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: Rh  Rh* 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 forces w noise The pdf satisfies the following equations within the outer segment F=0. whereand m mass of the molecule g viscosity coefficient T absolute temperature k Boltzmann constant

Longitudinal diffusion in rod outer segments Method: projection 3D  1D Conclusion: standard linear diffusion

Longitudinal diffusion in cone outer segments Method: projection 3D  1D  diameter of disc connecting two adjacent compartments D Diffusion constant d min diameter at the tip CONCLUSION 1-the diffusion coefficient is not a constant value, but change with longitudinal position 2-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

Numerical Simulations

Comparison across species of spread of excitation SpeciesCOS structurecGMP diffusion Length (  m) Base radius (  m) Tip radius (  m)   m  D l (base) (  m 2 /sec) D l (tip) (  m 2 /sec) D l (at L/2) (  m 2 /sec) con (at L/2) (  m) Striped bass, single cone Tiger salamander, single cone Human, peripheral retina SpeciesROS structurecGMP diffusion length (  m) diameter (  m) No. incisures D aq (  m 2 /sec) D l (experiment) (  m 2 /sec) DlDl (theory) (  m 2 /sec) rod (  m) Tiger salamander Striped bass Human, peripheral retina Guinea pig Rat 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

Access to all global variable Membrane potential V(t) Total Calcium and cGMP

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 The joint probability of a trajectory and the number of bound sites in the volume

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 J=flux, K1 redined forward binding, k-1 backward rate

Steady state Parabolic variance

Push-Pull mechanism Fact: cGMP is regulated by 1 PDE* and another molecule  total number of cGMP fluctuate Continuum model 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 (RGS9  PDE*) 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) N* colored noise= fluctuation of independent PDE K, a,b, sigma, gamma constant W=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

Fokker Planck Operator Find P0