Advertisement

Journal of Mathematical Biology

, Volume 31, Issue 6, pp 563–600 | Cite as

A stochastic model for adhesion-mediated cell random motility and haptotaxis

  • Richard B. Dickinson
  • Robert T. Tranquillo
Article

Abstract

The active migration of blood and tissue cells is important in a number of physiological processes including inflammation, wound healing, embryogenesis, and tumor cell metastasis. These cells move by transmitting cytoplasmic force through membrane receptors which are bound specifically to adhesion ligands in the surrounding substratum. Recently, much research has focused on the influence of the composition of extracellular matrix and the distribution of its components on the speed and direction of cell migration. It is commonly believed that the magnitude of the adhesion influences cell speed and/or random turning behavior, whereas a gradient of adhesion may bias the net direction of the cell movement, a phenomenon known as haptotaxis. The mechanisms underlying these responses are presently not understood.

A stochastic model is presented to provide a mechanistic understanding of how the magnitude and distribution of adhesion ligands in the substratum influence cell movement. The receptor-mediated cell migration is modeled as an interrelation of random processes on distinct time scales. Adhesion receptors undergo rapid binding and transport, resulting in a stochastic spatial distribution of bound receptors fluctuating about some mean distribution. This results in a fluctuating spatio-temporal pattern of forces on the cell, which in turn affects the speed and turning behavior on a longer time scale. The model equations are a system of nonlinear stochastic differential equations (SDE's) which govern the time evolution of the spatial distribution of bound and free receptors, and the orientation and position of the cell. These SDE's are integrated numerically to simulate the behavior of the model cell on both a uniform substratum, and on a gradient of adhesion ligand concentration.

Furthermore, analysis of the governing SDE system and corresponding Fokker-Planck equation (FPE) yields analytical expressions for indices which characterize cell movement on multiple time scales in terms of cell cytomechanical, morphological, and receptor binding and transport parameters. For a uniform adhesion ligand concentration, this analysis provides expressions for traditional cell movement indices such as mean speed, directional persistence time, and random motility coefficient. In a small gradient of adhesion, a perturbation analysis of the FPE yields a constitutive cell flux expression which includes a drift term for haptotactic directional cell migration. The haptotactic drift contains terms identified as contributions from directional orientation bias (taxis).

Keywords

Cell Movement Multiple Time Scale Movement Index Tumor Cell Metastasis Orientation Bias 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Akiyama, S. K., Yamada, K. M.: The interactions of plasma fibronectin with fibroblastic cells in suspension. J. Biol. Chem. 260, 4492–4500 (1985)Google Scholar
  2. Alt, W.: Biased random walk models for chemotaxis and related diffusion approximations. J. Math. Biol. 9(2), 147–77 (1980)Google Scholar
  3. Alt, W.: Modelling of motility in biological systems. In: ICIAM'87 Proceedings, pp. 15–30. Philadelphia: SIAM 1988Google Scholar
  4. Alt, W.: Correlation analysis of two-dimensional locomotion paths. In: Alt, W., Hoffmann, G. (eds.) Biological Motion, pp. 254–268. Berlin Heidelberg New York: Springer 1990Google Scholar
  5. Aznavoorian, S., Stracke, M. L., Knutzsch, H., Schiffman, E., Liotta, L.A.: Signal transductions for chemotaxis and haptotaxis by matrix molecules in tumor cells. J. Cell Biol. 110, 1427–1438 (1990)Google Scholar
  6. Bell, G. I. Models for the specific adhesion of cells to cells. Science 200, 618–627 (1978)Google Scholar
  7. Bell, G. L: Models of cell adhesion involving specific binding. In: Bongrand, P. (ed.) Physical Basis of Cell-cell Adhesion, pp. 227–258. Boca Raton: CRC Press 1988Google Scholar
  8. Bell, G. I., Dembo, M. Bongrand, P.: Cell adhesion. Competition between nonspecific repulsion and specific bonding. Biophys J. 45 (6), 1051–64 (1984)Google Scholar
  9. Brandley, B. K., Schnaar, R. L.: Covalent attachment of an Arg-Gly-Asp sequence peptide to derivatizable polyacrylamide surfaces: support of fibroblast adhesion and long-term growth. Anal. Biochem. 172(1), 270–8 (1988)Google Scholar
  10. Brandley, B. K., Schnaar, R. L.: Tumor cell haptotaxis on covalently immobilized linear and exponential gradients of a cell adhesion peptide. Dev. Biol. 135(1), 74–86 (1989)Google Scholar
  11. Bretscher, M. S.: Endocytosis: relation to capping and cell locomotion. Science 224(4650), 681–6 (1984)Google Scholar
  12. Bretscher, M. S.: Fibroblasts on the move. J. Cell Biol. 106, 235–237 (1988)Google Scholar
  13. Bretscher, M. S.: Endocytosis and recycling of the fibronectin receptor in CHO cells. EMBO J. 8(5), 1341–1348 (1989)Google Scholar
  14. Buck, C. A., Horwitz, H. F.: Integrin, a transmembrane glycoprotein complex mediating cell-substratum adhesion. J. Cell Sci., Suppl. 8, 231–250 (1987)Google Scholar
  15. Carter, S. B.: Principles of cell motility: the direction of cell movement and cancer invasion. Nature 208(5016), 1183–1187 (1965)Google Scholar
  16. Chen, W. T.: Mechanism of retraction of the trailing edge during fibroblast movement. J. Cell Biol. 90(1), 187–200 (1981)Google Scholar
  17. Dembo, M., Torney, D. C., Saxman, K., Hammer, D.: The reaction-limited kinetics of membrane-to-surface adhesion and detachment. Proc. R. Soc. Lond., Ser. B 234(1274), 55–83 (1988)Google Scholar
  18. Dembo, M., Tuckerman, L., Goad, W.: Motion of polymorphonuclear leukocytes: theory of receptor redistribution and the frictional force on a moving cell. Cell Motil. 1(2), 205–35 (1981)Google Scholar
  19. Dickinson, R. B.: Quantitative Analysis and Mathematical Models of Blood and Tissue Cell Invasion and Migration. Ph.D thesis, University of Minnesota (1992)Google Scholar
  20. Dickinson, R. B., Tranquillo, R. T.: Transport equations and indices for random and biased cell migration based on single cell properties (submitted)Google Scholar
  21. DiMilla, P., Barbee, K., Lauffenburger, D. A.: A mathematical model for the effects of adhesion and mechanics on cell migration speed. Biophys. J. 60, 15–37 (1991)Google Scholar
  22. Duband, J.-L., Nuckolls, G. H., Ishihara, A., Hasegawa, T., Yamada, K. M., Thiery, J. P., Jacobson, K.: Fibronectin receptor exhibits high lateral mobility in embryonic locomoting cells but is immobile in focal contacts and fibrillar streaks in stationary cells. J. Cell Biol. 107, 1385–1396 (1988)Google Scholar
  23. Dunn, G. A.: Characterizing a kinesis response: time averaged measures of cell speed and directional persistence. Agents Actions Suppl. 12, 14–33 (1983)Google Scholar
  24. Dunn, G. A., Brown, A. F.: A unified approach to analysing cell motility. J. Cell. Sci., Suppl. 8, 81–102 (1987)Google Scholar
  25. Erickson, C. A.: Control of directional migration of avian and murine neural crest. Expl. Biol. Med. 10, 194–208 (1985)Google Scholar
  26. Felder, S., Elson, E. L.: Mechanics of fibroblast locomotion: quantitative analysis of forces and motions at the leading lamellas of fibroblasts. J. Cell Biol. 111(6, Pt.2), 2513–2526 (1990)Google Scholar
  27. Furth, R. von: Die Brownshe Bewegung bei Berücksichtigung einer Persistenze der Bewegungsrichtung. Mit Anwendungen auf die Bewegung lebender Infusorien. Z. Phys. 11, 244–256 (1920)Google Scholar
  28. Gail, M. H., Boone, C. W.: The locomotion of mouse fibroblasts in tissue culture. Biophys. J. 10, 980–993 (1970)Google Scholar
  29. Gard, T. C.: Introduction to Stochastic Differential Equations. (Monogr. Textb. Pure Appl. Math., vol. 114) New York: Marcel Dekker, 1988Google Scholar
  30. Gardiner, C. W.: Handbook of Stochastic Methods. (Springer Ser. Synerget., vol. 13) Berlin Heidelberg New York: Springer 1983Google Scholar
  31. Goodman, S. L., Deutzmann, R., Mark, K. von der: Two distinct cell-binding domains in laminin can independently promote nonneuronal cell adhesion and spreading. Cell 60, 849–859 (1987)Google Scholar
  32. Goodman, S. L., Risse, G., Mark, K. von der: The E8 subfragment of laminin promotes locomotion of myoblasts over extracellular matrix. J. Cell Biol. 109, 799–809 (1989)Google Scholar
  33. Gruler, H.: Cell movement and symmetry of the cellular environment. Z. Naturforsch., Teil C 43(9–10), 754–64 (1988)Google Scholar
  34. Gruler, H., Bultmann, B. D.: Analysis of cell movement. Blood Cells 10(1), 61–77 (1984)Google Scholar
  35. Grigolini, P.: The projection approach to the Fokker-Planck equation: applications to phenomenological stochastic equations with colored noise. In: Moss, F., McClintock, P. V. E. (eds.) Theory of continuous Fokker-Planck equations, pp. 161–190. Cambridge: Cambridge University Press 1989Google Scholar
  36. Hall, R. L.: Amoeboid movement as a correlated walk. J. Math. Biol. 4, 327–335 (1977)Google Scholar
  37. Hammer, D. A., Lauffenburger, D. A.: A dynamical model for receptor-mediated cell adhesion to surfaces. Biophys J. 52(3), 475–87 (1987)Google Scholar
  38. Harris, A. K.: Behavior of cultured cells on substrata of variable adhesiveness. Exp. Cell Res. 77, 285–297 (1973)Google Scholar
  39. Harris, A. K., Wild, P., Stopak, D.: Silicone rubber substrata: a new wrinkle in the study of cell locomotion. Science 208, 177–179 (1980)Google Scholar
  40. Ishihara, A., Holifield, B., Jacobson, K.: Analysis of lateral redistribution of a plasma membrane glycoprotein-monoclonal antibody complex. J. Cell Biol. 106(2), 329–43 (1988)Google Scholar
  41. Jacobson, K., Ishihara, A., Inman, R.: Lateral diffusion of proteins in membranes. Ann. Rev. Physiol. 49, 163–175 (1987)Google Scholar
  42. Kampen, N. G. van: Stochastic Processes in Physics and Chemistry. Amsterdam: North-Holland 1981Google Scholar
  43. Keller, E. F., Segel, L. A.: Model for chemotaxis. J. Theor. Biol. 30, 225–234 (1971)Google Scholar
  44. Keller, H. U., Wissler, J. H., Ploem, J.: Chemotaxis is no special case of haptotaxis. Experientia 35, 1669–1671 (1979)Google Scholar
  45. Keller, H. U., Zimmermann, A., Cottier, H.: Crawling-like movements, adhesion to solid substrata and chemokinesis of neutrophil granulocytes. J. Cell Sci. 64, 89–106 (1983)Google Scholar
  46. Lackie, J., Smith, R.: Interactions of leukocytes and endothelium. In: Curtis, A., Pitts, J. (eds.) Cell Adhesion and Motility, pp. 235–272. Cambridge: Cambridge University Press 1980Google Scholar
  47. Lackie, J. M.: Cell Movement and Cell Behaviour. London: Allen & Unwin 1986Google Scholar
  48. Lackie, J. M., Brown, A. F.: Substratum adhesion and the movement of neutrophil leucocytes. In: Bagge, G. V. U., Born, R., Gaehtgens, P. (eds.) White blood cells: morphology and rheology as related to function, pp. 127–133. Den Hague: Martinus Nijhoff Publishers 1982Google Scholar
  49. Lauffenburger, D.: A simple model for the effects of receptor-mediated cell-substratum adhesion on cell migration. Chem. Eng. Sci. 44(9), 1903–1914 (1989)Google Scholar
  50. Lauffenburger, D. A.: Chemotaxis: analysis for quantitative studies. Biotech. Prog. 1(3), 151–160 (1985)Google Scholar
  51. Lester, B. R., Weinstein, L. S., McCarthy, J. B., Sun, Z. Q., Smith, R. S., Furcht, L. T.: The role of G-protein in matrix-mediated motility of highly and poorly invasive melanoma cells. Int. J. Cancer 48(1), 113–20 (1991)Google Scholar
  52. Linderman, J. J., Lauffenburger, D. A.: Receptor/Ligand Sorting Along the Endocytic Pathway. (Lect. Notes Biomath., vol. 78) Berlin Heidelberg New York: Springer 1989Google Scholar
  53. Mardia, K. V.: Statistics of Directional Data. London: Academic Press 1972Google Scholar
  54. McCarthy, J. B., Basara, M. L., Palm, S. L., Sas, D. F., Furcht, L. T.: The role of cell adhesion proteins—laminin and fibronectin—in the movement of malignant and metastatic cells. Cancer Metastasis Rev. 4(2), 125–52 (1985)Google Scholar
  55. McCarthy, J. B., Furcht, L. T.: Laminin and fibronectin promote the haptotactic migration of B16 mouse melanoma cells. J. Cell Biol. 98(4), 1474–80 (1984)Google Scholar
  56. McCarthy, J. B., Palm, S. L., Furcht, L. T.: Migration by haptotaxis of a Schwarm cell tumor line to the basement membrane glycoprotein laminin. J. Cell Biol. 97(3), 772–777 (1983)Google Scholar
  57. McCarthy, J. B., Sas, D. F., Furcht, L. T.: Mechanisms of parenchymal cell migration into wounds. In: Clark, R. A. F., Henson, P. M. (eds.) The Molecular and Cellular Biology of Wound Repair, pp. 281–308. New York: Plenum Press 1988Google Scholar
  58. Othmer, H. G., Dunbar, S. R., Alt, W.: Models of dispersal in biological systems. J. Math. Biol. 26(3), 263–98 (1988)Google Scholar
  59. Rivero, M. A., Tranquillo, R. T., Buettner, H. M., Lauffenburger, D. A.: Transport models for chemotactic cell populations based on individual cell behavior. Chem. Eng. Sci. 44(12), 2881–2897 (1989)Google Scholar
  60. Rogers, S. L., Letourneau, P. C., Palm, S. L., McCarthy, J. B., Furcht, L. T.: Neurite extension by peripheral and central nervous system neurons in response to substratum-bound fibronectin and laminin. Dev. Biol. 98(1), 212–20 (1983)Google Scholar
  61. Schnitzer, M. J., Block, S. M., Berg, H. C., Purcell, E. M.: Strategies for chemotaxis. In: Armitage, J. P., Lackie, J. M. (eds.) Biology of the chemotactic response, pp. 15–34. Cambridge: Cambridge University Press 1990Google Scholar
  62. Sczekan, M. M., Juliano, R. L.: Internalization of the fibronectin receptor is a constitutive process. J. Cell. Phys. 142, 575–580 (1990)Google Scholar
  63. Taraboletti, G., Roberts, D. D., Liotta, L. A.: Thrombospondin-induced tumor cell migration: haptotaxis and chemotaxis are mediated by different molecular domains. J. Cell Biol. 105(5), 2409–15 (1987)Google Scholar
  64. Tranquillo, R. T., Lauffenburger, D. A.: Consequences of chemosensory phenomena for leukocyte chemotactic orientation. Cell Biophys. 8(1), 1–46 (1986)Google Scholar
  65. Tranquillo, R. T., Lauffenburger, D. A.: Stochastic model of leukocyte chemosensory movement. J. Math. Biol. 25(3), 229–62 (1987)Google Scholar
  66. Tranquillo, R. T., Zigmond, S. H., Lauffenburger, D. A.: Measurement of the chemotaxis coefficient for human neutrophils in the under-agarose migration assay. Cell Motil. Cytoskeleton 11(1), 1–15 (1988)Google Scholar
  67. Tranquillo, R. T.:Models of chemical gradient sensing by cells.. In: Alt, W., Hoffmann, G. (eds.) Biological Motion, pp. 415–441. Berlin Heidelberg New York: Springer 1990aGoogle Scholar
  68. Tranquillo, R. T.: Theories and models of gradient perception. In: Armitage, J. P., Lackie, M. (eds.) Biology of the Chemotactic Response, pp. 35–75. Cambridge: Cambridge University Press 1990bGoogle Scholar
  69. Tranquillo, R. T., Alt, W.: Glossary of terms concerning oriented movement. In: Alt, W., Hoffmann, G. (eds.) Biological Motion, pp. 510–517. Berlin Heidelberg New York: Springer 1990Google Scholar
  70. Tranquillo, R. T., Alt, W.: Stochastic model of chemotactic receptor-mediated dynamic morphology and migration of leukocytes (in preparation)Google Scholar
  71. Trinkaus, J. P.: Cells Into Organs: The Forces that Shape the Embryo. Englewood Cliffs, NJ: Prentice-Hall 1984Google Scholar
  72. Wilkinson, P. C.: Chemotaxis and inflammation, 2nd ed. New York: Churchhill, Livingstone 1982Google Scholar
  73. Wilkinson, P. C., Lackie, J. M., Forrester, J. V., Dunn, G. A.: Chemokinetic accumulation of human neutrophils on immune complex-coated substrata: analysis at a boundary. J. Cell Biol. 99(5), 1761–8 (1984)Google Scholar
  74. Yamada, K. M.: Cell surface interactions with extracellular materials. Annu. Rev. Biochem. 52, 761–799 (1983)Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Richard B. Dickinson
    • 1
  • Robert T. Tranquillo
    • 1
  1. 1.Department of Chemical Engineering and Materials ScienceUniversity of MinnesotaMinneapolisUSA

Personalised recommendations