Bubbles, clusters and denaturation in genomic DNA: modeling, parametrization, efficient computation

arXiv:1102.0259v1 [cond-mat.stat-mech] 1 Feb 2011 Journal of Nonlinear Mathematical Physics, Vol. X, No. XX (2011) 1–19 c⃝N. Theodorakopoulos BUBBLES, CLUSTERS AND DENATURATION IN GENOMIC DNA: MODELING, PARAMETRIZATION, EFFICIENT COMPUTATION NIKOS THEODORAKOPOULOS Theoretical and Physical Chemistry Institute, National Hellenic Research Foundation Vasileos Constantinou 48, 116 35 Athens, Greece and Fachbereich Physik, Universit¨at Konstanz, 78457 Konstanz, Germany ntheodor@eie.gr ; Nikos.Theodorakopoulos@uni-konstanz.de November 16, 2021 The paper uses mesoscopic, non-linear lattice dynamics based (Peyrard-Bishop-Dauxois, PBD) modeling to describe thermal properties of DNA below and near the denaturation temperature. Computationally efficient notation is introduced for the relevant statistical mechanics. Computed melting profiles of long and short heterogeneous sequences are presented, using a recently introduced reparametrization of the PBD model, and critically discussed. The statistics of extended open bubbles and bound clusters is formulated and results are presented for selected examples. Keywords: DNA melting; denaturation bubbles; Peyrard-Bishop-Dauxois (PBD) model 1. Introduction Thermal denaturation, i.e. separation of the two strands as a result of heating, is one of the oldest established physico-chemical results pertaining to the DNA molecule. The “melting”- of DNA was first observed [1] very soon after the determination of the double helical struc- ture. Theoretical descriptions in the sixties, proposed by Poland and Scheraga (PS) [2, 3], largely based on the concept of the helix-coil transition [4], were subsequently refined and developed in considerable detail, incorporating known enthalpic data on controlled oligomer thermodynamics and, when combined with appropriate software, can claim significant pre- dictive power in regard to actual experimental melting profiles [5,6]. Helix-coil models adopt a mesoscopic description of DNA, describing the open (coil) or closed (helix) state of an indi- vidual base pair in terms of a discrete, Ising-type variable. By virtue of their construction they cannot describe any dynamic phenomena. An interesting alternative was proposed two decades ago by Peyrard, Bishop and Daux- ois (PBD) [7,8]. In accordance with contemporary soft mode concepts related to structural phase transitions, they attempted a reduced, also mesoscopic, lattice-dynamics motivated description of the double helix. In their case, the relevant degree of freedom is a transverse displacement which represents how far an individual base pair is from equilibrium. This is locally determined by a nonlinear (typically Morse-like) potential which accounts for the effective hydrogen bonding linking bases together. The motion of neighboring base pairs is coupled in a way which favors double-helical ordering (DNA stacking interaction). As it turns out, this minimal nonlinear lattice-dynamics (also proposed in the context of wetting phenomena [9]) generates one of the simplest known one-dimensional models with short- range interactions which exhibit an exact phase transition in the thermodynamic limit. 1 2 N. Theodorakopoulos The PBD model of DNA does more than a “demonstration of principle” regarding the denaturation transition. As shown by Cule and Hwa [10] the randomness introduced by base- pair heterogeneity results in a structured rounding of the transition (multistep melting) in qualitative accordance with experimental observation. Recently, using improved computa- tional methods and a new, global set of model parameters, it has been possible to compute detailed, multipeak melting profiles of long DNA chains using only sequence information and salt concentration as input; although further parameter optimization is probably needed, agreement with experiment is impressive [11]. The model has also been used to calculate neutron scattering structure factors, also in very good agreement with recent experimental results [12]. Furthermore, as will be noted in this work, the low-frequency optical phonons of the model are in the range where Raman spectroscopy has located some vibrational activity in DNA [13]. Important issues remain open. One of the key advantages of the PBD approach is its potential ability to describe local openings (“denaturation bubbles”) of the double helix, such as those known to occur during the initial stage of the transription process. The rela- tively slow dynamics involved in these events would be consistent with the coarse graining implicit in the model. Bubble statistics has been extensively studied in the framework of the PBD model, and some specific correlation with transcription initiation sites has in fact been claimed [14] and debated [15, 16]. On a more modest - and yet quite fundamental - level, the formation of a single bubble has been observed by studying the melting of specially designed oligomers [17]. As the results reported here will show, although the process and its roug

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