1
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Brückner DB, Broedersz CP. Learning dynamical models of single and collective cell migration: a review. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2024; 87:056601. [PMID: 38518358 DOI: 10.1088/1361-6633/ad36d2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/07/2023] [Accepted: 03/22/2024] [Indexed: 03/24/2024]
Abstract
Single and collective cell migration are fundamental processes critical for physiological phenomena ranging from embryonic development and immune response to wound healing and cancer metastasis. To understand cell migration from a physical perspective, a broad variety of models for the underlying physical mechanisms that govern cell motility have been developed. A key challenge in the development of such models is how to connect them to experimental observations, which often exhibit complex stochastic behaviours. In this review, we discuss recent advances in data-driven theoretical approaches that directly connect with experimental data to infer dynamical models of stochastic cell migration. Leveraging advances in nanofabrication, image analysis, and tracking technology, experimental studies now provide unprecedented large datasets on cellular dynamics. In parallel, theoretical efforts have been directed towards integrating such datasets into physical models from the single cell to the tissue scale with the aim of conceptualising the emergent behaviour of cells. We first review how this inference problem has been addressed in both freely migrating and confined cells. Next, we discuss why these dynamics typically take the form of underdamped stochastic equations of motion, and how such equations can be inferred from data. We then review applications of data-driven inference and machine learning approaches to heterogeneity in cell behaviour, subcellular degrees of freedom, and to the collective dynamics of multicellular systems. Across these applications, we emphasise how data-driven methods can be integrated with physical active matter models of migrating cells, and help reveal how underlying molecular mechanisms control cell behaviour. Together, these data-driven approaches are a promising avenue for building physical models of cell migration directly from experimental data, and for providing conceptual links between different length-scales of description.
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Affiliation(s)
- David B Brückner
- Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria
| | - Chase P Broedersz
- Department of Physics and Astronomy, Vrije Universiteit Amsterdam, 1081 HV Amsterdam, The Netherlands
- Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-Maximilian-University Munich, Theresienstr. 37, D-80333 Munich, Germany
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2
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Beta C, Edelstein-Keshet L, Gov N, Yochelis A. From actin waves to mechanism and back: How theory aids biological understanding. eLife 2023; 12:e87181. [PMID: 37428017 DOI: 10.7554/elife.87181] [Citation(s) in RCA: 5] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/22/2023] [Accepted: 06/01/2023] [Indexed: 07/11/2023] Open
Abstract
Actin dynamics in cell motility, division, and phagocytosis is regulated by complex factors with multiple feedback loops, often leading to emergent dynamic patterns in the form of propagating waves of actin polymerization activity that are poorly understood. Many in the actin wave community have attempted to discern the underlying mechanisms using experiments and/or mathematical models and theory. Here, we survey methods and hypotheses for actin waves based on signaling networks, mechano-chemical effects, and transport characteristics, with examples drawn from Dictyostelium discoideum, human neutrophils, Caenorhabditis elegans, and Xenopus laevis oocytes. While experimentalists focus on the details of molecular components, theorists pose a central question of universality: Are there generic, model-independent, underlying principles, or just boundless cell-specific details? We argue that mathematical methods are equally important for understanding the emergence, evolution, and persistence of actin waves and conclude with a few challenges for future studies.
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Affiliation(s)
- Carsten Beta
- Institute of Physics and Astronomy, University of Potsdam, Potsdam, Germany
| | | | - Nir Gov
- Department of Chemical and Biological Physics, Weizmann Institute of Science, Rehovot, Israel
| | - Arik Yochelis
- Swiss Institute for Dryland Environmental and Energy Research, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion, Israel
- Department of Physics, Ben-Gurion University of the Negev, Be'er Sheva, Israel
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3
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Würthner L, Goychuk A, Frey E. Geometry-induced patterns through mechanochemical coupling. Phys Rev E 2023; 108:014404. [PMID: 37583206 DOI: 10.1103/physreve.108.014404] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/07/2022] [Accepted: 05/15/2023] [Indexed: 08/17/2023]
Abstract
Intracellular protein patterns regulate a variety of vital cellular processes such as cell division and motility, which often involve dynamic cell-shape changes. These changes in cell shape may in turn affect the dynamics of pattern-forming proteins, hence leading to an intricate feedback loop between cell shape and chemical dynamics. While several computational studies have examined the rich resulting dynamics, the underlying mechanisms are not yet fully understood. To elucidate some of these mechanisms, we explore a conceptual model for cell polarity on a dynamic one-dimensional manifold. Using concepts from differential geometry, we derive the equations governing mass-conserving reaction-diffusion systems on time-evolving manifolds. Analyzing these equations mathematically, we show that dynamic shape changes of the membrane can induce pattern-forming instabilities in parts of the membrane, which we refer to as regional instabilities. Deformations of the local membrane geometry can also (regionally) suppress pattern formation and spatially shift already existing patterns. We explain our findings by applying and generalizing the local equilibria theory of mass-conserving reaction-diffusion systems. This allows us to determine a simple onset criterion for geometry-induced pattern-forming instabilities, which is linked to the phase-space structure of the reaction-diffusion system. The feedback loop between membrane shape deformations and reaction-diffusion dynamics then leads to a surprisingly rich phenomenology of patterns, including oscillations, traveling waves, and standing waves, even if these patterns do not occur in systems with a fixed membrane shape. Our paper reveals that the local conformation of the membrane geometry acts as an important dynamical control parameter for pattern formation in mass-conserving reaction-diffusion systems.
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Affiliation(s)
- Laeschkir Würthner
- Arnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS), Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, D-80333 Munich, Germany
| | - Andriy Goychuk
- Arnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS), Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, D-80333 Munich, Germany
| | - Erwin Frey
- Arnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS), Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, D-80333 Munich, Germany
- Max Planck School Matter to Life, Hofgartenstraße 8, D-80539 Munich, Germany
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4
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Cusseddu D, Madzvamuse A. Numerical investigations of the bulk-surface wave pinning model. Math Biosci 2022; 354:108925. [PMID: 36397641 DOI: 10.1016/j.mbs.2022.108925] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/25/2022] [Revised: 10/12/2022] [Accepted: 10/15/2022] [Indexed: 11/15/2022]
Abstract
The bulk-surface wave pinning model is a reaction-diffusion system for studying cell polarisation. It is constituted by a surface reaction-diffusion equation, coupled to a bulk diffusion equation with a non-linear boundary condition. Cell polarisation arises as the surface component develops specific patterns. Since proteins diffuse much faster in the cell interior than on the membrane, in the literature, the bulk component is often assumed to be spatially homogeneous. Therefore, the model can be reduced to a single surface equation. However, in real applications a spatially non-uniform bulk component might be an important player to take into account. In this paper, we study, through numerical computations, the role of the bulk component and, more specifically, how different bulk diffusion rates might affect the polarisation response. We find that the bulk component is indeed a key factor in determining the surface polarisation response. Moreover, for certain geometries, it is the spatial heterogeneity of the bulk component that triggers the polarisation response, which might not be possible in a reduced model. Understanding how polarisation depends on bulk diffusivity might be crucial when studying models of migrating cells, which are naturally subject to domain deformation.
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Affiliation(s)
- Davide Cusseddu
- Grupo de Física-Matematica, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Lisboa, 1749-016, Portugal; Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Sussex House, Falmer, Brighton, BN1 9RH, United Kingdom.
| | - Anotida Madzvamuse
- Faculty of Science, Vancouver Campus, Mathematics Department, 121 - 1984 Mathematics Road, Vancouver, B.C., Canada V6T 1Z2; Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Sussex House, Falmer, Brighton, BN1 9RH, United Kingdom.
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5
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Buttenschön A, Edelstein-Keshet L. Cell Repolarization: A Bifurcation Study of Spatio-Temporal Perturbations of Polar Cells. Bull Math Biol 2022; 84:114. [PMID: 36058957 DOI: 10.1007/s11538-022-01053-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/21/2021] [Accepted: 06/30/2022] [Indexed: 11/02/2022]
Abstract
The intrinsic polarity of migrating cells is regulated by spatial distributions of protein activity. Those proteins (Rho-family GTPases, such as Rac and Rho) redistribute in response to stimuli, determining the cell front and back. Reaction-diffusion equations with mass conservation and positive feedback have been used to explain initial polarization of a cell. However, the sensitivity of a polar cell to a reversal stimulus has not yet been fully understood. We carry out a PDE bifurcation analysis of two polarity models to investigate routes to repolarization: (1) a single-GTPase ("wave-pinning") model and (2) a mutually antagonistic Rac-Rho model. We find distinct routes to reversal in (1) vs. (2). We show numerical simulations of full PDE solutions for the RD equations, demonstrating agreement with predictions of the bifurcation results. Finally, we show that simulations of the polarity models in deforming 1D model cells are consistent with biological experiments.
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Affiliation(s)
- Andreas Buttenschön
- Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada.
| | - Leah Edelstein-Keshet
- Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada
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6
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Abstract
Some dividing cells sense their shape by becoming polarized along their long axis. Cell polarity is controlled in part by polarity proteins, like Rho GTPases, cycling between active membrane-bound forms and inactive cytosolic forms, modeled as a "wave-pinning" reaction-diffusion process. Does shape sensing emerge from wave pinning? We show that wave pinning senses the cell's long axis. Simulating wave pinning on a curved surface, we find that high-activity domains migrate to peaks and troughs of the surface. For smooth surfaces, a simple rule of minimizing the domain perimeter while keeping its area fixed predicts the final position of the domain and its shape. However, when we introduce roughness to our surfaces, shape sensing can be disrupted, and high-activity domains can become localized to locations other than the global peaks and valleys of the surface. On rough surfaces, the domains of the wave-pinning model are more robust in finding the peaks and troughs than the minimization rule, although both can become trapped in steady states away from the peaks and valleys. We can control the robustness of shape sensing by altering the Rho GTPase diffusivity and the domain size. We also find that the shape-sensing properties of cell polarity models can explain how domains localize to curved regions of deformed cells. Our results help to understand the factors that allow cells to sense their shape-and the limits that membrane roughness can place on this process.
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Miller PW, Fortunato D, Muratov C, Greengard L, Shvartsman S. Forced and spontaneous symmetry breaking in cell polarization. NATURE COMPUTATIONAL SCIENCE 2022; 2:504-511. [PMID: 37309402 PMCID: PMC10260237 DOI: 10.1038/s43588-022-00295-0] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/20/2022] [Accepted: 07/12/2022] [Indexed: 06/14/2023]
Abstract
How does breaking the symmetry of an equation alter the symmetry of its solutions? Here, we systematically examine how reducing underlying symmetries from spherical to axisymmetric influences the dynamics of an archetypal model of cell polarization, a key process of biological spatial self-organization. Cell polarization is characterized by nonlinear and non-local dynamics, but we overcome the theory challenges these traits pose by introducing a broadly applicable numerical scheme allowing us to efficiently study continuum models in a wide range of geometries. Guided by numerical results, we discover a dynamical hierarchy of timescales that allows us to reduce relaxation to a purely geometric problem of area-preserving geodesic curvature flow. Through application of variational results, we analytically construct steady states on a number of biologically relevant shapes. In doing so, we reveal non-trivial solutions for symmetry breaking.
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Affiliation(s)
- Pearson W. Miller
- Center for Computational Biology, Flatiron Institute, New York, NY, USA
- These authors contributed equally: Pearson W. Miller, Daniel Fortunato
| | - Daniel Fortunato
- Center for Computational Mathematics, Flatiron Institute, New York, NY, USA
- These authors contributed equally: Pearson W. Miller, Daniel Fortunato
| | - Cyrill Muratov
- Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, USA
- Dipartimento di Matematica, Università di Pisa, Pisa, Italy
| | - Leslie Greengard
- Center for Computational Mathematics, Flatiron Institute, New York, NY, USA
- Courant Institute, New York University, New York, NY, USA
| | - Stanislav Shvartsman
- Center for Computational Biology, Flatiron Institute, New York, NY, USA
- Courant Institute, New York University, New York, NY, USA
- Department of Molecular Biology, Princeton University, Princeton, NJ, USA
- Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, NJ, USA
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8
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Krause AL, Gaffney EA, Maini PK, Klika V. Modern perspectives on near-equilibrium analysis of Turing systems. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2021; 379:20200268. [PMID: 34743603 PMCID: PMC8580451 DOI: 10.1098/rsta.2020.0268] [Citation(s) in RCA: 18] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 06/18/2021] [Indexed: 05/02/2023]
Abstract
In the nearly seven decades since the publication of Alan Turing's work on morphogenesis, enormous progress has been made in understanding both the mathematical and biological aspects of his proposed reaction-diffusion theory. Some of these developments were nascent in Turing's paper, and others have been due to new insights from modern mathematical techniques, advances in numerical simulations and extensive biological experiments. Despite such progress, there are still important gaps between theory and experiment, with many examples of biological patterning where the underlying mechanisms are still unclear. Here, we review modern developments in the mathematical theory pioneered by Turing, showing how his approach has been generalized to a range of settings beyond the classical two-species reaction-diffusion framework, including evolving and complex manifolds, systems heterogeneous in space and time, and more general reaction-transport equations. While substantial progress has been made in understanding these more complicated models, there are many remaining challenges that we highlight throughout. We focus on the mathematical theory, and in particular linear stability analysis of 'trivial' base states. We emphasize important open questions in developing this theory further, and discuss obstacles in using these techniques to understand biological reality. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.
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Affiliation(s)
- Andrew L. Krause
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
- Department of Mathematical Sciences, Durham University, Upper Mountjoy Campus, Stockton Rd, Durham DH1 3LE, UK
| | - Eamonn A. Gaffney
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
| | - Philip K. Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
| | - Václav Klika
- Department of Mathematics, FNSPE, Czech Technical University in Prague, Trojanova, 13, 12000 Praha, Czech Republic
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9
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Wang C, Li S, Ademiloye AS, Nithiarasu P. Biomechanics of cells and subcellular components: A comprehensive review of computational models and applications. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING 2021; 37:e3520. [PMID: 34390323 DOI: 10.1002/cnm.3520] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/10/2021] [Accepted: 08/10/2021] [Indexed: 06/13/2023]
Abstract
Cells are a fundamental structural, functional and biological unit for all living organisms. Up till now, considerable efforts have been made to study the responses of single cells and subcellular components to an external load, and understand the biophysics underlying cell rheology, mechanotransduction and cell functions using experimental and in silico approaches. In the last decade, computational simulation has become increasingly attractive due to its critical role in interpreting experimental data, analysing complex cellular/subcellular structures, facilitating diagnostic designs and therapeutic techniques, and developing biomimetic materials. Despite the significant progress, developing comprehensive and accurate models of living cells remains a grand challenge in the 21st century. To understand current state of the art, this review summarises and classifies the vast array of computational biomechanical models for cells. The article covers the cellular components at multi-spatial levels, that is, protein polymers, subcellular components, whole cells and the systems with scale beyond a cell. In addition to the comprehensive review of the topic, this article also provides new insights into the future prospects of developing integrated, active and high-fidelity cell models that are multiscale, multi-physics and multi-disciplinary in nature. This review will be beneficial for the researchers in modelling the biomechanics of subcellular components, cells and multiple cell systems and understanding the cell functions and biological processes from the perspective of cell mechanics.
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Affiliation(s)
- Chengyuan Wang
- Zienkiewicz Centre for Computational Engineering, Faculty of Science and Engineering, Swansea University, Bay Campus, Swansea, UK
| | - Si Li
- Zienkiewicz Centre for Computational Engineering, Faculty of Science and Engineering, Swansea University, Bay Campus, Swansea, UK
| | - Adesola S Ademiloye
- Zienkiewicz Centre for Computational Engineering, Faculty of Science and Engineering, Swansea University, Bay Campus, Swansea, UK
| | - Perumal Nithiarasu
- Zienkiewicz Centre for Computational Engineering, Faculty of Science and Engineering, Swansea University, Bay Campus, Swansea, UK
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10
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Liu Y, Rens EG, Edelstein-Keshet L. Spots, stripes, and spiral waves in models for static and motile cells : GTPase patterns in cells. J Math Biol 2021; 82:28. [PMID: 33660145 PMCID: PMC7929972 DOI: 10.1007/s00285-021-01550-0] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/19/2019] [Revised: 06/18/2020] [Accepted: 10/16/2020] [Indexed: 11/29/2022]
Abstract
The polarization and motility of eukaryotic cells depends on assembly and contraction of the actin cytoskeleton and its regulation by proteins called GTPases. The activity of GTPases causes assembly of filamentous actin (by GTPases Cdc42, Rac), resulting in protrusion of the cell edge. Mathematical models for GTPase dynamics address the spontaneous formation of patterns and nonuniform spatial distributions of such proteins in the cell. Here we revisit the wave-pinning model for GTPase-induced cell polarization, together with a number of extensions proposed in the literature. These include introduction of sources and sinks of active and inactive GTPase (by the group of A. Champneys), and negative feedback from F-actin to GTPase activity. We discuss these extensions singly and in combination, in 1D, and 2D static domains. We then show how the patterns that form (spots, waves, and spirals) interact with cell boundaries to create a variety of interesting and dynamic cell shapes and motion.
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Affiliation(s)
- Yue Liu
- Department of Mathematics, University of British Columbia, Vancouver, V6T 1Z2, BC, Canada. .,Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK.
| | - Elisabeth G Rens
- Department of Mathematics, University of British Columbia, Vancouver, V6T 1Z2, BC, Canada.,Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands
| | - Leah Edelstein-Keshet
- Department of Mathematics, University of British Columbia, Vancouver, V6T 1Z2, BC, Canada
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11
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DiNapoli KT, Robinson DN, Iglesias PA. Tools for computational analysis of moving boundary problems in cellular mechanobiology. WILEY INTERDISCIPLINARY REVIEWS. SYSTEMS BIOLOGY AND MEDICINE 2020; 13:e1514. [PMID: 33305503 DOI: 10.1002/wsbm.1514] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/03/2020] [Revised: 10/08/2020] [Accepted: 10/20/2020] [Indexed: 12/29/2022]
Abstract
A cell's ability to change shape is one of the most fundamental biological processes and is essential for maintaining healthy organisms. When the ability to control shape goes awry, it often results in a diseased system. As such, it is important to understand the mechanisms that allow a cell to sense and respond to its environment so as to maintain cellular shape homeostasis. Because of the inherent complexity of the system, computational models that are based on sound theoretical understanding of the biochemistry and biomechanics and that use experimentally measured parameters are an essential tool. These models involve an inherent feedback, whereby shape is determined by the action of regulatory signals whose spatial distribution depends on the shape. To carry out computational simulations of these moving boundary problems requires special computational techniques. A variety of alternative approaches, depending on the type and scale of question being asked, have been used to simulate various biological processes, including cell motility, division, mechanosensation, and cell engulfment. In general, these models consider the forces that act on the system (both internally generated, or externally imposed) and the mechanical properties of the cell that resist these forces. Moving forward, making these techniques more accessible to the non-expert will help improve interdisciplinary research thereby providing new insight into important biological processes that affect human health. This article is categorized under: Cancer > Cancer>Computational Models Cancer > Cancer>Molecular and Cellular Physiology.
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Affiliation(s)
- Kathleen T DiNapoli
- Department of Cell Biology, Johns Hopkins School of Medicine, Baltimore, Maryland, USA
| | - Douglas N Robinson
- Department of Cell Biology, Johns Hopkins School of Medicine, Baltimore, Maryland, USA
| | - Pablo A Iglesias
- Department of Cell Biology, Johns Hopkins School of Medicine, Baltimore, Maryland, USA
- Department of Electrical & Computer Engineering, Johns Hopkins University, Baltimore, Maryland, USA
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12
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Buttenschön A, Edelstein-Keshet L. Bridging from single to collective cell migration: A review of models and links to experiments. PLoS Comput Biol 2020; 16:e1008411. [PMID: 33301528 PMCID: PMC7728230 DOI: 10.1371/journal.pcbi.1008411] [Citation(s) in RCA: 28] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/25/2022] Open
Abstract
Mathematical and computational models can assist in gaining an understanding of cell behavior at many levels of organization. Here, we review models in the literature that focus on eukaryotic cell motility at 3 size scales: intracellular signaling that regulates cell shape and movement, single cell motility, and collective cell behavior from a few cells to tissues. We survey recent literature to summarize distinct computational methods (phase-field, polygonal, Cellular Potts, and spherical cells). We discuss models that bridge between levels of organization, and describe levels of detail, both biochemical and geometric, included in the models. We also highlight links between models and experiments. We find that models that span the 3 levels are still in the minority.
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Affiliation(s)
- Andreas Buttenschön
- Department of Mathematics, University of British Columbia, Vancouver, Canada
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13
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Kopfer KH, Jäger W, Matthäus F. A mechanochemical model for rho GTPase mediated cell polarization. J Theor Biol 2020; 504:110386. [DOI: 10.1016/j.jtbi.2020.110386] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/16/2020] [Revised: 06/15/2020] [Accepted: 06/17/2020] [Indexed: 01/13/2023]
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14
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Zmurchok C, Collette J, Rajagopal V, Holmes WR. Membrane Tension Can Enhance Adaptation to Maintain Polarity of Migrating Cells. Biophys J 2020; 119:1617-1629. [PMID: 32976760 PMCID: PMC7642449 DOI: 10.1016/j.bpj.2020.08.035] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/01/2020] [Revised: 08/20/2020] [Accepted: 08/25/2020] [Indexed: 12/31/2022] Open
Abstract
Migratory cells are known to adapt to environments that contain wide-ranging levels of chemoattractant. Although biochemical models of adaptation have been previously proposed, here, we discuss a different mechanism based on mechanosensing, in which the interaction between biochemical signaling and cell tension facilitates adaptation. We describe and analyze a model of mechanochemical-based adaptation coupling a mechanics-based physical model of cell tension coupled with the wave-pinning reaction-diffusion model for Rac GTPase activity. The mathematical analysis of this model, simulations of a simplified one-dimensional cell geometry, and two-dimensional finite element simulations of deforming cells reveal that as a cell protrudes under the influence of high stimulation levels, tension-mediated inhibition of Rac signaling causes the cell to polarize even when initially overstimulated. Specifically, tension-mediated inhibition of Rac activation, which has been experimentally observed in recent years, facilitates this adaptation by countering the high levels of environmental stimulation. These results demonstrate how tension-related mechanosensing may provide an alternative (and potentially complementary) mechanism for cell adaptation.
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Affiliation(s)
- Cole Zmurchok
- Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee
| | - Jared Collette
- Department of Biomedical Engineering, University of Melbourne, Melbourne, Australia
| | - Vijay Rajagopal
- Department of Biomedical Engineering, University of Melbourne, Melbourne, Australia
| | - William R Holmes
- Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee; Department of Mathematics, Vanderbilt University, Nashville, Tennessee; Quantitative Systems Biology Center, Vanderbilt University, Nashville, Tennessee.
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15
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Campbell EJ, Bagchi P. A computational study of amoeboid motility in 3D: the role of extracellular matrix geometry, cell deformability, and cell-matrix adhesion. Biomech Model Mechanobiol 2020; 20:167-191. [PMID: 32772275 DOI: 10.1007/s10237-020-01376-7] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2019] [Accepted: 08/01/2020] [Indexed: 12/24/2022]
Abstract
Amoeboid cells often migrate using pseudopods, which are membrane protrusions that grow, bifurcate, and retract dynamically, resulting in a net cell displacement. Many cells within the human body, such as immune cells, epithelial cells, and even metastatic cancer cells, can migrate using the amoeboid phenotype. Amoeboid motility is a complex and multiscale process, where cell deformation, biochemistry, and cytosolic and extracellular fluid motions are coupled. Furthermore, the extracellular matrix (ECM) provides a confined, complex, and heterogeneous environment for the cells to navigate through. Amoeboid cells can migrate without significantly remodeling the ECM using weak or no adhesion, instead utilizing their deformability and the microstructure of the ECM to gain enough traction. While a large volume of work exists on cell motility on 2D substrates, amoeboid motility is 3D in nature. Despite recent progress in modeling cellular motility in 3D, there is a lack of systematic evaluations of the role of ECM microstructure, cell deformability, and adhesion on 3D motility. To fill this knowledge gap, here we present a multiscale, multiphysics modeling study of amoeboid motility through 3D-idealized ECM. The model is a coupled fluid‒structure and coarse-grain biochemistry interaction model that accounts for large deformation of cells, pseudopod dynamics, cytoplasmic and extracellular fluid motion, stochastic dynamics of cell-ECM adhesion, and microstructural (pore-scale) geometric details of the ECM. The key finding of the study is that cell deformation and matrix porosity strongly influence amoeboid motility, while weak adhesion and microscale structural details of the ECM have secondary but subtle effects.
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Affiliation(s)
- Eric J Campbell
- Mechanical and Aerospace Engineering Department, Rutgers, The State University of New Jersey, Piscataway, NJ, 08854, USA
| | - Prosenjit Bagchi
- Mechanical and Aerospace Engineering Department, Rutgers, The State University of New Jersey, Piscataway, NJ, 08854, USA.
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16
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Moure A, Gomez H. Influence of myosin activity and mechanical impact on keratocyte polarization. SOFT MATTER 2020; 16:5177-5194. [PMID: 32459252 DOI: 10.1039/d0sm00473a] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
In cell migration, polarization is the process by which a stationary cell breaks symmetry and initiates motion. Although a lot is known about the mechanisms involved in cell polarization, the role played by myosin contraction remains unclear. In addition, cell polarization by mechanical impact has received little attention. Here, we study the influence of myosin activity on cell polarization and the initiation of motion induced by mechanical cues using a computational model for keratocytes. The model accounts for cell deformation, the dynamics of myosin and the signaling protein RhoA (a member of the Rho GTPases family), as well as the forces acting on the actomyosin network. Our results show that the attainment of a steady polarized state depends on the strength of myosin down- or up-regulation and that myosin upregulation favors cell polarization. Our results also confirm the existence of a threshold level for cell polarization, which is determined by the level of polarization of the Rho GTPases at the time the external stimuli vanish. In all, this paper shows that capturing the interactions between the signaling proteins (Rho GTPases for keratocytes) and the compounds of the motile machinery in a moving cell is crucial to study cell polarization.
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Affiliation(s)
- Adrian Moure
- School of Mechanical Engineering, Purdue University, West Lafayette, 47907 IN, USA.
| | - Hector Gomez
- School of Mechanical Engineering, Purdue University, West Lafayette, 47907 IN, USA. and Weldon School of Biomedical Engineering, Purdue University, West Lafayette, 47907 IN, USA and Purdue University Center for Cancer Research, Purdue University, West Lafayette, 47906 IN, USA
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Merchant B, Feng JJ. A Rho-GTPase based model explains group advantage in collective chemotaxis of neural crest cells. Phys Biol 2020; 17:036002. [DOI: 10.1088/1478-3975/ab71f1] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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18
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Zmurchok C, Holmes WR. Simple Rho GTPase Dynamics Generate a Complex Regulatory Landscape Associated with Cell Shape. Biophys J 2020; 118:1438-1454. [PMID: 32084329 DOI: 10.1016/j.bpj.2020.01.035] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/09/2019] [Revised: 01/27/2020] [Accepted: 01/30/2020] [Indexed: 02/08/2023] Open
Abstract
Migratory cells exhibit a variety of morphologically distinct responses to their environments that manifest in their cell shape. Some protrude uniformly to increase substrate contacts, others are broadly contractile, some polarize to facilitate migration, and yet others exhibit mixtures of these responses. Prior studies have identified a discrete collection of shapes that the majority of cells display and demonstrated that activity levels of the cytoskeletal regulators Rac1 and RhoA GTPase regulate those shapes. Here, we use computational modeling to assess whether known GTPase dynamics can give rise to a sufficient diversity of spatial signaling states to explain the observed shapes. Results show that the combination of autoactivation and mutually antagonistic cross talk between GTPases, along with the conservative membrane binding, generates a wide array of distinct homogeneous and polarized regulatory phenotypes that arise for fixed model parameters. From a theoretical perspective, these results demonstrate that simple GTPase dynamics can generate complex multistability in which six distinct stable steady states (three homogeneous and three polarized) coexist for a fixed set of parameters, each of which naturally maps to an observed morphology. From a biological perspective, although we do not explicitly model the cytoskeleton or resulting cell morphologies, these results, along with prior literature linking GTPase activity to cell morphology, support the hypothesis that GTPase signaling dynamics can generate the broad morphological characteristics observed in many migratory cell populations. Further, the observed diversity may be the result of cells populating a complex morphological landscape generated by GTPase regulation rather than being the result of intrinsic cell-cell variation. These results demonstrate that Rho GTPases may have a central role in regulating the broad characteristics of cell shape (e.g., expansive, contractile, polarized, etc.) and that shape heterogeneity may be (at least partly) a reflection of the rich signaling dynamics regulating the cytoskeleton rather than intrinsic cell heterogeneity.
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Affiliation(s)
- Cole Zmurchok
- Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee
| | - William R Holmes
- Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee; Department of Mathematics, Vanderbilt University, Nashville, Tennessee; Quantitative Systems Biology Center, Vanderbilt University, Nashville, Tennessee.
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19
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Lavi I, Meunier N, Voituriez R, Casademunt J. Motility and morphodynamics of confined cells. Phys Rev E 2020; 101:022404. [PMID: 32168566 DOI: 10.1103/physreve.101.022404] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/13/2019] [Accepted: 12/03/2019] [Indexed: 06/10/2023]
Abstract
We introduce a minimal hydrodynamic model of polarization, migration, and deformation of a biological cell confined between two parallel surfaces. In our model, the cell is driven out of equilibrium by an active cytsokeleton force that acts on the membrane. The cell cytoplasm, described as a viscous droplet in the Darcy flow regime, contains a diffusive solute that actively transduces the applied cytoskeleton force. While fairly simple and analytically tractable, this quasi-two-dimensional model predicts a range of compelling dynamic behaviours. A linear stability analysis of the system reveals that solute activity first destabilizes a global polarization-translation mode, prompting cell motility through spontaneous symmetry breaking. At higher activity, the system crosses a series of Hopf bifurcations leading to coupled oscillations of droplet shape and solute concentration profiles. At the nonlinear level, we find traveling-wave solutions associated with unique polarized shapes that resemble experimental observations. Altogether, this model offers an analytical paradigm of active deformable systems in which viscous hydrodynamics are coupled to diffusive force transducers.
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Affiliation(s)
- Ido Lavi
- Laboratoire Jean Perrin, CNRS/Sorbonne Université, 4 Place Jussieu, 75005 Paris, France
- Departament de Fsica de la Matria Condensada, Universitat de Barcelona, Avinguda Diagonal 647, 08028 Barcelona, Spain
| | | | - Raphael Voituriez
- Laboratoire Jean Perrin, CNRS/Sorbonne Université, 4 Place Jussieu, 75005 Paris, France
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS/Sorbonne Université, 4 Place Jussieu, 75005 Paris, France
| | - Jaume Casademunt
- Departament de Fsica de la Matria Condensada, Universitat de Barcelona, Avinguda Diagonal 647, 08028 Barcelona, Spain
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20
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Rens EG, Edelstein-Keshet L. From energy to cellular forces in the Cellular Potts Model: An algorithmic approach. PLoS Comput Biol 2019; 15:e1007459. [PMID: 31825952 PMCID: PMC6927661 DOI: 10.1371/journal.pcbi.1007459] [Citation(s) in RCA: 26] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/22/2019] [Revised: 12/23/2019] [Accepted: 10/05/2019] [Indexed: 11/30/2022] Open
Abstract
Single and collective cell dynamics, cell shape changes, and cell migration can be conveniently represented by the Cellular Potts Model, a computational platform based on minimization of a Hamiltonian. Using the fact that a force field is easily derived from a scalar energy (F = −∇H), we develop a simple algorithm to associate effective forces with cell shapes in the CPM. We predict the traction forces exerted by single cells of various shapes and sizes on a 2D substrate. While CPM forces are specified directly from the Hamiltonian on the cell perimeter, we approximate the force field inside the cell domain using interpolation, and refine the results with smoothing. Predicted forces compare favorably with experimentally measured cellular traction forces. We show that a CPM model with internal signaling (such as Rho-GTPase-related contractility) can be associated with retraction-protrusion forces that accompany cell shape changes and migration. We adapt the computations to multicellular systems, showing, for example, the forces that a pair of swirling cells exert on one another, demonstrating that our algorithm works equally well for interacting cells. Finally, we show forces exerted by cells on one another in classic cell-sorting experiments. Cells exert forces on their surroundings and on one another. In simulations of cell shape using the Cellular Potts Model (CPM), the dynamics of deforming cell shapes is traditionally represented by an energy-minimization method. We use this CPM energy, the Hamiltonian, to derive and visualize the corresponding forces exerted by the cells. We use the fact that force is the negative gradient of energy to assign forces to the CPM cell edges, and then extend the results to approximate interior forces by interpolation. We show that this method works for single as well as multiple interacting model cells, both static and motile. Finally, we show favorable comparison between predicted forces and real forces measured experimentally.
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Affiliation(s)
- Elisabeth G. Rens
- Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada
| | - Leah Edelstein-Keshet
- Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada
- * E-mail:
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21
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Cusseddu D, Edelstein-Keshet L, Mackenzie J, Portet S, Madzvamuse A. A coupled bulk-surface model for cell polarisation. J Theor Biol 2019; 481:119-135. [DOI: 10.1016/j.jtbi.2018.09.008] [Citation(s) in RCA: 25] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/03/2018] [Revised: 09/04/2018] [Accepted: 09/07/2018] [Indexed: 10/28/2022]
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22
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Merchant B, Edelstein-Keshet L, Feng JJ. A Rho-GTPase based model explains spontaneous collective migration of neural crest cell clusters. Dev Biol 2018; 444 Suppl 1:S262-S273. [DOI: 10.1016/j.ydbio.2018.01.013] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/24/2017] [Revised: 01/18/2018] [Accepted: 01/18/2018] [Indexed: 02/06/2023]
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Campbell EJ, Bagchi P. A computational model of amoeboid cell motility in the presence of obstacles. SOFT MATTER 2018; 14:5741-5763. [PMID: 29873659 DOI: 10.1039/c8sm00457a] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Locomotion of amoeboid cells is mediated by finger-like protrusions of the cell body, known as pseudopods, which grow, bifurcate, and retract in a dynamic fashion. Pseudopods are the primary mode of locomotion for many cells within the human body, such as leukocytes, embryonic cells, and metastatic cancer cells. Amoeboid motility is a complex and multiscale process, which involves bio-molecular reactions, cell deformation, and cytoplasmic and extracellular fluid motion. Additionally, cells within the human body are subject to a confined 3D environment known as the extra-cellular matrix (ECM), which resembles a fluid-filled porous medium. In this article, we present a 3D, multiphysics computational approach coupling fluid mechanics, solid mechanics, and a pattern formation model to simulate locomotion of amoeboid cells through a porous matrix composed of a viscous fluid and an array of finite-sized spherical obstacles. The model combines reaction-diffusion of activator/inhibitors, extreme deformation of the cell, pseudopod dynamics, cytoplasmic and extracellular fluid motion, and fully resolved extracellular matrix. A surface finite-element method is used to obtain the cell deformation and activator/inhibitor concentrations, while the fluid motion is solved using a combined finite-volume and spectral method. The immersed-boundary methods are used to couple the cell deformation, obstacles, and fluid. The model is able to recreate squeezing and weaving motion of cells through the matrix. We study the influence of matrix porosity, obstacle size, and cell deformability on the motility behavior. It is found that below certain values of these parameters, cell motion is completely inhibited. Phase diagrams are presented depicting such motility limits. Interesting dynamics seen in the presence of obstacles but absent in unconfined medium, such as freezing or cell arrest, probing, doubling-back, and tug-of-war are predicted. Furthermore, persistent unidirectional motion of cells that is often observed in an unconfined medium is shown to be lost in presence of obstacles, and is attributed to an alteration of the pseudopod dynamics. The same mechanism, however, allows the cell to find a new direction to penetrate further into the matrix without being stuck in one place. The results and analysis presented here show a strong coupling between cell deformability and ECM properties, and provide new fluid mechanical insights on amoeboid motility in confined medium.
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Affiliation(s)
- Eric J Campbell
- Mechanical and Aerospace Engineering Department, Rutgers, The State University of New Jersey, Piscataway, NJ 08854, USA.
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24
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Feng S, Zhou L, Zhang Y, Lü S, Long M. Mechanochemical modeling of neutrophil migration based on four signaling layers, integrin dynamics, and substrate stiffness. Biomech Model Mechanobiol 2018; 17:1611-1630. [PMID: 29968162 DOI: 10.1007/s10237-018-1047-2] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/21/2018] [Accepted: 06/24/2018] [Indexed: 01/09/2023]
Abstract
Directional neutrophil migration during human immune responses is a highly coordinated process regulated by both biochemical and biomechanical environments. In this paper, we developed an integrative mathematical model of neutrophil migration using a lattice Boltzmann-particle method built in-house to solve the moving boundary problem with spatiotemporal regulation of biochemical components. The mechanical features of the cell cortex are modeled by a series of spring-connected nodes representing discrete cell-substrate adhesive sites. The intracellular signaling cascades responsible for cytoskeletal remodeling [e.g., small GTPases, phosphoinositide-3-kinase (PI3K), and phosphatase and tensin homolog] are built based on our previous four-layered signaling model centered on the bidirectional molecular transport mechanism and implemented as reaction-diffusion equations. Focal adhesion dynamics are determined by force-dependent integrin-ligand binding kinetics and integrin recycling and are thus integrated with cell motion. Using numerical simulations, the model reproduces the major features of cell migration in response to uniform and gradient biochemical stimuli based on the quantitative spatiotemporal regulation of signaling molecules, which agree with experimental observations. The existence of multiple types of integrins with different binding kinetics could act as an adaptation mechanism for substrate stiffness. Moreover, cells can perform reversal, U-turn, or lock-on behaviors depending on the steepness of the reversal biochemical signals received. Finally, this model is also applied to predict the responses of mutants in which PTEN is overexpressed or disrupted.
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Affiliation(s)
- Shiliang Feng
- Center for Biomechanics and Bioengineering, Key Laboratory of Microgravity (National Microgravity Laboratory), and Beijing Key Laboratory of Engineered Construction and Mechanobiology, Institute of Mechanics, Chinese Academy of Sciences, Beijing, China
- School of Engineering Science, University of Chinese Academy of Sciences, Beijing, China
| | - Lüwen Zhou
- Center for Biomechanics and Bioengineering, Key Laboratory of Microgravity (National Microgravity Laboratory), and Beijing Key Laboratory of Engineered Construction and Mechanobiology, Institute of Mechanics, Chinese Academy of Sciences, Beijing, China
- School of Engineering Science, University of Chinese Academy of Sciences, Beijing, China
| | - Yan Zhang
- Center for Biomechanics and Bioengineering, Key Laboratory of Microgravity (National Microgravity Laboratory), and Beijing Key Laboratory of Engineered Construction and Mechanobiology, Institute of Mechanics, Chinese Academy of Sciences, Beijing, China
- School of Engineering Science, University of Chinese Academy of Sciences, Beijing, China
| | - Shouqin Lü
- Center for Biomechanics and Bioengineering, Key Laboratory of Microgravity (National Microgravity Laboratory), and Beijing Key Laboratory of Engineered Construction and Mechanobiology, Institute of Mechanics, Chinese Academy of Sciences, Beijing, China
- School of Engineering Science, University of Chinese Academy of Sciences, Beijing, China
| | - Mian Long
- Center for Biomechanics and Bioengineering, Key Laboratory of Microgravity (National Microgravity Laboratory), and Beijing Key Laboratory of Engineered Construction and Mechanobiology, Institute of Mechanics, Chinese Academy of Sciences, Beijing, China.
- School of Engineering Science, University of Chinese Academy of Sciences, Beijing, China.
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25
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Feng SL, Zhou LW, Lü SQ, Zhang Y. Dynamic seesaw model for rapid signaling responses in eukaryotic chemotaxis. Phys Biol 2018; 15:056004. [PMID: 29757152 DOI: 10.1088/1478-3975/aac45b] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/18/2023]
Abstract
Directed movement of eukaryotic cells toward spatiotemporally varied chemotactic stimuli enables rapid intracellular signaling responses. While macroscopic cellular manifestation is shaped by balancing external stimuli strength with finite internal delays, the organizing principles of the underlying molecular mechanisms remain to be clarified. Here, we developed a novel modeling framework based on a simple seesaw mechanism to elucidate how cells repeatedly reverse polarity. As a key feature of the modeling, the bottom module of bidirectional molecular transport is successively controlled by three upstream modules of signal reception, initial signal processing, and Rho GTPase regulation. Our simulations indicated that an isotropic cell is polarized in response to a graded input signal. By applying a reversal gradient to a chemoattractant signal, lamellipod-specific molecules (i.e. PIP3 and PI3K) disappear, first from the cell front, and then they redistribute at the opposite side, whereas functional molecules at the rear of the cell (i.e. PIP2 and PTEN) act oppositely. In particular, the model cell exhibits a seesaw-like spatiotemporal pattern for the establishment of front and rear and interconversion, consistent with those related experimental observations. Increasing the switching frequency of the chemotactic gradient causes the cell to stay in a trapped state, further supporting the proposed dynamics of eukaryotic chemotaxis with the underlying cytoskeletal remodeling.
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Affiliation(s)
- Shi Liang Feng
- Institute of mechanical engineering and mechanics, Ningbo University, Ningbo 315211, People's Republic of China. Center of Biomechanics and Bioengineering and Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, People's Republic of China
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26
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Ebata H, Yamamoto A, Tsuji Y, Sasaki S, Moriyama K, Kuboki T, Kidoaki S. Persistent random deformation model of cells crawling on a gel surface. Sci Rep 2018; 8:5153. [PMID: 29581462 PMCID: PMC5980085 DOI: 10.1038/s41598-018-23540-x] [Citation(s) in RCA: 18] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/03/2017] [Accepted: 03/15/2018] [Indexed: 12/19/2022] Open
Abstract
In general, cells move on a substrate through extension and contraction of the cell body. Though cell movement should be explained by taking into account the effect of such shape fluctuations, past approaches to formulate cell-crawling have not sufficiently quantified the relationship between cell movement (velocity and trajectory) and shape fluctuations based on experimental data regarding actual shaping dynamics. To clarify this relationship, we experimentally characterized cell-crawling in terms of shape fluctuations, especially extension and contraction, by using an elasticity-tunable gel substrate to modulate cell shape. As a result, an amoeboid swimmer-like relation was found to arise between the cell velocity and cell-shape dynamics. To formulate this experimentally-obtained relationship between cell movement and shaping dynamics, we established a persistent random deformation (PRD) model based on equations of a deformable self-propelled particle adopting an amoeboid swimmer-like velocity-shape relationship. The PRD model successfully explains the statistical properties of velocity, trajectory and shaping dynamics of the cells including back-and-forth motion, because the velocity equation exhibits time-reverse symmetry, which is essentially different from previous models. We discuss the possible application of this model to classify the phenotype of cell migration based on the characteristic relation between movement and shaping dynamics.
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Affiliation(s)
- Hiroyuki Ebata
- Laboratory of Biomedical and Biophysical Chemistry, Institute for Materials Chemistry and Engineering, Kyushu University, CE41-204, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan.
| | - Aki Yamamoto
- Laboratory of Biomedical and Biophysical Chemistry, Institute for Materials Chemistry and Engineering, Kyushu University, CE41-204, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan
| | - Yukie Tsuji
- Laboratory of Biomedical and Biophysical Chemistry, Institute for Materials Chemistry and Engineering, Kyushu University, CE41-204, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan
| | - Saori Sasaki
- Laboratory of Biomedical and Biophysical Chemistry, Institute for Materials Chemistry and Engineering, Kyushu University, CE41-204, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan
| | - Kousuke Moriyama
- Laboratory of Biomedical and Biophysical Chemistry, Institute for Materials Chemistry and Engineering, Kyushu University, CE41-204, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan
| | - Thasaneeya Kuboki
- Laboratory of Biomedical and Biophysical Chemistry, Institute for Materials Chemistry and Engineering, Kyushu University, CE41-204, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan
| | - Satoru Kidoaki
- Laboratory of Biomedical and Biophysical Chemistry, Institute for Materials Chemistry and Engineering, Kyushu University, CE41-204, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan.
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27
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Khatibi S, Rios KI, Nguyen LK. Computational Modeling of the Dynamics of Spatiotemporal Rho GTPase Signaling: A Systematic Review. Methods Mol Biol 2018; 1821:3-20. [PMID: 30062401 DOI: 10.1007/978-1-4939-8612-5_1] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
The Rho family of GTPases are known to play pivotal roles in the regulation of fundamental cellular processes, ranging from cell migration and polarity to wound healing and regulation of actin cytoskeleton. Over the past decades, accumulating experimental work has increasingly mapped out the mechanistic details and interactions between members of the family and their regulators, establishing detailed interaction circuits within the Rho GTPase signaling network. These circuits have served as a vital foundation based on which a multitude of mathematical models have been developed to explain experimental data, gain deeper insights into the biological phenomenon they describe, as well as make new testable predictions and hypotheses. Due to the diverse nature and purpose of these models, they often vary greatly in size, scope, complexity, and formulation. Here, we provide a systematic, categorical, and comprehensive account of the recent modeling studies of Rho family GTPases, with an aim to offer a broad perspective of the field. The modeling limitations and possible future research directions are also discussed.
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Affiliation(s)
- Shabnam Khatibi
- Department of Biochemistry and Molecular Biology, Monash University, Melbourne, VIC, Australia
| | - Karina Islas Rios
- Department of Biochemistry and Molecular Biology, Monash University, Melbourne, VIC, Australia
| | - Lan K Nguyen
- Department of Biochemistry and Molecular Biology, Monash University, Melbourne, VIC, Australia.
- Cancer Program, Biomedicine Discovery Institute, Monash University, Melbourne, VIC, Australia.
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28
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Nickaeen M, Novak IL, Pulford S, Rumack A, Brandon J, Slepchenko BM, Mogilner A. A free-boundary model of a motile cell explains turning behavior. PLoS Comput Biol 2017; 13:e1005862. [PMID: 29136638 PMCID: PMC5705165 DOI: 10.1371/journal.pcbi.1005862] [Citation(s) in RCA: 24] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/07/2017] [Revised: 11/28/2017] [Accepted: 10/31/2017] [Indexed: 01/14/2023] Open
Abstract
To understand shapes and movements of cells undergoing lamellipodial motility, we systematically explore minimal free-boundary models of actin-myosin contractility consisting of the force-balance and myosin transport equations. The models account for isotropic contraction proportional to myosin density, viscous stresses in the actin network, and constant-strength viscous-like adhesion. The contraction generates a spatially graded centripetal actin flow, which in turn reinforces the contraction via myosin redistribution and causes retraction of the lamellipodial boundary. Actin protrusion at the boundary counters the retraction, and the balance of the protrusion and retraction shapes the lamellipodium. The model analysis shows that initiation of motility critically depends on three dimensionless parameter combinations, which represent myosin-dependent contractility, a characteristic viscosity-adhesion length, and a rate of actin protrusion. When the contractility is sufficiently strong, cells break symmetry and move steadily along either straight or circular trajectories, and the motile behavior is sensitive to conditions at the cell boundary. Scanning of a model parameter space shows that the contractile mechanism of motility supports robust cell turning in conditions where short viscosity-adhesion lengths and fast protrusion cause an accumulation of myosin in a small region at the cell rear, destabilizing the axial symmetry of a moving cell. To understand shapes and movements of simple motile cells, we systematically explore minimal models describing a cell as a two-dimensional actin-myosin gel with a free boundary. The models account for actin-myosin contraction balanced by viscous stresses in the actin gel and uniform adhesion. The myosin contraction causes the lamellipodial boundary to retract. Actin protrusion at the boundary counters the retraction, and the balance of protrusion and retraction shapes the cell. The models reproduce a variety of motile shapes observed experimentally. The analysis shows that the mechanical state of a cell depends on a small number of parameters. We find that when the contractility is sufficiently strong, cells break symmetry and move steadily along either straight or circular trajectory. Scanning model parameters shows that the contractile mechanism of motility supports robust cell turning behavior in conditions where deformable actin gel and fast protrusion destabilize the axial symmetry of a moving cell.
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Affiliation(s)
- Masoud Nickaeen
- Richard D. Berlin Center for Cell Analysis and Modeling, Department of Cell Biology, University of Connecticut Health Center, Farmington, CT, United States of America
| | - Igor L. Novak
- Richard D. Berlin Center for Cell Analysis and Modeling, Department of Cell Biology, University of Connecticut Health Center, Farmington, CT, United States of America
| | - Stephanie Pulford
- Center for Engineering Learning & Teaching, University of Washington, Seattle, WA, United States of America
| | - Aaron Rumack
- Department of Computer Science, Cornell University, Ithaca, NY, United States of America
| | - Jamie Brandon
- Department of Mathematics, Adrian College, Adrian, MI, United States of America
| | - Boris M. Slepchenko
- Richard D. Berlin Center for Cell Analysis and Modeling, Department of Cell Biology, University of Connecticut Health Center, Farmington, CT, United States of America
| | - Alex Mogilner
- Courant Institute and Department of Biology, New York University, New York, NY, United States of America
- * E-mail:
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29
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Camley BA, Zhao Y, Li B, Levine H, Rappel WJ. Crawling and turning in a minimal reaction-diffusion cell motility model: Coupling cell shape and biochemistry. Phys Rev E 2017; 95:012401. [PMID: 28208438 DOI: 10.1103/physreve.95.012401] [Citation(s) in RCA: 43] [Impact Index Per Article: 6.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/25/2016] [Indexed: 11/07/2022]
Abstract
We study a minimal model of a crawling eukaryotic cell with a chemical polarity controlled by a reaction-diffusion mechanism describing Rho GTPase dynamics. The size, shape, and speed of the cell emerge from the combination of the chemical polarity, which controls the locations where actin polymerization occurs, and the physical properties of the cell, including its membrane tension. We find in our model both highly persistent trajectories, in which the cell crawls in a straight line, and turning trajectories, where the cell transitions from crawling in a line to crawling in a circle. We discuss the controlling variables for this turning instability and argue that turning arises from a coupling between the reaction-diffusion mechanism and the shape of the cell. This emphasizes the surprising features that can arise from simple links between cell mechanics and biochemistry. Our results suggest that similar instabilities may be present in a broad class of biochemical descriptions of cell polarity.
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Affiliation(s)
- Brian A Camley
- Department of Physics, University of California, San Diego, La Jolla, California 92093, USA
| | - Yanxiang Zhao
- Department of Mathematics, The George Washington University, Washington, DC 20052, USA
| | - Bo Li
- Department of Mathematics and Graduate Program in Quantitative Biology, University of California, San Diego, La Jolla, California 92093, USA
| | - Herbert Levine
- Department of Bioengineering, Center for Theoretical Biological Physics, Rice University, Houston, Texas 77005, USA
| | - Wouter-Jan Rappel
- Department of Physics, University of California, San Diego, La Jolla, California 92093, USA
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30
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Mak M, Spill F, Kamm RD, Zaman MH. Single-Cell Migration in Complex Microenvironments: Mechanics and Signaling Dynamics. J Biomech Eng 2016; 138:021004. [PMID: 26639083 DOI: 10.1115/1.4032188] [Citation(s) in RCA: 62] [Impact Index Per Article: 7.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/23/2015] [Indexed: 12/21/2022]
Abstract
Cells are highly dynamic and mechanical automata powered by molecular motors that respond to external cues. Intracellular signaling pathways, either chemical or mechanical, can be activated and spatially coordinated to induce polarized cell states and directional migration. Physiologically, cells navigate through complex microenvironments, typically in three-dimensional (3D) fibrillar networks. In diseases, such as metastatic cancer, they invade across physiological barriers and remodel their local environments through force, matrix degradation, synthesis, and reorganization. Important external factors such as dimensionality, confinement, topographical cues, stiffness, and flow impact the behavior of migrating cells and can each regulate motility. Here, we review recent progress in our understanding of single-cell migration in complex microenvironments.
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Modeling the Mechanosensitivity of Neutrophils Passing through a Narrow Channel. Biophys J 2016; 109:2235-45. [PMID: 26636935 DOI: 10.1016/j.bpj.2015.10.032] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/13/2015] [Revised: 09/20/2015] [Accepted: 10/13/2015] [Indexed: 11/21/2022] Open
Abstract
Recent experiments have found that neutrophils may be activated after passing through microfluidic channels and filters. Mechanical deformation causes disassembly of the cytoskeleton and a sudden drop of the elastic modulus of the neutrophil. This fluidization is followed by either activation of the neutrophil with protrusion of pseudopods or a uniform recovery of the cytoskeleton network with no pseudopod. The former occurs if the neutrophil traverses the narrow channel at a slower rate. We propose a chemo-mechanical model for the fluidization and activation processes. Fluidization is treated as mechanical destruction of the cytoskeleton by sufficiently rapid bending. Loss of the cytoskeleton removes a pathway by which cortical tension inhibits the Rac protein. As a result, Rac rises and polarizes through a wave-pinning mechanism if the chemical reaction rate is fast enough. This leads to recovery and reinforcement of the cytoskeleton at the front of the neutrophil, and hence protrusion and activation. Otherwise the Rac signal returns to a uniform pre-deformation state and no activation occurs. Thus, mechanically induced neutrophil activation is understood as the competition between two timescales: that of chemical reaction and that of mechanical deformation. The model captures the main features of the experimental observation.
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Chen W, Nie Q, Yi TM, Chou CS. Modelling of Yeast Mating Reveals Robustness Strategies for Cell-Cell Interactions. PLoS Comput Biol 2016; 12:e1004988. [PMID: 27404800 PMCID: PMC4942089 DOI: 10.1371/journal.pcbi.1004988] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/05/2016] [Accepted: 05/16/2016] [Indexed: 11/18/2022] Open
Abstract
Mating of budding yeast cells is a model system for studying cell-cell interactions. Haploid yeast cells secrete mating pheromones that are sensed by the partner which responds by growing a mating projection toward the source. The two projections meet and fuse to form the diploid. Successful mating relies on precise coordination of dynamic extracellular signals, signaling pathways, and cell shape changes in a noisy background. It remains elusive how cells mate accurately and efficiently in a natural multi-cell environment. Here we present the first stochastic model of multiple mating cells whose morphologies are driven by pheromone gradients and intracellular signals. Our novel computational framework encompassed a moving boundary method for modeling both a-cells and α-cells and their cell shape changes, the extracellular diffusion of mating pheromones dynamically coupled with cell polarization, and both external and internal noise. Quantification of mating efficiency was developed and tested for different model parameters. Computer simulations revealed important robustness strategies for mating in the presence of noise. These strategies included the polarized secretion of pheromone, the presence of the α-factor protease Bar1, and the regulation of sensing sensitivity; all were consistent with data in the literature. In addition, we investigated mating discrimination, the ability of an a-cell to distinguish between α-cells either making or not making α-factor, and mating competition, in which multiple a-cells compete to mate with one α-cell. Our simulations were consistent with previous experimental results. Moreover, we performed a combination of simulations and experiments to estimate the diffusion rate of the pheromone a-factor. In summary, we constructed a framework for simulating yeast mating with multiple cells in a noisy environment, and used this framework to reproduce mating behaviors and to identify strategies for robust cell-cell interactions. One of the riddles of Nature is how cells interact with one another to create complex cellular networks such as the neural networks in the brain. Forming precise connections between irregularly shaped cells is a challenge for biology. We developed computational methods for simulating these complex cell-cell interactions. We applied these methods to investigate yeast mating in which two yeast cells grow projections that meet and fuse guided by pheromone attractants. The simulations described molecules both inside and outside of the cell, and represented the continually changing shapes of the cells. We found that positioning the secretion and sensing of pheromones at the same location on the cell surface was important. Other key factors for robust mating included secreting a protein that removed excess pheromone from outside of the cell so that the signal would not be too strong. An important advance was being able to simulate as many as five cells in complex mating arrangements. Taken together we used our novel computational methods to describe in greater detail the yeast mating process, and more generally, interactions among cells changing their shapes in response to their neighbors.
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Affiliation(s)
- Weitao Chen
- Department of Mathematics, University of California, Irvine, Irvine, California, United States of America
| | - Qing Nie
- Department of Mathematics, University of California, Irvine, Irvine, California, United States of America
| | - Tau-Mu Yi
- Molecular, Cellular, and Developmental Biology, University of California, Santa Barbara, Santa Barbara, California, United States of America
- * E-mail: (TMY); (CSC)
| | - Ching-Shan Chou
- Department of Mathematics, The Ohio State University, Columbus, Ohio, United States of America
- * E-mail: (TMY); (CSC)
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Spill F, Andasari V, Mak M, Kamm RD, Zaman MH. Effects of 3D geometries on cellular gradient sensing and polarization. Phys Biol 2016; 13:036008. [PMID: 27345945 DOI: 10.1088/1478-3975/13/3/036008] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/06/2023]
Abstract
During cell migration, cells become polarized, change their shape, and move in response to various internal and external cues. Cell polarization is defined through the spatio-temporal organization of molecules such as PI3K or small GTPases, and is determined by intracellular signaling networks. It results in directional forces through actin polymerization and myosin contractions. Many existing mathematical models of cell polarization are formulated in terms of reaction-diffusion systems of interacting molecules, and are often defined in one or two spatial dimensions. In this paper, we introduce a 3D reaction-diffusion model of interacting molecules in a single cell, and find that cell geometry has an important role affecting the capability of a cell to polarize, or change polarization when an external signal changes direction. Our results suggest a geometrical argument why more roundish cells can repolarize more effectively than cells which are elongated along the direction of the original stimulus, and thus enable roundish cells to turn faster, as has been observed in experiments. On the other hand, elongated cells preferentially polarize along their main axis even when a gradient stimulus appears from another direction. Furthermore, our 3D model can accurately capture the effect of binding and unbinding of important regulators of cell polarization to and from the cell membrane. This spatial separation of membrane and cytosol, not possible to capture in 1D or 2D models, leads to marked differences of our model from comparable lower-dimensional models.
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Affiliation(s)
- Fabian Spill
- Department of Biomedical Engineering, Boston University, 44 Cummington Street, Boston MA 02215, USA. Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
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Abstract
The study of chemotaxis has benefited greatly from computational models that describe the response of cells to chemoattractant stimuli. These models must keep track of spatially and temporally varying distributions of numerous intracellular species. Moreover, recent evidence suggests that these are not deterministic interactions, but also include the effect of stochastic variations that trigger an excitable network. In this chapter we illustrate how to create simulations of excitable networks using the Virtual Cell modeling environment.
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Affiliation(s)
- Sayak Bhattacharya
- Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, MD, 21218, USA
| | - Pablo A Iglesias
- Department of Cell Biology, Johns Hopkins University School of Medicine, Baltimore, 21205, MD, USA.
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Giese W, Eigel M, Westerheide S, Engwer C, Klipp E. Influence of cell shape, inhomogeneities and diffusion barriers in cell polarization models. Phys Biol 2015; 12:066014. [DOI: 10.1088/1478-3975/12/6/066014] [Citation(s) in RCA: 35] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/27/2022]
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Camley BA, Brown FLH. Fluctuating hydrodynamics of multicomponent membranes with embedded proteins. J Chem Phys 2015; 141:075103. [PMID: 25149817 DOI: 10.1063/1.4892802] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
A simulation method for the dynamics of inhomogeneous lipid bilayer membranes is presented. The membrane is treated using stochastic Saffman-Delbrück hydrodynamics, coupled to a phase-field description of lipid composition and discrete membrane proteins. Multiple applications are considered to validate and parameterize the model. The dynamics of membrane composition fluctuations above the critical point and phase separation dynamics below the critical point are studied in some detail, including the effects of adding proteins to the mixture.
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Affiliation(s)
- Brian A Camley
- Department of Physics and Center for Theoretical Biological Physics, University of California, San Diego, La Jolla, California 92093, USA
| | - Frank L H Brown
- Department of Physics, University of California, Santa Barbara, Santa Barbara, California 93106, USA
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37
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Signaling networks and cell motility: a computational approach using a phase field description. J Math Biol 2013; 69:91-112. [PMID: 23835784 DOI: 10.1007/s00285-013-0704-4] [Citation(s) in RCA: 26] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/10/2012] [Revised: 05/27/2013] [Indexed: 10/26/2022]
Abstract
The processes of protrusion and retraction during cell movement are driven by the turnover and reorganization of the actin cytoskeleton. Within a reaction-diffusion model which combines processes along the cell membrane with processes within the cytoplasm a Turing type instability is used to form the necessary polarity to distinguish between cell front and rear and to initiate the formation of different organizational arrays within the cytoplasm leading to protrusion and retraction. A simplified biochemical network model for the activation of GTPase which accounts for the different dimensionality of the cell membrane and the cytoplasm is used for this purpose and combined with a classical Helfrich type model to account for bending and stiffness effects of the cell membrane. In addition streaming within the cytoplasm and the extracellular matrix is taken into account. Combining these phenomena allows to simulate the dynamics of cells and to reproduce the primary phenomenology of cell motility. The coupled model is formulated within a phase field approach and solved using adaptive finite elements.
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Shi C, Huang CH, Devreotes PN, Iglesias PA. Interaction of motility, directional sensing, and polarity modules recreates the behaviors of chemotaxing cells. PLoS Comput Biol 2013; 9:e1003122. [PMID: 23861660 PMCID: PMC3701696 DOI: 10.1371/journal.pcbi.1003122] [Citation(s) in RCA: 77] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/07/2013] [Accepted: 05/16/2013] [Indexed: 02/04/2023] Open
Abstract
Chemotaxis involves the coordinated action of separable but interrelated processes: motility, gradient sensing, and polarization. We have hypothesized that these are mediated by separate modules that account for these processes individually and that, when combined, recreate most of the behaviors of chemotactic cells. Here, we describe a mathematical model where the modules are implemented in terms of reaction-diffusion equations. Migration and the accompanying changes in cellular morphology are demonstrated in simulations using a mechanical model of the cell cortex implemented in the level set framework. The central module is an excitable network that accounts for random migration. The response to combinations of uniform stimuli and gradients is mediated by a local excitation, global inhibition module that biases the direction in which excitability is directed. A polarization module linked to the excitable network through the cytoskeleton allows unstimulated cells to move persistently and, for cells in gradients, to gradually acquire distinct sensitivity between front and back. Finally, by varying the strengths of various feedback loops in the model we obtain cellular behaviors that mirror those of genetically altered cell lines. Chemotaxis is the movement of cells in response to spatial gradients of chemical cues. While single-celled organisms rely on sensing and responding to chemical gradients to search for nutrients, chemotaxis is also an essential component of the mammalian immune system. However, chemotaxis can also be deleterious, since chemotactic tumor cells can lead to metastasis. Due to its importance, understanding the process by which cells sense and respond to chemical gradients has attracted considerable interest. Moreover, because of the complexity of chemotactic signaling, which includes multiple feedback loops and redundant pathways, this has been a research area in which computational models have had a significant impact in understanding experimental findings. Here, we propose a modular description of the signaling network that regulates chemotaxis. The modules describe different processes that are observed in chemotactic cells. In addition to accounting for these behaviors individually, we show that the overall system recreates many features of the directed motion of migrating cells. The signaling described by our modules is implemented as a series of equations, whereas movement and the accompanying cellular deformations are simulated using a mechanical model of the cell and implemented using level set methods, a method that allows simulations of cells as they change morphology.
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Affiliation(s)
- Changji Shi
- Department of Electrical and Computer Engineering, Whiting School of Engineering, Johns Hopkins University, Baltimore, Maryland, United States of America
| | - Chuan-Hsiang Huang
- Department of Cell Biology, School of Medicine, Johns Hopkins University, Baltimore, Maryland, United States of America
| | - Peter N. Devreotes
- Department of Cell Biology, School of Medicine, Johns Hopkins University, Baltimore, Maryland, United States of America
| | - Pablo A. Iglesias
- Department of Electrical and Computer Engineering, Whiting School of Engineering, Johns Hopkins University, Baltimore, Maryland, United States of America
- Biological Physics, Max Planck Institute for the Physics of Complex Systems, Dresden, Germany
- * E-mail:
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Abstract
Eukaryotic cell motility involves complex interactions of signalling molecules, cytoskeleton, cell membrane, and mechanics interacting in space and time. Collectively, these components are used by the cell to interpret and respond to external stimuli, leading to polarization, protrusion, adhesion formation, and myosin-facilitated retraction. When these processes are choreographed correctly, shape change and motility results. A wealth of experimental data have identified numerous molecular constituents involved in these processes, but the complexity of their interactions and spatial organization make this a challenging problem to understand. This has motivated theoretical and computational approaches with simplified caricatures of cell structure and behaviour, each aiming to gain better understanding of certain kinds of cells and/or repertoire of behaviour. Reaction–diffusion (RD) equations as well as equations of viscoelastic flows have been used to describe the motility machinery. In this review, we describe some of the recent computational models for cell motility, concentrating on simulations of cell shape changes (mainly in two but also three dimensions). The problem is challenging not only due to the difficulty of abstracting and simplifying biological complexity but also because computing RD or fluid flow equations in deforming regions, known as a “free-boundary” problem, is an extremely challenging problem in applied mathematics. Here we describe the distinct approaches, comparing their strengths and weaknesses, and the kinds of biological questions that they have been able to address.
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Affiliation(s)
- William R Holmes
- Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada.
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Signaling regulated endocytosis and exocytosis lead to mating pheromone concentration dependent morphologies in yeast. FEBS Lett 2012; 586:4208-4214. [PMID: 23108052 DOI: 10.1016/j.febslet.2012.10.024] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/28/2012] [Revised: 10/01/2012] [Accepted: 10/14/2012] [Indexed: 11/22/2022]
Abstract
Polarized cell morphogenesis requires actin cytoskeleton rearrangement for polarized transport of proteins, organelles and secretory vesicles, which fundamentally underlies cell differentiation and behavior. During yeast mating, Saccharomyces cerevisiae responds to extracellular pheromone gradients by extending polarized projections, which are likely maintained through vesicle transport to (exocytosis) and from (endocytosis) the membrane. We experimentally demonstrate that the projection morphology is pheromone concentration-dependent, and propose the underlying mechanism through mathematical modeling. The inclusion of membrane flux and dynamically evolving cell boundary into our yeast mating signaling model shows good agreement with experimental measurements, and provides a plausible explanation for pheromone-induced cell morphology.
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Du X, Doubrovinski K, Osterfield M. Self-organized cell motility from motor-filament interactions. Biophys J 2012; 102:1738-45. [PMID: 22768929 DOI: 10.1016/j.bpj.2012.03.052] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/09/2012] [Revised: 03/21/2012] [Accepted: 03/23/2012] [Indexed: 11/15/2022] Open
Abstract
Cell motility is driven primarily by the dynamics of the cell cytoskeleton, a system of filamentous proteins and molecular motors. It has been proposed that cell motility is a self-organized process, that is, local short-range interactions determine much of the dynamics that are required for the whole-cell organization that leads to polarization and directional motion. Here we present a mesoscopic mean-field description of filaments, motors, and cell boundaries. This description gives rise to a dynamical system that exhibits multiple self-organized states. We discuss several qualitative aspects of the asymptotic states and compare them with those of living cells.
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Affiliation(s)
- XinXin Du
- Physics Department, Princeton University, Princeton, New Jersey, USA.
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