CosmicMolecularQuanta

Does the inertia of a body depend on its energy content? Annalen der Physik, 18(1905), pp E = m c 2 On the electrodynamics of moving bodies (special relativity) Annalen der Physik, 17(1905), pp

"On a heuristic viewpoint concerning the production and transformation of light." (light quantum/photoelectric effect paper) (17 March 1905) Annalen der Physik, 17(1905), pp

"On the motion of small particles suspended in liquids at rest required by the molecular- kinetic theory of heat." (Brownian motion paper) (May 1905; received 11 May 1905) Annalen der Physik, 17(1905), pp

Brownian motion due to randomly directed impulses from collisions with thermally excited molecules Random walks and diffusion K = Boltzmann's constant 1.38  g cm 2 s -2 o K -1 bacteria colloids a ≈ 1 µ D ≈ to cm 2 /s T = temperature in o K µ = viscosity Leads to diffusion

More importantly : Shows how the kinematic* of small scale motion leads directly to macroscale properties *Kinematics: how things move Dynamics: why things move Random walk Diffusion Molecular Turbulence Motility of organisms Random walks and diffusion Microscopic Macroscopic Property of "individual" Property of "population"

x1x1 x2x2 xnxn X = x 1 + x x n Each change in direction is totally random  x i  = 0  X  = 0 Ensemble average For 1 realization The average position does not change Random walks and diffusion: theory Speed v Run duration 

Diffusion defined by If  is exponentially distributed then Random walks and diffusion: theory Uncorrelated = 0 Number of steps n = Total time t / run duration 

 Each path segment is of lenght v dt Particle moves with speed v Its position is updated every time step dt The probability of changing direction in a time step is Random walks and diffusion: modelling

Random walk of a single particle

Random walks of a cluster of particles

Random walks of a cluster of particles: diffusion

Motility of organisms erratic sinuous helical hop - sink some element of randomness

Motility of organisms: flagellate Bodo designis

Motility of organisms: adult copepod temora longicornis

How to analyse these swimming paths ? NGDR (net to gross distance ratio) Fractal dimension (  ) Diffusive analog of motility Ciliate Balanion comatum l L l  L 1/  NGDR = l / L l  (2Dt) 1/  (fractal dimension 2)

Diffusion by continuous motion GI Taylor Diffusion by continuous motion

 11 x1x1 xnxn x2x2 x3x3 XnXn  a measure of how fast the path becomes de-correlated with itself Diffusion by continuous motion

G.I. Taylor (1921); Diffusion by continuous random (Brownian) motion l 2 = 2 (L –  (1 – e –L/ )) Diffusion by continuous motion l 2 = 2 v 2  (t –  (1 – e –t/  )) or equivalently Diffusion by continuous motion

l L l L log(t) log(l) slope = 1 slope = 1/2  l = v t for t <<  Ballistic l = (2 v t) 1/2 for t >>  Diffusive Diffusion by continuous motion

Long term stochasticity Diffusion by continuous motion Short term coherency

Ciliate Balanion comatum Jakobsen et al (submitted) Visser & Kiørboe (submitted) v = 220 ± 10  m/s  = 0.3 ± 0.03 s = 65 ± 10  m D = 50 ± 10 x cm 2 /s Example similar motility for organisms from bacteria to copepods Diffusion by continuous motion

Motility length scale Swimming speed v Capture radius R Ambush predator feeding on motile prey simple example Two encounter rate models !! What does this mean for encounter rates ?

v R Gerritsen & Strickler 1977 Rothschild & Osborn 1988 Evans 1989 Randomly directed prey swimming ambush non turbulent Encounter rate: Ballistic model

Ballistic model variations

Gerritsen & Strickler 1977 Ballistic model variations moving prey

Rothschild & Osborn 1988 Ballistic model variations turbulence

Evans 1989 Ballistic model variations gaussian distribution

Z(t  ) = 4 / 3 C  R v Random walk diffusion Diffusion equation R c(R) = 0, c(r  ) = C time dependent steady state Encounter rate: Diffusive model

Ballistic Diffusive quantitatively and qualitatively different asymptotic limits scale of the process under consideration Two encounter rate models

Case a Case b /R = 0.1 /R = 2 Numerical simulation: basic setup Swimming speed and detection distance remain the same, only the tumbling rate changes Count the number of particles that are encountered each time step Those that are encountered are set to inactive Cyclic boundary conditions Reactivated when crossing boundaries

t(s) t  (mm 3 /s) ballistic diffusive(∞) diffusive(t) experimental /R = 0.1 /R = 2 Visser & Kiørboe (submitted) Clearance rate = Z/C Numerical simulation: results Case a Case b

Encounter rate cannot be faster than ballistic Ballistic – Diffusive transition (meso-diffusion) when R ≈ Maxwell – Cattaneo equations in 1 D Telegraph equation Kelvin 1860's Unsolved problem in 3 D, even for simple geometries Consequences for plankton motility ? Ballistic – Diffusive encounters

predator prey ballistic diffusive mean path lengthR prey R predator << Visser & Kiørboe: Fig 5 Is there an optimal behaviour for organisms ? predator to prey scaling ≈ 10:1

Visser & Kiørboe (submitted) Size of organism: esd (cm) motility length scale (cm) Marine Bacterium TW-3 Microscilla furvescens Bodo designis Spumella sp. Herterocapsa triquetra Balanion comatum Acartia tonsa Centropages typicus Temora longicornis Calanus helgolandicus protists copepod nauplii bacteria adult copepods = 7 d (r 2 = 0.90) Motility length vrs body size

Summary statements Random walk models are an simple way to link the behaviour of individuals to macroscopic effects at the population and environmental level. Continuous random walks appear to be a good model for the motility of plankton. Ballistic – Diffusive aspects of encounter processes are important. The wrong model can give wildly different estimates of encounter rate. Unanswered problem in classical physics Top down control on planktonic motility patterns ?