photo-transduction and related mathematical problems d. holcman, weizmann institute of science
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Photo-transduction
and related mathematical problems
D. Holcman, Weizmann Institute of Science
Retinal organization
Retina connection
• Cone > Bipolar cell > Ganglion cell
• Rod > Bipolar cell > Amacrine cell > Ganglion cell
Photo-response cone/rod
Actual state of art
• Initial phase of the transduction known
• The global recovery is still missing
• Difference of the two photoreceptors?
• How signal propagate from the outer-segment to the synapse?
• How the synapse is modulated?
Structures of Photoreceptors
Cone
Biochemistry of the photo-transduction
Compartment of photo-transduction
Steps of Photo-transduction
• 1-Arrival of a photon: RhRh*
• 2-Amplification from Rh*…PDE*
a single Rh^* activates 300 PDE
• 3-Destruction of cGMP messenger
• 4-Channels closed
• 5-hyper-polarization of the cell
• 6-Transmission like a wave capacitance to the Inner-Segment
• 7-Release of neurotransmitters
Order of magnitude
Number per compartment of • cGMP: 60 to 200• Channels 200 to 300• Open channels in dark= 6• Activated PDE=1• Free calcium =5
Photon close channels: Can closing 6 enough to generate a signal?
Longitudinal propagation of a signal
• cGMP holes propagate to close many channels: how much?
• Compute the propagation of the depleted area
A theory of longitudinal diffusion at a molecular level
Particle motion in the Outer Segment
F electrostatic forcesw noise ( ) 2X F X w
cD c
t
0( 0) ( )c x c x
The pdf satisfies the following equations within the outer segment F=0.
wherekT
Dm
and
m mass of the molecule g viscosity coefficientT absolute temperature k Boltzmann constant
Longitudinal diffusion in rod outer segments
• Method: projection 3D1D
2
2
( ) ( )k kl
k
c x t c x tD
t x
22
( ) 12
2 ( ) 2 1incs g
incs gl r
incs g n
D n lD D
r l n l
Conclusion:standard linear diffusion
Longitudinal diffusion in cone outer segments
• Method: projection 3D1D
2 2
2 2 2min
( ) 2 ( )
( )
c x t D c x t
t xd d x
max mind dd
L
diameter of disc connecting two adjacent compartments D Diffusion constantd min diameter at the tip
CONCLUSION 1-the diffusion coefficient is not a constant value, but change with longitudinal position2-No explicit solution (WKB asymptotic)
Matching theory and experience
Spread of excitation
cGMP =messenger that open channels
1-Compare spread of cGMP in rod/cone
2- Characterize the spread at time to peak tp of the photo-response
l prod D t
( ) ( )l pcon x D x t
Numerical Simulations
Comparison across species of spread of excitation
Species COS structure cGMP diffusion
Length(m)
Base radius(m)
Tip radius(m)
m
Dl (base)
(m2/sec)
Dl (tip)
(m2/sec)
Dl (at L/2)
(m2/sec)
con (at L/2)(m)
Striped bass,single cone
15 3.1 1.2 20.324 2.7 17.9 5.6 40.79
Tiger salamander,single cone
8.5 2.5 1.1 30.314 3.9 20.0 7.6 50.99
Human, peripheral retina1
7 1.5 0.75 30.244 6.6 25.8 11.6 60.68
Species ROS structure cGMP diffusion
length(m)
diameter(m)
No. incisures
Daq(m2/sec)
Dl
(experiment)
(m2/sec)
Dl
(theory)
(m2/sec)
rod
(m)
Tiger salamander 125.3 12.3 218 500 330-6021-11
18.5 84.7
7Striped bass 40 1.6 1 41.6 73.8
4 Human, peripheral retina
12 1.5 1 44.3 93.0
5Guinea pig 5 1.4 1 47.3
6Rat 25 1.7 1 39.3
1.our data, n=11
Conclusion on the longitudinal diffusion
1-Spread of Excitation depends on the geometry only but not on the size.
2-Geometry alone determines the longitudinal diffusion
2-Spread of excitation is similar across species for Cones and Rods
D. Holcman et al. Biophysical Journal, 2004l
Global model 2
12
02
23
2
( ) ( ) ( , )
( )1 ( )
( ) ( ). ( )
long
long p c
c cD x ca k PDE t x c
t x
cacaK
ca caD x x ca x c binding
t x
( , )
Dirac function
delay time
( ) ( ) ( , )
( ) , for 1,
( , ) random variable uniformly distributed = thermal activation o
k d kk photon
d
PDE x t t t t x x w x t
t t tt
w x t
f PDE
Access to all global variable
• Membrane potential V(t)
3
0
( ) ( , ) ( )L
rest revV t c x t dx V V Total Calcium and cGMP
0
0
( ) ( , )
( ) ( , )
L
c
L
ca
N t c x t dx
N t ca x t dx
Conclusion
• Presented here a global model
• Simulate photo-response from 1 to many
• Adaptation is not included
Noise in Photoreceptors
fluctuation of the membrane potential
G. Field. F.Rieke, Neuron 2002
Sources of Noise
• Definition: fluctuation of the membrane potential
Causes• Thermal activation of Rhodopsin• Local binding and unbinding of CGMP +
Push-pull mechanism (swimming noise) • PDE activity as a source of the noise in
chemical reactions: Push-Pull noise
Swimming noise
• Fluctuation of the number of open channels due the stochastic binding and unbinding.
Swimming noise
• Number of open channels (experimentally=6)• Variance= compute?
Model Rules:
1. cGMP bind and unbind to the channels, diffuse inside a compartment
2. When a channel is gated, no other cGMP can bind.3. cGMP stays bound during a given time.
Swimming noise
= number of unbound particles at time
= number of free sites in volume at time
= number of unbound binding sites at time
= number of bound particles at time .
= initial density of substrate
( ) Pr ( ) ( ) (0)p x S t y x x t x x S x t S x y
( )M t( )S x t
( )S t
( )MS t
0 ( )s x
The joint probability of a trajectory and the number of bound sites in the volume x
Fokker-Planck Equation for the joint pdf
• P(x,S,t)= proba to find a cGMP at position x at time t and S(0 or 1) channel are bound at position x
• Time evolution equation
21
21 0 1
1 0
( , , )( , , ) ( , , )
[ ( ) ] ( , , ) ( , 1, )( 1)
[ ( ) 1] ( , 1, )
p x S tJ x S t K p x S t S
t
k S x S p x S t K p x S t S
k S x S p x S t
J=flux, K1 redined forward binding, k-1 backward rate
Steady state
Parabolic variance
21 0 10 ( 0) ( ) ( 0) ( 1)D p x k S x p x K p x
21 0 10 ( 1) ( ) ( 0) ( 1)D p x k S x p x K p x
1( 0) ( 1)p x p x
L
21 0 1
1( 1) ( ) ( 1) ( 1) 0Dp x k S x p x K p x
L
20
0 1 0 1
1 1
2 2( ) (1 ) 1
1 1 4 1 1 4S M M
S S
M p pM k x M k x
N k N k
Push-Pull mechanism
Fact: cGMP is regulated by 1 PDE* and another moleculetotal number of cGMP fluctuate
Continuum model 2
1 12( )
p pD k x x p
t x
0
0x x L
p p
n n
0{ ( ) [0 ]} ( ) ( 0)
LPr x t L S t p x t dx
0Pr{ ( ) [0 ]} ( ) ( )
tx t L f s EN t s ds
1
( ) Pr{ ( ) }n
EN t n N t n
Steady state variance can be computed from the same analysis
Conclusion
• Simulation is needed
• Include cooperativity effect (up to 4 cGMP can be bound to a single channel)
• Derive the fluctuation of the number of open channels and the characteristic time
• Derive a Master equation to compute mean and variance of the cGMP due to the Push-pull.
Where we stand:Push-Pull noise, low frequency
• Molecular difference of the steady state noise (RGS9PDE*)
• Description of the noise: a problem of Mean First Passage Time in chemical reactions
Simplifies Model
• cGMP fluctuation due to the push-pull (no diffusion)
.
. .
*
* *
c kN c
N a bN w
N* colored noise= fluctuation of independent PDEK, a,b, sigma, gamma constantW=Brownian Characterization of the fluctuationin CGMP= Find the MFPT of c to a threshold as a function of the parameter
Mean First Passage Time
• Attractor (c,N*)= • p not the same for cones and rods
Kind of Smoluchowski limit
00
( , )pNkpN
.
0 0
1/ 2. .
1/ 20
(1 )
(1 )
x kn x kn x y
ry y w
n
... . .
1/ 2 1/ 21[ ] (1 )(1 ) ( )1 (1 )
xx x x x r k w
x x
Fokker Planck Operator
Find P0
2
2
(1 )(1 ) 1[ ( ) ]
2 1 (1 )
p k r x p p vv v p
t v x v x x
0 1
1( , , ) ( , , ) ( , , ) ..P x v t P x v t P x v t