Physical units do not imply biological calibration. This model combines two agar-diffusion anchors with explicitly illustrative membrane parameters.
Two pools, one mass balance
∂(φw C)/∂t = φw D ∂²C/∂x² − J − φw kd C
∂B/∂t = J
J = kex (Kmw φm C − B)
C is dissolved AHL concentration per liquid volume. B is membrane-associated AHL amount per total chamber volume. The membrane pool is stationary. Production is zero in every transport experiment.
φw = 1 − ρ vcell; φm = ρ vmem, using compatible volumes. One mL is 10¹² µm³. At equilibrium, B = Kmw φm C. The rapid-equilibrium delay factor is R = 1 + Kmw φm / φw; its corresponding diffusivity is D/R.
Geometry & initial conditions
A 6 mm long, cross-sectionally mixed chamber with a fixed 1 mm² area (6 µL). Cells are represented by a uniform density, not individually resolved obstacles. D represents movement through the chosen medium; additional crowding or tortuosity is not included.
A pulse initially places all AHL in the liquid in a Gaussian centered at x = 0.5 mm with width σ = 0.22 mm; both ends are sealed. It is normalized so the integrated initial amount is exactly mean dose × total chamber volume. A 10 nM mean dose is 60 fmol in this geometry, not a 10 nM peak.
Washout starts with a uniform, equilibrated free/membrane distribution at the same total dose. Both ends then contact ideal zero-AHL sinks. Even at zero exchange rate, that initial distribution is imposed as a prepared condition.
Where the molecule values come from
Tica et al. (2024) estimated distance-assay D values of 2.0 mm²/h for 3-oxo-C6-HSL and 1.2 mm²/h for 3-oxo-C12-HSL in agar. Applying these to a cell-containing model assumes the background medium remains comparable. These are not universal water diffusivities.
Other chain lengths use a log-linear interpolation or extrapolation: D(n) = 2 × 0.6^((n−6)/6). This convenient two-anchor curve is our assumption, not the publication’s fitted power law.
The default K(n) = 100 × 50^((n−6)/6) is illustrative throughout. K at C6 = 100 and at C12 = 5000 are not claimed measurements. Exchange = 1 h⁻¹, cell volume = 1 µm³, and accessible membrane volume = 0.03 µm³/cell also require calibration.
Membrane simulations (2021) and model-bilayer experiments (2012) support structure-dependent membrane association. They do not calibrate this model’s bacterial partition and exchange constants.
Concentration & omitted mechanisms
This is a dilute, linear, reversible partition model with no finite-capacity binding. Dose changes amplitudes, not intrinsic D. AHL aggregation experiments (2020) show that an additional pool may be needed at concentrations where aggregation occurs. The current tool does not predict that onset.
No cell growth, active efflux, intracellular uptake, reporter maturation, fluid flow, membrane movement, chemical interconversion, or organism-specific receptors. Only dissolved AHL degrades, using the entered loss rate. Transport data constrain products such as Kmw × accessible membrane volume; those parameters may not be individually identifiable.
Testing a prediction
Compare chemical concentration and total recovery across time and space. A reporter alone measures a combination of transport and biological response. Independent partition, recovery, and cell-free diffusion measurements help distinguish competing explanations. Nonproducing cells still may take up or degrade AHL; passive membrane preparations provide a complementary control.
Numerical method
120 finite volumes, ≤ 0.005 h steps, backward-Euler diffusion, and exact local reversible exchange. Source-free sealed runs conserve total AHL. Open runs separately track escaped signal; dissolved chemical loss is recorded as degraded. Comparison results use the same observation window and record missing threshold crossings as not reached. Parameter variation is sensitivity analysis, not a statistical confidence interval.