Phase diagram of the ground states of DNA condensates

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Phase diagram of the ground states of DNA condensates

Trinh X. Hoang, Hoa Lan Trinh, Achille Giacometti, Rudolf Podgornik, Jayanth R. Banavar, and Amos Maritan

Phys. Rev. E 92, 060701(R) – Published 28 December 2015

Abstract

The phase diagram of the ground states of DNA in a bad solvent is studied for a semiflexible polymer model with a generalized local elastic bending potential characterized by a nonlinearity parameter $x$ and effective self-attraction promoting compaction. $x = 1$ corresponds to the wormlike chain model. Surprisingly, the phase diagram as well as the transition lines between the ground states are found to be a function of $x$ . The model provides a simple explanation for the results of prior experimental and computational studies and makes predictions for the specific geometries of the ground states. The results underscore the impact of the form of the microscopic bending energy at macroscopic observable scales.

Received 24 June 2015

DOI:https://doi.org/10.1103/PhysRevE.92.060701

Physics Subject Headings (PhySH)

Phase diagrams

DNA

Polymers & Soft MatterPhysics of Living Systems

Authors & Affiliations

Trinh X. Hoang¹, Hoa Lan Trinh¹, Achille Giacometti², Rudolf Podgornik^3,4, Jayanth R. Banavar⁵, and Amos Maritan⁶

¹Institute of Physics, Vietnam Academy of Science and Technology, 10 Dao Tan, Ba Dinh, Hanoi, Vietnam
²Dipartimento di Scienze Molecolari e Nanosistemi, Universita' Ca' Foscari Venezia, I-30123 Venezia, Italy
³Department of Theoretical Physics, J. Stefan Institute and Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia
⁴Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA
⁵Department of Physics, University of Maryland, College Park, Maryland 20742, USA
⁶Dipartimento di Fisica e Astronomia, Università di Padova, CNISM and INFN, via Marzolo 8, I-35131 Padova, Italy

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Vol. 92, Iss. 6 — December 2015

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Images

Figure 1
Schematics of (a) toroidal, (b) rodlike, and (c) spheroidal models of DNA condensates. Ground state conformations obtained in Monte Carlo simulations of a $N = 64$ bead-and-spring model with $x = 1$ generalized elastic potential at (d) moderate and (e) low stiffness and $x = 2$ at (f) low stiffness. The conformation (f) is a torus with no hole.
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Figure 2
Ground state phase diagrams in the length-stiffness plane of a semiflexible chain with the generalized elastic potential for several values of $x$ , as indicated. The lines correspond to transitions from a toroidal to a rodlike condensate for $x < 3 / 2$ , and from a toroidal to a spheroidal configuration for $x > 3 / 2$ , being fits with slopes equal to $\frac{2 x - 1}{3}$ to numerical data points. An experimental data point (star) shown on the WLC model transition line corresponds to the average DNA length of $\sim 63.6$ kbp in 90 nm sized condensates found for a plasmid DNA condensation induced by cobalt(III) hexammine ( ${CoHex}^{3 +}$ ) with a coexistence of toroid phase and rodlike phase [14]. Projection from this point on the vertical axis yields $σ \approx 0.266 k_{B} T {(nm)}^{- 2}$ , or equivalently, an intermolecular contact interaction energy of $ɛ \approx - σ d \approx - 0.246 k_{B} T$ /bp, by using the DNA persistence length $β κ b = 50$ nm, diameter $b = 2$ nm, and the interaxial spacing $d = 2.8$ nm. The energy $ɛ$ roughly agrees with that for the ${CoHex}^{3 +}$ mediated DNA-DNA interaction obtained by the osmotic stress method ( $- 0.21 \pm 0.02 k_{B} T$ /bp) [29].
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Figure 3
Dependence of the toroidal mean radius $R$ and the toroidal thickness radius $Δ$ on the total contour length $L$ of DNA in toroidal condensates. Experimental data points are fitted to the theoretical results for the WLC model ( $x = 1$ ) as $R \sim L^{1 / 5}$ (solid line) and $Δ \sim L^{2 / 5}$ (dashed line) in (a) and (b), respectively. Experimental data are taken from Ref. [14] for condensation of plasmid DNA fragments of lengths 2700 bp (squares) and 1350 bp (circles) with $[DNA] = 10 μ g / mL$ . The condensation was induced by 150 $μ M {CoHex}^{3 +}$ and the sizes of the condensates were measured at 2 h and at 24 h after polyvalent salt addition. The volume of an average toroid was found to increase by $\sim 50 %$ , between 2 and 24 h. $R$ and $Δ$ are calculated based on the measured average toroid outer radius $R_{1}$ and inner radius $R_{2}$ given in the captions of Figs. 3 and 4 of Ref. [14] as $R = (R_{1} + R_{2}) / 2$ and $Δ = (R_{1} - R_{2}) / 2$ , respectively. The DNA contour length in a condensate is determined as $L = 8 π η R {(Δ / b)}^{2}$ by assuming a hexagonal close packing of DNA in the condensate with the volume fraction $η = (π / \sqrt{12}) {(b / d)}^{2}$ , where $d = 2.8$ nm is the DNA-DNA interaxial spacing, and $b = 2$ nm is the diameter of DNA.
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Figure 4
(a) Dependence of geometrical parameters $α$ and $γ$ on the chain length $L$ for toroidal (left) and rodlike (right) condensates. At $L = L_{c}$ the chain undergoes a transition from a toroidal to rodlike conformation. The data shown are obtained for $x = 1$ (solid), $x = 1.2$ (dashed), and $x = 1.6$ (dotted), as indicated. The dependence of the maximum values $α_{\max}$ and $γ_{\max}$ of the parameters $α$ and $γ$ on the exponent $x$ are shown in (b) and (c), respectively. The arrow indicates the result for the WLC model ( $x = 1$ ), for which $α_{\max} \approx 0.808$ and $γ_{\max} \approx 1.45$ . $α_{\max}$ increases from zero at $x = 0.5$ and reaches unity at $x \approx 1.264$ , whereas $γ_{\max}$ becomes zero only at $x = 1.5$ .
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Figure 5
Toroid thickness vs toroid diameter as obtained by experiments at various conditions (data points). The data points were extracted from Fig. 2 of Ref. [17]. For a given condensation condition (see legend), we take only the point with the maximum thickness to diameter ratio. The lines show the slopes corresponding to the maximum aspect ratios $α_{\max}$ as predicted from our model on changing the parameter $x$ . For the wormlike chain (WLC), $α_{\max} = 0.808$ (dotted line).
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