What Each Theory Explains — and Fails to Explain

The table below summarizes, for each theory, the phenomena it accounts for, the phenomena it cannot accommodate, and the specific tenets that have been most decisively contradicted by experiment. It is based on Martins et al. (2024)¹ and on the broader critical literature on each theory. “Refuted concepts” is used in the strong sense — claims that were falsified by direct evidence — and is distinguished from mere explanatory gaps in the “Does not explain” column.

Taken together, the pattern is consistent with the paper’s central claim: each theory illuminates a real facet of immune recognition, yet none survives as a complete account — the reductionist branch (two-signal, stranger, danger) struggles with context-independence and circularity, while the systemic branch (idiotypic, symmetrical, completeness) struggles with the empirical weakness of pervasive idiotypic regulation. A unifying “theory of everything” for immunology remains outstanding¹.

References

Here is what we found so far

Studying the changes in IgM repertoire in antiphospholipid syndrome (APS) patients, we found that a 0.5% of the IgM reactivities, normally found in healthy donors, are lost in APS patients. The APS specific IgM reactivties were several fold lower in number.The mimotope sequences of changed reactivities were mapping to idiotopes more often than expected, but those in healthy were also related to public reactivities while those in APS - not⁴.

Next, we tested the capacity of our optimized public IgM mimotope library to differentiate neurodegenerative diseases. The serum IgM (but not the IgG) distinguished a large cluster of public reactivities that were lost in Alzheimer’s and frontotemporal dementia, but not in other forms of dementia, and the respective mimotope sequences were non-randomly homologous to idiotopes⁵. Interestingly, the IgG reactivities better differentiated Alzheimer’s disease from frontotemporal dementia, but did not correlate with idiotypy.

Thus, IgOme maps show changes associated with autoimmune pathology with two recurrent features:

- Loss of public IgM reactivities and

- Non-random association with idiotypic reactivity.

What is IgOme?

The term IgOme was coined in a paper from the Jonathan Gershony’s lab¹. It consists of bulk mimotope selection from a random peptide phage display library, adsorption on non-specific monoclonals, and NGS of the regions coding for the peptide inserts. This technique can be coupled with bioinformatic analyses of the mimotope libraries for finding of clusters and motifs. It also helps infer properties of the global shape of the reactivity space.

Why this is more than a reframing

Three things follow that the binary picture cannot give:

• Polyreactivity becomes a measurable quantity, not a nuisance — an entropy, not an error bar.
• Mimotope arrays become samples from the space of peptides. A microarray of peptides probes the landscape at many points at once; the measured reactivities are an empirical sketch of the distribution. Scalable mimotope libraries make this sampling practical at the scale of the public repertoire².
• The repertoire becomes an ensemble of distributions, opening the door to genuinely statistical-mechanical questions about the population of antibodies as a whole.

None of this is settled. The epitope space is not obviously enumerable, the parameters are not yet estimated or measured, and whether the equilibrium reading is the right one in a dynamic immune system is an open question. But as a way to organise mimotope and microarray data — and as a bridge to the repertoire-physics program — treating specificity as a shape has been more productive than treating it as a switch.

Viewing specificity as a distribution of binding energies may seem hard to reconcile with negative selection. Each monoclonal antibody can select thousands of short peptides from a random peptide library with a biologically relevant affinity. These peptides are found to span the entire peptide space. If each individual reaction can have biological consequences, then the probability of an antibody surviving negative selection would become negligible. Indeed, up to 70% of the early immature B cells in the bone marrow are self-reactive³ and most of them are eliminated by negative selection.

How can these views be reconciled:

• The affinities for small peptides are below the threshold for negative selection, the epitopes above that threshold are less frequent.
• The tolerogenic signals have typically been found to depend on high avidity⁴ .
• The accessible self-antigens, presented at sufficient concentration and local density, are probably many orders of magnitude fewer than the random peptide species in a phage library.
• The frequency of self-reactive BCR rearrangements is surprisingly high.Indeed, up to 70% of the early immature B cells in the bone marrow are self-reactive³ and most of them are eliminated by negative selection.

Thus, falsifying this concept would require careful modeling and definition of the model’s parameters.

References

The graph, briefly

Nodes are reactivities (mimotopes, or microarray features); edges here encode a supra threshold correlation between the profiles. Write the adjacency matrix and degree matrix . The combinatorial Laplacian

has eigenpairs with . 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

Based on data from² and¹.

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 (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.

The one equation you need

Let index epitopes in some space of antigenic determinants, and let be the binding free energy of a given antibody for epitope (more negative means tighter binding). Borrowing the Boltzmann form, the relative probability that the antibody, in equilibrium and far from saturating any one target, is engaging epitope is

where is the inverse temperature and is the density of epitopes at energy . This is literally a canonical distribution, with a partition function over epitope space rather than over the internal states of one site. The antibody’s identity, in this view, is the function — the landscape — not any single favoured partner.

An antibody is not a key. It is a temperature-weighted preference over a landscape of locks.

That is the entire theoretical commitment. Everything below is consequence and calibration. The same canonical reading of serum binding has been developed independently and in depth by Prechl, who models the repertoire as a fused binding-energy “super-landscape” whose partition function and distribution shape are the objects of interest^3,4.

Specificity as a shape statistic

Once specificity is a distribution, we can measure it. Three scalars capture most of the intuition:

A monoclonal with one dominant epitope and a natural polyreactive IgM occupy opposite ends of these axes — not different categories, but different points on a continuum of distribution shapes. This is exactly the move Janin made when he applied the Random Energy Model to antigen–antibody recognition: specificity is the gap-to-width ratio of an energy distribution, not a binary attribute⁵. The same gap-versus-roughness criterion recurs across the molecular-recognition literature as the operational definition of intrinsic specificity⁶.

The monospecific–polyspecific boundary as a freezing transition

If the epitope energies are drawn from a Gaussian with mean and variance over epitopes, the Random Energy Model predicts a freezing transition at

Below this temperature () the distribution collapses onto the single best epitope — sharp specificity; above it, many epitopes contribute — cross-reactivity^5,7. Here is the number of effectively independent epitopes the antibody is sampled against: the number of distinct peptide features on a microarray, or the diversity of mimotope clusters recovered from a phage-display library. It enters through the extreme value of Gaussian draws,

so a larger probed library pushes the best binder further below the bulk and raises — freezing sets in more easily.

Two cautions keep this honest. First, the dependence on is only logarithmic: doubling array size barely moves , whereas enters linearly and does the real work. Second, the REM assumes uncorrelated energies, while real epitope landscapes are correlated through overlapping motifs and sequence families — so use as an order-of-magnitude guide, not a law, and feed it an effective diversity (e.g. number of sequence clusters), not the raw feature count.

Affinity and kinetics are independent inputs

A point the equilibrium picture alone hides: the energy gap fixes the dissociation constant,

so stronger binding (more negative ) gives a smaller — the direction most easily gotten wrong. But the rates are set by activation barriers, not by the gap:

Detailed balance pins only the difference of the barriers, , so the same is consistent with many pairs. Two antibodies of identical affinity can have wildly different residence times — kinetics carries information that affinity alone does not, and affinity maturation is in part the sculpting of slower-off-rate, deeper wells⁸. is therefore an equilibrium quantity related to kinetics through ; it is not itself a kinetic metric.

A further consequence worth stating because it is so often mis-pictured: association in solution is diffusive, not ballistic. A ligand does not fly over its barrier with Maxwell–Boltzmann velocity; it random-walks across it under heavy solvent friction (Kramers’ overdamped regime), with a diffusion-limited ceiling on of order – set by Smoluchowski⁸. Any reported on-rate above that ceiling is suspect.

What a microarray actually measures

This is where the paradigm earns its keep, because it makes the measurement precise. A mimotope array probes the landscape at many points at once^9,10, but what it reads out depends on antibody concentration. Each epitope independently follows its own Langmuir isotherm,

and the two regimes of this expression are directly testable:

• Sub-saturating for all : . The spot pattern is the Boltzmann affinity distribution. This is the regime in which the array faithfully reports specificity shape, and the one in which the shape statistics , , are meaningful.
• Saturating : for every epitope. The array flattens, erasing strong-versus-weak distinctions. A flat, polyreactive-looking profile can be an artifact of too much antibody, not true polyreactivity — a concrete prediction of the model and a warning for titration design.

A practical recipe follows. From an intensity vector at known : subtract background and set a scale; invert the Langmuir relation away from saturation,

then build and summarise specificity via , , .

When antibody is limiting and epitopes compete for it, the picture becomes a genuine network: solving mass conservation self-consistently couples the epitopes, and for polyclonal serum the shared epitopes turn the affinity distribution into Prechl’s cross-reactivity super-landscape^3,4 — the regime where statistical-mechanical models of antibody mixtures become necessary¹¹.

A note on “temperature”

Because the formalism is borrowed, “temperature” is used in three different senses, and conflating them is the main way to abuse the analogy.

1. Literal physical is real but a weak lever: across 4 °C–37 °C the accessible range is a factor of , far too small to move .
2. Effective temperature — fitting a measured signal as if — is a width parameter, not a thermodynamic temperature. An antibody’s breadth is set by the chemistry of its CDR loops, not thermal agitation^12,13, so two antibodies in the same tube can have very different . Prefer reporting , , directly.
3. Selection / detection stringency (wash stringency, panning rounds, the positivity threshold) acts like an inverse temperature on the recovered distribution — and it is the strongest knob, but it is the temperature of a non-equilibrium process where detailed balance fails. A sharpened profile may reflect real biology or merely turned-up stringency; the two are confounded unless stringency is fixed and reported.

The cleaner, dimensionless statement is that specificity is governed by

the spread of binding energies in units of . A “cold,” specific antibody is not one at low temperature — it is one whose landscape has large , with a few epitopes sitting many below the rest.

Why this is more than a reframing

Three things follow that the binary picture cannot give:

• Polyreactivity becomes a measurable quantity — an entropy or an effective count , not an error bar — and one with a known biochemical basis: polyreactive Fabs bind diverse epitopes with uniformly low affinity and characteristic CDR signatures^1,12, while even “promiscuous” binding is built from specific hydrogen bonds across multiple discrete binding modes rather than nonspecific stickiness¹⁴.
• Mimotope and microarray data become samples from , with an explicit model of when the sample is faithful (sub-saturation) and when it lies (saturation, or non-equilibrium selection)⁹.
• The repertoire becomes an ensemble of distributions, opening genuinely statistical-mechanical questions about the antibody population as a whole — and connecting to our observation that the autoimmune repertoire is restricted rather than simply redirected¹⁵.

None of the framing layer is settled. The epitope space is not obviously enumerable, is a modelling choice rather than a measured constant, the REM’s uncorrelated-energy assumption is violated by real landscapes, and whether the equilibrium reading is the right one in a dynamic immune system is open. The mapping — affinity distribution as a canonical distribution over epitopes, with specificity as its shape — is sound and computable; the effective-temperature and freezing pictures are framing-level hypotheses. But as a way to organise mimotope and microarray data, and as a bridge to the repertoire-physics programme, treating specificity as a shape has been more productive than treating it as a switch.

The physics each step rests on is derived in full in the companion tutorial, From the Boltzmann distribution to receptor–ligand kinetics.

References