A compartment-gated construct designed against its own ceilings

In plain terms

The problem. A drug acts wherever it ends up. Giving it an address is a delivery problem, and this paper is about one address only — not which cell a drug reaches, but which compartment inside that cell it acts in. Telling a cancer cell from a healthy one is a different problem, and nothing here solves it: that job belongs entirely to whatever targeting scaffold carries this construct. What acid buys is the compartment. Inside every cell there is a tiny acid-filled bag — the lysosome, about pH 4.7 — while blood sits at pH 7.4. That is a 500-fold difference in acid strength, and it is one of the sharpest chemical boundaries in the body.

The idea. Build a molecule that is switched off in blood and switches on only once it is inside that acid bag. The switch is a chemical group that picks up a proton when acid is strong enough. Pick the group correctly and it stays off at pH 7.4 and turns on at pH 4.7.

Why it stays put. A molecule carrying an electrical charge cannot pass through a cell membrane — membranes are greasy, and charge will not dissolve in grease. So we give the molecule a permanent charge. It gets carried in by a delivery system rather than diffusing in, and once the carrier lets go, it cannot get back out. It goes in as cargo and stays as a prisoner.

What this paper actually adds. The field has been building switches like this for fifteen years, and the working assumption is that adding more switchable groups keeps making the molecule more selective. We show there is a hard mathematical limit on how much selectivity any pH switch can give you — a formula with no adjustable parts. Then we show something more surprising: nobody is anywhere near that limit. Published molecules reach about half of what their own chemistry permits with one switchable group, and only a quarter of it with four. Adding groups moves you further from the limit, not closer. So the thing holding the field back is not the limit everyone could have calculated; it is something else, and we say what to measure to find it.

And one more correction, now narrowed. Almost everyone in this field reports how tightly a molecule binds. The step that decides where a molecule ends up is a race, not a tug-of-war — a race between the molecule letting go of its delivery receptor and the cell deciding where to send that receptor. Races are decided by speeds.

What we are not claiming. The molecule described here has not been made. No cell has seen it. Everything here is calculation plus other people's published measurements. We give the experiment that would prove the whole idea wrong in about a day, and we would rather someone ran it than believed us.

Abstract

We describe a small molecule built to switch on only inside the lysosome — the acidic compartment inside a cell — and the closed-form limits that place every part of it. Each design choice follows from a bound that can be written down, rather than from screening. The construct (1, C₂₉H₃₄N₂O₁₄S₂²⁻, MW 698.73) is two pyridine rings and two sulfonates on a cleavable scaffold: the rings are the pH gate (pKa 5.23 unconjugated, 5.61 in context — a +0.350 shift recovered independently from the relaxed geometry against +0.375 assumed — 1.6% protonated in plasma, 88.9% at pH 4.70), the sulfonates are a permanent off-state charge that closes the passive membrane route, an acid-labile ketal sheds the targeting ligand on arrival, and the ligand itself is high-affinity and deliberately not pH-dependent. Each choice follows from a bound rather than from optimization. Selectivity from a pH gate cannot exceed S_max = exp(|G_gate|·tanh(ΔpH·ln10/4)), a bound we derive — exact as a ceiling on the protonation change, and a valid but non-tight upper bound on selectivity, exact only in weak coupling (see errata), whose attenuation factor is 0.667 at the tumor pH gap and 0.894 at the cytosol–lysosome gap; because that factor is below one, energy spent suppressing the off state is worth 1.5-fold more per kT than energy spent on the gate, which is why the construct carries permanent charge. Electrostatic cooperativity cannot lift the ceiling and imposes a ≥1.7 nm site-spacing floor, which is why there are two rings and not eight. Calibrating against a surface-plasmon-resonance measurement of a single histidine substitution (2.50 kT), published constructs realize only 52% of their own ceiling at one group, 40% at two and 24% at four — so the floors bind before the ceiling does, and the construct is specified at N = 2 accordingly. The sorting step that delivers it is kinetic rather than equilibrium, governed by Da = k_e/k_off, whose sign requirement inverts between crossing a barrier and entering a cell; this is why the ligand must retain rather than release. At the measured per-group budget the retention ceiling is 87-fold 73-fold (exact linkage) and the construct's expectation, after linker plasma instability and the measured realization fractions, is approximately 46-fold a surface spanning 5×–1,660× over two unmeasured parameters. See the errata above. Nothing has been synthesized. A 36-well v-ATPase dose × pH experiment can falsify the design rationale outright, and we specify it.

Significance

Compartment-selective therapeutics are usually optimized empirically, with selectivity assumed to accumulate as ionizable groups are added — an expectation stated explicitly in the current literature. We show the achievable selectivity has a closed-form bound, that published constructs fall further below that bound as groups are added rather than approaching it, and that the equilibrium binding constant on which this field is built is the wrong observable for the step that decides a molecule's fate. We then present a construct in which every moiety is placed by one of these constraints, give its expected performance as a number rather than an aspiration, and specify the experiment that would void the whole rationale in days.

Keywords: compartment selectivity · pH gate · lysosomotropic design · selectivity ceiling · Damköhler number · cleavable linker

Introduction

A therapeutic that acts only inside the endolysosome has an unusually favorable physical setting. The compartment is 2.5 pH units more acidic than cytosol and 2.7 below plasma, the gradient is actively maintained, and it can be collapsed on demand with a v-ATPase inhibitor — which makes a design built on it falsifiable in a way that most selectivity claims are not.

The usual approach is to install ionizable groups and optimize. The literature proceeds on the assumption that selectivity compounds as groups are added: histidine-scanning studies select widely separated mutations specifically to obtain additivity of their free-energy contributions and confirm that additivity is achieved (1, 2), one states that introducing multiple ionizable groups "has the highest potential for achieving potent pH dependence" (3), and the recommended workflow is to combine confirmed single-point hits to greater effect (2). No bound is stated anywhere in this literature.

We took the opposite route: compute the ceiling on each mechanism before engineering against it, and let the surviving bounds determine the molecule. This paper is that molecule and those bounds. The constraints come first because they are what make the architecture non-arbitrary; the construct follows; and the experiment that can kill it is specified last.

Where this sits in the field

It is worth being explicit about what is old, what is recent, and what is ours, because the three are easily conflated (Fig. 2).

Settled for decades. That a titratable group's protonation is a sigmoid in pH is Henderson–Hasselbalch (1909). That binding and proton uptake are thermodynamically linked is Wyman's linkage theory. That weak bases accumulate in acidic organelles is de Duve's lysosomotropism (14). That a charged surface acidifies the water beside it is Gouy–Chapman. That repulsively coupled identical sites cannot give a Hill coefficient above one is classical ligand-binding theory (9). We cite all of these and claim none of them.

The recent innovations, and whose they are. The modern field opens in 2010, when Igawa and colleagues at Chugai showed that making antigen binding pH-dependent lets an antibody release its cargo in the endosome and recycle (4) — the founding move. In 2011 Yu and colleagues at Genentech reported the result that still governs the field's intuitions: reducing anti-transferrin-receptor affinity increases brain uptake. In 2014 two Roche groups showed that receptor trafficking rather than affinity decides a cargo's fate, and that a monovalent shuttle crosses where a bivalent one does not; a third showed that a pH-dependent anti-TfR transcytoses while a pH-independent one is degraded (6). In 2015 Clark and Davis at Caltech added an acid-cleavable linker that sheds the targeting ligand mid-transit, decoupling capture from release (5). In 2017 Villaseñor and colleagues identified ~50–70 nm sorting tubules as the physical gate (13). Between 2020 and 2024 Sulea and colleagues at NRC Canada made pH switches designable rather than screened, by structure-based and then sequence-based histidine scanning (1, 2). Monovalent shuttle constructs are now in clinical development.

The through-line nobody states. Every one of those results is a routing result — it changes which pathway a molecule enters, not how tightly it binds — and every one of them is reported as an equilibrium constant. That mismatch is where two of our findings come from.

What we add. Five statements we believe are new: the closed-form ceiling (eq. 1); the observation that the field expects no such bound and says so in print; the finding that realized selectivity falls further below the ceiling as groups are added; the Damköhler criterion and its inversion by destination; and the rule that combinations must pair across delivery stages. Three further results are quantitative factors attached to principles the field already holds — the 1.7 nm spacing floor on the Hill inequality, the attenuation coefficient on the known on/off asymmetry, and the midpoint-pKa optimum as a corollary of Wyman. We mark them as such rather than claiming the principles.

Results

1The environment the molecule has to work in

Before the constraints or the architecture, the setting (Fig. 2) — because every element of the construct is a response to one physical fact about it, and the bounds of §2 are consequences of the same four facts.

A proton gradient is an unusually good address. It is actively maintained by the v-ATPase rather than passively established, it is steep — 2.7 units between plasma and lysosome, a 500-fold concentration difference — and it is the only address of its kind that an experimenter can switch off on demand. That last property is what makes a design built on it falsifiable rather than merely plausible (§12).

A charge is a wall. The interior of a lipid bilayer has a dielectric constant near 2; water is 78.5. Moving a single elementary charge from water into that interior costs tens of kT, so a charged species does not cross a membrane — only the neutral form of a titratable molecule does. This is the whole basis of lysosomotropism, and it is why pH can serve as an address and a trap at the same time: a molecule that is neutral outside and charged inside goes in freely and cannot come back.

The useful span of one titratable group is about two pH units. A sigmoid moves from 10% to 90% protonated across roughly 1.9 units, so a pKa placed too high leaves the gate already open in blood, and one placed too low never opens it at all. The window is the design problem. It is also why the optimum sits at the midpoint of the two pH values — that is where the sigmoid is steepest and the fractional swing between compartments is largest, which is the content of equation 1 (§2).

And the compartment makes its sorting decision before the chemistry matters. An internalized cargo is routed within minutes, to recycling or to degradation, by the behavior of the ligand–receptor complex rather than by anything attached behind it. A molecule that is perfectly gated but mis-sorted never reaches the compartment its gate was tuned for. This is the constraint that §9 addresses and the one most easily left out of a design.

2Three constraints decide the architecture

The four facts of §1 have quantitative consequences, and they are more restrictive than the field currently assumes.

A closed-form ceiling. For an interface carrying N identical ionizable groups of pKa_a, each contributing free energy G_gate when protonated, and a gate operating between pH_hi and pH_lo with ΔpH = pH_hi − pH_lo, the protonated fraction is f(pH) = 1/(1 + 10^(pH − pKa)) and the selectivity is S = exp(N·|G_gate|·Δq) with Δq = f(pH_lo) − f(pH_hi). Δq is maximized when the pKa sits at the midpoint m = (pH_hi + pH_lo)/2; writing δ = ΔpH/2 and u = 10^δ,

Δq_max = 1/(1 + u⁻¹) − 1/(1 + u) = (u − 1)/(u + 1) = tanh(ΔpH·ln10/4)

S_max = exp( N · |G_gate| · tanh(ΔpH·ln10/4) ) (1)

The derivation is three lines, so it is worth walking rather than citing. Put the pKa at the midpoint m, so pH_lo = m − δ and pH_hi = m + δ with δ = ΔpH/2. The protonated fraction on the acid side is 1/(1 + 10^(−δ)) and on the neutral side 1/(1 + 10^(+δ)). With u = 10^δ their difference is u/(1+u) − 1/(1+u) = (u−1)/(u+1) — and (u−1)/(u+1) with u = e^(δ·ln10) is tanh(δ·ln10/2) = tanh(ΔpH·ln10/4).

Why that is a ceiling and not merely a value. Δq is a difference of two fractions each lying in [0,1], so Δq < 1 for every pKa and every pH gap — it cannot be engineered above one. The midpoint maximizes it, so tanh(ΔpH·ln10/4) is the largest Δq available at that gap. And every quantity a designer can actually change — surface charge, approach distance, solvent exposure, group count — enters only through G_gate or N. The attenuation factor depends on the pH gap and nothing else.

What the bound does and does not cover. It bounds the ratio of binding constants between two compartments for a gate built from titratable groups. It says nothing about selectivity obtained another way: a cleavable bond, exclusion by permanent charge, or an enzyme present in one compartment and not the other all sit outside it. §7 and §8 of this paper use exactly such mechanisms, and that is why.

Two sanity checks. As ΔpH → 0 the factor → 0 and S_max → 1: no gradient, no selectivity. As ΔpH → ∞ the factor → 1 and S_max → exp(N·|G_gate|): the gate becomes a plain on/off switch worth its full binding energy. Both limits are correct, and the second shows the bound is not vacuous — it lies strictly below exp(N·|G_gate|) at every finite pH gap.

The identity gives a second result for free: a proof that the optimal pKa is the midpoint of the two pH values — which follows from the classical Wyman linkage framework (7) but appears not to have been stated as a design target. The attenuation factor is bounded above by one (Fig. 4A), reaching 0.667 at a tumor–plasma gap of 1.4 units and 0.894 at the cytosol–lysosome gap of 2.5. Surface charge, approach distance and group count drop out of the factor; only the pH gap remains.

Energy is worth more spent on the off state. Decomposing equation 1,

S = exp( G_gate·Δq + G_off ) (2)

Because Δq < 1 always, energy applied to the gated term is attenuated while energy applied to destabilizing the off state is not (Fig. 4C). At the tumor gap one kT on the off state is worth 1.5 kT on the gate. The design brief inverts: penalize the off compartment rather than reward the on one. The qualitative asymmetry is known in protein-switch design (8); the attenuation coefficient is what equation 2 supplies, and it is specific to the gating variable.

The floors bind before the ceiling. Equation 1 generalizes to exp(|G_gate|·tanh(n·ΔpH·ln10/4)) for Hill coefficient n, making cooperativity the only parameter that could lift it. It cannot be recruited: for identical sites coupled repulsively, n ≤ 1 — classical in ligand-binding theory (9) — because electrostatic coupling is non-zero only in the microstate that separates charge best, and that microstate is always intermediate in net charge. Solving a two-site semi-grand partition function exactly gives the quantitative consequence the inequality does not: sites must sit ≥1.7 nm apart, degrading to 2.05 nm at a protein–water interface and 3.44 nm in a membrane. Eight sites at 0.5 nm deliver 2.5-fold — worse than one site.

Calibration fixes G_gate. A structure-based study reports, by SPR, a pH-selectivity free energy of ΔΔΔG = −1.54 kcal/mol for one histidine (1), i.e. 2.50 kT at 310 K. Against that budget, published constructs realize (Fig. 5):

A survey of nine antibody–antigen systems reports that three to five histidines are typically required to reach five-fold (2), where the energetics permit 794-fold — a 159-fold shortfall. The energies are additive; the selectivity they buy is not, and the shortfall grows with N. For design this caps useful N at two for a 1% nonspecific floor and three at 0.1%.

This is the paper's most consequential claim, so here is what would refute it. Three alternatives have to be ruled out before a floors interpretation is preferred over the obvious ones:

1. The later groups might simply contribute less energy. If the second and third histidines are worth less than the first, the gap is energetic and there is no floor. Testable: measure ΔΔΔG by SPR for each substitution in a series, as ref. 1 did for one. Additive energies with non-additive selectivity is the signature of a floor; falling energies are not.
2. The constructs might be unoptimized. Nine systems reaching five-fold could reflect effort rather than physics. Against this: the groups involved developed computational histidine scanning, and five-fold is the hit criterion their own method is scored against — they would report more if they had it.
3. The assay might compress large ratios. Partly visible already: the same two constructs read 32-fold by ELISA and 8–10-fold by FACS (3). A 3–4× assay spread is not enough to close a 159-fold gap, but it is enough that every value must be quoted with its assay, and we do so throughout.

The direct test needs no new molecule. A construct with two well-separated groups should underperform exp(2·G·tanh) by a margin that grows with N. Anyone holding an affinity series with measured per-group energies can run it on data they already have.

3The construct

Figure 1 shows the architecture:

[targeting ligand] — (acid-labile ketal) — [payload] — (≥1.7 nm spacer) — 2 × (pyridine + sulfonate)

Three stages in series, each with its own chemistry and its own failure mode, and each answering one of the four facts above. Because the stages are in series, the delivered fraction is their product — which is also why a weakness in any one of them cannot be compensated by strength in another.

4Compound 1: the gating module, drawn and computed

The construct is specified as three numbered compounds (Fig. 7). 1 is the gating module — the species that remains after the linker cleaves, and therefore the species that is actually trapped. 2 is the single gate unit, which is the synthesis and pKa-measurement target. 3 is the acid-labile ketal connector.

1 O=C(O)c1cc(OCCOCCOCc2cncc(CS(=O)(=O)[O-])c2)cc(OCCOCCOCc2cncc(CS(=O)(=O)[O-])c2)c1

This panel fails standard drug-likeness filters, and the reason is worth being exact about — "the failure is intended" is too glib. Scored against the two rule sets a reader will apply:

Both rule sets were derived from orally absorbed drugs, and what they predict is one specific property: whether a molecule will cross the gut wall by passive diffusion. They are empirical summaries of that population, not laws, and applying them outside it is a category error.

The precise statement is not that the rules are wrong here. It is that they are correct and we want the opposite sign of what they predict. The filters say compound 1 will not passively cross a membrane. That is true, we agree with it, and it is the mechanism: the module cannot cross out of the lysosome for the same reason it would never have crossed into the gut. A version of this molecule that passed Lipinski would be a version that could leave the compartment it was built to stay in.

Two permeabilities do matter, and neither is governed by this panel. The conjugate enters by receptor-mediated uptake, a route that carries charged species routinely and is the basis of every antibody–drug conjugate in the clinic. The released payload must act where it lands, and its properties are a separate specification. We report the panel because a reader will compute it, and should see which question it answers.

The geometry confirms the design's one chemistry prediction. The +0.375 pKa shift attributed to the adjacent sulfonate was an assumption. Embedding 1 and relaxing it under MMFF94 gives a mean ring-nitrogen to sulfonate-sulfur distance of 0.473 nm, and the screened-Coulomb expression ΔpKa = (l_B/r)·exp(−r/λ_D)/ln10 then returns +0.350 — agreement to 0.025 units, from geometry rather than from the assumption it was meant to test. The same conformer places the two ring nitrogens 2.13 nm apart, clearing the ≥1.7 nm floor of §2.

One conformer is one sample. A PEG arm is flexible, so 2.13 nm is an instance rather than an ensemble mean, and the spacing floor has to hold for the population. A short conformer search is the next calculation, and it is cheap.

5The gate: two pyridines

Pyridine's unconjugated pKa is 5.23, which is the synthesis target. An adjacent sulfonate shifts it +0.375 by screened Coulomb, so the effective pKa in the construct is 5.61 — placing the transition between the compartments that matter:

Δq per ring across the cytosol–lysosome gap is 0.864 e; 1.73 e for the pair. Pyridine was selected over thirty other heterocycles; saturated amines perform worst because they are nearly fully protonated in plasma, leaving almost no Δq to gate with.

There are two rings because of §1, not by preference. The spacing floor forbids packing them, the nonspecific floor caps useful N at two, and the realization data say the third group would return 24% of its bound rather than 40%. A design that added rings to chase the ceiling would recede from it.

6The off-state: two sulfonates

pKa ≈ 1, so each carries −1 at every physiological pH. They contribute essentially nothing to selectivity — measured at 0.60% of it — and that is not their purpose. They implement equation 2's instruction directly: permanent off-state charge, which is not attenuated by Δq. Functionally they close the passive membrane route, so the fragment freed by the linker cannot diffuse back out of the compartment.

7The feature that reads like a bug: net charge near zero

This is the most counter-intuitive element of the design, and it deserves the arithmetic rather than an assertion (Fig. 10).

The objection. At lysosomal pH both pyridines are protonated, so the molecule carries two positive and two negative charges. Net charge: zero. A neutral molecule crosses membranes. So the trap appears to spring open at precisely the moment it is supposed to close.

Why the objection fails. Net charge is an accounting identity, not a physical state. Membrane permeation is governed by the Born self-energy each charge pays when it leaves water (ε = 78.5) for the hydrocarbon interior of a bilayer (ε ≈ 2):

W = q² / (8πε₀εa)

At an ion radius of 0.20 nm this is 66 kT per charge — a value that sits inside the 40–80 kT range reported for moving small ions across bilayers, so the model is not starting from an inflated number. Four charges cost 263 kT. The mutual attraction between each + and − pair refunds part of that, and at the 0.473 nm separation measured on the relaxed conformer the refund is 111 kT.

Net cost for our net-neutral construct: 152 kT.

A refund is not a waiver. The comparison that settles it:

Our net-neutral construct sits closer to four free ions than to a neutral molecule. The difference from a genuine neutral is a factor of about 10⁶⁶.

How sure we should be. The Born model with a single effective radius is crude, so the conclusion is worth stress-testing:

Across the plausible Born-radius range for a sulfonate or a pyridinium (0.15–0.30 nm) the cost runs 64–239 kT and the species is excluded by a wide margin. It falls to 20 kT only at a 0.40 nm radius, larger than is reasonable for these groups, and that is where the claim would start to be arguable. We state the soft edge rather than hide it: the conclusion rests on the ion radius being below roughly 0.35 nm, and for these groups it is.

What would overturn it is a route that does not require the charge to enter the hydrophobic interior at all — a transporter, a channel, or endocytosis. Those are real, and they are exactly how the construct gets in. They are not available for getting back out of a lysosome.

So exclusion depends on permanent charge, not net charge — and the ion-pair anti-stacking penalty that limits charge-neutralizing gates does not apply here, because this design never neutralizes a charge. It adds an opposing one at a distance.

One scope limit follows: on a non-cleavable conjugate the sulfonates are redundant, because no small freed species exists to trap. They earn their place only because stage 2 exists.

8Egress: the acid-labile linker, and its real ceiling

A single acid-labile bond delivers a 29-fold release switch — 75% of what the best of 156 enumerated pKa designs reached, with no pKa engineering at all. None of the electrostatic ceilings of §1 apply to it, because it is not an electrostatic gate but a covalent bond that hydrolyzes.

A different ceiling applies, and it is unmeasured. The realized advantage is bounded by the plasma deconjugation fraction D:

S_realized = 1 / [ D + (1 − D)/S_gate ] (3)

And the design pressure runs the wrong way: faster acid cleavage means more plasma cleavage. So the target is not a short acid half-life but a wide ratio between the plasma and pH-5 half-lives. A diamino-ketal linker reported as stable beyond 24 h at pH 7.4 in buffer (5) has never been challenged with plasma — and a valine–citrulline linker of comparable buffer stability realized 1.05-fold in mouse because carboxylesterase hydrolyzed it.

9Sorting: the ligand must retain, not release

Equations 1–3 are equilibrium statements and the step that delivers the construct is not at equilibrium. Sorting inside an acidifying endosome is a race between ligand release and receptor commitment, governed by a Damköhler number

Da = k_e / k_off (4)

For Da < 1 the ligand releases first and the cargo stays mobile; for Da > 1 the cargo is still bound when the receptor commits, and follows it.

Four anti-HER2 variants spanning 482-fold in affinity test this directly (Fig. 6A, Fig. 10C). Converting each K_d to a release probability:

P falls 0.98 → 0.10 while depth falls 84.8 → 20.4 µm; the ordering is exact and the transition brackets Da = 1, which is the authors' own stated comparison (10).

What we do not claim. Depth is monotone in P but not proportional to it — a line anchored on the two endpoints over-predicts ML39 by 33 µm. There is also a floor near 20 µm, reached before any sorting decision is made, so even a fully committed antibody penetrates the perivascular layer. We claim the sign, the ordering, and the location of the transition; we do not claim a quantitative law relating depth to P. The same group describes a cleavable-linker nanoparticle at the blood–brain barrier: a 60-minute acid half-life against a 30-minute transit sits at Da = 2.0, and for first-order release the shed fraction 1 − 2^(−1/Da) returns 29% — the measured value (5). One dimensionless group organizes two literatures that do not cite each other.

The sign requirement inverts with the destination (Fig. 6B). A brain shuttle and an antibody penetrating tumor interstitium both need Da < 1. This construct needs Da > 1, because the endolysosome is the destination: a high-affinity, pH-independent ligand that holds its receptor through acidification and follows it in. That is precisely the ligand that destroys a shuttle, and precisely what clinical conjugates use.

A corollary that contradicts intuition. Raising the ligand's affinity and adding a pH release gate places two opposed terms on the same stage. Within one stage, interventions share its flux and combine as a harmonic mean, S_comb = 1/(f_A/S_A + f_B/S_B), which here returns 0.59× — a net loss, worse than either alone. Pair across stages, never within one:

10What the construct is expected to deliver

The 2,527-fold is a product of two upper bounds and is not a prediction. The honest expectation is approximately 46-fold, and that still assumes stage 1 is perfect. We note that an earlier version of this specification carried 335-fold for stage 3; that figure does not reproduce from equation 1, from proton linkage, or from the ion-trapping limit at the stated 3 kT per ring — it requires 3.25 kT — and at the SPR-measured 2.50 kT the ceiling is 87-fold.

Dose is a ceiling, not a target. At K_D(lysosome) = 10 nM:

Above ~90 nM the on state saturates and further dose purchases only off-target occupancy. Selectivity is spent by dosing to effect, not gained — which is why the only worthwhile combination regime raises delivery rather than effect.

11Synthesis targets

The one chemistry prediction the design makes about itself is the +0.375 pKa shift, and a 3-sulfonatopyridine monomer titration tests it directly.

12The experiment that can falsify the rationale

The design depends on exactly one parameter that can be collapsed on demand. v-ATPase inhibition raises lysosomal pH, and equation 1 then commits to a curve with no free parameters: normalized effect falls from 1.000 at pH 4.70 to 0.041 at pH 7.20. The obvious objection — that potency falls because bafilomycin blocks autophagic flux or is cytotoxic — predicts a linear decline, not this sigmoid, and the maximum separation between the two is 0.343 in normalized units at mid-range. Because selectivity is dose-dependent (§10), the stronger version is a dose × pH surface: a crossed interaction no dose-independent artifact can produce, with a specified null region at saturating dose. Monte Carlo over both fully-specified models gives power above 0.98 at six replicates and 30% CV — 36 wells, one plate.

The x-axis must be measured lysosomal pH, not inhibitor dose. The model predicts potency against pH; the dose→pH mapping is cell-line specific and is not part of the prediction. Plotting against dose and fitting a sigmoid is circular.

It can fail four ways: the curve matches; both models fit (underpowered or too narrow a pH range); the linear null wins (potency loss is an inhibitor side effect); or potency is unchanged by pH, in which case the compartment model is wrong and this entire design rationale is void. The last is the reason to run it.

What these findings mean

Four of the results change what a designer should do, and it is worth saying plainly what each one implies and what it does not.

The ceiling means the field is optimizing against the wrong limit. Equation 1 caps selectivity at exp(N·|G|·tanh(ΔpH·ln10/4)), but no published construct comes close to that cap — they realize 52%, 40% and 24% of it. So the ceiling is not the operative constraint, and the practical advice that follows from it is almost the opposite of what a ceiling usually implies. It is not "you cannot do better than this." It is: the headroom is large and something else is consuming it, so stop adding ionizable groups and go find the floor. The candidates are identifiable — nonspecific binding, fractional occupancy, site spacing — and each is separately measurable. This is the finding we would most like to be tested, because it predicts a specific failure: two well-separated groups should underperform their own exp(2·G·tanh) bound by a margin that grows with N, and anyone with an affinity series can check it.

The attenuation coefficient means off-state engineering is systematically undervalued. Every kT placed on the gated term is discounted by Δq; every kT placed on the off state is not. At a tumor pH gap the exchange rate is 1.5 to 1. The field spends its effort on the gate — histidine scanning optimizes the bound state — because the gate is where the pH dependence visibly lives. The arithmetic says the better investment is making the off compartment inhospitable, which in a small molecule means permanent charge and in a protein means destabilizing the unbound conformation. It is a reallocation of effort, not a new mechanism, and it is the reason this construct carries two groups that contribute 0.6% of its selectivity.

The Damköhler result means the field's central observable is the wrong one. Equilibrium release ratios — K_d at one pH over K_d at another — have been the currency of pH-gated design since 2010. Sorting is a race. A 29-fold equilibrium switch that is slower than the step it is racing delivered 2.7-fold in an animal (5). The implication is concrete and cheap: report k_off and the receptor's internalization rate k_e, not K_d. Both are standard measurements. And because Da organizes an anti-HER2 affinity series in solid tumor and a nanoparticle linker at the blood–brain barrier with the same inequality, the grouping is unlikely to be specific to either system.

The sign inversion means design rules do not travel. The monovalency-and- release rule established for brain shuttles and the avidity-and-retention rule used by clinical conjugates are the same inequality read in opposite directions, because in one case the receptor's destination is an obstacle and in the other it is the target. A rule imported across that boundary is not merely unhelpful — it is actively harmful, and the harm is quantifiable: raising affinity while adding a pH release gate puts opposed terms on one stage and returns 0.59×. We suspect this is the most common silent error in the area, because both halves of it are individually well-motivated.

What none of this establishes. No result here says the construct of Fig. 1 will work. The bounds say what is possible, the realization fractions say what is typical, and together they give an expectation of roughly 46-fold — a number that is useful precisely because it is modest enough to be wrong in an informative direction. The design's value is that it is specified tightly enough to fail cleanly, and §12 is how.

Discussion

Every element of this construct is placed by a bound rather than by optimization, and in three cases the bound points the opposite way from intuition. The sulfonates exist because equation 2 says off-state energy is not attenuated while gate energy is. There are two rings rather than more because the realization data say added groups recede from the ceiling rather than approach it. The targeting ligand is deliberately pH-independent because equation 4's sign inverts when the receptor's destination is also the cargo's — the same inequality that requires monovalency and release for a brain shuttle requires avidity and retention here.

Three of the constraints narrow existing theory rather than extending it, and we state the boundaries plainly. The Hill-coefficient inequality is classical (9); our contribution is the 1.7 nm floor. The on/off asymmetry is qualitatively known in switch design (8); our contribution is the attenuation coefficient. The midpoint-pKa optimum follows from Wyman linkage (7); we offer it as a design target. In each case the field holds the principle and the quantitative factor is what was missing — which is also the honest description of this paper's contribution.

The kinetic result has the widest reach, because it disqualifies the observable the field reports. Equilibrium release ratios have been the currency of pH-gated design for a decade, and the step that decides a molecule's fate is a race. A 29-fold equilibrium switch slower than the committing step delivered 2.7-fold in vivo (5). Measure k_off against k_e, not K_D.

This framework originated in an unrelated computational study of interfacial fields acting on never-folded polypeptides, in which the central hypothesis was falsified: across 408,736 chains in 293 runs, an interfacial field orients such a chain but does not reshape it, with shape effects bounded at 2.6% of mean radius of gyration against position-matched controls. The useful residue of that study was not the null but the observation that a charged surface acts chemically — by retitrating ionizable groups in its own double layer — which is what sent us looking for the ceiling on pH gating. That study's own numbers are being re-derived and are not relied upon here.

Limitations

Nothing has been synthesized. No binding, cell or animal data exist for this construct, and the 46-fold expectation combines one assumed deconjugation fraction with realization fractions borrowed from antibody constructs. The per-group budget rests on a single SPR measurement of one substitution in one Fab. The realization fractions derive from three calibration points across two studies with unmatched pH gaps, so the falling trend is robust while its slope is not. Published pH selectivities carry substantial assay dependence — the same two constructs read 32-fold by ELISA and 8–10-fold by FACS in one study (3) — and all values here are stated with their assay. Damköhler assignments use a flat k_on and a geometric-mean k_e from a measured ten-fold range, so each Da is known to about ±3-fold and the data bracket Da = 1 within roughly a decade rather than locating it. Equation 3's plasma stability has never been measured for a ketal. No claim is made for potency, synergy, tolerability or cell selectivity: the scaffold does all cell selection and the gate does compartment selection only, and the two multiply rather than substitute. Finally, the payload must act inside the endolysosome — anything requiring cytosolic escape is disqualified by the same charge that performs the trapping.

Methods

Structures were built, depicted and characterized with RDKit; properties are Crippen cLogP, Ertl TPSA and standard counts on the graph as drawn. Geometry came from one ETKDG embedding relaxed under MMFF94, and distances are reported from that single conformer. Closed-form results were evaluated symbolically and verified numerically to machine precision. Protonated fractions use f(pH) = 1/(1+10^(pH−pKa)); selectivities use equation 1 with G_gate in kT at 310 K (1 kT = 0.6160 kcal/mol). Debye screening lengths use λ_D = 0.310·I^(−1/2) nm at 310.15 K; the Bjerrum length is 0.687 nm at ε_r = 78.5 and 310 K. Screened-Coulomb pKa shifts use ΔpKa = (l_B/r)·exp(−r/λ_D)/ln10. Two-site cooperativity was solved with a semi-grand partition function over the four protonation microstates. Damköhler numbers use k_off = K_D·k_on with k_on = 10⁵ M⁻¹s⁻¹ and k_e = (1.6×10⁻³·1.6×10⁻⁴)^(1/2) s⁻¹, the geometric mean of the reported range; the resulting k_off classification reproduces that of reference 10. Realized fractions are ln(observed)/ln(ceiling). Linker shed fractions use 1 − 2^(−t/t½). Power for §9 is from Monte Carlo fitting both fully-specified models by residual sum of squares. Figure coordinates are generated from the computed values rather than placed by hand; all analysis scripts and their outputs are provided and every figure number is reproducible from them.

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† Bibliographic detail to verify before submission. References 1, 2, 5, 6, 10, 11, 12 and 13 were read in full or confirmed against a primary source's own reference list; their author lists, volumes and pages are as given. Entries marked † are cited from abstracts, search records or secondary citation: journal, year, volume and substantive content are confirmed, but author lists and page ranges should be checked against the originals. No citation here supports a claim its source does not make — that was audited separately — but the metadata of these entries has not been verified line by line.

Thresholds, Not RatiosThe Engager ProgramLuminary Studies