Inchworm movement of two rings switching ontoa thread by biased Brownian diffusion represent athree-body problemChristopher R. Bensona, Christopher Maffeob, Elisabeth M. Fatilaa,1, Yun Liua,2, Edward G. Sheetza,Aleksei Aksimentievb, Abhishek Singharoyc,3, and Amar H. Flooda,3

aDepartment of Chemistry, Indiana University, Bloomington, IN 47405; bDepartment of Physics, University of Illinois at Urbana–Champaign, Urbana, IL61801; and cSchool of Molecular Sciences, Arizona State University, Tempe, AZ 85281

Edited by J. Fraser Stoddart, Northwestern University, Evanston, IL, and approved April 17, 2018 (received for review November 13, 2017)

The coordinated motion of many individual components underpins theoperation of all machines. However, despite generations of experiencein engineering, understanding the motion of three or more coupledcomponents remains a challenge, known since the time of Newton asthe “three-body problem.” Here, we describe, quantify, and simulate amolecular three-body problem of threading twomolecular rings onto alinearmolecular thread. Specifically, we use voltage-triggered reductionof a tetrazine-based thread to capture two cyanostar macrocycles andform a [3]pseudorotaxane product. As a consequence of the noncova-lent coupling between the cyanostar rings, we find the threading oc-curs by an unexpected and rare inchworm-like motion where one ringfollows the other. The mechanism was derived from controls, analysisof cyclic voltammetry (CV) traces, and Brownian dynamics simulations.CVs from two noncovalently interacting rings match that of two co-valently linked rings designed to thread via the inchworm pathway,and they deviate considerably from the CV of a macrocycle designed tothread via a stepwise pathway. Time-dependent electrochemistry pro-vides estimates of rate constants for threading. Experimentally derivedparameters (energy wells, barriers, diffusion coefficients) helped deter-mine likely pathways of motion with rate-kinetics and Brownian dy-namics simulations. Simulations verified intercomponent coupling couldbe separated into ring–thread interactions for kinetics, and ring–ringinteractions for thermodynamics to reduce the three-body problem toa two-body one. Our findings provide a basis for high-throughput de-sign of molecular machinery with multiple components undergoingcoupled motion.

molecular machines | macrocycles | switching | kinetic modeling |Brownian dynamics

Controlled movement is closely tied to the growth of moderncivilization (1), for example, horse and cart, and combustion

engine. Useful mechanical motion typically involves multipleobjects. When three or more of those objects are mutuallycoupled together, the classical “three-body problem” emerges.Such three-body problems cannot be solved exactly despite theirimportance; for example, the moon’s position relative to thegravitational-coupled movements of earth and sun is criticalfor navigation. Solutions require approximations and numericalmethods to address coupled motions. In molecular science,modeling of coupled motions in n-body problems is currentlybeing applied to biomachines and is capable of revealing newmechanistic pathways (2). Methodologies pioneered by Szaboet al. (3) for modeling these pathways rely on identifying keyreaction coordinates from molecular dynamics or Brownian dy-namics simulations, and transforming reaction coordinate in-formation into kinetics computations. These methods wereinstrumental in discovering kinetic rates in biomachines, for ex-ample, ATP synthesis by ATP synthase (4), ribosomal proteininsertion by chaperones (5), membrane trafficking by transporters(6), and prokaryotic translation by helicases (7). These methodshave only recently been applied to abiological systems, for exam-ple, cyclodextrin switches (8). Here, we extend this approach to a

three-body problem involving an abiological switch—one involvingthreading of two rings onto a rod. Using a set of macrocyclic rings(Fig. 1 A and D), all of the evidence indicates that inchworm-likemovement best describes threading with two rings followingeach other along the rod (Fig. 1B) as a result of ππ interactionsbetween rings.Molecular machines and switches with three components are

well represented by [3]pseudorotaxanes (9–14), [3]catenanes(15–18), and [3]rotaxanes (19–25). Examples include unidirec-tional motion of two small rings around a larger ring (15) andmuscle-like contraction of two rings along a dumbbell in a [3]rotaxane (20). Thermodynamic bistability of two-ring systems arewell described. Studies on the impact of two rings on kinetics andmechanism are considerably rarer (11, 13, 15–17, 24, 25). Ki-netics can be studied when timescales of motion match tech-niques used for observing switching (11, 13, 24, 26–32). Here, wetake advantage of cyclic voltammetry (CV) to study redox-drivenkinetics and mechanism of switching on 1-ms to 10-s timescales.We have previously described switches composed of two rings

and a single tetrazine thread exemplifying the three-body problem.With copper(I)-phenanthroline rings (13), there are no inter-ringinteractions and the rings can be considered independent of eachother. Stepwise threading of a first ring onto a tetrazine anionproduced an observable [2]pseudorotaxane intermediate with a

Significance

This article describes a molecular realization of the classicalthree-body problem, where the motion of three or more bodiesis directed by a set of pairwise forces. Surprisingly, motion ofthe components of the three-body molecular systems is foundto be highly choreographed by differences in strength ofintercomponent interactions, promoting a rare inchworm-likeloading of molecular rings onto a molecular thread. Our workdemonstrates the utility of an integrative approach to designand develop functional molecular machines.

Author contributions: C.R.B., C.M., A.S., and A.H.F. designed research; C.R.B., C.M., E.M.F.,Y.L., E.G.S., and A.S. performed research; C.R.B., C.M., E.M.F., Y.L., A.A., A.S., and A.H.F.analyzed data; and C.R.B., C.M., A.A., A.S., and A.H.F. wrote the paper.

Conflict of interest statement: The authors report a patent awarded on “poly-cyanostil-bene macrocycles” that covers the composition of matter of cyanostar macrocycles.

This article is a PNAS Direct Submission.

Published under the PNAS license.1Present address: Department of Chemistry, Physics, Molecular Science and Nanotechnol-ogy, Louisiana Tech University, Ruston, LA 71272.

2Present address: Beckman Institute for Advanced Science and Technology, University ofIllinois at Urbana–Champaign, Urbana, IL 61801.

3To whom correspondence may be addressed. Email: singhar@asu.edu or aflood@indiana.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1719539115/-/DCSupplemental.

Published online May 7, 2018.

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rate constant of 55,000 M−1·s−1. The slower threading of the sec-ond ring (13,000 M−1·s−1) reflects steric interactions between rings.A second example involved threading of two planar cyanostar (CS,Fig. 1A) macrocycles onto a tetrazine thread (12, 33–37). Thethreading mechanism was not determined (12) because CV sig-natures did not conform to the expected production of anintermediate, as observed with copper(I)-phenanthroline. Here,we describe a detailed analysis of this unexpected outcome.Threading of two rings onto a rod-like thread can follow

multiple pathways. The threading of two rings as a single unit isleast likely unless they are affixed intimately together. Morelikely is the independent, stepwise threading of rings to producean observable intermediate with a slower second step (Fig. 1C)(13). Last, two rings can thread in an inchworm movement (Fig.1B) with one ring following the other but only if the motions ofthe two rings are coupled. An inchworm mechanism was pre-viously proposed to describe kinesin (38) and myosin (39), butlater ruled out in favor of hand-over-hand motion by single-

molecule studies (38, 39). The inchworm mechanism is pres-ently on the table for the dynein motor motion (40); however, ithas not yet been observed in abiological contexts.Here, we show that when two cyanostar rings are coupled

together by noncovalent (ππ) or covalent (σ) bonds, they threadlike an inchworm onto one side of a thread. The components(Fig. 1A) are cyanostar macrocycles (the rings) that thread ontoa disubstituted tetrazine (the rod): 3,6-bis(5-methyl-2-pyridine)-1,2,4,5-tetrazine (BPTz) (10–13). The cyanostar self-associatesinto π-stacked multimers with Ke = 225 M−1 (13 kJ·mol−1,CH2Cl2) (33), and threading is initiated by the tetrazine’s re-duction to the anion, BPTz–. Control compounds help distinguishinchworm from stepwise pathways for cyanostar threading. Inch-worm movements followed by a covalently linked bis-cyanostar(Fig. 1A) produce CVs with no intermediates. This behaviormatches the parent cyanostar. In contrast, a macrocycle designedto exist as a single ring by inhibiting self-association until it bindsan anion (Fig. 1D, cyanodimer), displays the characteristic

Fig. 1. (A) Structures and cartoon representations of cyanostar ring systems (parent cyanostar and dodecyl-linked bis-cyanostar) and a tetrazine thread(colored red): 3,6-bis(5-methyl-2-pyridine)-1,2,4,5-tetrazine (BPTz). (B and C) Representations of the (B) inchworm and (C) stepwise mechanisms for threadingcyanostar-based rings onto a tetrazine rod. (D) Structures and cartoon representations of cyanostar macrocycles, cyanosolo and cyanodimer, that thread ontoa tetrazine rod in a stepwise manner. RDS indicates a rate-determining step.

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intermediate of a stepwise mechanism. The inchworm mecha-nism is then interrogated using a consistent digital simulation ofthe CVs to generate rate constants for switching. With barriersand wells determined from CV simulations, kinetics and Brow-nian dynamics simulations provided mean first-passage times ofcyanostar motion at different stages of the pathway (3). Wetested and confirmed the idea that the kinetics description of thethree-body problem could be simplified to just two-componentcouplings between ring and thread because ring–ring interactionsmostly only influence thermodynamics. These findings provide abasis for high-throughput design (41) of molecular machineryusing simulations (42) to evaluate the synchronized movementsof multiple coupled components (18).

Results and DiscussionTime Dependence of Redox-Driven Switching. The switching processinvolves reduction of tetrazine to generate the anion, whichdrives threading with two macrocycles to produce the doublythreaded [3]pseudorotaxane. Consistently, a traditional switchingsignature is seen in the CV (12). The free tetrazine displays (Fig.2A) a reversible reduction with cathodic peak at Epc = −1.00 V(vs. Ag wire pseudoreference) and corresponding anodic peak atEpa = −0.90 V. With 2 eq of cyanostar, the CV recorded as afunction of scan rate (Fig. 2B) show changes indicative ofswitching kinetics (12). A slow scan rate (0.1 V·s−1) allows ade-quate time (∼7 s) for switching. Complete threading of twomacrocycles occurs after reduction to the anion BPTz– yet beforethe anion’s deactivation by oxidation at approximately −0.9 V.The intensity of the reoxidation peak decreases with a concom-itant growth in intensity of the new reoxidation peak at −0.10 V(Fig. 2B). This peak is assigned (12) to the threaded product,CS2•BPTz

–. In contrast, switching is incomplete at faster scanrates (5 V·s−1, Fig. 3B), which allows only 140 ms for switching.This time is defined by the scan rate when the voltage sweeps700 mV from −1.0 V out to the vertex at −1.3 V and back to −0.9 V.No intermediate peaks for the singly threaded [2]pseudorotaxane

are observed across all scan rates (0.1–10 V·s−1) and concentra-tions (0.5 and 2.5 mM).

Qualitative Studies with Control Compounds. A key difference inCVs recorded with cyanostar (Fig. 2B) relative to copper(I)-phenanthroline (13) is the absence of an observable intermediate.This outcome is attributed to the coupling between cyanostarsproducing dimers (CS)2 and higher-order multimers (CS)n by thefollowing equilibria:

CS+CS ⇌ ðCSÞ2 Ke = 225 M−1, [1a]

ðCSÞn +CS⇌ ðCSÞn+1 Ke = 225 M−1. [1b]

To test for the role of coupling between macrocycles asrepresented by dimer (CS)2 in Eq. 1, we used a covalently linkedpair of macrocycles. The covalently linked bis-cyanostar (Fig.1A) is expected to reflect an inter-ring coupled state. It is un-likely that the second macrocycle can reach over to the other endof the tetrazine, as it requires a fully extended dodecyl chain.There is a sizable energy cost of 24 kJ·mol−1 (43) associated withthe loss of configurational entropy in this extended chain. Thus,threading will proceed by the inchworm pathway. Both the co-valently linked and noncovalently linked rings display similarCVs (compare Fig. 2 B and C). In both cases, there is no in-termediate. We infer, therefore, that both the molecules threadby the same pathway, namely the inchworm.Cyanodimer (Fig. 1D) is an excellent control macrocycle. It

has been designed to exclude inter-ring interactions and onlyform a dimer when complexed with an anion. It cannot proceedvia an inchworm pathway and is predicted, instead, to switch by astepwise mechanism. As expected, it displays (Fig. 3C) an in-termediate that is indicative of the stepwise pathway when ex-amined under similar conditions as the parent cyanostar (Fig.3A) and bis-cyanostar (Fig. 3B). The oxidation peak position ofthe intermediate was confirmed using the CV (Fig. 3D) obtained ona model compound denoted cyanosolo (36). This macrocycle (Fig.1C) bears bulky groups, only threads a single ring, and matchesthe oxidation peak of the cyanodimer intermediate (Fig. 3C vs.Fig. 3D). Altogether, the CV of noncoupled cyanodimer is in

Fig. 2. Electrochemical CV traces of (A) BPTz alone and (B) the switching systemwith 2 eq of cyanostar added (1.0 mM BPTz, 2.0 mM cyanostar). (C) Matchingvariable scan rate CVs of BPTz (1.0 mM) with 1 eq of the bis-cyanostar (1.0 mM).Conditions: 0.1 M TBABPh4, CH2Cl2 N2, variable scan rate, 1.0-mm glassy carbonwireworking electrode, platinumwire counter electrode, AgOwire quasireference,298 K, all referenced to Epc BPTz = −1.00 V. Times defined by scan rates: 7 s(0.1 V·s−1), 3.5 s (0.2 V·s−1), 0.7 s (1 V·s−1), and 0.14 s (5 V·s−1).

Fig. 3. Experimental CVs of switching with signatures that show no intermedi-ates—parent cyanostar (A) and bis-cyanostar (B)—compared with the CV signa-tures that show a signal for the [2]pseudorotaxane intermediate—cyanodimer (C)and cyanosolo (D). Conditions: 0.2 V·s−1; all other conditions are the same as Fig. 2.

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complete contrast to the parent cyanostar, indicating that cou-pling is responsible for their differences in behavior.

Quantitative Mechanistic Model of the Inchworm Pathway. All CVsare consistent with the parent cyanostar threading along an inch-worm pathway (Fig. 4 B and C and SI Appendix, Figs. S5 and S6).This model satisfies the observation that the parent cyanostardisplays the same CV and inchworm pathway as the bis-cyanostar;both do not have intermediates. Both of these coupled macrocycleshave different CVs and behavior relative to the cyanodimer’sstepwise threading. Only the inchworm mechanism can generate aself-consistent set of rate constants for the kinetics of motion.To quantify kinetics, we used a standard approach (10, 11, 13,

26). The variation in CV peak intensities with scan rate is repro-duced using digital simulations based on possible mechanisms ofswitching. CV simulations employ experimentally determined (12)and estimated thermodynamics, diffusion coefficients, and chemicaland electron-transfer reactions. The absence of an intermediatearises when the second step is fast enough to consume intermedi-ates. This boundary condition needs to be fulfilled in a mechanisticmodel. The final model (Fig. 4A) includes microscopic states rea-sonably indicated from experiment (SI Appendix, section 7).In the inchworm model (Fig. 4A), two cyanostar macrocycles

come together when the thermodynamically favored dimer isformed according to Eq. 1. About ∼21% of the cyanostars insolution (2.5 mM) reside as the dimer relative to the monomer(44%), trimer (7%), tetramer (7%), and higher-order multimers(20%) (SI Appendix, Fig. S1). Rapid equilibration among theseallows the dimer to be reformed. On account of weak associationseen between diglyme and cyanostar (β2 = 30 M−1, ΔG =−8 kJ·mol−1) (35), we introduce a step in the mechanism in-volving weak association (ΔG = −2 kJ·mol−1, Eq. 2 and Fig. 4A)

with the reduced thread, CS2@BPTz–. Subsequently, macro-cycles thread one by one like an inchworm. This first threading(Eq. 3) makes a [2]pseudorotaxane intermediate, which rapidlyassociates with the second cyanostar (Eq. 4) and converts intoproduct by threading of the second ring (Eq. 5).This inchworm model successfully reproduced all experimental

CVs (Fig. 4 B and C and SI Appendix, Figs. S5 and S6). SimulatedCVs match experiment as a function of scan rate. Modeling showsthe first threading occurs with rate constant 7,000 ± 1,000 s−1 (Eq.3). The second threading has a rate constant of 7,000 s−1 (Eq. 5),which is a lower bound and is sufficient to ensure the intermediate isconsumed. Variation in concentration was used to test the model.We find a single unified set of rate constants reproduce the CVsrecorded at 2.5 and 0.5 mM.We also conducted digital simulations of the CVs using the

stepwise mechanism. We are not able to generate a single unifiedset of rate constants to reproduce the experimental CVs acrossboth scan rate and concentration. Rate constants for the first andsecond steps developed at the lower concentration of 0.5 mM(700 ± 200 and 7,000 s−1; SI Appendix, Fig. S5) are significantlydifferent, and do not match, those at the higher concentration of2.5 mM (2,500 ± 500 and 25,000 s−1; SI Appendix, Fig. S6).

Defining the Energy–Distance Coordinate. With the inchwormmechanism refined using digital simulations of the CVs, we de-veloped a mechanical model of motion to probe this three-bodysystem with statistical mechanical computations. This mechanicalmodel involved transforming the reaction coordinate space (Fig.4A) into a spatial basis (Fig. 5 and SI Appendix, Fig. S11). Themovement of two rings was considered relative to the frame ofthe thread. For simplicity, only translational walks along the Xaxis were considered, although rotations/deformations of thering can also be incorporated (SI Appendix, section 5). Thethreading pathway in this three-body problem involves interringinteractions coupled to ring–thread interactions. For this reason,even the most elementary translation-based inchworm reactioncoordinate requires monitoring two Cartesian variables, locationof the rings on the thread and inter-ring distance. Given thiscomplexity, we needed to simplify the problem.By analogy with dimensionality-reduction schemes applied to

biomolecular simulations (44), we test the idea that the three-bodyproblem can be reduced to one that involves computing two of thethree variables at a time. We take advantage of the fact that barriersfor threading (steps 3 and 5) depend mostly on interactions betweenring and thread: threading needs to overcome the sterics of thelarge pyridine groups on either end of the tetrazine presumably bythe reasonably frequent rotations of the cyanostar’s olefins seenpreviously by molecular dynamics (35) (see Inset to Fig. 4 and SIAppendix, section 5). Similarly, inter-ring interactions contributeminimally to barriers but instead contribute considerably to depthsof energy wells; for example, introduction of one vs. two ringson the thread stabilizes the complex and increases the well depthfrom −40 to −80 kJ·mol−1. Thus, a physical description of the rate-determining step for threading (step 3) can be reduced to a one-dimensional problem involving just two bodies at a time. Here, thereaction coordinate is cast in terms of only the ring positions usingblock diagrams (Fig. 5A). The validity of this simplified coordinatecan be tested by comparing rates determined from kinetics/Brow-nian dynamics simulations to experiment.Block diagrams were created for each step before kinetics mod-

eling (Fig. 5). One-half of the steps are described best as diffusive(step 1, ring dimerization; steps 2 and 4, ring–thread association)and others as thermally activated barrier crossings (steps 3 and 5,first and second threading). Diffusive steps involve translating thecenter of mass of the mobile rings using a one-dimensional randomwalk, while thermally activated steps are modeled as Brownianmotion. Fluctuations in the walk are determined by the diffusion

Fig. 4. (A) Reaction coordinate with free energies used in the CV simulationsshowing the inchwormmodel of switching involving threading of two cyanostarmacrocycles onto the BPTz– thread. (Inset) Threading likely occurs with onecyanostar (at Left) undergoing 90° olefin rotation (at Right) to accommodate thecross-sectional area of BPTz– (broken red oval). (B and C) Experimental CVs of theparent cyanostar recorded as a function of scan rate (B) compared with digitalsimulations of the CVs (C). The shaded regions are an inversion of the cathodicpeak added onto the primary data to help guide assessment of current ratios.

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coefficient following the dissipation-fluctuation theorem (45). De-tails of individual steps are in the SI Appendix, section 7.

Results from Kinetics Simulations. The kinetics simulations provideestimates of the mean first-passage time, τ, which is defined asthe diffusion-dominated movement of ring(s) from point I to J,and generated for all five steps. Here, τ is the inverse of the rateconstant, k: τ = k−1. Movement of the ring(s) along the X axisand onto the BPTz– thread (denoted by Cartesian variable x)between points xi and xj is modeled using the following re-lationship derived by Szabo et al. (3):

k−1 =Xj

l=i

8<:DðxiÞ e−βFðxlÞPl

k=ie−βFðxkÞ

9=;

−1

. [6]

Here, the one-dimensional space in x is discretized into points i, i +1, . . ., k, . . ., j, whereas D(xi) and F(xi) represent the diffusion and

free energy of the macrocycle at position xi (Fig. 5 A and B). Thequantity e−βFðxlÞ=

Plk=ie

−βFðxkÞ represents the probability of findingthe macrocycle at any point I along the pathway, where I ∈{i,. . . j}.On a flat free-energy profile, all of the points from I to J areequally probable, and the rate at which point I is visited is D−1 xI

2,that is, it depends entirely on the rate of diffusive motion of the ring.However, when the pathway is accompanied by a descent in freeenergy, the probability increases and the rate of this visit (τ−1) ishigher than that defined by diffusion alone, D−1 xI

2. Conversely,climbing the free-energy profile reduces the probability of visiting Iand slows down macrocycle movement relative to diffusion. Finally,the derivation of the mean first-passage time relation (Eq. 6) as-sumed Poisson-distributed statistics (3) of passage times. Conse-quently, successful transitions between endpoints occurs in 63% ofthe trajectories (1 − et/τ = 0.63) at time t = τ. The applicability of Eq.6 to threading is justified in SI Appendix, section 1.2.We assume the cyanostar’s diffusion coefficient, D, is in-

dependent of location on the thread, as has been demonstratedfor other molecular machines, but that it depends on the inter-cyanostar distance. Thus, for steps 1, 3, 4, and 5, D = 7.4 ×10−6 cm2/s represents a monomeric cyanostar (36), and for step2, D = 6.8 × 10−6 cm2/s represents a dimeric cyanostar (33, 34).A kinetics model of threading was made by combining ex-

perimentally determined free-energy profiles (Fig. 5A) with dif-fusion coefficients according to Eq. 6. The resulting kineticsmodel (Fig. 5B) for the inchworm mechanism provides the meanfirst-passage time for each step: step 1 (dimerization), 22 ns; step2 (diffusion–association), 40 ns; step 3 (first threading), 0.28 ms;step 4 (association), 44 ns; and step 5 (second threading), 4.6 μs.In addition to barrier-crossing events, step 1 also involves adiffusive search over a distance of 10 Å until the two rings di-merize. For step 1, the actual distance is estimated from theconcentration (1 mM) to be ∼700 Å. Diffusion across this dis-tance before dimerization takes an additional 6.4 μs (SI Appen-dix). Overall, step 3 dominates kinetics, and the entire processoccurs on the submillisecond timescale. This timescale is con-sistent with the kinetics for the rate-limiting step obtained byelectrochemistry, k3,expt = 7,000 ± 1,000 s−1, or τ3,expt = 0.12 ms.A similar kinetic description for the stepwise mechanism seen

with cyanodimer is accomplished employing a four-step pathway(SI Appendix, Fig. S12). Step 1 involves diffusion–association ofthe first macrocycle (80 ns). Step 2 describes the first threading(24 ms). Similar to step 1, step 3 also requires 80 ns. Finally, step4 describing threading of the second macrocycle takes 703 ms. Akinetic barrier of 57 kJ·mol−1 is attributed to step 2 and+59 kJ·mol−1 in step 4 based on digital simulations of the CVs.

Brownian Dynamics Simulations of Rate-Limiting Step of theInchworm Model. In the inchworm’s kinetic model, threading offirst and second cyanostars involve crossing energetic barriers.Direct observation of such crossing events is often computationallyintractable for barriers beyond 10 kBT (∼25 kJ·mol−1). Addressingthis issue, Brownian dynamics simulations provide complementaryestimates of mean first-passage times, τ, as a means of corrobo-rating results from kinetics simulations, as well as imparting in-sights on the macrocycle motion underlying the rate-determiningsteps. We combine Brownian dynamics with the principle of de-tailed balance along a transition pathway (46) to obtain threadingrates but employing modest computational resources (47) (SIAppendix, General Methods and Synthesis). An advantage of thisapproach is direct observation of individual cyanostar motionsmodeled as coarse-grained beads (SI Appendix, Fig. S11) repre-senting movements along a free-energy landscape (Movie S1).As expected of Brownian diffusion, these trajectories illustrateheterogeneity in threading dynamics (SI Appendix, Fig. S11) andhelp reveal the physical significance of thermal motions onthe nanoscale (Movie S2). In qualitative agreement with kinetics

Fig. 5. (A) Block diagrams describing each step in the inchworm mechanismneeded for the kinetics simulations. The block diagrams include sixth-orderpolynomial fits overlays to represent smooth free-energy profiles along the re-action coordinate cast in terms of cyanostar position relative to the BPTz thread.(B) Mean first-passage time of cyanostar moving along this reaction coordinatecomputed from Eq. 6 illustrates that step 3 (first threading event) is the slowest,determining the overall rate of cyanostar threading. (C) Snapshot overlaying 900Brownian particles (beads) determined after 5.0 ms of a Brownian dynamicssimulation of step 3. Each particle represents a cyanostar macrocycle, which startsat the top of a free energy barrier (red) and diffuses down into the valley on theright (blue). The spread in the location of the particles that are evolved over thesame period of time implies the stochasticity in motion at the nanoscale. (D)Location of a macrocycle (represented by a bead) at the end of two represen-tative 5.0-ms BD trajectories of macrocycle-threading. The bottom instance showsthe macrocycle in a state closest to threading, that is, reaching the energy min-ima, while the top instance shows themacrocycle still overcoming the barrier. Themultiple threading scenarios illustrate heterogeneity of the threading dynamicsand reveal the physical significance of thermal noise in this nanoscale process.

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simulations, we observed times for crossing barriers in steps 1–5 are, in order, 100 ns, 190 ns, 5.6 ms, 200 ns, and 5.9 μs.

ConclusionsThe classical three-body problem has been examined using a mo-lecular switch composed of two rings threading onto a rod. Exper-imental characterization provides a basis to evaluate a simplificationof the problem. The two rings are shown to thread in a rare inch-worm manner onto a rod when coupled together by noncovalentbonds. This pathway of motion was verified using control com-pounds: one that only threads by the inchworm pathway and an-other by the alternative stepwise pathway. Mechanistic models thatreproduce electrochemistry data support this conclusion. While thisis a formal three-body problem on account of coupling between allcomponents, the theoretical treatment offered by kinetics simula-tions and Brownian dynamics successfully reduced the complexity ofthe kinetics to a one-dimensional system governed by ring–threadinteractions during barrier crossing. Greater coupling is expected tointroduce deviations from this model, which may be handled bymore detailed simulations (48) and is expected to reveal novel

pathways of motion. Thus, this general approach suggests a meansto evaluate designs for three or more coupled bodies capable ofundergoing controlled mechanical motion on the molecular scale.

Supporting InformationSupporting information includes SI Appendix (general methodsand synthesis, mechanistic models, CV studies, digital simula-tions of CVs, molecular dynamics, and derivation of kineticsmodels) and Movies S1–S4 (Movie S1, full movie of the motiondepicted in Fig. 5C snapshot; Movies S2–S4, full movies showingthe motion of a single particle depicted in Fig. 5D snapshots).

ACKNOWLEDGMENTS. C.R.B., E.M.F., Y.L., E.G.S., and A.H.F. acknowledgesupport from the NSF Grant (CHE 1709909). C.M., A.A., and A.S. acknowledgesupport from the NIH Center for Macromolecular Modeling and BioinformaticsGrant (P41-GM104601) and the National Science Foundation (NSF) Center forthe Physics of Living Cells Grant (PHY-1430124). A.S. acknowledges startupaward funds from Arizona State University, NSF Grant (MCB1616590), andNIH Grant (R01-GM067887-11). This research used resources of the Oak RidgeLeadership Computing Facility, which is supported by the Office of Science, USDepartment of Energy (DE-AC05-00OR22725).

1. Benson CR, Share AI, Flood AH (2012) Bioinspired molecular machines. Bioinspirationand Biomimicry in Chemistry: Reverse-Engineering Nature (Wiley, New York), pp 71–119.

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