Nonlinear elasticity of single collapsed polyelectrolytes

Nonlinear elastic responses of short and stiff polyelectrolytes are investigated by dynamic simulations on a single-molecule level. When a polyelectrolyte condensate undergoes a mechanical unfolding, two types of force-extension curves—i.e., a force plateau and a stick-release pattern—are observed depending on the strength of the electrostatic interaction. We provide a physical interpretation of such force-extension behavior in terms of intramolecular structures of the condensates. We also describe charge distributions of counterions condensed onto a polyelectrolyte, which clarify formation of one-dimensional strongly correlated liquid at large Coulomb coupling regime. These findings may provide significant insights into the relationship between a molecular elasticity and a molecular mechanism of like-charge attractions observed in a wide range of charged biopolymer systems.

Figure 1

(Color online) The force-extension curves of a PE for three different values of the coupling parameter (a) $\Gamma =0.1$ (WLC), (b) $\Gamma =40$ (force plateau), and (c) $\Gamma =110$ (stick release). The dashed lines are Eq. (6) with bare persistence length $L_p^0$ as a reference. Arrows indicate stretching (red) and relaxing (blue). In the inset of (b), points with negative force values are removed.

Figure 2

The effective persistence length $L_p$ of a PE estimated from its force-extension curve as a function of $\Gamma$. The inset is the same plot for $L_p∕L_p^0$ on a semilogarithmic scale.

Figure 3

(Color online) Top: snapshots of the PE condensates for $\Gamma =40$ and $\Gamma =110$ at time steps $1.2\times {10}^7$. Bottom: (angle-averaged) pair-distribution functions of monomer and counterion, $g_{mc}\left(r\right)$ (straight line), and of two counterions, $g_{cc}\left(r\right)$, (dashed line) for $\Gamma =40$ (black) and for $\Gamma =110$ (red).

Figure 4

(Color online) Snapshots of a PE under stretching for $\Gamma =40$. Corresponding $f\text{−}x$ curve is Fig. 1b. Red and white spheres represent counterions and monomers, respectively. All the counterions are condensed onto the PE.

Figure 5

(Color online) Top: the number of the monomers remaining in the globule, $N_g$, as a function of $x∕⟨L⟩$, under pulling with $\Gamma =40$. Bottom: snapshot of a PE with $\Gamma =40$ at the intermediate stage of pulling, exhibiting a ball-chain structure, together with its schematic illustration.

Figure 6

(Color online) Snapshots of a PE during stretching for $\Gamma =110$. Corresponding $f\text{−}x$ curve is Fig. 1c. The ball-chain configuration is evident for a PE with $N=64$ monomers. Its force-extension curve is shown in Fig. 7a.

Figure 7

(a) Force-extension curve of a PE with $N=64$ for $\Gamma =110$ in stretching. (b) The $f\text{−}x$ curve averaged over the five independent runs with an identical initial state of a PE with $N=32$ for $\Gamma =110$.

Figure 8

(Color online) Top: snapshot of a PE condensate without surrounding counterions, which takes a helical configuration. Below: winding number $\rho$ as a function of the fractional extension $x∕⟨L⟩$. Arrows highlight the points where the force drops involve one or half turn decrease of the winding number, $\rho$.

Figure 9

(Color online) (a) 1D pair-distribution function, $g_{cc}\left(x\right)$, of counterions condensed onto a stretched PE with its fractional extension $u$ being nearly 0.9, for $\Gamma =40$ (solid line) and for $\Gamma =110$ (dashed line). (b) 1D structure factor $S\left(q_{\Vert }\right)$ of counterions, a direct Fourier transform of (a). (c) Self-diffusions of a single counterion for $\Gamma =40$ and 110, both of which exhibit subdiffusive behavior given, respectively, by $⟨\delta x^2\left(t\right)⟩\propto t^z$ with $z=0.92$ for $\Gamma =40$ and with $z=0.85$ for $\Gamma =110$.

Figure 10

(Color online) (a) Force-extension curves obtained for the pulling rate $\alpha =0.1$ (red) and $\alpha =4.0$ (blue), with $\Gamma =60$. (b) Excess work $\Delta W$ as a function of the pulling rate $\alpha$ for $\Gamma =60$. Data for $\alpha >1$ are 3-times averaged.