Conformations of confined biopolymers

Nanoscale and microscale confinement of biopolymers naturally occurs in cells and has been recently achieved in artificial structures designed for nanotechnological applications. Here, we present an extensive theoretical investigation of the conformations and shape of a biopolymer with varying stiffness confined to a narrow channel. Combining scaling arguments, analytical calculations, and Monte Carlo simulations, we identify various scaling regimes where master curves quantify the functional dependence of the polymer conformations on the chain stiffness and strength of confinement.

Figure 1

(Color online) Scaling regimes for confined biopolymer conformations as a function of polymer flexibility $\epsilon =L∕{\ell }_p$ and confinement strength $c=L∕L_d$. In the flexible regime $(\epsilon \gg 1)$ two scaling regimes are known [dark gray (red)]: free coil behavior for weak confinement and de Gennes scaling for intermediate confinement. In the parameter range most relevant for biopolymers [light gray (green)], one has to distinguish between weak confinement of stiff polymers and strong confinement for chains of arbitrary stiffness.

Figure 2

(Color online) Left: marginal probability distribution function of polymer configurations in a plane containing the tube axis for $c=4$ and $\epsilon =2$. The probability density increases from dark to light colors. Right: normalized tangent-tangent correlation function $\varphi \left(s\right)=⟨\mathbf{t}\left(s\right)\bullet \mathbf{t}\left(0.5\right)⟩$ for $\epsilon =0.1$ and various confinements ranging from $c=3$ to $c=20$. MC simulations are represented by symbols, whereas the solid lines show the analytical approximations of $\varphi \left(s\right)$ as from Eq. (5). Dashed line: exponential decay of an unconfined chain.

Figure 3

Reduced mean-square end-to-end distance $\mathrm{\Delta }{R}^2$ as a function of the collision parameter $c$ for a series of flexibilities $\epsilon$ indicated in the graph. The universal scaling curve (solid line) asymptotes the analytical result (long-dashed line) in the limit of strong confinement. Short-dashed lines are guides to the eye for $\epsilon >1$. Inset: orientational order parameter $S$ as a function of $c$ for $\epsilon =0.01$, 0.1, 1, 10.

Figure 4

Scaling plot for the mean-square end-to-end distance $⟨R^2⟩$ versus $c∕\epsilon ={\ell }_p∕L_d$ for various flexibilities $\epsilon$. Universal scaling curve as estimated from the simulation (solid line) and obtained in the WBR limit (dashed line).