Crossover behavior in static and dynamic properties of a single DNA molecule from three to quasi-two dimensions

We studied the conformation and dynamics of a single DNA molecule in a thin slit by a fluorescent microscope. In a slit thinner than the Flory radius in three dimensions, the length of the major axis, the translational self-diffusion coefficient and the rotational relaxation time in a dilute solution show the apparent dependence on the thickness of the slit. The observed dependence is in agreement with that predicted by blob theory, despite the number of blobs is very small. The radial distribution of the segments around the center of mass of a single molecule was also studied and compared with that calculated for a Gaussian and an excluded volume chain. The influence of the polymer concentration on the geometrical confinement by slits was also studied in a semidilute solution near the overlap concentration $c^{\ast }$. The confinement effect is found to be not so serious near $c^{\ast }$ and is only significant in the so-called “two-dimensional pancake” region.

Figure 1

(Color online) Fluorescent image of a single DNA molecule (left) and its binarized image (right). The ellipsoid drawn as a dotted line is the best-fitted one to the binarized image. $\mathbf{u}$ is the unit vector parallel to the major axis. $L$ is the length of the major axis.

Figure 2

(Color online) Blob picture of a polymer confined in a slit. A sphere (online yellow) represents a blob with the size $d$. $d$ is the thickness of the slit, and $L$ is the lateral size of the polymer.

Figure 3

(Color online) Distribution of the length of the major axis $L$, $P\left(L\right)$, in a slit with different thickness $d$. The solid lines are the best-fitted ones with Eq. (12). The best-fitted values of $L_{\text{min}}$ are respectively 2.33, 2.97, 2.41, 1.82, and $2.28\mu \text{m}$ for $d=1$, 2, 3, 4, and $5\mu \text{m}$.

Figure 4

(Color online) Dependence of the average major length of a DNA molecule $\overline{L}$ on the thickness of a slit $d$. Both $\overline{L}$ and $d$ are represented in the normalized scale by the Flory radius in three dimensions, $R_F=4.45\mu \text{m}$. The fluorescent images in a thin (left: $d=1\mu \text{m}$) and a thick cell (right: $d=10\mu \text{m}$) are also presented. The exponent of the best-fitted power law drawn as a solid line in the region $d/R_F\le 1$ is $-0.26\pm 0.04$. The dotted line shows the power law of $\overline{L}\propto d^{-0.25}$ for a guide to the eyes.

Figure 5

(Color online) Dependence of the translational self-diffusion coefficient $D$ of a DNA molecule on the thickness of a slit $d$. Both $D$ and $d$ are represented in the normalized scale by the value of $D$ at $d=20\mu \text{m}$, $D_0=0.22\mu {\text{m}}^2/\text{s}$, and the Flory radius in three dimensions $R_F=4.45\mu \text{m}$. The exponent of the best-fitted power law drawn as a solid line in the region $d/R_F\le 1$ is $0.57\pm 0.07$. The dotted line shows the power law of $D\propto d^{0.67}$ for the guide of eyes. Dependence of MSD on the lag time $\Delta t$ at $d=1\mu \text{m}$ is plotted in the inset.

Figure 6

(Color online) Dependence of the hydrodynamic radius $R_{\text{H}}$ (filled circles: online red) and the ratio of $R_{\text{H}}/\overline{L}$ (empty circles: online blue) of a DNA molecule on the thickness of a slit $d$. Theoretical ratio of $R_{\text{H}}/\overline{L}=0.255$ in a three-dimensional free space is drawn as a dotted line. The exponent of the best-fitted power law to $R_{\text{H}}/\overline{L}$ in the region of $d/R_F\le 1$ drawn in a solid line is $-0.31\pm 0.05$.

Figure 7

(Color online) Dependence of the rotational relaxation time $\tau$ of a DNA molecule on the thickness of a slit $d$. Both $\tau$ and $d$ are represented in the normalized scale by the experimentally obtained value of ${\tau }_0$ at $d=20\mu \text{m}$, ${\tau }_0=2.3\text{s}$, and the Flory radius in three dimensions $R_F=4.45\mu \text{m}$. The exponent of the best-fitted power law drawn as a solid line in the region of $d/R_F\le 1$ is $-1.07\pm 0.10$. The dotted line shows the power law of $\tau \propto d^{-1.17}$ for the guide of eyes. The time correlation function of the unit vector parallel to the major axis at $d=1\mu \text{m}$ is plotted in the inset as a semilogarithmic plot.

Figure 8

(Color online) Radial distribution function of segments around the center of mass of a single DNA molecule in a slit with different thickness. The thick solid curve is calculated from the coordinate data of segments in the two-dimensional projection of the excluded volume chains in three-dimension obtained by computer simulation. The dotted curve is the distribution function of the segments in the two-dimensional Gaussian chain.

Figure 9

(Color online) Dependence of (a) the length of the major axis $\overline{L}$ and (b) the translational self-diffusion coefficient $D$ of a single DNA molecule on the thickness of a slit $d$. The open squares (online blue): dilute solutions. The filled circles (online red): semidilute solutions. The solid line in (a) indicates $\overline{L}\propto d^{-1/2}$ and that in (b) indicates $D\propto d^{2/3}$ for the guide of eyes. The regions marked by (I), (II) and (III) are calculated from the phase diagram [20, 35].