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October  2020, 13(5): 1007-1027. doi: 10.3934/krm.2020035

Stability of a non-local kinetic model for cell migration with density dependent orientation bias

Nadia Loy , and  Luigi Preziosi

Department of Mathematical Sciences "G. L. Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy

* Corresponding author: Nadia Loy

Received  January 2020 Revised  May 2020 Published  August 2020

Fund Project: This work was partially supported by Istituto Nazionale di Alta Matematica, Ministry of Education, Universities and Research, through the MIUR grant Dipartimenti di Eccellenza 2018-2022, Project no. E11G18000350001, and the Scientific Reseach Programmes of Relevant National Interest project n. 2017KL4EF3. NL also acknowledges Compagnia di San Paolo that funds her Ph.D. scholarship

The aim of the article is to study the stability of a non-local kinetic model proposed in [17], that is a kinetic model for cell migration taking into account the non-local sensing performed by a cell in order to decide its direction and speed of movement. We show that pattern formation results from modulation of one non-dimensional parameter that depends on the tumbling frequency, the sensing radius, the mean speed in a given direction, the uniform configuration density and the tactic response to the cell density. Numerical simulations show that our linear stability analysis predicts quite precisely the ranges of parameters determining instability and pattern formation. We also extend the stability analysis to the case of different mean speeds in different directions.

Keywords: Kinetic model, non-linear Bolztmann equation, non-local interactions, stability analysis, cell migration, pattern formation.
Mathematics Subject Classification: Primary: 35Q20, 35B36, 92B05; Secondary: 45K05, 92C15.
Citation: Nadia Loy, Luigi Preziosi. Stability of a non-local kinetic model for cell migration with density dependent orientation bias. Kinetic and Related Models, 2020, 13 (5) : 1007-1027. doi: 10.3934/krm.2020035
References:
[1]

N. J. Armstrong, K. J. Painter and J. A. Sherratt, A continuum approach to modelling cell-cell adhesion, J. Theoret. Biol., 243 (2006), 98-113.  doi: 10.1016/j.jtbi.2006.05.030.

[2]

V. Bitsouni and R. Eftimie, Non-local parabolic and hyperbolic models for cell polarisation in heterogeneous cancer cell populations, Bull. Math. Biol., 80 (2018), 2600-2632.  doi: 10.1007/s11538-018-0477-4.

[3]

A. Buttenschön, Integro-partial Differential Equation Models for Cell-cell Adhesion and its Application, Ph.D thesis, University of Alberta, 2018.

[4]

A. Buttenschön and T. Hillen, Non-local adhesion models for microorganisms on bounded domains, SIAM J. Appl. Math., 80(1), (2020) 382–401. doi: 10.1137/19M1250315.

[5]

A. Buttenschön and T. Hillen, Non-local cell adhesion models: Steady states and bifurcations, 2020, arXiv: 2001.00286.

[6]

A. Buttenschön, T. Hillen, A. Gerisch and K. J. Painter, A space-jump derivation for non-local models of cell-cell adhesion and non-local chemotaxis, J. Math. Biol., 76 (2018), 429-456.  doi: 10.1007/s00285-017-1144-3.

[7]

J. Carrillo, F. Hoffmann and R. Eftimie, Non-local kinetic and macroscopic models for self-organised animal aggregations, Kin. Rel. Models, 8 (2015), 413-441.  doi: 10.3934/krm.2015.8.413.

[8]

A. Colombi, M. Scianna and L. Preziosi, Coherent modelling switch between pointwise and distributed representations of cell aggregates, J. Math. Biol., 74 (2017), 783-808.  doi: 10.1007/s00285-016-1042-0.

[9]

A. Colombi, M. Scianna and A. Tosin, Differentiated cell behavior: A multiscale approach using measure theory, J. Math. Biol., 71 (2015), 1049-1079.  doi: 10.1007/s00285-014-0846-z.

[10]

R. Eftimie, Hyperbolic and kinetic models for self-organized biological aggregations and movement: A brief review, J. Math. Biol., 65 (2012), 35-75.  doi: 10.1007/s00285-011-0452-2.

[11]

R. Eftimie, G. de Vries, M. A Lewis and F. Lutscher, Modeling group formation and activity patterns in self-organizing collectives of individuals, Bull. Math. Biol., 69 (2007), 1537-1565.  doi: 10.1007/s11538-006-9175-8.

[12]

R. Eftimie, M. Perez and P.-L. Buono, Pattern formation in a nonlocal mathematical model for the multiple roles of the TGF-$\beta$ pathway in tumour dynamics, Math. Biosci., 289 (2017), 96-115.  doi: 10.1016/j.mbs.2017.05.003.

[13]

R. Eftimie, G. Vries and M. Lewis, Complex spatial group patterns result from different animal communication mechanisms, PNAS USA, 104 (2007), 6974-6979.  doi: 10.1073/pnas.0611483104.

[14]

F. Filbet and K. Yang, Numerical simulation of kinetic models for chemotaxis, SIAM J. Sci. Comput., 36 (2014), B348–B366. doi: 10.1137/130910208.

[15]

T. Hillen, K. J. Painter and C. Schmeiser, Global existence for chemotaxis with finite sampling radius, Discr. Cont. Dyn. Sys. - B, 7 (2007), 125-144.  doi: 10.3934/dcdsb.2007.7.125.

[16]

N. Loy and L. Preziosi, Modelling physical limits of migration by a kinetic model with non-local sensing, J. Math. Biol., 80 (2020), 1759-1801.  doi: 10.1007/s00285-020-01479-w.

[17]

N. Loy and L. Preziosi, Kinetic models with non-local sensing determining cell polarization and speed according to independent cues, J. Math. Biol., 80 (2019), 373-421.  doi: 10.1007/s00285-019-01411-x.

[18]

H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.  doi: 10.1007/BF00277392.

[19]

H. Othmer and T. Hillen, The diffusion limit of transport equations Ⅱ: Chemotaxis equations, SIAM J. Appl. Math., 62 (2002), 1222-1250.  doi: 10.1137/S0036139900382772.

[20]

J. K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (4) (2002), 501-543. 

[21]

K. J. Painter, N. J. Armstrong and J. A. Sherratt, The impact of adhesion on cellular invasion processes in cancer and development, J. Theoret. Biol., 264 (2010), 1057-1067.  doi: 10.1016/j.jtbi.2010.03.033.

[22]

K. J. Painter, M. J. Bloomfield, J. A. Sherratt and A. Gerisch, A nonlocal model for contact attraction and repulsion in heterogeneous cell populations, Bull. Math. Biol., 77 (2015), 1132-1165.  doi: 10.1007/s11538-015-0080-x.

[23]

B. Perthame and S. Yasuda, Stiff-response-induced instability for chemotactic bacteria and flux-limited keller-segel equation, Nonlinearity, 31 (2018), 4065-4089.  doi: 10.1088/1361-6544/aac760.

[24]

R. G. Plaza, Derivation of a bacterial nutrient-taxis system with doubly degenerate cross-diffusion as the parabolic limit of a velocity-jump process, J. Math. Biol., 78 (2019), 1681-1711.  doi: 10.1007/s00285-018-1323-x.

[25]

C. Schmeiser and A. Nouri, Aggregated steady states of a kinetic model for chemotaxis, Kin. Rel. Models, 10 (2017), 313-327.  doi: 10.3934/krm.2017013.

[26]

D. W. Stroock, Some stochastic processes which arise from a model of the motion of a bacterium, Zeit. Wahr. Ver. Geb., 28 (1974), 305-315.  doi: 10.1007/BF00532948.

[27]

A. Tosin and P. Frasca, Existence and approximation of probability measure solutions to models of collective behaviors, Netw. Heter. Media, 6 (2011), 561-596.  doi: 10.3934/nhm.2011.6.561.

show all references

References:
[1]

N. J. Armstrong, K. J. Painter and J. A. Sherratt, A continuum approach to modelling cell-cell adhesion, J. Theoret. Biol., 243 (2006), 98-113.  doi: 10.1016/j.jtbi.2006.05.030.

[2]

V. Bitsouni and R. Eftimie, Non-local parabolic and hyperbolic models for cell polarisation in heterogeneous cancer cell populations, Bull. Math. Biol., 80 (2018), 2600-2632.  doi: 10.1007/s11538-018-0477-4.

[3]

A. Buttenschön, Integro-partial Differential Equation Models for Cell-cell Adhesion and its Application, Ph.D thesis, University of Alberta, 2018.

[4]

A. Buttenschön and T. Hillen, Non-local adhesion models for microorganisms on bounded domains, SIAM J. Appl. Math., 80(1), (2020) 382–401. doi: 10.1137/19M1250315.

[5]

A. Buttenschön and T. Hillen, Non-local cell adhesion models: Steady states and bifurcations, 2020, arXiv: 2001.00286.

[6]

A. Buttenschön, T. Hillen, A. Gerisch and K. J. Painter, A space-jump derivation for non-local models of cell-cell adhesion and non-local chemotaxis, J. Math. Biol., 76 (2018), 429-456.  doi: 10.1007/s00285-017-1144-3.

[7]

J. Carrillo, F. Hoffmann and R. Eftimie, Non-local kinetic and macroscopic models for self-organised animal aggregations, Kin. Rel. Models, 8 (2015), 413-441.  doi: 10.3934/krm.2015.8.413.

[8]

A. Colombi, M. Scianna and L. Preziosi, Coherent modelling switch between pointwise and distributed representations of cell aggregates, J. Math. Biol., 74 (2017), 783-808.  doi: 10.1007/s00285-016-1042-0.

[9]

A. Colombi, M. Scianna and A. Tosin, Differentiated cell behavior: A multiscale approach using measure theory, J. Math. Biol., 71 (2015), 1049-1079.  doi: 10.1007/s00285-014-0846-z.

[10]

R. Eftimie, Hyperbolic and kinetic models for self-organized biological aggregations and movement: A brief review, J. Math. Biol., 65 (2012), 35-75.  doi: 10.1007/s00285-011-0452-2.

[11]

R. Eftimie, G. de Vries, M. A Lewis and F. Lutscher, Modeling group formation and activity patterns in self-organizing collectives of individuals, Bull. Math. Biol., 69 (2007), 1537-1565.  doi: 10.1007/s11538-006-9175-8.

[12]

R. Eftimie, M. Perez and P.-L. Buono, Pattern formation in a nonlocal mathematical model for the multiple roles of the TGF-$\beta$ pathway in tumour dynamics, Math. Biosci., 289 (2017), 96-115.  doi: 10.1016/j.mbs.2017.05.003.

[13]

R. Eftimie, G. Vries and M. Lewis, Complex spatial group patterns result from different animal communication mechanisms, PNAS USA, 104 (2007), 6974-6979.  doi: 10.1073/pnas.0611483104.

[14]

F. Filbet and K. Yang, Numerical simulation of kinetic models for chemotaxis, SIAM J. Sci. Comput., 36 (2014), B348–B366. doi: 10.1137/130910208.

[15]

T. Hillen, K. J. Painter and C. Schmeiser, Global existence for chemotaxis with finite sampling radius, Discr. Cont. Dyn. Sys. - B, 7 (2007), 125-144.  doi: 10.3934/dcdsb.2007.7.125.

[16]

N. Loy and L. Preziosi, Modelling physical limits of migration by a kinetic model with non-local sensing, J. Math. Biol., 80 (2020), 1759-1801.  doi: 10.1007/s00285-020-01479-w.

[17]

N. Loy and L. Preziosi, Kinetic models with non-local sensing determining cell polarization and speed according to independent cues, J. Math. Biol., 80 (2019), 373-421.  doi: 10.1007/s00285-019-01411-x.

[18]

H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.  doi: 10.1007/BF00277392.

[19]

H. Othmer and T. Hillen, The diffusion limit of transport equations Ⅱ: Chemotaxis equations, SIAM J. Appl. Math., 62 (2002), 1222-1250.  doi: 10.1137/S0036139900382772.

[20]

J. K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (4) (2002), 501-543. 

[21]

K. J. Painter, N. J. Armstrong and J. A. Sherratt, The impact of adhesion on cellular invasion processes in cancer and development, J. Theoret. Biol., 264 (2010), 1057-1067.  doi: 10.1016/j.jtbi.2010.03.033.

[22]

K. J. Painter, M. J. Bloomfield, J. A. Sherratt and A. Gerisch, A nonlocal model for contact attraction and repulsion in heterogeneous cell populations, Bull. Math. Biol., 77 (2015), 1132-1165.  doi: 10.1007/s11538-015-0080-x.

[23]

B. Perthame and S. Yasuda, Stiff-response-induced instability for chemotactic bacteria and flux-limited keller-segel equation, Nonlinearity, 31 (2018), 4065-4089.  doi: 10.1088/1361-6544/aac760.

[24]

R. G. Plaza, Derivation of a bacterial nutrient-taxis system with doubly degenerate cross-diffusion as the parabolic limit of a velocity-jump process, J. Math. Biol., 78 (2019), 1681-1711.  doi: 10.1007/s00285-018-1323-x.

[25]

C. Schmeiser and A. Nouri, Aggregated steady states of a kinetic model for chemotaxis, Kin. Rel. Models, 10 (2017), 313-327.  doi: 10.3934/krm.2017013.

[26]

D. W. Stroock, Some stochastic processes which arise from a model of the motion of a bacterium, Zeit. Wahr. Ver. Geb., 28 (1974), 305-315.  doi: 10.1007/BF00532948.

[27]

A. Tosin and P. Frasca, Existence and approximation of probability measure solutions to models of collective behaviors, Netw. Heter. Media, 6 (2011), 561-596.  doi: 10.3934/nhm.2011.6.561.

Figure 1.  Stability diagram for a localized sensing kernel $ \gamma_R = \delta(\lambda-R) $. The red dashed line delimits the unstable region, in (a) when $ b'(\rho_{\infty})<0 $ and in (b) when $ b'(\rho_{\infty})>0 $. The blue dotted lines evidentiate the dimensionless wave numbers $ k_{max}R $ with local maxima of the growth rate (given by (34)) in the unstable regime. The lowest curves correspond to the most dangerous wave numbers in both cases
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Figure 2.  Stability diagram for a uniform (a) and a ramp (b) sensing kernel. The blue dotted line represents $ k_{max}R $ given respectively by (a) (41) and (b) (44). The unstable region is the one to the left of the red dashed line, $ {\it i.e.} $ the values of $ k_{max}R $ also satisfy (40) in (a) and (43) in (b)
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Figure 3.  Temporal evolution of $ \rho(t,x) $ from $ \rho_0(x) = 0.2\left(1+0.1\sin(\pi x/5)\right) $ (a) in the unstable case $ \mathcal{V}_b\approx -0.1125 $ and (b) in the stable case $ \mathcal{V}_b\approx -0.3376 $. (c) Wavelength of the most unstable mode as obtained from the simulation (black line) and from Eq.(34) (green line)
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Figure 4.  Evolution of the density distribution in the adhesion case starting from the initial condition $ \rho_0(x) = 0.2\left(1+0.1\sin(\pi x/5)\right) $, so that $ \rho_{\infty}\approx 0.2127 $, for a delta kernel (top row) and a Heaviside kernel (bottom row). In the left column the values of $ \mathcal{V}_b $ correspond to stable cases, while in the right column $ \mathcal{V} $ correspond to the unstable case. Specifically, in (a) $ \mathcal{V}_b\approx 1.1364 $, in (b) $ \mathcal{V}_b\approx 0.7812 $, in (c) $ \mathcal{V}_b\approx 0.5681 $, and in (d) $ \mathcal{V}_b\approx 0.4032 $
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Figure 5.  Comparison of unstable evolutions for the same value of $ \mathcal{V} = 0.0625 $, (given by $ V = 0.25, R = 0.04, \mu = 100 $) starting form the initial condition $ \rho_0(x) = 0.2\left(1+0.1\sin(\pi x/5)\right) $, so that $ \rho_{\infty}\approx 0.2127 $. In (a) $ \gamma_R(\lambda) = \delta(\lambda-R) $, in (b) $ \gamma_R(\lambda) = H(R-\lambda) $ and in (c) $ \gamma_R = \left(1-\frac{\lambda}{R}\right)_+ $
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Figure 6.  Evolution in the asymmetric case for a localised sensing kernel. The initial condition is always $ \rho_0(x) = 0.2\left(1+0.01\sin(\pi x/5)\right) $, so that $ \rho_{\infty}\approx 0.2013 $. (a)-(c): Adhesion, $ \mu = 3, R = 0.44, V^+ = 0.5, V^- = 1, \mathcal{V}_b\approx 0.5682 $. (d)-(f): Volume filling, $ \mu = 200, R = 0.4, V^+ = 0.25,V^- = 0.5,\rho_{th} = 1, \mathcal{V}_b\approx -0.018 $. (g): Volume filling, $ \mu = 200, R = 0.4, V^+ = V^- = 0.375,\rho_{th} = 1, \mathcal{V}_b\approx -0.018 $. (h)-(i): Adhesion, $ \mu = 1, R = 0.44, V^+ = 0.5, V^- = 1, \mathcal{V}_b\approx 1.7045 $
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