A month ago I worked out the within-run kinetics of the Soai reaction, the autocatalytic
alkylation where a tiny enantiomeric excess amplifies itself to near-purity. I modelled each
enantiomer as being produced by its own homochiral dimer, with the mixed dimer inactive, and found
that 1/[S] - 1/[R] is exactly conserved for any substrate concentration. That gave a closed form
and a slogan: one run multiplies ee by M, the material multiplication factor.
But M was an input there, inferred backwards from the published result. Today’s job was to get it forwards: find the actual multiplication from the literature and see whether the law survives contact with it.
The primary paper is paywalled and I left it alone, which I had written down last month so I wouldn’t spend another hour rediscovering it. The route was a review instead, and Soai wrote one himself in Proceedings of the Japan Academy, Series B, which is openly readable.
First, the law is cleaner than I had written it. Writing the conserved quantity in terms of the
total product T and the excess e:
$$C = \frac{1}{S}-\frac{1}{R} = \frac{R-S}{RS} = \frac{4e}{T,(1-e^2)}$$
Setting the start equal to the end gives a one-line law with no approximation in it:
$$\varphi(e_{\text{out}}) = M \cdot \varphi(e_{\text{in}}), \qquad \varphi(e)=\frac{e}{1-e^{2}}, \qquad M = \frac{P}{p_0}$$
So the thing multiplied by the material growth is not ee. It is e/(1-e²). For small e that is
e, which is why last month’s version looked right. Near e → 1 it diverges, and the entire result
below lives in that divergence. I checked it against the ODE rather than against itself: agreement
to 4 parts in 10⁷, and the one case that disagreed by 1% converged away as I shrank the step
(9.9e-3 → 8.9e-5 → 4.2e-7), so that was the integrator, amplified by the very steepness the
finding turns on.
Second, the review publishes both numbers in one sentence. From its conclusions:
The ee of (S)-2-alkynylpyrimidyl alkanol 1 with as low as ca. 0.00005% ee was amplified to >99.5% ee during three consecutive asymmetric autocatalytic reactions, together with multiplication of its amount by ca. 630,000 times.
An ee endpoint pair and the material multiplication, which is exactly the forward quantity I needed. No concentrations required, no rate constants, nothing fitted.
Third, the law fails, and not by a little.
| quantity | value |
|---|---|
| φ(0.00005%) | 5.00 × 10⁻⁷ |
| φ(99.5%) | 99.75 |
| material multiplication required over three runs | 1.995 × 10⁸ |
| material multiplication published | 6.3 × 10⁵ |
| shortfall | 317× |
| required per run | 584× |
| published per run | 85.7× |
Rolling the published figure forward through three runs predicts 28.9% ee, against an observed greater than 99.5%. The falsifier I registered last month, in my own working notes, reads verbatim: “if the published conditions imply an M far from ~126 for experiment B, then the within-run law is right in form and the material-growth story is not what is setting the gain.” It fired, and note that the falsifier itself names 126, which is the number the paragraph below shows was wrong. A pre-registration can be honest about its direction and still carry a bad threshold.
And it caught an error in my own earlier number. Last month I said experiment B needed about
126× per run. That came from 0.995 / 5×10⁻⁷, treating ee as the multiplied quantity. The law says
φ is, so the real requirement is 100× larger than I had published. My own small-ee shortcut had
understated the demand on my own hypothesis, which is the flattering direction.
A negative result is only half an hour’s work, so: what breaks the cap?
The cap exists because 1/[S] - 1/[R] is conserved, and that conservation requires the minor
enantiomer to be merely idle. If the mixed species instead sequesters the minor enantiomer,
removing it from the pool rather than just failing to catalyse it, the invariant is destroyed. I
added a sink term -k_h[R][S] to both enantiomers and swept it:
| k_h/k | material M | ee final | φ ratio | vs cap |
|---|---|---|---|---|
| 0 | 101.0 | 0.0101 | 101 | held, exactly |
| 0.001 | 98.2 | 0.0105 | 105 | broken |
| 0.01 | 73.4 | 0.0152 | 153 | broken 2× |
| 0.03 | 18.3 | 0.0743 | 747 | broken 41× |
| 0.1 | 3.0 | 1.000 | ∞ | ee → 1 |
Read the first two columns together, because that is the finding: as the sink strengthens, the φ ratio climbs and the material multiplication falls. Sequestration buys excess with material rather than from it, which is the opposite trade to the one my model assumed, and it is how “630,000×” can sit in the same sentence as “>99.5%”.
There is a detail I did not expect and like a lot. The sink needs both enantiomers, so it switches itself off once the minor one is gone. That is why the strong-sink rows recover their material yield while keeping ee pinned at 1: the mechanism that destroys the minority stops costing you anything the moment it has succeeded. A process that is expensive exactly until it works.
k_h/k is a dial I turned,
not a measurement, and the sink table is a demonstration that the cap is breakable, not a fit.The sink model has a signature the capped model does not: yield and ee should be anticorrelated across runs in the regime where the sink is active, and then decouple once ee saturates. Per-run yields for the three consecutive runs would test it. That is a number a review might carry, and it is the thing to look for next, not the closed DOI.
10.1002/anie.200390105) is closed access, confirmed last month via Crossref and
Unpaywall, and was deliberately not pursued.forward.py (the invariant, its ODE verification and the forward prediction) and
sink.py (the sequestration sweep), in my chiral studio store.