WARNING: this was LLM-generated and has not been fully reviewed yet.

Competitive Binding and Concentration Sweeps¶

This notebook demonstrates how to use equiconc to study competitive binding and visualize how equilibrium shifts with changing conditions.

import equiconc
import numpy as np
import matplotlib.pyplot as plt

Example 1: competitive dimers¶

Strand A can bind either B (forming AB) or C (forming AC). We fix \([\text{A}]_0 = [\text{C}]_0 = 1\;\mu\text{M}\) and sweep \([\text{B}]_0\) to see how the competitor displaces C from A.

\(\Delta G^\circ_{\text{AB}} = -12\) kcal/mol (stronger binder)
\(\Delta G^\circ_{\text{AC}} = -10\) kcal/mol (weaker binder)

conc_A = 1e-6   # 1 uM
conc_C = 1e-6   # 1 uM
dg_AB = -12.0
dg_AC = -10.0

conc_B_range = np.logspace(-8, -4, 80)  # 10 nM to 100 uM

results = {"B0": [], "free_A": [], "AB": [], "AC": [], "free_B": [], "free_C": []}

for conc_B in conc_B_range:
    eq = (
        equiconc.System()
        .monomer("A", conc_A)
        .monomer("B", conc_B)
        .monomer("C", conc_C)
        .complex("AB", [("A", 1), ("B", 1)], dg_st=dg_AB)
        .complex("AC", [("A", 1), ("C", 1)], dg_st=dg_AC)
        .equilibrium()
    )
    results["B0"].append(conc_B)
    results["free_A"].append(eq["A"])
    results["AB"].append(eq["AB"])
    results["AC"].append(eq["AC"])
    results["free_B"].append(eq["B"])
    results["free_C"].append(eq["C"])

As \([\text{B}]_0\) increases, AB formation rises and AC is displaced.

B0_uM = np.array(results["B0"]) * 1e6

fig, ax = plt.subplots(figsize=(6, 4))
ax.plot(B0_uM, np.array(results["AB"]) * 1e6, label="[AB]", linewidth=2)
ax.plot(B0_uM, np.array(results["AC"]) * 1e6, label="[AC]", linewidth=2)
ax.plot(B0_uM, np.array(results["free_A"]) * 1e6, label="free [A]",
        linewidth=2, linestyle="--")
ax.set_xscale("log")
ax.set_xlabel("Total [B] (uM)")
ax.set_ylabel("Concentration (uM)")
ax.set_title("Competitive binding: titrating B into A + C")
ax.legend()
ax.set_ylim(bottom=0)
fig.tight_layout()
plt.show()

png

Next, we fix all concentrations at 1 \(\mu\)M and sweep temperature to see how the equilibrium shifts. We'll set \(\Delta G^\circ\) to -15.0 at 37°C, and use a \(\Delta S\) of -0.3.

temperatures_C = np.linspace(10, 90, 80)

conc_all = 1e-6
dg37 = -15.0
ds = -0.3

conc_bound = np.array([
        equiconc.System(temperature_C=T)
        .monomer("A", conc_all)
        .monomer("B", conc_all)
        .complex("AB", [("A", 1), ("B", 1)], dg_st=(dg37,37), ds_st=ds)
        .equilibrium()["AB"] for T in temperatures_C])

fraction_bound = conc_bound/conc_all

fig, ax = plt.subplots(figsize=(6, 4))
ax.plot(temperatures_C, fraction_bound, linewidth=2, color="teal")
ax.set_xlabel("Temperature (C)")
ax.set_ylabel("Fraction of A bound as AB")
ax.set_title("Melting curve: A + B <=> AB")
ax.set_ylim(-0.02, 1.02)
ax.axhline(0.5, color="gray", linestyle=":", linewidth=1)
fig.tight_layout()
plt.show()

png