Marginally compact phase and ordered ground states in a model polymer with side spheres

We present the results of a quantitative study of the phase behavior of a model polymer chain with side spheres using two independent computer simulation techniques. We find that the mere addition of side spheres results in key modifications of standard polymer behavior. One obtains a marginally compact phase at low temperatures; the structures in this phase are reduced in dimensionality and are ordered, they include strands assembled into sheets and a variety of helices, and at least one of the transitions on lowering the temperature to access these ordered states is found to be first order. Our model serves to partially bridge conventional polymer phases with biomolecular phases.

Figure 1

Cross plot of $\theta$ and $\mu$ showing local structure in three unstructured phases of a standard polymer chain, where $\theta$ is the bond-bending angle and $\mu$ is the torsion angle. (a) Low-temperature compact globular phase in the absence of side spheres for a chain of length $N=60$. (b) and (c) Infinite-temperature coil phase for $N=20$ for a chain with (b) no side spheres and (c) large side spheres of size ${\sigma }_{SC}/\sigma =2.8$. All three panels show $100000$ points.

Figure 2

Specific heat per bead ($C_v/N{k}_B$) as a function of reduced temperature ($T^{*}=k_{B}T/\varepsilon$) for chains of length $N=40$, 60, and 80. (a) Specific heat curve for $R_c/\sigma =1.6$ and ${\sigma }_{SC}/\sigma =0$; the ground state is an unstructured compact globule. The specific heat curve suggests the existence of two phase transitions. The transition at the higher temperature, between the coil and the collapsed globule (the $\theta$ transition), is signaled by the shoulder. The more pronounced lower-temperature peak is suggestive of a second continuous transition between two unstructured phases, the collapsed globule and the compact globule. (b) Specific heat curve for $R_c/\sigma =1.6$ and ${\sigma }_{SC}/\sigma =1.0$; the ground state is the (7,45) precessing helix [shown in Fig. 3. (c) Specific heat curve for $R_c/\sigma =1.6$ and ${\sigma }_{SC}/\sigma =1.5$; the ground state here is the (1,5) dual helix [shown in Fig. 3]. In all cases, the inset shows the canonical probability distribution of the energy in the vicinity of the transition to the ground state (the lower transition temperature when there are two transitions) for the longest chain studied ($N=80$) at three temperatures: the transition temperature corresponding to the peak in the specific heat (purple curve), a temperature 1% below the transition temperature (blue curve), and a temperature 1% above the transition temperature (red curve). The numbers of attractive contacts in the lowest-energy conformations in our simulations are 443, 316, and 259 for ${\sigma }_{SC}/\sigma =0$, 1, and 1.5, respectively. The key message is that the low-temperature transition in the absence of side spheres appears to be continuous, whereas the addition of large enough side spheres results in a structured helical ground state and a first-order transition to it upon lowering the temperature. The low-temperature first-order transition weakens upon increasing the side sphere size [compare (c) versus (b)] because of the decrease in conformational entropy in the restricted coil phase.

Figure 3

Sketches of several ordered structures in the marginally compact phase of a tangent chain for different attraction ranges and side chain diameters. The structures are mathematical idealizations of those observed in the simulations having the same number of attractive contacts. Small variations around the idealized structures are permitted without any change in energy (the number of contacts). The helical structures shown are robust and reproducible over a range of chain lengths with two different simulation techniques and are likely ground states in the thermodynamic limit. The sheet structure is not observed in our simulations of a short chain but is mathematically constructed to maximize the number of contacts for the parameters shown. The side spheres are explicitly shown only for the sheet. For the helices, the side spheres stick out tangentially. In all cases, there are no steric clashes. The ground state for (a) $R_c/\sigma =1.2$ and ${\sigma }_{SC}/\sigma =1.0$ is a (1,2) sheet with six contacts per interior sphere and a repeat of $(\theta ,\mu )$ values of $({60}^{\circ },{180}^{\circ })$; for (b) $R_c/\sigma =1.4$ and ${\sigma }_{SC}/\sigma =1.0$, a (7,39) helix with $(5+5+8+\cdots )$ contacts per main chain sphere (degenerate with the sheet and a dual helix) corresponding to repeat $(\theta ,\mu )$ values of $({90}^{\circ },{20}^{\circ })$, $({90}^{\circ },{20}^{\circ })$, and $({170}^{\circ },{10}^{\circ })$; for (c) $R_c/\sigma =1.6$ and ${\sigma }_{SC}/\sigma =1.0$, a (7,45) helix with $(6+10+10+\cdots )$ contacts and a repeat of $(\theta ,\mu )$ values of $({60}^{\circ },{180}^{\circ })$, $({160}^{\circ },{30}^{\circ })$, and $({160}^{\circ },0^{\circ })$; for (d) $R_c/\sigma =1.6$ and ${\sigma }_{SC}/\sigma =1.5$, a dual helix with six contacts per bulk main chain sphere and a repeat of $(\theta ,\mu )$ values of $({120}^{\circ },{40}^{\circ })$; and for (e) $R_c/\sigma =1.8$ and ${\sigma }_{SC}/\sigma =1.0$, a (7,38) helix with $(12+16+\cdots )$ contacts and a repeat of $(\theta ,\mu )$ values of $({70}^{\circ },{15}^{\circ })$ and $({160}^{\circ },{15}^{\circ })$. The notation used is $(\text{number}\text{of}\text{approximate}\text{full}\text{turns}\text{per}\text{period}\text{of}\text{the}\text{helix},\text{number}\text{of}\text{main}\text{chain}\text{spheres}\text{per}\text{period})$. The sheet picture does not depict the turns connecting the zigzag strands and the double helix picture does not show the single turn.

Figure 4

Phase diagram in the reduced temperature: side chain diameter plane for the tangent polymer chain of length $N=60$. The blue diamonds indicate continuous transitions whereas the red circles are first-order transitions. Both parallel tempering and Wang-Landau simulations yield consistent results. The shaded region indicates a part of the phase diagram where the relatively modest chain length and potential equilibration issues make the results possibly unreliable. In the unshaded region, the error estimates are smaller than the size of the points. The coil is the canonical high-temperature phase. For small side spheres, there are two globule phases, a collapsed globule at intermediate temperatures (with the familiar $\theta$ transition between the coil and the collapsed globule) as well as a distinct low-temperature phase of a compact globule (with a second continuous transition between the compact and collapsed globule phases). The penetrated helix phase (PE) has, as its ground state, a single or a dual helix with a large enough radius to allow penetration of a rod (an essentially straight chain segment) within it (see Fig. 5), the precessing helix phase (PR) has a helical ground state with a period of 45 main chain spheres yielding a rotation of approximately seven turns [see Fig. 3], and the dual helix is comprised of two symmetric helices connected by a turn [see Fig. 3 (turn not shown)].

Figure 5

Mathematical idealizations of two structures, penetrated helical structures, corresponding to a single rod (an essentially straight chain segment) within (a) a dual helix and (b) a single helix, for $R_c/\sigma =1.6$ and (a) ${\sigma }_{SC}/\sigma =0.7$ and (b) ${\sigma }_{SC}/\sigma =0.8$. The point corresponding to both $\theta$ and $\mu$ equal to ${180}^{\circ }$ denotes the rod, whereas the small-$\mu$ values correspond to the helical jackets. These structures are seen consistently in our computer simulations, they can be sustained in the long chain limit, and they are probably ground-state structures.