Dynamics of a double-stranded DNA segment in a shear flow

We study the dynamics of a double-stranded DNA (dsDNA) segment, as a semiflexible polymer, in a shear flow, the strength of which is customarily expressed in terms of the dimensionless Weissenberg number Wi. Polymer chains in shear flows are well known to undergo tumbling motion. When the chain lengths are much smaller than the persistence length, one expects a (semiflexible) chain to tumble as a rigid rod. At low Wi, a polymer segment shorter than the persistence length does indeed tumble as a rigid rod. However, for higher Wi the chain does not tumble as a rigid rod, even if the polymer segment is shorter than the persistence length. In particular, from time to time the polymer segment may assume a buckled form, a phenomenon commonly known as Euler buckling. Using a bead-spring Hamiltonian model for extensible dsDNA fragments, we first analyze Euler buckling in terms of the oriented deterministic state (ODS), which is obtained as the steady-state solution of the dynamical equations by turning off the stochastic (thermal) forces at a fixed orientation of the chain. The ODS exhibits symmetry breaking at a critical Weissenberg number ${\mathrm{Wi}}_{\text{c}}$, analogous to a pitchfork bifurcation in dynamical systems. We then follow up the analysis with simulations and demonstrate symmetry breaking in computer experiments, characterized by a unimodal to bimodal transformation of the probability distribution of the second Rouse mode with increasing Wi. Our simulations reveal that shear can cause strong deformation for a chain that is shorter than its persistence length, similar to recent experimental observations.

Figure 1

The tumbling frequency as a function of the Weissenberg number Wi for a series of chain lengths $N=7$, 15, 31, and 63. The conversion to Weissenberg numbers is based on the ground-state moment of inertia [see Eq. (11)].

Figure 2

The squared value of the transverse component $R_2^s$ as a function of the Weissenberg number for a dsDNA chain of length $N=63$. The critical Weissenberg number is ${\mathrm{Wi}}_{\text{c}}\approx 19.4$.

Figure 3

(a) The appearance of the first two even Rouse modes in a window around the optimal value $\varphi =3\pi /4$ for $\theta =\pi /2$ at $\mathrm{Wi}=21.3$. (b) The end-to-end distance as a function of $\varphi$ for $\theta =\pi /2$ for some values of Wi around ${\mathrm{Wi}}_{\text{c}}\approx 19.4$.

Figure 4

Shapes of the dsDNA chain of length $N=63$ at $\mathrm{Wi}=100$, demonstrating Euler buckling in the ODS. The shapes are shown on the $x-y$ plane (i.e., $\theta =\pi /2$) for $0.6\pi \le \varphi <\pi$. In the direction of the arrow the shapes correspond to $\varphi =0.60\pi , 0.65\pi , 0.70\pi , 0.75\pi , 0.80\pi , 0.85\pi , 0.90\pi$, and $0.95\pi$, respectively. Interestingly, the orientation here defined by the first Rouse mode corresponds closely to the direction of the end-to-end vector.

Figure 5

Scatter plots of $R_2^s$ as a function of $\varphi$ around $\theta =\pi /2$ at four values of Wi of length $N=63$. In between the solid lines, representing $0.35\pi <\varphi <3\pi /4$, we see the probability distribution $P\left(R_2^s\right)$ of $R_2^s$ changing with increasing Wi. We take these issues up in Fig. 6. See also the text for details.

Figure 6

(a) Probability distribution $P\left(R_2^s\right)$ of $R_2^s$ corresponding to $0.35\pi <\varphi <3\pi /4$ in Fig. 5 for $N=63$, showing the unimodal distribution for $\text{Wi}=10$ gradually transforming into a bimodal distribution symmetric around $R_2^s=0$ for higher Wi. (b) The corresponding Binder cumulant $B$, as defined in Eq. (36), which changes from $\approx 0.3$ at $\text{Wi}=10$ to $\approx 0.53$ at $\text{Wi}=100$.

Figure 7

(a) Probability distribution $P\left(R_2^s\right)$ of $R_2^s$ corresponding to $0.35\pi <\varphi <3\pi /4$ in Fig. 5 for $N=31$, showing the unimodal distribution for $\text{Wi}=10$ slowly transforming into a bimodal distribution symmetric around $R_2^s=0$ for higher Wi. (b) The corresponding Binder cumulant $B$, as defined in Eq. (36), which changes from $\approx 0.23$ at $\text{Wi}=10$ to $\approx 0.58$ at $\text{Wi}=200$.

Figure 8

Simulation snapshots of a tumbling dsDNA chain of length $N=63$ at $\text{Wi}=100$, projected on the $x-y$ plane: (a) $U$ shape, (b) $S$ shape.