{"title":"细菌奔跑和翻滚动力学模型的推导:定量和强收敛结果","authors":"Alain Blaustein","doi":"10.1111/sapm.70060","DOIUrl":null,"url":null,"abstract":"<p>During the past century, biologists and mathematicians investigated two mechanisms underlying bacteria motion: the run phase during which bacteria move in straight lines and the tumble phase in which they change their orientation. When surrounded by a chemical attractant, experiments show that bacteria increase their run time as moving up concentration gradients, leading to a biased random walk toward favorable regions. This observation raises the following question, which has drawn intense interest from both biological and mathematical communities: what cellular mechanisms enable bacteria to feel concentration gradients? In this article, we investigate an asymptotic regime that was proposed to explain this ability thanks to internal mechanisms. More precisely, we derive the run-and-tumble kinetic equation with concentration's gradient-dependent tumbling rate from a more comprehensive model, which incorporates internal cellular mechanisms. Our result improves on previous investigations, as we obtain strong convergence toward the gradient-dependent kinetic model with quantitative and formally optimal convergence rates. The main ingredient consists in identifying a set of coordinates for the internal cellular dynamics in which concentration gradients arise explicitly. Then, we use relative entropy methods in order to capture quantitative measurement of the distance between the model incorporating cellular mechanisms and the one with concentration-gradient-dependent tumbling rate.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"154 5","pages":""},"PeriodicalIF":2.6000,"publicationDate":"2025-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70060","citationCount":"0","resultStr":"{\"title\":\"Derivation of the Bacterial Run-and-Tumble Kinetic Model: Quantitative and Strong Convergence Results\",\"authors\":\"Alain Blaustein\",\"doi\":\"10.1111/sapm.70060\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>During the past century, biologists and mathematicians investigated two mechanisms underlying bacteria motion: the run phase during which bacteria move in straight lines and the tumble phase in which they change their orientation. When surrounded by a chemical attractant, experiments show that bacteria increase their run time as moving up concentration gradients, leading to a biased random walk toward favorable regions. This observation raises the following question, which has drawn intense interest from both biological and mathematical communities: what cellular mechanisms enable bacteria to feel concentration gradients? In this article, we investigate an asymptotic regime that was proposed to explain this ability thanks to internal mechanisms. More precisely, we derive the run-and-tumble kinetic equation with concentration's gradient-dependent tumbling rate from a more comprehensive model, which incorporates internal cellular mechanisms. Our result improves on previous investigations, as we obtain strong convergence toward the gradient-dependent kinetic model with quantitative and formally optimal convergence rates. The main ingredient consists in identifying a set of coordinates for the internal cellular dynamics in which concentration gradients arise explicitly. Then, we use relative entropy methods in order to capture quantitative measurement of the distance between the model incorporating cellular mechanisms and the one with concentration-gradient-dependent tumbling rate.</p>\",\"PeriodicalId\":51174,\"journal\":{\"name\":\"Studies in Applied Mathematics\",\"volume\":\"154 5\",\"pages\":\"\"},\"PeriodicalIF\":2.6000,\"publicationDate\":\"2025-05-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70060\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Studies in Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1111/sapm.70060\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Studies in Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1111/sapm.70060","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Derivation of the Bacterial Run-and-Tumble Kinetic Model: Quantitative and Strong Convergence Results
During the past century, biologists and mathematicians investigated two mechanisms underlying bacteria motion: the run phase during which bacteria move in straight lines and the tumble phase in which they change their orientation. When surrounded by a chemical attractant, experiments show that bacteria increase their run time as moving up concentration gradients, leading to a biased random walk toward favorable regions. This observation raises the following question, which has drawn intense interest from both biological and mathematical communities: what cellular mechanisms enable bacteria to feel concentration gradients? In this article, we investigate an asymptotic regime that was proposed to explain this ability thanks to internal mechanisms. More precisely, we derive the run-and-tumble kinetic equation with concentration's gradient-dependent tumbling rate from a more comprehensive model, which incorporates internal cellular mechanisms. Our result improves on previous investigations, as we obtain strong convergence toward the gradient-dependent kinetic model with quantitative and formally optimal convergence rates. The main ingredient consists in identifying a set of coordinates for the internal cellular dynamics in which concentration gradients arise explicitly. Then, we use relative entropy methods in order to capture quantitative measurement of the distance between the model incorporating cellular mechanisms and the one with concentration-gradient-dependent tumbling rate.
期刊介绍:
Studies in Applied Mathematics explores the interplay between mathematics and the applied disciplines. It publishes papers that advance the understanding of physical processes, or develop new mathematical techniques applicable to physical and real-world problems. Its main themes include (but are not limited to) nonlinear phenomena, mathematical modeling, integrable systems, asymptotic analysis, inverse problems, numerical analysis, dynamical systems, scientific computing and applications to areas such as fluid mechanics, mathematical biology, and optics.