Allosteric control of olefin isomerization kinetics via remote metal binding and its mechanochemical analysis

Allosteric control of reaction thermodynamics is well understood, but the mechanisms by which changes in local geometries of receptor sites lower activation reaction barriers in electronically uncoupled, remote reaction moieties remain relatively unexplored. Here we report a molecular scaffold in which the rate of thermal E-to-Z isomerization of an alkene increases by a factor of as much as 104 in response to fast binding of a metal ion to a remote receptor site. A mechanochemical model of the olefin coupled to a compressive harmonic spring reproduces the observed acceleration quantitatively, adding the studied isomerization to the very few reactions demonstrated to be sensitive to extrinsic compressive force. The work validates experimentally the generalization of mechanochemical kinetics to compressive loads and demonstrates that the formalism of force-coupled reactivity offers a productive framework for the quantitative analysis of the molecular basis of allosteric control of reaction kinetics. Important differences in the effects of compressive vs. tensile force on the kinetic stabilities of molecules are discussed.

Allosteric control of reaction thermodynamics is well understood, but the mechanisms by which changes in local geometries of receptor sites lower activation reaction barriers in electronically uncoupled, remote reaction moieties remain relatively unexplored.Here we report a molecular scaffold in which the rate of thermal E-to-Z isomerization of an alkene increases by a factor of as much as 10 4 in response to fast binding of a metal ion to a remote receptor site.A mechanochemical model of the olefin coupled to a compressive harmonic spring reproduces the observed acceleration quantitatively, adding the studied isomerization to the very few reactions demonstrated to be sensitive to extrinsic compressive force.The work validates experimentally the generalization of mechanochemical kinetics to compressive loads and demonstrates that the formalism of force-coupled reactivity offers a productive framework for the quantitative analysis of the molecular basis of allosteric control of reaction kinetics.Important differences in the effects of compressive vs. tensile force on the kinetic stabilities of molecules are discussed.
The central role of allosteric regulation in enabling life as we know it 1 and the fundamental questions in information transfer 2 , conformational dynamics 3,4 and emergent properties 5,6 that allostery presents underlie continued effort to design synthetic molecules with allosteric behavior 7 .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 understood [8][9][10][11][12] .
In comparison, abiological examples of allosteric control of reaction kinetics are few 8,12 .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 catalysts 13,14 , 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 equivalents 8 .
Here we report a molecular scaffold which allows the rate of thermal E→Z isomerization of an alkene (stiff stilbene 15 , Fig. 1) to vary systematically by a factor of up to 10 4 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 loads 16 , 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 force 17 , and highlights important differences in the effects of compressive vs. tensile force on the kinetic stabilities of molecules.

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 protocol 17,18 , 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 crystallography 19 , whereas smaller macrocycles, E(2,2) and E(2,3) were generated as single diastereomers as evidenced Fig. 1 | The concept of allosteric reaction control and its implementation.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 PtCl 2 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 PtCl 2 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.
by 1 H NMR spectroscopy 19 .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 macrocycles 20,21 .This preference was rationalized by a combination of rapid epimerization of Z-stiff stilbene 22 and the strong dependence of the quantum yields of Z→E photoisomerization on the strain energy of the product [23][24][25] .Macrocycles E(m,n) react rapidly and quantitatively with (NBD)PtCl 2 (nbd = norbornadiene) at or above -30 °C to form the ligated macrocycles [E(m,n)]PtCl 2 (Fig. 1c).The PtCl 2 fragment was chosen as effector for the high kinetic and thermodynamic stability of the Pt-P bond 26 .
We measured the activation barriers of E→Z isomerization of stiff stilbene in the absence of metal ions and upon coordination to PtCl 2 (Supplementary Figs 1-24).In p-xylene-d 10 solutions at ≥69 °C the Ptfree macrocycles E(m,n) isomerized exclusively to their Z(m,n) isomers with the first-order kinetics over three half-lives as determined by 31 P 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 reactivity [27][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)]PtCl 2 in DMF-d 7 at or above -10 °C led to first-order decay over three half-lives to form exclusively the corresponding [Z(m,n)]PtCl 2 macrocycles.We previously demonstrated 20 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)]PtCl 2 and [(S,R)-E(2,2)]PtCl 2 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.
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 Ar C-C = C-C Ar torsion, θ, relative to that of free E-SS (θ ≈ 180 o ), in all macrocycles (Fig. 2a).In all Pt complexes this torsion was reduced further by 4-8 o 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 −6 o and 2 o , depending on the conformer (Supplementary Fig. 26), suggesting low strain of either tension or compression, regardless of the presence of Pt.
Across all conformers, geometric parameters of the SS(OCH 2 ) 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 MeO C … C OMe 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 mechanochemistry [30][31][32][33] , suggesting that the observed variation in ΔG ǂ of isomerization across the series, including the allosteric effect of Pt binding, could be amenable to analysis within the formalism of mechanochemical kinetics 34 .
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 freedom 33 .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 solutions 35 or material under plastic deformations 36 .The measured kinetics of reactions in overstretched polymer chains 33 or non-macromolecular mimics reported to date 37 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 inhibition 21 ) 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 force 38,39 .
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 uncertain 16 .Within the standard mechanochemical model it is outright aphysical 34 .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 The measured (ΔG ‡ expt ) and calculated (ΔG ‡ calc ) activation free energies of stiff stilbene E → Z isomerization in E(m,n) macrocycles and their platinum complexes [E(m,n)]PtCl2 in kcal/mol.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 ensembleaverage 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 ΔG ǂ 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, 〈f E 〉, a soft potential reduces ΔG ǂ 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 ΔG ǂ on the ensemble-average force experienced by E-SS, 〈f E 〉 decreases as well until the ΔG vs. 〈f E 〉 correlation becomes complianceindependent 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 , f E , needed to reproduce the geometry of the corresponding moiety in each thermally accessible conformer of each E macrocycle.We then averaged resulting single-conformer f E in proportion to the Boltzmann weight of each corresponding conformer in the reactant ensembles.The correlation between these ensemble 〈f E 〉 and ΔG ǂ 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 10 4 -fold acceleration at 300 K) arises from only a small increase in the strain of the E-SS moiety, as evidenced by 〈f E 〉 of stiff stilbene in Pt-coordinated complexes exceeding that of the Pt-free precursors by only 30   Maximum f E that a compressive potential can apply decreases with increasing compliance, resulting in force-dependent activation free energy of isomerization, ΔG ǂ , 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, 〈f ts 〉, compared to that in the E isomer, 〈f E 〉.The plotted data is tabulated in Supplementary Tables 6-8.
(7-15%).The mechanochemical formalism allows the underlying molecular basis to be understood and generalized by analyzing the ΔG ǂ vs. 〈f E 〉 correlations in Fig. 2c in terms of 4 components: the ensembleaverage 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.
The strain energies of both E-SS(OMe) 2 and its compressing potential increase monotonically with 〈f E 〉 (red curves).Conversely, those of the TS vary non-monotonically: they are non-zero at 〈f E 〉 = 0 and decrease with increasing 〈f E 〉 up to 0.42 nN.This seemingly counterintuitive dependence is the direct consequence of the finite compliance of the potential.At 〈f E 〉 = 0 the equilibrium length of the potential equals that of MeO C … C OMe 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 (〈f E 〉 = 0), stretches the shorter TS (Fig. 2d).As the equilibrium distance of the constraining potential contracts with increasing 〈f E 〉, the corresponding 〈f ts 〉 decreases before becoming compressive.Above this switchover force, all 4 strain energies increase monotonically.
The approximately planar geometry of strain-free E-SS makes its MeO C … C OMe 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 MeO C … C OMe and O … O distances at f E = 0.35-0.6nN (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 where ΔG ǂ X 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 ΔΔG ǂ isom , all combinations of the two 〈f E 〉 values corresponding to effector-free and effector-bound reactant were considered to identify the minimum necessary increase in the total reactant strain energy, ΔE strain .For macrocycles (blue dots), ΔΔG ǂ bind describes the reaction free energy of PtCl 2 transfer between free and SS-coupled BiPHEP (eq.1).c 〈f E 〉 of E-SS prior to effector binding giving the highest achievable ΔΔG ǂ isom /ΔE strain 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.applied force, yielding a range of 〈f E 〉 (0.35-0.6 nN) where a small differential compression of the E isomer causes disproportionately large reduction in ΔG ǂ .Note that because force acting on the transition state, 〈f TS 〉, cannot be directly experimentally controlled, but is uniquely defined by 〈f E 〉 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 (〈f E 〉), rather than 〈f TS 〉.
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., 〈f E 〉<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, ΔE strain , 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 ΔE strain 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 ΔΔG ǂ bind for transfer of PtCl 2 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, ΔΔG ǂ bind , 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.ðBiPHEPÞPtCl 2 + Eðn,mÞ"BiPHEP + Eðn,mÞPtCl 2 ð1Þ Figure 3b illustrates the relationship between ΔΔG ǂ bind , ΔE strain and ΔΔG ǂ isom for E stiff stilbene estimated by mechanochemical analysis.It plots the minimum effector-induced increase in the reactant strain energy, ΔE strain , needed to reduce the isomerization barrier by ΔΔG ǂ isom below that of free E-SS(OMe) 2 for compressive potential of different compliances.The dashed red line corresponds to full ΔE strain contributing to barrier lowering (ΔE strain = −ΔΔG ǂ isom ).Values below this line reflect the contribution of relaxation of the pre-strained TS to ΔΔG ǂ isom .For E-SS, this contribution can exceed 50% for weak accelerations (ΔΔG ǂ isom > −3 kcal/mol), provided the substrate is significantly strained prior to effector binding (as quantified by its restoring force, e.g., 〈f E 〉 in Fig. 3c).At every ΔΔG ǂ isom , a stiffer potential requires a smaller effector-induced increase in the reactant strain energy, ΔE strain , than a softer one, which is opposite to what is calculated for tensile force 40 .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 ΔΔG ǂ bind ) 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., ΔΔG ǂ isom >−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 ΔΔG ǂ isom / ΔΔG ǂ bind 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 ΔΔG ǂ isom /ΔΔG ǂ bind ratio is within 90% of ΔΔG ǂ isom /ΔE strain predicted by the mechanochemical model.In these macrocycles the reduction in the effector affinity, ΔΔG ǂ bind , 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 ΔG ǂ 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 scaffolds 5 .In addition, the formalism applied here might inform approaches to improving the efficiencies of switching 23 , energy storage 41 and work-generation 15 in increasingly diverse macrocyclic molecular switches [42][43][44] and machines 24,[45][46][47] for a range of potential applications [48][49][50] .Equally important, E→Z isomerization of stiff stilbene reported here is only the 2nd reaction 17 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 ovendried glassware employing standard Schlenk or glovebox techniques unless noted otherwise.Nitrogen-flushed plastic syringes and ovendried 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/Seal TM 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-d 10 was degassed via three freeze-pumpthaw 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.
Isomerization of (S,R)-E(2,3), (S,S)-E(3,3), and (S,R)-E (3,3).Firstorder 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-d 10 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-d 10 in an oil bath at 131 °C, which was removed periodically, cooled immediately to 0 °C, and analyzed by 31 P NMR spectroscopy at 25 °C (Supplementary Table 1, entry 24).

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 described 18 , 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 LANL2DZ 52 .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 ΔG ǂ53 .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, q sp o , 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, q sp o = q i (f) -fλ, where q i (f) is the MeO C … C OMe 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, MeO C … C OMe and its restoring force of each conformer of SS(OMe) 2 for each combination of λ and q sp o (primary correlation matrix).All conformers coupled to the potential of the same λ and q sp o 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 q sp o 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 ΔG ǂ of SS(OMe) 2 is uniquely defined by specifying λ and q sp o , the change in ΔG ǂ caused by a given change in q sp o (which represents the contraction of the BIPHEP containing strap upon PtCl 2 binding in our experiments and more generally, the mechanism by with effector binding lowers the kinetic barrier), ΔΔG ǂ isom , (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 Δq sp o values needed to achieve each ΔΔG ǂ value from 0 to −37 kcal/mol in 0.1 kcal/mol increments and for each ΔΔG ǂ selected the combination of two q sp o (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.
a at 298 K unless noted otherwise.b free ligand in p-xylene-d 10 ; metal complex in DMF-d 7 .c at 298 K unless specified otherwise, in vacuum at 1 atm; the two values for E(2,n), n = 2 or 3 correspond to the (S,R) and (S,S) diastereomers where * indicates the more stable diastereomer.

Fig. 2 |
Fig. 2 | Summary of the mechanochemical model of the measured isomerization kinetics.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 Ar C-C = C-C Ar 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 ΔG ǂ 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 f E that a compressive potential can apply decreases with increasing compliance, resulting in force-dependent activation free energy of isomerization, ΔG ǂ , 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, 〈f ts 〉, compared to that in the E isomer, 〈f E 〉.The plotted data is tabulated in Supplementary Tables 6-8.

Fig. 3 |
Fig. 3 | Mechanochemical analysis of allosteric acceleration.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, 〈f E 〉. b the minimum energy needed to reduce the isomerization barrier by ΔΔG ǂ isom = ΔG ǂ

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