Non-linear Regression

The rate at which a protein – with one binding site – binds to its ligand is related to the concentrations of the free and bound receptors, and to the concentration of the free ligand. The equation to describe these relationships is similar to the Michaelis-Menten equation, which describes enzyme kinematics. The binding kinetics equation is:

Where SB is defined as specific binding of the ligand to the protein, Bmax is the total protein concentration (or free and bound protein concentration), [L] is the free ligand concentration, and Kd is the dissociation rate constant (the ratio between the backward and forward reaction rate constants).


Usually, Kd is the one unknown variable in the equation that is of interest to me. To determine its value, I'll run a saturation binding assay in which I mix the protein with a dilution series of its ligand and measure the amount of protein-ligand complexes. The Kd can then be determined by Scatchard plotting or non-linear regression, the latter method of which is considered to be the gold standard.


I work with a protein called corticosteroid binding globulin (aka transcortin, CBG, and serpin peptidase inhibitor clade A member 6). One of its ligands in most mammals is cortisol, and the interplay between this protein-ligand pair plays significant roles in modulating the stress response, the immune system, and energy balance, amongst other systems. I use tritiated cortisol as a tracer molecule in a saturation binding assay to determine the binding kinematics of cortisol to CBG.


After running the assay, I get specific binding data in counts (of beta decays) per minute dependent on the free ligand concentration. An example data set is shown:

Dilution SB     LA        1134   46.28B        798    22.48C        656    10.95D        493    5.24E        305    2.50F        198    1.25

A plot of the data (plot()) shows an exponential relationship between SB and [L].

In a Scatchard plot, I plot specific binding per free ligand concentration on the y-axis against specific binding on the x-axis. This transformation results in a roughly linear plot that is suitable for linear regression analysis.

I'll use the lm() command to run a linear regression and abline() to plot it on the above scatterplot. These are the results:

Call:lm(formula = sbl.518 ~ sb.518.3, data = dfl.518)
Residuals:      1       2       3       4       5       6  19.697 -17.888 -14.008  -3.400  -2.666  18.264 
Coefficients:             Estimate Std. Error t value Pr(>|t|)    (Intercept) 168.76344   15.59183  10.824 0.000413 ***sb.518.3     -0.14458    0.02312  -6.253 0.003335 ** ---Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 
Residual standard error: 17.72 on 4 degrees of freedomMultiple R-squared: 0.9072, Adjusted R-squared: 0.884 F-statistic:  39.1 on 1 and 4 DF,  p-value: 0.003335 

The absolute value of the slope of the line (the sb.518.3 coefficient) is inversely proportional to the Kd of CBG, or Kd = 1/0.1445 = 6.92. I can also calculate Bmax as the y-axis intercept (Intercept) divided by the absolute value of the slope. The adjusted R-squared value is not very high (0.884), so this suggests to me that this Kd is a ballpark estimate of the real Kd.

The other way to determine Kd is by non-linear regression. This method is considered to be 'better' than Scatchard analysis because I can define the equation of the line to be fitted to my data, rather than transforming my data to fit a linear line. This can be done in R using the nls() command:

nls.518 <- nls(SB ~ Bmax * L / (Kd + L), data = dfl.518, start = list(Bmax = 1167, Kd = 6.92)

So in the nls() command, I define the equation to be used to fit my data as SB ~ Bmax * L / (Kd + L), and I also define starting values for Bmax and Kd as the values I derived from my Scatchard analysis. R needs these values as the starting point to its analysis. My results are:

Formula: SB ~ Bmax * L / (Kd + L)
Parameters:
            Estimate Std. Error t value Pr(>|t|)  
Bmax         982.2157    17.2532   56.93 1.19e-05 ***
kd             5.3959     0.2556   21.11 0.000233 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 9.391 on 3 degrees of freedom
Number of iterations to convergence: 3
Achieved convergence tolerance: 5.493e-06 

So by using non-linear regression analysis, I've determined the Kd and Bmax to be 5.40 and 982.2, respectively, which are quite different than the values I obtained using Scatchard plotting. The p-values are highly significant, and the residuals are fantastic.

So what does it all mean? One practical interpretation of Kd is that at a ligand concentration equal to the Kd, half of the receptors are bound to its ligand. The lower the Kd, the higher the affinity of the receptor to its ligand, and vice versa.