Modeling DNA thermal denaturation at the mesoscopic level

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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

Abstract
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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 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.

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