Scaling exponents and probability distributions of DNA end-to-end distance

Abstract

Correlation length exponent $\nu$ for long linear DNA molecules was determined by direct measurement of the average end-to-end distance as a function of the contour length $s$ by means of atomic force microscopy (AFM). Linear DNA, up to 48'502 base pairs (bp), was irreversibly deposited from a solution onto silanized mica and imaged in air. Under the adsorption conditions used, the DNA is trapped onto the surface without any two-dimensional equilibration. The measured exponent is $\nu =0.589\pm 0.006$, in agreement with the theoretical 3D value of $\nu =0.5880\pm 0.0010$. The persistence length ${\ell }_p$ of DNA was estimated to be 44$\pm$3 nm, in agreement with the literature values. The distribution of the end-to-end distances for a given contour length $s$ and the exponents characterizing the distribution were determined for different $s$. For $s$ smaller or comparable to ${\ell }_p$, a delta function like distribution was observed, while for larger $s$, a probability distribution of the type $x^{d-1}{x}^g{e}^{-b{x}^{\delta }}$ was observed with $g=0.33\pm 0.22$ and $\delta =2.58\pm 0.76$. These values are compared to the theoretical exponents for Self-Avoiding Walk (SAW): namely $g=\frac{\gamma -1}{\nu }$ and $\delta =(1-\nu {)}^{-1}$. So for $d=2$, $g\approx 0.44$ and $\delta =4$, while for $d=3$, $g\approx 0.33$ and $\delta \approx 2.5$. The derived entropic exponent $\gamma$ is $\gamma =1.194\pm 0.129$. The present data indicate that the DNA behaves on large length scales like a 3 dimensional SAW.