Introduction

Binding Curves (or Binding Assays) are an approach to determining the equilibrium constant for the interaction of two molecules, commonly used in biochemistry¹. Typically, the a small molecule (commonly called the “ligand”) attaches to a larger molecule such as a protein. If we denote the protein as P and the ligand as L we can express the dissociation as:

\(PL \rightleftharpoons P + L\)

and where the total protein is the bound plus unbound.

\(P_{tot} = PL + P\)

With the dissociation constant:

\(K_d = \frac{[P][L]}{[PL]}\)

Note that since L is in excess, it is assumed that [L] added is the same as [L] in the equilibrium expression.

The concentration of the ligand (which is in excess) is varied and the fraction bound \(\frac{[PL]}{P_{tot}}\) is plotted versus the L².

Substituting \(P_{tot} = P + PL\) into the \(K_d\) expression:

we get

\(\large K_d = \frac{([P_{tot}]-[PL])[[L]}{[PL]} = \frac{([P_{tot}][L]-[PL]{L]}}{[PL]}\)

\(K_d[PL] + [PL][L] = [P_{tot}][L]\)

\([PL](K_d +[L]) = [P_{tot}][L]\)

and we get expression for fraction bound as a function of \(K_d\) and L.

\(\large\frac{[PL]}{P_{tot}} = \frac{[L]}{K_d + [L]}\)

note as L increases the fraction bound goes from 0 to 1.

The key to binding assay is to find an experimental method to determine the concentration of PL, the bound form.

⊕Although it is called a “binding assay” the equilibrium is discussed in terms of the dissociation, with \(K_d\)

Myoglobin Binding

knitr::include_graphics('250px-Myoglobin.png')

#  Nonlinear Regression / Binding Curve 
#  simulate noisy data for myoglobin  KD = 0.26  [L] in kPa
#  data generated with random noise added 

library(nls2)

## Loading required package: proto

 KD <- 0.26              # initializing variables
 L <- 0.0  
 y <- 0.0
 
 L <-  seq(.1,1.5,0.1)   #  set of 15 ligand concentrations 
 
 rnd <- rnorm(15,0,0.04)   #   set of 15 random values

 y <-  L/(KD+L) + rnd     #  binding curve fuction with noise
 
 plot(L,y, main = "Simulated binding data", xlab = "pO2 Kpa",
      ylab = "Fractional Binding")

tryfit <- nls2(y ~ L/(KD+L),  
              start = c(KD = 5))

summary(tryfit)

## 
## Formula: y ~ L/(KD + L)
## 
## Parameters:
##    Estimate Std. Error t value Pr(>|t|)    
## KD  0.23870    0.01357   17.59  6.1e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.04085 on 14 degrees of freedom
## 
## Number of iterations to convergence: 6 
## Achieved convergence tolerance: 1.192e-06

plot(L,y, main = "Myoglobin Binding Curve", xlab = "pO2 Kpa", ylab = "Fractional Binding")
  
lines(L,predict(tryfit), col = "blue")    # add a line fitted to points