Chemotaxis and Random Motility:

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Chemotaxis and Random Motility A Computational
Study
Cameron F. Abrams Ehsan Jabbarzadeh Department
of Chemical Engineering Drexel University Philadel
phia, PA USA
16 November 2004 Philadelphia Computational
Orchestra Greater Philadelphia Bioinformatics
Alliance
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Chemotaxis Cell motion in the direction
of concentration gradients
Oriented human PMNs after 30 min in a gradient
from 0 on the left to 10-5 M FMMM on the right.
Bar, 15 µm (Zigmond 1977)
Schematic of E. Coli motility (Adapted from Berg)
polymorphonuclear leukocytes N-formylmethionylm
ethionylmethionone
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Quantitative Characterization of Cell Motion
Discrete
Continuum
-

mobility (diffusivity) and drift (Keller Segel,
1971)
persistence and speed (e.g. Zygourakis, 1996)
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Chemotaxis is an important component in
angiogenesis (among many others)
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Angiogenesis in porous media is desirable, yet
poorly understood and difficult
Vascularization of porous tissue engineering
scaffolds Pore-size limitations (60
µm) Porous domain structure, interconnectedness
Pore surface chemical properties
SEM image of microvessels penetrating e-PTFE bar
30 µm (Kidd 2001)
Will a simple model will help reign in complexity?
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Modeling angiogenesis is fundamentally modeling
chemotactic cell migration
Tong Yuan 2002

• Elements
• ri(t) biased random walk
• c(r,t) chemoattractant concentration field

Stokes Lauffenburger 1991
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Our idea A computational model of chemotactic
cell locomotion in porous domains with
time-varying concentration fields
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Our questions of interest How does a temporally
variable source of chemoattractant affect the
chemotactic response of mobile cells? How can
two genetically identical cells send and receive
chemical signals? How do reactive surfaces
confound the chemotactic migration of cells?
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Chemoattractant concentration field Transport
model
diffusion
point sources
global boundary condition R 2 mm
reactive-flux boundary condition
initial condition
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Discrete random walk model
cell position update process
unit vector direction of current step
velocity randomly selected from a Gaussian process
persistence
chemotaxis
randomness
Thresholds
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Simulation domain and case studies
2 mm
2 mm
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Continuous vs. Pulsed Secretion
Rate
Time
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Basic chemotactic response Point source
continuously secretes attractant
With increased ß, cell speed increases
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Pulsed secretion reveals sensitivity to
diffusivity
I. Initial lag II. More rapid decay more
frequent threshhold violation III. Lower peak
c less frequent threshhold viol. IV. Faster
moving pulses more freq. threshhold viol.
IV
II
I
III
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Pulsed secretion allows to mobile cells
to communicate
Cells secreting with random intervals
Cells secreting with different rates
Cells secreting at a uniform rate
Cell A
Cell A
Cell A
Cell B
Cell B
Cell B
Lost signal
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Cell speed depends on diffusivity
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Boundary-condition-induced autocrine drift
Global zero-boundary biases concentration gradient
toward center
not secreting
secreting in pulses
Minor effect, but must be accounted for!
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Simulation domain Single obstacle
2 mm
immobile point source
2 mm
obstacle with reactive surface
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Reactivity of obstacle surfaces
confounds chemotaxis
1.0
k 0.0
0.001
0.01
0.1
launch posn
target point source
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Contact time shows a maximum with reaction rate
Single obstacle, stationary continuous point
source, launched at ? 180º
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Simulation domain Single obstacle, dual walkers
2 mm
2 mm
obstacle with reactive surface
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Locomotion paths in paracrine response can be
tuned by obstacle surface reactivity
1.0
0.1
0
0.01
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Multiple obstacles can be placed to
create channels
k 1.0 reactive surfaces
k 0 unreactive surfaces
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Porous domains Creation protocol
following Dziubla, PhD thesis, Drexel University,
2002

• Removal of circular holes with a given radius
  from a solid square box by picking points at
  random.
• Circles are allowed to overlap each other to
  allow interconnection.
• The process is continued until desired porosity.

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Channelling is observed in complicated porous
domains
f 0.70
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Surface reactivity can accelerate locomotion
Difference 6 hours (2)
f 0.70
k 0
k 1
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• Conclusions
• A computational model of chemotaxis
  couplingreaction-diffusion of signal and biased
  randomwalks has been developed
• Pulsed secretion can reveal a sensitivity of
  cell speed on signal diffusivity
• Paracrine signaling via pulses results in
  successful dual taxis
• Surface reactivity of obstacles can
  tunemigration paths and speed

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• Ongoing and future work
• Porous domains with tuned reactivity
  Canreactivity engineering help to overcome
  tortuosity?
• Three-dimensions New solver required!
• Angiogenesis simulation in porous constructs
• Funding NSF-BES 0331191