^{1,a)}, Naoko Makita

^{2}and Kenichi Yoshikawa

^{3}

### Abstract

We consider how the DNA coil-globule transition progresses via the formation of a toroidal ring structure. We formulate a theoretical model of this transition as a phenomenon in which an unstable single loop generated as a result of thermal fluctuation is stabilized through association with other loops along a polyelectrolyte chain. An essential property of the chain under consideration is that it follows a wormlike chain model. A toroidal bundle of loop structures is characterized by a radius and a winding number. The statistical properties of such a chain are discussed in terms of the free energy as a function of the fraction of unfolded segments. We also present an actual experimental observation of the coil-globule transition of single giant DNA molecules, T4 DNA, with spermidine , where intrachain phase segregation appears at a NaCl concentration of more than . Both the theory and experiments lead to two important points. First, the transition from a partially folded state to a completely folded state has the characteristics of a continuous transition, while the transition from an unfolded state to a folded state has the characteristics of a first-order phase transition. Second, the appearance of a partially folded structure requires a folded structure to be less densely packed than in the fully folded compact state.

I. INTRODUCTION

II. THEORETICAL CONSIDERATION

A. General formalism

B. Free energy of a ring structure

1. Statistical probability of the first loop

2. Fluctuations of the end point

3. Free energy of the second loop

4. Total free energy of a ring structure

III. RESULT AND DISCUSSION

IV. CONCLUSION

V. EXPERIMENTAL

## Figures

Schematic representation of a rings-on-a-string chain. A ring structure (an aggregate of loops) is characterized by the winding number and the mean loop length . By regarding the ring as a torus, we can introduce other shape parameters, i.e., torus radius and torus thickness , as in the figure. The inset electron micrograph of the T4 DNA (total length ) segregated structure is reprinted with permission from Ref. 16.

Schematic representation of a rings-on-a-string chain. A ring structure (an aggregate of loops) is characterized by the winding number and the mean loop length . By regarding the ring as a torus, we can introduce other shape parameters, i.e., torus radius and torus thickness , as in the figure. The inset electron micrograph of the T4 DNA (total length ) segregated structure is reprinted with permission from Ref. 16.

Theoretical characterization of the folding transition of DNA. The figures correspond to solutions with ionic strengths of (a) , (b) , and (c) , respectively. The top row shows the equilibrium fractions of an unfolded part in a folded chain vs , the variation of the normal surface energy of the chain segments in the solution. corresponds to an equiprobable point between unfolded and folded chains. The second row shows the relative probability densities of the folded chain at as functions of the fraction of unfolded segments. The bottom row shows estimated histograms of long-axis lengths of T4 DNA chains at the equiprobable point, where a black bar represents the folded chain, and a light gray bar represents the unfolded chain.

Theoretical characterization of the folding transition of DNA. The figures correspond to solutions with ionic strengths of (a) , (b) , and (c) , respectively. The top row shows the equilibrium fractions of an unfolded part in a folded chain vs , the variation of the normal surface energy of the chain segments in the solution. corresponds to an equiprobable point between unfolded and folded chains. The second row shows the relative probability densities of the folded chain at as functions of the fraction of unfolded segments. The bottom row shows estimated histograms of long-axis lengths of T4 DNA chains at the equiprobable point, where a black bar represents the folded chain, and a light gray bar represents the unfolded chain.

Experimental results regarding the folding transition of T4 DNA by spermidine through single-DNA observation with highly sensitive fluorescence microscopy. The figures correspond to (a) , (b) , and (c) NaCl solutions, respectively. The top row shows observed fractions of coils (○), globules (●), and partially folded chains (×) vs the concentration of spermidine [SPD]. The bottom row shows histograms of the long-axis lengths (maximum breadth of a fluorescent region on a focal plane) for coils (light gray bars) and folded chains (black bars). We inset these histograms with the pseudo-three-dimensional expression of the fluorescent intensity for typical fluorescent images of DNA. Each histogram corresponds to the condition proximate to the equiprobable point between coils and folded chains. The fractions of coils are (a) 23%, (b) 34%, and (c) 63%, respectively.

Experimental results regarding the folding transition of T4 DNA by spermidine through single-DNA observation with highly sensitive fluorescence microscopy. The figures correspond to (a) , (b) , and (c) NaCl solutions, respectively. The top row shows observed fractions of coils (○), globules (●), and partially folded chains (×) vs the concentration of spermidine [SPD]. The bottom row shows histograms of the long-axis lengths (maximum breadth of a fluorescent region on a focal plane) for coils (light gray bars) and folded chains (black bars). We inset these histograms with the pseudo-three-dimensional expression of the fluorescent intensity for typical fluorescent images of DNA. Each histogram corresponds to the condition proximate to the equiprobable point between coils and folded chains. The fractions of coils are (a) 23%, (b) 34%, and (c) 63%, respectively.

Experimental results regarding the folding transition of T4 DNA at high-salt concentration, showing the seemingly continuous nature of the transition. The left histograms show the distributions of the long-axis lengths of coils (light gray bars), partially folded chains (gray bars) and globules (black bars) of T4 DNA The solution contains NaCl and spermidine, with the concentration indicated in the figure. Figures on the right are typical fluorescent images of DNA, the pseudo-three-dimensional expression of their fluorescent intensity, and the corresponding schematic representations of DNA conformation, representing the classes indicated by , , , , and in the histograms on the left, respectively.

Experimental results regarding the folding transition of T4 DNA at high-salt concentration, showing the seemingly continuous nature of the transition. The left histograms show the distributions of the long-axis lengths of coils (light gray bars), partially folded chains (gray bars) and globules (black bars) of T4 DNA The solution contains NaCl and spermidine, with the concentration indicated in the figure. Figures on the right are typical fluorescent images of DNA, the pseudo-three-dimensional expression of their fluorescent intensity, and the corresponding schematic representations of DNA conformation, representing the classes indicated by , , , , and in the histograms on the left, respectively.

Theoretical expectation regarding the dependence of (a) the equilibrium fraction of unfolded segments , (b) the equilibrium ring size , and (c) the equilibrium winding number of a ring on the salt concentration. The normal surface energy is set to induce equiprobability between unfolded and folded chains.

Theoretical expectation regarding the dependence of (a) the equilibrium fraction of unfolded segments , (b) the equilibrium ring size , and (c) the equilibrium winding number of a ring on the salt concentration. The normal surface energy is set to induce equiprobability between unfolded and folded chains.

Theoretical prediction regarding the dependence of the folded chain conformation on elasticity (persistence length ). The entire solution stays at the equiprobable point between unfolded and folded chains. The concentration of salt is . The figures show (a) the distribution and free energy profile of a folded chain, (b) the average size of a ring , and (c) the average winding number of a ring vs the fraction of string segments . The lines correspond to (solid lines), (dot-dashed lines), and (dashed lines).

Theoretical prediction regarding the dependence of the folded chain conformation on elasticity (persistence length ). The entire solution stays at the equiprobable point between unfolded and folded chains. The concentration of salt is . The figures show (a) the distribution and free energy profile of a folded chain, (b) the average size of a ring , and (c) the average winding number of a ring vs the fraction of string segments . The lines correspond to (solid lines), (dot-dashed lines), and (dashed lines).

Theoretical prediction regarding the dependence of the folded chain conformation on the packing density . The entire solution stays at the equiprobable point of unfolded and folded chains. The concentration of salt is . The figures show (a) the distribution and free energy profile of a folded chain, (b) the average size of a ring , and (c) the average winding number of a ring vs the fraction of string segments . The lines correspond to (solid lines), (dot-dashed lines), and (dashed lines).

Theoretical prediction regarding the dependence of the folded chain conformation on the packing density . The entire solution stays at the equiprobable point of unfolded and folded chains. The concentration of salt is . The figures show (a) the distribution and free energy profile of a folded chain, (b) the average size of a ring , and (c) the average winding number of a ring vs the fraction of string segments . The lines correspond to (solid lines), (dot-dashed lines), and (dashed lines).

Experimental result regarding the histograms of the long-axis lengths and the hydrodynamic radii of DNA globules in the presence of (a) NaCl and spermidine and (b) NaCl and spermidine. The panels on the bottom show typical fluorescent images of the globules and the trajectories of their center positions over (61 frames). From such a time trace, the hydrodynamic radius is obtained based on the Einstein-Stokes relationship.

Experimental result regarding the histograms of the long-axis lengths and the hydrodynamic radii of DNA globules in the presence of (a) NaCl and spermidine and (b) NaCl and spermidine. The panels on the bottom show typical fluorescent images of the globules and the trajectories of their center positions over (61 frames). From such a time trace, the hydrodynamic radius is obtained based on the Einstein-Stokes relationship.

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