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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. Grosberg

Department 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 theliterature 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 openingsas fluctuations of melting over a short DNA length, and the other based on an attempt to directly describe the DNAdynamics by guessing a few generalized coordinates with appropriate Hamiltonian and, in particular, ignoring the roleof water. Authors review both approaches, forcefully advocating the former one, based on extrapolation of the DNAmelting theory.

Basic understanding of DNA melting, under the name helix–coil transition, dates back to the 1960s and 70s,a summary can be found, e.g., in [2, Chapter 7]. The simplest model is equivalent to 1D Ising system. This model is,of course, far too primitive. One necessary improvement is the so-called loop factor, peculiar polymer entropy of thetwo unwound chains which must connect at both ends of the molten part. As a function of loop length �, the partitionfunction of a molten loop behaves as eε�/�c , and the scenario of helix–coil transition in a homopolymer turns outto depend quite sensitively on the power c, the subject of great interest in theoretical physics community [3]. Thesecond necessary improvement is due to the difference in stabilities between AT and GC pairs. Complete theory mustinclude both heterogeneous sequence and the loop factor [4], and when the rigidity of the helical parts is properlyconsidered [5], this theory accounts for experimentally observed character of the melting curves. The same basicprinciples also allowed to develop the theory of forced helix–coil transition, under the name of unzipping [6], whichbecame 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 representsa major annoyance to computational treatment of melting curves as a function of the specific sequence. At the sametime, of course, the presence of the loop factor is a very attractive feature of this theory, because it does reflect thenature of DNA: there are loops indeed; accordingly, the long range interaction between melting base pairs is not justa computational annoyance, but a physical reality. For instance, if one considers melting of a DNA confined in asufficiently 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.

http://dx.doi.org/10.1016/j.plrev.2014.03.0071571-0645/© 2014 Elsevier B.V. All rights reserved.

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“screened”; according to the recent work [7], such effects are not observed yet, but will soon become accessible toexperimentation.

For the reasons not entirely clear to the present writer, the theory of helix–coil transition appears to be largelyforgotten in the present day biophysics community. For instance, although helix–coil transition is quite prominentlypresented in the modern online lecture course on “Statistical Physics in Biology” [8], the most common moderntextbooks on molecular biophysics [9,10] only briefly mention helix–coil transition (under the name of melting), anddo not provide any explanations. Other recent biophysics books [11,12] (which are otherwise very good in the opinionof this author) do not mention this topic at all. A more elementary, perhaps undergraduate level, book [13] does havea decent presentation of the simplest helix–coil transition, but, justifiably for its level, does not cover more advancedtopics, such as heteropolymers, loop factors etc. Among the recent biophysics books only the lesser known one [14]mentions the loop factor.

It is a sad fact that present day students and younger scientists mostly do not know anything about helix–coiltransition, never heard of the beautiful science associated with that concept, and, accordingly, ignore its findings. Inpractice, for many students, the only thing to be known about DNA melting is the web address of one of the manysites 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 evenin most) of practical biophysical situations the small scale events are of the greatest interest, while classical helix–coiltransition theory mostly considers regimes with long molten loops. As it is explained in the review [1], helix–coiltransition theory operates with phenomenological parameters, such as the free energy difference per base pair betweenhelical and coiled states and free energy of a “boundary” between helical and coiled states. Both parameters arefundamentally free energies, because “helical” and “coiled” are macrostates. Clearly, such description can be validonly under certain restrictive conditions: (a) interconversions between microstates of “helical state” are fast, helicalmacrostate equilibrates fast; (b) same for “coiled macrostate”; (c) transitions from helical to coiled macrostates areinfrequent; (d) transition process from coiled to helical macrostate or back is fast, the system does not waste much timein transit. In particular, the necessary condition is obviously � � 1. A priori, this places under doubt the applicability ofhelix–coil transition theory to the problems of short scale fluctuations, such as fluctuational openings of double helix.Whether helix–coil transition theory results, valid in � � 1 regime, can be extrapolated to the border line of theirapplicability at � ∼ 1, is by no means obvious, it is only for experiment to decide. The review [1] nicely summarizesthe investigation of this very non-trivial question. The conclusion, which appears difficult to dispute, is that helix–coiltransition theory, even in the questionable limit � ∼ 1, gives correctly at least an order of magnitude estimate forprobability of fluctuational openings.

Of course, this does not invalidate the goal to develop another theory which would give a more detailed descriptionof the � ∼ 1 regime. Authors of the review [1] are quite skeptical about the possibility to develop such theory andpoint out that even sufficiently complete molecular dynamics simulations are beyond the reach at present. Whetherthis skepticism is right or not, it seems important to emphasize that whatever theory or simulation of the future willbe offered to treat this problem – will have to obey the correspondence principle, and will have to return to the wellestablished two-state helix–coil results in the � � 1 limit.

As a reminder, correspondence principle was first formulated by N. Bohr in quantum mechanics [15], as a statementthat quantum dynamics at large quantum numbers should approach the predictions of classical mechanics. Later on,it came to be recognized in theoretical physics as a quite general requirement that any novel theory must agree withthe 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 doublehelix do not meet the correspondence principle criteria, as they fail to reproduce the helix–coil transition results in the� � 1 range – even if only the statistical mechanics equilibrium is considered. In this regard, it is instructive to lookat the work [16] which offers a computational method to find melting curves based on nonlinear lattice dynamics.Of course, since melting curve is an equilibrium property, the water friction (“the tyranny of water”) is not relevant,but other aspects of water presence, such as its contribution to the interactions between bases, are taken into accountby adjusting the parameters of the Hamiltonian in their dependence on the ionic strength; that dependence alreadysuggests, in accordance with the claim in [1], that these parameters are free energies, not energies. The resultingmethod may be useful for practical computation as an efficient interpolation in some range, but its applicability rangeremains unclear; surely, the method becomes invalid when loops are long, � � 1, precisely because it ignores the loop

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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 [17–19]. 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 helix–coil 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.

References

[1] Frank-Kamenetskii MD, Prakash S. Fluctuations in the DNA double helix: a critical review. Phys Life Rev 2014.http://dx.doi.org/10.1016/j.plrev.2014.01.005 [in this issue].

[2] Grosberg A, Khokhlov A. Statistical physics of macromolecules. American Institute of Physics Press; 1994.[3] Kafri Y, Mukamel D, Peliti L. Why is the DNA denaturation transition first order? Phys Rev Lett 2000;85:4988–91.[4] Grosberg A, Shakhnovich EI. Theory of phase transitions of the coil-globule type in a heteropolymer chain with disordered sequence of links.

Sov Phys JETP 1986;64:1284–90.[5] Cule D, Hwa T. Denaturation of heterogeneous DNA. Phys Rev Lett 1997;79:2375–8.[6] Nelson D. Statistical physics of unzipping DNA. In: Skjeltorp A, Belushkin A, editors. Forces, growth and form in soft condensed matter: at

the interface between physics and biology. Kluwer Academic Publishers; 2004. p. 65–92.[7] Reisner W, Larsen N, Silahtaroglu A, Kristensen A, Tommerup N, Tegenfeldt J, et al. Single-molecule denaturation mapping of DNA in

nanofluidic channels. Proc Natl Acad Sci USA 2010;107:13294–9.[8] Kardar M, Mirny L. Statistical physics in biology. MIT OpenCourseWare: Massachusetts Institute of Technology, http://ocw.mit.edu/

courses/physics/8-592j-statistical-physics-in-biology-spring-2011; 2011.[9] Nelson P. Biological physics: energy, information, life. New York: W.H. Freeman Co; 2003.

[10] Phillips R, Kondev J, Theriot J, Garcia H. Physical biology of the cell. Garland Science; 2012.[11] Bialek W. Biophysics: searching for principles. Princeton University Press; 2012.[12] Schiessel H. Biophysics for beginners: a journey through the cell nucleus. Pan Stanford Publishing; 2013.[13] Dill K, Bromberg S. Molecular driving forces: statistical thermodynamics in biology, chemistry, physics, and nanoscience. Garland Science;

2010.[14] Sneppen K, Zocchi G. Physics in molecular biology. Cambridge University Press; 2005.[15] Bohr N. Über die serienspektra der element. Z Phys 1920;2:423–78.[16] Theodorakopoulos N. Melting of genomic DNA: predictive modeling by nonlinear lattice dynamics. Phys Rev E 2010;82:021905.[17] Wiggins P, van der Heijden T, Moreno-Herrero F, Spakowitz A, Phillips R, Widom J, et al. High flexibility of DNA on short length scales

probed by atomic force microscopy. Nat Nanotechnol 2006;1:137–41.[18] Vafabakhsh R, Ha T. Extreme bendability of DNA less than 100 base pairs long revealed by single-molecule cyclization. Science

2012;337:1097–101.[19] Nelson P. Spare the (elastic) rod. Science 2012;337:1045–6.[20] Vologodskii A, Frank-Kamenetskii M. Strong bending of the DNA double helix. Nucleic Acids Res 2013;41:6785–92.