# American Institute of Mathematical Sciences

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  December 2020 Early access  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 and 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. [2] E. A. Codling, M. 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. [3] C. A. Copos, S. Walcott, J. C. del Alamo, E. Bastounis, A. Mogilner and R. D. Guy, Mechanosensitive adhesion explains stepping motility in amoeboid cells, Biophysical Journal, 112 (2017), 2672-2682. [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. [5] J. C. Dallon, L. C. Despain, E. J. Evans, C. 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. [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. [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. [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. [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. [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. [11] S. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, 2nd edition, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511626630. [12] D. Minassian, A mean value theorem for one-sided derivatives, Amer. Math. Monthly, 114 (2007), 28. [13] C. S. Patlak, Random walk with persistence and external bias, The Bulletin of Mathematical Biophysics, 15 (1953), 311-338.  doi: 10.1007/BF02476407. [14] F. Ulrich and C.-P. Heisenberg, Trafficking and cell migration, Traffic, 10 (2009), 811. [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. [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.

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##### 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. [2] E. A. Codling, M. 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. [3] C. A. Copos, S. Walcott, J. C. del Alamo, E. Bastounis, A. Mogilner and R. D. Guy, Mechanosensitive adhesion explains stepping motility in amoeboid cells, Biophysical Journal, 112 (2017), 2672-2682. [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. [5] J. C. Dallon, L. C. Despain, E. J. Evans, C. 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. [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. [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. [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. [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. [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. [11] S. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, 2nd edition, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511626630. [12] D. Minassian, A mean value theorem for one-sided derivatives, Amer. Math. Monthly, 114 (2007), 28. [13] C. S. Patlak, Random walk with persistence and external bias, The Bulletin of Mathematical Biophysics, 15 (1953), 311-338.  doi: 10.1007/BF02476407. [14] F. Ulrich and C.-P. Heisenberg, Trafficking and cell migration, Traffic, 10 (2009), 811. [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. [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.
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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