Working with a different kind of AI

Enzyme-linked immunosorbent assay

Source: Tizard, Ian R. Veterinary immunology. 9^th ed. St. Louis, Missouri: Elsevier; 2013. 499 p.

This talk is a stats talk, but in order to understand the data we do need some context- and also I did promise in my abstract that I would explain what things like “optical density” are and “what is being diluted”.

The data came from running an ELISA, and this slide shows a basic overview of what it does:

On the right we see a 96-well plate. On the left we see 5 steps on what happens in each well in the standard/control case:

The first step is to coat each well with the antigen (here: recombinant protein/microbe vaccine). It is adsorbed to the bottom of the plastic wells.
Animal blood serum/saliva - which may contain antibodies- are added to the well. If antibodies are present, then antigen-antibody complexes form.
Next, a secondary antibody, antiglobulin is added that assists with the detection of these complexes.
For easier detection, the antiglobulin is covalently linked to an enzyme.
Finally, when the enzyme binds to its substrate, a reaction occurs to create a coloured product.

Disruption

Between step 2 and 3, a reagent is added that disrupts the antibody-antigen binding.

As a result, the proportion of colour-making antibodies is reduced.

Serial dilution

Dilute the amount of serum added

\(\log_2(800) = 9.64, \log_2(1600) = 10.64....\)

Quantify the change in colour, more blue detected = higher OD value

When wanting to understand the antibody specificity and binding ability we need to do multiple ELISA runs for each animal, where we measure the amount of antibodies found for a series of samples where we dilute the amount of sera added from one well to the next. This is talked of as serial dilution ELISA and is the reason we need so many wells:

There are 12 columns and 8 rows in a 96 plate well -

Each row is used for a given animal/replicate ; while 11 of the columns are used for 11 dilutions. The 12th column is used as a positive control.

Here we have dilutions starting from 800. 800 is the least diluted and we double the dilution from each well to the next, that is, a dilution from 1/800 to 1/1600 is a 2-fold difference between each dilution.

The range of dilutions depend on medium and type of antibody- and there is a limit to how “undiluted” you can go. Bear this is mind as it will become relevant later on.

To help with the plotting and analysis, we transform the dilutions so we have a difference of 1 between each doubling of the dilution. We use the log transformation with base 2, which gives us 9.64, 10.64 etc. for dilutions of 1/800, 1/1600 etc.

-----

Optical density = you shine a beam of light through the sample and work out how much light reaches the detector and how much light has been absorbed or reflected by the sample. This is used to measure the amount of antibodies - once a certain amount of light is filtered out you know the population is dense enough.

Our data

Treated curve shows response when the antibody binding is disrupted, control is without disruption.

Lower OD values mean lower density of antibodies.

Area under the curve

Compute the area under a curve (AUC) by dividing it into trapezoids that sit as closely as possible to the line, and adding up their areas to get an approximate value.

\[ \approx \ \frac{\bigtriangleup x}{2}[f(x_0) + 2f(x_1) + \\ ... + 2f(x_{n-1}) + f(x_n)] \]

Different kind of AI

Our AI refers to a measure of curve comparison called

avidity index

which is defined as

\[ \text{AI} = \frac{\text{AUC}_\text{treated}}{\text{AUC}_\text{control}}. \]

The smaller the value, the bigger the disruption.

AI based on raw data

We compute the AUC values based on the actual observed values as is, eg. for animal D100 we have:

\[\text{AUC}_t\]	\[\text{AUC}_c\]	\[\text{AI}_{raw}\]
0.984	6.66	0.148

4PL models

Model relationship between the dilution levels and their corresponding dose responses using four-parametric logistic (4PL) regression models that are defined as follows:

\[ y(x;b,l,u,e) = l + \frac{u-l}{1+ \exp(b(\log(x) -\log(e)}, \]

where \(y\) and \(x\) are the response and dose variables, respectively.

Parameters of 4PL model

Schematic curve

Our fitted data

4PL curve superimposed over raw data

AI based on fitted data

When computing AI value based on AUC from the fitted values (green for treated, blue for control), we get a similar result to before (0.148 vs. 0.144)

\[\text{AUC}_t\]	\[\text{AUC}_c\]	\[\text{AI}_{fit}\]
0.990	6.88	0.144

So far so good – but…

One curve is biologically speaking a shifted version of the other, and we’re only capturing a small part of the curves.

We’re seeing the bits between the yellow lines

Our data extrapolated

Going beyond the pale

AI based on extrapolated data

Same procedure as before…

\[\text{AUC}_t\]	\[\text{AUC}_c\]	\[\text{AI}_{extra}\]
6.031	15.677	0.385

…get a rather different result: 0.385 vs 0.148 (raw) and 0.144 (fitted).

Overview

Results for ALL the data
\[\text{Animal ID}\]	\(\text{AI}_{raw}\)	\(\text{AI}_{fit}\)	\(\text{AI}_{extra}\)
D097	0.261	0.282	0.496
D098	0.192	0.191	0.466
D099	0.152	0.144	0.442
D100	0.148	0.144	0.385

Not only are the results on a different scale, they paint a different picture.

Closing remarks

Fitting curve is a step up to working with raw data.
Still concerned about dependence of AI calculations on range of dilutions.
However: can we trust the results from the extrapolated curves?
Other idea: first fit a curve to control data, then shift until it fits the treated best.

The present AI methodology tend to depend on the titre to an uncomfortable degree.

We’re never going to be able to capture either curve in its entirety, so extrapolation will always be necessary. This is risky, however.

Thanks for listening!

\[ \ \]

Co-authors: Peter Janssen, Sofia Khanum, Neil Wedlock, Juliana Yeung

Any questions?