Boids Model Applied to Cell Segregation L G Brunnet, J Belmonti, G L Thomas, R M C de Almeida, H Chaté Instituto de Física, Univ. Federal do Rio Grande do Sul, Brasil and CEA-Saclay, France XXVIII Encontro Nacional de Física da Matéria Condensada - 10-14 Maio, 2005 p.1/18
Contents Motivation Boids Generalization of Vicsek s Model Cell segregation using boids Time evolution, finite size scaling and fluctuations Conclusions XXVIII Encontro Nacional de Física da Matéria Condensada - 10-14 Maio, 2005 p.2/18
Motivation - General 1 What is the advantage of being part of a community? What is the relevant information exchanged? What is the role of space in collective behavior? Why did it take so long for nature to start with multicellular systems? Bacteria produce shapes of snow flakes, vortices and spirals due to chemiotaxis. XXVIII Encontro Nacional de Física da Matéria Condensada - 10-14 Maio, 2005 p.3/18
Motivation - General 2 Dictyostelium Discoideum under starvation form clusters following camp gradients. D. discoideum amoebae towards a point source of the chemoattractant camp. From G. Gerisch, Max-Planck-Institut fur Biochemie. (film ax320x.mov - From R. Firtel, University of California, San Diego ) XXVIII Encontro Nacional de Física da Matéria Condensada - 10-14 Maio, 2005 p.4/18
Motivation - Focus Hydras: Two different kinds of tissues, external and internal. Differential adhesiveness and regeneration. Segregation during regeneration Burst-2.mpg - film from R. Almeida et all., UFRGS, Brazil. XXVIII Encontro Nacional de Física da Matéria Condensada - 10-14 Maio, 2005 p.5/18
Boids I Collective motion without a leader (T. Vicsek, 1995). Defining fixed local rules among individuals may produce emergent non-trivial collective behavior, despite of the local noise. θ t+1 i = arg j i v j + ηξ t i ξ t i [ π, π], η [0, 1] XXVIII Encontro Nacional de Física da Matéria Condensada - 10-14 Maio, 2005 p.6/18
Boids II Once the velocity direction is defined all positions are updated in that direction with a unitary step. i = v i t+1 t + x t i x t+1 v i = v 0 = cte Self-propelling objects, non-equilibrium, no energy conservation. XXVIII Encontro Nacional de Física da Matéria Condensada - 10-14 Maio, 2005 p.7/18
Vicsek s results - I Transition ordered flight to random flight. Order parameter: Average velocity over the population. φ = v v 0 XXVIII Encontro Nacional de Física da Matéria Condensada - 10-14 Maio, 2005 p.8/18
Vicsek s results - II T Vicsek et all., PRL 75 (1995) 1226. XXVIII Encontro Nacional de Física da Matéria Condensada - 10-14 Maio, 2005 p.9/18
Vicsek s Model Generalization θ t+1 i = arg(α j i v j + β j i f i,j + N i η u i ) f ij r o r c r e r a r ij G. Gregoire, H. Chaté and Yuhai Tu, Physica D 181 (2003) 157. XXVIII Encontro Nacional de Física da Matéria Condensada - 10-14 Maio, 2005 p.10/18
Gregoire-Chaté-Tu results - I XXVIII Encontro Nacional de Física da Matéria Condensada - 10-14 Maio, 2005 p.11/18
Gregoire-Chaté-Tu results - II XXVIII Encontro Nacional de Física da Matéria Condensada - 10-14 Maio, 2005 p.12/18
Cell segregation using boids Two kinds of boids, 1 and 2, representing two different kinds of cells. Three different force parameters: β11 > β12 = β21 > β22 and null inertial term α = 0. Typical values: β11 = 75 (solid); β12 = β21 = 40 (solid-liquid); β22 = 30 (liquid). (Film 400 boids ) 11 22 12 21 Β Β Β Β XXVIII Encontro Nacional de Física da Matéria Condensada - 10-14 Maio, 2005 p.13/18
Segregation Time Evolution Four different sample sizes (N) with 1/4 of boids 1 and 3/4 of boids 2: N = 400, 800, 1600, 3200. Measure of the average fraction of equal neighbors less the different ones. γ 1,2 = n eq n diff n eq + n diff XXVIII Encontro Nacional de Física da Matéria Condensada - 10-14 Maio, 2005 p.14/18
Time evolution γ 1 0,4 0,2 0 2400-800 1800-600 1200-400 600-200 300-100 γ 1 = A+B ln (t) -0,2-0,4 10 2 10 3 10 4 10 5 10 6 10 7 t Segregation saturates at the same value for different sample sizes. Before saturation, a log(t) can be fitted somewhere ;(. XXVIII Encontro Nacional de Física da Matéria Condensada - 10-14 Maio, 2005 p.15/18
Time evolution and fluctuations 0,8 0,6 0,4 0,2 0-0,2-0,4 0 γ 1 (100 300) γ 2 (100 300) γ 1 (200 600) γ 2 (200 600) 1 10 6 2 10 6 3 10 6 4 10 6 t XXVIII Encontro Nacional de Física da Matéria Condensada - 10-14 Maio, 2005 p.16/18
Fluctuations and sample size 10-1 <γ 2 > <γ> 2 y x -1 10-2 10-3 10 2 10 3 10 4 N XXVIII Encontro Nacional de Física da Matéria Condensada - 10-14 Maio, 2005 p.17/18
Conclusions There is segregation in a proper parameter range. (Not if all boids are in the liquid phase!) During the growing time segregation seems to follow log(t) but... Saturation at large t is independent of the sample size. Saturation happens well below the ideal value for zero noise. Fluctuations (of γ) scale with the inverse of the system size. Boids do have a path dynamical quantities. No pinning effect, no problems during collisions. XXVIII Encontro Nacional de Física da Matéria Condensada - 10-14 Maio, 2005 p.18/18