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

Modeling DNA thermal denaturation at the mesoscopic level

Marina Dolfin 1, and  Mirosław Lachowicz 2,

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.
Keywords: integro--differential equations., DNA thermal denaturation, mesoscopic model.
Mathematics Subject Classification: Primary: 92B05, 35R09; Secondary: 35R0.
Citation: Marina Dolfin, Mirosław Lachowicz. Modeling DNA thermal denaturation at the mesoscopic level. Discrete & 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).  doi: 10.1103/PhysRevE.79.031903.  Google Scholar

[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.  doi: 10.1016/j.aml.2011.09.043.  Google Scholar

[3]

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

[4]

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

[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.  doi: 10.1103/PhysRevE.84.031918.  Google Scholar

[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.  doi: 10.1016/j.physleta.2012.05.062.  Google Scholar

[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.   Google Scholar

[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).  doi: 10.1142/S0218202512005885.  Google Scholar

[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.  doi: 10.1016/j.camwa.2011.09.043.  Google Scholar

[10]

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

[11]

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

[12]

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

[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.   Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[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.  doi: 10.1063/1.2827471.  Google Scholar

[20]

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

[21]

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

[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.  doi: 10.1016/j.aml.2012.04.021.  Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[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, (1970).   Google Scholar

[30]

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

[31]

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

[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.  doi: 10.1186/1471-2199-9-96.  Google Scholar

[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.  doi: 10.1186/1472-6750-8-47.  Google Scholar

[34]

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

[35]

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

[36]

C. J. Thompson, Mathematical Statistical Mechanics,, Princeton University, (1979).   Google Scholar

[37]

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

[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.   Google Scholar

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).  doi: 10.1103/PhysRevE.79.031903.  Google Scholar

[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.  doi: 10.1016/j.aml.2011.09.043.  Google Scholar

[3]

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

[4]

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

[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.  doi: 10.1103/PhysRevE.84.031918.  Google Scholar

[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.  doi: 10.1016/j.physleta.2012.05.062.  Google Scholar

[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.   Google Scholar

[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).  doi: 10.1142/S0218202512005885.  Google Scholar

[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.  doi: 10.1016/j.camwa.2011.09.043.  Google Scholar

[10]

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

[11]

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

[12]

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

[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.   Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[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.  doi: 10.1063/1.2827471.  Google Scholar

[20]

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

[21]

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

[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.  doi: 10.1016/j.aml.2012.04.021.  Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[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, (1970).   Google Scholar

[30]

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

[31]

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

[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.  doi: 10.1186/1471-2199-9-96.  Google Scholar

[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.  doi: 10.1186/1472-6750-8-47.  Google Scholar

[34]

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

[35]

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

[36]

C. J. Thompson, Mathematical Statistical Mechanics,, Princeton University, (1979).   Google Scholar

[37]

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

[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.   Google Scholar

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