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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

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 & 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.  doi: 10.1007/s002850000053.  Google Scholar

[2]

P. Armitage, The statistical theory of bacterial populations subject to mutation,, J. Royal Statist. Soc. B, 14 (1952), 1.   Google Scholar

[3]

M. S. Bartlett, An Introduction to Stochastic Processes With Special Reference to Methods and Applications,, Second edition Cambridge University Press, (1966).   Google Scholar

[4]

A. V. Bobylev, The theory of the spatially Uniform Boltzmann equation for Maxwell molecules,, Sov. Sci. Review C, 7 (1988), 111.   Google Scholar

[5]

C. Cercignani, The Boltzmann Equation and its Applications,, Applied Mathematical Sciences, (1988).  doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[6]

A. Chakraborti, Distributions of money in models of market economy,, Int. J. Modern Phys. C, 13 (2002), 1315.   Google Scholar

[7]

A. Chakraborti and B. K. Chakrabarti, Statistical mechanics of money: Effects of saving propensity,, Eur. Phys. J. B, 17 (2000), 167.   Google Scholar

[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).   Google Scholar

[9]

S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy,, J. Stat. Phys., 120 (2005), 253.  doi: 10.1007/s10955-005-5456-0.  Google Scholar

[10]

K. S. Crump and D. G. Hoel, Mathematical models for estimating mutation rates in cell populations,, Biometrika, 61 (1974), 237.   Google Scholar

[11]

A. Drăgulescu and V. M. Yakovenko, Statistical mechanics of money,, Eur. Phys. Jour. B, 17 (2000), 723.   Google Scholar

[12]

B. Düring, D. Matthes and G. Toscani, Kinetic equations modelling wealth redistribution: A comparison of approaches,, Phys. Rev. E, 78 (2008).  doi: 10.1103/PhysRevE.78.056103.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/BF02179298.  Google Scholar

[15]

B. Hayes, Follow the money,, American Scientist, 90 (2002), 400.   Google Scholar

[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.  doi: 10.1016/S0378-4371(03)00430-8.  Google Scholar

[17]

M. E. Jones, S. M. Thomas and A. Rogers, Luria-Delbrück fluctuation experiments: Design and analysis,, Genetics, 136 (1994), 1209.   Google Scholar

[18]

E. Kashdan and L. Pareschi, Mean field dynamics and the continuous Luria-Delbrück distribution,, Mathematical Biosciences, 240 (2012), 223.  doi: 10.1016/j.mbs.2012.08.001.  Google Scholar

[19]

D. G. Kendall, Stochastic processes and population growth,, Journal of the Royal Statistical Society, 11 (1949), 230.   Google Scholar

[20]

A. L. Koch, Mutation and growth rates from Luria-Delbrück fluctuation tests,, Mutat. Res., 95 (1982), 129.   Google Scholar

[21]

D. E. Lea and C. A. Coulson, The distribution of the numbers of mutants in bacterial populations,, J. Genetics, 49 (1949), 264.   Google Scholar

[22]

S. E. Luria and M. Delbrück, Mutations of bacteria from virus sensitivity to virus resistance,, Genetics, 28 (1943), 491.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[25]

D. Matthes and G. Toscani, On steady distributions of kinetic models of conservative economies,, J. Stat. Phys., 130 (2008), 1087.  doi: 10.1007/s10955-007-9462-2.  Google Scholar

[26]

A. G. Pakes, Remarks on the Luria-Delbrück distribution,, J. Appl. Prob., 30 (1993), 991.   Google Scholar

[27]

L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic equations & Monte Carlo Methods,, Oxford University Press, (2013).   Google Scholar

[28]

F. Slanina, Inelastically scattering particles and wealth distribution in an open economy,, Phys. Rev. E, 69 (2004).   Google Scholar

[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.   Google Scholar

[30]

G. Toscani, Kinetic models of opinion formation,, Commun. Math. Sci., 4 (2006), 481.   Google Scholar

[31]

Q. Zheng, Progress of a half century in the study of the Luria-Delbrück distribution,, Math. Biosciences, 162 (1999), 1.  doi: 10.1016/S0025-5564(99)00045-0.  Google Scholar

show all references

References:
[1]

W. P. Angerer, An explicit representation of the Luria-Delbrück distribution,, J. Math. Biol., 42 (2001), 145.  doi: 10.1007/s002850000053.  Google Scholar

[2]

P. Armitage, The statistical theory of bacterial populations subject to mutation,, J. Royal Statist. Soc. B, 14 (1952), 1.   Google Scholar

[3]

M. S. Bartlett, An Introduction to Stochastic Processes With Special Reference to Methods and Applications,, Second edition Cambridge University Press, (1966).   Google Scholar

[4]

A. V. Bobylev, The theory of the spatially Uniform Boltzmann equation for Maxwell molecules,, Sov. Sci. Review C, 7 (1988), 111.   Google Scholar

[5]

C. Cercignani, The Boltzmann Equation and its Applications,, Applied Mathematical Sciences, (1988).  doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[6]

A. Chakraborti, Distributions of money in models of market economy,, Int. J. Modern Phys. C, 13 (2002), 1315.   Google Scholar

[7]

A. Chakraborti and B. K. Chakrabarti, Statistical mechanics of money: Effects of saving propensity,, Eur. Phys. J. B, 17 (2000), 167.   Google Scholar

[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).   Google Scholar

[9]

S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy,, J. Stat. Phys., 120 (2005), 253.  doi: 10.1007/s10955-005-5456-0.  Google Scholar

[10]

K. S. Crump and D. G. Hoel, Mathematical models for estimating mutation rates in cell populations,, Biometrika, 61 (1974), 237.   Google Scholar

[11]

A. Drăgulescu and V. M. Yakovenko, Statistical mechanics of money,, Eur. Phys. Jour. B, 17 (2000), 723.   Google Scholar

[12]

B. Düring, D. Matthes and G. Toscani, Kinetic equations modelling wealth redistribution: A comparison of approaches,, Phys. Rev. E, 78 (2008).  doi: 10.1103/PhysRevE.78.056103.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/BF02179298.  Google Scholar

[15]

B. Hayes, Follow the money,, American Scientist, 90 (2002), 400.   Google Scholar

[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.  doi: 10.1016/S0378-4371(03)00430-8.  Google Scholar

[17]

M. E. Jones, S. M. Thomas and A. Rogers, Luria-Delbrück fluctuation experiments: Design and analysis,, Genetics, 136 (1994), 1209.   Google Scholar

[18]

E. Kashdan and L. Pareschi, Mean field dynamics and the continuous Luria-Delbrück distribution,, Mathematical Biosciences, 240 (2012), 223.  doi: 10.1016/j.mbs.2012.08.001.  Google Scholar

[19]

D. G. Kendall, Stochastic processes and population growth,, Journal of the Royal Statistical Society, 11 (1949), 230.   Google Scholar

[20]

A. L. Koch, Mutation and growth rates from Luria-Delbrück fluctuation tests,, Mutat. Res., 95 (1982), 129.   Google Scholar

[21]

D. E. Lea and C. A. Coulson, The distribution of the numbers of mutants in bacterial populations,, J. Genetics, 49 (1949), 264.   Google Scholar

[22]

S. E. Luria and M. Delbrück, Mutations of bacteria from virus sensitivity to virus resistance,, Genetics, 28 (1943), 491.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[25]

D. Matthes and G. Toscani, On steady distributions of kinetic models of conservative economies,, J. Stat. Phys., 130 (2008), 1087.  doi: 10.1007/s10955-007-9462-2.  Google Scholar

[26]

A. G. Pakes, Remarks on the Luria-Delbrück distribution,, J. Appl. Prob., 30 (1993), 991.   Google Scholar

[27]

L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic equations & Monte Carlo Methods,, Oxford University Press, (2013).   Google Scholar

[28]

F. Slanina, Inelastically scattering particles and wealth distribution in an open economy,, Phys. Rev. E, 69 (2004).   Google Scholar

[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.   Google Scholar

[30]

G. Toscani, Kinetic models of opinion formation,, Commun. Math. Sci., 4 (2006), 481.   Google Scholar

[31]

Q. Zheng, Progress of a half century in the study of the Luria-Delbrück distribution,, Math. Biosciences, 162 (1999), 1.  doi: 10.1016/S0025-5564(99)00045-0.  Google Scholar

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