繊毛 鞭毛運動の生体力学シミュレーション 石川拓司 東北大学大学院工学研究科

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1 繊毛 鞭毛運動の生体力学シミュレーション 濱田 T 石川 G 石川拓司 東北大学大学院工学研究科

2 Today s Topics Ciliate Bacteria Red Blood Cells Slender body Fluid mechanics Fluid-solid interaction Mechanics of ciliary flow and motion

3 Mechanics of Biological Cells Deformation, Swimming, Pumping Red Blood Cell: Deformation Cilia, Flagella: Swimming Pumping Volvox Suresh et al (2005) 2 Bacteria: Swimming w x wave Airway E. coli body W

4 Hydrodynamics of a Single Cell Flow field Characteristic length (cell size): 1-100mm Characteristic velocity: RBC : radius x shear rate Micro-organism: velocity 1-10 body length / sec Re = Stokes flow (Inertia-free) Force-Torque condition of a cell Force is almost free Torque is not free for algae cells e x c x g bottom-heaviness g

5 Paramecium caudatum Approximately 250mm in length 50mm in width Ciliate: Paramecia Experimental Setup Digital video camera Light source Light source Macro lens Inner dish Paramesium swim between flat plats with 70mm gap. Test fluid Outer dish 70mm

6 How to model Paramecium? Squirmer model assumed to propel itself by generating tangential velocities on its surface. Surface velocity is given as a B.C. Velocity field around Paramecium 1.5 PIV u s experimental results

7 Flow Field: Boundary Element Method u i Numerical Methods Paramecium: Force-free, Torque-free Ishikawa et al., J. Fluid Mech. (2006) ( x) u i ( x) 1 8 m N 1 A J ij ( x y) q j ( y) da q : single-layer potential A : surface of a particle u : velocity J : Green function y

8 Comparison between Exp & Sim Paramecium caudatum Squirmer model Boundary Element Method Ishikawa et al, J Fluid Mech (2006) Ishikawa & Hota, J Exp Biol (2006)

9 Bacterial Model Escherichia coli is used as a model bacterium In experiments: E. coli : MG1655 (wildtype) In simulations: - spheroidal body + single flagellum - swims by executing a helical wave down its flagellum

10 Numerical Methods Boundary element method N 1 u ( x) u ( x) i i 8 m 1 Basic equations J : Green function for a half-space 680 triangle elements per bacterium Flow Field : Stokes flow Bacterial Motion : force-free torque-free B.C. : relative rotational velocity between the cell body and flagella Ishikawa et al. Biophys. J (2007) A J ij ( x y) q j ( y) da y u : velocity q : single-layer potential A : surface of a particle

11 Comparison between Exp. & Sim. Trajectories of a real E. coli and its model. Experiment Simulation Agree quantitatively Giacche et al., Phys. Rev. E (2010)

12 Flagella Motions Flagella bundling Bundling Tumbling E. coli swimming (H. Berg) Synchronization Mechanism: Fluid mechanics Kanehl & Ishikawa, Phys. Rev. E (2014)

13 Basic Equations for RBC Fluid Mechanics : Stokes flow A v m in n m out v : velocity q : traction force n : outward normal vector J, K : Green functions : viscous ratio Solved by a Boundary Element Method. Foessel et al., J. Fluid Mech. (2011)

14 Basic Equations for RBC Solid Mechanics : 2D Hyperelastic Material Weak form of the virtual work principle :virtual displacement, strain q : load t : tension 5120 linear triangular mesh Constitutive law: Skalak law Reference shape: RBC Solved by a Finite Element Method. Walter et al., J. Fluid Mech. (2011)

15 Oblate Capsule Jeffery Orbit: Rigid Oblate # Trajectory is invariant under time reversal Ca = 0.3 # Reorientation does not occur under simple shear flow Oblate Capsule Ca = 1.0 Tracer initially placed on the revolution axis Omori et al., Phys. Rev. Lett. (2012)

16 Single RBC Mechanics Phase diagram of RBC motion (Omori et al., Phys. Rev. Lett., 2012) Tank treading Swinging Tank treading Ca Transition Swinging l Tumbling

17 Comparison with Exp Comparison with Former Experiments Tank treading Tank treading Fischer et al., 1978 Swinging Ca Transition Tumbling Swinging Abkarian et al., 2007 Good Agreement l

18 Summary Ciliate Bacteria Red Blood Cells Mechanics of ciliary flow and motion