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Dynamics of concentrated swimming micro-organisms. John O Kessler. Bacillus subtilis , from individuals to great, concentrated populations: What we see, what we suspect, what we think we know, and at least some of what we ought to know. Physics Dept, University of Arizona, Tucson, AZ

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dynamics of concentrated swimming micro organisms

Dynamics of concentrated swimming micro-organisms

John O Kessler

Bacillus subtilis, from individuals to great, concentrated populations: What we see, what we suspect, what we think we know, and at least some of what we ought to know.

Physics Dept, University of Arizona, Tucson, AZ


DOE W31-109-ENG38; NSF PHY 0551742

ANL SO 2007

movie 1


Martin Bees................. Glasgow

Luis Cisneros.............. Arizona

Ricardo Cortez.............. Tulane

Chris Dombrowski....... Arizona

Ray Goldstein.............. DAMTP


Igor Aronson............... Argonne

Andrey Sokolov........... Argonne



Bacillus subtilis TEM

(near cell division)

Width apprx 0.7mm

Pic by C. Dombrowski

& D. Bentley


The plan: Start with single swimmers, proceed to pairs, small groups, finally arrive at phenomena at high concentration—dynamics, self-organization, modification of themselves and their environment

Note that Re<<1, BUT

boundary conditions that change with flow imply nonlinearity

and irreversibility; blinking Stokeslets. Flow generated by a

swimmer is bounded by other moving swimmers. Swimming

exerts force on the fluid; it is a source of energy (bio to mecha-

nical). The collectively generated flow modifies trajectories.

Constraints affect “behavior” of individual organisms

Flippancy: longitudinal symmetry in propulsion

Swim velocity Vx~ -Ux direction when dUx/dy = 0

Intracellular Brownian fluctuations, AND biochemistry:

polymer exudates, autoinducers modify gene expression

(quorum sensing; + feedback); antibiotics; consumption.

Electrostatics: pH taxis (Sokolov/Aronson)


Transverse flows toward axis of a self-propelled “organism”. This quadrupole-like flow field attracts neighbors and nearby surfaces.


Extending rod/rotating helix



self propelled swimmer
Self-propelled swimmer
  • R(1)V(1)=R(2)V(2)
  • V(1)=velocity of head relative to fluid
  • V(2)=W-V(1)=velocity of tail relative to fluid
  • V(1)=WR(2)/[R(1)+R(2)]
  • W=(helix pitch) X (freq of rotation)


W–V(1)=V(2)=velocity relative to fluid


Elongating rod, rotating helix or whatever, resistance R(2).


Note: “early” computational models of swimmers and the associated flows:

Ramia et al (Biophys Jnl, 1993)

Phan-Thien and Nasseri 1997

(also PhanThien)

and Lighthill...!

and Fauci; Hopkins;....

Single swimming bacterium ~~”trapped” near edgeNote backup(=reversal) 1st collision,turnaround at second
one dimensional traffic chain
One-Dimensional Traffic Chain

Control volume




Force balance:

Helix pitch


Leads to:

Rotation freq.

u= 70 mm/s


f=100/s, l =3 mm , R1 =2R2 , w= 300μm/s, v1 = 30 mm/s:

10 “ “ 30 30

including an efficiency factor = 0.1


signaling by consumption
Signaling by consumption

B. subtilis require oxygen. A population suspended in water, bounded by glass, except at one interface with air, accumu-lates there.

WATER & B. subtilis



Flat glass “microslide”


movie 2_ OxTx.cin
  • movie 3_BlpTrb10.cin
close packing and o 2 consumption
Close packing and O2 consumption
  • Concentration of cells: n ~ 1011 cells/cm3
  • Volume of a cell: v ~ 1.5  10-12 cm3
  • Consumption rate: r ~ 106 mol/seccell
  • Saturation concentration of O2: Cs ~ 1017 mol/cm3
  • On a volume V: Nb = nV; Nm = Cs(V-Nbv)
  • Consumption time:t = Nm/(Nbr) = Cs(1-nv)/(nr)~ 1sec
  • On typical ZBN experiments L ~ 5  10-3 cm
  • Diffusion of O2: D = 210-5cm2/sec
  • Diffusion time:L2D-1 ~ 1 sec
  • Collective velocity: u ~ 5  10-3 cm/sec
  • Advection time:Lu-1~ 1 sec
close packing and o 2 consumption21
Close packing and O2 consumption
  • In the absence of transport, close packed structures consume suspended O2 in seconds
  • Then oxygen is delivered “just in time” for consumption (Pe ~ 1)
  • These time scales also correspond to the lifetime of vortices (sub-collective scale)
What is the distribution of the fluid’s

velocity in the interior of, and around the

periphery of a phalanx?

five self propelled model bacteria note the almost vanishing internal flow
Five self-propelled model bacteria. Note the almost vanishing internal flow


Ricardo Cortez

Tulane, Maths



There is not much flow in the interior. The

push by the flagella is counteracted by the drag of the heads. Fluid is pushed forward by the leading heads, backward by the trailing (propelling) tails, the bundles of flagella.

Do lateral flows stabilize the phalanx?

tail pusher phalanx
Tail pusher phalanx

mistake: a should be ~ 0.0001 cm; 1.8 really = 2

well what about magnitudes
Well, what about magnitudes?

(Analogy with Re)

Ratio of work by n moving “organisms”/volume to the collective shear stress:

The Bs (Bacterial shear) number definition is:


These parameters are typical for the Zooming BioNematic(ZBN)

Note that Bs is not viscosity-dependent

chemical exudates communication
Chemical exudates?Communication?

Also need to consider: Quorum sensing,

diffusion sensing, efficiency sensing.

Biofilm production, crowding out (via

production of antibiotics), topology...

S Park, P Wolanin et al, PNAS 03; B L Bassler (lots); B A Hense et al.,

Nature Reviews Microbiology 2007.

movie 4 (also note Lévy flights and superdiffusion; Kate Remick). movie 5

3 regions zbn bioconvection and dynamically concentrated biofilm
3 Regions: ZBN, bioconvection and dynamically concentrated biofilm

The ~chaotic (ZBN) transport region sweeps

auto-connected groups of bacteria (biofilms)

away from the “action” = “upwards”, into

deeper region. Similarly, in deeper layers of

fluid, dominated by bioconvection + ZBN,

that dynamic also concentrates the biofilm,

now downwards, toward shallower fluid.

movie 6

Can collective efforts alter the environment? Moving grains, pasting them together ? (Az “desert” surface)
  • gravity + oxygen taxis in natural environment?

movie 7