Four ways of looking at secondary structure: a) Molecular configurations; b) Ramachandran plot; c) Histogram (-code) of Ramachandran numbers; and (d) as a function of residue number (for the Σ-sheet we have chosen a single polymer).

<p>Panel (c) provides a compact assay-by-geometry of the residues within molecular structures, while panel (d) shows that one can use to identify the spatial connectivity of domains of secondary structure within a polymer.</p

Dihedral angles converted to Ramachandran numbers can be recovered only approximately, but the error incurred during this back-mapping can be made much smaller than the standard error (typically 1Å) associated with structures in the protein databank.

<p>Here we show the root-mean-squared-deviation (RMSD) in dihedral angles (a) and in protein <i>α</i>-carbon spatial coordinates (b) generated upon taking 8560 protein structures obtained from SCOP [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0160023#pone.0160023.ref036" target="_blank">36</a>], converting their dihedral angles to Ramachandran numbers, and recovering approximately those dihedral angles using Eqs (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0160023#pone.0160023.e022" target="_blank">7</a>) and (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0160023#pone.0160023.e023" target="_blank">8</a>). The parameter <i>σ</i> indicates the grid resolution used to calculate <i>R</i>; see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0160023#pone.0160023.e009" target="_blank">Eq (1)</a>.</p

The Ramachandran Number: An Order Parameter for Protein Geometry - Fig 6

<p>(a) -codes for the SCOP protein dataset reveal at a glance several geometric properties of the set. Each column represents a histogram of the indicated protein class, normalized so that the largest value is unity. A feature common to all classes is the prominence of <i>α</i>-helices ( ≈0.36). Another common feature is the presence of loops that connect ordered secondary structure ( ≈0.62). Moreover, <i>α</i>-helical regions are prominently visible in ‘all-<i>β</i>’ proteins. (b) The -code for a peptoid nanosheet shows two dominant rotational states, which coexist within a single secondary structure (see Figs <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0160023#pone.0160023.g005" target="_blank">5</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0160023#pone.0160023.g007" target="_blank">7</a>).</p

Potential pathologies of are avoided by the sparse occupancy of the Ramachandran plot.

<p>(a) We construct by slicing across the Ramachandran plot, which can cause points distant in dihedral angle space to be grouped together, the more so as we approach the negative-sloping diagonal (near = 0.5). This grouping can be inferred by superposing the standard deviation (error bars) in polymer end-to-end distance on top of the mean value (smooth line) for hypothetical structures built from the relevant part of the Ramachandran diagram. (b) However, many structures distant in dihedral angle space but close in do not arise in proteins; the Ramachandran diagram is in general relatively sparsely occupied. Consequently, can resolve the major types of protein secondary structure, which can be inferred from the fact that lines parallel to the negative-sloping diagonal (marked), along which varies only slowly, can touch each region of known secondary structure (colored) individually. This sensitivity allows to function as an order parameter for protein geometry. [Data in (a) were calculated for a 5-residue peptoid; values are shown at discrete intervals of 0.01.].</p

Physical trends within the Ramachandran plot suggest a way of describing regions of it with a single number.

<p>(a) First, the sense of residue twist changes from right-handed (‘D’) to left-handed (‘L’) as one moves from the bottom left of the Ramachandran plot to the top right. Second, contours (colored) of end-to-end polymer distance <i>R</i>_e (here calculated for a 20-residue glycine) have a negative slope, resulting in the general trend shown in panel (b). Panel (c) indicates one method of indexing the Ramachandran plot so as to move from the region of right-handed twist to the region of left-handed twist with <i>R</i>_e changing as slowly as possible. This method provides the basis for the construction of the Ramachandran number, .</p

The indexing system defined by Eqs (6) and (1) collapses the Ramachandran plot into a single line, the Ramachandran number .

<p>This number can act as an order parameter to distinguish secondary structures of different geometry, as shown (the overlap between distributions exists in the original Ramachandran plot representation; see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0160023#pone.0160023.g003" target="_blank">Fig 3(b)</a>). Top: interpolates between regions of right-handed and left-handed twist, with polymer extension <i>R</i>_e varying smoothly throughout.</p

The Ramachandran plot is an important way of describing protein secondary structure.

<p>(a) The state of a residue within a peptide (top) and a peptoid (bottom) can be largely specified by the two dihedral angles <i>ϕ</i> and <i>ψ</i>. (b) Regular protein secondary structures, such as <i>α</i>-helices and <i>β</i>-sheets, correspond to single diffuse regions on a plot drawn in terms of <i>ϕ</i> and <i>ψ</i>, called a Ramachandran plot (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0160023#sec006" target="_blank">Methods</a>). (c) Peptoid Σ-sheets [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0160023#pone.0160023.ref016" target="_blank">16</a>] harbor a secondary-structure motif in which backbone residues alternate between two regions on the Ramachandran plot. In order to describe each region in terms of a single number, so that the state of each residue in a backbone can be compactly indicated, we describe in this paper the development and properties of a structurally meaningful combination of <i>ϕ</i> and <i>ψ</i> that we call the Ramachandran number, . [Panel (a) was adapted from an image found on Wikimedia Commons (<a href="https://commons.wikimedia.org/wiki/File%3AProtein_backbone_PhiPsiOmega_drawing.jpg" target="_blank">link</a>) by Dcrjsr (CC BY 3.0 (<a href="http://creativecommons.org/licenses/by/3.0" target="_blank">link</a>)). The contours in (b) and (c) represent regions within which 70% of a secondary structure resides; see Section 4.1.].</p

Implicit-Solvent Coarse-Grained Simulation with a Fluctuating Interface Reveals a Molecular Mechanism for Peptoid Monolayer Buckling

Peptoid polymers form extended two-dimensional nanostructures via an interface-mediated assembly process: the amphiphilic peptoids first adsorb to an air–water interface as a monolayer, then buckle and collapse into free-floating bilayer nanosheets when the interface is compressed. Here, we investigate the molecular mechanism of monolayer buckling by developing a method for incorporating interface fluctuations into an implicit-solvent coarse-grained model. Representing the interface with a triangular mesh controlled by surface tension and surfactant adsorption, we predict the direction of buckling for peptoids with a segregated arrangement of charged side chains and predict that peptoids with with an alternating charge pattern should buckle less easily than peptoids with a segregated charge pattern

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

Ion-Specific Control of the Self-Assembly Dynamics of a Nanostructured Protein Lattice

Self-assembling proteins offer a potential means of creating nanostructures with complex structure and function. However, using self-assembly to create nanostructures with long-range order whose size is tunable is challenging, because the kinetics and thermodynamics of protein interactions depend sensitively on solution conditions. Here we systematically investigate the impact of varying solution conditions on the self-assembly of SbpA, a surface-layer protein from <i>Lysinibacillus sphaericus</i> that forms two-dimensional nanosheets. Using high-throughput light scattering measurements, we mapped out diagrams that reveal the relative yield of self-assembly of nanosheets over a wide range of concentrations of SbpA and Ca²⁺. These diagrams revealed a localized region of optimum yield of nanosheets at intermediate Ca²⁺ concentration. Replacement of Mg²⁺ or Ba²⁺ for Ca²⁺ indicates that Ca²⁺ acts both as a specific ion that is required to induce self-assembly and as a general divalent cation. In addition, we use competitive titration experiments to find that 5 Ca²⁺ bind to SbpA with an affinity of 67.1 ± 0.3 μM. Finally, we show <i>via</i> modeling that nanosheet assembly occurs by growth from a negligibly small critical nucleus. We also chart the dynamics of nanosheet size over a variety of conditions. Our results demonstrate control of the dynamics and size of the self-assembly of a nanostructured lattice, the constituents of which are one of a class of building blocks able to form novel hybrid nanomaterials