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## Uniqueness and traveling waves in a cell motility model

 1 Department of Mathematics and Statistics, The College of New Jersey, 2000 Pennington Road, Ewing, NJ 08628, USA 2 Institute of Natural Sciences and Department of Mathematics, Key Laboratory of Scientific and Engineering Computing; Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, China

* Corresponding author

Received  March 2017 Revised  May 2018 Published  November 2018

Fund Project: The work of MSM was completed at Pennsylvania State University, supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program. He also received partial support from NSF grants DMS-1106666 and DMS- 1405769. The work of PZ was partially supported by the National Natural Science Foundation of China grant 11471214 and the One Thousand Plan of China for young scientists

We study a non-linear and non-local evolution equation for curves obtained as the sharp interface limit of a phase-field model for crawling motion of eukaryotic cells on a substrate. We establish uniqueness of solutions to the sharp interface limit equation in the subcritical parameter regime. The proof relies on a Grönwall estimate for a specially chosen weighted $L^2$ norm.

As persistent motion of crawling cells is of central interest to biologists, we next study the existence of traveling wave solutions. We prove that traveling wave solutions exist in the supercritical parameter regime provided the non-linearity of the sharp interface limit equation possesses certain asymmetry (related, e.g., to myosin contractility).

Finally, we numerically investigate traveling wave solutions and simulate their dynamics. Due to non-uniqueness of solutions of the sharp interface limit equation we numerically solve a related, singularly perturbed PDE system which is uniquely solvable. Our simulations predict instability of traveling wave solutions and capture both bipedal wandering cell motion as well as rotating cell motion; these behaviors qualitatively agree with recent experimental and theoretical findings.

Citation: Matthew S. Mizuhara, Peng Zhang. Uniqueness and traveling waves in a cell motility model. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018315
##### References:
 [1] E. Barnhart, K.-C. Lee, G. M. Allen, J. A. Theriot and A. Mogilner, Balance between cell-substrate adhesion and myosin contraction determines the frequency of motility initiation in fish keratocytes, PNAS, 112 (2015), 5045-5050. [2] E. L. Barnhart, G. M. Allen, F. Jülicher and J. A. Theriot, Bipedal locomotion in crawling cells, Biophys J., 98 (2010), 933-942. [3] E. L. Barnhart, K. Lee, K. Keren, A. Mogilner and J. A. Theriot, An adhesion-dependent switch between mechanisms that determine motile cell shape, PLoS Biol, 9 (2011), e1001059. [4] J. W. Barrett, H. Garcke and R. Nürnberg, The approximation of planar curve evolutions by stable fully implicit finite element schemes that equidistribute, Numer. Methods Partial Differential Equations, 27 (2011), 1-30. doi: 10.1002/num.20637. [5] L. Berlyand, M. Potomkin and V. Rybalko, Sharp interface limit in a phase field model of cell motility, Network and Heteregeneous Media, 12 (2017), 551-590. doi: 10.3934/nhm.2017023. [6] A. Bonami, D. Hilhorst and E. Logak, Modified motion by mean curvature: local existence and uniqueness and qualitative properties, Differential Integral Equations, 13 (2000), 1371-1392. [7] K. A. Brakke, The Motion of a Surface by Its Mean Curvature, Princeton University Press and University of Tokyo Press, 1978. [8] B. A. Camley, Y. Zhao, B. Li, H. Levine and W.-J. Rappel, Crawling and turning in a minimal reaction-diffusion cell motility model: Coupling cell shape and biochemistry, Phys. Rev. E, 95 (2017), 012401. [9] B. A. Camley, J. Zimmermann, H. Levine and W.-J. Rappel, Collective signal processing in cluster chemotaxis: Roles of adaptation, amplification, and co-attraction in collective guidance, PLoS Comput Biol., 12 (2016), e1005008. [10] M. P. D. Carmo, Differential Geometry of Curves and Surfaces, Pearson, 1976. [11] X. Chen, The Hele-Shaw problem and area-preserving curve-shortening motions, Arch. Rational Mech. Anal., 123 (1993), 117-151. doi: 10.1007/BF00695274. [12] X. Chen, D. Hilhorst and E. Logak, Mass conserving Allen-Cahn equation and volume preserving mean curvature flow, Interfaces Free Bound., 12 (2010), 527-549. doi: 10.4171/IFB/244. [13] Y. G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom, 33 (1991), 749-786. doi: 10.4310/jdg/1214446564. [14] K. Deckelnick and G. Dziuk, On the approximation of the curve shortening flow, in Calculus of Variations, Applications and Computations (Pont-à-Mousson, 1994), Longman Sci. Tech., 326 (1995), 100–108. [15] K. Deckelnick, G. Dziuk and C. M. Elliott, Computation of geometric partial differential equations and mean curvature flow, Acta Numerica, 14 (2005), 139-232. doi: 10.1017/S0962492904000224. [16] C. M. Elliott and H. Garcke, Existence results for diffusive surface motion laws, Adv. Math. Sci. Appl., 7 (1997), 467-490. [17] J. Escher and G. Simonett, The volume preserving mean curvature flow near spheres, Proc. Amer. Math. Soc., 126 (1998), 2789-2796. doi: 10.1090/S0002-9939-98-04727-3. [18] L. C. Evans and J. Spruck, Motion of level sets by mean curvature I, J. Differential Geom., 33 (1991), 635-681. doi: 10.4310/jdg/1214446559. [19] M. Gage, On an area-preserving evolution equation for plane curves, Contemp. Math., 51 (1986), 51-62. doi: 10.1090/conm/051/848933. [20] M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), 69-96. doi: 10.4310/jdg/1214439902. [21] L. Giomi and A. DeSimone, Spontaneous division and motility in active nematic droplets, Phys. Rev. Lett., 112 (2014), 147802. [22] D. Golovaty, The volume-preserving motion by mean curvature as an asymptotic limit of reaction-diffusion equations, Quart. Appl. Math, 55 (1997), 243-298. doi: 10.1090/qam/1447577. [23] M. A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom., 26 (1987), 285-314. doi: 10.4310/jdg/1214441371. [24] R. J. Hawkins, R. Poincloux, O. Bénichou, M. Piel, P. Chavrier and R. Voituriez, Spontaneous contractility-mediated cortical flow generates cell migration in three-dimensional environments, Biophysical Journal, 101 (2011), 1041-1045. [25] M. F. Krummel, F. Bartumeus and A. Gérard, T cell migration, search strategies and mechanisms, Nature Reviews Immunology, 16 (2016), 193-201. [26] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, The American Mathematical Society, 1968. [27] J. Löber, F. Ziebert and I. S. Aranson, Modeling crawling cell movement on soft engineered substrates, Soft Matter, 9 (2014), 1365-1373. [28] J. Löber, F. Ziebert and I. S. Aranson, Collisions of deformable cells lead to collective migration, Scientific Reports, 5. [29] S. S. Lou, A. Diz-Muñoz, O. D. Weiner, D. A. Fletcher and J. A. Theriot, Myosin light chain kinase regulates cell polarization independently of membrane tension or rho kinase, J. Cell Biol., 209 (2015), 275-288. [30] W. Marth, S. Praetorius and A. Voigt, A mechanism for cell motility by active polar gels, Journal of The Royal Society Interface, 12 (2015), 20150161. [31] B. Merriman, J. Bence and S. Osher, Diffusion generated motion by mean curvature motion, in AMS Select Lectures in Mathematics: The Computational Crystal Grower's Workshop (ed. J. Taylor), Am. Math. Soc., 1993. [32] K. Mikula and D. Sevčovič, Computational and qualitative aspects of evolution of curves driven by curvature and external force, Comput. Visual Sci., 6 (2004), 211-225. doi: 10.1007/s00791-004-0131-6. [33] M. S. Mizuhara, L. Berlyand, V. Rybalko and L. Zhang, On an evolution equation in a cell motility model, Physica D, 318/319 (2016), 12-25. doi: 10.1016/j.physd.2015.10.008. [34] A. Mogilner, Mathematics of cell motility: Have we got its number?, J. Math. Biol., 58 (2009), 105-134. doi: 10.1007/s00285-008-0182-2. [35] A. Mogilner and K. Keren, The shape of motile cells, Curr. Biol., 19 (2009), R762-R771. [36] R. R. Mohan, A. E. K. Hutcheon, R. Choi, J. Hong, J. Lee, R. R. Mohan, R. A. Jr., J. D. Zieske and S. E. Wilson, Apoptosis, necrosis, proliferation, and myofibroblast generation in the stroma following LASIK and PRK, Exp. Eye. Res., 76 (2003), 71-87. [37] S. Osher and J. A. Sethian, Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2. [38] F. Raynaud, M. E. Ambühl, C. Gabella, A. Bornert, I. F. Sbalzarini, J.-J. Meister and A. B. Verkhovsky, Minimal model for spontaneous cell polarization and edge activity in oscillating, rotating and migrating cells, Nature Physics, 12 (2016), 367-373. [39] P. Recho, T. Putelat and L. Truskinovsky, Mechanics of motility initation and motility arrest in crawling cells, Journal of the Mechanics and Physics of Solids, 84 (2015), 469-505. doi: 10.1016/j.jmps.2015.08.006. [40] S. J. Ruuth and B. T. R. Wetton, A simple scheme for volume-preserving motion by mean curvature, J. Sci. Comput., 19 (2003), 373-384. doi: 10.1023/A:1025368328471. [41] D. Shao, W. J. Rappel and H. Levine, Computational model for cell morphodynamics, Phys. Rev. Lett., 105 (2010), 108104. [42] P. Smereka, Semi-implicit level set methods for curvature and surface diffusion motion, J. Sci. Comput., 19 (2003), 439-456. doi: 10.1023/A:1025324613450. [43] E. Tjhung, A. Tiribocchi, D. Marenduzzo and M. E. Cates, A minimal physical model captures the shapes of crawling cells, Nature communications, 6. [44] L. N. Trefethen and M. Embree, Spectra and Pseudospectra, Princeton University Press, 2005. [45] A. K. Wilson, G. Gorgas, W. D. Claypool and P. De Lanerolle, An increase or a decrease in myosin ii phosphorylation inhibits macrophage motility., The Journal of Cell Biology, 114 (1991), 277-283. [46] F. Ziebert and I. S. Aranson, Effects of adhesion dynamics and substrate compliance on the shape and motility of crawling cells, PLoS ONE, 8 (2013), e64511. [47] F. Ziebert, S. Swaminathan and I. S. Aranson, Model for self-polarization and motility of keratocyte fragments, J. R. Soc. Interface, 9 (2012), 1084-1092.

show all references

##### References:
 [1] E. Barnhart, K.-C. Lee, G. M. Allen, J. A. Theriot and A. Mogilner, Balance between cell-substrate adhesion and myosin contraction determines the frequency of motility initiation in fish keratocytes, PNAS, 112 (2015), 5045-5050. [2] E. L. Barnhart, G. M. Allen, F. Jülicher and J. A. Theriot, Bipedal locomotion in crawling cells, Biophys J., 98 (2010), 933-942. [3] E. L. Barnhart, K. Lee, K. Keren, A. Mogilner and J. A. Theriot, An adhesion-dependent switch between mechanisms that determine motile cell shape, PLoS Biol, 9 (2011), e1001059. [4] J. W. Barrett, H. Garcke and R. Nürnberg, The approximation of planar curve evolutions by stable fully implicit finite element schemes that equidistribute, Numer. Methods Partial Differential Equations, 27 (2011), 1-30. doi: 10.1002/num.20637. [5] L. Berlyand, M. Potomkin and V. Rybalko, Sharp interface limit in a phase field model of cell motility, Network and Heteregeneous Media, 12 (2017), 551-590. doi: 10.3934/nhm.2017023. [6] A. Bonami, D. Hilhorst and E. Logak, Modified motion by mean curvature: local existence and uniqueness and qualitative properties, Differential Integral Equations, 13 (2000), 1371-1392. [7] K. A. Brakke, The Motion of a Surface by Its Mean Curvature, Princeton University Press and University of Tokyo Press, 1978. [8] B. A. Camley, Y. Zhao, B. Li, H. Levine and W.-J. Rappel, Crawling and turning in a minimal reaction-diffusion cell motility model: Coupling cell shape and biochemistry, Phys. Rev. E, 95 (2017), 012401. [9] B. A. Camley, J. Zimmermann, H. Levine and W.-J. Rappel, Collective signal processing in cluster chemotaxis: Roles of adaptation, amplification, and co-attraction in collective guidance, PLoS Comput Biol., 12 (2016), e1005008. [10] M. P. D. Carmo, Differential Geometry of Curves and Surfaces, Pearson, 1976. [11] X. Chen, The Hele-Shaw problem and area-preserving curve-shortening motions, Arch. Rational Mech. Anal., 123 (1993), 117-151. doi: 10.1007/BF00695274. [12] X. Chen, D. Hilhorst and E. Logak, Mass conserving Allen-Cahn equation and volume preserving mean curvature flow, Interfaces Free Bound., 12 (2010), 527-549. doi: 10.4171/IFB/244. [13] Y. G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom, 33 (1991), 749-786. doi: 10.4310/jdg/1214446564. [14] K. Deckelnick and G. Dziuk, On the approximation of the curve shortening flow, in Calculus of Variations, Applications and Computations (Pont-à-Mousson, 1994), Longman Sci. Tech., 326 (1995), 100–108. [15] K. Deckelnick, G. Dziuk and C. M. Elliott, Computation of geometric partial differential equations and mean curvature flow, Acta Numerica, 14 (2005), 139-232. doi: 10.1017/S0962492904000224. [16] C. M. Elliott and H. Garcke, Existence results for diffusive surface motion laws, Adv. Math. Sci. Appl., 7 (1997), 467-490. [17] J. Escher and G. Simonett, The volume preserving mean curvature flow near spheres, Proc. Amer. Math. Soc., 126 (1998), 2789-2796. doi: 10.1090/S0002-9939-98-04727-3. [18] L. C. Evans and J. Spruck, Motion of level sets by mean curvature I, J. Differential Geom., 33 (1991), 635-681. doi: 10.4310/jdg/1214446559. [19] M. Gage, On an area-preserving evolution equation for plane curves, Contemp. Math., 51 (1986), 51-62. doi: 10.1090/conm/051/848933. [20] M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), 69-96. doi: 10.4310/jdg/1214439902. [21] L. Giomi and A. DeSimone, Spontaneous division and motility in active nematic droplets, Phys. Rev. Lett., 112 (2014), 147802. [22] D. Golovaty, The volume-preserving motion by mean curvature as an asymptotic limit of reaction-diffusion equations, Quart. Appl. Math, 55 (1997), 243-298. doi: 10.1090/qam/1447577. [23] M. A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom., 26 (1987), 285-314. doi: 10.4310/jdg/1214441371. [24] R. J. Hawkins, R. Poincloux, O. Bénichou, M. Piel, P. Chavrier and R. Voituriez, Spontaneous contractility-mediated cortical flow generates cell migration in three-dimensional environments, Biophysical Journal, 101 (2011), 1041-1045. [25] M. F. Krummel, F. Bartumeus and A. Gérard, T cell migration, search strategies and mechanisms, Nature Reviews Immunology, 16 (2016), 193-201. [26] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, The American Mathematical Society, 1968. [27] J. Löber, F. Ziebert and I. S. Aranson, Modeling crawling cell movement on soft engineered substrates, Soft Matter, 9 (2014), 1365-1373. [28] J. Löber, F. Ziebert and I. S. Aranson, Collisions of deformable cells lead to collective migration, Scientific Reports, 5. [29] S. S. Lou, A. Diz-Muñoz, O. D. Weiner, D. A. Fletcher and J. A. Theriot, Myosin light chain kinase regulates cell polarization independently of membrane tension or rho kinase, J. Cell Biol., 209 (2015), 275-288. [30] W. Marth, S. Praetorius and A. Voigt, A mechanism for cell motility by active polar gels, Journal of The Royal Society Interface, 12 (2015), 20150161. [31] B. Merriman, J. Bence and S. Osher, Diffusion generated motion by mean curvature motion, in AMS Select Lectures in Mathematics: The Computational Crystal Grower's Workshop (ed. J. Taylor), Am. Math. Soc., 1993. [32] K. Mikula and D. Sevčovič, Computational and qualitative aspects of evolution of curves driven by curvature and external force, Comput. Visual Sci., 6 (2004), 211-225. doi: 10.1007/s00791-004-0131-6. [33] M. S. Mizuhara, L. Berlyand, V. Rybalko and L. Zhang, On an evolution equation in a cell motility model, Physica D, 318/319 (2016), 12-25. doi: 10.1016/j.physd.2015.10.008. [34] A. Mogilner, Mathematics of cell motility: Have we got its number?, J. Math. Biol., 58 (2009), 105-134. doi: 10.1007/s00285-008-0182-2. [35] A. Mogilner and K. Keren, The shape of motile cells, Curr. Biol., 19 (2009), R762-R771. [36] R. R. Mohan, A. E. K. Hutcheon, R. Choi, J. Hong, J. Lee, R. R. Mohan, R. A. Jr., J. D. Zieske and S. E. Wilson, Apoptosis, necrosis, proliferation, and myofibroblast generation in the stroma following LASIK and PRK, Exp. Eye. Res., 76 (2003), 71-87. [37] S. Osher and J. A. Sethian, Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2. [38] F. Raynaud, M. E. Ambühl, C. Gabella, A. Bornert, I. F. Sbalzarini, J.-J. Meister and A. B. Verkhovsky, Minimal model for spontaneous cell polarization and edge activity in oscillating, rotating and migrating cells, Nature Physics, 12 (2016), 367-373. [39] P. Recho, T. Putelat and L. Truskinovsky, Mechanics of motility initation and motility arrest in crawling cells, Journal of the Mechanics and Physics of Solids, 84 (2015), 469-505. doi: 10.1016/j.jmps.2015.08.006. [40] S. J. Ruuth and B. T. R. Wetton, A simple scheme for volume-preserving motion by mean curvature, J. Sci. Comput., 19 (2003), 373-384. doi: 10.1023/A:1025368328471. [41] D. Shao, W. J. Rappel and H. Levine, Computational model for cell morphodynamics, Phys. Rev. Lett., 105 (2010), 108104. [42] P. Smereka, Semi-implicit level set methods for curvature and surface diffusion motion, J. Sci. Comput., 19 (2003), 439-456. doi: 10.1023/A:1025324613450. [43] E. Tjhung, A. Tiribocchi, D. Marenduzzo and M. E. Cates, A minimal physical model captures the shapes of crawling cells, Nature communications, 6. [44] L. N. Trefethen and M. Embree, Spectra and Pseudospectra, Princeton University Press, 2005. [45] A. K. Wilson, G. Gorgas, W. D. Claypool and P. De Lanerolle, An increase or a decrease in myosin ii phosphorylation inhibits macrophage motility., The Journal of Cell Biology, 114 (1991), 277-283. [46] F. Ziebert and I. S. Aranson, Effects of adhesion dynamics and substrate compliance on the shape and motility of crawling cells, PLoS ONE, 8 (2013), e64511. [47] F. Ziebert, S. Swaminathan and I. S. Aranson, Model for self-polarization and motility of keratocyte fragments, J. R. Soc. Interface, 9 (2012), 1084-1092.
Sketch of functions $w_B$, $w_F$ and $w_{new}$ from the proof of Theorem 3.2
(Left) Plot of $I_2(V, \lambda)$ with $\tilde{\Phi}_\beta(V)$ and $\beta = 100$ (Right) Traveling wave profile for $\tilde{\Phi}_\beta(V)$, $\beta = 100$, $V \approx 2.15$ (in positive $y$ direction), $\lambda \approx 9.75$
(Left) Plot of $I_2(V, \lambda)$ with $\Phi_\beta(V)$ derived from (4)-(6) with $W$ as in (37); $\beta = 100$. (Right) Traveling wave profile with $V\approx 1.7$, $\lambda \approx 0$
(Left) Sketch of rotating cell; intervals in red (dotted) represent unstable velocities (Right) Graph of the isoperimetric inequality $Q$ over time for various $\varepsilon$
(Left) Plot of $V-\Phi_\beta(V)$ tracked for a point on a curve evolving by system (38)-(39) shows approximate hysteresis jumps. (Right) Trajectory of center of curve in (38)-(39) when $\varepsilon = .01$; after short transience period, convergence to "zig-zag" motion
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