Is DNA a Good Model Polymer?

Abstract

The details surrounding the cross-over from wormlike-specific to universal polymeric behavior has been the subject of debate and confusion even for the simple case of a dilute, unconfined wormlike chain. We have directly computed the polymer size, form factor, free energy and Kirkwood diffusivity for unconfined wormlike chains as a function of molecular weight, focusing on persistence lengths and effective widths that represent single-stranded and double-stranded DNA in a high ionic strength buffer. To do so, we use a chain-growth Monte Carlo algorithm, the Pruned-Enriched Rosenbluth Method (PERM), which allows us to estimate equilibrium and near-equilibrium dynamic properties of wormlike chains over an extremely large range of contour lengths. From our calculations, we find that very large DNA chains (≈ 1,000,000 base pairs depending on the choice of size metric) are required to reach flexible, swollen non-draining coils. Furthermore, our results indicate that the commonly used model polymer λ-DNA (48,500 base pairs) does not exhibit "ideal" scaling, but exists in the middle of the transition to long-chain behavior. We subsequently conclude that typical DNA used in experiments are too short to serve as an accurate model of long-chain, universal polymer behavior.

Figure 1

Summary of the classical scaling arguments for a real semiflexible chain in dilute suspension. Three regimes are predicted based on the interplay between the contour length (L), the chain stiffness (l_p) and the chain width (w). Very short chains (L ⪡ l_p) are rod-like and long “thin” chains are nearly Gaussian (L ⪢ l_p and L ⪡ l_T). Long chains (L ⪢ l_T) are swollen coils.

Figure 2

Ionic strength dependence of the Kuhn length (dashed blue line) and the effective width (dot-dashed green line) and the monomer anisotropy (solid red line) for dsDNA. The vertical gray dotted line indicates an ionic strength of 165 mM (≈ 5× TBE). The schematic illustrates two chains with similar b but different w, demonstrating the decrease in monomer anisotropy as the effective width increases more rapidly than the Kuhn length.

Figure 3

PERM data for the radius of gyration of ssDNA (open red triangles) and dsDNA (open blue circles) compared to experimental data for dsDNA from light and neutron scattering (filled green squares).^,– The experimental data were obtained from many different references, at varying ionic strength (all ≥ 100 mM), with varying information about the molecular weight. To obtain a consistent value, the molecular weight of dsDNA was assumed to obey the relation 660 Da = 0.34 nm = 1 bp. Single stranded DNA was assumed to follow 1 base = 0.65 nm.

Figure 4

Power-law exponent of the end-to-end distance of a semiflexible chain with excluded volume as calculated by results from renormalization group theory. As outlined in Sec. 3.1, ν = 1 corresponds to rod-like behavior, ν = 0.5 to a Gaussian chain and ν = 0.588 to a swollen chain. Results are shown for five different values of w/b (from top to bottom): 1.0, 0.217 (ssDNA), 0.043 (dsDNA), 3.16 × 10⁻³, 3.16 × 10⁻⁴ and 0 (no excluded volume). (Inset) PERM results for the excess free energy per Kuhn length due to excluded volume interactions in a dilute solution of dsDNA.

Figure 5

Form factor of dsDNA for ql_p < 1 for various contour lengths (from left to right): 865 kbp, 218 kbp, 48.5 kbp (λ), 13.8 kbp, 3.45 kbp, 865 bp. Symbols correspond to PERM calculations (with excluded volume), dashed lines to Eq. 21 and dotted lines to a semi-empirical expression by Sharp and Bloomfield. Solid lines correspond to the form factor of an ideal wormlike chain with the same stiffness (see online supporting information).

Figure 6

PERM calculations of dsDNA for the value of the exponent ν in R ~ L^ν (blue circles), S ~ L^ν (red triangles) and X ~ L^ν (green diamonds) for dsDNA (w/b = 0.0434) ~ and RG calcula tions for ν for R (blue dashed line) and for S (red solid line). The dashed gray lines correspond to the values of ν = 0.5 (Gaussian) and ν = 0.5876 (swollen).

Figure 7

Schematic of diffusive behaviors of wormlike chains. Rod-like diffusion dominates for very short, stiff chains. Polymer coils can exhibit either free-draining (weak HI) or non-draining (strong HI) behavior. Partial draining behavior is also possible for chains with relatively open structures. In this case, the polymer conformation is not sufficient to describe the diffusive behavior of the chain since the strength of the HI (i.e. the hydrodynamic radius) can vary independently.

Figure 8

Parameterization of the hydrodynamic diameter for the DWLC. (A) Curves (black, blue, red) show the relative diffusivity of a CWLC without excluded volume at different values of b/d. The touching bead DWLC model (symbols) shows excellent agreement with the CWLC chain (curves). (B) Experimental data for the diffusivity from dynamic light scattering (filled green triangles),^–,– sedimentation (filled green circles),^–,,, and single molecule methods (open green triangles).^, The dsDNA data fits the DWLC simulation (now with excluded volume interactions) for b/d = 36 or d = 2.9 nm. Notice that since the diffusivity is scaled by the Zimm diffusion (ideal chain diffusion), the asymptotic value does not approach 1. Additionally, it appears that the single molecule data give poorer agreement with the simulation data, presumably due to the impact of intercalating dyes.

Figure 9

(A) PERM results for the ratio of the radius of gyration S to the hydrodynamic radius R_H as a function of the molecular weight for both ssDNA (open red triangles) and dsDNA (open blue circles). λ-DNA and T4-DNA are shown for reference (arrow, open orange squares). The horizontal dashed line corresponds to S/R_H = 1.562. (B) The diffusion coefficient from PERM, rescaled by Eq. 25, as a function of the number of base pairs of DNA. Double-stranded DNA is shown to transition from rod-like behavior (green dashed line) to non-draining behavior (purple dotted line) over several orders of magnitude in molecular weight. The diffusion coefficient without excluded volume (closed blue circles) is shown for reference.