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August  2022, 21(8): 2643-2660. doi: 10.3934/cpaa.2022065

Wellposedness of a DNA replication model based on a nucleation-growth process

J. Nieto 1,, and  M. O. Vásquez 2,

1. 

Dpto. Matemática Aplicada and Research Unit "Modeling Nature" (MNat), Universidad de Granada, 18071 Granada, Spain
2. 

Universidad de Cartagena, Cartagena, Colombia

* Corresponding author

Received  July 2021 Revised  October 2021 Published  August 2022 Early access  April 2022

Fund Project: The first author is supported by JJAA (Spain) projects P18-RT-242 & A-FQM-311-UGR18, and MICINN (Spain) project RTI2018-098850-B-I00

In this paper, we analyze a nonlinear equation modeling the mechanical replication of the DNA molecule based on a Kolmogorov-Jhonson-Mehl-Avrami (KJMA) type model inspired on the mathematical analogy between the DNA replication process and the crystal growth. There are two different regions on the DNA molecule deep into a duplication process, the connected regions where the base pairs have been already duplicated, called eyes or islands and the regions not yet duplicated, called holes. The Cauchy problem associated with this model will be analyzed, where some dependences and nonlinearities on the replication velocity and the origins of replication are introduced.

Keywords: DNA replication model, partial differential equations, nucleation-and-growth model, Cauchy problem, crystal growth, KJMA.
Mathematics Subject Classification: Primary: 35Q92, 35A01; Secondary: 00A71, 45K05.
Citation: J. Nieto, M. O. Vásquez. Wellposedness of a DNA replication model based on a nucleation-growth process. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2643-2660. doi: 10.3934/cpaa.2022065
References:
[1]

M. Avrami, Granulation, phase change, and microstuture kinetics of phase change. III, J. Chem. Phys., 9 (1941), 177-184.  doi: 10.1063/1.1750872.

[2]

M. U. Bäbler and M. Morbidelli, Analysis of the aggregation-fragmentation population balance equation with application to coagulation, J. Colloid Interface Sci., 316 (2007), 428-441.  doi: 10.1016/j.jcis.2007.08.029.

[3]

E. Ben-Naim and P. L. Krapivsky, Nucleation and growth in one dimension, Phys. Rev. E, 54 (1996), 3562-3568.  doi: 10.1103/PhysRevE.54.3562.

[4]

J. J. Blow, P. J. Gillespie, D. Francis and D. A. Jackson, Replication origins in Xenopus egg extract Are 5–15 kilobases apart and are activated in clusters that fire at different times, J. Cell Biol., 152 (2001), 15-25.  doi: 10.1083/jcb.152.1.15.

[5]

J. Calvo, M. Doumic and B. Perthame, Long-time Asymptotics for polymerizations models, Commun. Math. Phys., 363 (2018), 111-137.  doi: 10.1007/s00220-018-3218-5.

[6]

M. Doumic and P. Gabriel, Eigenelements of a general aggregation-fragmentation model, Math. Models Methods Appl. Sci., 20 (2010), 757-783.  doi: 10.1142/S021820251000443X.

[7]

M. G. Gauthier, P. Norio and J. Bechhoefer, Modeling inhomogeneous DNA replication kinetics, PLoS ONE, 7 (2012), e32053, 13 pp. doi: 10.1371/journal.pone.0032053.

[8]

D. Ghosh, J. Saha and J. Kumar, Existence and uniqueness of steady-state solution to a singular coagulation-fragmentation equation, J. Comput. Appl. Math., 380 (2020), 112992, 11 pp. doi: 10.1016/j.cam.2020.112992.

[9]

J. Herrick, S. Jun, J. Bechhoefer and A. Bensimon, Kinetic model of DNA replication in eukaryotic organisms, J. Mol. Biol., 320 (2002), 741-750.  doi: 10.1016/s0022-2836(02)00522-3.

[10]

W. A. Johnson and R. F Mehl, Reaction kinetics in processes of nucleation and growth, Trans. AIME, 135 (1939), 416-442. 

[11]

A. Kolmogorov, A statistical theory for the recrystallization of metals, Bull. Acad. Sci. USSR, Ser. Math., 3 (1937), 335-359. 

[12]

S. Jun, H. Zhang and J. Bechhoefer, Nucleation and growth in one dimension. I. The generalized Kolmogorov-Johnson-Mehl-Avrami model, Phys. Review E, 71 (2005), 011908, 8pp. doi: 10.1103/PhysRevE.71.011908.

[13]

S. Jun and J. Bechhoefer, Nucleation and growth in one dimension. II. Application to DNA replication kinetics, Phys. Rev. E, 71 (2005), 011909, 8pp. doi: 10.1103/PhysRevE.71.011909.

[14]

S. Jun, J. Herrick, A. Bensimon and J. Bechhoefer, Persistence length of chromatin determines origin spacing in Xenopus Early-Embryo DNA replication: Quantitative comparisons between theory and experiment, Cell Cycle, 3 (2004), 211-217.  doi: 10.4161/cc.3.2.655.

[15]

P. Lauren\c{c}ot, Stationary solution to coagulation-fragmentation equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 36, (2019), 1903–1939. doi: 10.1016/j.anihpc.2019.06.003.

[16]

C. Lécot C., P. L'Ecuyer, R. El Haddad and A. Tarhini, Quasi-Monte Carlo simulation of coagulation-fragmentation, Math. Comput. Simul., 161 (2019), 113-124.  doi: 10.1016/j.matcom.2019.02.003.

[17]

K. Sekimoto, Evolution of the domain structure during the nucleation-and-growth process with non-conserved order parameter, Int. J. Mod. Phys. B, 5 (1991), 1843-1869.  doi: 10.1142/S0217979291000717.

[18]

K. Sekimoto, Evolution of the domain structure during the nucleation-and-growth process with non-conserved order parameter, Phys. A, 135 (1986), 328-346.  doi: 10.1016/0378-4371(86)90146-9.

[19]

K. Sekimoto, Kinetics of magnetization switching in a 1-D system-size distribution of unswitched domains, Phys. A, 125 (1984), 261-269.  doi: 10.1016/0378-4371(84)90014-1.

[20]

K. Sekimoto, Kinetics of magnetization switching in a 1-D system. II. Long time behavior of switched domains, Physica A, 128 (1984), 132-149. 

[21]

M. Tomellini, M. Fanfoni and M. Volpe, Spatially correlated nuclei: How the Johnson-Mehl-Avrami-Kolmogorov formula is modified in the case of simultaneous nucleation, Phys. Rev. B, 62 (2000), 4 pp. doi: 10.1103/PhysRevB.62.11300.

[22]

S. C. Yang and J. Bechhoefer, How Xenopus laevis embryos replicate reliably: Investigating the random-completion problem, Phys. Rev. E, 78 (2008), 041917, 15 pp. doi: 10.1103/PhysRevE.78.041917.

show all references

References:
[1]

M. Avrami, Granulation, phase change, and microstuture kinetics of phase change. III, J. Chem. Phys., 9 (1941), 177-184.  doi: 10.1063/1.1750872.

[2]

M. U. Bäbler and M. Morbidelli, Analysis of the aggregation-fragmentation population balance equation with application to coagulation, J. Colloid Interface Sci., 316 (2007), 428-441.  doi: 10.1016/j.jcis.2007.08.029.

[3]

E. Ben-Naim and P. L. Krapivsky, Nucleation and growth in one dimension, Phys. Rev. E, 54 (1996), 3562-3568.  doi: 10.1103/PhysRevE.54.3562.

[4]

J. J. Blow, P. J. Gillespie, D. Francis and D. A. Jackson, Replication origins in Xenopus egg extract Are 5–15 kilobases apart and are activated in clusters that fire at different times, J. Cell Biol., 152 (2001), 15-25.  doi: 10.1083/jcb.152.1.15.

[5]

J. Calvo, M. Doumic and B. Perthame, Long-time Asymptotics for polymerizations models, Commun. Math. Phys., 363 (2018), 111-137.  doi: 10.1007/s00220-018-3218-5.

[6]

M. Doumic and P. Gabriel, Eigenelements of a general aggregation-fragmentation model, Math. Models Methods Appl. Sci., 20 (2010), 757-783.  doi: 10.1142/S021820251000443X.

[7]

M. G. Gauthier, P. Norio and J. Bechhoefer, Modeling inhomogeneous DNA replication kinetics, PLoS ONE, 7 (2012), e32053, 13 pp. doi: 10.1371/journal.pone.0032053.

[8]

D. Ghosh, J. Saha and J. Kumar, Existence and uniqueness of steady-state solution to a singular coagulation-fragmentation equation, J. Comput. Appl. Math., 380 (2020), 112992, 11 pp. doi: 10.1016/j.cam.2020.112992.

[9]

J. Herrick, S. Jun, J. Bechhoefer and A. Bensimon, Kinetic model of DNA replication in eukaryotic organisms, J. Mol. Biol., 320 (2002), 741-750.  doi: 10.1016/s0022-2836(02)00522-3.

[10]

W. A. Johnson and R. F Mehl, Reaction kinetics in processes of nucleation and growth, Trans. AIME, 135 (1939), 416-442. 

[11]

A. Kolmogorov, A statistical theory for the recrystallization of metals, Bull. Acad. Sci. USSR, Ser. Math., 3 (1937), 335-359. 

[12]

S. Jun, H. Zhang and J. Bechhoefer, Nucleation and growth in one dimension. I. The generalized Kolmogorov-Johnson-Mehl-Avrami model, Phys. Review E, 71 (2005), 011908, 8pp. doi: 10.1103/PhysRevE.71.011908.

[13]

S. Jun and J. Bechhoefer, Nucleation and growth in one dimension. II. Application to DNA replication kinetics, Phys. Rev. E, 71 (2005), 011909, 8pp. doi: 10.1103/PhysRevE.71.011909.

[14]

S. Jun, J. Herrick, A. Bensimon and J. Bechhoefer, Persistence length of chromatin determines origin spacing in Xenopus Early-Embryo DNA replication: Quantitative comparisons between theory and experiment, Cell Cycle, 3 (2004), 211-217.  doi: 10.4161/cc.3.2.655.

[15]

P. Lauren\c{c}ot, Stationary solution to coagulation-fragmentation equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 36, (2019), 1903–1939. doi: 10.1016/j.anihpc.2019.06.003.

[16]

C. Lécot C., P. L'Ecuyer, R. El Haddad and A. Tarhini, Quasi-Monte Carlo simulation of coagulation-fragmentation, Math. Comput. Simul., 161 (2019), 113-124.  doi: 10.1016/j.matcom.2019.02.003.

[17]

K. Sekimoto, Evolution of the domain structure during the nucleation-and-growth process with non-conserved order parameter, Int. J. Mod. Phys. B, 5 (1991), 1843-1869.  doi: 10.1142/S0217979291000717.

[18]

K. Sekimoto, Evolution of the domain structure during the nucleation-and-growth process with non-conserved order parameter, Phys. A, 135 (1986), 328-346.  doi: 10.1016/0378-4371(86)90146-9.

[19]

K. Sekimoto, Kinetics of magnetization switching in a 1-D system-size distribution of unswitched domains, Phys. A, 125 (1984), 261-269.  doi: 10.1016/0378-4371(84)90014-1.

[20]

K. Sekimoto, Kinetics of magnetization switching in a 1-D system. II. Long time behavior of switched domains, Physica A, 128 (1984), 132-149. 

[21]

M. Tomellini, M. Fanfoni and M. Volpe, Spatially correlated nuclei: How the Johnson-Mehl-Avrami-Kolmogorov formula is modified in the case of simultaneous nucleation, Phys. Rev. B, 62 (2000), 4 pp. doi: 10.1103/PhysRevB.62.11300.

[22]

S. C. Yang and J. Bechhoefer, How Xenopus laevis embryos replicate reliably: Investigating the random-completion problem, Phys. Rev. E, 78 (2008), 041917, 15 pp. doi: 10.1103/PhysRevE.78.041917.

Figure 1.  Analogy between DNA replication and crystal growth. We identify the growing islands with the growing eyes in the DNA case, a nucleation point (N) or zero size island with an origin of replication and holes with non replicated DNA regions. The arrows indicate the symmetric bidirectional growth of the islands or the DNA synthesis
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Figure 2.  Several characteristics curves $ Y(s) = \ln \left( e^{Y(0)} -2vs \right) $
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