Emergent Collective Behaviour in Volvox

Home , Volvox

Raymond E Goldstein

Department of Applied Mathematics and Theoretical Physics

University of Cambridge

www.damtp.cam.ac.uk/user/gold www.youtube.com/Goldsteinlab The Size-Complexity Relation

Amoebas, Ciliates, Brown Seaweeds Green Algae and Plants Red Seaweeds Fungi Animals

Bell & Mooers (1997) ? Bonner (2004) The Recent Literature

Phil. Trans. 22, 509-518 (1700)

(1758) Green Algae as Model Organisms Multicellularity Flow Fields Hydrodynamic Bound States

Cytoplasmic Streaming Tracer Statistics

Synchronisation

Phototaxis

REG, et al., Annual Review of Fluid Mechanics (2014) Advection & Diffusion If a fluid has a typical velocity U, varying on L a length scale L, with a molecular species of tadvection = diffusion constant D. Then there are two times: U We define the Péclet number as the ratio: L2 tdiffusion = tdiffusion UL D Pe = = This is like the Reynolds tadvection D number comparing UL inertia to viscous dissipation: Re = ν If U=100 µm/s, L=10 µm, Re ~ 10-3, Pe ~ 1 At the scale of an individual cell, diffusion dominates advection.

The opposite holds for multicellularity…

Solari, Ganguly, Michod, Kessler, Goldstein, PNAS (2006) Short, Solari, Ganguly, Powers, Kessler & Goldstein, PNAS (2006) Volvox In Its Own Frame

Tracking microscope in vertical orientation Laser sheet illumination of microspheres

Drescher, Goldstein, Michel, Polin, and Tuval, PRL 105, 168101 (2010) Rushkin, Kantsler, Goldstein, PRL 105, 188101 (2010)

Microscopy & Micromanipulation

micro- manipulator micro- manipulator Volvox on a Micropipette Life Cycles of the Green and Famous division maturation inversion of gonidia

hatching of juveniles

cytodifferentiation and expansion Systematics of Volvox

Upswimming speed Spinning frequency

HOVERING

Settling speed

Reorientation time Phototaxis Volvox Eyespots

Top view at anterior pole anterior at view Top Planar Cell Polarity in Volvox carteri

Hugo Wioland The Mathematics of Turning angular direction axis surface surface velocity of gravity direction normal fluid velocity 1 3 Ω(t) = gˆ ×kˆ − nˆ ×u(θ,φ,t)dS 3 ∫ τ bh 8πR Based on Reciprocal Theorem bottom-heaviness (Stone & Samuel) relaxation time

In the Volvox frame of dIˆ reference, light direction = −Ω×Iˆ evolves according to: dt Adaptive Flagellar Dynamics and the Fidelity of Multicellular Phototaxis

Drescher, Goldstein, Tuval, PNAS 107, 11171 (2010) See also Ueki, Matsunaga, Inouye, and Hallmann (2010) Step Response of Flagellar Beating Dynamic PIV Measurements – Step Response Adaptive, two-variable model (s − h) − p p = τ r s − h h = τ a

p=“photoresponse” amplitude h=“hidden” biochemistry

Adaptive dynamics also play u = u0 (1− β p) a role in sperm chemotaxis: Friedrich and Jülicher (2007,09) Simple modulation of flow Angular Dependence of the Transient Response

anterior is sensitive posterior is not

u(θ,t) = u0 (θ )[1− β (θ ) p(θ,t)] Frequency-Dependent Response

Two-variable model

Data

ωτ R(ω) = a 2 2 2 2 1/ 2 [(1+ω τ r )(1+ω τ a )] Peak of frequency-response coincides accurately with the range of rotational frequencies within which accurate phototaxis occurs: TUNING Multicellular Phototaxis as Dynamic Phototropism

Light direction Reduced model A Test of the Theory Interacting Volvox Dual-View Apparatus Free of Thermal Convection

White LED & shutter

Capable of imaging protists from 10 μm to 1 mm, with tracking precision of ~1 micron, @ 20 fps.

Drescher, Leptos, Goldstein, Review of Scientific Instruments 80, 014301 (2009) Walzing Volvox: Orbiting “Bound State”

Drescher, et al. PRL (2009) Dual Views with PIV Model for Mutually-Advected Stokeslets

x i = u(xi ) + vi 1 1 p = p ×(zˆ ×p )+ (∇×u)×p i τ i i 2 i

Blake (1971): Flow field of a Stokeslet near a no-slip wall Squires (2001): Attractive interaction for spheres falling away from a wall

r tF dx 3 x = τ = = πη + = − x ; 2 ; F 6 R(U V ) 5/ 2 h ηh dτ π (x2 + 4) Formation of the Bound State

K. Drescher, K. Leptos, T. Ishikawa, T.J. Pedley, R.E. Goldstein (preprint) Numerical Studies (Bottom-Heavy Squirmers with Swirl)

Drescher, et al. PRL (2009) The Minuet Bound State

Side view

Chamber bottom Simplest Model of Minuet Bound State

Assumptions of a simple model: 1. Volvox hover 2. Each Volvox produces a Stokeslet 3. When the Volvox axis k is tilted from vertical, stokeslets move in horizontal direction 4. Direction of Volvox axis changes as given by Pedley & Kessler (1992)

The Diffusional Bottleneck

Metabolic requirements scale with surface Diffusion to an somatic cells: ~R2 absorbing sphere

Currents  R  2 C = C∞ 1−  Im = 4π β R  r 

Id = 4πDC∞ R

2- PO4 and O2 estimates yield bottleneck radius ~50-200 µm (~Pleodorina, start of germ-soma differentiation)

DC Organism radius R R = ∞ b β Flagellar-Driven Flows and Scaling Laws Specified shear stress f at surface  R3  ∞  = −  −  θ − θ ur U c 3 P1(cos ) ∑ Al (r)Pl (cos )  r  l=2   R3  ∞  = −  +  1 θ + 1 θ uθ U d 3 P1 (cos ) ∑ Bl (r)Pl (cos )  2r  l=2 

π f R U = 8η

Short, Solari, Ganguly, Powers, Kessler & Goldstein, PNAS (2006) Metabolite Exchange  2 Acrivos & Taylor (1962) u ⋅∇c = D∇ c heat transport from a solid sphere: current ~ RPe1/3 Magar, Goto & Pedley (2003) prescribed tangential velocity in a model of “squirmers” 2 1/ 2 2Ru  R   4ηD  Pe = θ ≈   ; R =   ≈ 10µm

2 ∂C ∂ C Boundary layer scale: ur ≈ D 2 ∂y ∂y −1/ 2 ε UR  ε C C ~   ~ Pe−1/ 2 U ≈ D R  D  R ε ε 2 The Peclet number scales as: 2 1/ 2 2Ru  R   4ηD  = θ ≈   =   ≈ µ < Pe   ; Ra   10 m Rb D  Ra   π f  Bottleneck Bypassed (!) ∂C C = − 2 Ω ≈ π 2 ∞ ≈ π 1/ 2 2 Ia DR ∫ d 4 DR 4 DC∞ RPe ~ R (!) ∂r Ra Advantage of Size Collaborators Postdocs: Faculty: Marco Polin (to Warwick) Cristian Solari – U. Buenos Aires Idan Tuval (now Mallorca) Adriana I. Pesci – DAMTP Kyriacos Leptos Timothy J. Pedley - DAMTP Vasily Kantsler (to Warwick) John O. Kessler – Arizona Jorn Dunkel (to MIT) Rick Michod – Arizona Ph.D. students: Jerry P. Gollub – Haverford/Cambridge Sujoy Ganguly (now Dresden) Jeffrey S. Guasto – Haverford/MIT Knut Drescher (now Princeton) Staff: Kirsty Wan David Page-Croft G.K. Batchelor Douglas Brumley Colin Hitch Lab (DAMTP) Visiting students: John Milton Nicholas Michel (Ecole Poly.) Silvano Furlan (Pisa)