Introduction

The central role of allosteric regulation in enabling life as we know it1 and the fundamental questions in information transfer2, conformational dynamics3,4 and emergent properties5,6 that allostery presents underlie continued effort to design synthetic molecules with allosteric behavior7. Allosteric control of reaction thermodynamics, particularly binding affinities of small ligands, has been realized in thousands of synthetic molecules, and the design principles of such receptors are well understood8,9,10,11,12.

In comparison, abiological examples of allosteric control of reaction kinetics are few8,12. The most productive approach to date has been to combine a known catalyst with a regulating moiety for on/off control, with the catalyst in the on state retaining largely unchanged the properties of the non-allosteric equivalent. Despite clever applications of such catalysts13,14, they are thought to offer only limited insights into how electronically uncoupled changes in local geometries of remote receptor sites lower activation reaction barriers below those of non-allosteric equivalents8.

Here we report a molecular scaffold which allows the rate of thermal E→Z isomerization of an alkene (stiff stilbene15, Fig. 1) to vary systematically by a factor of up to 104 in response to subtle structural perturbations triggered by fast binding of a metal ion to a remote receptor site. Representing the latter as a harmonic compressive potential constraining E stiff stilbene, which metal binding increases, reproduced measured kinetics accurately. Our work provides an experimentally validated generalization of mechanochemical kinetics to compressive loads16, and it demonstrates that the formalism of force-coupled reactivity constitutes a productive and quantitative approach to the analysis of allosteric control of reaction kinetics. We are unaware of previous use of a mechanochemical formalism to support quantitative analysis of allosteric accelerations, which here provides insight into how molecular structure enables allostery. The work also expands the very limited range of reactions demonstrated to be sensitive to externally imposed compressive force17, and highlights important differences in the effects of compressive vs. tensile force on the kinetic stabilities of molecules.

Fig. 1: The concept of allosteric reaction control and its implementation.
figure 1

a A substrate reaction is accelerated allosterically if it is coupled to a remote receptor site such that binding of another molecule (effector) to this site suitably strains the substrate (in this example by imposing a compressive load, represented by a pair of square arrows, on the substrate that contracts during the reaction). This strain comes at the expense of reduced affinity of the effector for the receptor. b, c Here, allosteric acceleration of E-stiff stilbene isomerization (red) by binding of PtCl2 moiety to a bidentate phosphine receptor (blue) is demonstrated using a series of strained macrocycles, E(m,n). These are synthesized by photoisomerization of strain-free Z(m,n) analogs (b); the two diastereomers of Z macrocycles are in rapid equilibrium and only the S configuration of the stiff stilbene is shown; conversely, epimerization of E-stiff stilbene in macrocycles is negligibly slow. Photoisomerization of Z(3,3) yields a separable mixture of (S,S)-E(3,3) and (S,R)-E(3,3) diastereomers which differ in the configuration of E-stiff stilbene; (S,R)-E(2,3) and (S,R)-E(2,2) are formed as single diastereomers. In the presence of a source of the effector (nbd = norbornadiene), rapid coordination of PtCl2 to the phosphine on the opposite side of E-stiff stilbene accelerate its thermal isomerization of the Z isomer (c), as illustrated on the example of the S,S-diastereomer.

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Results and discussion

We studied strained macrocycles comprising an (S)-BIPHEP moiety as the effector binding site (blue, Fig. 1) that bridges and constrains the separation of the C6,C6’ atoms of E-stiff stilbene (the substrate, red) on the opposite side of the macrocycle. Strain within the macrocycle is modulated by the length of the alkyl chains that tether the BIPHEP and E-stiff stilbene. Following the previously reported protocol17,18, we generated the strained E(m,n) macrocycles by photoisomerization of the relaxed Z(m,n) macrocycles (Fig. 1b): the axial chirality of both the stiff-stilbene and BIPHEP moieties makes all macrocycles diastereomeric. Z(3,3) yielded a separable mixture of (S,S)-E(3,3) and (S,R)-E(3,3) diastereomers, whose diselenide derivatives were previously characterized by X-ray crystallography19, whereas smaller macrocycles, E(2,2) and E(2,3) were generated as single diastereomers as evidenced by 1H NMR spectroscopy19. We assigned these diastereomers as (S,R), which we calculated (see below) to be less strained than the (S,S) analogs, based on the previously reported predominant generation of the least strained E diastereomers by photoisomerizations of Z-stiff stilbene macrocycles20,21. This preference was rationalized by a combination of rapid epimerization of Z-stiff stilbene22 and the strong dependence of the quantum yields of Z→E photoisomerization on the strain energy of the product23,24,25. Macrocycles E(m,n) react rapidly and quantitatively with (NBD)PtCl2 (nbd = norbornadiene) at or above –30 °C to form the ligated macrocycles [E(m,n)]PtCl2 (Fig. 1c). The PtCl2 fragment was chosen as effector for the high kinetic and thermodynamic stability of the Pt–P bond26.

We measured the activation barriers of E→Z isomerization of stiff stilbene in the absence of metal ions and upon coordination to PtCl2 (Supplementary Figs 1–24). In p-xylene-d10 solutions at ≥69 °C the Pt-free macrocycles E(m,n) isomerized exclusively to their Z(m,n) isomers with the first-order kinetics over three half-lives as determined by 31P NMR spectroscopy (Table 1). The isomerization rate increases as the size of the macrocycle decreases, e.g., E(2,2) > E(2,3) > E(3,3), which follows the trend in the relative strain energy of these macrocycles, e.g., (S,R)-E(2,2), (S,R)-E(2,3) and (S,R)-E(3,3) are 19.9, 10.7 and 9.9 kcal/mol less stable than the corresponding Z isomers. However, as demonstrated previously restoring forces of carefully chosen internal molecular coordinates, rather than relative energies, are often a better quantitative correlant of reactivity27,28,29. As discussed below, the mechanochemical formalism captures quantitatively the variation in isomerization kinetics across the whole range of the macrocycles, whether metalated or metal-free. Similarly, warming solutions of platinum-ligated macrocycles [E(m,n)]PtCl2 in DMF-d7 at or above –10 °C led to first-order decay over three half-lives to form exclusively the corresponding [Z(m,n)]PtCl2 macrocycles. We previously demonstrated20 that the isomerization kinetics of stiff stilbene varies little with reaction solvent, a conclusion supported by our own control experiments on (S,R)-E(2,3) (Supplementary Fig. 25). Consequently, we chose the solvents for kinetic measurements that maximize the solubility of the components, e.g., xylene for Pt-free macrocycles and DMF for Pt analogs. Given the range of activation energies across the full series of macrocycles, temperatures were chosen on a case-by-case basis to achieve rates that were practical for NMR studies (Table 1). The low activation free energies of isomerization of [(S,R)-E(2,3)]PtCl2 and [(S,R)-E(2,2)]PtCl2 macrocycles allowed us to measure the E→Z isomerization kinetics across a range of temperatures and extrapolated those results to the standard temperature of 298 K.

Table 1 Summary of isomerization kinetics
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In each studied macrocycle, platinum coordination lowered the activation free energies of isomerization, by between 6.7 ± 2.0 kcal/mol in the smallest (S,R)-E(2,2) macrocycle and 1.4 ± 0.1 kcal/mol in the largest E(3,3) macrocycles. These decreases in activation energies are dominated by reduced activation enthalpies [e.g., 6.8 ± 2.0 kcal/mol in (S,R)-E(2,2) and 6.0 ± 2.0 kcal/mol in (S,R)-E(2,3)], as expected for an elementary reaction with a non-polar transition state.

To understand the structural and energetic origin of the accelerated isomerization of stiff stilbene by metal coordination and to test the utility of a force-based approach to broader analysis of allosterically controlled reaction kinetics we optimized conformational ensembles of Z and E isomers of both metal-free and Pt-coordinated macrocycles at (u)BMK/def2SVP//(u)B3LYP/def2SVP level of DFT in vacuum. The calculations reproduced measured ΔG‡ to within 1–2.5 kcal/mol (Table 1).

The optimized geometries reveal compressively strained E-stiff stilbene (SS), as evidenced by the contraction of its ArC-C = C-CAr torsion, θ, relative to that of free E-SS (θ ≈ 180o), in all macrocycles (Fig. 2a). In all Pt complexes this torsion was reduced further by 4–8o compared to the metal-free congeners, which can be attributed to the contraction of the BiPHEP-containing strap needed to accommodate the preferred Pt-P bond distances and P-Pt-P bond angle. Conversely, the corresponding torsion angles of the conformers comprising the transition states of isomerization deviate from the strain-free value by between −6o and 2o, depending on the conformer (Supplementary Fig. 26), suggesting low strain of either tension or compression, regardless of the presence of Pt.

Fig. 2: Summary of the mechanochemical model of the measured isomerization kinetics.
figure 2

a, b E-SS(OMe)2 with compressive force acting on its methoxy C atoms reproduces the structures of the stiff stilbene moiety in the macrocycles. In (a) the underlying structural homology is illustrated by the correlation between the ArC-C = C-CAr torsion and the O…O distance in all conformations of the E macrocycles following the same correlation in force-coupled SS(OMe)2. In (b), the C…C and O…O distances in the conformers of the macrocycles are compared with the same distances in E-SS(OMe)2 as a function of the applied force. c Calculated ΔGǂ in the two diastereomeric series of the macrocycles are reproduced accurately using SS(OMe)2 coupled to a compressive potential with compliance of 2 and 0.3 Å/nN respectively. Maximum fE that a compressive potential can apply decreases with increasing compliance, resulting in force-dependent activation free energy of isomerization, ΔGǂ, terminating below 1 nN for compliances >2 Å/nN. d Force exerted on SS(OMe)2 by a compressive potential with a finite compliance decreases as the molecule progresses along the isomerization reaction path, resulting in lower ensemble-average force in the transition state, 〈fts〉, compared to that in the E isomer, 〈fE〉. The plotted data is tabulated in Supplementary Tables 6–8.

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Across all conformers, geometric parameters of the SS(OCH2)2 moieties (indicated in blue, red and green in Fig. 2a) closely resemble those of bis-(6,6’-dimethoxy)stiff stilbene, SS(OMe)2, with a compressive force of varying magnitude applied across the MeOC…COMe coordinate (Fig. 2a, b and Supplementary Fig. 26). A reactive moiety coupled to an external force acting across a pair of its atoms is a widely used model in polymer mechanochemistry30,31,32,33, suggesting that the observed variation in ΔGǂ of isomerization across the series, including the allosteric effect of Pt binding, could be amenable to analysis within the formalism of mechanochemical kinetics34.

The formalism of mechanochemical kinetics is designed to allow quantitative analysis and predictions of the effect of highly anisotropic molecular strain on reaction rates and selectivities when the strain results from interactions of intractably many molecular degrees of freedom33. This model was developed and validated for reactions of polymer chains stretched beyond their strain-free geometries by energy-dissipative environments such as elongational flow fields in rapidly flowing polymer solutions35 or material under plastic deformations36. The measured kinetics of reactions in overstretched polymer chains33 or non-macromolecular mimics reported to date37 can be analyzed productively by representing the arbitrarily large reactant and its surroundings as the small reactive site coupled to an infinitely compliant harmonic stretching potential. Such a soft potential applies identical force on every conformer of the reactive site (and any transition state), independent of its molecular geometry. In this model, acceleration (or occasional inhibition21) of reactions in stretched reactants is primarily determined by the difference in the potential energy of the stretching potential coupled to the reactant and the rate-determining transition state, which is proportional to this force38,39.

Under compressive load, the validity of the assumption that all conformers of a reacting molecule across all kinetically-significant states experience the same force is at best uncertain16. Within the standard mechanochemical model it is outright aphysical34. The reason is that the rate of a chemical reaction is sensitive to applied force only if the formation of the rate-determining transition state is accompanied by changes in the separation of the two atoms across which the force acts. A compressing potential soft enough for the applied force to remain approximately constant despite changes in the coupled molecular distance would require an equilibrium distance shorter than zero, which is obviously impossible.

Conversely, a physical compressive potential necessarily applies different force on each conformer of each kinetically-significant state. The stiffer this potential, the greater the variation of both the force among the conformers and the difference of the ensemble-average forces acting on the reactant and on the (rate-determining) transition state. As a result, the activation barrier of a reactant coupled to an external compressive potential depends both on its ensemble-average force and the potential compliance or, alternatively, the ensemble-average forces of the reactant and the rate-determining transition state.

To apply the formalism of mechanochemical kinetics to the macrocycles in this study, we first calculated ΔGǂ of E→Z isomerization of SS(OMe)2 coupled to harmonic potentials with compliances between 100 Å/nN (very soft) and 0.1 Å/nN (very stiff), solid lines in Fig. 2c. With its compliance fixed, the force applied by a potential on the coupled molecule is controlled by the equilibrium distance of the potential: the shorter the distance, the higher the force. At zero equilibrium distance this force reaches maximum, which increases as the potential stiffens (e.g., blue, magenta and grey lines). At the same ensemble-average force, 〈fE〉, a soft potential reduces ΔGǂ by more than a stiffer potential because the change in its strain energy between the reactant and the transition state is proportional to the potential’s compliance. As the latter decreases, its effect on the dependence of ΔGǂ on the ensemble-average force experienced by E-SS, 〈fE〉 decreases as well until the ΔG vs. 〈fE〉 correlation becomes compliance-independent at ~0.2 Å/nN.

To analyze mechanochemically the variation of the kinetic stability of E-stiff stilbene across both the metalated and unmetalled macrocycles, we first calculated the force on the methoxy C atoms of E-SS(OMe)2, fE, needed to reproduce the geometry of the corresponding moiety in each thermally accessible conformer of each E macrocycle. We then averaged resulting single-conformer fE in proportion to the Boltzmann weight of each corresponding conformer in the reactant ensembles. The correlation between these ensemble 〈fE〉 and ΔGǂ calculated for each macrocycle (symbols in Fig. 2c) follows closely the analogous correlations calculated for E-SS(OMe)2 coupled to potentials with compliances of 2 Å/nN or 0.3 Å/nN, for the two diastereomeric series (Fig. 2c).

These calculations reveal that the exceptionally large allosteric kinetic effect of Pt binding to our macrocycles (up to 104-fold acceleration at 300 K) arises from only a small increase in the strain of the E-SS moiety, as evidenced by 〈fE〉 of stiff stilbene in Pt-coordinated complexes exceeding that of the Pt-free precursors by only 30–80 pN (7–15%). The mechanochemical formalism allows the underlying molecular basis to be understood and generalized by analyzing the ΔGǂ vs. 〈fE〉 correlations in Fig. 2c in terms of 4 components: the ensemble-average strain energies of the substrate (E-SS(OMe)2 and the isomerization transition state) and those of the coupled constraining potential, which represents the rest of the molecule, including the effector binding site, Fig. 3a.

Fig. 3: Mechanochemical analysis of allosteric acceleration.
figure 3

a Strain energies of the substrate (solid lines) and the coupled compressive potential with compliance of 2 Å/nN (dashed lines) for the reactant (red) and the transition state (blue) as the function of the ensemble-average compressive force on E-SS, 〈fE〉. b the minimum energy needed to reduce the isomerization barrier by ΔΔGǂisom = ΔGǂallost−ΔGǂfree, where ΔGǂX is the activation free energy of isomerization of free E-SS (X = free) or E-SS in an effector-bound allosteric substrate (e.g., macrocycles in Fig. 1, X = allost). For each ΔΔGǂisom, all combinations of the two 〈fE〉 values corresponding to effector-free and effector-bound reactant were considered to identify the minimum necessary increase in the total reactant strain energy, ΔEstrain. For macrocycles (blue dots), ΔΔGǂbind describes the reaction free energy of PtCl2 transfer between free and SS-coupled BiPHEP (eq. 1). c 〈fE〉 of E-SS prior to effector binding giving the highest achievable ΔΔGǂisom/ΔEstrain ratio (see Supplementary Table 3 for the definition of all parameters of the mechanochemical model). The plotted data is tabulated in Supplementary Tables 7–9.

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The strain energies of both E-SS(OMe)2 and its compressing potential increase monotonically with 〈fE〉 (red curves). Conversely, those of the TS vary non-monotonically: they are non-zero at 〈fE〉 = 0 and decrease with increasing 〈fE〉 up to 0.42 nN. This seemingly counterintuitive dependence is the direct consequence of the finite compliance of the potential. At 〈fE〉 = 0 the equilibrium length of the potential equals that of MeOC…COMe in strain-free E-SS(OMe)2, which exceeds the same internuclear distance of the strain-free transition state by ~3.8 Å (e.g., the red structures in Fig. 4). Consequently, a potential that applies no force on the E isomer (〈fE〉 = 0), stretches the shorter TS (Fig. 2d). As the equilibrium distance of the constraining potential contracts with increasing 〈fE〉, the corresponding 〈fts〉 decreases before becoming compressive. Above this switchover force, all 4 strain energies increase monotonically.

Fig. 4: The computed effect of compressive force on the geometry of stiff stilbene.
figure 4

Overlay of the minimum-energy conformers of E-SS(OMe)2 (a) and its transition state (b) in the absence of force (red) and coupled to compressive force of 0.5, 1 and 1.5 nN (light orange, blue and green, respectively). The view is along the C = C bond that isomerizes. Note the much greater bending of the indanone core in the E isomer compared to the TS at the same force.

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The approximately planar geometry of strain-free E-SS makes its MeOC…COMe coordinate quite stiff at applied compressive force <0.3 nN, but the bending of the indanone core at higher force (Fig. 4a) causes a sharp contraction of both MeOC…COMe and O…O distances at fE = 0.35–0.6 nN (Fig. 2b), accompanied by the steeply increasing molecular strain energy. Because the TS is stiffer and less anharmonic (Fig. 4b and Supplementary Fig. 27), its strain energy is less sensitive to applied force, yielding a range of 〈fE〉 (0.35–0.6 nN) where a small differential compression of the E isomer causes disproportionately large reduction in ΔGǂ. Note that because force acting on the transition state, 〈fTS〉, cannot be directly experimentally controlled, but is uniquely defined by 〈fE〉 and the compliance of the coupled potential, both of which are experimentally controllable, we find it more productive to discuss the properties of the transition state in terms of the force acting on the reactant (〈fE〉), rather than 〈fTS〉.

This analysis suggests two limiting mechanisms of allosteric acceleration of a reaction. In one, both the reactant and the transition state in the absence of the effector are strained and effector binding does not alter the strain of the reactant but allows relaxation of the transition state. For SS this regime is accessible only at moderate substrate strain (e.g., 〈fE〉<0.3 nN for E-SS(OMe)2, the TS stabilization region in Fig. 3a). In the other limit, the effector-free substrate may be strained or unstrained but effector binding must increase the strain energy, ΔEstrain, of the reactant considerably more than that of the transition state. In SS, this regime is only accessible for substrates that are already moderately strained in the absence of effector (the reactant destabilization in Fig. 3a), as they are in our macrocycles.

In allosteric reactants such ΔEstrain increases come at the expense of reduced binding affinity of the effector relative to free receptor (or any molecular architecture where receptor and substrate are conformationally uncoupled). For example, in our macrocycles, calculated ΔΔGǂbind for transfer of PtCl2 between E(n,m) and free BiPHEP (eq. 1) becomes increasingly unfavorable as the macrocycle shrinks (and hence gets more strained). Consequently, the highest practically achievable allosteric acceleration is likely limited by the acceptable reduction of the affinity of the coupled receptor for the chosen effector, and the attendant minimum necessary increase in the concentration of the effector. The ability to estimate the minimum effector-binding free energy, ΔΔGǂbind, that must be sacrificed to achieve desired acceleration of a substrate reaction is thus a primary determinant in the design of practical allosteric kinetic control.

$$({{{{{\rm{BiPHEP}}}}}}){{{{{{\rm{PtCl}}}}}}}_{2}+{{{{{\rm{E}}}}}}({{{{{\rm{n}}}}}},{{{{{\rm{m}}}}}})\rightleftharpoons {{{{{\rm{BiPHEP}}}}}}+{{{{{\rm{E}}}}}}({{{{{\rm{n}}}}}},{{{{{\rm{m}}}}}}){{{{{{\rm{PtCl}}}}}}}_{2}$$
(1)

Figure 3b illustrates the relationship between ΔΔGǂbind, ΔEstrain and ΔΔGǂisom for E stiff stilbene estimated by mechanochemical analysis. It plots the minimum effector-induced increase in the reactant strain energy, ΔEstrain, needed to reduce the isomerization barrier by ΔΔGǂisom below that of free E-SS(OMe)2 for compressive potential of different compliances. The dashed red line corresponds to full ΔEstrain contributing to barrier lowering (ΔEstrain = −ΔΔGǂisom). Values below this line reflect the contribution of relaxation of the pre-strained TS to ΔΔGǂisom. For E-SS, this contribution can exceed 50% for weak accelerations (ΔΔGǂisom > −3 kcal/mol), provided the substrate is significantly strained prior to effector binding (as quantified by its restoring force, e.g., 〈fE〉 in Fig. 3c). At every ΔΔGǂisom, a stiffer potential requires a smaller effector-induced increase in the reactant strain energy, ΔEstrain, than a softer one, which is opposite to what is calculated for tensile force40. This suggests that in general, increasing the stiffness of the receptor and the molecular segments connecting it to the substrate sites would reduce the loss of the effector affinity (i.e., smaller ΔΔGǂbind) to achieve a desired barrier lowering. One scenario in which a softer compressive potential (and hence the molecular segment it represents) may be advantageous is at low target barrier reductions (e.g., ΔΔGǂisom >−3 kcal/mol for E-SS(OMe)2), where it decreases how much the substrate needs to be pre-strained to achieve the same allosteric coupling efficiency, i.e., the same ΔΔGǂisom/ΔΔGǂbind ratio (e.g., grey vs. green curves in Fig. 3c).

Our macrocycles broadly follow these trends. For both diastereomers of E(2,2), (S,S)-E(2,3) and (S,R)-E(3,3), the ΔΔGǂisom/ΔΔGǂbind ratio is within 90% of ΔΔGǂisom/ΔEstrain predicted by the mechanochemical model. In these macrocycles the reduction in the effector affinity, ΔΔGǂbind, accounts for ~78% of observed barrier lowering, vs. 65–75% predicted by the model. These values confirm that the structure of these macrocycles provides a scaffold for remarkably efficient transmission of molecular strain across ~1 nm and ~100 non-H atoms and is amenable to both full atomistic description and simple quantitative analysis with a mechanochemical model.

The reported macrocycles represent one of the very few examples of synthetic molecules which exploit allostery to lower the activation barrier of a reaction below that in the equivalent non-allosteric reactant. The unusually large kinetic allosteric effects in these macrocycles result from the effector binding increasing the strain energy of the reactant without a concomitant increase in the strain energy of the corresponding transition state. The formalism of mechanochemical kinetics, generalized to compressive load, explains the observed trends quantitatively and suggests broad molecular-design approaches to achieving efficient allosteric acceleration of reactions. For example, the dependence of ΔGǂ on force shown in Fig. 2c quantifies the importance of molecular compliance as a design feature to enhance or suppress the fraction of the effector binding energy that reduced the kinetic barrier. As a result, the allosteric sensitivity depends non-monotonically on applied force: higher force regimes do not necessarily lead to greater allosteric regulation. Consequently, the mechanochemical formalism may prove valuable for guiding the design of synthetic allosteric catalysts and for quantitative tests of molecular models of allosterically controlled enzymatic activity as resulting from structural transmission of molecular strain across suitably stiff portions of the biomolecular scaffolds5. In addition, the formalism applied here might inform approaches to improving the efficiencies of switching23, energy storage41 and work-generation15 in increasingly diverse macrocyclic molecular switches42,43,44 and machines24,45,46,47 for a range of potential applications48,49,50. Equally important, E→Z isomerization of stiff stilbene reported here is only the 2nd reaction17 whose kinetics has been demonstrated to be sensitive to compressive force. As such, it constitutes a valuable tractable model reaction on which to develop quantitative descriptions of mechanochemical kinetics across the full range of externally applied force.

Methods

All reactions were performed under a nitrogen atmosphere in oven-dried glassware employing standard Schlenk or glovebox techniques unless noted otherwise. Nitrogen-flushed plastic syringes and oven-dried stainless steel cannulas were employed for reagent transfer. NMR spectra were obtained at 25 °C unless noted otherwise. 31P NMR spectra were referenced using absolute frequency referencing in Mnova software or from trimethylphosphine oxide internal standard. NMR probe temperature were determined from a single scan of neat ethylene glycol (high temperature) or neat methanol (low temperature) with accuracy of ±1 °C51. Anhydrous solvents were obtained either from Sigma-Aldrich in Sure/SealTM containers or were dried and degassed using an Innovative Technologies PureSolv solvent purification system. All deuterated solvents were obtained from Cambridge Isotope Laboratory and were dried over activated 3 Å molecular sieves. Freshly opened anhydrous p-xylene-d10 was degassed via three freeze-pump-thaw cycles and stored in a glovebox. All other reagents were purchased from major chemical suppliers and were used as received. All macrocycles possess an S-configuration about the MeOBiphep axis. Error values associated with individual rate constants refer to the standard deviation of the linear regression.

Synthesis of (P–P)PtCl2 complexes

[(S,S)-E(3,3)]PtCl2 19

A solution of (COD)PtCl2 (26.3 mg, 7.0 × 10–2 mmol) in CD2Cl2 (0.2 mL) was added dropwise via syringe to a solution of (S,S)-E(3,3) (63.1 mg, 7.0 × 10–2 mmol) in CD2Cl2 (0.5 mL) in an NMR tube, which was sealed under nitrogen. The reaction mixture was monitored by 31P NMR spectroscopy to achieve ≥95% conversion (1 h). The solution was diluted with diethyl ether (20 mL) and the resulting precipitate was filtered and washed with diethyl ether. Recrystallization from a saturated dichloromethane solution layered with diethyl ether at –20 °C gave pure [(S,S)-E(3,3)]PtCl2 (69 mg, 82%) as colorless crystals. 1H NMR (500 MHz, CD2Cl2): δ 7.78 (s, br, 4H), 7.66–7.59 (m, 4H), 7.50–7.38 (m, 8H), 7.25 (t, J = 7.5 Hz, 4H), 7.18 (d, J = 8.1 Hz, 2H), 6.95 (s, 2H), 6.88 (td, J = 8.3, 2.3 Hz, 2H), 6.75–6.63 (m, 4H), 6.26 (d, J = 8.4 Hz, 2H), 4.31–4.19 (m, 4H), 3.38–3.28 (m, 2H), 3.17–2.80 (m, 10H), 1.65–1.49 (m, 4H). 13C{1H} NMR (126 MHz, CD2Cl2): δ 156.9 (d, J = 5.4 Hz), 156.9 (d, J = 5.4 Hz), 155.4, 144.9, 141.2, 135 2 (d, J = 5.4 Hz), 135.1 (d, J = 4.7 Hz), 134.7, 131.6, 130.7, 129.1 (d, J = 4.4 Hz), 129.0 (d, J = 6.0 Hz), 128.9 (d, J = 6.0 Hz), 128.7 (d, J = 5.2 Hz), 128.6 (d, J = 4.4 Hz), 128.1 (d, J = 5.0 Hz), 127.2 (d, J = 5.7 Hz), 127.1 (d, J = 5.7 Hz), 126.4 (d, J = 7.6 Hz), 126.3 (d, J = 7.6 Hz), 126.0, 124.8, 124.7 (d, J = 4.3 Hz), 124.0 (d, J = 5.5 Hz), 123.5 (d, J = 5.5 Hz), 118.2, 113.0, 110.7, 65.7, 65.0, 36.1, 31.8, 27.4. 31P NMR (162 MHz, CD2Cl2) δ 8.78 (s, JPtP = 3664 Hz). HRMS-ESI (m/z) calcd (found) for C60H52ClO4P2Pt [M–Cl]+: 1129.2681 (1129.2685). Complexes [(R,S)-E(3,3)]PtCl2 and [(R,S)-E(2,3)]PtCl2 were synthesized from (S,R)E(3,3) and (S,R)-E(2,3), respectively, employing analogous procedures.

[(S,R)-E(3,3)]PtCl2 19

Pale yellow crystals, 58%. 1H NMR (500 MHz, CD2Cl2): δ 7.70 (br s, 4H), 7.51 (m, 4H), 7.38–7.29 (m, 8 H), 7.17 (t, J = 7.0 Hz, 4 H), 7.13 (d, J = 8.0 Hz, 2 H), 7.03 (d, J = 1.5 Hz, 2 H), 6.69−6.67 (m, 4 H), 6.56 (dd, J = 10.5, 8.0 Hz, 2 H), 5.70 (d, J = 8.5 Hz, 2 H), 4.29 (ddd, J = 12, 6.5, 4.5 Hz, 2H), 3.99 (ddd, J = 12.5, 7.0, 5.0 Hz, 2 H), 3.20 (td, J = 11.5, 5.0 Hz, 2 H), 3.00-2.75 (m, 10 H), 1.76 (m, 2 H), 1.61 (m, 2 H). 13C{1H} NMR (126 MHz, CD2Cl2): δ 157.0 (d, J = 5.4 Hz), 156.9 (d, J = 5.4 Hz), 156.4, 144.7, 141.9, 135.1 (d, J = 6.0 Hz), 135.0 (d, J = 6.0 Hz), 134.9, 131.5, 130.6, 129.1 (d, J = 6.0), 128.9 (d, J = 8.0 Hz), 128.8 (d, J = 8.0 Hz), 128.7 (d, J = 7.0 Hz), 128.5 (d, J = 5.7 Hz), 128.2 (d, J = 7.4 Hz), 128.1 (d, J = 7.4 Hz), 127.2 (d, J = 7.4 Hz), 127.1, (d, J = 7.4 Hz), 126.4 (d, J = 9.4 Hz), 126.3 (d, J = 9.4 Hz), 125.9, 124.8 (d, J = 5.8 Hz), 124.7 (d, J = 5.8 Hz), 124.4 (d, J = 7.4 Hz) 123.9 (d, J = 7.8 Hz), 118.7, 113.0, 111.7, 67.0, 66.7, 35.7, 31.6, 30.6. 31P NMR (162 MHz, CD2Cl2): δ 8.79 (s, JPtP = 3652 Hz). HRMS-ESI (m/z) calcd (found) for C60H52ClO4P2Pt [M–Cl]+: 1129.2681 (1129.2672).

[(S,R)-E(2,3)]PtCl2 19

White solid, 69%. 1H NMR (500 MHz, CD2Cl2): δ 7.84 (qd, J = 13.4, 7.8 Hz, 4H), 7.63 (ddd, J = 31.8, 11.8, 7.4 Hz, 5H), 7.54–7.30 (m, 13H), 7.19 (dt, J = 17.0, 7.8 Hz, 4H), 7.00 (s, 1H), 6.86 (d, J = 8.5 Hz, 3H), 6.69 (ddq, J = 30.5, 21.8, 12.3, 9.9 Hz, 3H), 6.20 (d, J = 8.6 Hz, 1H), 5.48 (d, J = 8.4 Hz, 1H), 4.34 (dd, J = 12.2, 6.1 Hz, 2H), 4.04 (dt, J = 42.9, 10.6 Hz, 3H), 3.21–2.76 (m, 10H), 2.63 (q, J = 10.7 Hz, 1H), 2.40 (t, J = 10.9 Hz, 1H), 2.12 (td, J = 11.2, 6.0 Hz, 1H), 1.69 (dq, J = 13.7, 7.4 Hz, 1H), 1.42 (dt, J = 16.9, 9.4 Hz, 1H). 13C{1H} NMR (126 MHz, CD2Cl2): δ 158.09, 157.99, 157.17, 157.11, 157.01, 155.59, 146.68, 145.81, 144.09, 142.02, 135.77, 135.68, 135.58, 135.52, 135.44, 135.37, 135.29, 134.90, 132.27, 132.26, 131.95, 131.94, 131.05, 131.03, 130.99, 130.98, 129.29, 129.25, 129.19, 129.02, 128.82, 128.74, 128.51, 128.27, 128.18, 128.14, 128.05, 127.61, 127.58, 127.52, 127.49, 126.81, 126.12, 125.43, 125.36, 124.90, 124.83, 124.48, 124.44, 123.99, 123.94, 120.38, 118.90, 117.99, 113.80, 113.09, 109.34, 71.13, 68.47, 66.21, 63.60, 37.32, 36.07, 32.39, 32.14, 30.97. 31P NMR (202 MHz, CD2Cl2): δ 9.44, 8.44 (ABq, JPP = 18 Hz, JPtP = 3670, 3662 Hz). HRMS-ESI (m/z) calcd (found) for C59H50ClO4P2Pt [M–Cl]+: 1115.2524 (1115.2508).

[(S,R)-E(2,2)]PtCl2 19

A solution of (NBD)PtCl2 (6.2 mg, 1.7 × 10–2 mmol) in DMF-d7 (0.3 mL) was added dropwise via syringe to a septum-capped NMR tube containing a solution of (S,R)-E(2,2) (15.9 mg, 1.8 × 10–2 mmol) in DMF-d7 (0.3 mL) cooled at −78 °C. The contents of the tube were mixed thoroughly at −78 °C and placed in the probe of an NMR spectrometer pre-cooled at −30 °C. 31P NMR analysis after 10 min revealed quantitative formation of [(S,R)-E(2,2)]PtCl2 (9.37 ppm, s). Thermally unstable [(S,R)-E(2,2)]PtCl2 was characterized in solution without isolation. 1H NMR (700 MHz, DMF-d7, −30 °C): 7.64 (m, 6 H), 7.52 (d, J = 7.0 Hz, 2 H), 7.50 (t, J = 7 Hz, 8H), 7.39 (d, J = 8.4 Hz, 2 H), 7.01 (t, J = 7.7 Hz, 2H), 6.98 (d, J = 7.7 Hz, 2H), 6.78 (d, J = 1.5 Hz, 2H), 6.89 (d, J = 8.4 Hz, 2H), 6.59 (t, J = 8.4, Hz, 2H), 4.13 (m, 4 H), 3.80 (m, 2 H), 3.25 (m, 2 H), 3.01 (m 4 H), 2.86 (dd, J = 7.0, 13 Hz, 2 H), 2.81 (dd, J = 5.6, 12 Hz, 2 H). 31P NMR (283 MHz, CD2Cl2): δ 9.37 (JPtP = 3650 Hz).

Isomerization of [(S,R)-E(2,3)]PtCl2 18

A solution of [(S,R)-E(2,3)]PtCl2 (8.6 mM) containing free (S,R)-E(2,3) (~8.6 mM) was generated via addition of DMF-d7 (0.60 mL) via syringe to a septum-capped NMR tube containing (NBD)PtCl2 (1.85 mg, 5.17 × 10–3 mmol, 8.6 mM) and (S,R)-E(2,3) (9.12 mg, 1.03 × 10–2 mmol) at 25 °C. The tube was shaken until the mixture was homogeneous, inserted into the probe of an NMR spectrometer preheated at 80 °C, and analyzed periodically by 31P NMR spectroscopy. The concentration of [(S,R)-E(2,3)]PtCl2 was determined by integrating the resonances corresponding to [(S,R)-E(2,3)]PtCl2 at δ 6.39 and 5.39 (ABq, JPP = 19 Hz) and [Z(2,3)]PtCl2 at δ 4.94 (s) assuming quantitative E to Z isomerization; this assumption was supported by the absence of any additional resonances in the 31P NMR spectrum throughout complete conversion of [(S,R)-E(2,3)]PtCl2 to [Z(2,3)]PtCl2. Furthermore, upon complete (≥95%) isomerization of [(S,R)-E(2,3)]PtCl2 to [Z(2,3)]PtCl2, no significant isomerization of free (S,R)-E(2,3) was observed, as judged by 31P NMR spectroscopy. The first-order rate constant for the conversion of [(S,R)-E(2,3)]PtCl2 to [Z(2,3)]PtCl2 was obtained from a linear plot of ln{[(S,R)-E(2,3)]PtCl2}17 versus time through 3 half-lives (Supplementary Table 1, entry 1). First-order rate constants for the conversion of [(S,R)-E(2,3)]PtCl2 to [Z(2,3)]PtCl2 were determined as a function of temperature from 64 to 80 °C employing similar procedures (Supplementary Table 1, entries 1–5). A plot of ln(k/T) versus reciprocal temperature provided the activation parameters for isomerization (Supplementary Table 2).

Isomerization of [(S,S)-E(3,3)]PtCl2 and [(S,R)-E(3,3)]PtCl2

First-order rate constants for the isomerization [(S,S)-E(3,3)]PtCl2 at 125 °C in DMF-d7 (Supplementary Table 1, entry 6) and for the isomerization of [(S,R)-E(3,3)]PtCl2 at 121 °C in DMF-d7 (Supplementary Table 1, entry 7) were determined employing procedures similar to those used to determine the first-order rate constants for the isomerization of [E(2,3)]PtCl2. Both [(S,S)-E(3,3)]PtCl2 and [(S,R)-E(3,3)]PtCl2 isomerize to form [Z(3,3)]PtCl219.

Isomerization of [(S,R)-E(2,2)]PtCl2

A solution of [(S,R)-E(2,2)]PtCl2 containing free (S,R)-E(2,2) was generated by addition of a solution of (NBD)PtCl2 (1.85 mg, 5.17 × 10–3 mmol) in DMF-d7 (0.60 mL) via syringe to a septum-capped NMR tube containing (S,R)-E(2,2) (8.97 mg, 1.03 × 10–2 mmol) at −30 °C. The contents of the tube were mixed thoroughly at −30 °C, the tube was placed in the probe of an NMR spectrometer pre-cooled at 6 °C, at which time the 31P NMR spectrum revealed quantitative formation of [(S,R)-E(2,2)]PtCl2. The solution was analyzed periodically by 31P NMR spectrometry by integrating the resonances corresponding to [(S,R)-E(2,2)]PtCl2 (δ 6.27, s) and [Z(2,2)]PtCl2 (δ 5.09, s). First-order rate constants (Supplementary Table 1, entries 8–12) and activation parameters (Supplementary Table 2) for the isomerization [(S,R)-E(2,2)]PtCl2 to [Z(2,2)]PtCl2 were determined employing procedures similar to those used to determine the first-order rate constants for the isomerization of [(S,R)-E(2,3)]PtCl2.

Isomerization of (S,R)-E(2,2)

A septum-capped NMR tube containing a solution (S,R)-E(2,2) (4.4 mg, 5.17 × 10–3 mmol, 8.61 mM) in p-xylene-d10 (0.6 mL) was placed in the probe of an NMR spectrometer preheated at 84 °C and analyzed periodically by 31P NMR spectroscopy. The concentration of (S,R)-E(2,2) was determined by integrating the resonances corresponding to E(2,2) at δ –7.04 (s) and Z(2,2) at δ –9.34 (s) assuming quantitative E to Z isomerization; this assumption was supported by the absence of any additional resonances in the 31P NMR spectrum throughout complete conversion of (S,R)-E(2,2) to Z(2,2). The first-order rate constant for the conversion of (S,R)-E(2,2) to Z(2,2) was obtained from the linear plot of ln[(S,R)-E(2,2)] versus time through 3 half-lives (Supplementary Table 1, entry 13). First-order rate constants for the conversion of (S,R)-E(2,2) to Z(2,2) were determined as a function of temperature from 69 to 84 °C employing a similar procedure (Supplementary Table 1, entries 13–16). A plot of ln(k/T) versus reciprocal temperature provided the activation parameters for isomerization (Supplementary Table 2).

Isomerization of (S,R)-E(2,3), (S,S)-E(3,3), and (S,R)-E(3,3)

First-order rate constants and activation parameters for the isomerization of (S,R)-E(2,3) were obtained employing procedures similar to that used to determine the first-order rate constants and activation parameters for the isomerization of (S,R)-E(2,2) (Supplementary Table 1, entries 17–22; Supplementary Table 2). The first-order rate constant for the isomerization of (S,R)-E(3,3) in p-xylene-d10 Supplementary Table 1, entry 23) at 126 °C was obtained employing a procedure similar to that used to determine the first-order rate constants for the isomerization of (S,R)-E(2,2). The first-order rate constant for the isomerization of (S,S)-E(3,3) was obtained by heating an NMR tube containing a solution of (S,S)-E(3,3) in p-xylene-d10 in an oil bath at 131 °C, which was removed periodically, cooled immediately to 0 °C, and analyzed by 31P NMR spectroscopy at 25 °C (Supplementary Table 1, entry 24).

Solvent effect on ligand isomerization

Separate solutions of (S,R)-E(2,3) (8.6 mM) in DMF-d7 (0.60 mL) and p-xylene-d10 were heated together at 120 °C, removed periodically, cooled immediately to 0 °C, and analyzed at identical time points by 31P NMR spectroscopy at 25 °C. The first-order rate constant for the isomerization of (S,R)-E(2,3) in DMF-d7 (kobs = 9.8 ± 0.3 × 10–5 s–1) and p-xylene-d10 (kobs = 9.3 ± 0.3 × 10–5 s–1; Table 1, entry 22, Fig. S25) differed by ~5%.

DFT calculations

All calculations were performed with Gaussian 16.C in vacuum; the Berny algorithm was applied to locate stationary points and tight convergence criteria and ultrafine integration grids were used in optimizations and frequency calculations. Reactant conformers of the macrocycles were generated and optimized as previously described18, except for the used model chemistry (B3LYP/def2SVP); we previously demonstrated that the def2SVP basis set leads to faster convergence of geometry optimizations of complexes of heavy transition metals and yields kinetic barriers for reactions of such complexes in better agreement with experiment than LANL2DZ52. Free energies of individual conformers were sums of the single-point electronic energy calculated at the (u)BMK/def2SVP level and the corresponding thermodynamic corrections calculated statistically-mechanically with all frequencies <500 cm−1 replaced with 500 cm−1 to avoid artifactually large contributions of vibrations to ΔGǂ53. Further details are provided in the Supplementary Information.

The mechanochemical model of allosteric isomerization

For each conformer of E-SS(OMe)2 and TS-SS(OMe)2 we calculated the equilibrium distance, qspo, of every compressive potential with compliance, λ, from 100 Å/nN to 0.1 Å/nN in 0.1 Å/nN increments needed to exert force, f, from −0.2 nN to 1 nN, or the highest compressive force at which the conformer exists, in 1 pN increments (f > 0 corresponds to compression) according to formula, qspo = qi(f) – fλ, where qi(f) is the MeOC…COMe distance of conformer i of SS(OMe)2 in equilibrium with force f, derived quantum-chemically (see the preceding section). Linear interpolation of this data yielded the values of relative free energy, MeOC…COMe and its restoring force of each conformer of SS(OMe)2 for each combination of λ and qspo (primary correlation matrix). All conformers coupled to the potential of the same λ and qspo comprise a conformational ensemble, characterized by ensemble-average parameters, such as force, molecular strain energy and the potential energy of the constraint. The activation free energy of SS isomerization coupled to a potential with given λ and qspo is derived from the corresponding E and TS ensembles (Supplementary Tables 7 and 8). A subset of this data is plotted in Figs. 2c, d and 3a.

Whereas isomerization ΔGǂ of SS(OMe)2 is uniquely defined by specifying λ and qspo, the change in ΔGǂ caused by a given change in qspo (which represents the contraction of the BIPHEP containing strap upon PtCl2 binding in our experiments and more generally, the mechanism by with effector binding lowers the kinetic barrier), ΔΔGǂisom, (Fig. 3b, c) is generally not. Consequently, to identify the highest achievable efficiency of allosteric control for potential of each compliance, we calculated by interpolation of the master correlation matrix all Δqspo values needed to achieve each ΔΔGǂ value from 0 to −37 kcal/mol in 0.1 kcal/mol increments and for each ΔΔGǂ selected the combination of two qspo (before and after binding) that corresponding to the smallest increase in the energy of the potential (Supplementary Table 9). This data is plotted in Fig. 3b, c.