Stretching of homopolymers and contact order

Mechanical stretching of self-interacting homopolymers is studied through molecular dynamics simulations in which the polymers are pulled with constant speed. At temperatures below the compactification temperature, the force-extension curves show a plateau that corresponds to the situation in which the polymer adopts “ball-string” configurations. The dependence of rupture distances on contact order and the effects of temperature are similar to those found in the case of model proteins. The dependence of behavior on the pulling speed is logarithmic. In the entropic limit, above the compactification temperature, the rupturing of contacts shows a monotonic decrease of extension with the contact order. The attainment of this limit depends on the system size and the pulling speed.

Figure 1

Examples of folded homopolymer conformations as obtained by annealing. The system sizes are indicated.

Figure 2

The temperature dependence of the heat capacity $C$ and the averaged radius of gyration $⟨R_g⟩$ for the homopolymers of sizes that are indicated. ($C$ for $N=120$ is not shown to avoid overcrowding of the figure.) The error bars are of the order of the thickness of the lines. The square symbols show the unfolding temperature $T_u$. The data shown are obtained by averaging over $n_t$ annealing trajectories where $n_t=50$, 50, and 25 for $N=40$, 80, and 120, respectively. Each trajectory starts from a random open conformation and the reduced temperature $\tilde{T}$ is decreased in steps of 0.1. After an equilibration time of $1000\tau$ the energy and the radius of gyration are averaged over a time between 2000 and $\mathrm{10\hspace{0.17em}000}\tau$ depending on the temperature.

Figure 3

The $F\text{−}d$ curves for an $N=40$ homopolymer system. The values of $\tilde{T}$ are indicated. The pulling speed is $0.005\mathrm{\AA }∕\tau$. The upper panel shows pulling with a stiff cantilever $\left(k_c=100ϵ∕{\mathrm{\AA }}^2\right)$ whereas the lower panel corresponds to the soft cantilever $\left(k_c=0.1ϵ∕{\mathrm{\AA }}^2\right)$.

Figure 4

Similar to Fig. 3 but for an $N=160$ homopolymer system.

Figure 5

Examples of the $F\text{−}d$ curves obtained at $\tilde{T}$ of 2 and 4 for the $N=160$ system pulled by a soft spring. Inset: the force vs the end-to-end distance $d_{1,N}$ at $\tilde{T}=4$ is fitted by the $\mathrm{WLC}$ equation, $F=(\tilde{T}∕4p)[{(1-d_{1,N}∕L)}^{-2}-1+4d_{1,N}∕L]$, using the contour length $L=840\mathrm{\AA }$ and the persistence length $p=12.3\mathrm{\AA }$. The finite persistence length is due to local steric repulsion in the chain. The end-to-end distance was computed as $d_{1,N}=d+d_0-F∕k$, where $d_0$ is the initial end-to-end distance and $k$ is the cantilever spring constant.

Figure 6

Snapshots of a $N=160$ system stretched by distances of 100 and 200 $\mathrm{\AA }$ are shown in the left and right hand panels, respectively. The pulling springs are soft and the pulling speed $v_p=0.005\mathrm{\AA }∕\tau$. The value of $\tilde{T}$ is 0, 0.2, 0.5, and 2.0 as one goes from top to bottom.

Figure 7

Scenarios of mechanical unfolding averaged over 20 homopolymers with $N=40$ for the temperatures indicated. For each homopolymer, one stretching trajectory is considered. The pulling speed is $0.005\mathrm{\AA }∕\tau$.

Figure 8

Scenarios of mechanical unfolding of the $N=40$ homopolymers at three indicated values of the pulling speed. The crosses, black circles, and open squares correspond to $v_p$ of 0.0005, 0.005, and $0.05\mathrm{\AA }∕\tau$, respectively.

Figure 9

The logarithmic dependence of $d_u$ on $v_p$ for $N=40$. The results corresponding to a given value of $\mid j-i\mid$ are averaged along the chain, i.e., averaged along the vertical lines in Fig. 8. The lines have slopes $s$ that are indicated. The inset shows the absolute values of the slope for selected values of $\mid j-i\mid$.

Figure 10

Scenarios of mechanical unfolding for the three indicated system sizes and at a fixed value of $\tilde{T}$ of 2.

Figure 11

Scenarios of mechanical unfolding for the four indicated system sizes and adjusted indicated values of $\tilde{T}$. The legend corresponds to the data points in a top-to-bottom way.

Figure 12

A scaled representation of the stretching scenarios of the three sets of data shown in Fig. 12. The legend corresponds to the data points in a top-to-bottom way.