Stretching short biopolymers by fields and forces

We study the mechanical properties of semiflexible polymers when the contour length of the polymer is comparable to its persistence length. We compute the exact average end-to-end distance and shape of the polymer for different boundary conditions, and show that boundary effects can lead to significant deviations from the well-known long-polymer results. We also consider the case of stretching a uniformly charged biopolymer by an electric field, for which we compute the average extension and the average shape, which is shown to be trumpetlike. Our results also apply to long biopolymers when thermal fluctuations have been smoothed out by a large applied field or force.

Figure 1

A diagram of different ways to stretch single molecules. (a) Magnetic tweezer apparatus. The bead is paramagnetic, and the force exerted is proportional to the spatial gradient of the magnetic field. (b) Stretching using a micropipette. The tip is imaged, and its bending is precalibrated with the corresponding force exerted. (c) Laser tweezers and (d) stretching using a uniform electric field.

Figure 2

The Green’s function $G(s,s^{\prime })$, with ${\kappa }_F=1$, plotted for $s^{\prime }∕L=0.25$ (long dashes), 0.5 (solid line), and 0.75 (small dashes).

Figure 3

The force-extension curve for a $100\mathrm{nm}$ strand of DNA, plotted against force in piconewtons (solid line) using Eq. (16). The dashed line shows the force-extension curve for a long polymer, plotted using the approximate interpolation formula of [4]

Figure 4

The set of parameter values that satisfy the small fluctuation approximation is shown by the shaded area. The $x$-axis is the reduced force, ${\kappa }_F^2{\xi }^2\equiv F\xi ∕k_{B}T$. The $y$-axis is the polymer length in terms of the $3d$-persistence length, $L∕\xi$.

Figure 5

(a) The relative extension of a polymer $X∕L$ pulled by a constant force in the presence of axis-clamping is plotted against the reduced force $F\xi ∕\left(k_{B}T\right)$, for a short polymer (solid line) for $d=2$. The dot-dashed line is the extension without the axis-clamping condition, Eq. (16), while the dashed curve is the long chain result [5]. The reduced polymer length is $L∕\xi =0.4$ ($\approx 20\mathrm{nm}$ for DNA and $\approx 6\mu \mathrm{m}$ for actin). In terms of force, $F\xi ∕\left(k_{B}T\right)=1$ means $0.082\mathrm{pN}$ for DNA, and $0.27\mathrm{fN}$ for actin. Inset shows the range of allowed parameter values (shaded area). (b) The shape of a polymer stretched with a laser tweezer is plotted parametrically in units of polymer contour length $L$ for a short polymer $(L∕\xi =0.4)$. The solid curve is for $F\xi ∕\left(k_{B}T\right)=1$, and the dotted curve is for $F\xi ∕\left(k_{B}T\right)=1000$. RMS stands for root mean square.

Figure 6

(a) Relative extension $X∕L$ is plotted against the reduced electric field $\beta \lambda E{\xi }^2$ for a polymer of size $L∕\xi =0.4$ for $d=2$. Inset shows parameter values allowed by self-consistency (shaded area). For actin, if $\lambda \approx 1e∕\mathrm{nm}$, $\beta \lambda E{\xi }^2=1$ is about ${10}^{-3}\mathrm{V}∕\mathrm{cm}$, for DNA with $0.6e∕\mathrm{nm}$ it is about $170\mathrm{V}∕\mathrm{cm}$. Note that we are assuming that the screened charge density is about one-tenth of the bare charge density. This is consistent with experiments on DNA [12]. (b) The shape of a short polymer stretched by a constant electric field in two dimensions is plotted parametrically in units of polymer contour length $L∕\xi =0.4$. The solid curve is for $\beta \lambda E{\xi }^2=1$, and the dotted curve is for $\beta \lambda E{\xi }^2=1000$.

Figure 7

The relative extension of a $15\mu \mathrm{m}$ actin molecule in a constant electric field compared to the case when it is stretched by a constant force due to a single charge on the free tip of the polymer. When this charge is chosen to be exactly $1∕4$ of the total charge on the polymer, the two curves agree well over most of the range of applicability.