December  2020, 25(12): 4839-4852. doi: 10.3934/dcdsb.2020129

A continuous-time stochastic model of cell motion in the presence of a chemoattractant

Department of Mathematics, Brigham Young University, Provo, UT 84602-6539, USA

* Corresponding author

Received  October 2019 Revised  January 2020 Published  April 2020

We consider a force-based model for cell motion which models cell forces using Hooke's law and a random outreach from the cell center. In previous work this model was simplified to track the centroid by setting the relaxation time to zero, and a formula for the expected velocity of the centroid was derived. Here we extend that formula to allow for chemotaxis of the cell by allowing the outreach distribution to depend on the spatial location of the centroid.

Citation: J. C. Dallon, Lynnae C. Despain, Emily J. Evans, Christopher P. Grant. A continuous-time stochastic model of cell motion in the presence of a chemoattractant. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4839-4852. doi: 10.3934/dcdsb.2020129
References:
[1]

S. Alonso, M. Stange and C. Beta, Modeling random crawling, membrane deformation and intracellular polarity of motile amoeboid cells, PloS ONE, 13 (2018), e0201977. doi: 10.1371/journal.pone.0201977.  Google Scholar

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E. A. CodlingM. J. Plank and S. Benhamou, Random walk models in biology, Journal of the Royal Society Interface, 5 (2008), 813-834.  doi: 10.1098/rsif.2008.0014.  Google Scholar

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J. C. Dallon, M. Scott and W. V. Smith, A force based model of individual cell migration with discrete attachment sites and random switching terms, Journal of Biomechanical Engineering, 135 (2013), 071008. doi: 10.1115/1.4023987.  Google Scholar

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J. C. DallonL. C. DespainE. J. EvansC. P. Grant and W. V. Smith, A continuous-time model of centrally coordinated motion with random switching, Journal of Mathematical Biology, 74 (2017), 727-753.  doi: 10.1007/s00285-016-1040-2.  Google Scholar

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B. M. Gumbiner, Cell adhesion: The molecular basis of tissue architecture and morphogenesis, Cell, 84 (1996), 345-357.  doi: 10.1016/S0092-8674(00)81279-9.  Google Scholar

[8]

T. Hillen and K. J. Painter, A users guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[9]

O. Kallenberg, Foundations of Modern Probability, Second edition, Probability and its Applications (New York), Springer-Verlag, New York, 2002, http://opac.inria.fr/record=b1098179. doi: 10.1007/978-1-4757-4015-8.  Google Scholar

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E. F. Keller and L. A. Segel, Model for chemotaxis, Journal of Theoretical Biology, 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[11]

S. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, 2nd edition, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511626630.  Google Scholar

[12]

D. Minassian, A mean value theorem for one-sided derivatives, Amer. Math. Monthly, 114 (2007), 28. Google Scholar

[13]

C. S. Patlak, Random walk with persistence and external bias, The Bulletin of Mathematical Biophysics, 15 (1953), 311-338.  doi: 10.1007/BF02476407.  Google Scholar

[14]

F. Ulrich and C.-P. Heisenberg, Trafficking and cell migration, Traffic, 10 (2009), 811. Google Scholar

[15]

M. Winkler, Global classical solvability and generic infinite-time blow-up in quasilinear Keller-Segel systems with bounded sensitivities, Journal of Differential Equations, 266 (2019), 8034-8066.  doi: 10.1016/j.jde.2018.12.019.  Google Scholar

[16]

C. Yoon and Y.-J. Kim, Bacterial chemotaxis without gradient-sensing, Journal of Mathematical Biology, 70 (2015), 1359-1380.  doi: 10.1007/s00285-014-0790-y.  Google Scholar

show all references

References:
[1]

S. Alonso, M. Stange and C. Beta, Modeling random crawling, membrane deformation and intracellular polarity of motile amoeboid cells, PloS ONE, 13 (2018), e0201977. doi: 10.1371/journal.pone.0201977.  Google Scholar

[2]

E. A. CodlingM. J. Plank and S. Benhamou, Random walk models in biology, Journal of the Royal Society Interface, 5 (2008), 813-834.  doi: 10.1098/rsif.2008.0014.  Google Scholar

[3]

C. A. CoposS. WalcottJ. C. del AlamoE. BastounisA. Mogilner and R. D. Guy, Mechanosensitive adhesion explains stepping motility in amoeboid cells, Biophysical Journal, 112 (2017), 2672-2682.   Google Scholar

[4]

J. C. Dallon, M. Scott and W. V. Smith, A force based model of individual cell migration with discrete attachment sites and random switching terms, Journal of Biomechanical Engineering, 135 (2013), 071008. doi: 10.1115/1.4023987.  Google Scholar

[5]

J. C. DallonL. C. DespainE. J. EvansC. P. Grant and W. V. Smith, A continuous-time model of centrally coordinated motion with random switching, Journal of Mathematical Biology, 74 (2017), 727-753.  doi: 10.1007/s00285-016-1040-2.  Google Scholar

[6]

P. Friedl and D. Gilmour, Collective cell migration in morphogenesis, regeneration and cancer, Nature Reviews Molecular Cell Biology, 10 (2009), 445-457.  doi: 10.1038/nrm2720.  Google Scholar

[7]

B. M. Gumbiner, Cell adhesion: The molecular basis of tissue architecture and morphogenesis, Cell, 84 (1996), 345-357.  doi: 10.1016/S0092-8674(00)81279-9.  Google Scholar

[8]

T. Hillen and K. J. Painter, A users guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[9]

O. Kallenberg, Foundations of Modern Probability, Second edition, Probability and its Applications (New York), Springer-Verlag, New York, 2002, http://opac.inria.fr/record=b1098179. doi: 10.1007/978-1-4757-4015-8.  Google Scholar

[10]

E. F. Keller and L. A. Segel, Model for chemotaxis, Journal of Theoretical Biology, 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[11]

S. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, 2nd edition, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511626630.  Google Scholar

[12]

D. Minassian, A mean value theorem for one-sided derivatives, Amer. Math. Monthly, 114 (2007), 28. Google Scholar

[13]

C. S. Patlak, Random walk with persistence and external bias, The Bulletin of Mathematical Biophysics, 15 (1953), 311-338.  doi: 10.1007/BF02476407.  Google Scholar

[14]

F. Ulrich and C.-P. Heisenberg, Trafficking and cell migration, Traffic, 10 (2009), 811. Google Scholar

[15]

M. Winkler, Global classical solvability and generic infinite-time blow-up in quasilinear Keller-Segel systems with bounded sensitivities, Journal of Differential Equations, 266 (2019), 8034-8066.  doi: 10.1016/j.jde.2018.12.019.  Google Scholar

[16]

C. Yoon and Y.-J. Kim, Bacterial chemotaxis without gradient-sensing, Journal of Mathematical Biology, 70 (2015), 1359-1380.  doi: 10.1007/s00285-014-0790-y.  Google Scholar

Figure 1.  Panel a) and c) depicts the state space $ \mathit{x_e} $ given as an example. The circle represents the cell centroid. The black "x"s represent attached I-sites (labeled 0 and 1) and the red "x" denotes the detached I-site (labeled 2). The sets $ A $ and $ B $ represent the example sets used in the measure for case 1 and case 2. Panels a) and b) are for case 1 where I-site 1 dettaches. Panel a) is the initial configuration and panel b) is the new state. Likewise, panels c), and d) are for cases 2 where I-site 2 attaches. The sets $ \bar{A} $ and $ \bar{B} $ are the sets $ A $ and $ B $ transformed by $ S^{-1}\circ F^{-1} $. In panel a), $ \bar{A} = A $ and $ \bar{B} = B+(.5, .5) $. In panel c), $ \bar{A} = A-(.5, .5) $ and $ \bar{B} = 3B-(1.5, 1.5) $ and their intersection contains $ \mathbf{ \pmb{\mathsf{ η}}} = (1, 0) $. In panel d), I-site 2 has attached at location $ \mathbf {x}_2 = \mathbf{c}+\mathbf{ \pmb{\mathsf{ η}}} = (.5, .5)+(1, 0) $
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