Data fitting theory

Law of mass action

During the association phase the complex [LA] increases as a function of time. The differential rate equation describes the relation between the parameters:

[LA] complex formation

In the equation for complex formation the k_a is the association rate constant which describes the rate of complex formation, i.e. the number of LA complexes formed per second in a one molar solution of L and A. The units of k_a are M^-1s^-1 and are typically between 1.10³ and 1.10⁷ in biological systems.

k_d is the dissociation rate constant which describes the stability of the complex, i.e. the fraction of complexes that decays per second. The unit of k_d is s^-1 and is typically between 1.10^-1 and 1.10^-6 in biological systems. A k_d of 1.10^-2 s^-1 = 0.01 s^-1. This means that 1 percent of the complexes decay per second.

Integration

Although the differential rate equation describes the one to one Langmuir reaction accurately, it is not useful as a model to fit to a curve. By integrating the differential rate equation, sensorgrams can be analysed directly using non-linear methods (6). This is because integrated rate equations describe the whole curve as opposed to the differential rate equations, which describe the slope of a curve.

The classical Langmuir surface adsorption (7) can be described with the following integrated rate equation(6)

Integrated rate equation

Numerical integration

While simple rate equations are easily integrated, the more complex (e.g. with mass transport parameters) are not. The more complex rate equations are resolved by numeric calculations. Using numerical integration (a.k.a. non-linear regression) has also the advantage that global analysis can be linked in. This means that several curves are fitted simultaneously giving a single and more robust answer for the rate constants (8).

Non-linear regression

Non-linear regression is a more general approach to data fitting because all models that define Y as a function of X can be fitted. The process will find the values for the variables giving the closest fit, reducing the sum-of-squares (SS) to a minimum. The SS is calculated from the vertical distances between the curve and the measured points.

Sum-of-squares

CHI²-calculation

Assumptions with non-linear regression

The results of non-linear regression are meaningful only if the following assumptions are true (or nearly true):

• The model is correct. Non-linear regression adjusts the variables in the chosen equation to minimize the sum-of squares. It does not attempt to find a better equation.
• The variability of values around the curve follows a Gaussian distribution. Although no biological variable follows a Gaussian distribution exactly, it is sufficient that the variation is approximately Gaussian.
• The SD of the variability is the same everywhere, regardless of the value of X. If the SD is not constant but rather is proportional to the value of Y, the data should be weighted to minimize the sum-of-squares of the relative distances.
• The model assumes that X is known exactly. This is rarely the case, but it is sufficient to assume that any imprecision in measuring X is very small compared to the variability in Y.
• The errors are independent. The deviation of each value from the curve should be random, and should not be correlated with the deviation of the previous or next point.

References