The Slope the Threshold Hides

I.The First Known Thing — the law of the small and the large

seismology · the frequency–magnitude distribution of an earthquake catalogue

Earthquakes are not counted one at a time. They are counted by size, and the count has a shape. In 1944 Beno Gutenberg and Charles Richter, tabulating the earthquakes of California, found that the number of events at or above a given magnitude falls off as a straight line when the logarithm of the count is plotted against the magnitude. The relation is written log N = a − bM, and since magnitude is itself a logarithm of radiated energy, the line says that for every step up in size there are roughly ten times fewer events. The intercept a is the productivity of a region — how many earthquakes it makes. The slope b is something else. It is the ratio of small to large, and it is close to one nearly everywhere on Earth, in every catalogue, at every scale from a mine to a subduction zone. The near-constancy of b is one of the few laws seismology has, and the field has spent seventy years learning to read its small departures.

the floor read from the catalogue itself

The straight line does not go on forever downwards, and the reason is not the Earth. Below some magnitude a seismic network no longer records every event: the small ones are lost in noise, or fall between stations, or are dropped by the detector. The observed count therefore rolls off beneath the line, and the point where it does so is called the magnitude of completeness, written M_c. The field's central methodological discovery is that M_c can be estimated from the catalogue alone, without any separate calibration of the instruments, because the law supplies the expectation: the catalogue is complete down to the magnitude at which the observed count stops following the Gutenberg–Richter line and starts falling away from it. Stefan Wiemer and Max Wyss set out the standard procedures in 2000 — find the maximum curvature of the frequency–magnitude curve, or fit the line to successively lower cut-offs and take the lowest at which the fit still holds — and a catalogue is not used for anything until its M_c has been mapped, in space and in time, since a new station lowers the floor and an old one failing raises it.

and what the slope turned out to mean

Once the floor is known, b is estimated from the events above it — Keiiti Aki showed in 1965 that the maximum-likelihood value is fixed by the mean magnitude above the cut-off and nothing else — and the estimate is then a property of the source rather than of the instrument. This matters because b carries information that the count does not. In 2005 Danijel Schorlemmer, Wiemer and Wyss showed from global catalogues that b is lowest where faults are in compression, highest where they are in extension, and intermediate where they slide past one another, and read this as differential stress: a crust under high stress makes proportionally more large ruptures, and the slope of its catalogue steepens or flattens accordingly. Laboratory rock samples under load show the same. Two regions with identical productivity can sit under different stress, and the productivity will not tell you; the slope will. The three quantities are therefore kept strictly apart. a is how much. b is the ratio of small to large, and is a property of the source. M_c is where the instrument stops, and is a property of the network. A count above any single threshold is a function of all three and reports none of them.

II.The Second Known Thing — the hit rate

drug discovery · hit finding, screening and library design

Walk to a hall where nothing shakes. A programme that has chosen its target must next find molecules that act on it, and the instrument for that is the screen: a library of compounds — a few hundred fragments, or a million drug-like molecules — tested against the target, one well at a time, for a signal that crosses a line. The line is a threshold. A compound is a hit if it inhibits by more than some percentage at some concentration, or if its confirmed potency is better than some value; ten micromolar is the usual mark for a high-throughput screen, a few hundred micromolar for a fragment screen. Everything the campaign reports downstream — the hit rate, the hit list, the triage — is defined relative to that line. The field's quality metrics, of which the Z-factor is the best known, measure whether the assay separates its positive and negative controls cleanly enough for the line to be trusted. They say nothing about what lies beneath it.

the count that became a property of the target

The hit rate was originally a statistic about a campaign. It has since become a statistic about a target. In 2005 Philip Hajduk and colleagues at Abbott reported that the fraction of a fragment library that binds a protein, as measured by NMR, correlates with whether that protein has ever yielded a drug-like ligand, and proposed the hit rate as a druggability index. The idea has held up and is now general practice: a fragment screen is run, in part, to learn whether the target is worth pursuing, and a low hit rate is taken as a warning. Structural methods estimate the same thing from the pocket alone, and the screening hit rate is what they are validated against. The count above the line has become the field's measure of a target's ligandability.

and what the count is made of

But the count above the line is one number, and the field has long known, in pieces, that at least two things go into it. The first is the shape of the distribution of potencies beneath the line. Lower the threshold and there are more hits; the relation is steep, and everyone who has run a fragment campaign knows that the weak binders vastly outnumber the strong. Mike Hann's model of molecular complexity gave a reason in 2001: the probability that a molecule fits a pocket at all falls as the molecule grows, so a library has many weak partial matches for every strong one. The second is the sensitivity of the assay. Every format has a floor — a potency below which a real binder produces no signal the detector will report — and the floor moves with the concentration screened, the readout, the protein construct and the day. A hit rate is therefore a count of everything above a line, taken by an instrument that stops hearing somewhere below that line, of a distribution whose steepness is not reported. When two targets return different hit rates, or one target returns two, the field cannot say from the number whether the source differed or the floor did. It has a count. It has not, so far as any published screen shows, fitted a slope.

III.The Bridge

no screening review cites Gutenberg

The halls share nothing. The statistics of earthquake catalogues and the design of screening campaigns have no common literature and no reviews spanning them. Power laws have been noticed in chemistry, in the sizes of scaffold families and the degree distribution of target networks, but not in the frequency of hits against potency, and the screening vocabulary — hit rate, Z-factor, ligandability, false negative — was built without the seismologists' habit of reading two parameters and a floor out of one distribution. But both halls are taking a census of events of graded size with an instrument that hears the large and loses the small, and one of them worked out, because it had no choice, how to tell the source from the instrument using the census alone.

X: a threshold count is the product of a slope and a floor, and reports neither

Call it slope–floor separation. Where events come in graded sizes and the small outnumber the large by a regular ratio, a census taken by an imperfect instrument has three parameters, not one: how many events the source makes, how steeply their number falls with size, and where the instrument stops hearing. The slope is a property of the source and is read from the census above the floor; the floor is a property of the instrument and is read from where the census falls away from the slope; and the two readings are separable because the regularity of the ratio supplies an expectation that the roll-off can be measured against. From which follows the half that the screening hall has not ported: a count above a single threshold is a function of all three parameters and carries none of them, so a hit rate cannot say whether a target is rich in binders or merely rich in weak ones, nor whether a low hit rate is a hard target or a deaf assay — and a ligandability index built from the count inherits the assay's floor as if it were a property of the protein. The seismologists' habit reverses the order of operations. First find the floor from the roll-off. Then fit the slope above it. Then, and only then, compare the count — and compare it at a threshold above every floor in the comparison, not at the threshold that convention supplies.

IV.The Question That Becomes Possible

neither discipline would ask this alone

The data for a slope already exists wherever a screen has been followed by dose–response on its actives. A confirmatory assay returns a potency for every confirmed hit, which is a frequency–potency catalogue for that target on that library; the public repositories hold thousands of such pairs, and every fragment campaign that reports a dissociation constant for each hit holds another. What is not done is to plot the logarithm of the number of hits at or better than each potency against that potency, find the roll-off, fit the line above it, and treat the slope as a quantity in its own right.

the slope hypothesis

Is the slope of a screen's hit catalogue — the fall in the number of actives per unit of log potency, fitted above the assay's own roll-off — a property of the target that is stable across libraries and assay formats, and does it predict which hit sets go on to yield an optimisable series where the hit rate at a fixed threshold does not?

why it would matter

Because the two things a hit rate conflates come apart in a direction that can be checked. Two targets returning the same hit rate at ten micromolar may have very different slopes: the steep one has a great many weak binders and almost no strong ones, the shallow one has fewer hits at the line but proportionally more well beneath it, and it is the second that a medicinal chemist would rather have, since the shallow slope says that potency is available in the pocket. Under the mechanism a shallow slope should mark the targets whose hits optimise, and the steep slope should mark the ones that give a hundred fragments and no lead — a pattern the field describes anecdotally, as targets that are ligandable but not druggable, and has no number for. The floor does something as well. Since the roll-off locates the assay's effective limit of detection from the screen's own output, the same fit reports whether a low hit rate was the target's doing or the assay's; a target that returns few hits on a screen whose roll-off sits far above the threshold has not been screened at the threshold at all. And the seismic hall offers a prior that cuts against the candidate, which is why it is worth stating. Earthquake b is close to one nearly everywhere, and its departures are small and were hard to establish. If potency slopes prove as uniform, the slope is a property of chemical space rather than of targets, Hann's model has explained it, and there is nothing to read.

V.How It Could Be Proven — or Broken

a candidate must be breakable, or it is merely poetry

An intercognix candidate earns nothing until it names its own test. This one is retrospective first, and the data for it sits in public bioassay repositories and in the fragment campaigns that were generous enough to publish an affinity for every hit.

the honest status

Note the narrowness. The claim is about screens that report a potency for each hit, and about a single slope and a single floor read from that distribution; it says nothing about screens that report only a percentage at one concentration, from which no catalogue can be built. It proposes no new assay and no new library. It takes the field's existing ligandability index and says only that the index is a count, that a count has a slope and a floor inside it, and that the two are readable and should be read before the count is compared. And a caution about the analogy's centre: an earthquake catalogue is a census of one source over time, whereas a screen is a census of one source over a chosen set of probes, and the probes were chosen — so the slope may be as much a portrait of the library as of the target, which is exactly the third way the candidate breaks. Until a set of hit catalogues has been fitted above their roll-offs and set against what their targets went on to become, this remains exactly what its stamp says: a candidate. A structural echo between two halls of the library, waiting for someone with the tools of both to walk the corridor.