An exhaustive survey of regular peptide conformations using a new metric for backbone handedness (h).

The Ramachandran plot is important to structural biology as it describes a peptide backbone in the context of its dominant degrees of freedom-the backbone dihedral angles φ and ψ (Ramachandran, Ramakrishnan & Sasisekharan, 1963). Since its introduction, the Ramachandran plot has been a crucial tool to characterize protein backbone features. However, the conformation or twist of a backbone as a function of φ and ψ has not been completely described for both cis and trans backbones. Additionally, little intuitive understanding is available about a peptide's conformation simply from knowing the φ and ψ values of a peptide (e.g., is the regular peptide defined by φ = ψ =  - 100°  left-handed or right-handed?). This report provides a new metric for backbone handedness (h) based on interpreting a peptide backbone as a helix with axial displacement d and angular displacement θ, both of which are derived from a peptide backbone's internal coordinates, especially dihedral angles φ, ψ and ω. In particular, h equals sin(θ)d∕|d|, with range [-1, 1] and negative (or positive) values indicating left(or right)-handedness. The metric h is used to characterize the handedness of every region of the Ramachandran plot for both cis (ω = 0°) and trans (ω = 180°) backbones, which provides the first exhaustive survey of twist handedness in Ramachandran (φ, ψ) space. These maps fill in the 'dead space' within the Ramachandran plot, which are regions that are not commonly accessed by structured proteins, but which may be accessible to intrinsically disordered proteins, short peptide fragments, and protein mimics such as peptoids. Finally, building on the work of (Zacharias & Knapp, 2013), this report presents a new plot based on d and θ that serves as a universal and intuitive alternative to the Ramachandran plot. The universality arises from the fact that the co-inhabitants of such a plot include every possible peptide backbone including cis and trans backbones. The intuitiveness arises from the fact that d and θ provide, at a glance, numerous aspects of the backbone including compactness, handedness, and planarity.

Structure based sequence analysis of viral and cellular protein assemblies.

It is well accepted that, in general, protein structural similarity is strongly related to the amino acid sequence identity. To analyze in great detail the correlation, distribution and variation levels of conserved residues in the protein structure, we analyzed all available high-resolution structural data of 5245 cellular complex-forming proteins and 293 spherical virus capsid proteins (VCPs). We categorized and compare them in terms of protein structural regions. In all cases, the buried core residues are the most conserved, followed by the residues at the protein-protein interfaces. The solvent-exposed surface shows greater sequence variations. Our results provide evidence that cellular monomers and VCPs could be two extremes in the quaternary structural space, with cellular dimers and oligomers in between. Moreover, based on statistical analysis, we detected a distinct group of icosahedral virus families whose capsid proteins seem to evolve much slower than the rest of the protein complexes analyzed in this work.

The Ramachandran Number: An Order Parameter for Protein Geometry.

Three-dimensional protein structures usually contain regions of local order, called secondary structure, such as α-helices and ß-sheets. Secondary structure is characterized by the local rotational state of the protein backbone, quantified by two dihedral angles called ϕ and ψ. Particular types of secondary structure can generally be described by a single (diffuse) location on a two-dimensional plot drawn in the space of the angles ϕ and ψ, called a Ramachandran plot. By contrast, a recently-discovered nanomaterial made from peptoids, structural isomers of peptides, displays a secondary-structure motif corresponding to two regions on the Ramachandran plot [Mannige et al., Nature 526, 415 (2015)]. In order to describe such 'higher-order' secondary structure in a compact way we introduce here a means of describing regions on the Ramachandran plot in terms of a single Ramachandran number, [Formula: see text], which is a structurally meaningful combination of ϕ and ψ. We show that the potential applications of [Formula: see text] are numerous: it can be used to describe the geometric content of protein structures, and can be used to draw diagrams that reveal, at a glance, the frequency of occurrence of regular secondary structures and disordered regions in large protein datasets. We propose that [Formula: see text] might be used as an order parameter for protein geometry for a wide range of applications.

Predicting the outcome of the growth of binary solids far from equilibrium.

The growth of multicomponent structures in simulations and experiments often results in kinetically trapped, nonequilibrium objects. In such cases we have no general theoretical framework for predicting the outcome of the growth process. Here we use computer simulations to study the growth of two-component structures within a simple lattice model. We show that kinetic trapping happens for many choices of growth rate and intercomponent interaction energies, and that qualitatively distinct kinds of kinetic trapping are found in different regions of parameter space. In a region in which the low-energy structure is an "antiferromagnet" or "checkerboard," we show that the grown nonequilibrium structure displays a component-type stoichiometry that is different from the equilibrium one but is insensitive to growth rate and solution conditions. This robust nonequilibrium stoichiometry can be predicted via a mapping to the jammed random tiling of dimers studied by Flory, a finding that suggests a way of making defined nonequilibrium structures in experiment.

Design, Synthesis, Assembly, and Engineering of Peptoid Nanosheets.

Two-dimensional (2D) atomically defined organic nanomaterials are an important material class with broad applications. However, few general synthetic methods exist to produce such materials in high yields and to precisely functionalize them. One strategy to form ordered 2D organic nanomaterials is through the supramolecular assembly of sequence-defined synthetic polymers. Peptoids, one such class of polymer, are designable bioinspired heteropolymers whose main-chain length and monomer sequence can be precisely controlled. We have recently discovered that individual peptoid polymers with a simple sequence of alternating hydrophobic and ionic monomers can self-assemble into highly ordered, free-floating nanosheets. A detailed understanding of their molecular structure and supramolecular assembly dynamics provides a robust platform for the discovery of new classes of nanosheets with tunable properties and novel applications. In this Account, we discuss the discovery, characterization, assembly, molecular modeling, and functionalization of peptoid nanosheets. The fundamental properties of peptoid nanosheets, their mechanism of formation, and their application as robust scaffolds for molecular recognition and as templates for the growth of inorganic minerals have been probed by an arsenal of experimental characterization techniques (e.g., scanning probe, electron, and optical microscopy, X-ray diffraction, surface-selective vibrational spectroscopy, and surface tensiometry) and computational techniques (coarse-grained and atomistic modeling). Peptoid nanosheets are supramolecular assemblies of 16-42-mer chains that form molecular bilayers. They span tens of microns in lateral dimensions and freely float in water. Their component chains are highly ordered, with chains nearly fully extended and packed parallel to one another as a result of hydrophobic and electrostatic interactions. Nanosheets form via a novel interface-catalyzed monolayer collapse mechanism. Peptoid chains first assemble into a monolayer at either an air-water or oil-water interface, on which peptoid chains extend, order, and pack into a brick-like pattern. Upon mechanical compression of the interface, the monolayer buckles into stable bilayer structures. Recent work has focused on the design of nanosheets with tunable properties and functionality. They are readily engineerable, as functional monomers can be readily incorporated onto the nanosheet surface or into the interior. For example, functional hydrophilic "loops" have been displayed on the surfaces of nanosheets. These loops can interact with specific protein targets, serving as a potentially general platform for molecular recognition. Nanosheets can also bind metal ions and serve as 2D templates for mineral growth. Through our understanding of the formation mechanism, along with predicted features ascertained from molecular modeling, we aim to further design and synthesize nanosheets as robust protein mimetics with the potential for unprecedented functionality and stability.

Landscape of kinetically trapped binary assemblies.

For two-component assemblies, an inherent structure diagram (ISD) is the relationship between set inter-subunit energies and the types of kinetic traps (inherent structures) one may obtain from those energies. It has recently been shown that two-component ISDs are apportioned into regions or plateaux within which inherent structures display uniform features (e.g., stoichometries and morphologies). Interestingly, structures from one of the plateaux were also found to be robust outcomes of one type of non-equilibrium growth, which indicates the usefulness of the two-component ISD in predicting outcomes of some types of far-from-equilibrium growth. However, little is known as to how the ISD is apportioned into distinct plateaux. Also, while each plateau displays classes of structures that are morphologically distinct, little is known about the source of these distinct morphologies. This article outlines an analytic treatment of the two-component ISD and shows that the manner in which any ISD is apportioned arises from a single unitless order parameter. Additionally, the analytical framework allows for the characterization of local properties of the trapped structures within each ISD plateau. This work may prove to be useful in the design of novel classes of robust nonequilibrium assemblies.

Modeling sequence-specific polymers using anisotropic coarse-grained sites allows quantitative comparison with experiment.

Certain sequences of peptoid polymers (synthetic analogs of peptides) assemble into bilayer nanosheets via a nonequilibrium assembly pathway of adsorption, compression, and collapse at an air-water interface. As with other large-scale dynamic processes in biology and materials science, understanding the details of this supramolecular assembly process requires a modeling approach that captures behavior on a wide range of length and time scales, from those on which individual side chains fluctuate to those on which assemblies of polymers evolve. Here, we demonstrate that a new coarse-grained modeling approach is accurate and computationally efficient enough to do so. Our approach uses only a minimal number of coarse-grained sites but retains independently fluctuating orientational degrees of freedom for each site. These orientational degrees of freedom allow us to accurately parametrize both bonded and nonbonded interactions and to generate all-atom configurations with sufficient accuracy to perform atomic scattering calculations and to interface with all-atom simulations. We have used this approach to reproduce all available experimental X-ray scattering data (for stacked nanosheets and for peptoids adsorbed at air-water interfaces and in solution), in order to resolve the microscopic, real-space structures responsible for these Fourier-space features. By interfacing with all-atom simulations, we have also laid the foundation for future multiscale simulations of sequence-specific polymers that communicate in both directions across scales.

Peptoid nanosheets exhibit a new secondary-structure motif.

A promising route to the synthesis of protein-mimetic materials that are capable of complex functions, such as molecular recognition and catalysis, is provided by sequence-defined peptoid polymers--structural relatives of biologically occurring polypeptides. Peptoids, which are relatively non-toxic and resistant to degradation, can fold into defined structures through a combination of sequence-dependent interactions. However, the range of possible structures that are accessible to peptoids and other biological mimetics is unknown, and our ability to design protein-like architectures from these polymer classes is limited. Here we use molecular-dynamics simulations, together with scattering and microscopy data, to determine the atomic-resolution structure of the recently discovered peptoid nanosheet, an ordered supramolecular assembly that extends macroscopically in only two dimensions. Our simulations show that nanosheets are structurally and dynamically heterogeneous, can be formed only from peptoids of certain lengths, and are potentially porous to water and ions. Moreover, their formation is enabled by the peptoids' adoption of a secondary structure that is not seen in the natural world. This structure, a zigzag pattern that we call a Σ('sigma')-strand, results from the ability of adjacent backbone monomers to adopt opposed rotational states, thereby allowing the backbone to remain linear and untwisted. Linear backbones tiled in a brick-like way form an extended two-dimensional nanostructure, the Σ-sheet. The binary rotational-state motif of the Σ-strand is not seen in regular protein structures, which are usually built from one type of rotational state. We also show that the concept of building regular structures from multiple rotational states can be generalized beyond the peptoid nanosheet system.

Heterogeneity of functional groups in a metal-organic framework displays magic number ratios.

Multiple organic functionalities can now be apportioned into nanoscale domains within a metal-coordinated framework, posing the following question: how do we control the resulting combination of "heterogeneity and order"? Here, we report the creation of a metal-organic framework, MOF-2000, whose two component types are incorporated in a 2:1 ratio, even when the ratio of component types in the starting solution is varied by an order of magnitude. Statistical mechanical modeling suggests that this robust 2:1 ratio has a nonequilibrium origin, resulting from kinetic trapping of component types during framework growth. Our simulations show how other "magic number" ratios of components can be obtained by modulating the topology of a framework and the noncovalent interactions between component types, a finding that may aid the rational design of functional multicomponent materials.

Growth of equilibrium structures built from a large number of distinct component types.

We use simple analytic arguments and lattice-based computer simulations to study the growth of structures made from a large number of distinct component types. Components possess 'designed' interactions, chosen to stabilize an equilibrium target structure in which each component type has a defined spatial position, as well as 'undesigned' interactions that allow components to bind in a compositionally-disordered way. We find that high-fidelity growth of the equilibrium target structure can happen in the presence of substantial attractive undesigned interactions, as long as the energy scale of the set of designed interactions is chosen appropriately. This observation may help explain why equilibrium DNA 'brick' structures self-assemble even if undesigned interactions are not suppressed [Ke et al. Science, 338, 1177, (2012)]. We also find that high-fidelity growth of the target structure is most probable when designed interactions are drawn from a distribution that is as narrow as possible. We use this result to suggest how to choose complementary DNA sequences in order to maximize the fidelity of multicomponent self-assembly mediated by DNA. We also comment on the prospect of growing macroscopic structures in this manner.

Dynamic New World: Refining Our View of Protein Structure, Function and Evolution.

Proteins are crucial to the functioning of all lifeforms. Traditional understanding posits that a single protein occupies a single structure ("fold"), which performs a single function. This view is radically challenged with the recognition that high structural dynamism-the capacity to be extra "floppy"-is more prevalent in functional proteins than previously assumed. As reviewed here, this dynamic take on proteins affects our understanding of protein "structure", function, and evolution, and even gives us a glimpse into protein origination. Specifically, this review will discuss historical developments concerning protein structure, and important new relationships between dynamism and aspects of protein sequence, structure, binding modes, binding promiscuity, evolvability, and origination. Along the way, suggestions will be provided for how key parts of textbook definitions-that so far have excluded membership to intrinsically disordered proteins (IDPs)-could be modified to accommodate our more dynamic understanding of proteins.

Origination of the Protein Fold Repertoire from Oily Pluripotent Peptides.

While the repertoire of protein folds that exists today underlies most of life's capabilities, our mechanistic picture of protein fold origination is incomplete. This paper discusses a hypothetical mechanism for the emergence of the protein fold repertoire from highly dynamic and collapsed peptides, exemplified by peptides with high oil content or hydrophobicity. These peptides are called pluripotent to emphasize their capacity to evolve into numerous folds transiently available to them. As evidence, the paper will discuss previous simulation work on the superior fold evolvability of oily peptides, trace ("fossil") evidence within proteomes seen today, and a general relationship between protein dynamism and evolvability. Aside from implications on the origination of protein folds, the hypothesis implies that the vanishing utility of a random peptide in protein origination may be relatively exaggerated, as some random peptides with a certain composition (e.g., oily) may fare better than others. In later sections, the hypothesis is discussed in the context of existing discussions regarding the spontaneous origination of biomolecules.

Development and use of an atomistic CHARMM-based forcefield for peptoid simulation.

Peptoids are positional isomers of peptides: peptoid sidechains are attached to backbone nitrogens rather than α-carbons. Peptoids constitute a class of sequence-specific polymers resistant to biological degradation and potentially as diverse, structurally and functionally, as proteins. While molecular simulation of proteins is commonplace, relatively few tools are available for peptoid simulation. Here, we present a first-generation atomistic forcefield for peptoids. Our forcefield is based on the peptide forcefield CHARMM22, with key parameters tuned to match both experimental data and quantum mechanical calculations for two model peptoids (dimethylacetamide and a sarcosine dipeptoid). We used this forcefield to demonstrate that solvation of a dipeptoid substantially modifies the conformations it can access. We also simulated a crystal structure of a peptoid homotrimer, H-(N-2-phenylethyl glycine)3 -OH, and we show that experimentally observed structural and dynamical features of the crystal are accurately described by our forcefield. The forcefield presented here provides a starting point for future development of peptoid-specific simulation methods within CHARMM.

Two modes of protein sequence evolution and their compositional dependencies.

Protein sequence evolution has resulted in a vast repertoire of molecular functionality crucial to life. Despite the central importance of sequence evolution to biology, our fundamental understanding of how sequence composition affects evolution is incomplete. This report describes the utilization of lattice model simulations of directed evolution, which indicate that, on average, peptide and protein evolvability is strongly dependent on initial sequence composition. The report also discusses two distinct regimes of sequence evolution by point mutation: (a) the "classical" mode where sequences "crawl" over free energy barriers towards acquiring a target fold, and (b) the "quantum" mode where sequences appear to "tunnel" through large energy barriers generally insurmountable by means of a crawl. Finally, the simulations indicate that oily and charged peptides are the most efficient substrates for evolution at the "classical" and "quantum" regimes, respectively, and that their respective response to temperature is commensurate with analogies made to barrier crossing in classical and quantum systems. On the whole, these results show that sequence composition can tune both the evolvability and the optimal mode of evolution of peptides and proteins.

A universal trend among proteomes indicates an oily last common ancestor.

Despite progresses in ancestral protein sequence reconstruction, much needs to be unraveled about the nature of the putative last common ancestral proteome that served as the prototype of all extant lifeforms. Here, we present data that indicate a steady decline (oil escape) in proteome hydrophobicity over species evolvedness (node number) evident in 272 diverse proteomes, which indicates a highly hydrophobic (oily) last common ancestor (LCA). This trend, obtained from simple considerations (free from sequence reconstruction methods), was corroborated by regression studies within homologous and orthologous protein clusters as well as phylogenetic estimates of the ancestral oil content. While indicating an inherent irreversibility in molecular evolution, oil escape also serves as a rare and universal reaction-coordinate for evolution (reinforcing Darwin's principle of Common Descent), and may prove important in matters such as (i) explaining the emergence of intrinsically disordered proteins, (ii) developing composition- and speciation-based "global" molecular clocks, and (iii) improving the statistical methods for ancestral sequence reconstruction.

Periodic table of virus capsids: implications for natural selection and design.

BACKGROUND: For survival, most natural viruses depend upon the existence of spherical capsids: protective shells of various sizes composed of protein subunits. So far, general evolutionary pressures shaping capsid design have remained elusive, even though an understanding of such properties may help in rationally impeding the virus life cycle and designing efficient nano-assemblies. PRINCIPAL FINDINGS: This report uncovers an unprecedented and species-independent evolutionary pressure on virus capsids, based on the the notion that the simplest capsid designs (or those capsids with the lowest "hexamer complexity", C(h)) are the fittest, which was shown to be true for all available virus capsids. The theories result in a physically meaningful periodic table of virus capsids that uncovers strong and overarching evolutionary pressures, while also offering geometric explanations to other capsid properties (rigidity, pleomorphy, auxiliary requirements, etc.) that were previously considered to be unrelatable properties of the individual virus. SIGNIFICANCE: Apart from describing a universal rule for virus capsid evolution, our work (especially the periodic table) provides a language with which highly diverse virus capsids, unified only by geometry, may be described and related to each other. Finally, the available virus structure databases and other published data reiterate the predicted geometry-derived rules, reinforcing the role of geometry in the natural selection and design of virus capsids.

Geometric considerations in virus capsid size specificity, auxiliary requirements, and buckling.

Spherical capsids are shells of protein subunits that protect the genomes of many viral strains. Although nature displays a range of spherical capsid sizes (reflected by the number of subunits in the formation), specific strains display stringent requirements for forming capsids of specific sizes, a requirement that appears crucial to infectivity. Despite its importance in pathogenicity, little is known regarding the determinants of capsid size. Still less is known about exactly which capsids can undergo maturation events such as buckling transitions--postcapsid-assembly events that are crucial to some virus strains. We show that the exclusive determinant of capsid size is hexamer shape, as defined by subunit-subunit dihedral angles. This conclusion arises from considering the dihedral angle patterns within hexamers belonging to natural canonical capsids and geometric capsid models (deltahedra). From simple geometric models and an understanding of endo angle propagation discussed here, we then suggest that buckling transitions may be available only to capsids of certain size (specifically, T < 7 capsids are precluded from such transformations) and that T > 7 capsids require the help of auxiliary mechanisms for proper capsid formation. These predictions, arising from simple geometry and modeling, are backed by a body of empirical evidence, further reinforcing the extent to which the evolution of the atomistically complex virus capsid may be principled around simple geometric design/requirements.

Tilable nature of virus capsids and the role of topological constraints in natural capsid design.

Virus capsids are highly specific assemblies that are formed from a large number of often chemically identical capsid subunits. In the present paper we ask to what extent these structures can be viewed as mathematically tilable objects using a single two-dimensional tile. We find that spherical viruses from a large number of families-eight out of the twelve studied-qualitatively possess properties that allow their representation as two-dimensional monohedral tilings of a bound surface, where each tile represents a subunit. This we did by characterizing the extent to which individual spherical capsids display subunit-subunit (1) holes, (2) overlaps, and (3) gross structural variability. All capsids with T numbers greater than 1 from the Protein Data Bank, with homogeneous protein composition, were used in the study. These monohedral tilings, called canonical capsids due to their platonic (mathematical) form, offer a mathematical segue into the structural and dynamical understanding of not one, but a large number of virus capsids. From our data, it appears as though one may only break the long-standing rules of quasiequivalence by the introduction of subunit-subunit structural variability, holes, and gross overlaps into the shell. To explore the utility of canonical capsids in understanding structural aspects of such assemblies, we used graph theory and discrete geometry to enumerate the types of shapes that the tiles (and hence the subunits) must possess. We show that topology restricts the shape of the face to a limited number of five-sided prototiles, one of which is the "bisected trapezoid" that is a platonic representation of the most ubiquitous capsid subunit shape seen in nature (the trapezoidal jelly-roll motif). This motif is found in a majority of seemingly unrelated virus families that share little to no host, size, or amino acid sequence similarity. This suggests that topological constraints may exhibit dominant roles in the natural design of biological assemblies, while having little effect on amino acid sequence similarity.