Exact theory of kinkable elastic polymers

The importance of nonlinearities in material constitutive relations has long been appreciated in the continuum mechanics of macroscopic rods. Although the moment (torque) response to bending is almost universally linear for small deflection angles, many rod systems exhibit a high-curvature softening. The signature behavior of these rod systems is a kinking transition in which the bending is localized. Recent DNA cyclization experiments by Cloutier and Widom have offered evidence that the linear-elastic bending theory fails to describe the high-curvature mechanics of DNA. Motivated by this recent experimental work, we develop a simple and exact theory of the statistical mechanics of linear-elastic polymer chains that can undergo a kinking transition. We characterize the kinking behavior with a single parameter and show that the resulting theory reproduces both the low-curvature linear-elastic behavior which is already well described by the wormlike chain model, as well as the high-curvature softening observed in recent cyclization experiments.

Figure 1

(a) The discretized KWLC is a chain of wormlike and kinklike vertices. In this illustration $N=4$; thus there are four vertices, of which one is kinklike. When a vertex $i$ is wormlike $({\sigma }_i=1)$, the energy is given by the normal wormlike chain energy; if it is kinklike $({\sigma }_i=0)$, the energy is $ϵ$, independent of ${\theta }_i$. (b) The continuum version of this theory. Although the number of vertices is now infinite, the continuum limit maintains a finite average kink density.

Figure 2

(a) Diagrammatic representation of the kink expansion for the tangent partition function. The dashed curve represents the KWLC theory and the solid curves represent the WLC theory. It is convenient to collect the terms by kink number as shown. (b) Detail of the two-kink term, showing the relation to the underlying discrete model. ${\vec{u}}_i$ and ${\vec{v}}_i$ are the tangent vectors flanking kink number $i$.

Figure 3

The tangent propagator and the tangent free energy as functions of the deflection angle for the illustrative values $L=0.2\xi$ and $\zeta \xi ={10}^{-2}$. The solid curves are KWLC and the dashed curves are WLC with the same value of $\xi$. In the absence of kinking, the WLC distribution $\left(H\right)$ is essentially zero away from small deflection. For the small value of $\zeta$ chosen above, WLC and KWLC are indistinguishable in the top panel. The presence of kinks adds a background level to the propagator which is independent of $\theta$, but is thermally inaccessible—too small to distinguish from zero in the top panel, but visible in the free energy in the lower panel. The tangent free energy gives an intuitive picture of the system interpreted as as single-state system with an effective bending modulus which saturates due to kinking. Most thermally driven experiments measure the polymer distribution as it is pictured in the top panel and are therefore insensitive to the high-curvature constitutive relation. But experiments which do probe this regime, short-contour-length cyclization for example, will be extremely sensitive to the difference between the theories due to the large free energy difference at large deflection.

Figure 4

The bending moment $\tau$ and average kink number $⟨m⟩$ as functions of the tangent deflection angle for illustrative values $L=0.2\xi$ and $\zeta \xi ={10}^{-2}$. The solid curves are KWLC and the dashed curves are WLC with the same bend persistence length. At small $\theta$, the normalized bending moment exhibits a linear spring dependence and the chain is unkinked. The limiting linear behavior of the short rod limit is the dotted curve, labeled $T=0$ corresponding to the mechanical limit of WLC. For large deflection, the chain kinks and the moment drops to zero. This correspondence between kinking and the moment is clearly illustrated in the short length limit depicted above.

Figure 5

Force-extension characteristic for KWLC compared to WLC and rigid rod for $L=4\xi$ and $\zeta \xi =4$. At low extension, the force extension of KWLC (solid curve) approaches WLC (dashed curve) with a persistence length equal to the effective persistence length of KWLC. At high extension, the kink modes are frozen out and the KWLC force-extension characteristic approaches WLC (dotted curve) with a persistence length equal to the bend persistence length of KWLC. Rigid rod (dot dashed curve) has been plotted for comparison. The extension of rigid rod corresponds to alignment only.

Figure 6

Left: Semilog plot of the best fit of the WLC model $(\zeta =0)$ to experimental data on the force-extension relation of a single molecule of lambda DNA. Right: Best fit of the KWLC model to the same data, taking $\zeta \xi =0.05$. The fits are equally good, even though this value of $\zeta$ is larger than the one that we will argue fits cyclization data. Thus, force-extension measurements can only set a weak upper bound on the value of $\zeta$. (Data kindly supplied by Vincent Croquette; see [4].)

Figure 7

The structure factor and the role of effective persistence length. The solid curve is the structure factor for KWLC with contour length $L=4\xi$, and kink parameter $\zeta =4∕\xi$. For comparison, we have plotted the structure factor for WLC of the same contour length for identical bend persistence lengths (dashed) and identical effective persistence length (dotted). At short length scales (large wave number) the KWLC structure factor approaches that for WLC with an identical bend persistence length. At long length scales (small wave number), the KWLC structure factor approaches that for WLC with a persistence length equal to its effective persistence length $\xi *$. We have also plotted the structure factor for rigid rod (dot dashed curve) for comparison.

Figure 8

The diagrammatic representation of the kink number expansion for cyclized polymers. The dashed curve represents the KWLC theory which is the sum of the $m$ kink contributions. In the interval between the kinks, the polymer is described by WLC, represented by the solid curves. For each $m$ kink contribution, we sum over the kink position. In order to meet the tangent alignment conditions for cyclized polymers, we close the chain at a kink for kink number one or greater.

Figure 9

The KWLC looping $J$ factor, $J_L^{*}$, as a function of contour length plotted for various values of the kinking parameter $\zeta$. The numbers labeling the curves indicate the value of the dimensionless quantity $\xi \zeta$. WLC is the curve labeled 0. For large contour length $L$, the effect of kinking can be accounted for by computing $J_L$ for the effective persistence length, $\xi *$. But as the contour length shrinks to a persistence length, the effect of kinking becomes dominant, even for small $\zeta$. At short contour length the looping $J$ factor is one kink dominated and diverges in contrast to the WLC looping $J$ factor which approaches zero precipitously for short contour length.

Figure 10

(Color) The KWLC cyclization $J$ factor as a function of contour length $L$ for various values of the kink parameter $\zeta$. As discussed in the text, our theory does not include the twist induced $10.5\mathrm{bp}$ modulation of the $J$ factor. The numbers labeling the curves indicate the value of the dimensionless parameter $\xi \zeta$. The WLC theory corresponds to $\zeta =0$. For large contour length $L$, the effect of kinking can be accounted for by computing the $J$ factor using the effective persistence length, $\xi \zeta$. As the contour length $L$ falls below a persistence length, kinking dramatically increases the $J$ factor, even for small $\zeta$. For small $L$, the chain is two kink dominated and diverges, in contrast to the WLC theory which precipitously falls to zero at small $L$. Experimental cyclization data for DNA are plotted for comparison, assuming $\xi =50\mathrm{nm}$. (Data sources: CW [7], SB [41], SLB [42], and VV [43].) At contour length $L=0.6\xi$, the experimentally measured $J$ factor is $\approx {10}^4$ times larger than predicted by the WLC theoretical curve. The KWLC with $\zeta \xi ={10}^{-2}$ correctly captures this behavior, while matching the WLC theory at large contour length.

Figure 11

The kink-number distribution compared for cyclized chains (solid curves) and unconstrained chains (dotted curves) as a function of contour length $L$. To illustrate constraint-driven kinking, we have chosen the illustrative value $\zeta \xi ={10}^{-2}$. At large contour length $L$, the cyclization constraint is irrelevant but as the arc length shrinks to roughly a persistence length, the bending energy required to cyclize the chain becomes significant and there is a dramatic transition to the two kink state which dominates at short contour length. The contributions of one and $m>2$ kink states are secondary.

Figure 12

A schematic diagram of the coordinate transformation exploited to compute the circular convolution. The crosses represent chain ends and dots represent kinks. The center line represents a periodic coordinate system. For regular convolutions, we set the chain end to be the zero and we compute the convolution in the $s$ coordinate system. For the circular convolution, it is more convenient to choose $L_1$, the first kink arc length position as the zero and sum over the chain end position as represented by the $s^{\prime }$ coordinate system.