Reading a reactivity graph
Probing antibody repertoires with mid-range peptide libraries (n=103 - 104) as microarrays yields data that can also be presented as a graph of cross-reactivities. Cross-reactivity is detected by measuring the correlation between the binding profiles across serum samples from individuals (usually grouped by diagnosis)(1). Obviously, thus measured, the cross-reactivity is related to a set of reactivities found in the patients studied with respect to the peptide library used. The IgM reactivities to known self and viral tumor (associated) antigens in patients with brain tumors(1) showed a much more pronounced correlation with ABO blood group than in patients with neurodegenerative diseases, as probed with an optimized IgM IgOme library(2).
The graph, briefly
Nodes are reactivities (mimotopes, or microarray features); edges here encode a supra threshold correlation between the profiles. Write the adjacency matrix \(A\) and degree matrix \(D\). The combinatorial Laplacian
\[ L \;=\; D - A \]
has eigenpairs \(L\,\mathbf{v}_k = \lambda_k\,\mathbf{v}_k\) with \(0 = \lambda_1 \le \lambda_2 \le \dots\). The smallest non-trivial eigenvectors give a low-dimensional embedding in which graph structure becomes geometric — the coordinate system we then feed to UMAP (itself using spectral clustering) for visualization.
Blood Group Reactivity Correlation Sometimes Dominates
The first thing a reactivity graph tells you is often the thing you already knew. The signal you want lives in the residual.
Regress it out, or model it?
Two roads diverge here:
- Regress it out. Project out the blood-group component — e.g. remove \(\mathbf{v}_2\) (and any other dominant background modes) before clustering. Clean, but risks discarding real signal correlated with the background.
- Model it. Treat blood-group reactivity as an explicit covariate in a generative model of the graph, so disease structure is estimated conditional on it rather than after subtracting it. More honest, more work.
My current preference is to model rather than subtract, because the “background” and the disease signal are unlikely to be orthogonal. But I do not have a clean demonstration that one beats the other across cohorts — that is exactly the kind of sensitivity analysis this notebook exists to record.