Transport and collective dynamics in suspensions of swimming microorganisms M. D. Graham Patrick Underhill, Juan Hernandez-Ortiz, Chris Stoltz Dept. of Chemical and Biological Engineering Univ. of Wisconsin-Madison
Flowing Complex Fluids Research Group Collective motions of self-propelled agents: flocks, herds, schools Freefoto.com www.tanzaniaparks.com www.eeb.uconn.edu?
Flowing Complex Fluids Research Group Some microscopic swimmers B. subtilis E. coli (http://textbookofbacteriology.net/bacillus.html) (www.agen.ufl.edu/~chyn) catalytic nanorod Human sperm (http://uuhsc.utah.edu/andrology/photo_gallery.html)v (Kline et al, Angew. Chem. 2005) Chlamydomonas (http://remf.dartmouth.edu/images/algaesem/source/2.html) Microbot (www.sciencemuseum.org.uk/antenna/nano/seeforyourself/111.asp)
Collective swimming in bacterial suspensions: experimental observations Kessler, Goldstein et al. 04 B. subtilis in sessile and pendant drops µpiv data 1 cm Sessile drop: chemotaxis to upper (oxygen-rich) surface generates unstable density gradient ==> bioconvection Pendant drop: gravitationally stable, but large scale motions persist
Collective swimming in bacterial suspensions: experimental observations Weibel group, UW E. coli on agar Wu & Libchaber PRL 00 E. coli in a soap film, 10 µm tracers Large scale motions Enhanced transport Molecular diffusivity for 4 and 10 µm tracers
Collective swimming in bacterial suspensions: experimental observations Monolayers of B. subtilis in soap film dilute concentrated real time slowed 10x Aranson et al, 2006
Do krill mix the ocean? An organism of a given body size! cannot inject energy into a flow at length scales larger than itself. Observation: dramatically increased turbulence intensity during dusk ascent of krill. The mechanism by which small krill can create turbulence with scales of 1 to 10 m remains a mystery.
Studies of populations of swimmers Experiments Kessler, Goldstein, Libchaber, Tang, Breuer, Wu, Yodh, Phenomenological theory Ramaswamy, Aranson, Marchetti, Lubensky, Micromechanical theory Shelley, Koch, Particle-level simulation and theory Pedley, Ishikawa, Nott, Shelley,Cates, Beryland,
What principles underlie swimming of populations of microorganisms? fluid mechanical framework scaling ideas for velocities and diffusivities theory of velocity correlations simulations in periodic and slit domains Can small scale swimming generate large scale correlations?
Simple swimmer model Pushers (p=-1) Pullers (p=+1) Body: bead-rod dumbbell/chain, length,, bead friction Phantom flagellum exerts equal and opposite forces on body and fluid along axis of body Excluded volume Low-Re hydrodynamics: regularized point-forces (Hernández et al. PRL 2007) Neutrally buoyant free swimmer is a force dipole effective volume fraction: B. subtilis (http://textbookofbacteriology.net/bacillus.html) Chlamydomonas (Algae) (www.ucmp.berkeley.edu) isolated swimmer speed: dipole strength: Hernández et al PRL 2005, cf Ishikawa & Pedley 2006, Saintillan and Shelley 2007,
Fast method for point particles in Stokes flow Near-field details easy to regularize Exponentially decaying (short range) Easy to calculate, i.e. analytical expression O(N particles ) Long-range interactions Correct boundary conditions at walls, Solution through O(N mesh ) CFD method Generalizes to Stokesian dynamics, immersed boundary method Jendrejack et al 03, Hernandez et al 07, Anekal et al. 08
Hydrodynamic interactions matter simulations of monolayers of dumbbell pushers slit -- no slip walls Steric interactions only Steric and hydrodynamic interactions Hydrodynamic interactions suppress: polar nematic ordering large density variations (flocking)! consistent with expt
Simple dilute theory: velocity and diffusion velocities Tracers tracer velocity is sum of disturbances due to each swimmer assuming independent swimmers Swimmers swimmer velocity doubles by " e =0.1 diffusivities Smooth: like dilute gas Run& tumble Simulations follow dilute scalings for " e <10-2
Simple dilute theory: comparison to expt. swimmers Leptos et al. 2009: Chlammy (pullers) fluorescent tracers tracers Tracers tracer velocity is sum of disturbances due to each swimmer assuming independent swimmers Swimmers Smooth: like dilute gas Run& tumble tracer diffusivity
Simple dilute theory: correlations Homogeneous isotropic distribution of point force dipoles in Stokes flow Spatial Fluid Structure independent point dipoles characteristic fluid velocity Even independent swimmers produce long ranged spatial correlations in the fluid Underhill et al PRL 2008
Stress: Simple dilute theory: correlations Stress autocorrelation: Stress fluctuation spectrum For dilute isotropic, homogeneous run-and-tumblers (Poisson process, mean run time # r =(6D r ) -1 ): Chen et al. 2007: stress fluctuations in dilute E. coli suspensions are not Lorentzian: stochastic behavior within runs? Underhill et al PRL 2008
Finite concentration Simulation: mean square displacements Trajectories transition from ballistic at short lag times to diffusive at long lag times " e = 0.1 tracer trajectory pushers +1 +2 pullers independent Diffusivity changes with system size only for pushers Underhill et al 2008
System size dependence: pushers velocity and tracer diffusivity tracer correlation time velocity diffusivity N=51200 puller diffusivity N=100 Divergence with increasing box size? Underhill et al 2008
Fluid spatial correlations Spatial structure in the fluid also increases N=3200 pushers independent pushers independent theory pullers increasing system size Pushers drive long-range, L-dependent velocity correlations Underhill et al 2008
Velocity correlation for interacting swimmers independent swimmers: O(" e ) swimmer correlations: O(" e2 ) Point dipole approximation, large L limit Pushers and pullers have different h (and possibly g too), which leads to different fluid correlation h is an orientational correlation function: correlations are interesting only when it is nonuniform Underhill et al 2009
Simple model for spatial and orientational correlations Hard sphere gas: Decaying orientation correlation: Velocity correlation explicitly system-size-dependent Logarithmic singularity of velocity fluctuations with L Small swimmers generate large scale motion
Theory and simulations Velocity fluctuations N=6400, " e =0.1 Correlation function pushers independent Apparent logarithmic singularity Theory gives correct structure for correlations
Polymer solution d>=0 Why are pushers different? Simple mechanism Puller suspension d>0 Pusher suspension d<0 Underhill et al 2008
Why are pushers different? mean field theory Linear stability analysis of homogeneous isotropic state Allow for rotational diffusion D r of swimmers Growth rate in long wave limit: Pure shear, no conc. flucs Comparison: prepare homogeneous isotropic state, impose small rapid strain, follow time evolution MFT works quantitatively for pullers, not so well for pushers N=25600, " e =0.1 k=2$/l pushers pullers Underhill et al 2008, cf Ramaswamy 2004, Saintillan & Shelley 2007, Koch 2009
Summary Experiments indicate collective dynamics and large scale motions in suspensions of swimming microbes A simple model reproduces key features of the experiments: large scale flows, enhanced transport hydrodynamic interactions are important Many phenomena have been neglected: near field HI, details of propulsion, tumbling, Current work: run-and-tumble bacteria stress fluctuations/microrheology theory for orientation correlations confinement comparisons to experiments coupling to chemical transport -- chemotaxis Biological & technological implications of collective swimming? Implications of transport differences between pushers and pullers?
Collaborators/support Juan-Pablo Hernandez Patrick Underhill Chris Stoltz Prof. Doug Weibel NSF: CTS/DMS, UW-NSEC