clophfit.fitting.models#

Model functions for fitting macromolecule titration data.

This module provides mathematical models for fitting pH titration and ligand binding experiments. The models are based on standard biochemical equilibria and are used throughout the clophfit package for curve fitting.

Mathematical Background#

1-Site Binding Model

For ligand binding (e.g., Cl⁻ binding to a protein):

\[\begin{split}S(x) = S_0 + (S_1 - S_0) \\frac{x/K}{1 + x/K}\end{split}\]

where: - S(x) is the observed signal at ligand concentration x - S₀ is the signal when fully unbound - S₁ is the signal when fully bound - K is the dissociation constant

For pH titrations (Henderson-Hasselbalch):

\[\begin{split}S(pH) = S_0 + (S_1 - S_0) \\frac{10^{(K-pH)}}{1 + 10^{(K-pH)}}\end{split}\]

where K is the pKa value.

pH-Dependent Kd Model

Models infinite cooperativity between protonation and ligand binding:

\[\begin{split}K_d(pH) = K_{d1} \\frac{1 + 10^{(pKa - pH)}}{10^{(pKa - pH)}}\end{split}\]

This describes how the dissociation constant changes with pH due to protonation effects.

Functions#

`binding_1site`(…)	Single site binding model function.
`kd`(kd1, pka, ph)	Calculate pH-dependent dissociation constant with infinite cooperativity.

Module Contents#

clophfit.fitting.models.binding_1site(x: float, K: float, S0: float, S1: float, *, is_ph: bool = False) → float#

clophfit.fitting.models.binding_1site(x: clophfit.clophfit_types.ArrayF, K: float, S0: float, S1: float, *, is_ph: bool = False) → clophfit.clophfit_types.ArrayF

Single site binding model function.

This is the core model used throughout ClopHfit for fitting titration curves. It implements either a standard binding isotherm (for ligand concentration) or the Henderson-Hasselbalch equation (for pH titrations).

Parameters:

x (float | ArrayF) – Independent variable values. - For ligand binding: concentration in mM (e.g., [Cl⁻]) - For pH titrations: pH values
K (float) – Equilibrium constant. - For ligand binding: Kd in mM (dissociation constant) - For pH titrations: pKa (acid dissociation constant) Higher K means weaker binding (higher concentration needed for half-maximal signal).
S0 (float) – Signal plateau value for the first state. - For ligand binding: signal when fully unbound (no ligand) - For pH titrations: signal in protonated state (low pH) Units depend on the measurement (e.g., fluorescence intensity, absorbance).
S1 (float) – Signal plateau value for the second state. - For ligand binding: signal when fully bound (saturating ligand) - For pH titrations: signal in deprotonated state (high pH) Same units as S0.
is_ph (bool, optional) – Selects the equation form: - True: Henderson-Hasselbalch for pH titration (default) - False: Standard binding isotherm for ligand concentration

Returns:

Predicted signal value(s) at the given x value(s). Returns same type as input x (scalar if x is scalar, array if x is array).

Return type:

float | ArrayF

Examples

Standard chloride binding at half-saturation:

>>> binding_1site(x=10.0, K=10.0, S0=100, S1=200)
150.0

pH titration at the pKa:

>>> binding_1site(x=7.0, K=7.0, S0=100, S1=200, is_ph=True)
150.0

Array input for generating a titration curve:

>>> import numpy as np
>>> x_vals = np.array([0.5, 1.0, 2.0])
>>> binding_1site(x_vals, K=1.0, S0=0, S1=1)
array([0.33333333, 0.5       , 0.66666667])

Typical fluorescence titration:

>>> # GFP with pKa=7.0, fluorescence ratio changes from 0.1 to 2.5
>>> ph_range = np.array([5.0, 6.0, 7.0, 8.0, 9.0])
>>> signals = binding_1site(ph_range, K=7.0, S0=0.1, S1=2.5, is_ph=True)
>>> np.round(signals, 2)
array([2.48, 2.28, 1.3 , 0.32, 0.12])

Notes

Parameter Naming Convention

K, S0, and S1 use uppercase by convention from the lmfit library, where parameter names are case-sensitive and typically uppercase for clarity in fitting output.

Mathematical Details

For ligand binding (is_ph=False):

\[\begin{split}S(x) = S_0 + (S_1 - S_0) \\frac{x/K}{1 + x/K}\end{split}\]

This is a rectangular hyperbola with: - S(0) = S₀ - S(∞) = S₁ - S(K) = (S₀ + S₁)/2 (half-saturation)

For pH titrations (is_ph=True):

\[\begin{split}S(pH) = S_0 + (S_1 - S_0) \\frac{10^{(K-pH)}}{1 + 10^{(K-pH)}}\end{split}\]

This is the Henderson-Hasselbalch equation with: - S(pH << K) ≈ S₀ (protonated) - S(pH >> K) ≈ S₁ (deprotonated) - S(pH = K) = (S₀ + S₁)/2 (half-titration)

Typical Use Cases

Chloride binding to ClopHensor proteins (K ~ 5-50 mM)
pH titrations of fluorescent proteins (pKa ~ 5-9)
General 1:1 protein-ligand binding studies
Spectroscopic signal changes during titration

See also

kd: pH-dependent dissociation constant model
clophfit.fitting.core.fit_binding_glob: Global fitting across multiple datasets

clophfit.fitting.models.kd(kd1, pka, ph)#

Calculate pH-dependent dissociation constant with infinite cooperativity.

This model describes how the binding affinity of a ligand (e.g., Cl⁻) to a protein changes with pH when there is infinite cooperativity between protonation and ligand binding. It’s particularly useful for biosensors like ClopHensor where chloride binding is pH-dependent.

Parameters:

kd1 (float) – Dissociation constant at low pH (mM). This is the Kd when the protein is fully protonated, representing the intrinsic binding affinity in the protonated state. Typically measured at pH << pKa.
pka (float) – Acid dissociation constant (pKa). The pH at which the protein is 50% protonated. This determines how sensitive the Kd is to pH changes. For ClopHensor variants, this is typically around 7-8.
ph (ArrayF | float) – pH value(s) at which to calculate Kd. Can be a single value or an array for generating pH-dependent curves.

Returns:

Predicted Kd value(s) at the given pH (mM). Returns same type as input ph (scalar if ph is scalar, array if ph is array). Kd increases with pH, meaning weaker binding at higher pH.

Return type:

ArrayF | float

Examples

Calculate Kd at a single pH:

>>> kd(kd1=10.0, pka=8.4, ph=7.4)
11.0

The Kd at the pKa is exactly 2*kd1:

>>> kd(kd1=10.0, pka=8.4, ph=8.4)
20.0

Generate a pH-dependent Kd curve:

>>> import numpy as np
>>> ph_values = np.array([6.4, 7.4, 8.4, 9.4])
>>> kd_values = kd(kd1=10.0, pka=8.4, ph=ph_values)
>>> np.round(kd_values, 1)
array([ 10.1,  11. ,  20. , 110. ])

Typical ClopHensor parameters:

>>> # E2 variant: kd1=9mM, pKa=7.4
>>> ph_range = np.linspace(6, 9, 7)
>>> kds = kd(kd1=9.0, pka=7.4, ph=ph_range)
>>> np.round(kds, 1)
array([  9.4,  10.1,  12.6,  20.3,  44.8, 122.3, 367.3])

Notes

Mathematical Model

The model assumes infinite cooperativity between protonation and ligand binding:

\[\begin{split}K_d(pH) = K_{d1} \\frac{1 + 10^{(pKa - pH)}}{10^{(pKa - pH)}}\end{split}\]

This can be rewritten as:

\[\begin{split}K_d(pH) = K_{d1} \\left(10^{-(pKa - pH)} + 1\\right)\end{split}\]

Physical Interpretation

At pH << pKa: Kd ≈ kd1 (fully protonated, strong binding)
At pH = pKa: Kd = 2 * kd1 (half-protonated, intermediate binding)
At pH >> pKa: Kd → ∞ (deprotonated, no binding)

The model predicts that deprotonation completely abolishes binding, which is consistent with many pH-sensitive biosensors where a critical ionizable residue must be protonated for ligand coordination [1].

Applications

Analyzing ClopHensor chloride binding across pH ranges
Predicting sensor performance at physiological pH
Designing pH-optimized biosensor variants
Understanding structure-function relationships in pH-sensitive proteins

Limitations

Assumes infinite cooperativity (protonation either completely enables or disables binding)
Does not account for multiple ionizable groups with different pKa values
Simplified model may not capture all aspects of real protein behavior

clophfit.fitting.models#

Mathematical Background#

Functions#

Module Contents#

This Page