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