Cellular motility: mechanical bases and control theoretic view

Transcription

1 Cellular motility: mechanical bases and control theoretic view A. DeSimone SISSA International School for Advanced Studies, Trieste, Italy Joint work with F. Alouges, A. Lefebvre, L. Heltai PICOF, -4 April, Ecole Polytechniue

2 Motility Motility is the capability of exhibiting directed, purposeful movement. Motile cells provides fascinating examples of motility at microscopic scales (-5 µm) Tumor cells (crawling on a solid surface) Sperm cells (swimming in a fluid) Bacteria (swimming in a fluid)

3 Need better understanding of underlying mechanisms Understand and manipulate mechanisms and interactions with the surroundings to enhance (infertility) or reduce (metastasis) motility Engineer self-propelled artificial micro-motile systems (nano-robots inside the human body for diagnostics and therapy) We are currently unable to obtain self-propulsion at microscopic scales artificially. Maybe we can learn from Nature? (bio-mimetic or bio-inspired design) Bio-mimetic or bio-inspired design, even though using Nature as a template is sometimes naive (airplanes don t flap their wings) Focus on micrometer-scale swimmers for the rest of the talk. 3

4 Swimming: definition the ability to advance in water by performing a cyclic shape change (a stroke) in the absence of external propulsive forces 4

5 Microscopic bio-swimmers: bacteria and cells Euglenids Metaboly in Eutreptiella sp. Escherichia Coli (E. Coli) 5

6 Man made micro-swimmers: micro-robots red blood cell + flexible magnetic filament polymer film + muscle cells al. et H. Stone, Nature (5) al. et G. Whitesides, Science (7) 6

7 Reynolds Number (Re) Velocity (typical order of magnitude) V Re VLρ η Diameter (typical length scale) Mass density of the fluid Viscosity of the fluid L ρ η For water at room temperature ρ/η 6 (m s - ) -. Re is a dimensionless measure of relative importance of inertia vs. viscosity Orders of magnitude for swimmers: Men, tuna, sharks: Lm, V- ms - Re 6-7 Bacteria: Lx -6 m, V-x -6 ms - Re Neglect all inertial effects. Surrounding fluid modeled by Stokes, rather than by Navier-Stokes euations. 7

8 Life at low Reynolds numbers in a flow regime obeying Stokes euations, a scallop cannot advance through the reciprocal motion of its valves whatever forward motion will be produced by closing the valves, it will be exactly canceled by a backward motion upon reopening them G.I. Taylor, 95 (eukariotic flagella) (cilia) H. Berg, 973 (bacterial flagella) 8

9 How to beat the scallop theorem with minimal artificial swimmers The three-link swimmer of Purcell (977) PurcellSwimmer.flv The three-sphere swimmer of Najafi and Golestanian (4) 9

10 Theory: two conceptual ingredients Assume that shape changes are prescribed over time (data of the swimming problem). It s a simplifying assumption (How is the shape change driven? Molecular motors).. How does the surrounding medium react (namely, which forces F shape does it exert) to shape changes of the swimmer? (euations of motion of the surrounding medium). How does the swimmer move in repsonse to the forces that the surrounding medium applies to it? (euations of motion of the swimmer)

11 Outer Stokes problem Find u induced in the fluid surrounding swimmer Ω by its shape changes: Ω Ω u v Dir on Ω v Dir σ n DN Ω [v Dir ] viscous reactive force p.u. area on Ω Uniue solution u for any given v Dir, nice variational structure Not all of v Dir is a-priori known (only shape is given, not position)

12 Euation of motion Self-propulsion The motion of the swimmer induced by its shape changes is a-priori unknown. Euations of motion: m d dt (x c)f tot F tot neglecting swimmer s inertia F ext + F visc F visc self-propulsion Similarly, total viscous torue. Six ODEs to determine six scalar unknowns (translational and orientational velocities)

13 Mathematical swimming in a nutshell ξ ξ c 3 x (i) 3 i ξ ξ () () x x (3) () x x or c x () shape position Swimming is... ξ (t), ξ (t) periodic of period T executing cyclic shape changes producing Δc T cdt net positional change after one period under f () + f () + f (3) no external force (self-propulsion) 3

14 Positional change from shape change (axisym.) i ξ ξ v Dir Dirichlet data on Ω DN ξ [v Dir ]σ n surface force on Ω Ω i DNξ [vdir ] da ϕ ξ, c/ ) ξ + ϕ ( ξ, c/ ) ξ + ϕ (,/ c) c by transl. inv. ( 3 ξ Then c is uniuely determined and depends linearly on and ξ ξ dc dξ d V ξ ( ξ ) + V( ξ) dt dt dt nonlinear dep. on ξ! V i ( ) ϕ ( ξ)/ ϕ ( ξ) ξ i n+ 4

15 Swimming with one formula Δc T T dξ dξ cdt ( V + V ) dt dt dt (notice that Δc is entirely geometric) ω curl V dξ dξ Δc Δc ξ ξ c space of states ξ ω (ξ (t),ξ (t)) (ξ,ξ,c ) ξ space of shapes ξ ξ (ξ,ξ,c ) space of shapes Swimming rests on the differential form V dξ + V dξ not being exact V(ξ) summarizes hydrodynamic interactions and swimming capabilities of a swimmer Swimming Controllability ; Optimal Swimming Optimal Control Need to solve infinitely many outer Stokes problems to compute V(ξ)!!! 5

16 6 Controllability u ξ u ξ ) ( ) ( u V u V c ξ ξ ) ( ), ( ), ( : : : i i u i g u V u V c dt d ξ ξ The three-spere swimmer is globally controllable (Lie-bracket generating or totally nonholonomic control system) ), ( curl ),, ]](, [ det[ 3 V g g g g ξ ξ An affine control system without drift c 3 ξ ξ ε [g,g ] -ε g ε g ε g -ε g ), ( curl ) ( ) ( ], [ V g g g g g g Revisit scallop thm.

17 Swimming at max efficiency Rescale to unit time interval. Minimize expended power Ω σ n at given v dt Δc Ω DN ξ [ v ] v dadt G( ξ) ξ ξ dt Dir V ( ξ) ξ ω Dir curl V ( ξ) dξ stroke length enclosed area Euler-Lagrange e. d dt @ G! + curlv ( )? c (ξ,ξ,c ) ξ Optimal strokes exist (sub-riemannian geodesics). They can be computed numerically. ξ (ξ,ξ,c ) 8

18 Three sphere swimmers: a race RACE 9

19 Motility maps and optimal stroke ξ ξ ξ ξ Level curves of curl V(ξ,ξ ) Motility map Optimal stroke maximizes flux of curl V at given energy input, or minimizes power consumption at given flux of curl V

20 Summary for low Re swimming motility - Self-propulsion yields the key euation(s) giving positional change from shape change - Swimming problem as a problem of controllability, Lie-bracket generating swimmers - Optimal strokes of swimmers computable (solve many PDEs), subriemannian geodesics - The picture is ualitatively similar for general swimmers: metaboly of Eutreptiella - What is the biological function of metaboly (Swimming? Is it optimized?) - How is this motion accomplished (motor?)

21 Actin polymerization at the leading edge drives motility of crawling cells Long range order at the µm scale of the cell (the front is flat) emerging from uncoordinated growth at the nm scale of the individual filaments. How is this possible? Hypothesis: self-organized growth is orchestrated by long range mechanical interactions mediated by the membrane. (metastatic tumor cells, immune system response,.) L. Cardamone et al. PNAS, 8: 3978 () Motility at microscopic scales.. Antonio DeSimone, SISSA (Trieste, ITALY) 4

22 References on swimming motility D.A. Fletcher and J.A. Theriot, An introduction to celluler motility for the physical scientist, Phys. Biol., T-T (4). E.M. Purcell: Life at low Reynolds numbers, Am. J. Physics 45, 3- (977). S. Childress: Mechanics of swimming and flying, Cambridge University Press, 98. E. Lauga and T.R. Powers, The hydrodynamics of swimming microorganisms, Rep. Prog. Phys (9). G.I. Taylor, Analysis of the swimming of microscopic organisms, Proc. Roy. Soc. A 9 (95). A. Najafi and R. Golestanian: Simple swimmer at low Reynolds numbers: three linked spheres, PRE 69, 69 (4). A. Shapere and F. Wilczeck: Self-propulsion al low Reynolds numbers, PRL 58, 5 (987) G. Galdi, A. Bressan et al., M. Tucsnak et al., A. Munnier et al., R. Montgomery, J. Marsden et al. F. Alouges, A. DeSimone, A. Lefebvre: Optimal strokes of low Reynolds number swimmers: an example. Journal of Nonlinear Science 8, 77-3 (8). F. Alouges, A. DeSimone, A. Lefebvre: Biological fluid dynamics. Springer Encyclopedia of Complexity and Systems Science (9). F. Alouges, A. DeSimone, A. Lefebvre: Optimal strokes of axisymmetric microswimmers. European Physical Journal E 8, (9). F. Alouges, A. DeSimone, L. Heltai: Numerical strategies for stroke optimization at low Re numbers. M3AS, (). G. Dal Maso, A. DeSimone, M. Morandotti: An existence and uniueness result for the selfpropelled motion of micro-swimmers. SIAM J. Math. Analysis 43, (). 5