P:H-rich foci inside the droplet-like condensates. Therefore, complex phase behaviors of membraneless organelles, including rescue of aberrant phase transitions, demixing of condensates, and time evolution of material properties, can all be reconstituted and understood via a minimal macromolecular system.

Threshold Phenomena in the Spectrum of the Two-Particle Schrödinger Operator on a Lattice

For a broad class of short-range pairwise attraction potentials, we study threshold phenomena in the spectrum of the two-particle Schrodinger operator associated with the energy operator of the s–d exchange model. We prove that the bound state (eigenvalue) either exists or does not exist depending on the exchange interaction parameter, the system quasimomentum, and dimension of the lattice.

Filamentous Viruses Grafted with Thermoresponsive Block Polymers: Liquid Crystal Behaviors of a Rodlike Colloidal Model with “True” Attractive Interactions

Understanding how attractive interactions among rigid polymers or rodlike particles influence their liquid crystal (LC) phase behavior is of fundamental and practical importance. This question has not been fully answered yet, mainly due to the shortage of model systems with “true” pairwise attractions on a single particle level while with excellent colloidal stability. Herein, we report on a well-defined rodlike system that fulfills such criteria, through covalently grafting the free end of the thermoresponsive PNIPAM block of poly(ethylene glycol)-block-poly(N-isopropylacrylamide) (PEG-b-PNIPAM) onto the classic rodlike model system—the fd or M13 virus—which is the hallmark in understanding the LC behaviors of rigid polymer or rodlike particles. Increasing temperature induces dehydration and collapse of the PNIPAM chains onto the virus surface and therefore introduces attractions among the viruses, while the outer hydrophilic PEG block offers steric stabilization to prevent interparticle aggregation, gel...

Designing nanoindentation simulation studies by appropriate indenter choices: Case study on single crystal tungsten

Atomic simulations are widely used to study the mechanics of small contacts for many contact loading processes such as nanometric cutting, nanoindentation, polishing, grinding and nanoimpact. A common assumption in most such studies is the idealisation of the impacting material (indenter or tool) as a perfectly rigid body. In this study, we explore this idealisation and show that active chemical interactions between two contacting asperities lead to significant deviations of atomic scale contact mechanics from predictions by classical continuum mechanics. We performed a testbed study by simulating velocity-controlled, fixed displacement nanoindentation on single crystal tungsten using five types of indenter (i) a rigid diamond indenter (DI) with full interactions, (ii) a rigid indenter comprising of the atoms of the same material as that of the substrate i.e. tungsten atoms (TI), (iii) a rigid diamond indenter with pairwise attraction turned off, (iv) a deformable diamond indenter and (v) an imaginary, ideally smooth, spherical, rigid and purely repulsive indenter (RI). Corroborating the published experimental data, the simulation results provide a useful guideline for selecting the right kind of indenter for atomic scale simulations.

PRISM Theory Study of Amphiphilic Block Copolymer Solutions with Varying Copolymer Sequence and Composition

We present a comparison of Polymer Reference Interaction Site Model (PRISM) theory and molecular dynamics (MD) simulations for studying amphiphilic block copolymers in solution. We use a generic coarse-grained model to represent amphiphilic A–B block copolymers in implicit solvent with the solvophobicity of the B segments captured using effective B–B pairwise attraction modeled using the Lennard-Jones potential. We study the assembly of the amphiphilic A–B block copolymer as a function of solvophobicity for varying copolymer sequences (diblock and triblock) and composition (solvophobic-rich or solvophilic-rich). The PRISM theory equation along with the atomic Percus–Yevick closure is solved to obtain the intermolecular pair correlations in real space, g(r), and structure factors in Fourier space, S(k), for block copolymer solutions at increasing values of solvophobicity. The real-space intermolecular pair correlation functions and structure factors from PRISM theory and from MD simulations are compared di...

Dipole codes attractively encode glue functions

Dipole words are sequences of magnetic dipoles, in which alike elements repel and opposite elements attract. Magnetic dipoles contrast with more general sets of bonding types, called glues, in which pairwise bonding strength is specified by a glue function. We prove that every glue function g has a set of dipole words, called a dipole code, that attractively encodes g: the pairwise attractions (positive or non-positive bond strength) between the words are identical to those of g. Moreover, we give such word sets of asymptotically optimal length. Similar results are obtained for a commonly used subclass of glue functions.

Geometry of flexible filament cohesion: better contact through twist?

Cohesive interactions between filamentous molecules have broad implications for a range of biological and synthetic materials. While long-standing theoretical approaches have addressed the problem of inter-filament forces from the limit of infinitely rigid rods, the ability of flexible filaments to deform intra-filament shape in response to changes in inter-filament geometry has a profound affect on the nature of cohesive interactions. In this paper, we study two theoretical models of inter-filament cohesion in the opposite limit, in which filaments are sufficiently flexible to maintain cohesive contact along their contours, and address, in particular, the role played by helical-interfilament geometry in defining interactions. Specifically, we study models of featureless, tubular filaments interacting via: (1) pair-wise Lennard-Jones (LJ) interactions between surface elements and (2) depletion-induced filament binding stabilized by electrostatic surface repulsion. Analysis of these models reveals a universal preference for cohesive filament interactions for non-zero helical skew, and further, that in the asymptotic limit of vanishing interaction range relative to filament diameter, the skew-dependence of cohesion approaches a geometrically defined limit described purely by the close-packing geometry of twisted tubular filaments. We further analyze non-universal features of the skew-dependence of cohesion at small-twist for both potentials, and argue that in the LJ model the pair-wise surface attraction generically destabilizes parallel filaments, while in the second model, pair-wise electrostatic repulsion in combination with non-pairwise additivity of depletion leads to a meta-stable parallel state.

Two-Phase Fluid Simulation Using a Diffuse Interface Model with Peng--Robinson Equation of State

In this paper, two-phase fluid systems are simulated using a diffusive interface model with the Peng--Robinson equation of state (EOS), a widely used realistic EOS for hydrocarbon fluid in the petroleum industry. We first utilize the gradient theory of thermodynamics and variational calculus to derive a generalized chemical equilibrium equation, which is mathematically a second-order elliptic partial differential equation (PDE) in molar density with a strongly nonlinear source term. To solve this PDE, we convert it to a time-dependent parabolic PDE with the main interest in its final steady state solution. A Lagrange multiplier is used to enforce mass conservation. The parabolic PDE is then solved by mixed finite element methods with a semi-implicit time marching scheme. Convex splitting of the energy functional is proposed to construct this time marching scheme, where the volume exclusion effect of an EOS is treated implicitly while the pairwise attraction effect of EOS is calculated explicitly. This sch...

Stability and minimum size of colloidal clusters on a liquid-air interface

A vertical force applied to each of two colloids, trapped at a liquid-air interface, induces their logarithmic pairwise attraction. I recently showed [Phys. Rev. E 79, 011407 (2009)] that in clusters of size R much larger than the capillary length λ, the attraction changes to that of a power law and is much stronger due to a many-body effect, and I derived two equations that describe the equilibrium coarse-grained meniscus profile and colloid density in such clusters. In this paper, this theory is shown also to describe small clusters with R≪ λ provided the number N of colloids therein is sufficiently large. An analytical solution for a small circular cluster with an arbitrary short-range power-law pairwise repulsion is found. The energy of a cluster is obtained as a function of its radius R and colloid number N. As in large clusters, the attraction force and energy universally scale with the distance L between colloids as L(-3) and L(-2), respectively, for any repulsion forces. The states of an equilibrium cluster, predicted by the theory, are shown to be stable with respect to small perturbations of the meniscus profile and colloid density. The minimum number of colloids in a circular cluster, which sustains the thermal motion, is estimated. For standard parameters, it can be very modest, e.g., in the range 20-200, which is in line with experimental findings on reversible clusterization on a liquid-air interface.

How small can an equilibrium colloidal cluster on a liquid-air interface be?

A vertical force on a pair of spherical colloids trapped at a liquid-air interface induces their logarithmic pairwise attraction. However, in clusters of size R much larger than the capillary length λ, the many-body interaction dramatically changes the capillary attraction: it is much stronger and no longer logarithmic [V. M. Pergamenshchik, Phys. Rev. E 79, 011407 (2009)]. In this paper, this theory is shown to describe also small clusters with R ≪ λ if they consist of sufficiently large numbers of colloids. The analytical solution for a small circular cluster with a short range powerlaw repulsion with arbitrary exponent n is found. Irrespective of n, the attraction force and the energy scale with the distance L between colloids as L−3 and L−2 for clusters of any size. We estimate the minimum size of a cluster in which the predicted many-body capillary attraction can prevent evaporation of colloids due to their thermal motion. For standard parameters, the minimum number of colloids in a stable nonevaporating cluster can be as low as 20–200. These numbers are in accord with the experimental findings on reversible clusterization on a liquid-air interface.

Spatially controlled reversible colloidal self-assembly

We studied the localized self-assembly of colloidal crystals on a topographically patterned substrate. A competition between particle and pattern interactions provided the ability to reversibly assemble quasi-two-dimensional colloidal crystals on a periodic landscape. The assembly process was visualized and controlled in real-space and real-time using video microscopy. Independent measurements and computer simulations were used to quantify all interactions controlling self-assembly. Steady-state studies characterized spatially inhomogeneous, coexisting fluid and crystal microstructures at various stages of assembly. Microstructures arise from a balance of local sedimentation equilibria within potential energy features and a tunable pairwise depletion attraction between colloids. Transient colloidal crystal self-assembly occurred via a quasiequilibrium process as characterized by continuously evolving spatial profiles of local density, bond orientational order, and self-diffusivities.

Modularity clustering is force-directed layout

Two natural and widely used representations for the community structure of networks are clusterings, which partition the vertex set into disjoint subsets, and layouts, which assign the vertices to positions in a metric space. This paper unifies prominent characterizations of layout quality and clustering quality, by showing that energy models of pairwise attraction and repulsion subsume Newman and Girvan's modularity measure. Layouts with optimal energy are relaxations of, and are thus consistent with, clusterings with optimal modularity, which is of practical relevance because the two representations are complementary and often used together.

Strong collective attraction in colloidal clusters on a liquid-air interface

It is shown that in a cluster of many colloids, trapped at a liquid-air interface, the well-known vertical-force-induced pairwise logarithmic attraction changes to a strongly enhanced power-law attraction. In large two-dimensional clusters, the attraction energy scales as the inverse square of the distance between colloids. The enhancement is given by the ratio eta = (square of the capillary length) / (interface surface area per colloid) and can be as large as 10;{5} . This explains why a very small vertical force on colloids, which is too weak to bring two of them together, can stabilize many-body structures on a liquid-air interface. The profile of a cluster is shown to consist of a large slow collective envelope modulated by a fast low-amplitude perturbation due to individual colloids. A closed equation for the slow envelope, which incorporates an arbitrary power-law repulsion between colloids, is derived. For example, this equation is solved for a large circular cluster with the hard-core colloid repulsion. It is suggested that the predicted effect is responsible for mysterious stabilization of colloidal structures observed in experiments on a surface of isotropic liquid and nematic liquid crystal.

Repulsive and attractive Casimir forces in a glide-symmetric geometry

We describe a three-dimensional geometry in which both attractive and repulsive Casimir forces arise using ordinary metallic materials, as computed via an exact numerical method (no uncontrolled approximations). The geometry consists of a zipperlike, glide-symmetric structure formed of interleaved metal brackets attached to parallel plates---because of the interleaving pattern, a net repulsive force can arise from a combination of attractive interactions. Depending on the separation, the perpendicular force between the plates and brackets varies from attractive (large separations) to repulsive (intermediate distances) and back to attractive (close separations), with one point of stable equilibrium in the perpendicular direction. This geometry was motivated by a simple intuition of attractive interactions between surfaces, and so we also consider how a rough proximity-force approximation of pairwise attractions compares to the exact calculations.

Finiteness of the discrete spectrum of the Schrödinger operator of three particles on a lattice

We consider a system of three quantum particles interacting by pairwise short-range attraction potentials on a three-dimensional lattice (one of the particles has an infinite mass). We prove that the number of bound states of the corresponding Schrodinger operator is finite in the case where the potentials satisfy certain conditions, the two two-particle sub-Hamiltonians with infinite mass have a resonance at zero, and zero is a regular point for the two-particle sub-Hamiltonian with finite mass.

Sequential quenching of randomly deposited ellipsoids: Anisotropy and spatial patterns

We studied the sequential quenching of prolate ellipsoids on a homogeneous surface by using our proposed pair potential whose repulsive part was that of hard ellipses and attractive part was the r −6 pairwise attraction. Both the strength and range parameters for the attraction were functions of the orientations of the pair of ellipses and related by ε ∝ σ − λ . The parameter λ determines the relative strength of the side-by-side and end-to-end attractions and thus plays an important role in determining the alignment of the particles. We adopted the value λ = 2.19 by using point-energy additivity to compute the minimum energies for both of these configurations for a pair of ellipsoids of revolution with aspect ratio of 2:1. We investigated the effect of temperature and the parameter λ on the alignments of ellipses. Both radial distribution function, g ( r ) , and orientational correlation function, G ( r ) , show the expected longer ranges of orientational correlation at lower temperatures and show higher degree of orientational order for λ = 3.5 than λ = 2.19 and 0.10. This can also be seen in the examples of configurations showing that for λ = 3.5 , ellipses are more aligned than λ = 2.19 and 0.1.