Ignacio Franco, George C. Schatz and Mark A. Ratner- Single-molecule pulling and the folding of...

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Single-molecule pulling and the folding of donor-acceptor oligorotaxanes:Phenomenology and interpretation

Ignacio Franco,a George C. Schatz, and Mark A. RatnerDepartment of Chemistry, Northwestern University, Evanston, Illinois 60208-3113, USA

Received 2 July 2009; accepted 17 August 2009; published online 23 September 2009

The thermodynamic driving force in the folding of a class of oligorotaxanes is elucidated by meansof molecular dynamics simulations of equilibrium isometric single-molecule force spectroscopy byatomic force microscopy experiments. The oligorotaxanes consist of cyclobisparaquat-p-phenylene rings threaded onto an oligomer of 1,5-dioxynaphthalenes linked by polyethers. Thesimulations are performed in a high dielectric medium using MM3 as the force field. The resultingforce versus extension isotherms show a mechanically unstable region in which the moleculeunfolds and, for selected extensions, blinks in the force measurements between a high-force and alow-force regime. From the force versus extension data the molecular potential of mean force isreconstructed using the weighted histogram analysis method and decomposed into energetic andentropic contributions. The simulations indicate that the folding of the oligorotaxanes isenergetically favored but entropically penalized, with the energetic contributions overcoming theentropy penalty and effectively driving the folding. In addition, an analogy between thesingle-molecule folding/unfolding events driven by the atomic force microscope AFM tip and the

thermodynamic theory of first-order phase transitions is discussed. General conditions on themolecule and the AFM cantilever for the emergence of mechanical instabilities and blinks in theforce measurements in equilibrium isometric pulling experiments are also presented. In particular,it is shown that the mechanical stability properties observed during the extension are intimatelyrelated to the fluctuations in the force measurements. 2009 American Institute of Physics.doi:10.1063/1.3223729

I. INTRODUCTION

Unfolding and unbinding events in individual macromol-ecules can be computationally studied by means of pullingmolecular dynamics MD simulations.13 These calculations

simulate recent single-molecule experiments in which a mol-ecule typically a protein or DNA attached to a surface ismechanically unfolded by pulling it with optical tweezers oran atomic force microscope AFM tip attached to acantilever.47 Such experiments show stress maxima thathave been linked to the breaking of hydrogen bonds, provid-ing insight into the secondary and tertiary structures of themacromolecules, and the dynamical processes involved inmuscle contraction, protein folding, transcription, shapememory materials, and many others.

In this paper we present simulations of single-moleculeAFM pulling for a class of donor-acceptor oligorotaxanes, aswell as a detailed interpretation of the observed phenomenol-ogy. The pseudorotaxanes8,9 considered consist of a variablenumber of cyclobisparaquat-p-phenylene tetracationic cy-clophanes CBPQT4+ threaded onto a linear chain com-posed of three naphthalene units linked by polyethers withpolyether caps at each end see Fig. 1. These moleculeshave been recently synthesized by Basu et al.10 and are oli-gomeric analogs of polyrotaxanes previously developed byStoddart and co-workers.11 The relevant interactions that

drive the folding of this class of molecules have, in the ab-sence of applied stress, been recently characterized.12 Theinterest in this paper is to investigate folding/unfolding path-ways for these molecules under stress, and to determine,from a thermodynamic perspective, what drives the folding.

In this sense, the AFM pulling setup is a valuable tool sinceit allows inducing unfolding/refolding events at the single-molecule level while simultaneously performing thermody-namic measurements.

The AFM pulling of the rotaxanes is simulated usingconstant temperature MD in a high dielectric medium usingMM3 as the force field.1315 We focus on the isometric ver-sion of these experiments in which the distance between thesurface and the cantilever is controlled while the force ex-erted is allowed to vary. Closely related experimental effortsare currently underway by Stoddart and co-workers. Tobridge the several orders of magnitude gap between the pull-

ing speeds that are experimentally employed1103 m /s and those that are computationally acces-sible, in the simulations the unfolding events are drivenslowly enough that quasistatic behavior is recovered and theresults become independent of the pulling speed.

A central quantity of interest in this analysis is the mo-lecular potential of mean force16 PMF along the extensioncoordinate. The PMF is the Helmholtz free energy profilealong a given reaction coordinate. As such, it succinctly cap-tures the thermodynamic changes experienced by the mol-ecule during the folding. Here, the PMF is reconstructedaElectronic mail: [email protected].

THE JOURNAL OF CHEMICAL PHYSICS 131, 124902 2009

0021-9606/2009/13112 /124902/13/$25.00 2009 American Institute of Physics131, 124902-1

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from the equilibrium force measurements using the weighted

histogram analysis method WHAM.

1719

Related ap-proaches based on nonequilibrium force measurements thatexploit the Jarzynski2022 and related equalities have beenpresented previously.2327 Further insight into the thermody-namic driving force behind the folding is obtained by decom-posing the PMF into energetic and entropic contributions.

In addition, an analogy between the phenomenology ob-served during the pulling of the rotaxanes and that expectedfor a system undergoing a first-order phase transition is pre-sented. In the process of exploring such an analogy, severalgeneral features of the isometric pulling experiments will berevealed. Specifically, the dependence of the force versusextension curves on the cantilever spring constant is charac-

terized, and general conditions on the molecule and on thecantilever necessary for the emergence of mechanical insta-bility and blinks in the force measurements are isolated. Inparticular, it is shown that the mechanical stability propertiesobserved during the pulling are intimately related to the fluc-tuations in the force measurements. These results comple-ment previous studies done by Kreuzer and Payne28 in whichthe soft-spring and stiff-spring limits of the AFM pullingwere identified with the Gibbs and Helmholtz ensembles forthe isolated molecule, those of Kirmizialtin et al.29 in whichthe origin of the dynamical bistability observed in constantforce experiments5 and its relationship with the topographyof the PMF was clarified, and those of Friddle et al.30 in

which the dependence of the rupture force on the cantileverspring constant was analyzed for a simple model potentialmeant to represent bond breaking see Ref. 31 for an analysisof the bond survival probability. The conditions isolated be-low apply to any system subject to equilibrium isometricpulling using AFM tips attached to harmonic cantilevers.

This manuscript is organized as follows. In Sec. II A theprotocol employed to simulate the isometric pulling of therotaxanes is described. Sections II B and II C summarize,respectively, the strategy used to reconstruct the PMF fromthe force measurements and the procedure employed to de-compose it into energetic and entropic contributions. Ourmain results are presented in Sec. III. Specifically, in Sec.III A the basic features observed during the mechanical un-folding of the oligorotaxanes are presented and the resultingPMF discussed. In Sec. III B the origin of the mechanicalinstabilities and blinks in the force measurements observedduring the extension is clarified, and minimum conditions forthe emergence of these effects are presented. Our main con-clusions are summarized in Sec. IV.

II. THEORETICAL METHODOLOGY

A. Pulling simulations

Unfolding of the rotaxanes is computationally studied bymeans of pulling MD simulations. The simulations are analo-

gous to single-molecule force spectroscopy experiments in

which a molecule attached to a surface is mechanically un-folded by pulling it with an AFM tip attached to a cantilever.The general setup of this class of experiments is shown inFig. 2. The stretching computation begins by attaching oneend of the molecule to a stiff isotropic harmonic potentialthat mimics the molecular attachment to the surface. Simul-taneously, the opposite molecular end is connected to adummy atom via a virtual harmonic spring. The position ofthe dummy atom is the simulation analog of the cantileverposition and is controlled throughout. In turn, the varyingdeflection of the virtual harmonic spring measures the forceexerted during the pulling. The stretching is caused by mov-ing the dummy atom away from the molecule at a constantspeed. The pulling direction is defined by the vector connect-ing the two terminal atoms of the complex. Since cantileverpotentials are typically stiff in the direction perpendicular tothe pulling, in the simulations the terminal atom that is beingpulled is forced to move along the pulling direction by intro-ducing appropriate additional harmonic restraining poten-tials. This type of simulations does not take into account anyeffects that may arise due to the interaction between the mol-ecule and the AFM tip or the surface that cannot be ac-counted for by simple position restraints, nor any variationsin the cantilever position due to thermal fluctuations orsolvent-induced viscous drags.

The potential energy function of the molecule plus re-straints is of the form

ULr = U0r + VLr,t, 1

where U0r is the molecular potential energy plus potentialrestraints not varied during the simulation, r denotes the po-sition of the N atoms in the macromolecule, and

VLr,t =k

2r Lt2 2

is the potential due to the cantilever. Here r is the molecu-lar end-to-end distance function, L =L0 +vt is the distancefrom the cantilever to the surface at time t, v is the pulling

FIG. 1. Structure of the 3rotaxane. Oxygen atoms are depicted in red, thenaphthalene units in black, and the polyether carbons in orange. TheCBPQT4+ rings are depicted in blue with the pyridinium N+ ions in brown.

FIG. 2. Schematic of an isometric single-molecule force spectroscopy ex-periment using an AFM. In it, one end of the molecular system is attached toa surface and the other end to an AFM tip attached to a cantilever. Duringthe pulling, the distance between the surface and the cantilever Lt =L0+vt is controlled and varied at a constant speed v. The deflection of thecantilever from its equilibrium position measures the instantaneous appliedforce on the molecule by the cantilever Ft = kt Lt, where k is thecantilever spring constant and t is the fluctuating molecular end-to-endextension.

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speed, and k is the cantilever spring constant. The force ex-erted by the cantilever on the molecule at time t is given by

Ft = VL =VL

L= kr Lt , 3

where the gradient is with respect to the r L coordi-nate. In writing Eqs. 2 and 3 the vector nature of vt,Lt, r, and Ft has been obviated since these quantities

are collinear in the current setup.The simulations are performed using TINKER 4.2 Ref.

32 for which a pulling routine was developed. The stretch-ing is performed using a soft cantilever potential with springconstant k=0.011 N /m=1.1 pN /. Since thermal fluctua-tions in the force depend on k as Fk/ Ref. 1 where=1 /kBT is the inverse temperature, such a soft-spring con-stant provides high resolution in the force measurements.The macromolecules are described using the MM3 forcefield, which we have found adequately reproduces the com-plexation energies of model rotaxane systems.12 As a simplemodel for the solvent, we use a continuum high dielectricmedium in which the overall dielectric constant of the forcefield is set to that of water at room temperature 78.3. This isto be compared to the MM3 dielectric constant for vacuumof 1.5.33 Any solvent effects that cannot be described usingthis continuum description are absent. The dynamics ispropagated using a modified Beeman algorithm with a 1 fsintegration time step, and the system is coupled to a heatbath at 300 K using a NosHoover chain as the thermostat.Initial minimum energy structures for the pulling simulationswere obtained using simulated annealing, as described inRef. 12. These initial structures were allowed to equilibratethermally for 1 ns and subsequently stretched. The simula-tions do not include counterions for the tetracationic cyclo-

phanes since, at 300 K and for the high dielectric mediumemployed, their interaction with the main molecular back-bone is small and their effect on the folding is negligible.

B. Reconstructing the PMF using WHAM

Under reversible conditions, knowledge of the force ex-erted during the pulling immediately yields the associatedchange in the Helmholtz free energy A for the molecule pluscantilever. This is because the change in A is determined bythe reversible work exerted during the pulling

A =L0L

FLdL = 1

lnZL

ZL0 , 4

where FL is the average force at extension L and

ZL = dr exp ULr 5is the configurational partition function of the molecule pluscantilever. Equation 4 follows from standard thermody-namic integration considerations19 and assumes that at eachpoint during the extension the state of the molecule pluscantilever potential is well described by a canonical en-semble. This property is expected for a system in weak con-

tact with a thermal bath and is enforced in the simulations bythe nature of the thermostat employed.

From the force versus extension measurements it is alsopossible to extract the molecular PMF as a function ofthe end-to-end distance by properly removing the bias dueto the cantilever potential. The quantity is theHelmholtz free energy profile along the coordinate for theisolated cantilever-free molecule and is of central interest

since it succinctly characterizes the thermodynamics of theunfolding. Below we summarize how to estimate this quan-tity from the force measurements employing the WHAM.

The PMF is defined by34

= 1

ln p0

p0 , 6

where and are arbitrary constants and

p0 =dr rexp U0r

dr exp U0r=

Z0

Z0= r .

7

Here,

Z0 = dr exp U0r 8is the configurational partition function of the molecule and

Z0 = dr rexp U0r . 9In the context of the pulling experiments p0 is the prob-ability density that the molecular end-to-end distance func-tion r adopts the value in the unbiased cantilever-free

ensemble. The quantity p0 determines the PMF up to aconstant and can be estimated from the force measurements,as we now describe.

At this point it is convenient to discretize the timevariable during the pulling experiment into M steps:t1 , . . . , ti , . . . , tM. The potential of the system plus cantileverat time ti is Ui =U0 + Vi, where Vi = kr , ti Lti

2/2 is the

bias due to the cantilever potential. In the ith biased measure-ment knowledge of the force Fi and Lti gives the molecularend-to-end distance. The probability density of observing thevalue at this time is given by

pi

=

dr rexp Uir

dr exp Uir=

Zi

Zi=

r

i,

10

where Zi =dr expUir is the configurational partitionfunction for the system plus cantilever at the ith extensionand

Zi = dr rexp Uir . 11Knowledge of pi allows the unbiased probability densityp0 to be reconstructed since, by virtue of Eqs. 7 and10, these two quantities are related by

124902-3 Single-molecule pulling and the folding of oligorotaxanes J. Chem. Phys. 131, 124902 2009



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p0 = exp+ ViZi

Z0pi , 12

where we have exploited the fact thatZ0=exp+ViZi. In the pulling experiments, foreach Lti only a few force measurements are typically per-formed and the probability density pi is not properlysampled. Nevertheless, the experiments do provide measure-

ments at a wealth of values ofLti that can be combined toestimate p0,

p0 = i=1

M

wi exp+ ViZi

Z0pi, 13

where the wi are some normalized iwi =1 set of weights.In the limit of perfect sampling any set of weights shouldyield the same p0. In practice, for finite sampling it isconvenient to employ a set that minimizes the variance in thep0 estimate from the series of independent estimates ofbiased distributions. Such a set of weights is precisely pro-vided by the WHAM prescription1719,35

wi =

exp ViZ0

Zigi

i=1M exp Vi

Z0

Zigi

, 14

where gi =1+2i is the statistical inefficiency35 and i is the

integrated autocorrelation time.17,35 The difficulty in estimat-ing the i for each measurement generally leads to furthersupposing that the gis are approximately constant and factorout of Eq. 14, so that

p0 =i=1

Mpi

i=1M exp ViZ0/Zi

. 15

Neglecting the gis from Eq. 15 does not imply that theresulting estimate of p0 is incorrect, but simply that theweights selected do not precisely minimize the variance inthe estimate. Experience with this method indicates that ifthe gis do not differ by more than an order of magnitudetheir effect on is small.18

The computation of the PMF then proceeds as follows.Suppose that Ni force measurements are done for each ex-tension i. Knowledge of the force Fi

j j = 1 , . . . ,Ni and ofLti gives the molecular end-to-end distance i

j in each ofthese measurements. The numerator in Eq. 15 is then esti-mated by constructing a histogram with all the available

data,

i=1

M

pi i=1

M

j=1

Ni Cij

Ni, 16

where is the bin size and Cij=1 if

ij/2,+/2 and zero otherwise. Estimating the

denominator in Eq. 15 requires knowledge of the Zis, theconfigurational partition functions of the system plus canti-lever at all extensions Lti considered. There are two waysto obtain these quantities. The most direct one is to employthe Helmholtz free energies for the system plus cantileverobtained through the thermodynamic integration in Eq. 4,

as they determine the Zis up to a constant multiplicativefactor. Alternatively, it is also possible to determine the ratioof the configurational partition functions between the ith bi-ased system and its unbiased counterpart using p0,

Zi

Z0= dexp Vip0 . 17

Equations 15 and 17 can be solved iteratively. Starting

from a guess for the Zi /Z0, p0 is estimated using Eq. 15and normalized. The resulting p0 is then used to obtain anew set of Zi /Z0 through Eq. 17, and the process is re-peated until self-consistency. This latter approach is the usualprocedure to solve the WHAM equations. Note, however,that this procedure is not required if reversible force versusextension data is available.

The reconstruction of the PMF from the force versusextension measurements using WHAM is simple to imple-ment. Further, the procedure is versatile in the sense that itmakes no assumption about the stiffness of the cantileverpotential and permits the combination of data from severalpulling runs. In addition, as we describe below, it can also be

employed to decompose the PMF into entropic and energeticcontributions from data readily available from a MD run.

C. Energy-entropy decomposition of the PMF

Insight into the thermodynamic driving force responsiblefor the folding of oligorotaxanes can be obtained by decom-posing the changes in the PMF into entropic S and poten-tial energy U0 contributions,

= U0 U0 TS S . 18

Here,

U0 = 1Z0 dr rU0rexp U0r 19

is the average potential energy when the end-to-end distanceadopts the value and

TS = 1

0 dr r ln 20

is the corresponding configurational entropy, where=0 expU0r /Z0 and 0 is a constant irrelevantfactor with the same dimensions as Z0. Equation 18 fol-lows from the above definitions.

The energy-entropy decomposition of the PMF can be

obtained by estimating U0 directly and employing the pre-viously reconstructed to obtain TS using Eq. 18.Here, U0 is estimated by combining all measurements per-formed during the pulling simulations using WHAM. Themeasurable quantity in the simulation in the presence of thebias due to the cantilever is

U0r ri =dr rU0rexp Uir

Zi,

21

the average molecular potential energy when the end-to-enddistance adopts the value and the cantilever extension is




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Lti. This average in the biased ensemble is related to theunbiased average U0 by

U0 = exp+ ViZi

Z0U0r ri . 22

As in the estimation of p0, all data collected during pull-ing can be combined with appropriate weights wi to estimateU0,

U0 = i=1

M

wi exp+ ViZi

Z0U0r ri . 23

If one adopts the weights wi that define the WHAM prescrip-tion Eq. 14, then

U0 =iU0r ri

iexp ViZ0/Zi, 24

where we have assumed identical statistical inefficiencies forall i. Last, recalling the definition of p0 =Z0 /Z0 Eq.7 and its WHAM prescription Eq. 15, one arrives at auseful expression for U0,

U0 =iU0r ri

i ri. 25

Equation 25 provides a mean to calculate the potential en-ergy as a function of the end-to-end distance by combiningall the data obtained during the pulling. The remarkable sim-plicity of Eq. 25 is noteworthy: In order to obtain U0one just needs to generate histograms for the values of observed using all data collected during the pulling and then,within each bin, perform a simple average of the internalenergy of the configurations that fall into it. It does not re-quire knowledge of the properties of the bias potential.

This method of performing an energy-entropy decompo-sition of the PMF requires knowledge of U0 for the con-figurations observed during pulling. This quantity, althoughreadily available in a MD run, is not experimentally acces-sible. As a result, in experiments an energy-entropy decom-position of the PMF requires estimating the changes in en-tropy by performing the pulling at varying temperatures.

III. RESULTS AND DISCUSSION

A. Phenomenology of the pulling experiments

1. The approach to equilibrium

One of the challenges in simulating AFM pulling experi-

ments using MD is to bridge the large disparity between thepulling speeds v that are employed experimentally and thosethat can be accessed computationally. While typical experi-ments often use v 1103 m /s, current computationalcapabilities require pulling speeds that are several orders ofmagnitude faster. Here, this difficulty is circumvented bystriving for pulling speeds that are slow enough that revers-ible behavior is recovered. Under such conditions the resultsbecome independent ofv, and the simulations comparable toexperimental findings.

Consider the pulling of the 3rotaxane shown in Fig. 1the simulation details are specified in Sec. II A. Figure 3shows the effect of decreasing the pulling speed on the force

versus extension characteristics. In the simulations, the sys-tem is first extended black lines to a given L and thencontracted gray lines. For pulling speeds and equal retract-ing speeds between 1 and 0.01 /ps panels AC hysteresisand other nonequilibrium effects during the pulling are un-avoidable. However, for v=0.001 /ps=105 m /s the sys-tem behaves reversibly, and the F-L curves obtained duringextension and contraction essentially coincide. Note the sec-ond law of thermodynamics at play in the simulations: Thework required to stretch the moleculethe area under thecurvedecreases with the pulling speed until reversible be-havior is attained. We have observed that pulling speeds of103 /ps are generally sufficient to recover reversiblebehavior for the unfolding in a high dielectric medium of then rotaxanes considered. For simulations in vacuum slowerpulling speeds are required.

It is worth noting that this seemingly simple phenomeno-logical behavior is not recovered when a Berendsen thermo-stat is employed instead of a NosHoover chain. In fact, wehave observed that the Berendsen thermostat leads to a spu-rious violation of the second law during the pulling, presum-ably because the ensemble generated by it does not satisfy

the equipartition theorem.36

2. Reversible unfolding, mechanical instability,and blinks in the force measurements

Figure 4 details the force exerted upper panel and themolecular end-to-end distance lower panel during the re-versible pulling and contraction of the 3rotaxane. Snap-shots of typical structures encountered during the pulling areincluded in Fig. 5. As the system is stretched the oligorotax-ane undergoes a conformational transition from a foldedglobular state to an extended coil. In the process, the forceinitially increases approximately linearly with L, then drops,

0 50 100 150 200

0

50

100

150

L(A)

F

(pN)

0 50 100 150 200

0

50

100

150

L(A)

F

(pN)

0 25 50 75 100 125 150

0

20

40

60

80

L(A)

F

(pN)

0 25 50 75 100 125 150

0

20

40

60

80

L(A)

F

(pN)

B

D

A

C

FIG. 3. The approach to equilibrium in the pulling simulations. The figureshows force vs extension profiles for the 3rotaxane immersed in a highdielectric medium at 300 K for different pulling speeds v of a 100, b101, c 102, and d 103 /ps. The harmonic cantilever employed has asoft-spring constant ofk=0.011 N /m, L is the distance between the surfaceand the cantilever, and F is the instantaneous applied force. The black linescorrespond to the extension process; the gray ones correspond to thecontraction.




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and subsequently increases again. The drop in the force isdue to the unfolding of the rotaxane during the stretching.

Figure 6 shows the change in the Helmholtz free energyA for the molecule plus cantilever obtained from integratingdirectly the F-L curve in Fig. 4. The net change in free en-ergy for the complete thermodynamic cycle is zero, as ex-pected for a quasistatic process. The free energy profile forthe system plus cantilever has three distinct regions, labeledIIII in the figure. Regions I and III correspond to the foldedand extended states, respectively, while region II is where thefolding/unfolding event occurs. From a thermodynamicsperspective,37 regions I and III are mechanically stablephases since A is a convex function of the extension L, i.e.,

2

A/

L2

T=

FL/

LT

0. By contrast, region II, wherethe unfolding occurs, is mechanically unstable since A is aconcave function of L and hence 2A /L2T= FL /LT0.

The dynamical behavior of the system in the unstableregion is illustrated in Fig. 7, which shows the evolution ofthe radius of gyration Rg and the molecular end-to-end dis-tance when L =70.0 . In this region, the 3rotaxane under-goes transitions between a folded globular state, partiallyfolded structures, and an extended coil. The right panels inFig. 7 show the probability density of the distribution of Rg

2

and values obtained from a 20 ns trajectory. The systemexhibits a clear dynamical bistability along the coordinate

and at least a tristability along Rg2. Figure 8 shows some

representative structures encountered during this dynamics.The bistability along the end-to-end distance leads to a blink-

ing in the force measurements from a high force to a lowforce regime during the pulling cf. the unstable region inFig. 4. The blinking in the force measurements in the iso-metric experiments is analogous to the blinking in the mo-lecular extension observed during the constant force revers-ible pulling of single RNA molecules.5 Note that for the

50 100 150 100 50

0

20

40

60

80

L (A)

F

(pN)

0 50 100 150 200 2500

25

50

75

t (ns)

(A)

1 3 45 6 72

FIG. 4. Time dependence of the force F and the molecular end-to-enddistance during the pulling of the 3rotaxane under equilibrium conditionsv=0.001 /ps, k=0.011 N /m. Typical structures labels 17 observedduring the extension are shown in Fig. 5.

FIG. 5. Snapshots of the 3rotaxane during its extension. The numericallabels shown here are employed in Figs. 4, 6, and 9.

20 40 60 80 100 120 140

0

10

20

30

40

50

60

70

L (A)

A(kcal/mol)

I II III

7

65

4

3

21

FIG. 6. Changes in the Helmholtz free energy A of the 3rotaxane pluscantilever obtained from the data shown in Fig. 4. The blue line correspondsto the pulling and the red one corresponds to the contraction. The dottedlines provide an estimate of errors in the thermodynamic integration due toforce fluctuations at each pulling step. The difference in the degree of con-vexity of regions I and III is evidenced by extrapolating the data in eachregion outside of its domain through fitting to quartic polynomials blackdashed lines. The labels correspond to the structures shown in Fig. 5.

0 2 2 6 8 10 12 14 1 6 18 20

100

200

300

400

time (ns)

Rg2

(A2)

0 2 4 6 8 10 12 1 4 16 18 2 0

40

50

60

70

time (ns)

(A)

0 0.01 0.02

100

200

300

400

p(Rg2

) (A2

)

0 0.05 0.1

40

50

60

70

p() (A1)

1a

2a

3a

2a

3a

1a

FIG. 7. Time dependence and probability density distribution of the radiusof gyration Rg

2 and the end-to-end molecular extension when the 3ro-taxane plus cantilever is constrained to reside in the unstable region II ofFig. 6 with L =70.0 . Typical structures encountered in this regime labels1a3a are shown in Fig. 8.




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cantilever stiffness employed this multistable behavior in theradius of gyration, or the end-to-end distance, is not observedwhen L is set to be in the stable regions I or III of the freeenergy profile. The origin of the bistability along the end-to-end distance, the multistability in the radius of gyration, andits relation to the thermodynamic instability in the pulling arediscussed in Sec. III B.

3. The PMF and the thermodynamic driving forcein the folding

Figure 9 shows the PMF along the end-to-end distance for the 3rotaxane reconstructed from the force measure-ments as described in Sec. II B, as well as its decompositioninto entropic and potential energy contributions. The thermo-dynamic native state of the oligorotaxane, i.e., the minimumin , corresponds to a globular folded structure with 0=7 . The structure of the molecular free energy profile as afunction of is similar to that observed for the molecule pluscantilever as a function of L. Namely, it consists of a foldedand an unfolded stable branch, where is a convex func-

tion, and an unstable concave region around =4452 where the conformational transition occurs. As discussed inSec. III B, this characteristic structure of the PMF is respon-

sible for many of the interesting features observed duringpulling. Figure 10 shows the probability distribution for Rg

2 atvalues of in the three main regions of the PMF, each col-lected during a 20 ns trajectory. In the stable regions =7.0 and 64.9 , the evolution of the radius of gyrationreveals only one stable structural state. By contrast, in theunstable region =47 the distribution ofRg

2 values indi-cates that there is an inherent bistability in the molecularpotential along the Rg

2 coordinate. Such molecular bistabilitygives rise to the region of concavity in the PMF.

What drives the folding of the oligorotaxanes from athermodynamic perspective? The energy-entropy decomposi-tion of the PMF indicates that the folding of the rotaxane isenergetically favored but entropically penalized, with the en-ergetic contributions overcoming the entropy penalty and ef-fectively driving the folding. Further, while the potential en-ergy and the entropy of the molecule show considerablechanges during the unfolding, these two effects largely can-cel one another, leading to only modest changes in the Helm-holtz free energy. Note that while is a monotonic func-tion of for 0, the energy and entropy contributionsshow a clear nonmonotonic dependence on . For instance,there is a reduction in the systems entropy for both justbefore the conformational transition and for large extensionsdue to a reduction in the available degenerate conformational

space.It is important to stress that the PMF and its decompo-sition into energy and entropy contributions correspond to amolecule in which the terminal atoms are constrained tomove along the pulling coordinate. As a consequence, anyadditional contributions that may arise by relaxing these con-straints are simply not manifest in this picture of the folding.

4. Dependence of the PMF on the number of threadedrings

Figure 11 compares the PMF extracted from force mea-surements for the 3rotaxane and the associated 2rotaxaneand bare molecular thread. In the 2rotaxane system the

FIG. 8. Structures observed in the unstable region of the pulling simula-tions. The numerical labels are employed in Fig. 7.

0 10 20 30 40 50 60 70 8010

0

10

20

30

40

50

(A)

en

ergy(kcal/mol)

1

2 3

5

76

4

TS()

Uo

()

()

FIG. 9. Potential U0 and entropic TS contributions to the PMF Eq.18 of the 3rotaxane along the end-to-end distance . The results areaverages of three different pulling simulations. The error bars correspond totwice the standard deviation obtained from a bootstrapping analysis. Thesolid lines result from a spline interpolation of the available data points.Typical structures labels 17 are shown in Fig. 5.

0 100 200 300 400 5000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Rg

2(A

2)

p(R

g2)(A2)

FIG. 10. Probability density distribution of the radius of gyration pRg2 for

the 3rotaxane when is fixed at 7.0 dots, 47 open circles, and 64.9 crosses. Note the bistability along Rg

2 when is fixed at the concaveregion of the PMF.




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CBPQT4+ ring encircles the central naphthalene unit of the

underlying chain. The PMF in all cases is qualitatively simi-lar: It consists of two stable convex regions and an unstableconcave region where the unfolding occurs. Further, for allspecies the folding along is energetically driven and en-tropically penalized not shown. Note that in the high di-electric medium employed, the number of threaded ringsdoes not have a dramatic effect on the net change in the freeenergy undergone during folding nor, as expected, on theelastic properties of the extended state. It does, however,have an appreciable effect on the stability and elasticity ofthe folded conformation. Specifically, as the number of ringsis increased the range of values for which the folded con-formation is stable increases. That is, a larger L is required

during the pulling to unfold the rotaxanes with respect to thebare chain.

B. Interpretation of the pulling experiments

1. Analogy with first-order phase transitions

At first glance, the phenomenological behavior observedby the molecule plus cantilever during the extension is analo-gous to that expected for a system undergoing a first-orderphase transition.37 Namely, during the extension the com-bined system goes from one stable thermodynamic phase toanother regions I and III in Fig. 6 by varying an externallycontrollable parameter, in this case L. In the transition region

the Helmholtz free energy of the combined system is con-cave and, hence, fails to satisfy the thermodynamic stabilitycriteria region II in Fig. 6. As a consequence, the observedF-L isotherm Fig. 4 exhibits a region of mechanical insta-bility where FL /LT0. These observations largely par-allel the behavior of unstable P-V isotherms for a van derWaals fluid.

In addition, the phenomenology seems to indicate thatclose to the transition region the Helmholtz potential for themolecule plus cantilever is bistable along the end-to-end dis-tance and has the characteristic form exhibited during a first-order phase transition schematically shown in Fig. 12. ForL region I the stable global minimum corresponds to the

molecule in a folded state, and as L is increased the equilib-rium state shifts from one local minimum to the other. ForL region II the two minima become approximately equaland a transition from the folded state to the unfolded stateoccurs. Beyond this extension the unfolded molecular phasebecomes the global potential minimum and the extended

phase becomes absolutely stable. This interpretation is con-sistent with the observed behavior in which the system blinksbetween a folded and an extended state around the transitionregion, recall Fig. 7.

Note, however, that this analogy can never be com-pletely faithful since the modeled single-molecule pullingexperiment does not deal with a macroscopic system. Thiscontrasts with the elastic behavior observed for polymerchains38,39 and leads to salient differences between the twoprocesses. First, while in a true first-order phase transitionthe physical system exhibits coexistence between the stablephases, in this single-molecule version no coexistence is pos-sible. At the transition point, and for the value of k em-ployed, the system instead exhibits frequent jumps betweenthe stable phases cf. Fig. 7. Coexistence is replaced byergodicity in this single-molecule manifestation of bistabil-ity. Second, the variable that is varied during the extension Lis not a true thermodynamic variable since it is neither ex-tensive nor intensive. Third, the system considered has noclear thermodynamic limit in which a true discontinuity ofthe derivatives of the partition function can develop and,further, the system is far from the thermodynamic limit sincethe fluctuations observed in the F-L curves are comparable tothe average value, leading to a nonequivalence between sta-tistical ensembles.40,41 Additionally, the mechanical proper-

ties measured during the pulling are for the molecule pluscantilever, and hence the extension behavior is expected todepend on the nature of the cantilever.

Nevertheless, the similarities between the two processesare still intriguing and, in view of these observations, it isnatural to ask: i What is the origin of the mechanical insta-bility during the pulling?; ii how does the dynamical bista-bility along the end-to-end distance arise?, iii how does thenature of the cantilever affect the observed phenomenologi-cal behavior?; and iv which features of the observed behav-ior arise due to the molecule, and which are due to the can-tilever? These questions are addressed in Secs.III B 2III B 6 below.

0 10 20 30 40 50 60 70 805

0

5

10

15

20

25

30

(A)

()(kcal/mol)

Chain

[2]Rotaxane

[3]Rotaxane

FIG. 11. PMF as a function of the molecular extension extracted fromforce measurements using WHAM for rotaxanes with different numbers ofthreaded rings. The error bars correspond to twice the standard deviationobtained from a bootstrapping analysis.

FIG. 12. Schematic variation in the Helmholtz potential of the moleculeplus cantilever as a function of for different extensions L suggested by thephenomenological observations around the region of mechanical instability.The labels IIII correspond to the different stability regions in Fig. 6.




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The key quantity in the remainder of this analysis is thePMF along the end-to-end distance Eq. 6. Its utility relieson the fact that the configurational partition function of themolecule plus cantilever at extension L Eq. 5 can be ex-

pressed as

ZL = dexp + VL , 26where we have neglected the constant and L-independentmultiplicative factor that arises in the transformation since itis irrelevant for the present purposes. The above relation im-plies that the extension process can be viewed as thermalmotion along a one-dimensional effective potential deter-mined by the PMF and the bias due to the cantileverVL,

UL = + VL. 27

Further, by means of the PMF it is possible to estimate theforce versus extension characteristics for the composite sys-tem for any value of the force constant k. This is because theaverage force exerted on the system at extension L can beexpressed as

FL = 1

lnZL/Z0

L

= dVL

Lexp + VL

dexp + VL, 28

where, for convenience, we have introduced Z0 in the loga-rithm just to make the argument dimensionless. Since isa property of the isolated molecule it is independent of k.Hence, Eq. 28 can be employed to estimate the F-L iso-therms for arbitrary k.

2. Dependence of the F-L isotherms on the cantileverspring constant

Figure 13 shows representative F-L curves for the 3ro-taxane estimated using Eq. 28 for different cantileverspring constants, as well as the ratio between the thermalfluctuations and the average in the force measurements. Note

that in all cases the system is far from the thermodynamiclimit since the force fluctuations are comparable in magni-tude to the average. Further, the fluctuations increase withthe spring constant and, for example, in the case of k=5k0where k0 =1.1 pN / the region of instability in the F-Lcurve would be mostly masked by the fluctuations.

Note that the instability properties of the F-L isothermsdepend intricately on the cantilever spring constant em-ployed. This dependence is succinctly conveyed in Fig. 14which shows the values of the extension and the averageforce at the critical points in the F-L isotherms. In the figure,F+ and L+ or F and L denote the values of the force andthe cantilever extension when the F-L curve exhibits a maxi-mum or minimum. These values enclose the unstable re-gion in the F-L isotherm. In turn, is the average end-to-end distance at the critical points, which is approximatelyindependent ofk except for very small k. The critical forcesF+ and F show a smooth and strong dependence on thecantilever spring constant. For large k the system persistentlyshows a region of instability in the isotherms. However, asthe cantilever spring is made softer, F+ and F approach eachother, and for small k the mechanical instability in the F-Lisotherms is no longer present.

20 40 60 80 100 1200

10

20

30

40

50

60

70

80

L (A)

FL

(pN)

20 40 60 80 100 1200

0.5

1

1.5

2

L (A)

(F

2L

FL2

)1/2/FL

k0/2

k0

2k0

5k0

10k0

FIG. 13. Left panel: dependence of the F-L isotherms on the cantilever spring constant k during the extension of the 3rotaxane. Right panel: ratio betweenthe thermal fluctuations and the average in the force measurements. The cantilever spring constants employed are all expressed in terms of the cantileverspring constant used in the pulling simulations k0 =1.1 pN/.

0 5 10 15 20 2540

50

60

70

80

90

100

k (pN/A)

criticalextens

ion(A)

L+

L

+

0 5 10 15 20 2515

20

25

30

35

k (pN/A)

criticalforce

(pN)

F+

F

FIG. 14. Dependence of the instability of the F-L isotherms on the cantile-ver spring constant k for the 3rotaxane. The right panel shows the criticalvalues of the force as a function of k. Here F+ and F correspond to thevalues of the force when the F-L curves exhibit a maximum and minimum,respectively. The left panel shows the extension lengths L that enclose theunstable region in the F-L isotherms, as well as the average end-to-enddistance at the critical points .




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3. Disappearance of the mechanical instability in thesoft-spring limit

The disappearance of the instability regions in the F-Lmeasurements when the system is pulled with a soft cantile-ver is a general feature of the extension of any molecularsystem provided that no covalent bond breaking occurs dur-ing the process. To see this, consider the soft-spring approxi-mation of the configurational partition function of the system

plus cantilever Eq. 26,ZL

Z0

exp kL2/2

Z0 dexp expkL

= exp kL2/2expkL , 29

where the notation f stands for the unbiased cantilever-free average of f. In writing Eq. 29 we have supposed thatin the region of relevant in which the integrand is non-negligible k2 /VL1 and, hence, that the cantilever po-tential is well approximated by VL kL2 /2 kL. This ap-proximation is valid provided that no bond breaking isinduced during pulling and permits the introduction of a cu-

mulant expansion42 in the configurational partition function.Specifically, the average expkL can be expressed as

expkL = n=0

kLn

n!n = exp

n=1

kLn

n!n ,

30

where n is the nth order cumulant. In view of Eqs.2830 the slope of the F-L curves can be expressed as

FLL

= k k2n=0

kLn

n!n+2. 31

It then follows that to lowest order in k,FL

L k 0. 32

That is, no unstable region in the F-L curve can arise in thesoft-spring limit, irrespective of the specific form of .Instabilities, however, can arise in higher orders in the ex-pansion. For instance, to second order in k instabilities ariseprovided that the cantilever spring constant satisfies

k1

2, 33

where 2= 2

2

.

4. Emergence of the mechanical instability in thestiff-spring limit

A mechanically unstable region in the F-L curves canonly develop if the PMF of the unbiased molecule by itselfhas a region of concavity. To understand how this resultarises consider the configurational partition function of themolecule plus cantilever at extension L Eq. 26 for large k.In this regime, most contributions to the integral will comefrom the region where =L. Consequently, exp canbe expanded around this point to give

exp = exp L1 A1L L

2A2L L

2 + , 34where

A1L =

L

L , A2L =

2L

L2 L

L 2

. 35

Here, the notation is such that L /L = /=L. In-troducing Eq. 34 into Eq. 26, integrating explicitly thedifferent terms, and performing an expansion around1 / k=0 one obtains

ZL = 2k

exp L1 12k

A2L +O1/k2 .

36

Using Eq. 28, the derivative of the force can be obtainedfrom this approximation to the partition function. To lowestorder in 1 /k it is given by

FLL

=

2L

L2+

1

2k

2A2L

L2+O1/k2 , 37

from which it follows that to zeroth order in 1 / k,

FLL

=

2L

L2+O1/k. 38

That is, an unstable region in the force versus extension re-quires a region of concavity in the PMF.

5. Relation between force fluctuationsand mechanical stability

The mechanical stability properties observed during theAFM pulling experiments are intimately related to the fluc-tuations in the force measurements. To see this, note that theaverage slope of the F-L curves can be expressed in terms ofthe fluctuations in the force

FLL

= 2VLL2

L

F2L FL2

= k F2L FL

2 , 39

where we have employed Eq. 28. The sign of FL /Ldetermines the mechanical stability during the extension andhence Eq. 39 relates the thermal fluctuations in the forcemeasurements with the stability properties of the F-L curves.Specifically, for the stable branches for which FL /L0the force fluctuations satisfy

F2L FL2

k

. 40a

In turn, at the critical points where the derivative changessign the force fluctuations satisfy a strong constraint




11/13

F2L FL2 =

k

. 40b

Last, in the unstable branches the force fluctuations are largerthan in the stable branches and satisfy the inequality

F2L FL2

k

. 40c

These inequalities can be employed to determine criticalpoints in the force by measuring the fluctuations even insituations where the fluctuations mask the presence of criti-

cal points.Figure 15 illustrates these general observations in thespecific case of the pulling of 3rotaxane. As shown, fork= k0 and k=2k0 there is a region where the force fluctua-tions become larger than k/ and, consequently, unstablebehavior in F-L develops. For k=0.4k0 or less the fluctua-tions in the force are never large enough to satisfy Eq. 40cand no critical points in the F-L curves develop, as can beconfirmed in Fig. 14.

6. Origin of the blinking in the force measurements

The observation of force measurements that blink be-tween a high-force and a low force regime recall Fig. 7requires the composite system to be bistable along the end-to-end distance for some L, i.e., the effective potential ULEq. 27 must have a double minimum. Since the PMF ofthe molecule by itself is not bistable along see Fig. 9, thebistability must be introduced by the cantilever potential.Figure 16 shows UL for selected L for the 3rotaxane. ForL in the stable regions of the free energy L =40 andL =90 the effective potential exhibits a single minimumalong the coordinate. However, for L =70 a bistability inthe potential develops. The secondary minimum is the causefor the blinking in the force measurements observed at this L

cf. Fig. 7. Further, the relative energy between the twoinduced minima around the transition region can be manipu-lated by varying L.

What are the minimum requirements for the emergenceof bistability along ? A necessary condition is that the ef-fective potential UL is concave for some region along ,that is

2UL

2=

2

2+ k 0. 41

For Eq. 41 to be satisfied it is required that both i thePMF of the isolated molecule has a region of concavitywhere 2 /20 and ii the cantilever employed is suf-ficiently soft such that

k min22

. 42Equation 42 imposes an upper bound on k0 for bistabil-ity to be observable. If k is stiff the inequality would be

violated for all and bistability would not be manifest. Note,however, that there is no lower bound for k that preventsbistability along . This is in stark contrast with the behaviorof the mechanical instabilities which disappear in the soft-spring limit and persist when stiff springs are employed seeSecs. III B 2III B 5.

Figure 17 illustrates these general observations duringthe pulling of the 3rotaxane. Specifically, the figure shows,for cantilevers of varying stiffness, the probability density ofobserving the value F in the force measurements pLF up-per panels and the associated probability density pL thatthe molecule adopts the extension lower panels when thecantilever is fixed at L. For soft springs the system satisfiesinequality 42 for a range of L values and there is a clearbistability in both probability distributions. As k is increasedthis bistability becomes less prominent and for k=5k0 is nolonger observed.

7. Final remarks

What then should be the picture of the pulling? Clearly,a picture like the one shown in Fig. 12, although appropriateto describe the phenomenology observed in Sec. III Aaround the transition region and the basis of an influentialmodel of molecular extensibility43, is not accurate since themolecule itself does not show a double well structure along

20 30 40 50 60 70 80 90 100 110 1200

2

4

6

8

10

12

14

L (A)

(F

2

F2)1/2

(pN)

k= k0

k=2 k0

k=0.4 k0

FIG. 15. Standard deviation in the force measurements F=F2L FL2 asa function of L for three different cantilever spring constants k during thepulling of the 3rotaxane. In each case, the dotted line indicates the value ofk/ which sets the limit between the stable and unstable branches in theextension, see Eq. 40.

0 20 40 60 800

10

20

30

40

(A)

energy(kcal/mol)

0 20 40 60 800

10

20

30

40

(A)0 20 40 60 80

0

10

20

30

40

(A)

L=40 A L=70 A L=90 A

FIG. 16. Effective potential UL = + VL for the molecule plus can-

tilever for different values of the extension L. In the panels, the blue opencircles correspond to UL, the open black circles to the PMF , and thesolid red lines to the cantilever potential VL. The cantilever spring con-stant employed is the same as the one used in the pulling simulations pre-sented in Sec. III A. Note the bistability in the effective potential for L=70 .




12/13

in the free energy. Note, however, that the molecular freeenergy is inherently bistable along some natural unfoldingpathway. If one were able to determine such a pathway, apicture reminiscent to the one shown in Fig. 12 wouldemerge. However, such natural unfolding pathway is not nec-essarily accessible during the pulling experiment.

The analogy between first-order phase transition and theisometric single-molecule pulling experiments, even whensuggestive, is limited by the fact that the qualitative featuresobserved during pulling strongly depend on the cantileverconstant employed. This is because the properties that aremeasured are those of the combined cantilever plus moleculesystem. However, as previously noted,28 in the stiff-springlimit the measured F-L curves resemble those which wouldbe obtained from a calculation in the Helmholtz ensemble ofthe isolated molecule. In this limit, the instability behavior inthe force versus extension will only be due to molecularproperties recall Eq. 38, and the associated bistability isthat observed along the radius of gyration when is fixed inthe concave region of the PMF Fig. 10. It is in this limitthat the phenomenology observed during the isometricsingle-molecule pulling is closest to a first-order phase tran-sition. Experimentally, this limit is impractical since employ-

ing a stiff spring reduces the sensitivity in the force measure-ments.

IV. CONCLUSIONS

In this contribution we have simulated the equilibriumisometric AFM pulling of a series of donor-acceptor oligoro-taxanes by means of constant temperature MD in a high di-electric medium using MM3 as the force field. For this sys-tem, hysteresis and other nonequilibrium effects that mayarise during pulling can be overcome by reducing the pullingspeed. The resulting equilibrium force versus extension iso-therms shows a mechanically unstable region in which the

molecule unfolds and, for selected extensions, blinks in theforce measurements between a high-force and a low-forceregime.

From the force versus extension data, the Helmholtz freeenergy profile for the oligorotaxanes along the extension co-ordinate was reconstructed using the WHAM and decom-posed into energetic and entropic contributions. The simula-tions reveal an unfolding pathway for the oligorotaxanes andindicate that the folding is energetically driven but entropi-cally penalized. Even when the energy and entropy contribu-tions to the PMF can vary widely along the extension, theseeffects largely cancel one another leading to only modestchanges in the free energy profile. Further, we have observedthat for the rotaxanes studied increasing the number ofthreaded rings stabilizes the folded conformation, making itresilient to unfolding over a wider range of end-to-end dis-tances. As expected, the elastic properties of the extendedconformation are not affected by the number of threadedrings.

The PMF of the rotaxanes exhibits only one minimumalong the end-to-end distance. It consists of two stable con-vex regions characterizing the folded and unfolded confor-mations and a region of concavity where the unfolding oc-

curs. This thermodynamically unstable region arises becausethe molecule is inherently bistable. The inherent molecularbistability is not resolved along the end-to-end coordinatecf. Fig. 9 but, at least in the cases considered, it is manifestalong the radius of gyration recall Fig. 10.

Clearly, modifying the nature of the solvent employedduring the pulling can have a quantitative effect on the de-termined free energy profile since the solvent can vary therelative stability of the different molecular conformations en-countered along the extension. Nevertheless, except for verygood solvents, the qualitative features of the topography ofthe PMF described above are expected to remain intact. Also,hydrodynamic effects that may appear when considering an

F

(pN)

20 40 60 80 10012020

0

20

40

60

80

100

120

20 40 60 80 10012020

0

20

40

60

80

100

120

20 40 60 80 10012020

0

20

40

60

80

100

120

20 40 60 80 10012020

0

20

40

60

80

100

120

L (A)

(A)

20 40 60 80 1001200

20

40

60

L (A)20 40 60 80 100120

0

20

40

60

L (A)20 40 60 80 100120

0

20

40

60

L (A)20 40 60 80 100120

0

20

40

60

0

0.05

0.1

0.15

>0.2

0

0.1

0.2

> 0.3

k0/2 k

02k

05k

0

k0/2 k

02k

05k

0

FIG. 17. The upper panels show the probability density distribution of the force measurements pLF = F Fr ,LL in pN1 during the extension of the

3rotaxane using cantilevers of varying stiffness. The force function Fr ,L is defined by Eq. 3. The lower panels show the associated spatial probabilitydensity distributions pL in

1 Eq. 10 for the 3rotaxane plus cantilever. The color code is given in the far right. The spring constants are expressedin terms ofk0=1.1 pN /, the value employed in the pulling simulations presented in Sec. III A. Note how bistability along , and hence blinks in the force

measurements, arises for soft cantilevers and decays for stiff ones in accordance with Eq. 42.




13/13

explicit solvent e.g., drag forces acting on the AFM cantile-ver due to viscous friction with the surrounding solvent44are immaterial for the equilibrium pulling.

In addition, general conditions on the molecule and onthe cantilever stiffness required for the emergence of me-chanical instability and blinking in the force measurementsin isometric pulling experiments have been presented. Spe-cifically, we have shown that for such effects to arise the

PMF of the molecule needs to exhibit a region of concavityalong the end-to-end distance. Further, the stiffness of thecantilever employed needs to satisfy specific constraints. Forthe blinks in the force measurements to arise the cantileverspring constant needs to be sufficiently soft so that Eq. 42is satisfied. In turn, for the mechanical instability to be ob-servable the fluctuations in the force measurements have tobe sufficiently large to satisfy Eq. 40c for some L. In prac-tice, this implies that in the stiff-spring limit the mechanicalinstability can emerge, while in the soft-spring limit the me-chanical instability decays regardless of the details of themolecular system, provided that no covalent bond breakingoccurs during pulling.

In this light, it becomes apparent that the analogy pre-sented between first-order phase transitions and the mechani-cal properties observed during the extension is limited by thefact that the observed behavior depends on the cantileverconstant employed. The analogy is the closest in the stiff-spring limit where the mechanical properties recorded arethose that would have been obtained in the Helmholtz en-semble of the isolated molecule.

ACKNOWLEDGMENTS

The authors thank Professor J. Fraser Stoddart, Dr. WallyF. Paxton, and Dr. Subhadeep Basu for insightful remarks

and the AFOSR-MURI program of the DOD, NSF GrantNo. CHE-083832, and the Network for ComputationalNanoscience for support of this work. I.F. thanks Dr. GemmaC. Solomon and Dr. Neil Snider for discussions on an earlierversion of this manuscript.

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