DNA denaturation bubbles at criticality

arXiv:0802.2194v1 [cond-mat.stat-mech] 15 Feb 2008 DNA denaturation bubbles at criticality Nikos Theodorakopoulos1,2 1Theoretical and Physical Chemistry Institute, National Hellenic Research Foundation, Vasileos Constantinou 48, 116 35 Athens, Greece 2Fachbereich Physik, Universit¨at Konstanz, 78457 Konstanz, Germany (Dated: June 16, 2021) The equilibrium statistical properties of DNA denaturation bubbles are examined in detail within the framework of the Peyrard-Bishop-Dauxois model. Bubble formation in homogeneous DNA is found to depend crucially on the presence of nonlinear base-stacking interactions. Small bubbles extending over less than 10 base pairs are associated with much larger free energies of formation per site than larger bubbles. As the critical temperature is approached, the free energy associated with further bubble growth becomes vanishingly small. An analysis of average displacement profiles of bubbles of varying sizes at different temperatures reveals almost identical scaled shapes in the absence of nonlinear stacking; nonlinear stacking leads to distinct scaled shapes of large and small bubbles. PACS numbers: 87.10.-e, 87.14.gk, 87.15.Zg I. INTRODUCTION The nonlinear dynamics and statistical physics of DNA denaturation have been widely investigated [1]. Recent work has attempted to extend mesoscopic scale model- ing in order to describe with sufficient accuracy how se- quence details determine the statistical and dynamical properties of local fluctuations. Such local fluctuations, known as “denaturation bubbles” are believed to be in- strumental in the initiation of the transcription process at physiological temperatures. The possibility of sponta- neous, sequence-specific formation of a mid-size bubble has been the subject of considerable research interest - and some debate - [2, 3, 4, 5, 6]. A related - but nonethe- less distinct - question which might be relevant to the pro- cess of transcription concerns the growth of a bubble to much larger sizes. Since this is by definition - at least in an asymptotic sense - a scale-free phenomenon, it is best addressed at the level of the underlying phase transition; moreover, at least its salient features should be evident within the context of the homogeneous (polynucleotide) chain. It should be recalled that nonlinear lattice dynam- ics based DNA modeling of the Peyrard-Bishop-Dauxois (PBD [7]) type predicts either an (effectively) first-order or a (strict) second-order phase transition, depending on whether non-linear base-stacking effects are taken into account or not [8]. In the case of second-order transition, it has been determined that domain walls (DW) become entropically stable at the critical temperature [9, 10]; re- cent numerical evidence from Monte-Carlo simulations [11] suggests that the average bubble size also becomes critical. The first-order transition case is slightly more complicated. Entropic effects are not sufficient to enable spontaneous DW formation at the critical temperature. This appears to rule out DWs as agents of thermal denat- uration. Bubbles are natural - and in fact have always been - prime suspects for this role. Very recent work [12] has demonstrated that large bubbles may form in this case as well, and that their probability distribution cannot be described by a simple exponential. Detailed data in the vicinity of the critical temperature are not available[13]. It is however known from previous work [8, 14] that most of the physics of nonlinear base-stacking is generated by an effective thermal barrier which modi- fies the on-site Morse-like potential. The point at which the effects of the thermal barrier become important de- fines a natural crossover between different types of be- havior. As this note will show in some detail, such a crossover is also present in bubble statistics. Bubbles which extend over a few sites are entirely non-critical; the onset of criticality is reflected only in the statistics of large bubbles; the asymptotic properties of the latter are such that the free-energy barrier toward bubble growth is lowered - approaching zero at the transition temper- ature, whereas the average bubble size always remains finite. It will be further shown that this dichotomy be- tween large and small bubbles is not restricted to the statistics; reduced shapes of large and small-size bub- bles - which scale uniformly in the absence of nonlinear stacking interactions - reveal distinct differences when nonlinear stacking is included. The paper is organized as follows: Section II includes model definitions, notation and general properties of bub- bles. Section III presents numerical results on bubble statistics based on direct matrix multiplication. Sec- tion IV formulates an alternative procedure based on an associated eigenvalue problem which provides emphasis on asymptotic properties. Section V discusses bubbles shapes. The final section includes a brief summary and discussion of some key points. 2 II. DEFINITIONS A. Model Th

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