Disorder and denaturation transition in the generalized Poland–Scheraga model
Annales Henri Lebesgue, Volume 3 (2020) , pp. 299-339.

Metadata

KeywordsDNA denaturation, disordered polymer pinning model, critical behavior, disorder relevance, two-dimensional renewal processes

Abstract

We investigate the generalized Poland–Scheraga model, which is used in the bio-physical literature to model the DNA denaturation transition, in the case where the two strands are allowed to be non-complementary (and to have different lengths). The homogeneous model was recently studied from a mathematical point of view in [BGK18, GK17], via a 2–dimensional renewal approach, with a loop exponent 2+α (α>0): it was found to undergo a localization/delocalization phase transition (which corresponds to the denaturation transition) of order ν=min(1,α) -1 , together with (in general) other phase transitions. In this paper, we turn to the disordered model, and we address the question of the influence of disorder on the denaturation phase transition, that is whether adding an arbitrarily small amount of disorder (i.e. inhomogeneities) affects the critical properties of this transition. Our results are consistent with Harris’ predictions for d-dimensional disordered systems (here d=2). First, we prove that when α<1 (i.e. ν>d/2), then disorder is irrelevant: the quenched and annealed critical points are equal, and the disordered denaturation phase transition is also of order ν=α -1 . On the other hand, when α>1, disorder is relevant: we prove that the quenched and annealed critical points differ. Moreover, we discuss a number of open problems, in particular the smoothing phenomenon that is expected to enter the game when disorder is relevant.


References

[AB18] Alexander, Kenneth S.; Berger, Quentin Pinning of a renewal on a quenched renewal, Electron. J. Probab., Volume 23 (2018), 6, 48 pages | MR 3771743 | Zbl 1390.60341

[Ale08] Alexander, Kenneth S. The effect of disorder on polymer depinning transitions, Commun. Math. Phys., Volume 279 (2008) no. 1, pp. 117-146 | Article | MR 2377630 | Zbl 1175.82034

[AZ09] Alexander, Kenneth S.; Zygouras, Nikos Quenched and annealed critical points in polymer pinning models, Commun. Math. Phys., Volume 291 (2009) no. 3, pp. 659-689 | Article | MR 2534789 | Zbl 1188.82154

[BB01] Borovkov, Aleksandr Alekseevich; Borovkov, Konstantin A. On probabilities of large deviations for random walks. I. Regularly varying distribution tails, Theory Probab. Appl., Volume 46 (2001) no. 2, pp. 193-213 | Article | Zbl 1006.60020

[BBB + 99] Blake, R. D.; Bizzaro, J. W.; Blake, J. D.; Day, G. R.; Delcourt, Scott G.; Knowles, J.; Marx, K. A.; SantaLucia, J. Jr Statistical mechanical simulation of polymeric DNA melting with MELTSIM, Bioinformatics, Volume 15 (1999) no. 5, pp. 370-375 | Article

[BD98] Blake, R. D.; Delcourt, Scott G. Thermal stability of DNA, Nucleic Acids Res., Volume 26 (1998) no. 14, pp. 3323-3332 | Article

[Ber19a] Berger, Quentin Notes on random walks in the Cauchy domain of attraction, Probab. Theory Relat. Fields, Volume 175 (2019) no. 1-2, pp. 1-44 | Article | MR 4009704 | Zbl 07109856

[Ber19b] Berger, Quentin Strong renewal theorems and local large deviations for multivariate random walks and renewals, Electron. J. Probab., Volume 24 (2019), 46, 47 pages | MR 3949271 | Zbl 1412.60063

[BGK18] Berger, Quentin; Giacomin, Giambattista; Khatib, Maha DNA melting structures in the generalized Poland–Scheraga model, ALEA, Lat. Am. J. Probab. Math. Stat., Volume 15 (2018) no. 2, pp. 993-1025 | Article | MR 3852243 | Zbl 1398.92195

[BGL19] Berger, Quentin; Giacomin, Giambattista; Lacoin, Hubert Disorder and critical phenomena: the α=0 copolymer model, Probab. Theory Relat. Fields, Volume 174 (2019) no. 3-4, pp. 787-819 | Article | MR 3980305 | Zbl 07081457

[BGT87] Bingham, Nicolas Hugh; Goldie, Charles M.; Teugels, Jozef L. Regular variations, Encyclopedia of Mathematics and Its Applications, Volume 27, Cambridge University Press, 1987 | MR 898871

[BH02] Bundschuh, Ralf; Hwa, Terence Statistical mechanics of secondary structures formed by random RNA sequences, Phys. Rev. E, Volume 65 (2002) no. 3, 031903 | Article

[BL11] Berger, Quentin; Lacoin, Hubert The effect of disorder on the free-energy for the Random Walk Pinning Model: smoothing of the phase transition and low temperature asymptotics, J. Stat. Phys., Volume 142 (2011) no. 2, pp. 322-341 | Article | MR 2764128 | Zbl 1226.82023

[BL12] Berger, Quentin; Lacoin, Hubert Sharp critical behavior for pinning models in a random correlated environment, Stochastic Processes Appl., Volume 122 (2012) no. 4, pp. 1397-1436 | Article | MR 2914757 | Zbl 1266.82080

[BL17] Berger, Quentin; Lacoin, Hubert The high-temperature behavior of the directed polymer in dimension 1+2, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 53 (2017) no. 1, pp. 430-450 | Article | MR 3606747 | Zbl 1362.82055

[BL18] Berger, Quentin; Lacoin, Hubert Pinning on a defect line: characterization of marginal disorder relevance and sharp asymptotics for the critical point shift, J. Inst. Math. Jussieu, Volume 17 (2018) no. 2, pp. 305-346 | Article | MR 3773271 | Zbl 1405.60139

[BP15] Berger, Quentin; Poisat, Julien On the critical curve of the pinning and copolymer models in correlated Gaussian environment, Electron. J. Probab., Volume 20 (2015), 71, 35 pages | MR 3361259 | Zbl 1323.82022

[BS10] Birkner, Matthias; Sun, Rongfeng Annealed vs quenched critical points for a random walk pinning model, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 46 (2010) no. 2, pp. 414-441 | Article | Numdam | MR 2667704 | Zbl 1206.60087

[BS11] Birkner, Matthias; Sun, Rongfeng Disorder relevance for the random walk pinning model in dimension 3, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 47 (2011) no. 1, pp. 259-293 | Article | Numdam | MR 2779405 | Zbl 1217.60085

[BT10] Berger, Quentin; Toninelli, Fabio L. On the critical point of the Random Walk Pinning Model in dimension d=3, Electron. J. Probab., Volume 15 (2010), 21, pp. 654-683 | Article | MR 2650777 | Zbl 1226.82027

[CCP19] Cheliotis, Dimitris; Chino, Yuki; Poisat, Julien The random pinning model with correlated disorder given by a renewal set, Ann. Henri Lebesgue, Volume 2 (2019), pp. 281-329 | Article | MR 3978390 | Zbl 07099980

[CdH13] Caravenna, Francesco; den Hollander, Frank A general smoothing inequality for disordered polymers, Electron. Commun. Probab., Volume 18 (2013), 76, 15 pages | Article | MR 3109631 | Zbl 1329.60329

[Com07] Comets, Francis Weak disorder for low dimensional polymers: the model of stable laws, Markov Process. Relat. Fields, Volume 13 (2007) no. 4, pp. 681-696 | MR 2381596 | Zbl 1144.60059

[DGLT09] Derrida, Bernard; Giacomin, Giambattista; Lacoin, Hubert; Toninelli, Fabio L. Fractional moment bounds and disorder relevance for pinning models, Commun. Math. Phys., Volume 287 (2009) no. 3, pp. 867-887 | Article | MR 2486665 | Zbl 1226.82028

[DR14] Derrida, Bernard; Retaux, Martin The depinning transition in presence of disorder: a toy model, J. Stat. Phys., Volume 156 (2014) no. 2, pp. 268-290 | Article | Zbl 1312.82005

[EON11] Einert, T. R.; Orland, Henri; Netz, Roland R. Secondary structure formation of homopolymeric single-stranded nucleic acids including force and loop entropy: implications for DNA hybridization, Eur. Phys. J. E, Volume 34 (2011) no. 6, 55, 15 pages | Article

[Fis84] Fisher, Michael E. Walks, walls, wetting, and melting, J. Stat. Phys., Volume 34 (1984) no. 5-6, pp. 667-730 | Article | MR 751710 | Zbl 0589.60098

[Gia07] Giacomin, Giambattista Random polymer models, Imperial College Press; World Scientific, 2007 | Zbl 1125.82001

[Gia08] Giacomin, Giambattista Renewal convergence rates and correlation decay for homogeneous pinning models, Electron. J. Probab., Volume 13 (2008), 18, pp. 513-529 | Article | MR 2386741 | Zbl 1190.60086

[Gia11] Giacomin, Giambattista Disorder and critical phenomena through basic probability models. École d’Été de Probabilités de Saint-Flour XL – 2010, Lecture Notes in Mathematics, Volume 2025, Springer, 2011 | Zbl 1230.82004

[GK17] Giacomin, Giambattista; Khatib, Maha Generalized Poland–Sheraga denaturation model and two dimensional renewal processes, Stochastic Processes Appl., Volume 127 (2017) no. 2, pp. 526-573 | Article | Zbl 1354.60118

[GLT10] Giacomin, Giambattista; Lacoin, Hubert; Toninelli, Fabio L. Marginal relevance of disorder for pinning models, Commun. Pure Appl. Math., Volume 63 (2010) no. 2, pp. 233-265 | Article | MR 2588461 | Zbl 1189.60173

[GLT11] Giacomin, Giambattista; Lacoin, Hubert; Toninelli, Fabio L. Disorder relevance at marginality and critical point shif, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 47 (2011) no. 1, pp. 148-175 | Article | Zbl 1210.82036

[GO03] Garel, Thomas; Orland, Henri On the role of mismatches in DNA denaturation (2003) (https://arxiv.org/abs/cond-mat/0304080)

[GO04] Garel, Thomas; Orland, Henri Generalized Poland–Scheraga model for DNA hybridization, Biopolymers, Volume 75 (2004) no. 6, pp. 453-467 | Article

[GT06a] Giacomin, Giambattista; Toninelli, Fabio L. The localized phase of disordered copolymers with adsorption, ALEA, Lat. Am. J. Probab. Math. Stat., Volume 1 (2006), pp. 149-180 | MR 2249653 | Zbl 1134.82006

[GT06b] Giacomin, Giambattista; Toninelli, Fabio L. Smoothing effect of quenched disorder on polymer depinning transitions, Commun. Math. Phys., Volume 266 (2006) no. 1, pp. 1-16 | Article | MR 2231963 | Zbl 1113.82032

[Kha16] Khatib, Maha Le modèle de Poland–Scheraga généralisé/une approche de renouvellement bidimensionel pour la dénaturation de l’ADN (2016) (Ph. D. Thesis)

[Lac10a] Lacoin, Hubert The martingale approach to disorder irrelevance for pinning models, Electron. Commun. Probab., Volume 15 (2010), pp. 418-427 | Article | MR 2726088 | Zbl 1221.82058

[Lac10b] Lacoin, Hubert New bounds for the free energy of directed polymers in dimension 1+1 and 1+2, Commun. Math. Phys., Volume 294 (2010) no. 2, pp. 471-503 | Article | MR 2579463 | Zbl 1227.82098

[Mar02] Martin, James B. Linear growth for greedy lattice animals, Stochastic Processes Appl., Volume 98 (2002) no. 1, pp. 43-66 | Article | MR 1884923 | Zbl 1060.60045

[NG06] Neher, Richard A.; Gerland, Ulrich Intermediate phase in DNA melting, Phys. Rev. E, Volume 73 (2006) no. 4, 030902R

[PBG + 92] Peng, Chung-Kang; Buldyrev, Sergey V.; Goldberger, Ary L.; Havlin, Shlomo; Sciortino, Francesco; Simons, M.; Stanley, H. Eugene Long-range correlations in nucleotide sequences, Nature, Volume 356 (1992), pp. 168-170 | Article

[Pin81] Pinelis, Iosif Froimovich A problem on large deviations in a space of trajectories, Theory Probab. Appl., Volume 26 (1981), pp. 69-84 | Article | MR 605637

[PS70] Poland, Douglas; Scheraga, Harold A. Theory of helix-coil transitions in biopolymers. Statistical mechanical theory of order-disorder transitions in biological macromolecules, Academic Press Inc., 1970

[SW11] Shneer, Seva; Wachtel, Vitali I. A unified approach to the heavy-traffic analysis of the maximum of random walks, Theory Probab. Appl., Volume 55 (2011) no. 2, pp. 332-341 | Article | Zbl 1229.60059

[TN08] Tamm, Mikail V.; Nechaev, Serguei K. Unzipping of two random heteropolymers: Ground-state energy and finite-size effects, Phys. Rev. E, Volume 78 (2008), 011903, 12 pages

[Ton08] Toninelli, Fabio L. A replica-coupling approach to disordered pinning models, Commun. Math. Phys., Volume 280 (2008) no. 2, pp. 389-401 | Article | MR 2395475 | Zbl 1207.82026

[Wat12] Watbled, Frédérique Sharp asymptotics for the free energy of 1+1 dimensional directed polymers in an infinitely divisible environment, Electron. Commun. Probab., Volume 17 (2012), 53, 9 pages | MR 2999981 | Zbl 1306.60154

[Wei16] Wei, Ran On the long-range directed polymer model, J. Stat. Phys., Volume 165 (2016) no. 2, pp. 320-350 | MR 3554384 | Zbl 1355.82070