Figure 1

(Color online) Schematic phase diagram for hard rods highlighting the rich behavior and the new (with respect to hard spheres) liquid crystal phases exhibited at intermediate temperatures.Reuse & Permissions

Figure 2

Sketch of the zero-temperature phase diagram of a tube in the continuum, subject to a self-attraction promoting compaction. There are two phases when the tube length $L$ is long compared to the range of attractive interaction $R$. One obtains the semicrystalline phase (with parallel/antiparallel alignment between different stretches of the tube which then fill the space with hexagonal symmetry, as depicted in the figure) when the tube thickness $\Delta$ is small compared to $R$ and a swollen phase when $\Delta$ is large compared to $R$. There are interesting finite size effects in the semicrystalline phase. In the thin tube limit, on decreasing the length there is a crossover from the semicrystalline phase with overall cylindrical symmetry to a featureless compact phase with spherical symmetry when $L∕R\sim {(\Delta ∕R)}^{-2}$. There is an unusual finite size effect when $\Delta \sim R$ near the confluence of three phases (at $L=2\pi R$, $\Delta =R$ for a chain in the continuum): the semicrystalline phase, the featureless compact phase, and the swollen phase. A marginally compact phase is obtained in this regime and displays a dramatic entropy reduction, with the choice structure being a helix with a well-defined pitch to radius ratio (see Fig. 4). Other structures such as hairpins and sheets are present in the marginally compact phase for discrete chains (see Fig. 3) [22].Reuse & Permissions

Figure 3

(Color online) Building blocks of biomolecules and ground state structures associated with the marginally compact phase of a short tube corresponding to a discrete chain of tethered disks of radius $\Delta$. The axis in the middle indicates the direction along which the tube thickness $\Delta$ increases. The top row shows some of the building blocks of biomolecules, while the bottom row depicts the corresponding structures obtained as the ground state conformations of a short tube. (A1) is an $\alpha$ helix of a naturally occurring protein, while (A2) and (A3) are the helices obtained in our calculations—(A2) has a regular contact map (i.e., a matrix whose elements, corresponding to residue pairs, are either 0 or 1 depending on whether the two given residues are in contact or not) whereas (A3) is a distorted helix in which the distance between successive atoms along the helical axis is not constant but has period $2$. (B1) is a helix of strands in the alkaline protease of Pseudomonas aeruginosa, whereas (B2) shows the corresponding structure obtained in our computer simulations. (C1) shows the “kissing” hairpins of RNA and (C2) the corresponding conformation obtained in our simulations. Finally (D1) and (D2) are two instances of quasiplanar hairpins. The first structure is from the same protein as before (the alkaline protease of Pseudomonas aeruginosa) while the second is a typical conformation found in our simulations. The sheetlike structure (D3) is obtained for a longer tube (see [21] for more details). The biomolecular structures in the top row are shown in the $C^{\alpha }$ representation for proteins, and in the $P$ representation for RNA kissing hairpins.Reuse & Permissions

Figure 4

(Color online) (a) Space filling optimal helix, with a pitch to radius ratio $c^{*}\approx 2.512$ (drawn using MATHEMATICA). As explained in Appendix , this optimal value is determined by requiring that the radius of curvature of the helical curve is equal to half the minimum distance of closest approach between different turns of the helix. The corresponding tube (that can be thought of as being inflated uniformly around the curve) is optimally space filling since it stops growing when reaching its maximum thickness both locally (the radius of curvature) and nonlocally (half the minimum distance of closest approach between different turns) at the same time (see Appendixes ). Such an optimality criterion is shared by some of the conformations selected as ground states in our simulations in the marginally compact phase such as helices or planar hairpin and sheets shown in Fig. 3, when it is properly translated for the case of a discrete chain [see below and Eq. (A5) in Appendix ]. It can be shown [21] that the planarity of hairpins and sheets is a consequence of this optimal space filling criterion. The same geometrical feature is strikingly found to hold, within 3%, for $\alpha$ helices occurring in the native state of natural proteins [38]. (b) Plot of the ratio $f_i={\rho }_{NL}\left(i\right)∕{\rho }_L\left(i\right)$ of the nonlocal radius of curvature ${\rho }_{NL}\left(i\right)\equiv {\min }_{j<k}r({\mathbf{r}}_i,{\mathbf{r}}_j,{\mathbf{r}}_k)$ (with $\left\{j,k\right\}\ne \left\{i-2,i-1\right\},\left\{i-1,i+1\right\},\left\{i+1,i+2\right\}$) over the radius of curvature ${\rho }_L\left(i\right)\equiv r({\mathbf{r}}_{i-1},{\mathbf{r}}_i,{\mathbf{r}}_{i+1})$ as a function of the residue index $i$ for the native state structure of sperm whale myoglobin (Protein Data Bank code 1mbn), where ${\mathbf{r}}_i$ refers to the spatial coordinates of the $C^{\alpha }$ atom of the $i$th residue, $1⩽i⩽153$ [see Appendix for the definition of the triplet radius $r({\mathbf{r}}_i,{\mathbf{r}}_j,{\mathbf{r}}_k)$]. In correspondence with the eight $\alpha$ helices present in the myoglobin fold, shown as the solid (red) parts in the plot, the values of $f_i$ oscillate around unity, demonstrating that helices in natural proteins are optimally space filling in the sense described above.Reuse & Permissions

Figure 5

Histogram of local thicknesses computed for all residues of different protein native structures, when the virtual chain formed by the backbone $C^{\alpha }$ atoms is viewed as a discretized thick tube. At a given residue the local thickness is simply the minimum triplet radius over all triplets containing that residue (see Appendix for the definition of triplet radius and for an explanation of how such a quantity arises within a tube description).Reuse & Permissions

Figure 6

Sketch of the local coordinate system. For each $C^{\alpha }$ atom $i$ (except the first and the last ones), the axes of a right-handed local coordinate system are defined as follows. The tangent vector ${\widehat{t}}_i$ is parallel to the segment joining $i-1$ with $i+1$. The normal vector ${\widehat{n}}_i$ joins $i$ to the center of the circle passing through $i-1$, $i$, and $i+1$ and it is perpendicular to ${\widehat{t}}_i$. ${\widehat{t}}_i$ and ${\widehat{n}}_i$ along with the three contiguous $C^{\alpha }$ atoms lie in a plane shown in the figure. The binormal vector ${\widehat{b}}_i$ is perpendicular to this plane. The vectors ${\widehat{t}}_i$, ${\widehat{n}}_i$, ${\widehat{b}}_i$ are normalized to unit length.Reuse & Permissions

Figure 7

(Color online) Phase diagram of ground state conformations. The ground state conformations were obtained by means of Monte Carlo simulations of chains of $24C^{\alpha }$ atoms. $e_R$ and $e_W$ denote the local radius of curvature energy penalty and the solvent mediated interaction energy, respectively (see Appendix ). Over 600 distinct local minima were obtained in our simulations in different parts of parameter space starting from a randomly generated initial conformation. The temperature is set initially at a high value and then decreased gradually to zero. (a), (b), (c), (e), (f), (g), and (h) are the Molscript representations of the ground state conformations which are found in different parts of the parameter space as indicated by the arrows. The helices and strands are assigned when local or nonlocal hydrogen bonds are formed according to the rules described in Appendix . Conformations (i), (j), (k), (l), and (m) are competitive local minima. In the shaded phase (orange line) on the top right that does not correspond to any conformation, the ground state is a two-stranded $\beta$ hairpin (not shown). Two distinct topologies of a three-stranded $\beta$ sheet are found corresponding to conformations shown in (b) and (c), respectively. The region (white online) in the left of the phase diagram has large attractive values of $e_W$ and the ground state conformations are compact globular structures with a crystalline order induced by hard sphere packing considerations [54] and not by hydrogen bonding [conformation (d)].Reuse & Permissions

Figure 8

Simplified one-dimensional sketches of energy landscape. The quantity plotted on the horizontal axis schematically represents a distance between different conformations in the phase space and the barriers in the plots indicate the energy needed by the chain in order to travel between two neighboring local minima. (a) Rugged energy landscape for a homopolymer chain with an attractive potential promoting compaction as, e.g., in a string and bead model. There are many distinct maximally compact ground state conformations with roughly the same energy, separated by high energy barriers (the degeneracy of ground state energies would be exact in the case of both lattice models and off-lattice models with discontinuous square-well potentials). (b) Presculpted energy landscape for a homopolymer chain in the marginally compact phase. The number of minima is greatly reduced and the width of their basin increased by the introduction of geometrical constraints. (c) Funnel energy landscape for a protein sequence. As folding proceeds from the top to the bottom of the funnel, its width, a measure of the entropy of the chain, decreases cooperatively with the energy gain. Such a distinctive feature, crucial for fast and reproducible folding, arises from careful sequence design in models whose homopolymer energy landscape is similar to (a). In contrast, funnel-like properties already result from considerations of geometry and symmetry in the marginally compact phase (b), thereby making the goals of the design procedure the relatively easy task of stabilization of one of the presculpted funnels followed by the more refined task of fine-tuning the putative interactions of the protein with other proteins and ligands.Reuse & Permissions

Figure 9

(Color online) Aggregated structures formed by three chains of length 12. We show the lowest energy conformations obtained in long simulations for three 12-residue chains confined within a cubic box of side $L=80\mathrm{\AA }$ at $T=0.19$ (a), $T=0.18$ (b), and $T=0.16$ (c). The conformations shown in $\left(a^{\prime }\right)$, $\left(b^{\prime }\right)$, and $\left(c^{\prime }\right)$ are the same as in (a), (b), and (c), respectively, but viewed from a different angle. The parameters used in the model are $e_W=-0.08$ and $e_R=0.2$ which correspond to having a single helix ground state in the case of a single chain. The simulations start with random extended conformations for all chains and are carried out with pivot and crank-shaft moves that are accepted or rejected based on the METROPOLIS criterion. Moves that bring the residues out of the box are not allowed. The bundle of three helices (d) is a putative ground state of the system and was obtained in a simulation at a very low temperature $\left(T=0.05\right)$ starting with isolated single helices. This conformation has the lowest energy among those shown but is not the equilibrium conformation at intermediate temperatures. Indeed, a simulation run at $T=0.18$ starting with conformation (d) leads to the helix bundle being converted into the $\beta$-helix-like conformation shown in (b) which is the dominant equilibrium conformation at this temperature. (e) The aspecific heat as function of temperature for the system of three 12-residue peptides. The data shown were obtained using the weighted histogram technique [71] based on long equilibrium simulations at various temperatures between 0.16 and 4. The small shoulder (I) corresponds to a condensation of separated peptides into a disordered globule. The large peak (II) corresponds to a transition from disordered globule to the $\beta$-helix-like phase. (f) The energy as a function of time (in Monte Carlo steps) during a long simulation at a temperature corresponding to the maximum of the specific heat, $T=0.195$. The simulation shows several transitions between the disordered globule phase and the $\beta$-helix-like phase.Reuse & Permissions

Figure 10

(Color online) Aggregated structures formed by five and ten chains of length 12 with $e_W=-0.08$, $e_R=0.2$. We show the lowest energy conformations obtained in long simulations for five chains at $T=0.18$ (a), and for ten chains at $T=0.2$ (b). The five-chain system is confined within a cubic box of side $L=80\mathrm{\AA }$ whereas the ten-chain system is confined within a cubic box of side $L=100\mathrm{\AA }$. The conformations shown in $\left(a^{\prime }\right)$ and $\left(b^{\prime }\right)$ are the same as those in (a) and (b) but viewed from a different angle.Reuse & Permissions

Figure 11

(Color online) Contour plots of the effective free energy (a) at high temperature $\left(T=0.22\right)$ and (b) at the folding transition temperature $T_f=0.2$ for a single 24-residue homopolymeric chain, with $e_W=-0.08$, $e_R=0.3$. The effective free energy, defined as $F(N_l+N_{nl},N_W)=-\ln P(N_l+N_{nl},N_W)$, is obtained as a function of the total number of hydrogen bonds $N_l+N_{nl}$ and the total number of hydrophobic contacts $N_W$ from the histogram $P(N_l+N_{nl},N_W)$ collected in equilibrium Monte Carlo simulations at constant temperature. The spacing between consecutive levels in each contour plot is $1$ and corresponds to a free energy difference of $k_B\tilde{T}$, where $\tilde{T}$ is the temperature in physical units. The darker the color, the lower the free energy value. There is just one free energy minimum corresponding to the denatured state at a temperature higher than the folding transition temperature (a) whereas one can discern the existence of three distinct minima at the folding transition temperature (b). Typical conformations from each of the minima are shown in the figure.Reuse & Permissions

Figure 12

(Color online) Pauling and Ramachandran revisited: The top row depicts the “classic” structures of an $\alpha$ helix (a) and a pleated $\beta$ sheet (b). The main-chain backbone atoms and the $C^{\beta }$ atoms of the side chain groups are shown [color codes are different for (a) and (b)]. Hydrogen bonds, which stabilize the structures, are shown as dashed lines. In the bottom row we show the Ramachandran plot (c) describing how the torsional degrees of freedom $\left(\psi ,\varphi \right)$, the backbone dihedral angles within an all atom representation, are constrained by steric effects. The shaded (colored online) areas in the plot correspond to allowed regions in conformational space. The structures (a) and (b) stabilized by hydrogen bonding indeed lie squarely within the sterically accessible regions. An example of a dipeptide conformation disallowed because of steric hindrance is shown in (d).Reuse & Permissions

Figure 13

Schematic phase diagram of an EM in the temperature $\left(T\right)$ bending rigidity $\left({\kappa }_b\right)$ plane [117]. The different phases are $\mathrm{S}=$ swollen, $\mathrm{G}=$ globule, and $\mathrm{AG=}$ asymmetric globule.Reuse & Permissions

Figure 14

Phase diagram for a thick polymer chain in the temperature-thickness $(T,\Delta )$ plane obtained with Monte Carlo simulations (see [22] for details). The different phases are $\mathrm{S}=$ swollen, $\mathrm{G}=$ globule, and $\mathrm{AG=}$ asymmetric globule.Reuse & Permissions

Figure 15

(Color online) Statistical analysis of several quantities computed for residues classified as participating in secondary structures in protein native state structures from the Protein Data Bank. Light (dark) gray [red (blue) online] histograms refer to residues participating in $\alpha$ helices ($\beta$ strand).Reuse & Permissions