This is a 2D, formula-driven counterpart to the 3D binding-visualization version of this sensor. Instead of animating antigen spheres in 3D, it draws the electrode's binding-site grid flat and spends the rest of the canvas on two real instrument views: a Nyquist plot and a Bode plot, both computed from the same underlying Randles-circuit-with-Warburg model used in real bioanalytical EIS instruments.
The equivalent circuit is a solution resistance Rs in series with a parallel combination of the double-layer capacitance Cdl and a faradaic branch made of the charge-transfer resistance Rct in series with a Warburg diffusion element Zw:
Z_w(ω) = σ·ω^(-1/2)·(1 − j) (semi-infinite linear diffusion)
Z_f(ω) = R_ct + Z_w(ω) (faradaic branch)
Z(ω) = R_s + [ jωC_dl + 1/Z_f(ω) ]^-1 (full Randles + Warburg)
At high frequency the Warburg term vanishes and the plot reduces exactly to the textbook Randles semicircle of diameter Rct, apex at ωpeak = 1/(RctCdl). At low frequency, when σ > 0, diffusion of the redox probe to the electrode becomes rate-limiting and the curve straightens into the classic 45° Warburg line (Z' and −Z'' grow together) instead of closing back to the real axis — the real diagnostic signature of a mass-transport-limited assay, which the semicircle-only model can't show. Set σ = 0 to see the pure semicircle case.
Antigen binding still follows Langmuir kinetics and blocks the electrode exactly as before:
dθ/dt = k_on·[Ag]·(1-θ) − k_off·θ (Langmuir binding kinetics)
R_ct(θ) = R_ct0 / (1 − θ) (blocking-layer model)
- Analyte concentration — sets [Ag], the driving term in the binding ODE; higher concentration saturates θ faster and further.
- Bare-electrode Rct0 — the baseline charge-transfer resistance before any binding.
- Double-layer Cdl — capacitance of the electrode/electrolyte interface; sets the semicircle's apex frequency.
- Binding rate kon — a speed multiplier on the association kinetics.
- Warburg diffusion σ — how strongly probe diffusion limits the reaction at low frequency; 0 gives a closed semicircle, higher values pull the low-frequency tail into the 45° diffusion line.
- Toggle Nyquist / Bode to switch between the classic Z' vs −Z'' plot and a log-frequency magnitude/phase view — the two standard ways EIS data is actually reported.