Rigidity of melting DNA

The temperature dependence of DNA flexibility is studied in the presence of stretching and unzipping forces. Two classes of models are considered. In one case the origin of elasticity is entropic due to the polymeric correlations, and in the other the double-stranded DNA is taken to have an intrinsic rigidity for bending. In both cases single strands are completely flexible. The change in the elastic constant for the flexible case due to thermally generated bubbles is obtained exactly. For the case of intrinsic rigidity, the elastic constant is found to be proportional to the square root of the bubble number fluctuation.

Figure 1

Schematic of dsDNA as two directed walks in the $(1+1)$ dimension. The direction in which the forces act are indicated by arrows. (a) Flexible model: The polymers cannot cross each other. The bound segments can bend left or right freely. This freedom can be restricted by introducing a statistical weight $b$. Here, $b$ is associated with the left degree of freedom. (b) Rigid model: The polymers can cross here. The bound segments cannot bend to the right and they are at least two bonds long. $v$ is the weight associated with the bubble opening or closure.

Figure 2

Flexible model: Phase diagram of the system under a stretching force. The phase boundary separates the bound phase from the unbound phase. In the region $4/3>y>1$ there exists a critical $g_{sc}$ for every value of $y$. Above $y=4/3$ the system is, by default, in the bound state. The solid blue line is the analytical curve, Eq. (11), and the red squares represent the numerically obtained critical points [see discussion following Eq. (17)].

Figure 3

Flexible model: Variation of $n_c$ with $g_s$ for different values of $g_u$ and $y$. A nonzero $n_c$ indicates a bound state. For $g_u=0$, there is no unbound state for $y\ge 4/3$. The $g_u=0$ curves are from Eqs. (5), (10a), and (10b). The cyan line with triangles is for the melting point $y=y_c$ and shows the quadratic dependence on $g_s$. The black curve with filled diamonds represents $y=1.4>y_c$. The red curve, for $g_u=0$, shows a continuous transition, while the blue curve, for nonzero $g_u$, Eq. (21), shows a discontinuity. The symbols on these analytically obtained curves, in the infinite-chain limit, are to make them distinct.

Figure 4

Flexible model: Plot of the average extension as a function of the stretching force with a fixed $y=1.20$. These are obtained from Eqs. (3), (10a), and (10b). The average extension varies continuously around the critical point. The symbols on these analytically obtained curves are to make them distinct.

Figure 5

Flexible model: Elastic modulus curve for $y=1.20$. The solid brown line is the analytically obtained curve (see text). The dashed cyan line is for ${\kappa }_b=4{\text{sech}}^2\left(2g_s\right)$, Eq. (18a), for the case with no bubbles. Other curves are plots of $\kappa$ for different system lengths. Solid lines through the data points are guides for the eye.

Figure 6

Flexible model: A magnified version of Fig. 5 around the same crossing point of all the curves. The chain length for each curve is given in the legend. Solid lines through the data points are guides for the eye.

Figure 7

Flexible model: The elastic modulus, Eq. (15), as a function of $y$ for zero stretching force.

Figure 8

Flexible model: The elastic modulus as a function of $g_s$ at $y=4/3=y_c$ [Eq. (16)] and $y=1.40>y_c$ [from Eqs. (9a) and (9b)].

Figure 9

Flexible model: Three-dimensional phase diagram of the system, Eq. (22), in the presence of both stretching and unzipping forces. All points on the surface except the curve for $g_{uc}=0$ represent first-order phase transition points. The unzipping line for $g_s=0$ is shown by the thick black line.

Figure 10

Flexible model: $x$ vs $g_s$ plot, with $g_u=0.4$ and $y=1.20$. Inset: Magnification of the critical region. The dotted magenta line is the analytical curve. See Sec. 3d2. All other curves are for finite system sizes shown in the legend. The discontinuity at $g_{sc}$ indicates a first-order transition. Solid lines through the data points are guides for the eye.

Figure 11

Flexible model: $\kappa$ vs $g_s$ plot with $g_u=0.4$ and $y=1.20$. Inset: Magnification of the critical region. The dotted magenta lines are the analytical curves. See Sec. 3d2. All other curves are for finite system sizes shown in the legend. The peak height increases proportionally with $N$, signaling a $\delta \text{-function}$ peak which is not shown in the analytical curve. Solid lines through the data points are guides for the eye.

Figure 12

Rigid model: $n_c$ increases from zero to nonzero values continuously at $y>1$ before approaching the saturation value, 1. This indicates a binding-unbinding transition. Solid lines through the data points are guides for the eye.

Figure 13

Rigid model: $C_c$ vs $y$ plots for different $N$'s. The peak height increases with increasing $N$ but eventually saturates, creating a finite discontinuity. The discontinuity occurs at $y=1.18$, where all the curves meet. Solid lines through the data points are guides for the eye.

Figure 14

Rigid model: Continuous stretching of the DNA to its full extent by both positive and negative forces. The $y$ value is fixed at 1.20. Solid lines through the data points are guides for the eye.

Figure 15

Rigid model: $\kappa$ shows an increasing peak around $g_s=0.02$ with increasing system length $N$ and a fixed $y=1.20$. Inset: Maximum values of $\kappa ,{\kappa }_{\text{max}}$, are plotted vs $1/N$. A linear fit with the first four points (solid blue line) gives the estimate for $N\to \infty ,{\kappa }_{\text{max}}=4.95$. This indicates that there is a finite discontinuity in $\kappa$. The dashed black line is a plot of the function $\kappa =2{\text{sech}}^2\left(g_s\right)$, the unbound-state elastic constant. For $g_s>0.02,\kappa$ matches the unbound-state modulus. Solid lines through the data points are guides for the eye.

Figure 16

Rigid model: $C_c$ vs $g_s$ curves for finite lengths as indicated. The $y$ value is fixed at 1.20. Inset: $n_c$ decreases continuously from a finite value to 0, indicating a continuous phase transition. Peak heights in $C_c$ vs $g_s$ curves saturate, indicating a finite discontinuity. Around $g_s=0.02$ all curves meet at the critical point. Solid lines through the data points are guides for the eye.

Figure 17

Rigid model: Numerical phase diagram for the binding-unbinding transition. The solid line through the data points is a guide for the eye.

Figure 18

Rigid model: $n_b$ vs $g_s$ and $l_b$ vs $g_s$ plot for $y=1.20$. As the critical point is approached, $n_b$ becomes very small but $l_b$ becomes as large as $N$. Solid lines through the data points are guides for the eye.

Figure 19

Rigid model: $n_b$ vs $g_s$ and $C_b$ vs $g_s$ plot for $y=1.20$. Around the critical point $n_b$ is very small but its fluctuation $C_b$ is very large. Solid lines through the data points are guides for the eye.

Figure 20

Rigid model: $\frac{1}{x}\sqrt{n_b}$ vs $y$ curves for different system lengths collapse to a single master curve in the bound region. $g_u=g_s=0$. $y_c=1.18$. Solid lines through the data points are guides for the eye.

Figure 21

Rigid model: $\frac{1}{\kappa }\sqrt{C_b}$ vs $y$ curves for different system lengths collapse to a single master curve in the bound region. $g_u=g_s=0$. $y_c=1.18$. Solid lines through the data points are guides for the eye.

Figure 22

$n_c$ vs $y$ plot for $b=0.5$. $n_c$ becomes finite at $y=1.142$. Numerical data are also consistent with the analytical result. Solid lines through the data points are guides for the eye.

Figure 23

$C_c$ vs $y$ plot for $b=0.5$ showing a finite discontinuity at $y=1.142$. Numerical data also show very similar behavior. Solid lines through the data points are guides for the eye.