# Three-dimensional simulation of obstacle-mediated chemotaxis

- 358 Downloads
- 1 Citations

## Abstract

Amoeboid cells exhibit a highly dynamic motion that can be directed by external chemical signals, through the process of chemotaxis. Here, we propose a three-dimensional model for chemotactic motion of amoeboid cells. We account for the interactions between the extracellular substances, the membrane-bound proteins, and the cytosolic components involved in the signaling pathway that originates cell motility. We show two- and three-dimensional simulations of cell migration on planar substrates, flat surfaces with obstacles, and fibrous networks. The results show that our model reproduces the main features of chemotactic amoeboid motion. Our simulations unveil a complicated interplay between the geometry of the cell’s environment and the chemoattractant dynamics that tightly regulates cell motion. The model opens new opportunities to simulate the interactions between extra- and intra-cellular compounds mediated by the matrix geometry.

## Keywords

Amoeboid motion Chemotaxis Phase-field modeling 3D cell migration## Notes

### Acknowledgements

A.M. and H.G. were partially supported by the European Research Council (Contract # 307201) and by Consellería de Cultura, Educación e Ordenación Universitaria (Xunta de Galicia). A.M. was partially supported by the UDC-Inditex Ph.D. student grant program.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## Supplementary material

## References

- Allena R, Aubry D (2012) ‘Run-and-tumble’ or ‘look-and-run’? A mechanical model to explore the behavior of a migrating amoeboid cell. J Theor Biol 306:15–31MathSciNetCrossRefGoogle Scholar
- Andrew N, Insall RH (2007) Chemotaxis in shallow gradients is mediated independently of ptdins 3-kinase by biased choices between random protrusions. Nat Cell Biol 9(2):193CrossRefGoogle Scholar
- Bausch AR, Ziemann F, Boulbitch AA, Jacobson K, Sackmann E (1998) Local measurements of viscoelastic parameters of adherent cell surfaces by magnetic bead microrheometry. Biophys J 75(4):2038–2049CrossRefGoogle Scholar
- Bell GI (1978) Models for the specific adhesion of cells to cells. Science 200:618–627CrossRefGoogle Scholar
- Biben T, Kassner K, Misbah C (2005) Phase-field approach to three-dimensional vesicle dynamics. Phys Rev E 72(041):921Google Scholar
- Bosgraaf L, Van Haastert PJM (2009a) Navigation of chemotactic cells by parallel signaling to pseudopod persistence and orientation. PLoS ONE 4:e6842CrossRefGoogle Scholar
- Bosgraaf L, Van Haastert PJM et al (2009b) The ordered extension of pseudopodia by amoeboid cells in the absence of external cues. PLoS ONE 4:e5253CrossRefGoogle Scholar
- Buenemann M, Levine H, Rappel WJ, Sander LM (2010) The role of cell contraction and adhesion in dictyostelium motility. Biophys J 99(1):50–58CrossRefGoogle Scholar
- Camley BA, Zhao Y, Li B, Levine H, Rappel WJ (2013) Periodic migration in a physical model of cells on micropatterns. Phys Rev Lett 111(158):102Google Scholar
- Casquero H, Bona-Casas C, Gomez H (2017) NURBS-based numerical proxies for red blood cells and circulating tumor cells in microscale blood flow. Comput Methods Appl Mech Eng 316:646–667MathSciNetCrossRefGoogle Scholar
- Chen BC, Legant WR, Wang K, Shao L, Milkie DE, Davidson MW, Janetopoulos C, Wu XS, Hammer JA III, Liu Z, English BP, Mimori-Kiyosue Y, Romero DP, Ritter AT, Lippincott-Schwartz J, Fritz-Laylin L, Dyche Mullins R, Mitchell DM, Bembenek JN, Reymann AC, Böhme R, Grill SW, Wang JT, Seydoux G, Serdar Tulu U, Kiehart DP, Betzig E (2014) Lattice light-sheet microscopy: imaging molecules to embryos at high spatiotemporal resolution. Science 346(1257):998Google Scholar
- Choi CK, Vicente-Manzanares M, Zareno J, Whitmore LA, Mogilner A, Horwitz AR (2008) Actin and \(\alpha \)-actinin orchestrate the assembly and maturation of nascent adhesions in a myosin II motor-independent manner. Nat Cell Biol 10:1039–1050CrossRefGoogle Scholar
- Chung J, Hulbert GM (1993) A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-\(\alpha \) method. J Appl Mech 60:371–375MathSciNetCrossRefGoogle Scholar
- Dawes AT, Edelstein-Keshet L (2007) Phosphoinositides and Rho proteins spatially regulate actin polymerization to initiate and maintain directed movement in a one-dimensional model of a motile cell. Biophys J 92:744–768CrossRefGoogle Scholar
- del Álamo JC, Meili R, Alonso-Latorre B, Rodríguez-Rodríguez J, Aliseda A, Firtel RA, Lasheras JC (2007) Spatio-temporal analysis of eukaryotic cell motility by improved force cytometry. Proc Natl Acad Sci USA 104:13,343–13,348CrossRefGoogle Scholar
- Elliott CM, Stinner B, Venkataraman C (2012) Modelling cell motility and chemotaxis with evolving surface finite elements. J R Soc Interface 9:3027–3044CrossRefGoogle Scholar
- Friedl P, Wolf K (2003) Tumour-cell invasion and migration: diversity and escape mechanisms. Nat Rev Cancer 3:362–374CrossRefGoogle Scholar
- Friedl P, Wolf K (2009) Plasticity of cell migration: a multiscale tuning model. J Cell Biol 188:11–19CrossRefGoogle Scholar
- Fuller D, Chen W, Adler M, Groisman A, Levine H, Rappel WJ (2010) External and internal constraints on eukaryotic chemotaxis. Proc Natl Acad Sci USA 107:9656–9659CrossRefGoogle Scholar
- Gamba A, de Candia A, Di Talia S, Coniglio A, Bussolino F, Serini G (2005) Diffusion-limited phase separation in eukaryotic chemotaxis. Proc Natl Acad Sci USA 102:16,927–16,932CrossRefGoogle Scholar
- Geiger B, Spatz JP, Bershadsky AD (2009) Environmental sensing through focal adhesions. Nat Rev Mol Cell Biol 10:21–33CrossRefGoogle Scholar
- Goldstein RE (1996) Traveling-wave chemotaxis. Phys Rev Lett 77:775CrossRefGoogle Scholar
- Gomez H, van der Zee K (2017) Computational phase-field modeling. Encyclopedia of Computational Mechanics, accepted for publicationGoogle Scholar
- Hecht I, Skoge ML, Charest PG, Ben-Jacob E, Firtel RA, Loomis WF, Levine H, Rappel WJ (2011) Activated membrane patches guide chemotactic cell motility. PLoS Comput Biol 7(e1002):044MathSciNetGoogle Scholar
- Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195MathSciNetCrossRefGoogle Scholar
- Insall RH (2010) Understanding eukaryotic chemotaxis: a pseudopod-centred view. Nat Rev Mol Cell Biol 11:453–458CrossRefGoogle Scholar
- Janetopoulos C, Ma L, Devreotes PN, Iglesias PA (2004) Chemoattractant-induced phosphatidylinositol 3,4,5-trisphosphate accumulation is spatially amplified and adapts, independent of the actin cytoskeleton. Proc Natl Acad Sci USA 101:8951–8956CrossRefGoogle Scholar
- Jansen K, Whiting C, Hulbert G (2000) Generalized-\(\alpha \) method for integrating the filtered Navier–Stokes equations with a stabilized finite element method. Comput Methods Appl Mech Eng 190:305–319MathSciNetCrossRefGoogle Scholar
- Jurado C, Hasserick JR, Lee J (2005) Slipping or gripping? fluorescent speckle microscopy in fish keratocytes reveals two different mechanisms for generating a retrograde flow of actin. Mol Biol Cell 16:507–518CrossRefGoogle Scholar
- Lämmermann T, Sixt M (2009) Mechanical modes of ‘amoeboid’ cell migration. Curr Opin Cell Biol 21:636–644CrossRefGoogle Scholar
- Lämmermann T, Bader BL, Monkley SJ, Worbs T, Wedlich-Söldner R, Hirsch K, Keller M, Förster R, Critchley DR, Fässler R et al (2008) Rapid leukocyte migration by integrin-independent flowing and squeezing. Nature 453(7191):51CrossRefGoogle Scholar
- Levchenko A, Iglesias PA (2002) Models of eukaryotic gradient sensing: application to chemotaxis of amoebae and neutrophils. Biophys J 82:50–63CrossRefGoogle Scholar
- Levine H, Kessler DA, Rappel WJ (2006) Directional sensing in eukaryotic chemotaxis: a balanced inactivation model. Proc Natl Acad Sci USA 103:9761–9766CrossRefGoogle Scholar
- Li X, Lowengrub J, Rätz A, Voigt A (2009) Solving PDEs in complex geometries: a diffuse domain approach. Commun Math Sci 7:81–107MathSciNetCrossRefGoogle Scholar
- Liu WK, Liu Y, Farrell D, Zhang L, Wang XS, Fukui Y, Patankar N, Zhang Y, Bajaj C, Lee J, Hong J, Chen X, Hsu H (2006) Immersed finite element method and its applications to biological systems. Comput Methods Appl Mech Eng 195:1722–1749MathSciNetCrossRefGoogle Scholar
- MacDonald G, Mackenzie JA, Nolan M, Insall RH (2016) A computational method for the coupled solution of reaction-difusion equations on evolving domains and manifolds: application to a model of cell migration and chemotaxis. J Comput Phys 309:207–226MathSciNetCrossRefGoogle Scholar
- Marée AFM, Grieneisen VA, Edelstein-Keshet L (2012) How cells integrate complex stimuli: the effect of feedback from phosphoinositides and cell shape on cell polarization and motility. PLoS Comput Biol 8(e1002):402MathSciNetGoogle Scholar
- Marth W, Voigt A (2014) Signaling networks and cell motility: a computational approach using a phase field description. J Math Biol 69:91–112MathSciNetCrossRefGoogle Scholar
- Meinhardt H (1999) Orientation of chemotactic cells and growth cones: models and mechanisms. J Cell Sci 112:2867–2874Google Scholar
- Mori Y, Jilkine A, Edelstein-Keshet L (2008) Wave-pinning and cell polarity from a bistable reaction-diffusion system. Biophys J 94:3684–3697CrossRefGoogle Scholar
- Moure A, Gomez H (2016) Computational model for amoeboid motion: coupling membrane and cytosol dynamics. Phys Rev E 94(042):423MathSciNetGoogle Scholar
- Moure A, Gomez H (2017) Phase-field model of cellular migration: three-dimensional simulations in fibrous networks. Comput Methods Appl Mech Eng 320:162–197MathSciNetCrossRefGoogle Scholar
- Neilson MP, Veltman DM, van Haastert PJM, Webb SD, Mackenzie JA, Insall RH (2011) Chemotaxis: a feedback-based computational model robustly predicts multiple aspects of real cell behaviour. PLoS Biol 9(e1000):618Google Scholar
- Novak IL, Slepchenko BM, Mogilner A (2008) Quatitative analysis of G-actin transport in motile cells. Biophys J 95:1627–1638CrossRefGoogle Scholar
- Petrášek Z, Hoege C, Mashaghi A, Ohrt T, Hyman AA, Schwille P (2008) Characterization of protein dynamics in asymmetric cell division by scanning fluorescence correlation spectroscopy. Biophysical J 95(11):5476–5486CrossRefGoogle Scholar
- Ribeiro FO, Gómez-Benito MJ, Folgado J, Fernandes PR, García-Aznar JM (2017) Computational model of mesenchymal migration in 3D under chemotaxis. Comput Methods Biomech Biomed Eng 20:59–74CrossRefGoogle Scholar
- Rubinstein B, Fournier MF, Jacobson K, Verkhovsky AB, Mogilner A (2009) Actin-myosin viscoelastic flow in keratocyte lamellipod. Biophys J 97:1853–1863CrossRefGoogle Scholar
- Shao D, Levine H, Rappel WJ (2012) Coupling actin flow, adhesion, and morphology in a computational cell motility model. Proc Natl Acad Sci USA 109:6851–6856CrossRefGoogle Scholar
- Shi C, Huang CH, Devreotes PN, Iglesias PA (2013) Interaction of motility, directional sensing, and polarity modules recreates the behaviors of chemotaxing cells. PLoS Comput Biol 9(e1003):122MathSciNetGoogle Scholar
- Skoge M, Yue H, Erickstad M, Bae A, Levine H, Groisman A (2014) Cellular memory in eukaryotic chemotaxis. Proc Natl Acad Sci USA 111:14,448–14,453CrossRefGoogle Scholar
- Song L, Nadkarni SM, Bödeker HU, Beta C, Bae A, Franck C, Rapper WJ, Loomis WF, Bodenschatz E (2006) Dictyostelium discoideum chemotaxis: threshold for directed motion. Eur J Cell Biol 85:981–989CrossRefGoogle Scholar
- Strychalski W, Copos CA, Lewis OL, Guy RD (2015) A poroelastic immersed boundary method with applications to cell biology. J Comput Phys 282:77–97MathSciNetCrossRefGoogle Scholar
- Subramanian KK, Narang A (2004) A mechanistic model for eukaryotic gradient sensing: spontaneous and induced phosphoinositide polarization. J Theor Biol 231:49–67MathSciNetCrossRefGoogle Scholar
- Sunyer R, Conte V, Escribano J, Elosegui-Artola A, Labernadie A, Valon L, Navajas D, García-Aznar JM, Muñoz JJ, Roca-Cusachs P et al (2016) Collective cell durotaxis emerges from long-range intercellular force transmission. Science 353(6304):1157–1161. https://doi.org/10.1126/science.aaf7119 CrossRefGoogle Scholar
- Swaney KF, Huang CH, Devreotes PN (2010) Eukaryotic chemotaxis: a network of signaling pathways controls motility, directional sensing, and polarity. Annu Rev Biophys 39:265–289CrossRefGoogle Scholar
- Swanson JA, Taylor DL (1982) Local and spatially coordinated movements in dictyostelium discoideum amoebae during chemotaxis. Cell 28(2):225–232CrossRefGoogle Scholar
- Teigen KE, Li X, Lowengrub J, Wang F, Voigt A (2009) A diffuse-interface approach for modeling transport, diffusion and adsorption/desorption of material quantities on a deformable interface. Commun Math Sci 4:1009–1037MathSciNetzbMATHGoogle Scholar
- Tinevez JY, Schulze U, Salbreux G, Roensch J, Joanny JF, Paluch E (2009) Role of cortical tension in bleb growth. Proc Natl Acad Sci 106(44):18,581–18,586CrossRefGoogle Scholar
- Tjhung E, Tiribocchi A, Marenduzzo D (2015) A minimal physical model captures the shapes of crawling cells. Nat Commun 6:5420CrossRefGoogle Scholar
- Tweedy L, Meier B, Stephan J, Heinrich D, Endres RG (2013) Distinct cell shapes determine accurate chemotaxis. Sci Rep 3:2606CrossRefGoogle Scholar
- Ura S, Pollitt AY, Veltman DM, Morrice NA, Machesky LM, Insall RH (2012) Pseudopod growth and evolution during cell movement is controlled through SCAR/WAVE dephosphorylation. Curr Biol 22(7):553–561CrossRefGoogle Scholar
- Van Haastert PJM (2010) A stochastic model for chemotaxis based on the ordered extension of pseudopods. Biophys J 99:3345–3354Google Scholar
- Van Haastert PJM, Devreotes PN (2004) Chemotaxis: signalling the way forward. Nat Rev Mol Cell Biol 5:626–634CrossRefGoogle Scholar
- Vermolen FJ, Gefen A (2013) A phenomenological model for chemico-mechanically induced cell shape changes during migration and cell–cell contacts. Biomech Model Mechanobiol 12:301–323CrossRefGoogle Scholar
- Wessels D, Brincks R, Kuhl S, Stepanovic V, Daniels KJ, Weeks G, Lim CJ, Spiegelman G, Fuller D, Iranfar N, Loomis WF, Soll DR (2004) RasC plays a role in transduction of temporal gradient information in the cyclic-AMP wave of Dictyostelium discoideum. Eukaryot Cell 3:646–662CrossRefGoogle Scholar
- Ziebert F, Aranson IS (2016) Computational approaches to substrate-based cell motility. npj Comput Mater 2:16,019CrossRefGoogle Scholar