{"title":"芬克-瓦特基模型化学主方程的精确解析解","authors":"Tomasz Bednarek, Jakub Jędrak","doi":"arxiv-2409.08875","DOIUrl":null,"url":null,"abstract":"The Finke-Watkzy model is the reaction set consisting of autocatalysis, A + B\n--> 2B and the first order process A --> B. It has been widely used to describe\nphenomena as diverse as the formation of transition metal nanoparticles and\nprotein misfolding and aggregation. It can also be regarded as a simple model\nfor the spread of a non-fatal but incurable disease. The deterministic rate\nequations for this reaction set are easy to solve and the solution is used in\nthe literature to fit experimental data. However, some applications of the\nFinke-Watkzy model may involve systems with a small number of molecules or\nindividuals. In such cases, a stochastic description using a Chemical Master\nEquation or Gillespie's Stochastic Simulation Algorithm is more appropriate\nthan a deterministic one. This is even more so because for this particular set\nof chemical reactions, the differences between deterministic and stochastic\nkinetics can be very significant. Here, we derive an analytical solution of the\nChemical Master Equation for the Finke-Watkzy model. We consider both the\noriginal formulation of the model, where the reactions are assumed to be\nirreversible, and its generalization to the case of reversible reactions. For\nthe former, we obtain analytical expressions for the time dependence of the\nprobabilities of the number of A molecules. For the latter, we derive the\ncorresponding steady-state probability distribution. Our findings may have\nimplications for modeling the spread of epidemics and chemical reactions in\nliving cells.","PeriodicalId":501304,"journal":{"name":"arXiv - PHYS - Chemical Physics","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Exact analytical solution of the Chemical Master Equation for the Finke-Watkzy model\",\"authors\":\"Tomasz Bednarek, Jakub Jędrak\",\"doi\":\"arxiv-2409.08875\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The Finke-Watkzy model is the reaction set consisting of autocatalysis, A + B\\n--> 2B and the first order process A --> B. It has been widely used to describe\\nphenomena as diverse as the formation of transition metal nanoparticles and\\nprotein misfolding and aggregation. It can also be regarded as a simple model\\nfor the spread of a non-fatal but incurable disease. The deterministic rate\\nequations for this reaction set are easy to solve and the solution is used in\\nthe literature to fit experimental data. However, some applications of the\\nFinke-Watkzy model may involve systems with a small number of molecules or\\nindividuals. In such cases, a stochastic description using a Chemical Master\\nEquation or Gillespie's Stochastic Simulation Algorithm is more appropriate\\nthan a deterministic one. This is even more so because for this particular set\\nof chemical reactions, the differences between deterministic and stochastic\\nkinetics can be very significant. Here, we derive an analytical solution of the\\nChemical Master Equation for the Finke-Watkzy model. We consider both the\\noriginal formulation of the model, where the reactions are assumed to be\\nirreversible, and its generalization to the case of reversible reactions. For\\nthe former, we obtain analytical expressions for the time dependence of the\\nprobabilities of the number of A molecules. For the latter, we derive the\\ncorresponding steady-state probability distribution. Our findings may have\\nimplications for modeling the spread of epidemics and chemical reactions in\\nliving cells.\",\"PeriodicalId\":501304,\"journal\":{\"name\":\"arXiv - PHYS - Chemical Physics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - PHYS - Chemical Physics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.08875\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - PHYS - Chemical Physics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.08875","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
芬克-瓦茨模型是由自催化反应、A + B--> 2B 和一阶过程 A --> B 组成的反应集合。它被广泛用于描述过渡金属纳米粒子的形成、蛋白质的错误折叠和聚集等各种现象。它也可以被视为一种非致命但无法治愈的疾病传播的简单模式。该反应集的确定性速率方程很容易求解,文献中也用其来拟合实验数据。然而,芬克-瓦特奇模型的某些应用可能涉及分子或个体数量较少的系统。在这种情况下,使用化学主方程或 Gillespie 随机模拟算法进行随机描述比确定性描述更为合适。对于这组特殊的化学反应,确定性动力学和随机动力学之间的差异可能会非常大,这一点更为重要。在这里,我们推导出 Finke-Watkzy 模型的化学主方程的解析解。我们既考虑了假设反应是可逆的该模型的原始公式,也考虑了其对可逆反应情况的概括。对于前者,我们得到了 A 分子数概率随时间变化的分析表达式。对于后者,我们推导出了相应的稳态概率分布。我们的发现可能会对流行病的传播和活细胞中的化学反应建模产生影响。
Exact analytical solution of the Chemical Master Equation for the Finke-Watkzy model
The Finke-Watkzy model is the reaction set consisting of autocatalysis, A + B
--> 2B and the first order process A --> B. It has been widely used to describe
phenomena as diverse as the formation of transition metal nanoparticles and
protein misfolding and aggregation. It can also be regarded as a simple model
for the spread of a non-fatal but incurable disease. The deterministic rate
equations for this reaction set are easy to solve and the solution is used in
the literature to fit experimental data. However, some applications of the
Finke-Watkzy model may involve systems with a small number of molecules or
individuals. In such cases, a stochastic description using a Chemical Master
Equation or Gillespie's Stochastic Simulation Algorithm is more appropriate
than a deterministic one. This is even more so because for this particular set
of chemical reactions, the differences between deterministic and stochastic
kinetics can be very significant. Here, we derive an analytical solution of the
Chemical Master Equation for the Finke-Watkzy model. We consider both the
original formulation of the model, where the reactions are assumed to be
irreversible, and its generalization to the case of reversible reactions. For
the former, we obtain analytical expressions for the time dependence of the
probabilities of the number of A molecules. For the latter, we derive the
corresponding steady-state probability distribution. Our findings may have
implications for modeling the spread of epidemics and chemical reactions in
living cells.