GOx/HRP protocell cascade: spatial arrangement controls receiver output. Steady-state 2D reaction-diffusion model.
HRP/GOx ratio: —. Recommended: keep HRP at ~10x GOx for pseudo-first-order kinetics.
12 GOx + 12 HRP alternating in a 4x6 grid. Every HRP is equidistant from its nearest GOx neighbours.
12 GOx block (left) + 12 HRP block (right). Front-row HRP consume H₂O₂ before it reaches the back row.
Equivalent to a horizontal line ROI in Fiji drawn through the HRP row. Each point = resorufin signal at one HRP droplet.
Predicts a W-shaped profile: high signal near each GOx cluster, suppressed at the midpoint where the two H₂O₂ fields overlap but HRP depletion creates a saddle.
Steady-state form (∂C/∂t = 0):
where C = [H₂O₂], D = diffusion coefficient, k = first-order HRP consumption rate, S(x,y) = GOx source term.
2D finite difference on a uniform Cartesian grid (80 x 80 cells). Gauss-Seidel iterative solver, 600 iterations. Field of view is fixed at 3500 µm (matching experimental microscope FOV).
Droplet radius in cells: R_CELLS = round(diameter / (2 × Δx)). Current R_CELLS = 3, droplet diameter ≈ 262 µm on grid.
Dirichlet: C = 0 at all four edges. Physical justification: open dish geometry where H₂O₂ escapes to bulk solution. The 3500 µm field provides ample buffer between the droplet array and the boundary.
Each droplet stamps a disc of radius R_CELLS grid cells as either a uniform source (GOx) or a first-order sink (HRP). R_CELLS scales with the droplet diameter slider. All droplets at a given diameter share an identical precomputed disc footprint. This is a well-mixed droplet approximation. Source strength and sink rate both scale with droplet volume (d/d_ref)³ (reference diameter = 250 µm), reflecting the total number of enzyme molecules contained in the droplet at a given concentration.
Da relates reaction rate to diffusion rate. Da << 1: diffusion-dominated (uniform field). Da >> 1: reaction-dominated (steep gradients, depletion). The HRP concentration slider sets Da via: k = [HRP] · k_cat_eff, then Da = k · Δx² / D.
t½ is derived from the steady-state solution using a diffusion timescale heuristic: t½ = d² / (2D) × 1/(1 + Da₀), where d = centre-to-centre distance, D = diffusion coefficient, Da₀ = local Damköhler number at the HRP site. This captures: (1) t½ increases with distance squared (diffusion-limited), (2) t½ decreases with higher Da (faster consumption pulls the front forward).
1. Steady-state only: no true temporal dynamics; the time-course is a heuristic exponential approach.
2. Resorufin treated as non-diffusing (remains at HRP site where produced).
3. Amplex Red depletion not modelled (assumed excess).
4. Droplet interior assumed well-mixed: no intra-droplet concentration gradients.
5. 2D approximation of a 3D system (thin-layer geometry assumed).
6. No droplet-droplet steric exclusion in the solver (overlap possible at very close spacing).
| Parameter | Value | Unit |
|---|---|---|
| Field of view | 3500 | µm |
| Grid cell Δx | 43.75 | µm |
| Field size (80 cells) | 3500 | µm |
| Droplet diameter (slider) | — | µm |
| R_CELLS (droplet radius) | — | cells |
| Droplet diameter (on grid) | — | µm |
| Diffusion coeff. D | — | µm²/s |
| [GOx] effective | — | µM |
| [HRP] effective | — | µM |
| GOx source flux q | — | AU (relative) |
| HRP Da (Damköhler) | — | dimensionless |
| k (sink rate) | — | s⁻¹ |
| Centre-to-centre dist. | — | µm |