The tyranny of correspondence principle

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  • JID:PLREV AID:476 /DIS [m3SC+; v 1.191; Prn:4/04/2014; 12:35] P.1 (1-3)Available online at www.sciencedirect.com

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    Comment

    The tyranny of correspondence principleComment on Fluctuations in the DNA double helix: A criticalreview by Maxim D. Frank-Kamenetskii and Shikha Prakash

    Alexander Y. GrosbergDepartment of Physics and Center for Soft Matter Research, New York University, 4 Washington Place, New York, NY 10003, USA

    Received 24 March 2014; accepted 24 March 2014

    Communicated by E. Shakhnovich

    The paper [1] by M.D. Frank-Kamenetskii and S. Prakash is a timely review of a controversy which exists in theterature about fluctuational openings in DNA double helix, and all related matters.

    Authors discuss two theoretical approaches to the problem of fluctuations in DNA, one viewing helix openingsfluctuations of melting over a short DNA length, and the other based on an attempt to directly describe the DNAnamics by guessing a few generalized coordinates with appropriate Hamiltonian and, in particular, ignoring the rolewater. Authors review both approaches, forcefully advocating the former one, based on extrapolation of the DNA

    elting theory.Basic understanding of DNA melting, under the name helixcoil transition, dates back to the 1960s and 70s,

    summary can be found, e.g., in [2, Chapter 7]. The simplest model is equivalent to 1D Ising system. This model is,course, far too primitive. One necessary improvement is the so-called loop factor, peculiar polymer entropy of theo unwound chains which must connect at both ends of the molten part. As a function of loop length , the partitionnction of a molten loop behaves as e/c , and the scenario of helixcoil transition in a homopolymer turns outdepend quite sensitively on the power c, the subject of great interest in theoretical physics community [3]. The

    cond necessary improvement is due to the difference in stabilities between AT and GC pairs. Complete theory mustclude both heterogeneous sequence and the loop factor [4], and when the rigidity of the helical parts is properlynsidered [5], this theory accounts for experimentally observed character of the melting curves. The same basicinciples also allowed to develop the theory of forced helixcoil transition, under the name of unzipping [6], whichcame important with the advent of single molecule experimentation.Loop factor amounts effectively to a sort of long range interaction between base pairs along DNA, which represents

    major annoyance to computational treatment of melting curves as a function of the specific sequence. At the sameme, of course, the presence of the loop factor is a very attractive feature of this theory, because it does reflect theture of DNA: there are loops indeed; accordingly, the long range interaction between melting base pairs is not justcomputational annoyance, but a physical reality. For instance, if one considers melting of a DNA confined in afficiently thin capillary, then entropy restrictions for the loops will be reduced and long range interactions will be

    DOI of original article: http://dx.doi.org/10.1016/j.plrev.2014.01.005.

    tp://dx.doi.org/10.1016/j.plrev.2014.03.00771-0645/ 2014 Elsevier B.V. All rights reserved.

  • JID:PLREV AID:476 /DIS [m3SC+; v 1.191; Prn:4/04/2014; 12:35] P.2 (1-3)2 A.Y. Grosberg / Physics of Life Reviews ()

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    recreened; according to the recent work [7], such effects are not observed yet, but will soon become accessible toperimentation.For the reasons not entirely clear to the present writer, the theory of helixcoil transition appears to be largely

    rgotten in the present day biophysics community. For instance, although helixcoil transition is quite prominentlyesented in the modern online lecture course on Statistical Physics in Biology [8], the most common modernxtbooks on molecular biophysics [9,10] only briefly mention helixcoil transition (under the name of melting), and

    not provide any explanations. Other recent biophysics books [11,12] (which are otherwise very good in the opinionthis author) do not mention this topic at all. A more elementary, perhaps undergraduate level, book [13] does have

    decent presentation of the simplest helixcoil transition, but, justifiably for its level, does not cover more advancedpics, such as heteropolymers, loop factors etc. Among the recent biophysics books only the lesser known one [14]entions the loop factor.It is a sad fact that present day students and younger scientists mostly do not know anything about helixcoil

    ansition, never heard of the beautiful science associated with that concept, and, accordingly, ignore its findings. Inactice, for many students, the only thing to be known about DNA melting is the web address of one of the manytes where melting temperature of a DNA oligomer of any particular sequence can be found for free in a few clicks.

    Perhaps at least a part of the reason for this truly unfortunate situation has to do with the fact that in many (or evenmost) of practical biophysical situations the small scale events are of the greatest interest, while classical helixcoil

    ansition theory mostly considers regimes with long molten loops. As it is explained in the review [1], helixcoilansition theory operates with phenomenological parameters, such as the free energy difference per base pair betweenlical and coiled states and free energy of a boundary between helical and coiled states. Both parameters arendamentally free energies, because helical and coiled are macrostates. Clearly, such description can be validly under certain restrictive conditions: (a) interconversions between microstates of helical state are fast, helicalacrostate equilibrates fast; (b) same for coiled macrostate; (c) transitions from helical to coiled macrostates arefrequent; (d) transition process from coiled to helical macrostate or back is fast, the system does not waste much timetransit. In particular, the necessary condition is obviously 1. A priori, this places under doubt the applicability oflixcoil transition theory to the problems of short scale fluctuations, such as fluctuational openings of double helix.hether helixcoil transition theory results, valid in 1 regime, can be extrapolated to the border line of theirplicability at 1, is by no means obvious, it is only for experiment to decide. The review [1] nicely summarizese investigation of this very non-trivial question. The conclusion, which appears difficult to dispute, is that helixcoilansition theory, even in the questionable limit 1, gives correctly at least an order of magnitude estimate forobability of fluctuational openings.Of course, this does not invalidate the goal to develop another theory which would give a more detailed descriptionthe 1 regime. Authors of the review [1] are quite skeptical about the possibility to develop such theory andint out that even sufficiently complete molecular dynamics simulations are beyond the reach at present. Whetheris skepticism is right or not, it seems important to emphasize that whatever theory or simulation of the future willoffered to treat this problem will have to obey the correspondence principle, and will have to return to the well

    tablished two-state helixcoil results in the 1 limit.As a reminder, correspondence principle was first formulated by N. Bohr in quantum mechanics [15], as a statement

    at quantum dynamics at large quantum numbers should approach the predictions of classical mechanics. Later on,came to be recognized in theoretical physics as a quite general requirement that any novel theory must agree withe well established previous ones in the range of applicability of the latter.In the opinion of this writer, presently known attempts to develop a theory of small scale fluctuations in double

    lix do not meet the correspondence principle criteria, as they fail to reproduce the helixcoil transition results in the 1 range even if only the statistical mechanics equilibrium is considered. In this regard, it is instructive to lookthe work [16] which offers a computational method to find melting curves based on nonlinear lattice dynamics.

    f course, since melting curve is an equilibrium property, the water friction (the tyranny of water) is not relevant,t other aspects of water presence, such as its contribution to the interactions between bases, are taken into accountadjusting the parameters of the Hamiltonian in their dependence on the ionic strength; that dependence already

    ggests, in accordance with the claim in [1], that these parameters are free energies, not energies. The resultingethod may be useful for practical computation as an efficient interpolation in some range, but its applicability range

    mains unclear; surely, the method becomes invalid when loops are long, 1, precisely because it ignores the loop

  • JID:PLREV AID:476 /DIS [m3SC+; v 1.191; Prn:4/04/2014; 12:35] P.3 (1-3)A.Y. Grosberg / Physics of Life Reviews () 3

    factor which, as stated above, is an experimentally detectable fact of life. This is to be contrasted with the work [5],which does introduce proper loop factors.

    Other small scale phenomena involving strong perturbations of a helix are also actively discussed in the literature,such as kinks [1719]. These problems also need to be considered through the lens of correspondence principle, whichwas undertaken in the recent review [20] examining the relation between kinks and helixcoil transition.

    So far this note concentrated almost exclusively on the equilibrium statistical mechanics, including fluctuations.Dynamics is, of course, an entirely different matter. In this regard it is hard to disagree with the authors [1] that waterfriction is the dominant factor. At the same time, it seems also desirable to move the discussion to the new level, byexamining, both experimentally and theoretically, a more detailed aspects of fluctuational dynamics of DNA, such as,for instance, two point correlation functions between openings of different bases, as they depend both on frequencyand on spatial (or genomic) distance.

    The author acknowledges useful discussion with M.D. Frank-Kamenetskii.

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    Sov Phys JETP 1986;64:128490.5] Cule D, Hwa T. Denaturation of heterogeneous DNA. Phys Rev Lett 1997;79:23758.6] Nelson D. Statistical physics of unzipping DNA. In: Skjeltorp A, Belushkin A, editors. Forces, growth and form in soft condensed matter: at

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