Hyperuniformity order metric of Barlow packings

The concept of hyperuniformity has been a useful tool in the study of density fluctuations at large length scales in systems ranging across the natural and mathematical sciences. One can rank a large class of hyperuniform systems by their ability to suppress long-range density fluctuations through the use of a hyperuniformity order metric Λ[over ¯]. We apply this order metric to the Barlow packings, which are the infinitely degenerate densest packings of identical rigid spheres that are distinguished by their stacking geometries and include the commonly known fcc lattice and hcp crystal. The "stealthy stacking" theorem implies that these packings are all stealthy hyperuniform, a strong type of hyperuniformity, which involves the suppression of scattering up to a wave vector K. We describe the geometry of three classes of Barlow packings, two disordered classes and small-period packings. In addition, we compute a lower bound on K for all Barlow packings. We compute Λ[over ¯] for the aforementioned three classes of Barlow packings and find that, to a very good approximation, it is linear in the fraction of fcc-like clusters, taking values between those of least-ordered hcp and most-ordered fcc. This implies that the value of Λ[over ¯] of all Barlow packings is primarily controlled by the local cluster geometry. These results highlight the special nature of anisotropic stacking disorder, which provides impetus for future research on the development of anisotropic order metrics and hyperuniformity properties.

Classical many-particle systems with unique disordered ground states

Classical ground states (global energy-minimizing configurations) of many-particle systems are typically unique crystalline structures, implying zero enumeration entropy of distinct patterns (aside from trivial symmetry operations). By contrast, the few previously known disordered classical ground states of many-particle systems are all high-entropy (highly degenerate) states. Here we show computationally that our recently proposed "perfect-glass" many-particle model [Sci. Rep. 6, 36963 (2016)10.1038/srep36963] possesses disordered classical ground states with a zero entropy: a highly counterintuitive situation . For all of the system sizes, parameters, and space dimensions that we have numerically investigated, the disordered ground states are unique such that they can always be superposed onto each other or their mirror image. At low energies, the density of states obtained from simulations matches those calculated from the harmonic approximation near a single ground state, further confirming ground-state uniqueness. Our discovery provides singular examples in which entropy and disorder are at odds with one another. The zero-entropy ground states provide a unique perspective on the celebrated Kauzmann-entropy crisis in which the extrapolated entropy of a supercooled liquid drops below that of the crystal. We expect that our disordered unique patterns to be of value in fields beyond glass physics, including applications in cryptography as pseudorandom functions with tunable computational complexity.

Can exotic disordered "stealthy" particle configurations tolerate arbitrarily large holes?

The probability of finding a spherical cavity or "hole" of arbitrarily large size in typical disordered many-particle systems in the infinite-system-size limit (e.g., equilibrium liquid states) is non-zero. Such "hole" statistics are intimately linked to the thermodynamic and nonequilibrium physical properties of the system. Disordered "stealthy" many-particle configurations in d-dimensional Euclidean space [Doublestruck R]^d are exotic amorphous states of matter that lie between a liquid and crystal that prohibit single-scattering events for a range of wave vectors and possess no Bragg peaks [Torquato et al., Phys. Rev. X, 2015, 5, 021020]. In this paper, we provide strong numerical evidence that disordered stealthy configurations across the first three space dimensions cannot tolerate arbitrarily large holes in the infinite-system-size limit, i.e., the hole probability has compact support. This structural "rigidity" property apparently endows disordered stealthy systems with novel thermodynamic and physical properties, including desirable band-gap, optical and transport characteristics. We also determine the maximum hole size that any stealthy system can possess across the first three space dimensions.

Ground states of stealthy hyperuniform potentials: I. Entropically favored configurations

Systems of particles interacting with "stealthy" pair potentials have been shown to possess infinitely degenerate disordered hyperuniform classical ground states with novel physical properties. Previous attempts to sample the infinitely degenerate ground states used energy minimization techniques, introducing algorithmic dependence that is artificial in nature. Recently, an ensemble theory of stealthy hyperuniform ground states was formulated to predict the structure and thermodynamics that was shown to be in excellent agreement with corresponding computer simulation results in the canonical ensemble (in the zero-temperature limit). In this paper, we provide details and justifications of the simulation procedure, which involves performing molecular dynamics simulations at sufficiently low temperatures and minimizing the energy of the snapshots for both the high-density disordered regime, where the theory applies, as well as lower densities. We also use numerical simulations to extend our study to the lower-density regime. We report results for the pair correlation functions, structure factors, and Voronoi cell statistics. In the high-density regime, we verify the theoretical ansatz that stealthy disordered ground states behave like "pseudo" disordered equilibrium hard-sphere systems in Fourier space. The pair statistics obey certain exact integral conditions with very high accuracy. These results show that as the density decreases from the high-density limit, the disordered ground states in the canonical ensemble are characterized by an increasing degree of short-range order and eventually the system undergoes a phase transition to crystalline ground states. In the crystalline regime (low densities), there exist aperiodic structures that are part of the ground-state manifold but yet are not entropically favored. We also provide numerical evidence suggesting that different forms of stealthy pair potentials produce the same ground-state ensemble in the zero-temperature limit. Our techniques may be applied to sample the zero-temperature limit of the canonical ensemble of other potentials with highly degenerate ground states.

Ground states of stealthy hyperuniform potentials. II. Stacked-slider phases

Stealthy potentials, a family of long-range isotropic pair potentials, produce infinitely degenerate disordered ground states at high densities and crystalline ground states at low densities in d-dimensional Euclidean space R^{d}. In the previous paper in this series, we numerically studied the entropically favored ground states in the canonical ensemble in the zero-temperature limit across the first three Euclidean space dimensions. In this paper, we investigate using both numerical and theoretical techniques metastable stacked-slider phases, which are part of the ground-state manifold of stealthy potentials at densities in which crystal ground states are favored entropically. Our numerical results enable us to devise analytical models of this phase in two, three, and higher dimensions. Utilizing this model, we estimated the size of the feasible region in configuration space of the stacked-slider phase, finding it to be smaller than that of crystal structures in the infinite-system-size limit, which is consistent with our recent previous work. In two dimensions, we also determine exact expressions for the pair correlation function and structure factor of the analytical model of stacked-slider phases and analyze the connectedness of the ground-state manifold of stealthy potentials in this density regime. We demonstrate that stacked-slider phases are distinguishable states of matter; they are nonperiodic, statistically anisotropic structures that possess long-range orientational order but have zero shear modulus. We outline some possible future avenues of research to elucidate our understanding of this unusual phase of matter.

Probing the limitations of isotropic pair potentials to produce ground-state structural extremes via inverse statistical mechanics

Inverse statistical-mechanical methods have recently been employed to design optimized short-range radial (isotropic) pair potentials that robustly produce novel targeted classical ground-state many-particle configurations. The target structures considered in those studies were low-coordinated crystals with a high degree of symmetry. In this paper, we further test the fundamental limitations of radial pair potentials by targeting crystal structures with appreciably less symmetry, including those in which the particles have different local structural environments. These challenging target configurations demanded that we modify previous inverse optimization techniques. In particular, we first find local minima of a candidate enthalpy surface and determine the enthalpy difference ΔH between such inherent structures and the target structure. Then we determine the lowest positive eigenvalue λ(0) of the Hessian matrix of the enthalpy surface at the target configuration. Finally, we maximize λ(0)ΔH so that the target structure is both locally stable and globally stable with respect to the inherent structures. Using this modified optimization technique, we have designed short-range radial pair potentials that stabilize the two-dimensional kagome crystal, the rectangular kagome crystal, and rectangular lattices, as well as the three-dimensional structure of the CaF(2) crystal inhabited by a single-particle species. We verify our results by cooling liquid configurations to absolute zero temperature via simulated annealing and ensuring that such states have stable phonon spectra. Except for the rectangular kagome structure, all of the target structures can be stabilized with monotonic repulsive potentials. Our work demonstrates that single-component systems with short-range radial pair potentials can counterintuitively self-assemble into crystal ground states with low symmetry and different local structural environments. Finally, we present general principles that offer guidance in determining whether certain target structures can be achieved as ground states by radial pair potentials.

Water revisited

Liquid water consists of a macroscopically connected, random network of hydrogen bonds, with frequent strained and broken bonds, that is continually undergoing topological reformation. Anomalous properties of water arise from the competition between relatively bulky ways of connecting molecules into local patterns characterized by strong bonds and nearly tetrahedral angles and more compact arrangements characterized by more strain and bond breakage. However, these alternatives constitute virtually a continuum of architectural possibilities rather than a discrete pair of options. The singular behavior of supercooled water near -45 degrees C and the "hydrophobic" attraction between nonpolar entities are due to the same underlying phenomenon, namely, the clumping tendency of relatively strain-free convex cages or polyhedra.

Distinctive features arising in maximally random jammed packings of superballs

Dense random packings of hard particles are useful models of granular media and are closely related to the structure of nonequilibrium low-temperature amorphous phases of matter. Most work has been done for random jammed packings of spheres and it is only recently that corresponding packings of nonspherical particles (e.g., ellipsoids) have received attention. Here we report a study of the maximally random jammed (MRJ) packings of binary superdisks and monodispersed superballs whose shapes are defined by |x1|2p+...+|xd|2p<or=1 with d=2 and 3, respectively, where p is the deformation parameter with values in the interval (0,infinity). As p increases from zero, one can get a family of both concave (0<p<0.5) and convex (p>or=0.5) particles with square symmetry (d=2), or octahedral and cubic symmetry (d=3). In particular, for p=1 the particle is a perfect sphere (circular disk) and for p-->infinity the particle is a perfect cube (square). We find that the MRJ densities of such packings increase dramatically and nonanalytically as one moves away from the circular-disk and sphere point (p=1). Moreover, the disordered packings are hypostatic, i.e., the average number of contacting neighbors is less than twice the total number of degrees of freedom per particle, and yet the packings are mechanically stable. As a result, the local arrangements of particles are necessarily nontrivially correlated to achieve jamming. We term such correlated structures "nongeneric." The degree of "nongenericity" of the packings is quantitatively characterized by determining the fraction of local coordination structures in which the central particles have fewer contacting neighbors than average. We also show that such seemingly "special" packing configurations are counterintuitively not rare. As the anisotropy of the particles increases, the fraction of rattlers decreases while the minimal orientational order as measured by the tetratic and cubatic order parameters increases. These characteristics result from the unique manner in which superballs break their rotational symmetry, which also makes the superdisk and superball packings distinctly different from other known nonspherical hard-particle packings.

Geometrical ambiguity of pair statistics: point configurations

Point configurations have been widely used as model systems in condensed-matter physics, materials science, and biology. Statistical descriptors, such as the n -body distribution function g(n), are usually employed to characterize point configurations, among which the most extensively used is the pair distribution function g(2). An intriguing inverse problem of practical importance that has been receiving considerable attention is the degree to which a point configuration can be reconstructed from the pair distribution function of a target configuration. Although it is known that the pair-distance information contained in g(2) is, in general, insufficient to uniquely determine a point configuration, this concept does not seem to be widely appreciated and general claims of uniqueness of the reconstructions using pair information have been made based on numerical studies. In this paper, we present the idea of the distance space called the D space. The pair distances of a specific point configuration are then represented by a single point in the D space. We derive the conditions on the pair distances that can be associated with a point configuration, which are equivalent to the realizability conditions of the pair distribution function g(2). Moreover, we derive the conditions on the pair distances that can be assembled into distinct configurations, i.e., with structural degeneracy. These conditions define a bounded region in the D space. By explicitly constructing a variety of degenerate point configurations using the D space, we show that pair information is indeed insufficient to uniquely determine the configuration in general. We also discuss several important problems in statistical physics based on the D space, including the reconstruction of atomic structures from experimentally obtained g(2) and a recently proposed "decorrelation" principle. The degenerate configurations have relevance to open questions involving the famous traveling salesman problem.

Optimal packings of superballs

Dense hard-particle packings are intimately related to the structure of low-temperature phases of matter and are useful models of heterogeneous materials and granular media. Most studies of the densest packings in three dimensions have considered spherical shapes, and it is only more recently that nonspherical shapes (e.g., ellipsoids) have been investigated. Superballs (whose shapes are defined by |x1|2p+|x2|2p+|x3|2p<or=1) provide a versatile family of convex particles (p>or=0.5) with both cubic-like and octahedral-like shapes as well as concave particles (0<p<0.5) with octahedral-like shapes. In this paper, we provide analytical constructions for the densest known superball packings for all convex and concave cases. The candidate maximally dense packings are certain families of Bravais lattice packings (in which each particle has 12 contacting neighbors) possessing the global symmetries that are consistent with certain symmetries of a superball. We also provide strong evidence that our packings for convex superballs (p>or=0.5) are most likely the optimal ones. The maximal packing density as a function of p is nonanalytic at the sphere point (p=1) and increases dramatically as p moves away from unity. Two more nontrivial nonanalytic behaviors occur at pc*=1.150 9... and po*=ln 3/ln 4=0.792 4... for "cubic" and "octahedral" superballs, respectively, where different Bravais lattice packings possess the same densities. The packing characteristics determined by the broken rotational symmetry of superballs are similar to but richer than their two-dimensional "superdisk" counterparts [Y. Jiao, Phys. Rev. Lett. 100, 245504 (2008)] and are distinctly different from that of ellipsoid packings. Our candidate optimal superball packings provide a starting point to quantify the equilibrium phase behavior of superball systems, which should deepen our understanding of the statistical thermodynamics of nonspherical-particle systems.

Optimal packings of superdisks and the role of symmetry

Almost all studies of the densest particle packings consider convex particles. Here, we provide exact constructions for the densest known two-dimensional packings of superdisks whose shapes are defined by |x{1}|{2p}+|x{2}|{2p}<or=1 and thus contain a large family of both convex (p>or=0.5) and concave (0<p<0.5) particles. Our candidate maximal packing arrangements are achieved by certain families of Bravais lattice packings, and the maximal density is nonanalytic at the "circular-disk" point (p=1) and increases dramatically as p moves away from unity. Moreover, we show that the broken rotational symmetry of superdisks influences the packing characteristics in a nontrivial way.

Modeling heterogeneous materials via two-point correlation functions. II. Algorithmic details and applications

In the first part of this series of two papers, we proposed a theoretical formalism that enables one to model and categorize heterogeneous materials (media) via two-point correlation functions S(2) and introduced an efficient heterogeneous-medium (re)construction algorithm called the "lattice-point" algorithm. Here we discuss the algorithmic details of the lattice-point procedure and an algorithm modification using surface optimization to further speed up the (re)construction process. The importance of the error tolerance, which indicates to what accuracy the media are (re)constructed, is also emphasized and discussed. We apply the algorithm to generate three-dimensional digitized realizations of a Fontainebleau sandstone and a boron-carbide/aluminum composite from the two-dimensional tomographic images of their slices through the materials. To ascertain whether the information contained in S(2) is sufficient to capture the salient structural features, we compute the two-point cluster functions of the media, which are superior signatures of the microstructure because they incorporate topological connectedness information. We also study the reconstruction of a binary laser-speckle pattern in two dimensions, in which the algorithm fails to reproduce the pattern accurately. We conclude that in general reconstructions using S(2) only work well for heterogeneous materials with single-scale structures. However, two-point information via S(2) is not sufficient to accurately model multiscale random media. Moreover, we construct realizations of hypothetical materials with desired structural characteristics obtained by manipulating their two-point correlation functions.

New duality relations for classical ground states

We derive new duality relations that link the energy of configurations associated with a class of soft pair potentials to the corresponding energy of the dual (Fourier-transformed) potential. We apply them by showing how information about the classical ground states of short-ranged potentials can be used to draw new conclusions about the nature of the ground states of long-ranged potentials and vice versa. They also lead to bounds on the T=0 system energies in density intervals of phase coexistence, the identification of a one-dimensional system that exhibits an infinite number of "phase transitions," and a conjecture regarding the ground states of purely repulsive monotonic potentials.

Modeling heterogeneous materials via two-point correlation functions: basic principles

Heterogeneous materials abound in nature and man-made situations. Examples include porous media, biological materials, and composite materials. Diverse and interesting properties exhibited by these materials result from their complex microstructures, which also make it difficult to model the materials. Yeong and Torquato [Phys. Rev. E 57, 495 (1998)] introduced a stochastic optimization technique that enables one to generate realizations of heterogeneous materials from a prescribed set of correlation functions. In this first part of a series of two papers, we collect the known necessary conditions on the standard two-point correlation function S2(r) and formulate a conjecture. In particular, we argue that given a complete two-point correlation function space, S2(r) of any statistically homogeneous material can be expressed through a map on a selected set of bases of the function space. We provide examples of realizable two-point correlation functions and suggest a set of analytical basis functions. We also discuss an exact mathematical formulation of the (re)construction problem and prove that S2(r) cannot completely specify a two-phase heterogeneous material alone. Moreover, we devise an efficient and isotropy-preserving construction algorithm, namely, the lattice-point algorithm to generate realizations of materials from their two-point correlation functions based on the Yeong-Torquato technique. Subsequent analysis can be performed on the generated images to obtain desired macroscopic properties. These developments are integrated here into a general scheme that enables one to model and categorize heterogeneous materials via two-point correlation functions. We will mainly focus on basic principles in this paper. The algorithmic details and applications of the general scheme are given in the second part of this series of two papers.

Random sequential addition of hard spheres in high Euclidean dimensions

Sphere packings in high dimensions have been the subject of recent theoretical interest. Employing numerical and theoretical methods, we investigate the structural characteristics of random sequential addition (RSA) of congruent spheres in d -dimensional Euclidean space R{d} in the infinite-time or saturation limit for the first six space dimensions (1< or =d < or =6) . Specifically, we determine the saturation density, pair correlation function, cumulative coordination number and the structure factor in each of these dimensions. We find that for 2< or =d <or =6 , the saturation density phi{s} scales with dimension as phi{s}=c{1}/2{d}+c{2}d/2{d} , where c{1}=0.202048 and c{2}=0.973872 . We also show analytically that the same density scaling is expected to persist in the high-dimensional limit, albeit with different coefficients. A byproduct of this high-dimensional analysis is a relatively sharp lower bound on the saturation density for any d given by phi{s}> or =(d+2)(1-S{0})2;{d+1} , where S{0}[0,1] is the structure factor at k=0 (i.e., infinite-wavelength number variance) in the high-dimensional limit. We demonstrate that a Palàsti-type conjecture (the saturation density in R{d} is equal to that of the one-dimensional problem raised to the d th power) cannot be true for RSA hyperspheres. We show that the structure factor S(k) must be analytic at k=0 and that RSA packings for 1< or =d< or =6 are nearly "hyperuniform." Consistent with the recent "decorrelation principle," we find that pair correlations markedly diminish as the space dimension increases up to six. We also obtain kissing (contact) number statistics for saturated RSA configurations on the surface of a d -dimensional sphere for dimensions 2< or =d< or =5 and compare to the maximal kissing numbers in these dimensions. We determine the structure factor exactly for the related "ghost" RSA packing in R{d} and demonstrate that its distance from "hyperuniformity" increases as the space dimension increases, approaching a constant asymptotic value of 12 . Our work has implications for the possible existence of disordered classical ground states for some continuous potentials in sufficiently high dimensions.

Exactly solvable disordered sphere-packing model in arbitrary-dimensional Euclidean spaces

We introduce a generalization of the well-known random sequential addition (RSA) process for hard spheres in d-dimensional Euclidean space Rd. We show that all of the n-particle correlation functions of this nonequilibrium model, in a certain limit called the "ghost" RSA packing, can be obtained analytically for all allowable densities and in any dimension. This represents the first exactly solvable disordered sphere-packing model in an arbitrary dimension. The fact that the maximal density phi (infinity)=1/2d of the ghost RSA packing implies that there may be disordered sphere packings in sufficiently high d whose density exceeds Minkowski's lower bound for Bravais lattices, the dominant asymptotic term of which is 1/2d. Indeed, we report on a conjectural lower bound on the density whose asymptotic behavior is controlled by 2-(0.778,65...)d , thus providing the putative exponential improvement on Minkowski's 100-year-old bound. Our results suggest that the densest packings in sufficiently high dimensions may be disordered rather than periodic, implying the existence of disordered classical ground states for some continuous potentials.

Lattice-based random jammed configurations for hard particles

A nontrivial subset of the jammed packings for rigid disks and spheres are those that can be obtained by sequential removal of particles from periodic crystalline arrays. This paper considers the enumeration problems presented by such packings that are based on the close-packed triangular disk lattice, and the face-centered and body-centered cubic sphere lattices. Three distinct categories of packings have been distinguished, depending on their behavior with respect to nonoverlap geometric constraints and/or externally imposed virtual displacements: locally jammed, collectively jammed, and strictly jammed. Each of these possesses an upper limiting vacancy concentration beyond which no packings of the types considered can exist. For each of the three lattices, specific vacancy clusters have been identified whose presence would destroy local jamming, and some of the corresponding patterns that would destroy collective jamming in the triangular disk lattice have also been found. Within the allowable range of vacancy concentration for each case, the number of distinct jammed packings is expected to rise exponentially with system size. By using the concept of local attrition factors, approximate enumerations have been constructed for the three lattice classes of locally jammed packings. In the interests of later extension of this work, we stress that at least some aspects of these enumeration problems might benefit from the formal transcription to a lattice-gas/Ising-model representation with vacancy interactions chosen to enforce the packing category of interest.

Supercooled liquids and the glass transition

Glasses are disordered materials that lack the periodicity of crystals but behave mechanically like solids. The most common way of making a glass is by cooling a viscous liquid fast enough to avoid crystallization. Although this route to the vitreous state-supercooling-has been known for millennia, the molecular processes by which liquids acquire amorphous rigidity upon cooling are not fully understood. Here we discuss current theoretical knowledge of the manner in which intermolecular forces give rise to complex behaviour in supercooled liquids and glasses. An intriguing aspect of this behaviour is the apparent connection between dynamics and thermodynamics. The multidimensional potential energy surface as a function of particle coordinates (the energy landscape) offers a convenient viewpoint for the analysis and interpretation of supercooling and glass-formation phenomena. That much of this analysis is at present largely qualitative reflects the fact that precise computations of how viscous liquids sample their landscape have become possible only recently.

Atomistic simulation of aging and rejuvenation in glasses

Slow structural relaxation ("aging") observed in many atomic, molecular, and polymeric glasses substantially alters their stress-strain relations and can produce a distinctive yield point. Using Monte Carlo simulation for a binary Lennard-Jones mixture, we have observed these phenomena and their cooling-rate dependences for the first time in an atomistic model system. We also observe that aging effects can be reversed by plastic deformation ("rejuvenation"), whereby the system is expelled from the vicinity of deep minima in its potential energy surface.

Diversity in liquid supercooling and glass formation phenomena illustrated by a simple model

The opportunity to map condensed-phase inherent structures (potential energy minima) approximately onto the vertices of a high-dimensional hypercube provides simple conceptual and numerical modeling for first-order melting-freezing transitions, as well as for liquid supercooling and glass formation phenomena. That approach is illustrated here by examination of three interaction examples that were selected to demonstrate the diversity of thermodynamic behavior possible within this hypercube modeling technique. Two of the cases behave, respectively, as "strong" and "fragile" glass formers, at least as judged by their heat capacities. The third presents a "degenerate glass," wherein full equilibration of the supercooling liquid (i.e., no kinetic arrest) leads to (a) residual entropy in the limit of absolute zero temperature, and (b) a linear temperature dependence of heat capacity in the same limit. None of the three cases displays a positive-temperature ideal (intrinsic) glass transition.

A Topographic View of Supercooled Liquids and Glass Formation

Various static and dynamic phenomena displayed by glass-forming liquids, particularly those near the so-called "fragile" limit, emerge as manifestations of the multidimensional complex topography of the collective potential energy function. These include non-Arrhenius viscosity and relaxation times, bifurcation between the alpha- and beta-relaxation processes, and a breakdown of the Stokes-Einstein relation for self-diffusion. This multidimensional viewpoint also produces an extension of the venerable Lindemann melting criterion and provides a critical evaluation of the popular "ideal glass state" concept.

Poly(L-alanine) as a universal reference material for understanding protein energies and structures

We present a proposition, the "poly(L-alanine) hypothesis," which asserts that the native backbone geometry for any polypeptide or protein of M residues has a closely mimicking, mechanically stable, image in poly(L-alanine) of the same number of residues. Using a molecular mechanics force field to represent the relevant potential energy hypersurfaces, we have carried out calculations over a wide range of M values to show that poly(L-alanine) possesses the structural versatility necessary to satisfy the proposition. These include poly(L-alanine) representatives of minima corresponding to secondary and supersecondary structures, as well as poly(L-alanine) images for tertiary structures of the naturally occurring proteins bovine pancreatic trypsin inhibitor, crambin, ribonuclease A, and superoxide dismutase. The successful validation of the hypothesis presented in this paper indicates that poly(L-alanine) will serve as a good reference material in thermodynamic perturbation theory and calculations aimed at evaluating relative free energies for competing candidate tertiary structures in real polypeptides and proteins.

A strategy for finding classes of minima on a hypersurface: implications for approaches to the protein folding problem

Locating the native structure of a given protein is a task made difficult by the complexity of the potential energy hypersurface and by the huge number of local minima it contains. We have explored a strategy (the "antlion" method) for hypersurface modification that suppresses all minima but that of the native structure. Transferrable penalty functions with general applicability for modifying a hypersurface to retain the desired minimum are identified, and two blocked oligopeptides (alanine dipeptide and tetrapeptide) are used for specific numerical illustration of the dramatic simplification that ensues. In addition, an intermediary role for neural networks to manage some aspects of the antlion strategy applied to large polypeptides and proteins is introduced.