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Age Structure Can Account for Delayed Logistic Proliferation of Scratch Assays

  • Ana Victoria Ponce BobadillaEmail author
  • Thomas Carraro
  • Helen M. Byrne
  • Philip K. Maini
  • Tomás Alarcón
Article

Abstract

Scratch assays are in vitro methods for studying cell migration. In these experiments, a scratch is made on a cell monolayer and recolonisation of the scratched region is imaged to quantify cell migration rates. Typically, scratch assays are modelled by reaction diffusion equations depicting cell migration by Fickian diffusion and proliferation by a logistic term. In a recent paper (Jin et al. in Bull Math Biol 79(5):1028–1050, 2017), the authors observed experimentally that during the early stage of the recolonisation process, there is a disturbance phase where proliferation is not logistic, and this is followed by a growth phase where proliferation appears to be logistic. The authors did not identify the precise mechanism that causes the disturbance phase but showed that ignoring it can lead to incorrect parameter estimates. The aim of this work is to show that a nonlinear age-structured population model can account for the two phases of proliferation in scratch assays. The model consists of an age-structured cell cycle model of a cell population, coupled with an ordinary differential equation describing the resource concentration dynamics in the substrate. The model assumes a resource-dependent cell cycle threshold age, above which cells are able to proliferate. By studying the dynamics of the full system in terms of the subpopulations of cells that can proliferate and the ones that can not, we are able to find conditions under which the model captures the two-phase behaviour. Through numerical simulations, we are able to show that the interplay between the resource concentration in the substrate and the cell subpopulations dynamics can explain the biphasic dynamics.

Keywords

Logistic growth model Cell proliferation Scratch assay Von Foerster–McKendrick type model Age-structured model 

Notes

Acknowledgements

AVPB would like to thank Gergely Rost, Maria Vittoria Barbarossa, Wang Jin and Matthew Simpson for helpful discussions. AVPB was supported by the Heidelberg Graduate School of Mathematical and Computational Methods for the Sciences, University of Heidelberg (DE) (DFG grant GSC 220 in the German Universities Excellence Initiative).

Supplementary material

11538_2019_625_MOESM1_ESM.pdf (399 kb)
Supplementary material 1 (pdf 398 KB)

References

  1. Adan A, Kiraz Y, Baran Y (2016) Cell proliferation and cytotoxicity assays. Curr Pharm Biotechnol 17(14):1213–1221Google Scholar
  2. Bairoch A (2018) The cellosaurus, a cell-line knowledge resource. J Biomol Tech JBT 29(3):25–38Google Scholar
  3. Baker RE, Simpson MJ (2010) Correcting mean-field approximations for birth-death-movement processes. Phys Rev E 82(4):041905MathSciNetGoogle Scholar
  4. Bangerth W, Hartmann R, Kanschat G (2007) deal. IIa general-purpose object-oriented finite element library. ACM Trans Math Softw (TOMS) 33(4):24zbMATHGoogle Scholar
  5. Billy F, Clairambault J (2013) Designing proliferating cell population models with functional targets for control by anti-cancer drugs. Discrete Cont Dyn Syst Ser B 18(4):865–889MathSciNetzbMATHGoogle Scholar
  6. Billy F, Clairambault J, Delaunay F, Feillet C, Robert N (2012) Age-structured cell population model to study the influence of growth factors on cell cycle dynamics. Math Biosci Eng 10(1):1–17MathSciNetzbMATHGoogle Scholar
  7. Bourseguin J, Bonet C, Renaud E, Pandiani C, Boncompagni M, Giuliano S, Pawlikowska P, Karmous-Benailly H, Ballotti R, Rosselli F et al (2016) FANCD2 functions as a critical factor downstream of MiTF to maintain the proliferation and survival of melanoma cells. Sci Rep 6:36539Google Scholar
  8. Browning AP, McCue SW, Simpson MJ (2017) A Bayesian computational approach to explore the optimal duration of a cell proliferation assay. Bull Math Biol 79(8):1888–1906MathSciNetzbMATHGoogle Scholar
  9. Celis JE (2009) Cell biology assays: essential methods. Butterworth-Heinemann, OxfordGoogle Scholar
  10. Chaffey GS, Lloyd DJB, Skeldon AC, Kirkby NF (2014) The effect of the G1-S transition checkpoint on an age structured cell cycle model. PLoS ONE 9(1):e83477Google Scholar
  11. Cohen SM, Ellwein LB (1990) Cell proliferation in carcinogenesis. Science 249(4972):1007–1011Google Scholar
  12. Cook CC, Kim A, Terao S, Gotoh A, Higuchi M (2013) Consumption of oxygen: a mitochondrial-generated progression signal of advanced cancer. Cell Death Dis 3(1):e258Google Scholar
  13. Cunningham D, You Z (2015) In vitro and in vivo model systems used in prostate cancer research. J Biol Methods 2(1):e17Google Scholar
  14. Darnell JE, Lodish HF, Baltimore D et al (1990) Molecular cell biology, vol 2. Scientific American Books New York, New YorkGoogle Scholar
  15. de la Cruz R, Guerrero P, Spill F, Alarcón T (2016) Stochastic multi-scale models of competition within heterogeneous cellular populations: simulation methods and mean-field analysis. J Theor Biol 407:161–183MathSciNetzbMATHGoogle Scholar
  16. Domschke P, Trucu D, Gerisch A, Chaplain MAJ (2017) Structured models of cell migration incorporating molecular binding processes. J Math Biol 75(6–7):1517–1561MathSciNetzbMATHGoogle Scholar
  17. Foster DA, Yellen P, Xu L, Saqcena M (2010) Regulation of G1 cell cycle progression: distinguishing the restriction point from a nutrient-sensing cell growth checkpoint (s). Genes Cancer 1(11):1124–1131Google Scholar
  18. Gabriel P, Garbett SP, Quaranta V, Tyson DR, Webb GF (2012) The contribution of age structure to cell population responses to targeted therapeutics. J Theor Biol 311:19–27MathSciNetzbMATHGoogle Scholar
  19. Gavagnin E, Ford MJ, Mort RL, Rogers T, Yates CA (2018) The invasion speed of cell migration models with realistic cell cycle time distributions. arXiv:1806.03140
  20. Gerlee P (2013) The model muddle: in search of tumour growth laws. Cancer Res 73(8):2407–2411Google Scholar
  21. Grossmann C, Roos HG, Stynes M (2007) Numerical treatment of partial differential equations, vol 154. Springer, BerlinzbMATHGoogle Scholar
  22. Jin W, Shah ET, Penington CJ, McCue SW, Chopin LK, Simpson MJ (2016) Reproducibility of scratch assays is affected by the initial degree of confluence: experiments, modelling and model selection. J Theor Biol 390:136–145zbMATHGoogle Scholar
  23. Jin W, Shah ET, Penington CJ, McCue SW, Maini PK, Simpson MJ (2017) Logistic proliferation of cells in scratch assays is delayed. Bull Math Biol 79(5):1028–1050MathSciNetzbMATHGoogle Scholar
  24. Jin W, McCue SW, Simpson MJ (2018) Extended logistic growth model for heterogeneous populations. J Theor Biol 445:51–61MathSciNetzbMATHGoogle Scholar
  25. Johnston ST, Shah ET, Chopin LK, McElwain DS, Simpson MJ (2015) Estimating cell diffusivity and cell proliferation rate by interpreting IncuCyte ZOOM assay data using the Fisher–Kolmogorov model. BMC Syst Biol 9(1):38Google Scholar
  26. Kermack WO, McKendrick AG (1932) Contributions to the mathematical theory of epidemics. II. The problem of endemicity. Proc R Soc Lond A 138(834):55–83zbMATHGoogle Scholar
  27. Keyfitz BL, Keyfitz N (1997) The McKendrick partial differential equation and its uses in epidemiology and population study. Math Comput Modell 26(6):1–9MathSciNetzbMATHGoogle Scholar
  28. Kramer N, Walzl A, Unger C, Rosner M, Krupitza G, Hengstschläger M, Dolznig H (2013) In vitro cell migration and invasion assays. Mutat Res Rev Mutat Res 752(1):10–24Google Scholar
  29. Kuzmin D (2010) A guide to numerical methods for transport equations. University Erlangen-Nuremberg, ErlangenGoogle Scholar
  30. Liang CC, Park AY, Guan JL (2007) In vitro scratch assay: a convenient and inexpensive method for analysis of cell migration in vitro. Nat Protoc 2(2):329Google Scholar
  31. Lim S, Kaldis P (2013) Cdks, cyclins and CKIs: roles beyond cell cycle regulation. Development 140(15):3079–3093Google Scholar
  32. Liu JC, Zacksenhouse M, Eisen A, Nofech-Mozes S, Zacksenhaus E (2017) Identification of cell proliferation, immune response and cell migration as critical pathways in a prognostic signature for HER2+: \(\text{ ER }\alpha \)-breast cancer. PLoS ONE 12(6):e0179223Google Scholar
  33. Maini PK, McElwain DS, Leavesley D (2004) Travelling waves in a wound healing assay. Appl Math Lett 17(5):575–580MathSciNetzbMATHGoogle Scholar
  34. Masuzzo P, Van Troys M, Ampe C, Martens L (2016) Taking aim at moving targets in computational cell migration. Trends Cell Biol 26(2):88–110Google Scholar
  35. Menyhárt O, Harami-Papp H, Sukumar S, Schäfer R, Magnani L, de Barrios O, Győrffy B (2016) Guidelines for the selection of functional assays to evaluate the hallmarks of cancer. Biochim Biophys Acta (BBA) Rev Cancer 1866(2):300–319Google Scholar
  36. Nardini JT, Bortz DM (2018) Investigation of a structured Fisher’s equation with applications in biochemistry. SIAM J Appl Math 78(3):1712–1736MathSciNetzbMATHGoogle Scholar
  37. Nyegaard S, Christensen B, Rasmussen JT (2016) An optimized method for accurate quantification of cell migration using human small intestine cells. Metab Eng Commun 3:76–83Google Scholar
  38. Ortmann B, Druker J, Rocha S (2014) Cell cycle progression in response to oxygen levels. Cell Mol Life Sci 71(18):3569–3582Google Scholar
  39. Romar GA, Kupper TS, Divito SJ (2016) Research techniques made simple: techniques to assess cell proliferation. J Investig Dermatol 136(1):e1–e7Google Scholar
  40. Sarapata EA, de Pillis L (2014) A comparison and catalog of intrinsic tumor growth models. Bull Math Biol 76(8):2010–2024MathSciNetzbMATHGoogle Scholar
  41. Savla U, Olson LE, Waters CM (2004) Mathematical modeling of airway epithelial wound closure during cyclic mechanical strain. J Appl Physiol 96(2):566–574Google Scholar
  42. Smith J, Winslow D, Rudland P (1984) Different growth factors stimulate cell division of rat mammary epithelial, myoepithelial, and stromal cell lines in culture. J Cell Physiol 119(3):320–326Google Scholar
  43. Topman G, Sharabani-Yosef O, Gefen A (2012) A standardized objective method for continuously measuring the kinematics of cultures covering a mechanically damaged site. Med Eng Phys 34(2):225–232Google Scholar
  44. Vittadello ST, McCue SW, Gunasingh G, Haass NK, Simpson MJ (2018) Mathematical models for cell migration with real-time cell cycle dynamics. Biophys J 114(5):1241–1253Google Scholar
  45. Walker D, Hill G, Wood S, Smallwood R, Southgate J (2004a) Agent-based computational modeling of wounded epithelial cell monolayers. IEEE Trans Nanobiosci 3(3):153–163Google Scholar
  46. Walker D, Southgate J, Hill G, Holcombe M, Hose D, Wood S, Mac Neil S, Smallwood R (2004b) The epitheliome: agent-based modelling of the social behaviour of cells. Biosystems 76(1–3):89–100Google Scholar
  47. Warne D, Baker R, Simpson M (2018) Using experimental data and information criteria to guide model selection for reaction-diffusion problems in mathematical biology. Bull Math Biol 81(6):1–45MathSciNetGoogle Scholar
  48. West GB, Brown JH, Enquist BJ (2001) A general model for ontogenetic growth. Nature 413(6856):628Google Scholar

Copyright information

© Society for Mathematical Biology 2019

Authors and Affiliations

  1. 1.Institute for Applied MathematicsHeidelberg UniversityHeidelbergGermany
  2. 2.Interdisciplinary Center for Scientific Computing (IWR)Heidelberg UniversityHeidelbergGermany
  3. 3.Wolfson Centre for Mathematical Biology, Mathematical InstituteUniversity of OxfordOxfordUK
  4. 4.ICREABarcelonaSpain
  5. 5.Centre de Recerca MatemàticaBellaterraSpain
  6. 6.Barcelona Graduate School of Mathematics (BGSMath)BarcelonaSpain
  7. 7.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterraSpain

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