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Ayensa-Jiménez J, Doweidar MH, Doblaré M, Gaffney EA. A Mathematical Modelling Study of Chemotactic Dynamics in Cell Cultures: The Impact of Spatio-temporal Heterogeneity. Bull Math Biol 2023; 85:98. [PMID: 37684435 PMCID: PMC10491576 DOI: 10.1007/s11538-023-01194-9] [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: 12/19/2022] [Accepted: 08/04/2023] [Indexed: 09/10/2023]
Abstract
As motivated by studies of cellular motility driven by spatiotemporal chemotactic gradients in microdevices, we develop a framework for constructing approximate analytical solutions for the location, speed and cellular densities for cell chemotaxis waves in heterogeneous fields of chemoattractant from the underlying partial differential equation models. In particular, such chemotactic waves are not in general translationally invariant travelling waves, but possess a spatial variation that evolves in time, and may even oscillate back and forth in time, according to the details of the chemotactic gradients. The analytical framework exploits the observation that unbiased cellular diffusive flux is typically small compared to chemotactic fluxes and is first developed and validated for a range of exemplar scenarios. The framework is subsequently applied to more complex models considering the chemoattractant dynamics under more general settings, potentially including those of relevance for representing pathophysiology scenarios in microdevice studies. In particular, even though solutions cannot be constructed in all cases, a wide variety of scenarios can be considered analytically, firstly providing global insight into the important mechanisms and features of cell motility in complex spatiotemporal fields of chemoattractant. Such analytical solutions also provide a means of rapid evaluation of model predictions, with the prospect of application in computationally demanding investigations relating theoretical models and experimental observation, such as Bayesian parameter estimation.
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Affiliation(s)
- Jacobo Ayensa-Jiménez
- Aragon Institute of Engineering Research, University of Zaragoza, Mariano Esquillor, s.n., 50018 Zaragoza, Spain
- Tissue Microenvironment Laboratory (TME Lab), Institute for Health Research Aragón, San Juan Bosco, 13, 50009 Zaragoza, Spain
| | - Mohamed H. Doweidar
- Aragon Institute of Engineering Research, University of Zaragoza, Mariano Esquillor, s.n., 50018 Zaragoza, Spain
- Mechanical Engineering Department, School of Engineering and Architecture (EINA), University of Zaragoza, María de Luna s.n., 50018 Zaragoza, Spain
| | - Manuel Doblaré
- Aragon Institute of Engineering Research, University of Zaragoza, Mariano Esquillor, s.n., 50018 Zaragoza, Spain
- Mechanical Engineering Department, School of Engineering and Architecture (EINA), University of Zaragoza, María de Luna s.n., 50018 Zaragoza, Spain
- Tissue Microenvironment Laboratory (TME Lab), Institute for Health Research Aragón, San Juan Bosco, 13, 50009 Zaragoza, Spain
- Nanjing Tech University, 30 South Puzhu Road, 211816 Nanjing, China
| | - Eamonn A. Gaffney
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG UK
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Wang Q, Wu H. Mathematical modeling of chemotaxis guided amoeboid cell swimming. Phys Biol 2021; 18. [PMID: 33853049 DOI: 10.1088/1478-3975/abf7d8] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/10/2021] [Accepted: 04/14/2021] [Indexed: 01/15/2023]
Abstract
Cells and microorganisms adopt various strategies to migrate in response to different environmental stimuli. To date, many modeling research has focused on the crawling-basedDictyostelium discoideum(Dd) cells migration induced by chemotaxis, yet recent experimental results reveal that even without adhesion or contact to a substrate, Dd cells can still swim to follow chemoattractant signals. In this paper, we develop a modeling framework to investigate the chemotaxis induced amoeboid cell swimming dynamics. A minimal swimming system consists of one deformable Dd amoeboid cell and a dilute suspension of bacteria, and the bacteria produce chemoattractant signals that attract the Dd cell. We use themathematical amoeba modelto generate Dd cell deformation and solve the resulting low Reynolds number flows, and use a moving mesh based finite volume method to solve the reaction-diffusion-convection equation. Using the computational model, we show that chemotaxis guides a swimming Dd cell to follow and catch bacteria, while on the other hand, bacterial rheotaxis may help the bacteria to escape from the predator Dd cell.
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Affiliation(s)
- Qixuan Wang
- Department of Mathematics, University of California, Riverside, CA, United States of America.,Interdisciplinary Center for Quantitative Modeling in Biology, University of California, Riverside, CA, United States of America
| | - Hao Wu
- Department of Polymer Science and Engineering, University of Massachusetts, Amherst, MA, United States of America
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Bitsouni V, Trucu D, Chaplain MAJ, Eftimie R. Aggregation and travelling wave dynamics in a two-population model of cancer cell growth and invasion. MATHEMATICAL MEDICINE AND BIOLOGY-A JOURNAL OF THE IMA 2018; 35:541-577. [DOI: 10.1093/imammb/dqx019] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2016] [Accepted: 11/14/2017] [Indexed: 12/25/2022]
Affiliation(s)
- Vasiliki Bitsouni
- Division of Mathematics, University of Dundee, Dundee, DD1 4HN, Scotland, UK
| | - Dumitru Trucu
- Division of Mathematics, University of Dundee, Dundee, DD1 4HN, Scotland, UK
| | - Mark A J Chaplain
- School of Mathematics and Statistics, Mathematical Institute (MI), North Haugh
- University of St Andrews, St Andrews, KY16 9SS, Scotland, UK
| | - Raluca Eftimie
- Division of Mathematics, University of Dundee, Dundee, DD1 4HN, Scotland, UK
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Domschke P, Trucu D, Gerisch A, Chaplain MAJ. Structured models of cell migration incorporating molecular binding processes. J Math Biol 2017; 75:1517-1561. [PMID: 28405746 DOI: 10.1007/s00285-017-1120-y] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/30/2016] [Revised: 03/07/2017] [Indexed: 10/19/2022]
Abstract
The dynamic interplay between collective cell movement and the various molecules involved in the accompanying cell signalling mechanisms plays a crucial role in many biological processes including normal tissue development and pathological scenarios such as wound healing and cancer. Information about the various structures embedded within these processes allows a detailed exploration of the binding of molecular species to cell-surface receptors within the evolving cell population. In this paper we establish a general spatio-temporal-structural framework that enables the description of molecular binding to cell membranes coupled with the cell population dynamics. We first provide a general theoretical description for this approach and then illustrate it with three examples arising from cancer invasion.
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Affiliation(s)
- Pia Domschke
- Fachbereich Mathematik, Technische Universität Darmstadt, Dolivostr. 15, 64293, Darmstadt, Germany.
| | - Dumitru Trucu
- Division of Mathematics, University of Dundee, Dundee, DD1 4HN, UK
| | - Alf Gerisch
- Fachbereich Mathematik, Technische Universität Darmstadt, Dolivostr. 15, 64293, Darmstadt, Germany
| | - Mark A J Chaplain
- School of Mathematics and Statistics, Mathematical Institute, University of St Andrews, St Andrews, KY16 9SS, UK
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Delitala M, Lorenzi T. Emergence of spatial patterns in a mathematical model for the co-culture dynamics of epithelial-like and mesenchymal-like cells. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2017; 14:79-93. [PMID: 27879121 DOI: 10.3934/mbe.2017006] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
Accumulating evidence indicates that the interaction between epithelial and mesenchymal cells plays a pivotal role in cancer development and metastasis formation. Here we propose an integro-differential model for the co-culture dynamics of epithelial-like and mesenchymal-like cells. Our model takes into account the effects of chemotaxis, adhesive interactions between epithelial-like cells, proliferation and competition for nutrients. We present a sample of numerical results which display the emergence of spots, stripes and honeycomb patterns, depending on parameters and initial data. These simulations also suggest that epithelial-like and mesenchymal-like cells can segregate when there is little competition for nutrients. Furthermore, our computational results provide a possible explanation for how the concerted action between epithelial-cell adhesion and mesenchymal-cell spreading can precipitate the formation of ring-like structures, which resemble the fibrous capsules frequently observed in hepatic tumours.
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Affiliation(s)
- Marcello Delitala
- Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy.
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Choi SH, Kim YJ. A Discrete Velocity Kinetic Model with Food Metric: Chemotaxis Traveling Waves. Bull Math Biol 2016; 79:277-302. [PMID: 27995380 DOI: 10.1007/s11538-016-0235-4] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/24/2016] [Accepted: 11/30/2016] [Indexed: 11/29/2022]
Abstract
We introduce a mesoscopic scale chemotaxis model for traveling wave phenomena which is induced by food metric. The organisms of this simplified kinetic model have two discrete velocity modes, [Formula: see text] and a constant tumbling rate. The main feature of the model is that the speed of organisms is constant [Formula: see text] with respect to the food metric, not the Euclidean metric. The uniqueness and the existence of the traveling wave solution of the model are obtained. Unlike the classical logarithmic model case there exist traveling waves under super-linear consumption rates and infinite population pulse-type traveling waves are obtained. Numerical simulations are also provided.
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Affiliation(s)
- Sun-Ho Choi
- Department of Applied Mathematics and the Institute of Natural Sciences, Kyung Hee University, Yongin, 446-701, South Korea
| | - Yong-Jung Kim
- Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon, 305-701, South Korea. .,National Institute of Mathematical Sciences, 70 Yuseong-daero, Yuseong-gu, Daejeon, 305-811, South Korea.
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Djomegni PMT. Travelling wave analysis in chemotaxis: case of starvation. SPRINGERPLUS 2016; 5:917. [PMID: 27386361 PMCID: PMC4927555 DOI: 10.1186/s40064-016-2507-8] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 03/31/2016] [Accepted: 06/02/2016] [Indexed: 11/24/2022]
Abstract
In this paper we investigate the existence of travelling wave solutions for a chemotaxis model under the scenarios of zero growth and constant growth rate. We use Lie symmetry analysis to generate generalized travelling wave solutions, a wider class of solutions than that obtained from the standard ansatz. Unlike previous approaches, we allow for diffusivity and signal degradation. We study the influence of cell growth, diffusivity and signal degradation on the behaviour of the system. We apply realistic boundary conditions to explicitly provide biologically relevant solutions. Our results generalize known results.
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Affiliation(s)
- P M Tchepmo Djomegni
- Department of Mathematical Sciences, University of South Africa, Johannesburg, 0003 South Africa
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9
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Symmetries and pattern formation in hyperbolic versus parabolic models of self-organised aggregation. J Math Biol 2014; 71:847-81. [DOI: 10.1007/s00285-014-0842-3] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/12/2014] [Revised: 09/18/2014] [Indexed: 11/26/2022]
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Bacterial chemotaxis without gradient-sensing. J Math Biol 2014; 70:1359-80. [PMID: 24865467 DOI: 10.1007/s00285-014-0790-y] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/04/2013] [Revised: 03/13/2014] [Indexed: 10/25/2022]
Abstract
Chemotaxis models are based on spatial or temporal gradient measurements by individual organisms. The key contribution of Keller and Segel (J Theor Biol 30:225-234, 1971a; J Theor Biol 30:235-248, 1971b) is showing that erratic measurements of individuals may result in an accurate chemotaxis phenomenon as a group. In this paper we provide another option to understand chemotactic behavior when individuals do not sense the gradient of chemical concentration by any means. We show that, if individuals increase their dispersal rate to find food when there is not enough food, an accurate chemotactic behavior may be obtained without sensing the gradient. Such a dispersal has been suggested by Cho and Kim (Bull Math Biol 75:845-870, 2013) and was called starvation driven diffusion. This model is surprisingly similar to the original Keller-Segel model. A comprehensive picture of traveling bands and fronts is provided.
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Franz B, Xue C, Painter KJ, Erban R. Travelling waves in hybrid chemotaxis models. Bull Math Biol 2013; 76:377-400. [PMID: 24347253 DOI: 10.1007/s11538-013-9924-4] [Citation(s) in RCA: 23] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/05/2013] [Accepted: 12/02/2013] [Indexed: 11/27/2022]
Abstract
Hybrid models of chemotaxis combine agent-based models of cells with partial differential equation models of extracellular chemical signals. In this paper, travelling wave properties of hybrid models of bacterial chemotaxis are investigated. Bacteria are modelled using an agent-based (individual-based) approach with internal dynamics describing signal transduction. In addition to the chemotactic behaviour of the bacteria, the individual-based model also includes cell proliferation and death. Cells consume the extracellular nutrient field (chemoattractant), which is modelled using a partial differential equation. Mesoscopic and macroscopic equations representing the behaviour of the hybrid model are derived and the existence of travelling wave solutions for these models is established. It is shown that cell proliferation is necessary for the existence of non-transient (stationary) travelling waves in hybrid models. Additionally, a numerical comparison between the wave speeds of the continuum models and the hybrid models shows good agreement in the case of weak chemotaxis and qualitative agreement for the strong chemotaxis case. In the case of slow cell adaptation, we detect oscillating behaviour of the wave, which cannot be explained by mean-field approximations.
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Affiliation(s)
- Benjamin Franz
- Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK,
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Travelling waves for a velocity-jump model of cell migration and proliferation. Math Biosci 2013; 244:98-106. [PMID: 23665453 DOI: 10.1016/j.mbs.2013.04.010] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/13/2013] [Revised: 04/13/2013] [Accepted: 04/24/2013] [Indexed: 11/22/2022]
Abstract
Cell invasion, characterised by moving fronts of cells, is an essential aspect of development, repair and disease. Typically, mathematical models of cell invasion are based on the Fisher-Kolmogorov equation. These traditional parabolic models cannot be used to represent experimental measurements of individual cell velocities within the invading population since they imply that information propagates with infinite speed. To overcome this limitation we study combined cell motility and proliferation based on a velocity-jump process where information propagates with finite speed. The model treats the total population of cells as two interacting subpopulations: a subpopulation of left-moving cells, L(x,t), and a subpopulation of right-moving cells, R(x,t). This leads to a system of hyperbolic partial differential equations that includes a turning rate, Λ⩾0, describing the rate at which individuals in the population change direction of movement. We present exact travelling wave solutions of the system of partial differential equations for the special case where Λ=0 and in the limit that Λ→∞. For intermediate turning rates, 0<Λ<∞, we analyse the travelling waves using the phase plane and we demonstrate a transition from smooth monotone travelling waves to smooth nonmonotone travelling waves as Λ decreases through a critical value Λcrit. We conclude by providing a qualitative comparison between the travelling wave solutions of our model and experimental observations of cell invasion. This comparison indicates that the small Λ limit produces results that are consistent with experimental observations.
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Saragosti J, Calvez V, Bournaveas N, Buguin A, Silberzan P, Perthame B. Mathematical description of bacterial traveling pulses. PLoS Comput Biol 2010; 6. [PMID: 20808878 PMCID: PMC2924242 DOI: 10.1371/journal.pcbi.1000890] [Citation(s) in RCA: 61] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/10/2010] [Accepted: 07/15/2010] [Indexed: 11/19/2022] Open
Abstract
The Keller-Segel system has been widely proposed as a model for bacterial waves driven by chemotactic processes. Current experiments on Escherichia coli have shown the precise structure of traveling pulses. We present here an alternative mathematical description of traveling pulses at the macroscopic scale. This modeling task is complemented with numerical simulations in accordance with the experimental observations. Our model is derived from an accurate kinetic description of the mesoscopic run-and-tumble process performed by bacteria. This can account for recent experimental observations with E. coli. Qualitative agreements include the asymmetry of the pulse and transition in the collective behaviour (clustered motion versus dispersion). In addition, we can capture quantitatively the traveling speed of the pulse as well as its characteristic length. This work opens several experimental and theoretical perspectives since coefficients at the macroscopic level are derived from considerations at the cellular scale. For instance, the particular response of a single cell to chemical cues turns out to have a strong effect on collective motion. Furthermore, the bottom-up scaling allows us to perform preliminary mathematical analysis and write efficient numerical schemes. This model is intended as a predictive tool for the investigation of bacterial collective motion.
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Affiliation(s)
- Jonathan Saragosti
- Physico-Chimie-Curie, Institut Curie, UMR CNRS 168, Paris, France
- * E-mail: (JS); (VC)
| | - Vincent Calvez
- Unité de Mathématiques Pures et Appliquées, École Normale Supérieure de Lyon, UMR CNRS 5669, Lyon, France
- NUMED, INRIA Rhône-Alpes, Montbonnot, Lyon, France
- * E-mail: (JS); (VC)
| | | | - Axel Buguin
- Physico-Chimie-Curie, Institut Curie, UMR CNRS 168, Paris, France
| | - Pascal Silberzan
- Physico-Chimie-Curie, Institut Curie, UMR CNRS 168, Paris, France
| | - Benoît Perthame
- Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, UMR CNRS 7598, Paris, France
- Institut Universitaire de France, Paris, France
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