Organized condensation of worm-like chains

We present results relevant to the equilibrium organization of DNA strands of arbitrary length interacting with a spherical organizing center, suggestive of DNA-histone complexation in nucleosomes. We obtain a rich phase diagram in which a wrapping state is transformed into a complex multi-leafed structure as the adhesion energy is reduced. The statistical mechanics of the “melting” of a rosette can be mapped onto an exactly soluble one-dimensional many-body problem. PACS numbers: 87.15.-v, 36.20.Ey Reversible coordinated condensation of long eukaryotic DNA strands into a highly compacted package (chromatin), and the controlled swelling of chromatin, are essential requirements for the successful duplication and replication of DNA [1]; strands of the order of a meter are orderly packed into micron-size nuclei without getting tangled-up. It is known that certain enzymes – the topo-isomerases – assist the condensation process by untangling knotted DNA, by releasing the inevitable torsional stresses that are produced during condensation, and by facilitating the “super-coiling” of DNA. It is also known that condensed DNA strands are wrapped around organizing proteins (globular complexes of histones) that carry a charge opposite to that of DNA. Gene transcription is believed to involve some kind of loosening of this “wrapping” state [1] . DNA condensation has recently attracted considerable experimental [2] and theoretical [3] interest from physicists, stimulated in particular by the development of new techniques of single-molecule micro-mechanics [4]. These new tools open up the possibility of detailed mechanical probing of DNA condensation. The study of DNA condensation simplifies significantly if we focus on the interaction of DNA with just one or a few "organizing centers", i.e., particles which either condense the DNA strand into a wrapped (continuously adsorbed) state or induce it into a centro-symmetric open structure. Even for this simple case we still encounter puzzling results. The in-vitro phase diagram of mixtures of short DNA strands with single octameric-histone (nucleosomal [5]) organizing-centers has been studied by Yager et al. [6]. They found an athermal first-order phase transition as a function of the DNA-histone interaction strength (controlled by changing the salt concentration) from a wrapped state to a dissociated state, consistent with the simple ("all or none") unwrapping transition proposed by Marky and Manning [7]. On the other hand, in a numerical study by Wallin and Linse [8] of the association of a long charged polymer (“polyelectrolyte”) with an oppositely charged sphere, a more gradual change was observed as the chain stiffness was increased, with one or more loops extending out of the sphere. Finally, recent work by Polach and Widom [9] found that the nucleosome wrapping state actually represents a dynamical equilibrium in which wrapped portions of the DNA strand also spent part of their time in a dissociated state. They measured the binding energy of DNA to the nucleosome to be about 0.15–0.2kBT per base-pair under standard conditions. In this paper we report on a model study of the finite-temperature conformation of DNA (or other semi-flexible polymers) interacting with a spherical organizing center, in an effort to further elucidate the nature of the unwrapping transition. We make use of the popular “Worm-Like Chain” (WLC) model [10] for DNA, which describes the molecule as a semi-flexible tube characterized by two elastic moduli, the bending and torsional stiffnesses. This model is able to describe, on a quantitative basis, the forceextension curve of DNA as measured by micro-mechanical methods [11,12] and it has been used to examine size-reduction of DNA loops by supercoiling, and the effect of thermal fluctuations on supercoiling [13], as well as the complexation of DNA with model nucleosomes [14]. In the present work we map the semi-flexible strand/spherical organizing center system onto an exactly soluble one-dimensional statistical mechanics problem, and exploit the solutions to formulate a phase diagram exhibiting a wide range of wrapped and open structures characterizing the chain-ball complexes. The elastic energy of a closed WLC loop of length L attached to an adhesive sphere at N sites can be expressed as: H = 12 ds κ 1 R(s)       2 + C dθ ds       2           ∫ − μN (1) In Eq. (1), κ is the bending stiffness and 1 R s ( ) the curvature of the chain at the point s along its contour. The stiffness is normally expressed as κ = kBTξ , with ξ the orientational persistence length of the chain (about 500Å for DNA under standard conditions). The torsional angle of the chain is θ and the torsional stiffness is C. (For DNA, C is comparable in magnitude to κ; we will set C = κ , and then argue later – see below Eq. (3) – that this term can be neglected altogether.) The diameter b of the chain (20Å for DNA) is assumed small compared to the diameter D of the spherical organizing center (about 110Å for nucleosomes). The last term of Eq. (1) describes the interaction between the sphere and the WLC, with μ ≅ λ Dδ the binding energy per adhesive site; λ is the binding energy per unit length, the length of adsorbed chain per site is of order Dδ , and δ is the range of the attractive interaction. We will assume short-range interactions with δ <>ξ    