4 Comparison between the macroscopic model and the particle dynamics
4.3 Profiles of the solutions
Here, we use the order parameter fitted in the microscopic simulations to com-pute the solutions of the macroscopic model for the same set of parameters. As explained in Appendix 7, simulations are performed on the same square domain for the microscopic and macroscopic model: [−S, S]×[−S, S] with side length 2S for S = 0.5. The numerical grid for the macroscopic simulations is such that
∆x = ∆y = 0.025 and the time step respect the CFL condition (see appendix 7) with δf = L2. If not differently stated, the macroscopic values of the model parameters are chosen such that:
νf =νd= 0.1, cu = 0.1, α= 0.1, κ= 0.1, dX = 5e−5, N = 1500, L= 0.2, and ε = 0.1. Fig. 3.8 shows the simulations performed for χ = 0.1 and different values of the noise intensitydand for external potentialcu = 0.01 (A) andcu = 0.1 (B). We recall that the noise in the space variables is fixed, i.e µ is chosen such that dµ=dX = 5e−5. As highlighten by the study of the macroscopic model (see section 3.2.3), buckled solutions are obtained for a certain set of parameters. Due to the randomness of the phenomena of the microscopic model (random noise, fiber linking unlinking, random initial configurations), the final solutions of the microscopic model will converge randomly to one of the buckled situations, with no preference for one or the other. Therefore, in order to enable the comparison between the microscopic and macroscopic buckled situations, we use the symmetry of the buckled situations and study the absolute value of the fiber orientations
|θ(x)|. Fig. 3.8 shows simulations for different noise parameterdas well as different external potential forces cu. For each set of parameter, we show the profiles of θ(x) for both microscopic and macroscopic models in unbuckled situations, and
|θ(x)| for both models in case of buckled situations. Red curves correspond to the solutions of the macroscopic model, black curves to the solutions of the microscopic one. As shown by Fig. 3.8, we obtain a very good agreement between the solutions of the microscopic and macroscopic models in the case χ= 0.1 (small amount of fiber links).
Figure 3.8: (A) Simulations of the microscopic model with ε = 0.1, external potentialcu = 0.01 and fiber linking/unlinking frequencies such that χ= 0.1 (first line). From A1 to A3: for increasing values of the microscopic noise d: d = 10−5, d = 10−4 and d = 10−3. For each, we show the profiles of the solutions averaged over the y direction and over 10 simulations for the microscopic model (black curves, second line). For each set of parameters, we superimpose the profiles of the solutions to the macroscopic model (red curve). For smalldand small external potential cu = 0.01 (A1,A2), fiber orientation angles reach π2 and −π2 on the left and right hand sides of the domain, with a zone of quasi horizontal fibers in the middle, as predicted by the macroscopic model. For increasing d (A3), the profile flattens and fibers no longer reach the orientation angles π2. The macroscopic model captures the same features for the same parameters. (B) Case cu = 0.1. In this case, all fibers reach orientation ±π2 due to the large intensity of the external potential for both the microscopic and macroscopic model.
As shown by Fig. 3.8 (A), for small noise intensity d and small external po-tential cu = 0.01, fibers reach π2 and −π2 on the left and right hand side of the domain, while fibers in the middle are quasi horizontally disposed, as predicted by the macroscopic model (Figs. 3.8 (A1-A2)). For increasing d, the profile of the fiber orientation flattens and fibers no longer reach the orientation angles π2 (Fig.
3.8 (A3)). The macroscopic model captures the same features for the same param-eters. In the case of a larger external potential, all the fibers are vertically disposed in the domain (except on the boundaries because of the Dirichlet conditions, see Fig. 3.8 (B)). This is also predicted by the macroscopic model. This tends to show that the macroscopic model captures the same features as the microscopic model in case of a small value ofχ. However, different solutions are obtained when comparing both models forχ= 1. As shown in Fig. 3.9, for the same parameters, the profile of the solution of the microscopic model is much more flattened than for the macroscopic model with χ= 1.
−−0.52 0 0.5
−1 0 1 2
x θ 0(x)
Macro Micro
Figure 3.9: Simulation of the microscopic model with ε = 0.1, external potential cu = 0.01, d = 10−4 and fiber linking/unlinking frequencies such that χ= 1. For the same parameters and χ = 1, the profile of the solution of the microscopic model (black curve) is much more flattened than for the macroscopic model (red curve).
This suggests that the noise must be much larger in the macroscopic model for both profiles to correspond. From this observation, we deduce that the amount of fiber links has a strong effect on the ’temperature’ of the system. This is due to the fact that fibers have a finite length in the microscopic model, whereas the large scale limit supposes that fiber length tends to 0. To highlight this phenomenon, we plot (in loglog scale) in Fig. 3.10 theL2 norm of the difference between the solution of the microscopic model and the one of the macroscopic model, as function of the
fiber length L. We explore the fiber lengths L ≈ 0.16, L ≈ 0.2 and L ≈ 0.22 (respective microscopic values: 0.05,0.06 and 0.07), and the respective number of fibers are chosen to Nf = 1800, Nf = 1500 and Nf = 1250. For each, three different noise intensities are tested d = 10−5,10−4 and d = 10−3. Figs. 3.10 (A) and (B) are obtained for χ = 0.1 and χ = 1 respectively. The external potential is cu = 0.01.
(A)
−3 −2.8 −2.6
−1
−0.5 0 0.5
L L2 norm
d=1e−005 d=0.0001 d=0.001
(B)
−30 −2.8 −2.6 0.5
1 1.5
L
L2 norm d=1e−005
d=0.0001 d=0.001
Figure 3.10: (A) L2-norm of the difference between the microscopic fiber orien-tation and the macroscopic one for cu = 0.01, χ = 0.1, plotted in loglog scale as function of the (microscopic) fiber length L for three different values of the noise intensity d: d = 10−5 (blue curve), d = 10−4 (black curve) and d = 10−3. (B) Same plots in the case χ= 1.
As shown by Fig. 3.10, the difference between the microscopic and macroscopic profiles decreases when the fiber length decreases. For χ = 0.1, we obtain a small error of order 0.6 whereas for χ = 1 the error is 1.64 for the smallest fiber length considered. These results are a first step towards the numerical proof of convergence of the microscopic model to the macroscopic one as the fiber length goes to zero.
5 Conclusion
In this paper, we have analyzed the macroscopic model derived from a microscopic model for fibers interacting through linking and unlinking interactions, alignment between cross-linked fibers and external rotation potential. We have shown that the starting Individual based dynamics can be written in form of a minimization problem under a given regime, showing the analogy between the models of [1]
and [2]. Under some regularity assumptions for the external potential, we were able to obtain existence of stationary solutions to the macroscopic derived model.
The numerical simulations of the macroscopic model showed the apparition of
a buckling phenomenon, giving a first insight into the mechanical properties of the system. We showed that the distribution of fiber mean local orientation of the microscopic model were in good agreement with the predictions of the macroscopic model and we were able to compute the order parameter of the microscopic model.
The numerical simulations of the microscopic model showed that in a well chosen regime, the microscopic and macroscopic models were in good agreement. We finally gave a first numerical analysis towards the proof of the convergence of the microscopic model to the macroscopic one.
We have seen that the fiber links density have a strong impact on the final structures that the macroscopic model does not capture. This is due to the fact that the scaling suppose that the linking/unlinking process is quasi instantaneous.
This assumption makes the action of the links vanish in the macroscopic model, and no memory effect of the fiber cross-links remain. Works are in progress to better understand the effects of the links on the temperature of the system. Further perspectives of this model include numerical simulations of the complete kinetic model, or the establishment of a hydrodynamic scaling based on a more realistic assumption for fiber linking/unlinking dynamics.
Many questions remain open concerning the macroscopic model. On an an-alytical viewpoint, unicity results for the stationary solutions of the macroscopic model are the subject of future work. On a numerical viewpoint, we plan to de-velop numerical techniques to enable the microscopic simulations to be performed under the kinetic regime. An other direct perspective of this work is to consider non homogeneous density. As shown in [2], this leads to a much more complex system of two coupled highly non linear equations requiring advanced numerical methods.
[1] D. Peurichard et al.Simple mechanical cues could explain adipose tissue mor-phology, submitted
[2] D. Peurichard, F. Delebecque, P. Degond Continuum model for linked fibers with alignment interactions, submitted
[3] D. Boal (2002),Mechanics of the Cell, Cambridge University Press, New York.
[4] P. Friedl, E.B.Brocker, (2000) The biology of cell locomotion within three di-mensional extracellular matrix, Cell Mol. Life Sci, 51:595-615.
[5] O. Chaudury, S. H. Parekh, D. A. Fletcher (2007)Reversible stress softening of actin networks, Nature, 445:295-298.
[6] O. Lieleg, J. Kayser, G. Brambilla, L. Cipelletti and A. R. Bausch (2011) Complex Slow Dynamics in Bundled Cytoskeletal Networks, Nature Materials, 10:236-242.
[7] B.A. DiDonna, A. J. Levine (2006)Filamin cross-linked semiflexible networks:
Fragility under strain, Phys Rev Lett. 97(6):068104
[8] D. Manoussaki, S. R. Lubkin, R. B. Vemon, J. D. Murray, (1996)A mechanical model for the formation of vascular networks in vitro, Acta Biotheoretica, 44(3-4):271-282
[9] A. Tosin, D. Ambrosi, L. Preziosi (2006),Mechanics, chemotaxis in the mor-phogenesis of vascular networks, Bull. Math. Biol., 68:1819-1836
[10] A. Chauviere, T. Hillen, L. Preziosi, (2007) Continuum model for mesenchy-mal motion in a fibrous network, Networks, Heterogeneous Media, 2:333-357 [11] T. Hillen, (2006), M5 mesoscopic and macroscopic models for mesenchymal
motion, J. Math. Biol. 53:585-616.
[12] W. Alt, M. Dembo, (1999)Cytoplasm dynamics, cell motion: two phase flow models,Mathematical Biosciences, 156(1):207-228
[13] R.B. Dickinson,(2000)A generalized transport model for biased cell migration in an anisotropic environment,J. Math. Biol., 40:97-135
[14] R. Alonso, J. Young, Y. Cheng, (2014) A particle interaction model for the simulation of biological, cross-linked fibers inspired from flocking theory, Cell.
mol. bioeng., 7(1):58-72
[15] C.P. Broedersz et al,(2010)Cross-link governed dynamics of biopolymer net-works, PRL 105, 238101
[16] J.A Astrom, P.B.S Kumar, I. Vattulaine, M. Karttunen (2005)Strain harden-ing in dense actin networks, Phy. Rev. E. 71:050901
[17] I. Pivkin, G. Karniadakis. (2008) Accurate coarse-grained modeling of red blood cells. Phys. Rev. Lett. 101(11):118105.
[18] G.A. Buxton, N. Clarke, P.J. Hussey (2009)Actin dynamics, the elasticity of cytoskeletal networks, Express Polymer Letters, 3(9):579-587.
[19] D.A Head, A.J. Levine, F.C MacKintosh,(2003) Distinct regimes of elastic response, deformation modes of cross-linked cytoskeletal, semiflexible polymer networks, Phys. Rev. E. 68:061907
[20] H. Krasher, J. Lammerding, H. Huang, R.T. Lee, R.D. Kamm (2003)A three-dimensional viscoelastic model for cell deformation with experimental verifi-cation, Biophys. J. ,85(5):3336-49
[21] L.A. Tabler, Y. Shi, L. Yang, P.V. Bayly (2011)A poroelastic model for cell crawling including mechanical coupling between cytoskeletal contraction, actin polymerization, J. Mech. Mat. Struct., 6:569-589
[22] J.F. Joanny, F. Jlicher, K. Kruse, J. Prost (2007)Hydrodynamic Theory for Multi-Component Active Polar Gels, New J. Phys. 9 442
[23] P. Degond, S. Motsch, (2008) Continuum limit of self-driven particles with orientation interaction, Math. Mod. Meth. App. S. 18:1193-1215
[24] M.E. Taylor, Partial Differential Equations III, App. Maths. Sc., vol 117 [25] D. Gilbarg, N.S. Trudinger, (1977)Elliptic Partial Differential Equations of
Second Order, Springer-Verlag