Semiflexible chains in confined spaces

We develop an analytical method for studying the properties of a noninteracting wormlike chain (WLC) in confined geometries. The mean-field-like theory replaces the rigid constraints of confinement with average constraints, thus allowing us to develop a tractable method for treating a WLC wrapped on the surface of a sphere, and fully encapsulated within it. The efficacy of the theory is established by reproducing the exact correlation functions for a WLC confined to the surface of a sphere. In addition, the coefficients in the free energy are exactly calculated. We also describe the behavior of a surface-confined chain under external tension that is relevant for single molecule experiments on histone-DNA complexes. The force-extension curves display spatial oscillations, and the extension of the chain, whose maximum value is bounded by the sphere diameter, scales as $f^{-1}$ at large forces, in contrast to the unconfined chain that approaches the contour length as $f^{-1∕2}$. A WLC encapsulated in a sphere, that is relevant for the study of the viral encapsulation of DNA, can also be treated using the mean-field approach. The predictions of the theory for various correlation functions are in excellent agreement with Langevin simulations. We find that strongly confined chains are highly structured by examining the correlations using a local winding axis. The predicted pressure of the system is in excellent agreement with simulations but, as is known, is significantly lower than the pressures seen for DNA packaged in viral capsids.

Figure 1

(Color online) Representative structures for a WLC confined to the surface of a sphere of radius $R=3a$. (a) shows $l_p=2.5a$ and (b) shows $l_p=20a$. An enlargement of the polymer in (b) diagrams the positions ${\mathbf{r}}_i$ and bond vectors ${\mathbf{u}}_i$.

Figure 2

(Color online) Free energy $\beta \Delta F=\beta F\left(R\right)-\beta F\left(\infty \right)$ as a function of $R$ for a surface-confined WLC. $\beta F\left(\infty \right)$ is determined from a simulation with $R=2\times {10}^{4}a$. The symbols are the simulation data, where the lines are the theoretical results of Eq. (15). Shown are $l_p∕a=20$ [solid (purple) line], 10 [dotted (blue) line], 5 [dashed (green) line], and 2.5 [dotted-dashed (red) line]. The inset shows $\ln \left(\beta \Delta F\right)$ as a function of $R$, displaying the good agreement between simulation and theory, particularly for large $R$.

Figure 3

(Color online) Linear extension under an external tension of a surface confined WLC as a function of the radius. In (a), $L=450a$ and $l_p=150a$. The applied tensions are $a\beta f=0.1$ [solid (blue) line] 0.05 [dashed (green) line], and 0.01 [dotted (red) line], displaying the oscillations in $⟨x_L-x_0⟩$ for stiff chains. In (b), the same values of the force are applied to a chain of length $L=450a$ and $l_p=15a$, showing that the FEC of a flexible chain is monotonic.

Figure 4

(Color online) $⟨{\mathbf{r}}^2\left(s\right)⟩$ vs $s$. (a) $L=100a$ and $l_p=100a$, with (from highest to lowest) $R∕a=5$, 10, and 20. The average monomer position is dominated by end-point effects for $s\lesssim 2R$ and $s\gtrsim L-2R$. (b) $L=200a$ and $R=5a$, with (from highest to lowest) $l_p∕a=100$, 50, and 20. With $R$ fixed, variations in $l_p$ change only the value of $\rho$, but do not alter the behavior of the monomers.

Figure 5

(Color online) (a) The mean-field parameters $\rho$ [solid (blue) line], ${\rho }_0$ [dotted (red) line], and ${\rho }_c$ [dashed (green) line] as a function of $l_p∕R$. Lines are determined from simulations with $L=100a$ for various $l_p$ and $R$. Points are from simulations with $L=200a$. (b) $\alpha =l_p∕l_0$ as a function of $l_p∕R$, determined by fitting Eq. (32) to the simulation results. Symbols are the fits for $L=100a$, the line is the fit for $L=200a$.

Figure 6

(Color online) Bending correlation function $⟨\mathbf{u}(L∕2)\cdot \mathbf{u}\left(s\right)⟩$ as a function of $s$ for a chain with $L=200a$. The points are simulation data, the solid lines are the theoretical results in Eq. (32). (a) has $l_p=20a$ and $R=8a$, with $\alpha \approx 2$, and (b) has $l_p=100a$ and $R=5$ with $\alpha \approx 5∕2$. The agreement between theory and simulations is excellent, except near the end points. Representative configurations are shown to the right-hand side, and clearly display the wrapping to the chain for a strongly confined WLC in (b).

Figure 7

(Color online) (a) Average correlations in the nearest-neighbor winding axis as a function of $a{l}_p∕R^2$, with the average taken over interior points only (i.e., $2R⩽s⩽L-2R$). The symbols are simulation data for $L=100a$ for various $l_p$ and $R$, with the lines the simulation data for $L=200a$. The inset shows the average nearest neighbor correlation for $L=200a$, $R=5a$, and $l_p∕a=100$ (blue, top), 50 (green, middile), and 20 (red, bottom). The correlations drop sharply near the end points. (b) $⟨\widehat{\mathbf{a}}(L∕2)\cdot \widehat{\mathbf{a}}\left(s\right)⟩$ as a function of $s$ for $L=200a$ and $R=5a$, for (from highest to lowest) $l_p∕a=100$, 50, and 20. The curves are a fit to the exponential $⟨\widehat{\mathbf{a}}(L∕2)\cdot \widehat{\mathbf{a}}\left(s\right)⟩\propto e^{-L∕2l_0}$. One representative configuration for $L=200a$ and $l_p=100a$, seen from two different viewpoints, clearly shows high correlations in the winding axis, except near the ends of the chain.

Figure 8

(Color online) $\beta PV$ as a function of $R$ for varying $l_p$ and $L$. In both, the dots are simulation results, and the solid line is the theoretical result, with $\rho$, ${\rho }_0$, and ${\rho }_c$ taken directly from the simulation results. (a) $L=100a$ and $l_p∕a=100$ [solid (red) line], 50 [long-dashed (green) line], 20 [dashed (blue) line], 10 [dotted (purple), line], and 5 [dashed-dotted (pink) line]. The inset is a log-log plot for this data. (b) $L=200a$ and $l_p∕a=100$ [solid (red) line], 50 [long-dashed (green) line], and 20 [dashed (blue) line].