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Mathematical Modeling to Resolve
the Photopolarization Mechanism
in Fucoid Algae
BE.400
December 12, 2002
Wilson Mok
Marie-Eve Aubin
Outline
 Biological background
 Model 1 : Diffusion – trapping of channels
 Model 2 : Static channels
 Model results
 Experimental setup
 Study on adaptation
Photopolarization in Fucoid Algae
(Kropf et al. 1999)
Signal Transduction
• Light
• Photoreceptor: rhodopsin-like protein
• cGMP
• Ca++
• Calcium channels
• F-actin
 Signal transduction pathway unknown
 The mechanism of calcium gradient
formation is still unresolved
Distribution of calcium
(Pu et al. 1998)
Model 1 : Diffusion - trapping of channels
Ca2+ channels
N
Blue light
N
N
Actin patch
Actin patch:
Involvement of microfilaments in cell
polarization as been shown
Model of Ca++ channel diffusion
suggested (Brawley & Robinson 1985)
(Kropf et al. 1999)
Model 1 : Bound & Unbound Channels
light
 We model one slice of the cell
 Reduce the system to 1D
 Divide the channels in two subpopulations:
1) unbound : free to move
2) bound : static
1)
2)
CU
 2CU
 DC
 kU ( x)CB  k B ( x)CU
2
t
x
CB
 k B ( x)CU  kU ( x)CB
t
Rate of binding
Rate of unbinding
Model 1 : Calcium Diffusion
We assume that the cell is a cylinder.
C
 2C
2
 D 2  klossC  P( x)(Cbulk  C )
t
x
R
where:
P( x)  KCc ( x)
Channel concentration
Flux on the illuminated side:
D
C
x
Flux on the shaded side:
D
C
x
x 0
 P(0)(Cbulk  C (0))
xL
 P( L)(C ( L)  Cbulk )
Model 2 : Static Channels
The players involved are similar to the
ones in rod cells.
In rod cells:
Activated rhodopsin
Electrical
response of
the cell
activate
G protein
activate
Reduce the
probability of opening
of Ca++ channels
Cyclic nucleotide
phosphodiesterase
[cGMP] 
=> similar process in Fucoid Algae ?
Model 2 : Static Channels
C
 2C
2
 D 2  klossC  P( x)(Cbulk  C )
t
x
R
where:
P( x)  K ( x)Cc
 Channels are immobile
 Permeability decreases with closing of channels
K
t
 kC ( x) K
Model 1 - results
linear distribution of light
Unbound channels distribution
#
Bound channels distribution
#
10 hrs
10 hrs
Total channels distribution
#
Calcium distribution
#
10 hrs
10 hrs
Model 1 - results
logarithmic distribution of light
Unbound channels distribution
Total channels distribution
Bound channels distribution
Calcium distribution
Distribution of calcium
linear distribution of light
logarithmic distribution of light
Model 1
linear distribution of light
Model 2
logarithmic distribution of light
Flux of calcium
linear distribution of light
logarithmic distribution of light
shaded side
Model 1
illuminated side
time
linear distribution of light
time
logarithmic distribution of light
shaded side
Model 2
illuminated side
time
time
Model 1 :
Rate of unbinding sensitivity analysis
(linear distribution of light)
Maximum Kunbind : 10-1 s-1
10-2 s-1
10-3 s-1
[Ca++]
[Ca++]
[Ca++]
position
10-4 s-1
[Ca++]
10-5 s-1
[Ca++]
Light distribution measurements
• Isolate 1 cell
• Attach it to a surface
• Use a high sensitive photodiode (e.g. Nano
Photodetector from EGK holdings) with pixels on
both sides what is coated with a previously
deposited thin transparent layer of
insulating polymer (e.g. parylene)
• Rotate the light vector
Light vector
 Identify best light distribution to improve this 1D model
Previous experimental data
Calcium indicator (Calcium Crimson)
Ca2+-dependent
fluorescence emission
spectra of the Calcium
Crimson indicator
Experimental Setup
to verify models accuracy
Calcium-specific vibrating probe : Flux measurement
Concluding remarks
 2 mathematical models which predict a
successful photopolarization were proposed:
 Diffusion-Trapping Channels Model
 Static Channels Model
 Generate more than quantitative predictions:
give insights on an unresolved mechanism
The experimental setup proposed would also
elucidate the adaptation of this sensory mechanism
Necessity for Adaptation
Sensitivity = increase of response per unit of intensity
of the stimulus (S = dr/dI )
Adaptation : change of sensitivity depending on the
level of stimulation
Dynamic range of photoresponse:
sunlight: 150 watts / m2
moonlight: 0.5 x 10-3 watts / m2
Adaptation
I ÷ IB = Weber fraction
Quantal effects
Acknowledgements
Professor Ken Robinson
Ali Khademhosseini
Professor Douglas Lauffenburger
Professor Paul Matsudaira
References
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