Cellular and Molecular Bioengineering

, Volume 6, Issue 4, pp 449–459 | Cite as

Master Equation-Based Analysis of a Motor-Clutch Model for Cell Traction Force



Microenvironmental mechanics play an important role in determining the morphology, traction, migration, proliferation, and differentiation of cells. A stochastic motor-clutch model has been proposed to describe this stiffness sensitivity. In this work, we present a master equation-based ordinary differential equation (ODE) description of the motor-clutch model, from which we derive an analytical expression to for a cell’s optimum stiffness (i.e., the stiffness at which the traction force is maximal). This analytical expression provides insight into the requirements for stiffness sensing by establishing fundamental relationships between the key controlling cell-specific parameters. We find that the fundamental controlling parameters are the numbers of motors and clutches (constrained to be nearly equal), and the time scale of the on–off kinetics of the clutches (constrained to favor clutch binding over clutch unbinding). Both the ODE solution and the analytical expression show good agreement with Monte Carlo motor-clutch output, and reduce computation time by several orders of magnitude, which potentially enables long time scale behaviors (hours–days) to be studied computationally in an efficient manner. The ODE solution and the analytical expression may be incorporated into larger scale models of cellular behavior to bridge the gap from molecular time scales to cellular and tissue time scales.


Adhesion Mechanosensing Master equation Durotaxis F-actin Myosin Cytoskeletal dynamics Multi-scale modeling Monte Carlo simulation 

Supplementary material

12195_2013_296_MOESM1_ESM.docx (37 kb)
Supplementary material 1 (DOCX 38 kb)
12195_2013_296_MOESM2_ESM.pptx (51 kb)
Supplementary material 2 (PPTX 52 kb)
12195_2013_296_MOESM3_ESM.pptx (124 kb)
Supplementary material 3 (PPTX 125 kb)
12195_2013_296_MOESM4_ESM.pptx (452 kb)
Supplementary material 4 (PPTX 452 kb)


  1. 1.
    Bangasser, B. L., S. S. Rosenfeld, and D. J. Odde. Determinants of maximal force transmission in a motor-clutch model of cell traction in a compliant microenvironment. Biophys. J. 105:581–592, 2013.CrossRefGoogle Scholar
  2. 2.
    Barnhart, E. L., K.-C. Lee, K. Keren, A. Mogilner, and J. A. Theriot. An adhesion-dependent switch between mechanisms that determine motile cell shape. PLoS Biol. 9:e1001059, 2011.CrossRefGoogle Scholar
  3. 3.
    Bell, G. I. Models for the specific adhesion of cells to cells. Science 200:618–627, 1978.CrossRefGoogle Scholar
  4. 4.
    Bridgman, P. C., S. Dave, C. F. Asnes, A. N. Tullio, and R. S. Adelstein. Myosin IIB is required for growth cone motility. J. Neurosci. 21:6159–6169, 2001.Google Scholar
  5. 5.
    Califano, J. P., and C. A. Reinhart-King. Substrate stiffness and cell area predict cellular traction stresses in single cells and cells in contact. Cell. Mol. Bioeng. 3:68–75, 2010.CrossRefGoogle Scholar
  6. 6.
    Chan, C. E., and D. J. Odde. Traction dynamics of filopodia on compliant substrates. Science 322:1687–1691, 2008.CrossRefGoogle Scholar
  7. 7.
    Craig, E. M., D. Van Goor, P. Forscher, and A. Mogilner. Membrane tension, myosin force, and actin turnover maintain actin treadmill in the nerve growth cone. Biophys. J. 102:1503–1513, 2012.CrossRefGoogle Scholar
  8. 8.
    Cuda, G., E. Pate, R. Cooke, and J. R. Sellers. In vitro actin filament sliding velocities produced by mixtures of different types of myosin. Biophys. J. 72:1767–1779, 1997.CrossRefGoogle Scholar
  9. 9.
    Dickinson, R. B., L. Caro, and D. L. Purich. Force generation by cytoskeletal filament end-tracking proteins. Biophys. J. 87:2838–2854, 2004.CrossRefGoogle Scholar
  10. 10.
    Dickinson, R. B., and R. T. Tranquillo. A stochastic model for adhesion-mediated cell random motility and haptotaxis. J. Math. Biol. 31:563–600, 1993.CrossRefMATHGoogle Scholar
  11. 11.
    Enculescu, M., M. Sabouri-Ghomi, G. Danuser, and M. Falcke. Modeling of protrusion phenotypes driven by the actin–membrane interaction. Biophys. J. 98:1571–1581, 2010.CrossRefGoogle Scholar
  12. 12.
    Gao, H., J. Qian, and B. Chen. Probing mechanical principles of focal contacts in cell-matrix adhesion with a coupled stochastic-elastic modelling framework. J. R. Soc. Interface 8:1217–1232, 2011.CrossRefGoogle Scholar
  13. 13.
    Gardel, M. L., B. Sabass, L. Ji, G. Danuser, U. S. Schwarz, and C. M. Waterman. Traction stress in focal adhesions correlates biphasically with actin retrograde flow speed. J. Cell Biol. 183:999–1005, 2008.CrossRefGoogle Scholar
  14. 14.
    Gillespie, D. T. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81:2340–2361, 1977.CrossRefGoogle Scholar
  15. 15.
    Isenberg, B. C., P. A. DiMilla, M. Walker, S. Kim, and J. Y. Wong. Vascular smooth muscle cell durotaxis depends on substrate stiffness gradient strength. Biophys. J. 97:1313–1322, 2009.CrossRefGoogle Scholar
  16. 16.
    Kanchanawong, P., G. Shtengel, A. M. Pasapera, E. B. Ramko, M. W. Davidson, H. F. Hess, and C. M. Waterman. Nanoscale architecture of integrin-based cell adhesions. Nature 468:580–584, 2010.CrossRefGoogle Scholar
  17. 17.
    Kaznessis, Y. N. Statistical Thermodynamics and Stochastic Kinetics: An Introduction for Engineers. Cambridge: Cambridge University Press, 2012.Google Scholar
  18. 18.
    Kolomeisky, A. B., and M. E. Fisher. Molecular motors: a theorist’s perspective. Annu. Rev. Phys. Chem. 58:675–695, 2007.CrossRefGoogle Scholar
  19. 19.
    Lo, C.-M., H.-B. Wang, M. Dembo, and Y.-L. Wang. Cell movement is guided by the rigidity of the substrate. Biophys. J. 79:144–152, 2000.CrossRefGoogle Scholar
  20. 20.
    Macdonald, A., A. R. Horwitz, and D. A. Lauffenburger. Kinetic model for lamellipodal actin-integrin “clutch” dynamics. Cell Adhes. Migr. 2:95–105, 2008.CrossRefGoogle Scholar
  21. 21.
    Mitchison, T., and M. Kirschner. Cytoskeletal dynamics and nerve growth. Neuron 1:761, 1988.CrossRefGoogle Scholar
  22. 22.
    Mogilner, A., and G. Oster. Cell motility driven by actin polymerization. Biophys. J. 71:3030–3045, 1996.CrossRefGoogle Scholar
  23. 23.
    Molloy, J. E., J. E. Burns, J. Kendrick-Jones, R. T. Tregear, and D. C. S. White. Movement and force produced by a single myosin head. Nature 378:209–212, 1995.CrossRefGoogle Scholar
  24. 24.
    Ng, M. R., A. Besser, G. Danuser, and J. S. Brugge. Substrate stiffness regulates cadherin-dependent collective migration through myosin-II contractility. J. Cell Biol. 199:545–563, 2012.CrossRefGoogle Scholar
  25. 25.
    Paszek, M. J., D. Boettiger, V. M. Weaver, and D. A. Hammer. Integrin clustering is driven by mechanical resistance from the glycocalyx and the substrate. PLoS Comput. Biol. 5:e1000604, 2009.MathSciNetCrossRefGoogle Scholar
  26. 26.
    Pathak, A., and S. Kumar. Biophysical regulation of tumor cell invasion: moving beyond matrix stiffness. Integr. Biol. 3(4):267–278, 2011.CrossRefGoogle Scholar
  27. 27.
    Peyton, S. R., and A. J. Putnam. Extracellular matrix rigidity governs smooth muscle cell motility in a biphasic fashion. J. Cell. Physiol. 204:198–209, 2005.CrossRefGoogle Scholar
  28. 28.
    Ricart, B. G., M. T. Yang, C. A. Hunter, C. S. Chen, and D. A. Hammer. Measuring traction forces of motile dendritic cells on micropost arrays. Biophys. J. 101:2620–2628, 2011.CrossRefMATHGoogle Scholar
  29. 29.
    Shao, D., H. Levine, and W.-J. Rappel. Coupling actin flow, adhesion, and morphology in a computational cell motility model. Proc. Natl Acad. Sci. U.S.A. 109(18):6851–6856, 2012.CrossRefGoogle Scholar
  30. 30.
    Shemesh, T., A. D. Bershadsky, and M. M. Kozlov. Physical model for self-organization of actin cytoskeleton and adhesion complexes at the cell front. Biophys. J. 102:1746–1756, 2012.CrossRefGoogle Scholar
  31. 31.
    Stroka, K. M., and H. Aranda-Espinoza. Neutrophils display biphasic relationship between migration and substrate stiffness. Cell Motil. Cytoskelet. 66:328–341, 2009.CrossRefGoogle Scholar
  32. 32.
    Thomas, T. W., and P. A. DiMilla. Spreading and motility of human glioblastoma cells on sheets of silicone rubber depend on substratum compliance. Med. Biol. Eng. Comput. 38:360–370, 2000.CrossRefGoogle Scholar
  33. 33.
    Ulrich, T. A., E. M. de Juan Pardo, and S. Kumar. The mechanical rigidity of the extracellular matrix regulates the structure, motility, and proliferation of glioma cells. Cancer Res. 69:4167–4174, 2009.CrossRefGoogle Scholar
  34. 34.
    Vallotton, P., G. Danuser, S. Bohnet, J-.J. Meister, and A. B. Verkhovsky. Tracking retrograde flow in keratocytes: news from the front. Mol. Biol. Cell 16:1223–1231, 2005.CrossRefGoogle Scholar
  35. 35.
    Walcott, S., D.-H. Kim, D. Wirtz, and S. X. Sun. Nucleation and decay initiation are the stiffness-sensitive phases of focal adhesion maturation. Biophys. J. 101:2919–2928, 2011.CrossRefGoogle Scholar
  36. 36.
    Walcott, S., and S. X. Sun. A mechanical model of actin stress fiber formation and substrate elasticity sensing in adherent cells. Proc. Natl Acad. Sci. U.S.A. 107:7757–7762, 2010.CrossRefGoogle Scholar

Copyright information

© Biomedical Engineering Society 2013

Authors and Affiliations

  1. 1.Department of Biomedical EngineeringUniversity of MinnesotaMinneapolisUSA

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