Two-dimensional structure in a generic model of triangular proteins and protein trimers

Motivated by the diversity and complexity of two-dimensional (2D) crystals formed by triangular proteins and protein trimers, we have investigated the structures and phase behavior of hard-disk trimers. In order to mimic specific binding interactions, each trimer possesses an “attractive” disk which can interact with similar disks on other trimers via an attractive square-well potential. At low density and low temperature, the fluid phase mainly consists of tetramers, pentamers, or hexamers. Hexamers provide the structural motif for a high-density, low-temperature periodic solid phase, but we also identify a metastable periodic structure based on a tetramer motif. At high density there is a transition between orientationally ordered and disordered solid phases. The connections between simulated structures and those of 2D protein crystals—as seen in electron microscopy—are briefly discussed.

Figure 1

Phase diagram of the model trimer system in the density-temperature $\left({\rho }^{*}\text{−}{T}^{*}\right)$ plane: (solid points and solid lines) approximate fluid-solid phase boundaries, assumed to be first order; (open points and dashed lines) boundaries between high-temperature unclustered states and low-temperature clustered states, as evidenced by maxima in the heat capacity along isobars; (dot-dashed line) close-packed density, ${\rho }_{\mathrm{cp}}^{*}=2∕3\sqrt{3}\simeq 0.3849$.

Figure 2

(Color online) Configuration snapshots from $NpT$ simulations: (a) normal fluid phase (fluid I) at $T^{*}=2$, $p^{*}=2.5$, ${\rho }^{*}=0.259$; (b) orientationally disordered $AB$ solid phase (solid I) at $T^{*}=2$, $p^{*}=12$, ${\rho }^{*}=0.345$; (c) clustered fluid phase (fluid II) at $T^{*}=0.25$, $p^{*}=0.75$, ${\rho }^{*}=0.222$; (d) metastable state at $T^{*}=0.25$, $p^{*}=2.6$, ${\rho }^{*}=0.290$; (e) orientationally ordered $AB$ solid phase (solid II) at $T^{*}=0.25$, $p^{*}=12$, ${\rho }^{*}=0.349$; (f) metastable orientationally ordered $AA$ solid at $T^{*}=0.25$, $p^{*}=20$, ${\rho }^{*}=0.356$. In each case the attractive disks are colored dark gray (red online), the repulsive disks are colored light gray, and all disks are drawn with diameter $1\sigma$.

Figure 3

Cluster distributions for systems along the isotherm $T^{*}=0.3$: (a) $p^{*}=0.5$, ${\rho }^{*}=0.188$; (b) $p^{*}=1$, ${\rho }^{*}=0.230$; (c) $p^{*}=1.5$, ${\rho }^{*}=0.252$; (d) $p^{*}=2$, ${\rho }^{*}=0.265$; (e) $p^{*}=2.5$, ${\rho }^{*}=0.280$; (f) $p^{*}=6$, ${\rho }^{*}=0.329$. In (a)–(e) the system is fluid, whilst in (f) the system is solid (II).

Figure 4

Equations of state along isotherms with $T^{*}=0.3$ (solid symbols, solid lines), and $T^{*}=1$ (open symbols, dashed lines): (circles) state points from $NpT$ simulations of $AB$ solid phase; (squares) state points from $NpT$ simulations of $AA$ solid phase ($T^{*}=1$ only); (diamonds) state points from $NpT$ simulations of $p3$ solid phase shown in Fig. 6 ($T^{*}=1$ only); (crosses) putative metastable state points; (triangles) approximate coexistence densities; (lines) fits to the fluid and solid branches (see text). The statistical errors in the $NpT$ simulation points are smaller than the symbols.

Figure 5

Configurational energy $U$ (left) and excess heat capacity $C_A$ (right) as functions of reduced temperature $T^{*}$ at the close-packed density ${\rho }^{*}=2∕3\sqrt{3}\simeq 0.3849$: (circles) simulation results; (lines) results derived from a Padé $[5,5]$ fit (see text).

Figure 6

(Color online) Illustrations of an alternative close-packed structure: (a) without an assignment of attractive disks; (b) and (c) mirror images of a possible structural motif for a periodic arrangement of attractive disks. The attractive disks are colored dark gray (red online), the repulsive disks are colored light gray, and all disks are drawn with diameter $1\sigma$.