# The Kinetic Theory of Mutation Rates

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## Abstract

**:**

## 1. Introduction

## 2. Basics of Kinetic Theory

**Remark**

**1.**

**Remark**

**2.**

## 3. The Kinetic Descriptions of Mutation Rates

**Remark**

**3.**

#### The Role of the Fourier Transform

## 4. The Quasi-Invariant Limit of the Growth of Mutant Cells

**Theorem**

**1.**

#### 4.1. The Bartlett Formulation

#### 4.2. Numerical Examples

Algorithm 1: Monte Carlo method for the kinetic model (18) under scaling (33). |

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Castellano, C.; Fortunato, S.; Loreto, V. Statistical physics of social dynamics. Rev. Mod. Phys.
**2009**, 81, 591–646. [Google Scholar] [CrossRef] - Düring, B.; Matthes, D.; Toscani, G. A Boltzmann type approach to the formation of wealth distribution curves. Riv. Mat. Univ. Parma
**2009**, 1, 199–261. [Google Scholar] [CrossRef] - Naldi, G.; Pareschi, L.; Toscani, G. Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences; Birkhäuser: Boston, MA, USA, 2010. [Google Scholar]
- Pareschi, L.; Toscani, G. Interacting Multiagent Systems: Kinetic Equations & Monte Carlo Methods; Oxford University Press: Oxford, UK, 2013. [Google Scholar]
- Cercignani, C. The Boltzmann Equation and Its Applications; Applied Mathematical Sciences, 67; Springer: New York, NY, USA, 1988. [Google Scholar]
- Luria, S.E.; Delbrück, M. Mutations of bacteria from virus sensitivity to virus resistance. Genetics
**1943**, 28, 491–511. [Google Scholar] [CrossRef] - Lea, D.E.; Coulson, C.A. The distribution of the numbers of mutants in bacterial populations. J. Genet.
**1949**, 49, 264–285. [Google Scholar] [CrossRef] - Angerer, W.P. An explicit representation of the Luria-Delbrück distribution. J. Math. Biol.
**2001**, 42, 145–174. [Google Scholar] [CrossRef] [PubMed] - Armitage, P. The statistical theory of bacterial populations subject to mutation. J. R. Statist. Soc. B
**1952**, 14, 1–40. [Google Scholar] [CrossRef] - Bartlett, M.S. An Introduction to Stochastic Processes with Special Reference to Methods and Applications, 2nd ed.; Cambridge University Press: London, UK; New York, NY, USA, 1966. [Google Scholar]
- Crump, K.S.; Hoel, D.G. Mathematical models for estimating mutation rates in cell populations. Biometrika
**1974**, 61, 237–252. [Google Scholar] [CrossRef] - Jones, M.E.; Thomas, S.M.; Rogers, A. Luria-Delbrück fluctuation experiments: Design and analysis. Genetics
**1994**, 136, 1209–1216. [Google Scholar] [CrossRef] - Koch, A.L. Mutation and growth rates from Luria-Delbrück fluctuation tests. Mutat. Res.
**1982**, 95, 129–143. [Google Scholar] [CrossRef] - Ma, W.T.; Sandri, G.v.H.; Sarkar, S. Analysis of the Luria and Delbrück distribution using discrete convolution powers. J. Appl. Prob.
**1992**, 29, 255–267. [Google Scholar] [CrossRef] - Mandelbrot, B. A population birth-and-mutation process, I: Explicit distributions for the number of mutants in an old culture of bacteria. J. Appl. Prob.
**1974**, 11, 437–444. [Google Scholar] [CrossRef] - Pakes, A.G. Remarks on the Luria-Delbrück distribution. J. Appl. Prob.
**1993**, 30, 991–994. [Google Scholar] [CrossRef] - Stewart, F.M.; Gordon, D.M.; Levin, B.R. Fluctuation analysis: The probability distribution of the number of mutants under different conditions. Genetics
**1990**, 124, 175–185. [Google Scholar] [CrossRef] [PubMed] - Zheng, Q. Progress of a half century in the study of the Luria–Delbrück distribution. Math. Biosci.
**1999**, 162, 1–32. [Google Scholar] [CrossRef] [PubMed] - Kashdan, E.; Pareschi, L. Mean field dynamics and the continuous Luria–Delbrück distribution. Math. Biosci.
**2012**, 240, 223–230. [Google Scholar] [CrossRef] - Toscani, G. A kinetic description of mutation processes in bacteria. Kinet. Relat. Models
**2013**, 6, 1043–1055. [Google Scholar] [CrossRef] - Kendall, D.G. Stochastic processes and population growth. J. R. Stat. Soc. B
**1949**, 11, 230–264. [Google Scholar] [CrossRef] - Sakamoto, S. Global solutions to an equation for mutation process in bacteria and its preservation of positive supports. J. Math. Anal. Appl.
**2022**, 507, 125771. [Google Scholar] [CrossRef] - Bernardi, B.; Pareschi, L.; Toscani, G.; Zanella, M. Effects of vaccination efficacy on wealth distribution in kinetic epidemic models. Entropy
**2022**, 24, 216. [Google Scholar] [CrossRef] [PubMed] - Chakraborti, A. Distributions of money in models of market economy. Int. J. Mod. Phys. C
**2002**, 13, 1315–1321. [Google Scholar] [CrossRef] - Chakraborti, A.; Chakrabarti, B.K. Statistical mechanics of money: Effects of saving propensity. Eur. Phys. J. B
**2000**, 17, 167–170. [Google Scholar] [CrossRef] - Chatterjee, A.; Chakrabarti, B.K.; Stinchcombe, R.B. Master equation for a kinetic model of trading market and its analytic solution. Phys. Rev. E
**2005**, 72, 026126. [Google Scholar] [CrossRef] - Cordier, S.; Pareschi, L.; Toscani, G. On a kinetic model for a simple market economy. J. Stat. Phys.
**2005**, 120, 253–277. [Google Scholar] [CrossRef] - Dimarco, G.; Pareschi, L.; Toscani, G.; Zanella, M. Wealth distribution under the spread of infectious diseases. Phys. Rev. E
**2020**, 102, 022303. [Google Scholar] [CrossRef] - Drǎgulescu, A.; Yakovenko, V.M. Statistical mechanics of money. Eur. Phys. J. B
**2000**, 17, 723–729. [Google Scholar] [CrossRef] - Düring, B.; Matthes, D.; Toscani, G. Kinetic equations modelling wealth redistribution: A comparison of approaches. Phys. Rev. E
**2008**, 78, 056103. [Google Scholar] [CrossRef] - Hayes, B. Follow the money. Am. Sci.
**2002**, 90, 400–405. [Google Scholar] [CrossRef] - Iglesias, J.R.; Gonçalves, S.; Pianegonda, S.; Vega, J.L.; Abramson, G. Wealth redistribution in our small world, Nonequilibrium statistical mechanics and nonlinear physics. Physica A
**2003**, 327, 12–17. [Google Scholar] [CrossRef] - Slanina, F. Inelastically scattering particles and wealth distribution in an open economy. Phys. Rev. E
**2004**, 69, 046102. [Google Scholar] [CrossRef] - Matthes, D.; Toscani, G. On steady distributions of kinetic models of conservative economies. J. Stat. Phys.
**2008**, 130, 1087–1117. [Google Scholar] [CrossRef] - Bobylev, A.V. The theory of the spatially Uniform Boltzmann equation for Maxwell molecules. Sov. Sci. Rev. C
**1988**, 7, 111–233. [Google Scholar] - Toscani, G. Kinetic models of opinion formation. Commun. Math. Sci.
**2006**, 4, 481–496. [Google Scholar] [CrossRef] - Dimarco, G.; Pareschi, L.; Zanella, M. Micro-macro stochastic Galerkin methods for nonlinear Fokker-Plank equations with random inputs. arXiv
**2022**, arXiv:2207.06494. [Google Scholar] - Pareschi, L.; Zanella, M. Structure preserving schemes for nonlinear Fokker–Planck equations and applications. J. Sci. Comput.
**2018**, 74, 1575–1600. [Google Scholar] [CrossRef] - Zheng, Q. New algorithms for Luria–Delbrück fluctuation analysis. Math. Biosci.
**2005**, 196, 198–214. [Google Scholar] [CrossRef] - Zheng, Q. Comparing mutation rates under the Luria–Delbrück protocol. Genetica
**2016**, 144, 351–359. [Google Scholar] [CrossRef] [PubMed] - Gualandi, S.; Toscani, G.; Vercesi, E. A kinetic description of the body size distributions of species. Math. Mod. Meth. Appl. Sci.
**2022**, 32, 2853–2885. [Google Scholar] [CrossRef] - Galton, F. Typical laws of heredity. Nature
**1877**, 15, 492–495. [Google Scholar] - Galton, F. Natural Inheritance; Mcmillan & Co.: London, UK, 1889. [Google Scholar]
- Stigler, S.M. The History of Statistics: The Measurement of Uncertainty before 1900; Harvard University Press: Cambridge, MA, USA, 1986. [Google Scholar]
- Furioli, G.; Pulvirenti, A.; Terraneo, E.; Toscani, G. Fokker–Planck equations in the modelling of socio-economic phenomena. Math. Mod. Meth. Appl. Sci.
**2017**, 27, 115–158. [Google Scholar] [CrossRef] - Iksanov, A. Renewal Theory for Perturbed Random Walks and Similar Processes; Probability and Its Applications; Springer: Berlin/Heidelberg, Germany, 2016. [Google Scholar]
- Bassetti, F.; Toscani, G. Explicit equilibria in a kinetic model of gambling. Phys. Rev. E
**2010**, 81, 066115. [Google Scholar] [CrossRef] - Gabetta, E.; Toscani, G.; Wennberg, B. Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation. J. Statist. Phys.
**1995**, 81, 901–934. [Google Scholar] [CrossRef] - Zolotarev, V.M. Metric distances in spaces of random variables and their measures. Math. USSR-Sb
**1976**, 30, 373. [Google Scholar] [CrossRef] - Zheng, Q. On Bartlett’s formulation of the Luria–Delbrück mutation model. Math. Biosci.
**2008**, 215, 48–54. [Google Scholar] [CrossRef] - Knuth, D.E. The Art of Computer Programming; Addison Wesley: Reading, MA, USA, 1997; Volume 1. [Google Scholar]
- Marsaglia, G. The incomplete Γ function as a continuous Poisson distribution. Comput. Math. Appl.
**1986**, 12, 1187–1190. [Google Scholar] [CrossRef]

**Figure 1.**Luria–Delbrück case. Distribution of mutant cells at ${T}_{f}=6.7$ with ${\beta}_{1}=3$, ${\beta}_{2}=2.5$ for $\epsilon =0.1$ (

**left**) and $\epsilon =0.01$ (

**right**) in the kinetic model (18) under scaling (33), with $\mathcal{L}\left(v\right)=\beta v$. The reference solution is computed using Lemma 2 in [18].

**Figure 2.**Lea–Coulson case. Distribution of mutant cells at ${T}_{f}=6.7$ with ${\beta}_{1}=3$ and ${\beta}_{2}=2.8$ for $\epsilon =0.1$ (

**left**) and $\epsilon =0.01$ (

**right**) in the kinetic model (18) under scaling (33), with $\mathcal{L}\left(v\right)$, a Poisson process of mean $\beta v$. The reference solution is computed using Lemma 2 in [18] and numerical quadrature.

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Pareschi, L.; Toscani, G. The Kinetic Theory of Mutation Rates. *Axioms* **2023**, *12*, 265.
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Pareschi L, Toscani G. The Kinetic Theory of Mutation Rates. *Axioms*. 2023; 12(3):265.
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Pareschi, Lorenzo, and Giuseppe Toscani. 2023. "The Kinetic Theory of Mutation Rates" *Axioms* 12, no. 3: 265.
https://doi.org/10.3390/axioms12030265