1
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Shu H, Jin HY, Wang XS, Wu J. Viral infection dynamics with immune chemokines and CTL mobility modulated by the infected cell density. J Math Biol 2024; 88:43. [PMID: 38491217 DOI: 10.1007/s00285-024-02065-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/04/2023] [Revised: 12/05/2023] [Accepted: 02/09/2024] [Indexed: 03/18/2024]
Abstract
We study a viral infection model incorporating both cell-to-cell infection and immune chemokines. Based on experimental results in the literature, we make a standing assumption that the cytotoxic T lymphocytes (CTL) will move toward the location with more infected cells, while the diffusion rate of CTL is a decreasing function of the density of infected cells. We first establish the global existence and ultimate boundedness of the solution via a priori energy estimates. We then define the basic reproduction number of viral infection R 0 and prove (by the uniform persistence theory, Lyapunov function technique and LaSalle invariance principle) that the infection-free steady state E 0 is globally asymptotically stable ifR 0 < 1 . WhenR 0 > 1 , then E 0 becomes unstable, and another basic reproduction number of CTL response R 1 becomes the dynamic threshold in the sense that ifR 1 < 1 , then the CTL-inactivated steady state E 1 is globally asymptotically stable; and ifR 1 > 1 , then the immune response is uniform persistent and, under an additional technical condition the CTL-activated steady state E 2 is globally asymptotically stable. To establish the global stability results, we need to prove point dissipativity, obtain uniform persistence, construct suitable Lyapunov functions, and apply the LaSalle invariance principle.
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Affiliation(s)
- Hongying Shu
- School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, 710062, China
| | - Hai-Yang Jin
- Department of Mathematics, South China University of Technology, Guangzhou, 510640, China
| | - Xiang-Sheng Wang
- Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA, 70503, USA
| | - Jianhong Wu
- Department of Mathematics and Statistics, York University, Toronto, ON, M3J 1P3, Canada.
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2
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Ridgway WJM, Dalwadi MP, Pearce P, Chapman SJ. Motility-Induced Phase Separation Mediated by Bacterial Quorum Sensing. PHYSICAL REVIEW LETTERS 2023; 131:228302. [PMID: 38101339 DOI: 10.1103/physrevlett.131.228302] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/03/2023] [Accepted: 10/09/2023] [Indexed: 12/17/2023]
Abstract
We study motility-induced phase separation (MIPS) in living active matter, in which cells interact through chemical signaling, or quorum sensing. In contrast to previous theories of MIPS, our multiscale continuum model accounts explicitly for genetic regulation of signal production and motility. Through analysis and simulations, we derive a new criterion for the onset of MIPS that depends on features of the genetic network. Furthermore, we identify and characterize a new type of oscillatory instability that occurs when gene regulation inside cells promotes motility in higher signal concentrations.
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Affiliation(s)
- Wesley J M Ridgway
- Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
| | - Mohit P Dalwadi
- Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
- Department of Mathematics, University College London, London WC1H 0AY, United Kingdom
- Institute for the Physics of Living Systems, University College London, London, United Kingdom
| | - Philip Pearce
- Department of Mathematics, University College London, London WC1H 0AY, United Kingdom
- Institute for the Physics of Living Systems, University College London, London, United Kingdom
| | - S Jonathan Chapman
- Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
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3
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Martínez-Calvo A, Wingreen NS, Datta SS. Pattern formation by bacteria-phage interactions. BIORXIV : THE PREPRINT SERVER FOR BIOLOGY 2023:2023.09.19.558479. [PMID: 37786699 PMCID: PMC10541591 DOI: 10.1101/2023.09.19.558479] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/04/2023]
Abstract
The interactions between bacteria and phages-viruses that infect bacteria-play critical roles in agriculture, ecology, and medicine; however, how these interactions influence the spatial organization of both bacteria and phages remain largely unexplored. Here, we address this gap in knowledge by developing a theoretical model of motile, proliferating bacteria that aggregate via motility-induced phase separation (MIPS) and encounter phage that infect and lyse the cells. We find that the non-reciprocal predator-prey interactions between phage and bacteria strongly alter spatial organization, in some cases giving rise to a rich array of finite-scale stationary and dynamic patterns in which bacteria and phage coexist. We establish principles describing the onset and characteristics of these diverse behaviors, thereby helping to provide a biophysical basis for understanding pattern formation in bacteria-phage systems, as well as in a broader range of active and living systems with similar predator-prey or other non-reciprocal interactions.
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4
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Feng T, Wu L. Global dynamics and pattern formation for predator-prey system with density-dependent motion. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:2296-2320. [PMID: 36899535 DOI: 10.3934/mbe.2023108] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/18/2023]
Abstract
In this paper, we concern with the predator-prey system with generalist predator and density-dependent prey-taxis in two-dimensional bounded domains. We derive the existence of classical solutions with uniform-in-time bound and global stability for steady states under suitable conditions through the Lyapunov functionals. In addition, by linear instability analysis and numerical simulations, we conclude that the prey density-dependent motility function can trigger the periodic pattern formation when it is monotone increasing.
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Affiliation(s)
- Tingfu Feng
- Department of Mathematics, Kunming University, Kunming, Yunnan, China
| | - Leyun Wu
- Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong, China
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5
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Sun GQ, Li L, Li J, Liu C, Wu YP, Gao S, Wang Z, Feng GL. Impacts of climate change on vegetation pattern: Mathematical modeling and data analysis. Phys Life Rev 2022; 43:239-270. [PMID: 36343569 DOI: 10.1016/j.plrev.2022.09.005] [Citation(s) in RCA: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/05/2022] [Accepted: 09/07/2022] [Indexed: 11/27/2022]
Abstract
Climate change has become increasingly severe, threatening ecosystem stability and, in particular, biodiversity. As a typical indicator of ecosystem evolution, vegetation growth is inevitably affected by climate change, and therefore has a great potential to provide valuable information for addressing such ecosystem problems. However, the impacts of climate change on vegetation growth, especially the spatial and temporal distribution of vegetation, are still lacking of comprehensive exposition. To this end, this review systematically reveals the influences of climate change on vegetation dynamics in both time and space by dynamical modeling the interactions of meteorological elements and vegetation growth. Moreover, we characterize the long-term evolution trend of vegetation growth under climate change in some typical regions based on data analysis. This work is expected to lay a necessary foundation for systematically revealing the coupling effect of climate change on the ecosystem.
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Affiliation(s)
- Gui-Quan Sun
- Department of Mathematics, North University of China, Taiyuan, 030051, China; Complex Systems Research Center, Shanxi University, Taiyuan, 030006, China; Southern Marine Science and Engineering Guangdong Laboratory (Zhuhai), Zhuhai, 519082, China.
| | - Li Li
- School of Computer and Information Technology, Shanxi University, Taiyuan, 030006, China
| | - Jing Li
- School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, 030006, China
| | - Chen Liu
- Center for Ecology and Environmental Sciences, Northwestern Polytechnical University, Xi'an, 710072, China
| | - Yong-Ping Wu
- College of Physics Science and Technology, Yangzhou University, Yangzhou, 225002, China
| | - Shupeng Gao
- School of Mechanical Engineering and School of Artificial Intelligence, Optics and Electronics (iOPEN), Northwestern Polytechnical University, Xian, 710072, China
| | - Zhen Wang
- School of Mechanical Engineering and School of Artificial Intelligence, Optics and Electronics (iOPEN), Northwestern Polytechnical University, Xian, 710072, China.
| | - Guo-Lin Feng
- College of Physics Science and Technology, Yangzhou University, Yangzhou, 225002, China; Laboratory for Climate Studies, National Climate Center, China Meteorological Administration, Beijing, 100081, China.
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6
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Zheng P. On a two-species competitive predator-prey system with density-dependent diffusion. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2022; 19:13421-13457. [PMID: 36654053 DOI: 10.3934/mbe.2022628] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/17/2023]
Abstract
This paper deals with a two-species competitive predator-prey system with density-dependent diffusion, i.e., \begin{eqnarray*}\label{1a} \left\{ \begin{split}{}&u_t=\Delta (d_{1}(w)u)+\gamma_{1}uF_{1}(w)-uh_{1}(u)-\beta_{1}uv,&(x,t)\in \Omega\times (0,\infty),\\&v_t=\Delta (d_{2}(w)v)+\gamma_{2}vF_{2}(w)-vh_{2}(v)-\beta_{2}uv,&(x,t)\in \Omega\times (0,\infty),\\&w_t=D\Delta w-uF_{1}(w)-vF_{2}(w)+f(w),&(x,t)\in \Omega\times (0,\infty), \end{split} \right. \end{eqnarray*} under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega\subset \mathbb{R}^{2}$, with the nonnegative initial data $\left( {u_{0}, v_{0}, w_{0}} \right) \in (W^{1,p}(\Omega))^{3}$ with $p>2$, where the parameters $D,\gamma_{1},\gamma_{2},\beta_{1},\beta_{2}>0$, $d_{1}(w)$ and $d_{2}(w)$ are density-dependent diffusion functions, $F_{1}(w)$ and $F_{2}(w)$ are commonly called the functional response functions accounting for the intake rate of predators as the functions of prey density, $h_{1}(u)$ and $h_{2}(v)$ represent the mortality rates of predators, and $f(w)$ stands for the growth function of the prey. First, we rigorously prove the global boundedness of classical solutions for the above general model provided that the parameters satisfy some suitable conditions by means of $L^{p}$-estimate techniques. Moreover, in some particular cases, we establish the asymptotic stabilization and precise convergence rates of globally bounded solutions under different conditions on the parameters by constructing some appropriate Lyapunov functionals. Our results not only extend the previous ones, but also involve some new conclusions.
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Affiliation(s)
- Pan Zheng
- College of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
- Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong 999077, China
- School of Mathematics and Statistics, Yunnan University, Kunming 650091, China
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7
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Chu J, Jin HY. Predator-prey systems with defense switching and density-suppressed dispersal strategy. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2022; 19:12472-12499. [PMID: 36654007 DOI: 10.3934/mbe.2022582] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/17/2023]
Abstract
In this paper, we consider the following predator-prey system with defense switching mechanism and density-suppressed dispersal strategy $ \begin{equation*} \begin{cases} u_t = \Delta(d_1(w)u)+\frac{\beta_1 uvw}{u+v}-\alpha_1 u, & x\in \Omega, \; \; t>0, \\ v_t = \Delta(d_2(w)v)+\frac{\beta_2 uvw}{u+v}-\alpha_2 v, & x\in \Omega, \; \; t>0, \\ w_t = \Delta w-\frac{\beta_3 uvw}{u+v}+\sigma w\left(1-\frac{w}{K}\right), & x\in \Omega, \; \; t>0, \\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = 0, & x\in\partial\Omega, \; \; t>0, \\ (u, v, w)(x, 0) = (u_0, v_0, w_0)(x), & x\in\Omega, \ \end{cases} \end{equation*} $ where $ \Omega\subset{\mathbb{R}}^2 $ is a bounded domain with smooth boundary. Based on the method of energy estimates and Moser iteration, we establish the existence of global classical solutions with uniform-in-time boundedness. We further prove the global stability of co-existence equilibrium by using the Lyapunov functionals and LaSalle's invariant principle. Finally we conduct linear stability analysis and perform numerical simulations to illustrate that the density-suppressed dispersal may trigger the pattern formation.
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Affiliation(s)
- Jiawei Chu
- Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong, China
| | - Hai-Yang Jin
- School of Mathematics, South China University of Technology, Guangzhou 510640, China
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8
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Fuest M. On the optimality of upper estimates near blow-up in quasilinear Keller–Segel systems. APPLICABLE ANALYSIS 2022; 101:3515-3534. [DOI: 10.1080/00036811.2020.1854234] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/27/2020] [Accepted: 11/04/2020] [Indexed: 09/02/2023]
Affiliation(s)
- Mario Fuest
- Institut für Mathematik, Universität Paderborn, Paderborn, Germany
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9
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Spatial patterns in ecological systems: from microbial colonies to landscapes. Emerg Top Life Sci 2022; 6:245-258. [PMID: 35678374 DOI: 10.1042/etls20210282] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/04/2022] [Revised: 05/10/2022] [Accepted: 05/12/2022] [Indexed: 11/17/2022]
Abstract
Self-organized spatial patterns are ubiquitous in ecological systems and allow populations to adopt non-trivial spatial distributions starting from disordered configurations. These patterns form due to diverse nonlinear interactions among organisms and between organisms and their environment, and lead to the emergence of new (eco)system-level properties unique to self-organized systems. Such pattern consequences include higher resilience and resistance to environmental changes, abrupt ecosystem collapse, hysteresis loops, and reversal of competitive exclusion. Here, we review ecological systems exhibiting self-organized patterns. We establish two broad pattern categories depending on whether the self-organizing process is primarily driven by nonlinear density-dependent demographic rates or by nonlinear density-dependent movement. Using this organization, we examine a wide range of observational scales, from microbial colonies to whole ecosystems, and discuss the mechanisms hypothesized to underlie observed patterns and their system-level consequences. For each example, we review both the empirical evidence and the existing theoretical frameworks developed to identify the causes and consequences of patterning. Finally, we trace qualitative similarities across systems and propose possible ways of developing a more quantitative understanding of how self-organization operates across systems and observational scales in ecology.
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10
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Irani E, Mokhtari Z, Zippelius A. Dynamics of Bacteria Scanning a Porous Environment. PHYSICAL REVIEW LETTERS 2022; 128:144501. [PMID: 35476466 DOI: 10.1103/physrevlett.128.144501] [Citation(s) in RCA: 10] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/01/2021] [Accepted: 03/09/2022] [Indexed: 06/14/2023]
Abstract
It has recently been reported that bacteria, such as Escherichia coli Bhattacharjee and Datta, Nat. Commun. 10, 2075 (2019).NCAOBW2041-172310.1038/s41467-019-10115-1 and Pseudomonas putida Alirezaeizanjani et al., Sci. Adv. 6, eaaz6153 (2020).SACDAF2375-254810.1126/sciadv.aaz6153, perform distinct modes of motion when placed in porous media as compared to dilute regions or free space. This has led us to suggest an efficient strategy for active particles in a disordered environment: reorientations are suppressed in locally dilute regions and intensified in locally dense ones. Thereby the local geometry determines the optimal path of the active agent and substantially accelerates the dynamics for up to 2 orders of magnitude. We observe a nonmonotonic behavior of the diffusion coefficient in dependence on the tumbling rate and identify a localization transition, either by increasing the density of obstacles or by decreasing the reorientation rate.
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Affiliation(s)
- Ehsan Irani
- Max Delbrück Center for Molecular Medicine in the Helmholtz Association (MDC), The Berlin Institute for Medical Systems Biology (BIMSB), 10115 Berlin, Germany
| | - Zahra Mokhtari
- Freie Universität Berlin, Department of Mathematics and Computer Science, Institute of Mathematics, Arnimallee 9, 14195 Berlin, Germany
| | - Annette Zippelius
- Georg-August-Universität Göttingen, Institut für Theoretische Physik, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
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11
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He C, Bayakhmetov S, Harris D, Kuang Y, Wang X. A Predictive Reaction-diffusion Based Model of E.coli Colony Growth Control. IEEE CONTROL SYSTEMS LETTERS 2021; 5:1952-1957. [PMID: 33829120 PMCID: PMC8021091 DOI: 10.1109/lcsys.2020.3046612] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
Bacterial colony formations exhibit diverse morphologies and dynamics. A mechanistic understanding of this process has broad implications to ecology and medicine. However, many control factors and their impacts on colony formation remain underexplored. Here we propose a reaction-diffusion based dynamic model to quantitatively describe cell division and colony expansion, where control factors of colony spreading take the form of nonlinear density-dependent function and the intercellular impacts take the form of density-dependent hill function. We validate the model using experimental E. coli colony growth data and our results show that the model is capable of predicting the whole colony expansion process in both time and space under different conditions. Furthermore, the nonlinear control factors can predict colony morphology at both center and edge of the colony.
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Affiliation(s)
- Changhan He
- School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
| | - Samat Bayakhmetov
- School of Biological and Health Systems Engineering, Arizona State University, Tempe, AZ 85287, USA
| | - Duane Harris
- School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
| | - Yang Kuang
- School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
| | - Xiao Wang
- School of Biological and Health Systems Engineering, Arizona State University, Tempe, AZ 85287, USA
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12
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Luo Y. Global existence and stability of the classical solution to a density-dependent prey-predator model with indirect prey-taxis. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2021; 18:6672-6699. [PMID: 34517551 DOI: 10.3934/mbe.2021331] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
Abstract
We study the existence of global unique classical solution to a density-dependent prey-predator population system with indirect prey-taxis effect. With two Lyapunov functions appropriately constructed, we then show that the solution can asymptotically approach prey-only state or coexistence state of the system under suitable conditions. Moreover, linearized analysis on the system at these two constant steady states shows their linear instability criterion. By numerical simulation we find that some density-dependent prey-taxis and predators' diffusion may either flatten the spatial one-dimensional patterns which exist in non-density-dependent case, or break the spatial two-dimensional distribution similarity which occurs in non-density-dependent case between predators and chemoattractants (released by prey).
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Affiliation(s)
- Yong Luo
- Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong
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13
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The spatial organization of microbial communities during range expansion. Curr Opin Microbiol 2021; 63:109-116. [PMID: 34329942 DOI: 10.1016/j.mib.2021.07.005] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/12/2021] [Revised: 06/26/2021] [Accepted: 07/05/2021] [Indexed: 12/28/2022]
Abstract
Microbes in nature often live in dense and diverse communities exhibiting a variety of spatial structures. Microbial range expansion is a universal ecological process that enables populations to form spatial patterns. It can be driven by both passive and active processes, for example, mechanical forces from cell growth and bacterial motility. In this review, we provide a taste of recent creative and sophisticated efforts being made to address basic questions in spatial ecology and pattern formation during range expansion. We especially highlight the role of motility to shape community structures, and discuss the research challenges and future directions.
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14
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Wang ZA, Xu J. On the Lotka-Volterra competition system with dynamical resources and density-dependent diffusion. J Math Biol 2021; 82:7. [PMID: 33491122 DOI: 10.1007/s00285-021-01562-w] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/24/2019] [Revised: 11/20/2020] [Accepted: 12/07/2020] [Indexed: 11/25/2022]
Abstract
In this paper, we consider the following Lotka-Volterra competition system with dynamical resources and density-dependent diffusion in a bounded smooth domain [Formula: see text] with homogeneous Neumann boundary conditions, where the parameters [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] ([Formula: see text]) are positive constants, m(x) is the prey's resource, and the dispersal rate function [Formula: see text] satisfies the the following hypothesis: [Formula: see text], [Formula: see text] on [Formula: see text] and [Formula: see text]. When m(x) is constant, we show that the system (*) with has a unique global classical solution when the initial datum is in functional space [Formula: see text] with [Formula: see text]. By constructing appropriate Lyapunov functionals and using LaSalle's invariant principle, we further prove that the solution of (*) converges to the co-existence steady state exponentially or competitive exclusion steady state algebraically as time tends to infinity in different parameter regimes. Our results reveal that once the resource w has temporal dynamics, two competitors may coexist in the case of weak competition regardless of their dispersal rates and initial values no matter whether there is explicit dependence in dispersal or not. When the prey's resource is spatially heterogeneous (i.e. m(x) is non-constant), we use numerical simulations to demonstrate that the striking phenomenon "slower diffuser always prevails" (cf. Dockery et al. in J Math Biol 37(1):61-83, 1998; Lou in J Differ Equ 223(2):400-426, 2006) fails to appear if the non-random dispersal strategy is employed by competing species (i.e. either [Formula: see text] or [Formula: see text] is non-constant) while it still holds true if both d(w) and [Formula: see text] are constant.
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Affiliation(s)
- Zhi-An Wang
- Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong.
| | - Jiao Xu
- School of Mathematics, South China University of Technology, Guangzhou, 510640, People's Republic of China
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15
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Fuest M. Blow-up profiles in quasilinear fully parabolic Keller–Segel systems. NONLINEARITY 2020; 33:2306-2334. [DOI: 10.1088/1361-6544/ab7294] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 09/02/2023]
Abstract
Abstract
We examine finite-time blow-up solutions
to
in a ball
,
, where D and S generalize the functions
with
. We show that if
as well as
and
is a nonnegative, radially symmetric classical solution to (
) blowing up at
, then there exists a so-called blow-up profile
satisfying
Moreover, for all
with
we can find
such that
for all
.
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16
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Fischer A, Schmid F, Speck T. Quorum-sensing active particles with discontinuous motility. Phys Rev E 2020; 101:012601. [PMID: 32069622 DOI: 10.1103/physreve.101.012601] [Citation(s) in RCA: 16] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/26/2019] [Indexed: 06/10/2023]
Abstract
We develop a dynamic mean-field theory for polar active particles that interact through a self-generated field, in particular one generated through emitting a chemical signal. While being a form of chemotactic response, it is different from conventional chemotaxis in that particles discontinuously change their motility when the local concentration surpasses a threshold. The resulting coupled equations for density and polarization are linear and can be solved analytically for simple geometries, yielding inhomogeneous density profiles. Specifically, here we consider a planar and circular interface. Our theory thus explains the observed coexistence of dense aggregates with an active gas. There are, however, differences from the more conventional picture of liquid-gas coexistence based on a free energy, most notably the absence of a critical point. We corroborate our analytical predictions by numerical simulations of active particles under confinement and interacting through volume exclusion. Excellent quantitative agreement is reached through an effective translational diffusion coefficient. We finally show that an additional response to the chemical gradient direction is sufficient to induce vortex clusters. Our results pave the way to engineer motility responses in order to achieve aggregation and collective behavior even at unfavorable conditions.
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Affiliation(s)
- Andreas Fischer
- Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudingerweg 7-9, 55128 Mainz, Germany
| | - Friederike Schmid
- Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudingerweg 7-9, 55128 Mainz, Germany
| | - Thomas Speck
- Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudingerweg 7-9, 55128 Mainz, Germany
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17
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Affiliation(s)
- Nan Luo
- Department of Biomedical Engineering, Duke University, Durham, North Carolina 27708, United States
| | - Shangying Wang
- Department of Biomedical Engineering, Duke University, Durham, North Carolina 27708, United States
| | - Lingchong You
- Department of Biomedical Engineering, Duke University, Durham, North Carolina 27708, United States
- Center for Genomic and Computational Biology, Duke University, Durham, North Carolina 27708, United States
- Department of Molecular Genetics and Microbiology, Duke University School of Medicine, Durham, North Carolina 27708, United States
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18
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Bettenworth V, McIntosh M, Becker A, Eckhardt B. Front-propagation in bacterial inter-colony communication. CHAOS (WOODBURY, N.Y.) 2018; 28:106316. [PMID: 30384658 DOI: 10.1063/1.5040068] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/14/2018] [Accepted: 09/09/2018] [Indexed: 06/08/2023]
Abstract
Many bacterial species exchange signaling molecules to coordinate population-wide responses. For this process, known as quorum sensing, the concentration of the respective molecules is crucial. Here, we consider the interaction between spatially distributed bacterial colonies so that the spreading of the signaling molecules in space becomes important. The exponential growth of the signal-producing populations and the corresponding increase in signaling molecule production result in an exponential concentration profile that spreads with uniform speed. The theoretical predictions are supported by experiments with different strains of the soil bacterium Sinorhizobium meliloti that display fluorescence when either producing or responding to the signaling molecules.
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Affiliation(s)
- Vera Bettenworth
- LOEWE-Zentrum für Synthetische Mikrobiologie (SYNMIKRO), Philipps-Universität Marburg
| | - Matthew McIntosh
- LOEWE-Zentrum für Synthetische Mikrobiologie (SYNMIKRO), Philipps-Universität Marburg
| | - Anke Becker
- LOEWE-Zentrum für Synthetische Mikrobiologie (SYNMIKRO), Philipps-Universität Marburg
| | - Bruno Eckhardt
- LOEWE-Zentrum für Synthetische Mikrobiologie (SYNMIKRO), Philipps-Universität Marburg
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19
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Abaurrea Velasco C, Abkenar M, Gompper G, Auth T. Collective behavior of self-propelled rods with quorum sensing. Phys Rev E 2018; 98:022605. [PMID: 30253508 DOI: 10.1103/physreve.98.022605] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/20/2018] [Indexed: 06/08/2023]
Abstract
Active agents-like phoretic particles, bacteria, sperm, and cytoskeletal filaments in motility assays-show a large variety of motility-induced collective behaviors, such as aggregation, clustering, and phase separation. The behavior of dense suspensions of engineered phoretic particles and of bacteria during biofilm formation is determined by two qualitatively different physical mechanisms: (i) volume exclusion (short-range steric repulsion) and (ii) quorum sensing (longer-range reduced propulsion due to alteration of the local chemical environment). To systematically characterize such systems, we study semi-penetrable self-propelled rods in two dimensions, with a propulsion force that decreases with increasing local rod density, by employing Brownian dynamics simulations. Volume exclusion and quorum sensing both lead to phase separation; however, the structure of the systems and the rod dynamics vastly differ. Quorum sensing enhances the polarity of the clusters, induces perpendicularity of rods at the cluster borders, and enhances cluster formation. For systems where the rods essentially become passive at high densities, formation of asters and stripes is observed. Systems of rods with larger aspect ratios show more ordered structures compared to those with smaller aspect ratios, due to their stronger alignment, with almost circular asters for strongly density-dependent propulsion force. With increasing range of the quorum-sensing interaction, the local density decreases, asters become less stable, and polar hedgehog clusters and clusters with domains appear.
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Affiliation(s)
- Clara Abaurrea Velasco
- Theoretical Soft Matter and Biophysics, Institute of Complex Systems and Institute for Advanced Simulation, Forschungszentrum Jülich, D-52425 Jülich, Germany
| | - Masoud Abkenar
- Theoretical Soft Matter and Biophysics, Institute of Complex Systems and Institute for Advanced Simulation, Forschungszentrum Jülich, D-52425 Jülich, Germany
| | - Gerhard Gompper
- Theoretical Soft Matter and Biophysics, Institute of Complex Systems and Institute for Advanced Simulation, Forschungszentrum Jülich, D-52425 Jülich, Germany
| | - Thorsten Auth
- Theoretical Soft Matter and Biophysics, Institute of Complex Systems and Institute for Advanced Simulation, Forschungszentrum Jülich, D-52425 Jülich, Germany
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20
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Xue X, Xue C, Tang M. The role of intracellular signaling in the stripe formation in engineered Escherichia coli populations. PLoS Comput Biol 2018; 14:e1006178. [PMID: 29864126 PMCID: PMC6002128 DOI: 10.1371/journal.pcbi.1006178] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/01/2018] [Revised: 06/14/2018] [Accepted: 05/05/2018] [Indexed: 11/18/2022] Open
Abstract
Recent experiments showed that engineered Escherichia coli colonies grow and self-organize into periodic stripes with high and low cell densities in semi-solid agar. The stripes develop sequentially behind a radially propagating colony front, similar to the formation of many other periodic patterns in nature. These bacteria were created by genetically coupling the intracellular chemotaxis pathway of wild-type cells with a quorum sensing module through the protein CheZ. In this paper, we develop multiscale models to investigate how this intracellular pathway affects stripe formation. We first develop a detailed hybrid model that treats each cell as an individual particle and incorporates intracellular signaling via an internal ODE system. To overcome the computational cost of the hybrid model caused by the large number of cells involved, we next derive a mean-field PDE model from the hybrid model using asymptotic analysis. We show that this analysis is justified by the tight agreement between the PDE model and the hybrid model in 1D simulations. Numerical simulations of the PDE model in 2D with radial symmetry agree with experimental data semi-quantitatively. Finally, we use the PDE model to make a number of testable predictions on how the stripe patterns depend on cell-level parameters, including cell speed, cell doubling time and the turnover rate of intracellular CheZ.
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Affiliation(s)
- Xiaoru Xue
- School of Mathematics and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, China
| | - Chuan Xue
- Department of Mathematics, Ohio State University, Columbus, Ohio, United States of America
| | - Min Tang
- School of Mathematics and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, China
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21
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Grafke T, Cates ME, Vanden-Eijnden E. Spatiotemporal Self-Organization of Fluctuating Bacterial Colonies. PHYSICAL REVIEW LETTERS 2017; 119:188003. [PMID: 29219541 DOI: 10.1103/physrevlett.119.188003] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/24/2017] [Indexed: 06/07/2023]
Abstract
We model an enclosed system of bacteria, whose motility-induced phase separation is coupled to slow population dynamics. Without noise, the system shows both static phase separation and a limit cycle, in which a rising global population causes a dense bacterial colony to form, which then declines by local cell death, before dispersing to reinitiate the cycle. Adding fluctuations, we find that static colonies are now metastable, moving between spatial locations via rare and strongly nonequilibrium pathways, whereas the limit cycle becomes almost periodic such that after each redispersion event the next colony forms in a random location. These results, which hint at some aspects of the biofilm-planktonic life cycle, can be explained by combining tools from large deviation theory with a bifurcation analysis in which the global population density plays the role of control parameter.
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Affiliation(s)
- Tobias Grafke
- Courant Institute, New York University, 251 Mercer Street, New York, New York 10012, USA
| | - Michael E Cates
- DAMTP, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
| | - Eric Vanden-Eijnden
- Courant Institute, New York University, 251 Mercer Street, New York, New York 10012, USA
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22
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Ghosh P, Levine H. Morphodynamics of a growing microbial colony driven by cell death. Phys Rev E 2017; 96:052404. [PMID: 29347664 DOI: 10.1103/physreve.96.052404] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/06/2017] [Indexed: 06/07/2023]
Abstract
Bacterial cells can often self-organize into multicellular structures with complex spatiotemporal morphology. In this work, we study the spatiotemporal dynamics of a growing microbial colony in the presence of cell death. We present an individual-based model of nonmotile bacterial cells which grow and proliferate by consuming diffusing nutrients on a semisolid two-dimensional surface. The colony spreads by growth forces and sliding motility of cells and undergoes cell death followed by subsequent disintegration of the dead cells in the medium. We model cell death by considering two possible situations: In one of the cases, cell death occurs in response to the limitation of local nutrients, while the other case corresponds to an active death process, known as apoptotic or programmed cell death. We demonstrate how the colony morphology is influenced by the presence of cell death. Our results show that cell death facilitates transitions from roughly circular to highly branched structures at the periphery of an expanding colony. Interestingly, our results also reveal that for the colonies which are growing in higher initial nutrient concentrations, cell death occurs much earlier compared to the colonies which are growing in lower initial nutrient concentrations. This work provides new insights into the branched patterning of growing bacterial colonies as a consequence of complex interplay among the biochemical and mechanical effects.
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Affiliation(s)
- Pushpita Ghosh
- Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, Hyderabad 500107, India
| | - Herbert Levine
- Center for Theoretical Biological Physics, Rice University, Texas 77005, USA
- Department of Bioengineering, Rice University, Texas 77005, USA
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23
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Phase separation driven by density-dependent movement: A novel mechanism for ecological patterns. Phys Life Rev 2016; 19:107-121. [PMID: 27478087 DOI: 10.1016/j.plrev.2016.07.009] [Citation(s) in RCA: 40] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/21/2016] [Revised: 06/30/2016] [Accepted: 07/08/2016] [Indexed: 11/21/2022]
Abstract
Many ecosystems develop strikingly regular spatial patterns because of small-scale interactions between organisms, a process generally referred to as spatial self-organization. Self-organized spatial patterns are important determinants of the functioning of ecosystems, promoting the growth and survival of the involved organisms, and affecting the capacity of the organisms to cope with changing environmental conditions. The predominant explanation for self-organized pattern formation is spatial heterogeneity in establishment, growth and mortality, resulting from the self-organization processes. A number of recent studies, however, have revealed that movement of organisms can be an important driving process creating extensive spatial patterning in many ecosystems. Here, we review studies that detail movement-based pattern formation in contrasting ecological settings. Our review highlights that a common principle, where movement of organisms is density-dependent, explains observed spatial regular patterns in all of these studies. This principle, well known to physics as the Cahn-Hilliard principle of phase separation, has so-far remained unrecognized as a general mechanism for self-organized complexity in ecology. Using the examples presented in this paper, we explain how this movement principle can be discerned in ecological settings, and clarify how to test this mechanism experimentally. Our study highlights that animal movement, both in isolation and in unison with other processes, is an important mechanism for regular pattern formation in ecosystems.
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Chakraborty C, George Priya Doss C, Zhu H, Agoramoorthy G. Rising Strengths Hong Kong SAR in Bioinformatics. Interdiscip Sci 2016; 9:224-236. [PMID: 26961385 PMCID: PMC7091071 DOI: 10.1007/s12539-016-0147-x] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/13/2015] [Revised: 12/07/2015] [Accepted: 01/08/2016] [Indexed: 12/18/2022]
Abstract
Hong Kong's bioinformatics sector is attaining new heights in combination with its economic boom and the predominance of the working-age group in its population. Factors such as a knowledge-based and free-market economy have contributed towards a prominent position on the world map of bioinformatics. In this review, we have considered the educational measures, landmark research activities and the achievements of bioinformatics companies and the role of the Hong Kong government in the establishment of bioinformatics as strength. However, several hurdles remain. New government policies will assist computational biologists to overcome these hurdles and further raise the profile of the field. There is a high expectation that bioinformatics in Hong Kong will be a promising area for the next generation.
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Affiliation(s)
- Chiranjib Chakraborty
- Department of Bio-informatics, School of Computer and Information Sciences, Galgotias University, Greater Noida, UP, 201306, India
- Department of Computer Sciences, Hong Kong Baptist University, Kowloon Tong, Hong Kong
| | - C George Priya Doss
- Medical Biotechnology Division, School of BioSciences and Technology, VIT University, Vellore, TN, 632014, India
| | - Hailong Zhu
- Department of Computer Sciences, Hong Kong Baptist University, Kowloon Tong, Hong Kong.
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25
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Zhong LX, Xu WJ, Chen RD, Qiu T, Shi YD, Zhong CY. Coupled effects of local movement and global interaction on contagion. PHYSICA A 2015; 436:482-491. [PMID: 32288092 PMCID: PMC7125621 DOI: 10.1016/j.physa.2015.05.023] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/11/2014] [Revised: 03/29/2015] [Indexed: 06/11/2023]
Abstract
By incorporating segregated spatial domain and individual-based linkage into the SIS (susceptible-infected-susceptible) model, we propose a generalized epidemic model which can change from the territorial epidemic model to the networked epidemic model. The role of the individual-based linkage between different spatial domains is investigated. As we adjust the timescale parameter τ from 0 to unity, which represents the degree of activation of the individual-based linkage, three regions are found. Within the region of 0 < τ < 0.02 , the epidemic is determined by local movement and is sensitive to the timescale τ . Within the region of 0.02 < τ < 0.5 , the epidemic is insensitive to the timescale τ . Within the region of 0.5 < τ < 1 , the outbreak of the epidemic is determined by the structure of the individual-based linkage. As we keep an eye on the first region, the role of activating the individual-based linkage in the present model is similar to the role of the shortcuts in the two-dimensional small world network. Only activating a small number of the individual-based linkage can prompt the outbreak of the epidemic globally. The role of narrowing segregated spatial domain and reducing mobility in epidemic control is checked. These two measures are found to be conducive to curbing the spread of infectious disease only when the global interaction is suppressed. A log-log relation between the change in the number of infected individuals and the timescale τ is found. By calculating the epidemic threshold and the mean first encounter time, we heuristically analyze the microscopic characteristics of the propagation of the epidemic in the present model.
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Affiliation(s)
- Li-Xin Zhong
- School of Finance and Coordinated Innovation Center of Wealth Management and Quantitative Investment, Zhejiang University of Finance and Economics, Hangzhou, 310018, China
- School of Economics and Management, Tsinghua University, Beijing, 100084, China
| | - Wen-Juan Xu
- School of Law, Zhejiang University of Finance and Economics, Hangzhou, 310018, China
| | - Rong-Da Chen
- School of Finance and Coordinated Innovation Center of Wealth Management and Quantitative Investment, Zhejiang University of Finance and Economics, Hangzhou, 310018, China
| | - Tian Qiu
- School of Information Engineering, Nanchang Hangkong University, Nanchang, 330063, China
| | - Yong-Dong Shi
- Research Center of Applied Finance, Dongbei University of Finance and Economics, Dalian, 116025, China
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26
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Hinz DF, Panchenko A, Kim TY, Fried E. Motility versus fluctuations in mixtures of self-motile and passive agents. SOFT MATTER 2014; 10:9082-9089. [PMID: 25300877 DOI: 10.1039/c4sm01562b] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
Many biological systems consist of self-motile and passive agents both of which contribute to overall functionality. However, little is known about the properties of such mixtures. Here we formulate a model for mixtures of self-motile and passive agents and show that the model gives rise to three different dynamical phases: a disordered mesoturbulent phase, a polar flocking phase, and a vortical phase characterized by large-scale counter rotating vortices. We use numerical simulations to construct a phase diagram and compare the statistical properties of the different phases with observed features of self-motile bacterial suspensions. Our findings afford specific insights regarding the interaction of microorganisms and passive particles and provide novel strategic guidance for efficient technological realizations of artificial active matter.
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Affiliation(s)
- Denis F Hinz
- Kamstrup A/S, Industrivej 28, Stilling, 8660 Skanderborg, Denmark
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27
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Matas-Navarro R, Golestanian R, Liverpool TB, Fielding SM. Hydrodynamic suppression of phase separation in active suspensions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:032304. [PMID: 25314443 DOI: 10.1103/physreve.90.032304] [Citation(s) in RCA: 24] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/19/2012] [Indexed: 05/15/2023]
Abstract
We simulate with hydrodynamics a suspension of active disks squirming through a Newtonian fluid. We explore numerically the full range of squirmer area fractions from dilute to close packed and show that "motility induced phase separation," which was recently proposed to arise generically in active matter, and which has been seen in simulations of active Brownian disks, is strongly suppressed by hydrodynamic interactions. We give an argument for why this should be the case and support it with counterpart simulations of active Brownian disks in a parameter regime that provides a closer counterpart to hydrodynamic suspensions than in previous studies.
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Affiliation(s)
- Ricard Matas-Navarro
- Department of Physics, University of Durham, Science Laboratories, South Road, Durham, DH1 3LE, UK
| | - Ramin Golestanian
- Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford, OX1 3NP, UK
| | | | - Suzanne M Fielding
- Department of Physics, University of Durham, Science Laboratories, South Road, Durham, DH1 3LE, UK
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28
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Brand HR, Pleiner H, Svenšek D. Reversible and dissipative macroscopic contributions to the stress tensor: active or passive? THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2014; 37:40. [PMID: 25260325 DOI: 10.1140/epje/i2014-14083-4] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/18/2014] [Accepted: 08/29/2014] [Indexed: 06/03/2023]
Abstract
The issue of dynamic contributions to the macroscopic stress tensor has been of high interest in the field of bio-inspired active systems over the last few years. Of particular interest is a direct coupling ("active term") of the stress tensor with the order parameter, the latter describing orientational order induced by active processes. Here we analyze more generally possible reversible and irreversible dynamic contributions to the stress tensor for various passive and active macroscopic systems. This includes systems with tetrahedral/octupolar order, polar and non-polar (chiral) nematic and smectic liquid crystals, as well as active fluids with a dynamic preferred (polar or non-polar) direction. We show that it cannot a priori be seen, neither from the symmetry properties of the macroscopic variables involved, nor from the structure of the cross-coupling contributions to the stress tensor, whether the system studied is active or passive. Rather, that depends on whether the variables that give rise to those cross-couplings in the stress tensor are driven or not. We demonstrate that several simplified descriptions of active systems in the literature that neglect the necessary counter term to the active term violate linear irreversible thermodynamics and lead to an unphysical contribution to the entropy production.
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Affiliation(s)
- H R Brand
- Theoretische Physik III, Universität Bayreuth, 95440, Bayreuth, Germany,
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29
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Liu C, Fu X, Huang JD. Synthetic biology: a new approach to study biological pattern formation. QUANTITATIVE BIOLOGY 2014. [DOI: 10.1007/s40484-013-0021-3] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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30
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Yang X, Marenduzzo D, Marchetti MC. Spiral and never-settling patterns in active systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:012711. [PMID: 24580261 DOI: 10.1103/physreve.89.012711] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/01/2013] [Indexed: 06/03/2023]
Abstract
We present a combined numerical and analytical study of pattern formation in an active system where particles align, possess a density-dependent motility, and are subject to a logistic reaction. The model can describe suspensions of reproducing bacteria, as well as polymerizing actomyosin gels in vitro or in vivo. In the disordered phase, we find that motility suppression and growth compete to yield stable or blinking patterns, which, when dense enough, acquire internal orientational ordering to give asters or spirals. We predict these may be observed within chemotactic aggregates in bacterial fluids. In the ordered phase, the reaction term leads to previously unobserved never-settling patterns which can provide a simple framework to understand the formation of motile and spiral patterns in intracellular actin systems.
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Affiliation(s)
- X Yang
- Physics Department, Syracuse University, Syracuse, New York 13244, USA
| | - D Marenduzzo
- SUPA, School of Physics and Astronomy, University of Edinburgh, Mayfield Road, Edinburgh EH93JZ, United Kingdom
| | - M C Marchetti
- Physics Department and Syracuse Biomaterials Institute, Syracuse University, Syracuse, New York 13244, USA
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31
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Svenšek D, Pleiner H, Brand HR. Collective stop-and-go dynamics of active bacteria swarms. PHYSICAL REVIEW LETTERS 2013; 111:228101. [PMID: 24329471 DOI: 10.1103/physrevlett.111.228101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/01/2013] [Revised: 09/22/2013] [Indexed: 06/03/2023]
Abstract
We set up a macroscopic model of bacterial growth and transport based on a dynamic preferred direction-the collective velocity of the bacteria. This collective velocity is subject to the isotropic-nematic transition modeling the density-controlled transformation between immotile and motile bacterial states. The choice of the dynamic preferred direction introduces a distinctive coupling of orientational ordering and transport not encountered otherwise. The approach can also be applied to other systems spontaneously switching between individual (disordered) and collective (ordered) behavior and/or collectively responding to density variations, e.g., bird flocks, fish schools, etc. We observe a characteristic and robust stop-and-go behavior. The inclusion of chirality results in a complex pulsating dynamics.
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Affiliation(s)
- Daniel Svenšek
- Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia
| | - Harald Pleiner
- Max-Planck-Institute for Polymer Research, Post Office Box 3148, 55021 Mainz, Germany
| | - Helmut R Brand
- Theoretische Physik III, Universität Bayreuth, 95440 Bayreuth, Germany
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32
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Abstract
Self-organization by bacterial cells often leads to the formation of a highly complex spatially-structured biofilm. In such a bacterial biofilm, cells adhere to each other and are embedded in a self-produced extracellular matrix (ECM). Bacillus substilis bacteria utilize localized cell-death patterns which focuses mechanical forces to form wrinkled sheet-like structures in three dimensions. A most intriguing feature underlying this biofilm formation is that vertical buckling and ridge location is biased to occur in region of high cell-death. Here we present a spatially extended model to investigate the role of the bacterial secreted ECM during the biofilm formation and the self-organization of cell-death. Using this reaction-diffusion model we show that the interaction between the cell's motion and the ECM concentration gives rise to a self-trapping instability, leading to variety of cell-death patterns. The resultant spot patterns generated by our model are shown to be in semi-quantitative agreement with recent experimental observation.
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Affiliation(s)
- Pushpita Ghosh
- Center for Theoretical Biological Physics, Rice University, Houston, TX 77005, USA
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33
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Pleiner H, Svenšek D, Brand HR. Active polar two-fluid macroscopic dynamics. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2013; 36:135. [PMID: 24287686 DOI: 10.1140/epje/i2013-13135-7] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/02/2013] [Accepted: 11/12/2013] [Indexed: 06/02/2023]
Abstract
We study the dynamics of systems with a polar dynamic preferred direction. Examples include the pattern-forming growth of bacteria as well as shoals of fish, flocks of birds and migrating insects. Due to the fact that the preferred direction only exists dynamically, but not statically, the macroscopic variable of choice is the macroscopic velocity associated with the motion of the active units, which are typically biological in nature. We derive the macroscopic equations for such a system and discuss novel static, reversible and irreversible cross-couplings connected to a second velocity as a variable. We analyze in detail how the macroscopic behavior of an active system with a polar dynamic preferred direction compares to other systems with two velocities including immiscible liquids and electrically neutral quantum liquids such as superfluid (4)He and (3)He . We critically discuss changes in the normal mode spectrum when comparing uncharged superfluids, immiscible liquids and active system with a polar dynamic preferred direction. We investigate the influence of a macroscopic hand (collective effects of chirality) on the macroscopic behavior of such active media.
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Affiliation(s)
- H Pleiner
- Max Planck Institute for Polymer Research, 55021, Mainz, Germany,
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Liu QX, Doelman A, Rottschäfer V, de Jager M, Herman PMJ, Rietkerk M, van de Koppel J. Phase separation explains a new class of self-organized spatial patterns in ecological systems. Proc Natl Acad Sci U S A 2013; 110:11905-10. [PMID: 23818579 PMCID: PMC3718087 DOI: 10.1073/pnas.1222339110] [Citation(s) in RCA: 77] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
The origin of regular spatial patterns in ecological systems has long fascinated researchers. Turing's activator-inhibitor principle is considered the central paradigm to explain such patterns. According to this principle, local activation combined with long-range inhibition of growth and survival is an essential prerequisite for pattern formation. Here, we show that the physical principle of phase separation, solely based on density-dependent movement by organisms, represents an alternative class of self-organized pattern formation in ecology. Using experiments with self-organizing mussel beds, we derive an empirical relation between the speed of animal movement and local animal density. By incorporating this relation in a partial differential equation, we demonstrate that this model corresponds mathematically to the well-known Cahn-Hilliard equation for phase separation in physics. Finally, we show that the predicted patterns match those found both in field observations and in our experiments. Our results reveal a principle for ecological self-organization, where phase separation rather than activation and inhibition processes drives spatial pattern formation.
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Affiliation(s)
- Quan-Xing Liu
- Department of Spatial Ecology, Royal Netherlands Institute for Sea Research, 4400 AC Yerseke, The Netherlands.
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35
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Localized cell death focuses mechanical forces during 3D patterning in a biofilm. Proc Natl Acad Sci U S A 2012; 109:18891-6. [PMID: 23012477 DOI: 10.1073/pnas.1212429109] [Citation(s) in RCA: 223] [Impact Index Per Article: 18.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/31/2022] Open
Abstract
From microbial biofilm communities to multicellular organisms, 3D macroscopic structures develop through poorly understood interplay between cellular processes and mechanical forces. Investigating wrinkled biofilms of Bacillus subtilis, we discovered a pattern of localized cell death that spatially focuses mechanical forces, and thereby initiates wrinkle formation. Deletion of genes implicated in biofilm development, together with mathematical modeling, revealed that ECM production underlies the localization of cell death. Simultaneously with cell death, we quantitatively measured mechanical stiffness and movement in WT and mutant biofilms. Results suggest that localized cell death provides an outlet for lateral compressive forces, thereby promoting vertical mechanical buckling, which subsequently leads to wrinkle formation. Guided by these findings, we were able to generate artificial wrinkle patterns within biofilms. Formation of 3D structures facilitated by cell death may underlie self-organization in other developmental systems, and could enable engineering of macroscopic structures from cell populations.
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