Equilibrium Concentrations of two ligands in case of Competitive binding

competitive_binding.py

# -*- coding: utf-8 -*-
"""Competitive Binding.ipynb
Created on Sep 11 15:40:05 2016
@author: Leela S. Dodda leela.dodda@yale.edu
@author: William L. Jorgensen Lab 
Automatically generated by Colaboratory.

Original file is located at
    https://colab.research.google.com/drive/1xm_e38URAOk4Aspxffkupbm_zMyKcFxv
"""

import numpy as np

def Concentrations(Ka,Kb,Ai,Bi,Pi):
  '''
  Input Variables:
  Ka = Binding COnstant of Ligand A
  Kb = Binding COnstant of Ligand B
  Ai = Initial Concentration of Ligand A
  Bi = Initial Concentration of Ligand B
  Pi = Initial Concentration of Protein

  Output:
  Af = Equilibrium Concentration of Ligand A
  Bf = Equilibrium Concentration of Ligand B
  Pf = Equilibrium Concentration of Protein

  Reference For This Method:

  ** An exact mathematical expression for describing competitive binding of
   two different ligands to a protein molecule **
   Zhi-Xin Wang
   FEBS Letters 360 (1995) 111-114
  '''
  a = Ka+Kb+ Ai+Bi - Pi
  b = Kb*(Ai-Pi) + Ka*(Bi-Pi) + Ka*Kb
  c = -1.0 * Ka * Kb*Pi
  num = (-2.0*a**3) + (9.0 *a*b) - (27.0*c)
  denom = 2.0*np.sqrt((a**2-3.0*b)**3)
  theta = np.arccos(num/denom)
  Pf = (-1.0*a/3.0) + ((2.0/3.0)*np.sqrt(a**2-3.0*b)*np.cos(theta/3.0))
  ratioA = Ka/Pf
  ratioB = Kb/Pf
  Af = Ai*(ratioA/(ratioA+1.0))
  Bf = Bi*(ratioB/(ratioB+1.0))
  return(Af,Bf,Pf)

Ka = 5.3e-9#Binding COnstant of Ligand A
Kb = 43.0e-9#Binding COnstant of Ligand B
Ai = 0.2e-6#Initial Concentration of Ligand A
Bi = 0.5e-6#Initial Concentration of Ligand B
Pi = 0.2e-6#Initial Concentration of Protein
Af,Bf,Pf = Concentrations(Ka,Kb,Ai,Bi,Pi)
print('Protein Concentration at Eq %.2E'%Pf)
print('Ligand A Concentration at Eq %.2E'%Af)
print('Ligand B Concentration at Eq %.2E'%Bf)