SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 1204-1226, June 2024. Abstract. Stochastic oscillations in individual cells are usually characterized by a nonmonotonic power spectrum with an oscillatory autocorrelation function. Here we develop an analytical approach to stochastic oscillations in a minimal hybrid model of stochastic gene expression including promoter state switching, protein synthesis and degradation, as well as a genetic feedback loop. The oscillations observed in our model are noise-induced since the deterministic theory predicts stable fixed points. The autocorrelated function, power spectrum, and steady-state distribution of protein concentration fluctuations are computed in closed form. Using the exactly solvable model, we illustrate sustained oscillations as a circular motion along a stochastic hysteresis loop induced by gene state switching. A triphasic stochastic bifurcation upon the increasing strength of negative feedback is observed, which reveals how stochastic bursts evolve into stochastic oscillations. In our model, oscillations tend to occur when the protein is relatively stable and when gene switching is relatively slow. Translational bursting is found to enhance the robustness and broaden the region of stochastic oscillations. These results provide deeper insights into R. Thomas’s two conjectures for single-cell gene expression kinetics.
SIAM 应用数学杂志》，第 84 卷第 3 期，第 1204-1226 页，2024 年 6 月。 摘要单个细胞中的随机振荡通常以具有振荡自相关函数的非单调功率谱为特征。在此，我们开发了一种分析方法，用于研究随机基因表达的最小混合模型中的随机振荡，该模型包括启动子状态切换、蛋白质合成和降解以及遗传反馈回路。在我们的模型中观察到的振荡是由噪声引起的，因为确定性理论预测了稳定的固定点。蛋白质浓度波动的自相关函数、功率谱和稳态分布是以封闭形式计算的。利用精确可解模型，我们说明了持续振荡是由基因状态切换引起的沿随机滞后环的圆周运动。随着负反馈强度的增加，会出现三相随机分岔，这揭示了随机突发是如何演变成随机振荡的。在我们的模型中，振荡往往发生在蛋白质相对稳定、基因转换相对缓慢的时候。我们发现，翻译猝发增强了随机振荡的稳健性，并扩大了随机振荡的区域。这些结果为 R. Thomas 关于单细胞基因表达动力学的两个猜想提供了更深入的见解。
{"title":"Exact Power Spectrum in a Minimal Hybrid Model of Stochastic Gene Expression Oscillations","authors":"Chen Jia, Hong Qian, Michael Q. Zhang","doi":"10.1137/23m1560914","DOIUrl":"https://doi.org/10.1137/23m1560914","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 1204-1226, June 2024. <br/> Abstract. Stochastic oscillations in individual cells are usually characterized by a nonmonotonic power spectrum with an oscillatory autocorrelation function. Here we develop an analytical approach to stochastic oscillations in a minimal hybrid model of stochastic gene expression including promoter state switching, protein synthesis and degradation, as well as a genetic feedback loop. The oscillations observed in our model are noise-induced since the deterministic theory predicts stable fixed points. The autocorrelated function, power spectrum, and steady-state distribution of protein concentration fluctuations are computed in closed form. Using the exactly solvable model, we illustrate sustained oscillations as a circular motion along a stochastic hysteresis loop induced by gene state switching. A triphasic stochastic bifurcation upon the increasing strength of negative feedback is observed, which reveals how stochastic bursts evolve into stochastic oscillations. In our model, oscillations tend to occur when the protein is relatively stable and when gene switching is relatively slow. Translational bursting is found to enhance the robustness and broaden the region of stochastic oscillations. These results provide deeper insights into R. Thomas’s two conjectures for single-cell gene expression kinetics.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508787","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Introduction to the SIAP Special Section on the Life Sciences","authors":"Simone Bianco, Jonathan E. Rubin","doi":"10.1137/23m1606824","DOIUrl":"https://doi.org/10.1137/23m1606824","url":null,"abstract":"","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141372131","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 1186-1203, June 2024. Abstract. We consider longitudinal shear flows over a superhydrophobic grating made up of a periodic array of grooves separated by infinitely thin slats, addressing the case where the liquid partially invades the grooves. We allow for curved menisci, specified via a depression angle at the contact line. We focus on the limit of small solid fractions where the length of the wetted portion of the slat is small compared with the period. Following an earlier analysis of the comparable flow over noninvaded grooves [O. Schnitzer, J. Fluid Mech., 820 (2017), pp. 580–603], this singular limit is treated using matched asymptotic expansions, with an outer region on the scale of a single period and an inner region on the scale of the wetted portion of the slat. The flow problem in both regions is solved using conformal mappings. Asymptotic matching yields a closed-form approximation for the slip length as a function of the solid fraction and depression angle.
{"title":"Longitudinal Shear Flow over a Superhydrophobic Grating with Partially Invaded Grooves and Curved Menisci","authors":"Ehud Yariv","doi":"10.1137/23m1616522","DOIUrl":"https://doi.org/10.1137/23m1616522","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 1186-1203, June 2024. <br/> Abstract. We consider longitudinal shear flows over a superhydrophobic grating made up of a periodic array of grooves separated by infinitely thin slats, addressing the case where the liquid partially invades the grooves. We allow for curved menisci, specified via a depression angle at the contact line. We focus on the limit of small solid fractions where the length of the wetted portion of the slat is small compared with the period. Following an earlier analysis of the comparable flow over noninvaded grooves [O. Schnitzer, J. Fluid Mech., 820 (2017), pp. 580–603], this singular limit is treated using matched asymptotic expansions, with an outer region on the scale of a single period and an inner region on the scale of the wetted portion of the slat. The flow problem in both regions is solved using conformal mappings. Asymptotic matching yields a closed-form approximation for the slip length as a function of the solid fraction and depression angle.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552943","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Siqing Li, Leevan Ling, Steven J. Ruuth, Xuemeng Wang
SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 1163-1185, June 2024. Abstract. We are interested in generating surfaces with arbitrary roughness and forming patterns on the surfaces. Two methods are applied to construct rough surfaces. In the first method, some superposition of wave functions with random frequencies and angles of propagation are used to get periodic rough surfaces with analytic parametric equations. The amplitude of such surfaces is also an important variable in the provided eigenvalue analysis for the Laplace–Beltrami operator and in the generation of pattern formation. Numerical experiments show that the patterns become irregular as the amplitude and frequency of the rough surface increase. For the sake of easy generalization to closed manifolds, we propose a second construction method for rough surfaces, which uses random nodal values and discretized heat filters. We provide numerical evidence that both surface construction methods yield comparable patterns to those observed in real-life animals.
{"title":"Realistic Pattern Formations on Surfaces by Adding Arbitrary Roughness","authors":"Siqing Li, Leevan Ling, Steven J. Ruuth, Xuemeng Wang","doi":"10.1137/22m1518001","DOIUrl":"https://doi.org/10.1137/22m1518001","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 1163-1185, June 2024. <br/> Abstract. We are interested in generating surfaces with arbitrary roughness and forming patterns on the surfaces. Two methods are applied to construct rough surfaces. In the first method, some superposition of wave functions with random frequencies and angles of propagation are used to get periodic rough surfaces with analytic parametric equations. The amplitude of such surfaces is also an important variable in the provided eigenvalue analysis for the Laplace–Beltrami operator and in the generation of pattern formation. Numerical experiments show that the patterns become irregular as the amplitude and frequency of the rough surface increase. For the sake of easy generalization to closed manifolds, we propose a second construction method for rough surfaces, which uses random nodal values and discretized heat filters. We provide numerical evidence that both surface construction methods yield comparable patterns to those observed in real-life animals.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259022","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Agustín Gabriel Yabo, Jean-Baptiste Caillau, Jean-Luc Gouzé
SIAM Journal on Applied Mathematics, Ahead of Print. Abstract. The study of living microorganisms using resource allocation models has been key in elucidating natural behaviors of bacteria, by allowing allocation of microbial resources to be represented through optimal control strategies. The approach can also be applied to research in microbial cell factories, to investigate the optimal production of value-added compounds regulated by an external control. The latter is the subject of this paper, in which we study batch bioprocessing from a resource allocation perspective. Based on previous works, we propose a simple bacterial growth model accounting for the dynamics of the bioreactor and intracellular composition, and we analyze its asymptotic behavior and stability. Using optimization and optimal control theory, we study the production of biomass and metabolites of interest for infinite- and finite-time horizons. The resulting optimal control problems are studied using Pontryagin’s maximum principle and numerical methods, and the solutions found are characterized by the presence of the Fuller phenomenon (producing an infinite set of switching points occurring in a finite-time window) at the junctions with a second-order singular arc. The approach, inspired by biotechnological engineering, aims to shed light upon the role of cellular composition and resource allocation during batch processing and, at the same time, poses very interesting and challenging mathematical problems.
{"title":"Optimal Bacterial Resource Allocation Strategies in Batch Processing","authors":"Agustín Gabriel Yabo, Jean-Baptiste Caillau, Jean-Luc Gouzé","doi":"10.1137/22m1506328","DOIUrl":"https://doi.org/10.1137/22m1506328","url":null,"abstract":"SIAM Journal on Applied Mathematics, Ahead of Print. <br/> Abstract. The study of living microorganisms using resource allocation models has been key in elucidating natural behaviors of bacteria, by allowing allocation of microbial resources to be represented through optimal control strategies. The approach can also be applied to research in microbial cell factories, to investigate the optimal production of value-added compounds regulated by an external control. The latter is the subject of this paper, in which we study batch bioprocessing from a resource allocation perspective. Based on previous works, we propose a simple bacterial growth model accounting for the dynamics of the bioreactor and intracellular composition, and we analyze its asymptotic behavior and stability. Using optimization and optimal control theory, we study the production of biomass and metabolites of interest for infinite- and finite-time horizons. The resulting optimal control problems are studied using Pontryagin’s maximum principle and numerical methods, and the solutions found are characterized by the presence of the Fuller phenomenon (producing an infinite set of switching points occurring in a finite-time window) at the junctions with a second-order singular arc. The approach, inspired by biotechnological engineering, aims to shed light upon the role of cellular composition and resource allocation during batch processing and, at the same time, poses very interesting and challenging mathematical problems.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141196806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 1140-1162, June 2024. Abstract. First hitting times (FHTs) describe the time it takes a random “searcher” to find a “target” and are used to study timescales in many applications. FHTs have been well-studied for diffusive search, especially for small targets, which is called the narrow capture or narrow escape problem. In this paper, we study the FHT to small targets for a one-dimensional superdiffusive search described by a Lévy flight. By applying the method of matched asymptotic expansions to a fractional differential equation we obtain an explicit asymptotic expansion for the mean FHT (MFHT). For fractional order [math] (describing a [math]-stable Lévy flight whose squared displacement scales as [math] in time [math]) and targets of radius [math], we show that the MFHT is order one for [math] and diverges as [math] for [math] and [math] for [math]. We then use our asymptotic results to identify the value of [math] which minimizes the average MFHT and find that (a) this optimal value of [math] vanishes for sparse targets and (b) the value [math] (corresponding to an inverse square Lévy search) is optimal in only very specific circumstances. We confirm our results by comparison to both deterministic numerical solutions of the associated fractional differential equation and stochastic simulations.
{"title":"First Hitting Time of a One-Dimensional Lévy Flight to Small Targets","authors":"Daniel Gomez, Sean D. Lawley","doi":"10.1137/23m1586239","DOIUrl":"https://doi.org/10.1137/23m1586239","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 1140-1162, June 2024. <br/> Abstract. First hitting times (FHTs) describe the time it takes a random “searcher” to find a “target” and are used to study timescales in many applications. FHTs have been well-studied for diffusive search, especially for small targets, which is called the narrow capture or narrow escape problem. In this paper, we study the FHT to small targets for a one-dimensional superdiffusive search described by a Lévy flight. By applying the method of matched asymptotic expansions to a fractional differential equation we obtain an explicit asymptotic expansion for the mean FHT (MFHT). For fractional order [math] (describing a [math]-stable Lévy flight whose squared displacement scales as [math] in time [math]) and targets of radius [math], we show that the MFHT is order one for [math] and diverges as [math] for [math] and [math] for [math]. We then use our asymptotic results to identify the value of [math] which minimizes the average MFHT and find that (a) this optimal value of [math] vanishes for sparse targets and (b) the value [math] (corresponding to an inverse square Lévy search) is optimal in only very specific circumstances. We confirm our results by comparison to both deterministic numerical solutions of the associated fractional differential equation and stochastic simulations.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259062","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Generalized Honeycomb-Structured Materials in the Subwavelength Regime","authors":"Borui Miao, Yi Zhu","doi":"10.1137/23m1559786","DOIUrl":"https://doi.org/10.1137/23m1559786","url":null,"abstract":"","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141102041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Inference of Interaction Kernels in Mean-Field Models of Opinion Dynamics","authors":"Weiqi Chu, Qin Li, M. A. Porter","doi":"10.1137/22m1544415","DOIUrl":"https://doi.org/10.1137/22m1544415","url":null,"abstract":"","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141111798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 1079-1095, June 2024. Abstract. Active Brownian particles (ABPs) are a model for nonequilibrium systems in which the constituent particles are self-propelled in addition to their Brownian motion. Compared to the well-studied mean first passage time (MFPT) of passive Brownian particles, the MFPT of ABPs is much less developed. In this paper, we study the MFPT for ABPs in a 1-D domain with absorbing boundary conditions at both ends of the domain. To reveal the effect of swimming on the MFPT, we consider an asymptotic analysis in the weak-swimming or small Péclet ([math]) number limit. In particular, analytical expressions for the survival probability and the MFPT are developed up to [math]. We explore the effects of the starting positions and starting orientations on the MFPT. Our analysis shows that if the starting orientations are biased towards one side of the domain, the MFPT as a function of the starting position becomes asymmetric about the center of the domain. The analytical results were confirmed by the numerical solutions of the full PDE model.
{"title":"Asymptotic Analysis and Simulation of Mean First Passage Time for Active Brownian Particles in 1-D","authors":"Sarafa A. Iyaniwura, Zhiwei Peng","doi":"10.1137/23m1593917","DOIUrl":"https://doi.org/10.1137/23m1593917","url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 1079-1095, June 2024. <br/> Abstract. Active Brownian particles (ABPs) are a model for nonequilibrium systems in which the constituent particles are self-propelled in addition to their Brownian motion. Compared to the well-studied mean first passage time (MFPT) of passive Brownian particles, the MFPT of ABPs is much less developed. In this paper, we study the MFPT for ABPs in a 1-D domain with absorbing boundary conditions at both ends of the domain. To reveal the effect of swimming on the MFPT, we consider an asymptotic analysis in the weak-swimming or small Péclet ([math]) number limit. In particular, analytical expressions for the survival probability and the MFPT are developed up to [math]. We explore the effects of the starting positions and starting orientations on the MFPT. Our analysis shows that if the starting orientations are biased towards one side of the domain, the MFPT as a function of the starting position becomes asymmetric about the center of the domain. The analytical results were confirmed by the numerical solutions of the full PDE model.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061644","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Tick-Borne Pathogen Co-infection by Co-feeding on Incompetent Hosts: Global Convergence and Impact of Developmental Delay","authors":"Xue Zhang, Jianhong Wu","doi":"10.1137/23m1577419","DOIUrl":"https://doi.org/10.1137/23m1577419","url":null,"abstract":"","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140966626","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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