Step 2 – Steady-State Dose–Response (Hill Curves)¶

Script:

• step_1c_fit_hill_curves.py

Biological question¶

Given a fixed construct, how does steady-state target expression depend on degron-controlled repressor abundance?

This defines the input–output transfer function of CasTuner:

• input: repressor abundance (from tBFP),
• output: target fold-change (mCherry or EGFP).

We model this with a Hill function to capture non-linearity and potential cooperativity.

Inputs¶

• Dose–response .fcs data:
  • cells grown at different dTAG-13 concentrations until steady state (e.g. 4 days),
• NFC controls for background subtraction,
• Metadata describing which files correspond to which:
  • construct,
  • guide set (targeting vs NTC),
  • dTAG dose.

Method¶

1. Gating and background subtraction

   • same gating strategy as in Step 1,
   • compute NFC-subtracted medians for BFP and mCherry/EGFP.

2. Normalisation

   • Repressor axis (input):
     • per construct, apply min–max normalisation to BFP: $$ x = \frac{BFP - BFP_\text{min}}{BFP_\text{max}-BFP_\text{min}} $$
       • this gives a dose-scaled “repressor activity” between 0 and 1.
   • Target axis (output):
     • compute fold-change in mCherry/EGFP relative to NTC controls at high dTAG.

3. Hill function fitting

We fit:

where:

• x – normalized BFP (0–1),
• K – effective midpoint (dose at half-maximal effect),
• n – Hill coefficient (steepness, cooperativity),
• y_min, y_max – asymptotic fold-change bounds.

Parameters are estimated by non-linear least squares (scipy.optimize.curve_fit).

1. Diagnostics

2. scatter plots of observed vs fitted fold-change,

3. per-construct curves overlaid on data points,
4. R² or MAE summaries.

Outputs¶

• parameters/Hill_parameters.csv with columns:
  • construct, assay type,
  • K, n, y_min, y_max,
  • fit diagnostics.
• Plots in plots/hill_curves/.

How to interpret¶

• K (midpoint):

  • small K → construct responds strongly already at low repressor levels,
  • large K → construct needs higher repressor levels to repress strongly.

• n (Hill coefficient):

  • n ≈ 1 → graded, almost linear response,
  • n >> 1 → switch-like behavior,
  • n < 1 → shallower-than-linear response.

These Hill parameters are reused directly in the ODE model to connect repressor trajectories to mCherry dynamics.