A kinetic description of mutation processes in bacteria

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December 2013, 6(4): 1043-1055. doi: 10.3934/krm.2013.6.1043

A kinetic description of mutation processes in bacteria

Giuseppe Toscani ^1,

1.	University of Pavia, Department of Mathematics, Via Ferrata 1, 27100 Pavia

Received September 2013 Revised September 2013 Published November 2013

Abstract
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The Luria--Delbrück mutation model has been mathematically formulated in a number of ways. Last, a mean field picture derived from a kinetic formulation has been derived by Kashdan and Pareschi in [18]. There, the Luria--Delbrück distribution appears as the solution of a Fokker-Planck like equation obtained as the quasi-invariant asymptotics of a linear Boltzmann equation for the number density of the number of mutated cells. This paper addresses the kinetic description for the Lea--Coulson formulation [21], as well as for the Kendall formulation [19], focusing on important modeling issues closely linked with the distribution of the number of mutants. The paper additionally emphasizes basic principles which not only help to unify existing results but also allow for a useful extensions.

Keywords: Luria-Delbrück distribution, Fokker-Planck equations., kinetic models, mutation rates.

Mathematics Subject Classification: Primary: 92D25, 60K35; Secondary: 76P0.

Citation: Giuseppe Toscani. A kinetic description of mutation processes in bacteria. Kinetic and Related Models, 2013, 6 (4) : 1043-1055. doi: 10.3934/krm.2013.6.1043

References:

[1]	W. P. Angerer, An explicit representation of the Luria-Delbrück distribution, J. Math. Biol., 42 (2001), 145-174. doi: 10.1007/s002850000053.
[2]	P. Armitage, The statistical theory of bacterial populations subject to mutation, J. Royal Statist. Soc. B, 14 (1952), 1-40.
[3]	M. S. Bartlett, An Introduction to Stochastic Processes With Special Reference to Methods and Applications, Second edition Cambridge University Press, London-New York, 1966.
[4]	A. V. Bobylev, The theory of the spatially Uniform Boltzmann equation for Maxwell molecules, Sov. Sci. Review C, 7 (1988), 111-233.
[5]	C. Cercignani, The Boltzmann Equation and its Applications, Applied Mathematical Sciences, 67. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.
[6]	A. Chakraborti, Distributions of money in models of market economy, Int. J. Modern Phys. C, 13 (2002), 1315-1321.
[7]	A. Chakraborti and B. K. Chakrabarti, Statistical mechanics of money: Effects of saving propensity, Eur. Phys. J. B, 17 (2000), 167-170.
[8]	A. Chatterjee, B. K. Chakrabarti and R. B. Stinchcombe, Master equation for a kinetic model of trading market and its analytic solution, Phys. Rev. E, 72 (2005), 026126.
[9]	S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy, J. Stat. Phys., 120 (2005), 253-277. doi: 10.1007/s10955-005-5456-0.
[10]	K. S. Crump and D. G. Hoel, Mathematical models for estimating mutation rates in cell populations, Biometrika, 61 (1974), 237-252.
[11]	A. Drăgulescu and V. M. Yakovenko, Statistical mechanics of money, Eur. Phys. Jour. B, 17 (2000), 723-729.
[12]	B. Düring, D. Matthes and G. Toscani, Kinetic equations modelling wealth redistribution: A comparison of approaches, Phys. Rev. E, 78 (2008), 056103, 12 pp. doi: 10.1103/PhysRevE.78.056103.
[13]	B. Düring, D. Matthes and G. Toscani, A Boltzmann type approach to the formation of wealth distribution curves, Riv. Mat. Univ. Parma, 1 (2009), 199-261.
[14]	E. Gabetta, G. Toscani and B. Wennberg, Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation, J. Statist. Phys., 81 (1995), 901-934. doi: 10.1007/BF02179298.
[15]	B. Hayes, Follow the money, American Scientist, 90 (2002), 400-405.
[16]	J. R. Iglesias, S. Gonçalves, S. Pianegonda, J. L. Vega and G. Abramson, Wealth redistribution in our small world, Nonequilibrium statistical mechanics and nonlinear physics (MEDYFINOL '02) (Colonia del Sacramento), Physica A, 327 (2003), 12-17. doi: 10.1016/S0378-4371(03)00430-8.
[17]	M. E. Jones, S. M. Thomas and A. Rogers, Luria-Delbrück fluctuation experiments: Design and analysis, Genetics, 136 (1994), 1209-1216.
[18]	E. Kashdan and L. Pareschi, Mean field dynamics and the continuous Luria-Delbrück distribution, Mathematical Biosciences, 240 (2012), 223-230. doi: 10.1016/j.mbs.2012.08.001.
[19]	D. G. Kendall, Stochastic processes and population growth, Journal of the Royal Statistical Society, B, 11 (1949), 230-264.
[20]	A. L. Koch, Mutation and growth rates from Luria-Delbrück fluctuation tests, Mutat. Res., 95 (1982), 129-143.
[21]	D. E. Lea and C. A. Coulson, The distribution of the numbers of mutants in bacterial populations, J. Genetics, 49 (1949), 264-285.
[22]	S. E. Luria and M. Delbrück, Mutations of bacteria from virus sensitivity to virus resistance, Genetics, 28 (1943), 491-511.
[23]	W. T. Ma, G. v. H. Sandri and S. Sarkar, Analysis of the Luria and Delbrück distribution using discrete convolution powers, J. Appl. Prob., 29 (1992), 255-267.
[24]	B. Mandelbrot, A population birth-and-mutation process, I: Explicit distributions for the number of mutants in an old culture of bacteria, J. Appl. Prob., 11 (1974), 437-444.
[25]	D. Matthes and G. Toscani, On steady distributions of kinetic models of conservative economies, J. Stat. Phys., 130 (2008), 1087-1117. doi: 10.1007/s10955-007-9462-2.
[26]	A. G. Pakes, Remarks on the Luria-Delbrück distribution, J. Appl. Prob., 30 (1993), 991-994.
[27]	L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic equations & Monte Carlo Methods, Oxford University Press, Oxford 2013.
[28]	F. Slanina, Inelastically scattering particles and wealth distribution in an open economy, Phys. Rev. E, 69 (2004), 046102.
[29]	F. M. Stewart, D. M. Gordon and B. R. Levin, Fluctuation analysis: The probability distribution of the number of mutants under different conditions, Genetics, 124 (1990), 175-185.
[30]	G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496.
[31]	Q. Zheng, Progress of a half century in the study of the Luria-Delbrück distribution, Math. Biosciences, 162 (1999), 1-32. doi: 10.1016/S0025-5564(99)00045-0.

show all references

References:

[1]	W. P. Angerer, An explicit representation of the Luria-Delbrück distribution, J. Math. Biol., 42 (2001), 145-174. doi: 10.1007/s002850000053.
[2]	P. Armitage, The statistical theory of bacterial populations subject to mutation, J. Royal Statist. Soc. B, 14 (1952), 1-40.
[3]	M. S. Bartlett, An Introduction to Stochastic Processes With Special Reference to Methods and Applications, Second edition Cambridge University Press, London-New York, 1966.
[4]	A. V. Bobylev, The theory of the spatially Uniform Boltzmann equation for Maxwell molecules, Sov. Sci. Review C, 7 (1988), 111-233.
[5]	C. Cercignani, The Boltzmann Equation and its Applications, Applied Mathematical Sciences, 67. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.
[6]	A. Chakraborti, Distributions of money in models of market economy, Int. J. Modern Phys. C, 13 (2002), 1315-1321.
[7]	A. Chakraborti and B. K. Chakrabarti, Statistical mechanics of money: Effects of saving propensity, Eur. Phys. J. B, 17 (2000), 167-170.
[8]	A. Chatterjee, B. K. Chakrabarti and R. B. Stinchcombe, Master equation for a kinetic model of trading market and its analytic solution, Phys. Rev. E, 72 (2005), 026126.
[9]	S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy, J. Stat. Phys., 120 (2005), 253-277. doi: 10.1007/s10955-005-5456-0.
[10]	K. S. Crump and D. G. Hoel, Mathematical models for estimating mutation rates in cell populations, Biometrika, 61 (1974), 237-252.
[11]	A. Drăgulescu and V. M. Yakovenko, Statistical mechanics of money, Eur. Phys. Jour. B, 17 (2000), 723-729.
[12]	B. Düring, D. Matthes and G. Toscani, Kinetic equations modelling wealth redistribution: A comparison of approaches, Phys. Rev. E, 78 (2008), 056103, 12 pp. doi: 10.1103/PhysRevE.78.056103.
[13]	B. Düring, D. Matthes and G. Toscani, A Boltzmann type approach to the formation of wealth distribution curves, Riv. Mat. Univ. Parma, 1 (2009), 199-261.
[14]	E. Gabetta, G. Toscani and B. Wennberg, Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation, J. Statist. Phys., 81 (1995), 901-934. doi: 10.1007/BF02179298.
[15]	B. Hayes, Follow the money, American Scientist, 90 (2002), 400-405.
[16]	J. R. Iglesias, S. Gonçalves, S. Pianegonda, J. L. Vega and G. Abramson, Wealth redistribution in our small world, Nonequilibrium statistical mechanics and nonlinear physics (MEDYFINOL '02) (Colonia del Sacramento), Physica A, 327 (2003), 12-17. doi: 10.1016/S0378-4371(03)00430-8.
[17]	M. E. Jones, S. M. Thomas and A. Rogers, Luria-Delbrück fluctuation experiments: Design and analysis, Genetics, 136 (1994), 1209-1216.
[18]	E. Kashdan and L. Pareschi, Mean field dynamics and the continuous Luria-Delbrück distribution, Mathematical Biosciences, 240 (2012), 223-230. doi: 10.1016/j.mbs.2012.08.001.
[19]	D. G. Kendall, Stochastic processes and population growth, Journal of the Royal Statistical Society, B, 11 (1949), 230-264.
[20]	A. L. Koch, Mutation and growth rates from Luria-Delbrück fluctuation tests, Mutat. Res., 95 (1982), 129-143.
[21]	D. E. Lea and C. A. Coulson, The distribution of the numbers of mutants in bacterial populations, J. Genetics, 49 (1949), 264-285.
[22]	S. E. Luria and M. Delbrück, Mutations of bacteria from virus sensitivity to virus resistance, Genetics, 28 (1943), 491-511.
[23]	W. T. Ma, G. v. H. Sandri and S. Sarkar, Analysis of the Luria and Delbrück distribution using discrete convolution powers, J. Appl. Prob., 29 (1992), 255-267.
[24]	B. Mandelbrot, A population birth-and-mutation process, I: Explicit distributions for the number of mutants in an old culture of bacteria, J. Appl. Prob., 11 (1974), 437-444.
[25]	D. Matthes and G. Toscani, On steady distributions of kinetic models of conservative economies, J. Stat. Phys., 130 (2008), 1087-1117. doi: 10.1007/s10955-007-9462-2.
[26]	A. G. Pakes, Remarks on the Luria-Delbrück distribution, J. Appl. Prob., 30 (1993), 991-994.
[27]	L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic equations & Monte Carlo Methods, Oxford University Press, Oxford 2013.
[28]	F. Slanina, Inelastically scattering particles and wealth distribution in an open economy, Phys. Rev. E, 69 (2004), 046102.
[29]	F. M. Stewart, D. M. Gordon and B. R. Levin, Fluctuation analysis: The probability distribution of the number of mutants under different conditions, Genetics, 124 (1990), 175-185.
[30]	G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496.
[31]	Q. Zheng, Progress of a half century in the study of the Luria-Delbrück distribution, Math. Biosciences, 162 (1999), 1-32. doi: 10.1016/S0025-5564(99)00045-0.

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