October  2014, 19(8): 2469-2482. doi: 10.3934/dcdsb.2014.19.2469

Modeling DNA thermal denaturation at the mesoscopic level

1. 

Dep. Ingegneria Civile Informatica Edile Ambientale e Matematica Applicata, University of Messina, Contrada Di Dio, Vill. S. Agata, 98166 Messina, Italy

2. 

Faculty of Mathematics, Informatics and Mechanics, Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa

Received  October 2013 Revised  April 2014 Published  August 2014

In this paper a mesoscopic approach is proposed to describe the process of breaking of hydrogen bonds during the DNA thermal denaturation, also known as DNA melting. A system of integro-differential equations describing the dynamic of the variable which characterizes the opening of the base pairs is proposed. In the derivation of the model non linear effects arising from the collective behavior, namely the interactions, of base pairs are taken into account. Solutions of the mesoscopic model show significative analogies with the experimental S-shaped curves describing the fraction of broken bonds as a function of temperature at the macroscopic level, althought we instead simulate the variation in time. With this respect a research perspective connecting the theoretical results to the experimental one is proposed.
Citation: Marina Dolfin, Mirosław Lachowicz. Modeling DNA thermal denaturation at the mesoscopic level. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2469-2482. doi: 10.3934/dcdsb.2014.19.2469
References:
[1]

A. E. Allahverdyan, Z. H. S. Gevorkian, C.-K. Hu and Th. M. Nieuwenhuizeni, How absorption influences DNA denaturation, Phys. Rev. E, 79 (2009), 031903, 15 pp. doi: 10.1103/PhysRevE.79.031903.

[2]

L. Arlotti, E. De Angelis, L. Fermo, M. Lachowicz and N. Bellomo, On a class of integro-differential equations modeling complex systems with nonlinear interactions, Appl. Math. Letters, 25 (2012), 490-495. doi: 10.1016/j.aml.2011.09.043.

[3]

A. Bar, A. Kabakcioglu and D. Mukamel, Macroscopic loop formation in circular DNA denaturation, Phys. Rev. E, 85 (2012), 051919.

[4]

A. Bar, A. Kabakcioglu and D. Mukamel, Constrained thermal denaturation of DNA underfixed linking number, Cent. Eur. J. Phys., 10 (2012), 582-586.

[5]

S. Behnia, A. Akhshani, M. Panahi, A. Mobaraki and M. Ghaderian, Multifractal analysis of thermal denaturation based on the Peyrard-Bishop-Dauxois model, Phys. Rev. E, 84 (2011), 1-8. doi: 10.1103/PhysRevE.84.031918.

[6]

S. Behnia, A. Akhshani, M. Panahi, A. Mobaraki and M. Ghaderian, Multifractal properties of denaturation process based on Peyrard-Bishop model, Phys. Letters A, 376 (2012), 2538-2547. doi: 10.1016/j.physleta.2012.05.062.

[7]

N. Bellomo and B. Carbonaro, Review: Toward a mathematical theory of living systems focusingon developmental biology and evolution: A review and perspectives, Phys. Life Reviews, 8 (2011), 1-18.

[8]

N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, On the asymptotic theory form microscopic to macroscopic growing tissue models: an overview with perspectives, Math. Models Meth. Appl. Sci., 22 (2012), 37 pp. doi: 10.1142/S0218202512005885.

[9]

V. Coscia, L. Fermo and N. Bellomo, On the mathematical theory of living systems II: The interplay between mathematics and system biology, Comp. Math. Appl., 62 (2011), 3902-3911. doi: 10.1016/j.camwa.2011.09.043.

[10]

T. Dauxois and M. Peyrad, Entropy-driven transition in a one-dimensional system, Phys. Rev. E, 51 (1995), 4027-4040. doi: 10.1103/PhysRevE.51.4027.

[11]

T. Dauxois, M. Peyrad and B. Bishop, Dynamics and thermodynamics of a nonlinear model for DNA denaturation, Phys. Rev. E, 47 (1993), 684-697. doi: 10.1103/PhysRevE.47.684.

[12]

R. Durrett, Probability Models for DNA Sequence Evolution, Springer, 2002. doi: 10.1007/978-1-4757-6285-3.

[13]

A. A. Evans and A. J. Levine, High-energy deformation of filaments with internal structureand localized torque-induced melting of DNA, Phys. Rev. E, 85 (2012), 1-9.

[14]

M. E. Fisher, Effect of excluded volume on phase transitions in biopolymers, J. Chem. Phys., 45 (1966), 1469-1473. doi: 10.1063/1.1727787.

[15]

R. M. Grey, Toeplitz and Circulant Matrices: A Review,, NOW the essence of knowledge, ().  doi: 10.1561/0100000006.

[16]

A. Hanke, M. G. Ochoa and R. Metzler, Denaturation Transition of Stretched DNA, Phys. Rev. Lett., 100 (2008), 1-4. doi: 10.1103/PhysRevLett.100.018106.

[17]

Y. Kafri, D. Mukamel and L. Peliti, Why is the DNA denaturation transition first order, Phys. Review Letters, 85 (2000), 4988-4991. doi: 10.1103/PhysRevLett.85.4988.

[18]

G. Kalosakas and S. Ares, Dependence on temperature and guanine-cytosine content of bubble length distributions in DNA, Chem. Phys., 130 (2009), 1-7. doi: 10.1063/1.3149859.

[19]

J. Y. Kim, J. H. Jeon and W. Sung, A breathing wormlike chain model on DNA denaturation and bubble:Effects of stacking interactions, J. Chem. Phys., 128 (2008), 1-6. doi: 10.1063/1.2827471.

[20]

, Lab Manual: Measuring DNA Melting Curves,, 2013. Available from: , (). 

[21]

M. Lachowicz, Microscopic, mesoscopic and macroscopic descriptions of complex systems, Probabilistic Engineering Mechanics, 26 (2010), 54-60. doi: 10.1016/j.probengmech.2010.06.007.

[22]

M. Lachowicz and A. Quartarone, A general framework for modeling tumor-immune system competition at the mesoscopic level, Appl. Math. Letters, 25 (2012), 2118-2122. doi: 10.1016/j.aml.2012.04.021.

[23]

M. Lachowicz, Individually-based Markov processes modeling nonlinear systems in mathematical biology, Nonlinear Analysis Real World Appl., 12 (2011), 2396-2407. doi: 10.1016/j.nonrwa.2011.02.014.

[24]

M. Lachowicz and D. Wrzosek, Nonlocal bilinear equations. Equilibrium solutions and diffusive limit, Math. Models Methods Appl. Sci., 11 (2001), 1393-1409. doi: 10.1142/S0218202501001380.

[25]

M. Lachowicz and T. V. Ryabukha, Equilibrium solutions for microscopic stochastic systems in population dynamics, Math. Biosci. Engin., 10 (2013), 777-786. doi: 10.3934/mbe.2013.10.777.

[26]

M. Peyrard, Nonlinear dynamics and statistical physics of DNA, Nonlinearity, 17 (2004), R1-R40. doi: 10.1088/0951-7715/17/2/R01.

[27]

M. Peyrard, S. Cuesta-López and G. James, Modelling DNA at the mesoscale: A challenge for nonlinear science?, Nonlinearity, 21 (2008), T91-T100. doi: 10.1088/0951-7715/21/6/T02.

[28]

M. Peyrard and B. Bishop, Statistical Mechanics of a Nonlinear Model for DNA Denaturation, Phys. Rev. Lett., 62 (1989), 1-70. doi: 10.1103/PhysRevLett.62.2755.

[29]

D. Poland and H. A. Scheraga, Theory of Helix-Coil Transitions in Biopolymers: Statistical Mechanical Theory of Order-Disorder Transitions in Biological Macromolecules, Academic, New York, 1970.

[30]

S. J. Rahi, M. P. Hertzberg and M. Kardar, Melting of persistent double-stranded polymers, Phys. Rev. E, 78 (2008), 1-8. doi: 10.1103/PhysRevE.78.051910.

[31]

S. J. Rahi, M. P. Hertzberg and M. Kardar, Denaturation of circular DNA: Supercoil mechanism, Phys. Rev. E, 84 (2008), 1-8.

[32]

R.G. Rutledge and D. Stewart, Critical evaluation of methods used to determine amplification efficiency refutes the exponential character of real-time PCR, BMC Molecular Biology, 9 (2008), 96-108. doi: 10.1186/1471-2199-9-96.

[33]

R. G. Rutledge and D. Stewart, A kinetic-based sigmoidal model for the polymerase chain reaction and its applications to high-capacity absolute quantitative real-time PCR, BMC Molecular Biology, 8 (2008), 47-75. doi: 10.1186/1472-6750-8-47.

[34]

P. Sadhukhan and S. M. Bhattacharjee, Entanglement entropy of a quantum unbinding transition and entropy of DNA,, preprint, (). 

[35]

N. Theodorakopoulos, Bubbles, Clusters and denaturation in genomic DNA: modeling, parametrization efficient computation, Journal of Nonlinear Mathematical Physics, 18 (2011), 429-447. doi: 10.1142/S1402925111001611.

[36]

C. J. Thompson, Mathematical Statistical Mechanics, Princeton University, 1979.

[37]

J. D. Watson and F. H. C. Crick, A Structure for Deoxyribose Nucleic Acid, Nature, 171 (1953), 737-738.

[38]

K. A. Velizhanin, C. C. Chien, Y. Dubi and M. Zwolak, Bubbles, Driving denaturation: Nanoscalethermal transport as a probe of DNA melting, Phys. Rev. E, 83 (2011), 1-4.

show all references

References:
[1]

A. E. Allahverdyan, Z. H. S. Gevorkian, C.-K. Hu and Th. M. Nieuwenhuizeni, How absorption influences DNA denaturation, Phys. Rev. E, 79 (2009), 031903, 15 pp. doi: 10.1103/PhysRevE.79.031903.

[2]

L. Arlotti, E. De Angelis, L. Fermo, M. Lachowicz and N. Bellomo, On a class of integro-differential equations modeling complex systems with nonlinear interactions, Appl. Math. Letters, 25 (2012), 490-495. doi: 10.1016/j.aml.2011.09.043.

[3]

A. Bar, A. Kabakcioglu and D. Mukamel, Macroscopic loop formation in circular DNA denaturation, Phys. Rev. E, 85 (2012), 051919.

[4]

A. Bar, A. Kabakcioglu and D. Mukamel, Constrained thermal denaturation of DNA underfixed linking number, Cent. Eur. J. Phys., 10 (2012), 582-586.

[5]

S. Behnia, A. Akhshani, M. Panahi, A. Mobaraki and M. Ghaderian, Multifractal analysis of thermal denaturation based on the Peyrard-Bishop-Dauxois model, Phys. Rev. E, 84 (2011), 1-8. doi: 10.1103/PhysRevE.84.031918.

[6]

S. Behnia, A. Akhshani, M. Panahi, A. Mobaraki and M. Ghaderian, Multifractal properties of denaturation process based on Peyrard-Bishop model, Phys. Letters A, 376 (2012), 2538-2547. doi: 10.1016/j.physleta.2012.05.062.

[7]

N. Bellomo and B. Carbonaro, Review: Toward a mathematical theory of living systems focusingon developmental biology and evolution: A review and perspectives, Phys. Life Reviews, 8 (2011), 1-18.

[8]

N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, On the asymptotic theory form microscopic to macroscopic growing tissue models: an overview with perspectives, Math. Models Meth. Appl. Sci., 22 (2012), 37 pp. doi: 10.1142/S0218202512005885.

[9]

V. Coscia, L. Fermo and N. Bellomo, On the mathematical theory of living systems II: The interplay between mathematics and system biology, Comp. Math. Appl., 62 (2011), 3902-3911. doi: 10.1016/j.camwa.2011.09.043.

[10]

T. Dauxois and M. Peyrad, Entropy-driven transition in a one-dimensional system, Phys. Rev. E, 51 (1995), 4027-4040. doi: 10.1103/PhysRevE.51.4027.

[11]

T. Dauxois, M. Peyrad and B. Bishop, Dynamics and thermodynamics of a nonlinear model for DNA denaturation, Phys. Rev. E, 47 (1993), 684-697. doi: 10.1103/PhysRevE.47.684.

[12]

R. Durrett, Probability Models for DNA Sequence Evolution, Springer, 2002. doi: 10.1007/978-1-4757-6285-3.

[13]

A. A. Evans and A. J. Levine, High-energy deformation of filaments with internal structureand localized torque-induced melting of DNA, Phys. Rev. E, 85 (2012), 1-9.

[14]

M. E. Fisher, Effect of excluded volume on phase transitions in biopolymers, J. Chem. Phys., 45 (1966), 1469-1473. doi: 10.1063/1.1727787.

[15]

R. M. Grey, Toeplitz and Circulant Matrices: A Review,, NOW the essence of knowledge, ().  doi: 10.1561/0100000006.

[16]

A. Hanke, M. G. Ochoa and R. Metzler, Denaturation Transition of Stretched DNA, Phys. Rev. Lett., 100 (2008), 1-4. doi: 10.1103/PhysRevLett.100.018106.

[17]

Y. Kafri, D. Mukamel and L. Peliti, Why is the DNA denaturation transition first order, Phys. Review Letters, 85 (2000), 4988-4991. doi: 10.1103/PhysRevLett.85.4988.

[18]

G. Kalosakas and S. Ares, Dependence on temperature and guanine-cytosine content of bubble length distributions in DNA, Chem. Phys., 130 (2009), 1-7. doi: 10.1063/1.3149859.

[19]

J. Y. Kim, J. H. Jeon and W. Sung, A breathing wormlike chain model on DNA denaturation and bubble:Effects of stacking interactions, J. Chem. Phys., 128 (2008), 1-6. doi: 10.1063/1.2827471.

[20]

, Lab Manual: Measuring DNA Melting Curves,, 2013. Available from: , (). 

[21]

M. Lachowicz, Microscopic, mesoscopic and macroscopic descriptions of complex systems, Probabilistic Engineering Mechanics, 26 (2010), 54-60. doi: 10.1016/j.probengmech.2010.06.007.

[22]

M. Lachowicz and A. Quartarone, A general framework for modeling tumor-immune system competition at the mesoscopic level, Appl. Math. Letters, 25 (2012), 2118-2122. doi: 10.1016/j.aml.2012.04.021.

[23]

M. Lachowicz, Individually-based Markov processes modeling nonlinear systems in mathematical biology, Nonlinear Analysis Real World Appl., 12 (2011), 2396-2407. doi: 10.1016/j.nonrwa.2011.02.014.

[24]

M. Lachowicz and D. Wrzosek, Nonlocal bilinear equations. Equilibrium solutions and diffusive limit, Math. Models Methods Appl. Sci., 11 (2001), 1393-1409. doi: 10.1142/S0218202501001380.

[25]

M. Lachowicz and T. V. Ryabukha, Equilibrium solutions for microscopic stochastic systems in population dynamics, Math. Biosci. Engin., 10 (2013), 777-786. doi: 10.3934/mbe.2013.10.777.

[26]

M. Peyrard, Nonlinear dynamics and statistical physics of DNA, Nonlinearity, 17 (2004), R1-R40. doi: 10.1088/0951-7715/17/2/R01.

[27]

M. Peyrard, S. Cuesta-López and G. James, Modelling DNA at the mesoscale: A challenge for nonlinear science?, Nonlinearity, 21 (2008), T91-T100. doi: 10.1088/0951-7715/21/6/T02.

[28]

M. Peyrard and B. Bishop, Statistical Mechanics of a Nonlinear Model for DNA Denaturation, Phys. Rev. Lett., 62 (1989), 1-70. doi: 10.1103/PhysRevLett.62.2755.

[29]

D. Poland and H. A. Scheraga, Theory of Helix-Coil Transitions in Biopolymers: Statistical Mechanical Theory of Order-Disorder Transitions in Biological Macromolecules, Academic, New York, 1970.

[30]

S. J. Rahi, M. P. Hertzberg and M. Kardar, Melting of persistent double-stranded polymers, Phys. Rev. E, 78 (2008), 1-8. doi: 10.1103/PhysRevE.78.051910.

[31]

S. J. Rahi, M. P. Hertzberg and M. Kardar, Denaturation of circular DNA: Supercoil mechanism, Phys. Rev. E, 84 (2008), 1-8.

[32]

R.G. Rutledge and D. Stewart, Critical evaluation of methods used to determine amplification efficiency refutes the exponential character of real-time PCR, BMC Molecular Biology, 9 (2008), 96-108. doi: 10.1186/1471-2199-9-96.

[33]

R. G. Rutledge and D. Stewart, A kinetic-based sigmoidal model for the polymerase chain reaction and its applications to high-capacity absolute quantitative real-time PCR, BMC Molecular Biology, 8 (2008), 47-75. doi: 10.1186/1472-6750-8-47.

[34]

P. Sadhukhan and S. M. Bhattacharjee, Entanglement entropy of a quantum unbinding transition and entropy of DNA,, preprint, (). 

[35]

N. Theodorakopoulos, Bubbles, Clusters and denaturation in genomic DNA: modeling, parametrization efficient computation, Journal of Nonlinear Mathematical Physics, 18 (2011), 429-447. doi: 10.1142/S1402925111001611.

[36]

C. J. Thompson, Mathematical Statistical Mechanics, Princeton University, 1979.

[37]

J. D. Watson and F. H. C. Crick, A Structure for Deoxyribose Nucleic Acid, Nature, 171 (1953), 737-738.

[38]

K. A. Velizhanin, C. C. Chien, Y. Dubi and M. Zwolak, Bubbles, Driving denaturation: Nanoscalethermal transport as a probe of DNA melting, Phys. Rev. E, 83 (2011), 1-4.

[1]

Elena Izquierdo-Kulich, José Manuel Nieto-Villar. Mesoscopic model for tumor growth. Mathematical Biosciences & Engineering, 2007, 4 (4) : 687-698. doi: 10.3934/mbe.2007.4.687

[2]

Olivier Bonnefon, Jérôme Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3057-3085. doi: 10.3934/dcdsb.2014.19.3057

[3]

Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Singular integro-differential equations with applications. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021051

[4]

Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Inverse problems on degenerate integro-differential equations. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022025

[5]

Michael Grinfeld, Harbir Lamba, Rod Cross. A mesoscopic stock market model with hysteretic agents. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 403-415. doi: 10.3934/dcdsb.2013.18.403

[6]

Jean-Michel Roquejoffre, Juan-Luis Vázquez. Ignition and propagation in an integro-differential model for spherical flames. Discrete and Continuous Dynamical Systems - B, 2002, 2 (3) : 379-387. doi: 10.3934/dcdsb.2002.2.379

[7]

Liang Zhang, Bingtuan Li. Traveling wave solutions in an integro-differential competition model. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 417-428. doi: 10.3934/dcdsb.2012.17.417

[8]

Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17

[9]

Yi Cao, Jianhua Wu, Lihe Wang. Fundamental solutions of a class of homogeneous integro-differential elliptic equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1237-1256. doi: 10.3934/dcds.2019053

[10]

Yubo Chen, Wan Zhuang. The extreme solutions of PBVP for integro-differential equations with caratheodory functions. Conference Publications, 1998, 1998 (Special) : 160-166. doi: 10.3934/proc.1998.1998.160

[11]

Ramasamy Subashini, Chokkalingam Ravichandran, Kasthurisamy Jothimani, Haci Mehmet Baskonus. Existence results of Hilfer integro-differential equations with fractional order. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 911-923. doi: 10.3934/dcdss.2020053

[12]

Tonny Paul, A. Anguraj. Existence and uniqueness of nonlinear impulsive integro-differential equations. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1191-1198. doi: 10.3934/dcdsb.2006.6.1191

[13]

Narcisa Apreutesei, Arnaud Ducrot, Vitaly Volpert. Travelling waves for integro-differential equations in population dynamics. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 541-561. doi: 10.3934/dcdsb.2009.11.541

[14]

Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977

[15]

Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065

[16]

Eitan Tadmor, Prashant Athavale. Multiscale image representation using novel integro-differential equations. Inverse Problems and Imaging, 2009, 3 (4) : 693-710. doi: 10.3934/ipi.2009.3.693

[17]

Patricio Felmer, Ying Wang. Qualitative properties of positive solutions for mixed integro-differential equations. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 369-393. doi: 10.3934/dcds.2019015

[18]

Sebti Kerbal, Yang Jiang. General integro-differential equations and optimal controls on Banach spaces. Journal of Industrial and Management Optimization, 2007, 3 (1) : 119-128. doi: 10.3934/jimo.2007.3.119

[19]

Ji Shu, Linyan Li, Xin Huang, Jian Zhang. Limiting behavior of fractional stochastic integro-Differential equations on unbounded domains. Mathematical Control and Related Fields, 2021, 11 (4) : 715-737. doi: 10.3934/mcrf.2020044

[20]

Martin Bohner, Osman Tunç. Qualitative analysis of integro-differential equations with variable retardation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 639-657. doi: 10.3934/dcdsb.2021059

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (126)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]