The amino-acid property basis#

This page states two properties of the vendored tables in mhcmatch.data.aa_tables. They are facts about the data tables, not about any peptide set, any label, or any benchmark — every number below is reproducible from a fresh install with no download, by the snippet at the bottom of the page. They are recorded here because both of them change what a physicochemical featurizer can possibly express, and both are easy to get wrong.

  1. The dominant eigenvector is a hydropathy axis.

  2. The Kidera factors are already orthogonal, so PCA on them is degenerate.

What is vendored#

aa_tables carries 148 residue-to-value tables: 102 components across 17 descriptor families (DESCRIPTORS), 45 hydrophobicity scales (HYDROPHOBICITY), and the Miyazawa-Jernigan partition energy (MJ_PARTITION). 142 of the 148 are complete and non-constant over the standard 20 residues; the 6 that are not are the pH-variant Wimley-White scales, which are undefined for some residues and are excluded from every statement here rather than imputed.

Provenance is in the module docstring: DESCRIPTORS and HYDROPHOBICITY are vendored from peptides 0.5.0 (GPL-3.0-or-later), MJ_PARTITION from tcren 2.8.0 (antigenomics/tcren), itself AAindex MIYS850101. The tables are generated, not hand-edited.

1. PC1 is hydropathy#

Column-standardize the 20 x 142 property matrix — residues x scales, not samples x features — and take its principal components. The first carries 32.79 % of the total variance and the second 18.43 %, so two axes hold just over half of everything the 142 scales measure. PC1’s residue order is

I F L W V M C Y A P G T H S Q N E K D R

which is a hydropathy ranking read off no hydropathy scale in particular. Against the 39 complete named hydrophobicity scales, the Spearman rank correlation of PC1 has median \(|\rho| = 0.894\), and 32 of the 39 reach \(|\rho| \ge 0.80\):

Spearman correlation of PC1 with named hydrophobicity scales (n = 20 residues)#

scale

Spearman rho

p

Fauchere

+0.9774

1.3e-13

Cowan7.5

+0.9774

1.3e-13

Eisenberg

+0.9699

1.7e-12

Roseman

+0.9643

7.9e-12

BlackMould

+0.9507

1.4e-10

Prabhakaran

-0.9383

9.8e-10

KyteDoolittle

+0.8943

1.1e-07

(Bold marks the strongest correlation; Fauchere and Cowan7.5 tie exactly. The sign of a principal component is arbitrary, so it is fixed here to make isoleucine positive; Prabhakaran is a polarity scale and its negative sign is the same statement.)

PC2 is not a second reading of the same axis — against Kyte-Doolittle it sits at \(\rho = +0.408\) (p = 0.074), i.e. not distinguishable from unrelated at this sample size.

What follows for the library. Any summed or averaged physicochemical feature is, up to a monotone reparametrization, a projection onto this one axis. So:

  • Adding hydrophobicity scales to mhcmatch.immuno.features() adds columns, not directions; a disagreement between two such features is a disagreement about scaling, not about chemistry.

  • A result that changes when the hydrophobicity scale is swapped is telling you something about the scale, not about the peptides.

  • The package ships this basis rather than recomputing it: mhcmatch.data.aa_tables.PROPERTY_PC1 and PROPERTY_PC2 are the frozen per-residue loadings, and the phys block of mhcmatch.complement.score() sums PC1 along the sequence. The basis is label-free, so it is independent of any fitted model.

2. PCA on the Kidera factors is a no-op#

The 10 Kidera factors are the orthogonal factor scores of a 1985 factor analysis of 188 physical properties. Orthogonality is not an approximation that decayed in transcription — it survives in the vendored table exactly. Over the 20 residues the largest absolute off-diagonal correlation between two Kidera factors is 0.0026, and the correlation matrix’s eigenvalues are

1.0043 1.0039 1.0024 1.0014 1.0004 0.9998 0.9988 0.9976 0.9965 0.9949

— a ratio of 1.0094 from largest to smallest, and a participation ratio of 10.00 out of 10 components. A covariance matrix whose eigenvalues are all equal is a sphere: its eigenbasis is defined only up to an arbitrary rotation, and PCA on it returns a rotation and reduces nothing.

Isotropy of three property bases over the 20 residues, uniform measure#

table

components

rank

max abs. off-diagonal correlation

largest / smallest non-zero eigenvalue

participation ratio

DESCRIPTORS["KIDERA"]

10

10

0.0026

1.01

10.00

DESCRIPTORS["VHSE"]

8

8

0.8201

50.43

4.79

all 142 complete scales

142

19

1.0000

101.23

5.99

(Nothing is bolded: this is a description of three bases, not a competition between them. A participation ratio equal to the rank means perfectly isotropic — Kidera is at 10.00 of 10 — and the further below the rank it falls, the more the variance concentrates in a few directions. The union of all 142 scales has rank 19, so 123 of those columns are exactly linearly dependent on the others, and its largest off-diagonal correlation of 1.0000 says at least one pair is the same scale twice under different names.)

The rotation destroys the hydropathy axis rather than finding it. Because the sphere has no preferred direction, the PC1 that a solver happens to return from the Kidera table alone correlates with Kyte-Doolittle at only \(\rho = -0.256\) (p = 0.28). Hydropathy is not spread across the Kidera factors — it sits almost entirely in KF4:

Each Kidera factor against Kyte-Doolittle (n = 20 residues)#

factor

Spearman rho

p

KF1

-0.3042

0.19

KF2

-0.4053

0.076

KF3

+0.2348

0.32

KF4

-0.7761

5.8e-05

KF5

+0.0370

0.88

KF6

-0.2635

0.26

KF7

-0.0355

0.88

KF8

+0.0298

0.90

KF9

+0.0015

0.99

KF10

-0.0468

0.84

KF4’s label is not this page’s invention. Rackovsky (Proc Natl Acad Sci U S A 2010, doi:10.1073/pnas.1001299107) independently names it “hydrophobicity” from a spectral analysis of localized vs. delocalized amino-acid interactions — corroborating the \(\rho = -0.7761\) measured above by an unrelated method. (KF10, “surrounding hydrophobicity” in the same table, is a different factor and is not what the -0.7761 above is about.)

Kidera factors reached epitope-immunogenicity prediction through ipred, introduced by Pogorelyy et al. (Genome Medicine 2018, doi:10.1186/s13073-018-0577-7) to link epitope physicochemistry to naive-repertoire precursor TCR frequency and immunogenicity: an EM-trained naive Bayes classifier over a multinomial-normal model of the ten Kidera-factor sums, returning a probability of T-cell immunogenicity, trained on the Chowell corpus — KF4 among its ten inputs, not singled out. Buckley et al. (Briefings in Bioinformatics 2022, doi:10.1093/bib/bbac141) benchmarked it against other CD8 T-cell immunogenicity predictors.

So: use the raw factors, or run PCA over the multi-family union where the collinearity is real. Running it over Kidera alone spends compute to scramble an interpretable basis.

Scope: what the degeneracy is not#

The no-op holds for the 20 x 10 alphabet table under the uniform measure over the 20 residue types. It is not a general licence to skip PCA anywhere Kidera factors appear:

  • Reweight the alphabet and it breaks. Under residue frequencies rather than one weight per residue type — for example mhcmatch.diffusion.PROTEOME_AA_FREQ — the same table gives an eigenvalue ratio of 3.72 and a participation ratio of 8.92 of 10, not 10.00 of 10. Isotropy is a property of the table and the measure.

  • Sum along a peptide and it breaks. A peptides x 10 summed-Kidera matrix inherits the composition statistics of the peptide set and is strongly anisotropic; PCA on that matrix does real work. Anyone reading “PCA on Kidera is degenerate” as covering that case will be wrong.

Reproduce#

No data download, no benchmark repo, numpy only:

import numpy as np
from mhcmatch.data import aa_tables as t

AA = list(t.AA20)

def spectrum(tables):
    """Eigenvalues of the residue-by-scale correlation matrix, uniform over residues."""
    A = np.array([[tab[a] for tab in tables] for a in AA])
    Z = (A - A.mean(0)) / A.std(0, ddof=0)
    ev = np.linalg.eigvalsh(Z.T @ Z / len(AA))[::-1]
    return ev[ev > 1e-9]

kidera = [t.DESCRIPTORS["KIDERA"][f"KF{i}"] for i in range(1, 11)]
ev = spectrum(kidera)
ev[0] / ev[-1]                          # 1.0094 -- a sphere
ev.sum() ** 2 / (ev ** 2).sum()         # 10.00 of 10 components

scales = [tab for fam in t.DESCRIPTORS.values() for tab in fam.values()]
scales += list(t.HYDROPHOBICITY.values()) + [t.MJ_PARTITION]
scales = [s for s in scales if all(a in s for a in AA) and len(set(s.values())) > 1]
len(scales)                             # 142

A = np.array([[s[a] for s in scales] for a in AA])
Z = (A - A.mean(0)) / A.std(0, ddof=0)
U, S, Vt = np.linalg.svd(Z / np.sqrt(len(AA)), full_matrices=False)
(S ** 2 / (S ** 2).sum())[:2]           # 0.3279, 0.1843

pc1 = Z @ Vt[0]
"".join(np.array(AA)[np.argsort(-pc1)])  # IFLWVMCYAPGTHSQNEKDR (up to overall sign)

The two claims are asserted as regression tests in tests/test_mhcmatch.py (test_property_matrix_pc1_is_a_hydropathy_axis, test_kidera_table_is_orthogonal_so_pca_on_it_is_degenerate, test_kidera_degeneracy_is_specific_to_the_uniform_alphabet_measure), so a table regenerated with different provenance cannot silently change them.