Rhodopsin Cascade: Reaction-Diffusion Across the Disc (2D)

Light Stimulus

Background light rate 0 photons/s

PDE catalytic gain (β) 18

Rhodopsin lifetime τ_R 0.30 s

PDE lateral diffusion D_E 0.40 µm²/s

Cascade State

Total active R* (disc)0.00

Total active PDE, E*0.0

cGMP level100%

Photocurrent (channels open)35.0 pA

Spread radius √(4·D_E·τ_E)0.00 µm

Photocurrent trace I(t)

How it works

Every rhodopsin, transducin and PDE molecule sits and diffuses within the disc membrane — the 3D companion simulation treats the whole disc as one well-mixed pool, but a real photon activates rhodopsin at one point and the activated PDE it produces has to spread out from there. This model solves that spreading directly as a 2D reaction-diffusion system on the disc surface, on a grid, with no-flux (Neumann) edges:

∂R*/∂t = -R*/τR + D_R·∇²R*                    (+source per photon, local patch)
∂E*/∂t = kRE·R* - E*/τE + D_E·∇²E*              (active PDE, spreads laterally)
dcGMP/dt = α - (β0 + β·mean(E*))·cGMP           (cytoplasm mixes fast: single well)
I/Idark = (cGMP / cGMP_dark)^n,  n ≈ 3          (Hill: channel cooperativity)

Flash one photon — deposits one R* unit at a random point on the disc; watch the pale-blue PDE patch spread outward from that exact spot rather than lighting the whole disc at once.
Background light rate — a Poisson stream of point photon absorptions scattered across the disc, each starting its own spreading patch.
PDE gain β — hydrolysis rate per unit of active PDE; unchanged from the 3D model's well-mixed value.
τR — how long rhodopsin stays active before shut-off; a longer τR both feeds more total PDE activation and lets the R* patch itself diffuse further before it decays.
PDE lateral diffusion D_E — the 2D-only control: how fast activated transducin/PDE complexes migrate through the disc membrane (real literature range for membrane-associated phototransduction proteins is roughly 0.1–1 µm²/s). Higher D_E spreads a single photon's effect over more of the disc, diluting its peak local density, exactly as it does in a real rod.

Because the grid uses no-flux boundaries, total R* and total E* mass across the whole disc obey the identical ODEs as the 3D well-mixed model regardless of D_E — diffusion only redistributes where the activation sits, it never creates or destroys it. What diffusion changes is the shape of the single-photon response and how quickly a sparse background of individual photon hits blurs into one uniform light-adapted disc.

Photon impact Active PDE, E* (heat) Open channel Closed channel