Myxobacteria - To Collide or Not To Collide?
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Myxobacteria - To Collide or Not To Collide
Collaboration with
Or: The Search for Collision Invariants
Christian Schmeiser, University of Vienna
Pierre Degond, Imperial College, London
Hui Yu, Imperial College, London
C o l l i s i o n B a s e d A p p r o a c h
Collision Based
Quadratic in f
Align
Collision Based
Biology
The behavior of the macroscopic quantities is described by
Collision Induced
Mass balance for (+) group
Reverse
Alignment
Mass balance for (-) group
Reversal
O. Sliusarenko, J. Neu, D. Zusman, Accordion waves in Myxococcus xanthus, 2006
1. Characterize
2. Extract information from Transport term
v dependence
WISH
CIs
The Kernel of
Tasks
1. Characterize
2. Extract information from Transport part
These are macroscopic quantities!
Reversal term could be added easily
PROBLEM!
We only have one Collision Invariant, but we need 3!
Mass conservation
Make special assumptions on Initial Conditions
SOLUTION?
Some bacteria here
Particle Model
(Micro)
Hydrodynamic Model
(Macro)
Kinetic Model
(Meso)
Now we get three Collision Invariants, i.e. Conserved Quantities
- Mass in upper half
- Mass in lower half
- Average angle
No bacteria here
Some bacteria here
x
v-space
Not very satisfying!
x, t dependence
Collision Based
1. Characterize
2. Extract information from Transport/Reversal term
Collision Based
WISH
CIs
v dependence
These are macroscopic quantities!
PROBLEM!
We only have one Collision Invariant, but we need 3!
Mass conservation
The Kernel of
SOLUTION?
P. Degond, S. Motsch. Continuum limit of self-driven particles with orientation interaction. Mathematical Models and Methods in Applied Sciences. 2008.
ORIGINAL
NEW
Tasks
1. Characterize
2. Extract information from Transport/Reversal part
Again: These are macroscopic quantities!
DOES IS WORK? YES!
x, t dependence
Are GCIs, since
With some more work we can find explicitly a third GCI,
What is the connection between the two approaches?
On the macroscopic level
From Mean Direction Based to Collision Based
Proof: Uses Laplace Method to determine the asymptotic behavior of certain integrals
Mean Direction Based
Lemma:
Myxobacteria
Mean Direction Based
Mass balance for (+) group
Mean Direction
The behavior of the macroscopic quantities is described by
Align
Mass balance for (-) group
Mean Direction
Alignment
Diffusion
Reverse
Mean Direction Based
The constants
depend on the diffusion-constant D and on the GCIs
Mean Direction
Diffusion
Alignment
Reversal
Reversal Rate
Reversal
Vicsek-type Model
T. Vicsek, et al. Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 1995
M e a n D i r e c t i o n B a s e d A p p r o a c h
Mean Direction Based
Simulation of the Macro Model
What can we say analytically about the Macro-Model?
Mean Direction Based
What does the Particle Model actually do?
Lemma: The Macro-Model is hyperbolic
Some preliminary results about the stability of waves
Analytical Results
Simulation Results
Talk available on homepage.univie.ac.at/angelika.manhart/