Optimizing rotary processes in synthetic molecular motors

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Oct 6, 2009 - The general concept of these molecular motors is visualized in. Scheme 1 (16, 17). The motor .... not depend on its history. This assumption is ...
Optimizing rotary processes in synthetic molecular motors Edzard M. Geertsemaa,1, Sense Jan van der Molenb, Marco Martensc, and Ben L. Feringaa,2 aStratingh

Institute for Chemistry, University of Groningen, Nijenborgh 4, 9747 AG, Groningen, The Netherlands; bKamerlingh Onnes Laboratorium, Leiden University, Niels Bohrweg 2, 2333 CA Leiden, The Netherlands; and cInstitute for Mathematical Sciences, State University of New York at Stony Brook, Stony Brook, NY 11794-3651 Edited by Jack Halpern, University of Chicago, Chicago, IL, and approved July 27, 2009 (received for review April 8, 2009)

Markov model 兩 unidirectional rotation

T

he development of nanoscale molecular devices that exhibit controlled movement upon energy input advances progressively (1–11). In the past decade, a spectrum of ingeniously designed molecular structures, ranging from artificial muscles (12) to molecular cars (13), was introduced. In the B.L.F. group, chiroptical switches have evolved into unidirectionally rotating molecular motors (14–17). In fact, an internal, 360° rotation was induced in one specific direction (i.e., clockwise or counterclockwise) upon input of photons and thermal energy. On the road toward applications, it was shown that the phase of liquid crystals can be controlled when doped with small portions of optically enriched molecular motor, and micro objects can be moved (18–20). In addition, molecular motors were recently grafted on gold (21) and quartz surfaces (22). The general concept of these molecular motors is visualized in Scheme 1 (16, 17). The motor consists of a lower part (‘‘stator,’’ red) and an upper part (‘‘rotor,’’ blue). The rotation takes place around a central double bond (‘‘axle,’’ black) in four stages, i.e., through transitions between four isomers. All four conversions in Scheme 1 are in principle reversible (i.e., equilibria are involved). For example, motor molecules with R ⫽ OCH3 have photochemical equilibria of 14/86 and 11/89 for the concentration ratios A/B and C/D, respectively. Importantly, the thermal conversions from unstable to stable forms are quantitative (i.e., 100%). Previous research focused on attempts to optimize molecular rotation kinetics through chemical design (24–27). However, it focused on single, thermal reaction steps only. Full and ongoing rotation of individual motor molecules can be realized when both light of appropriate energy (h␯) and thermal energy (denoted as ⌬) are supplied simultaneously. We therefore envisioned that quantification and optimization of continuous fourstage rotation should be the next challenge to take on. Remarkably, www.pnas.org兾cgi兾doi兾10.1073兾pnas.0903710106

S 2

R

S

rotor

Meax axle

hν, 365 nm stator

S (2'R)-(M)-trans-A (stable)

10°C, 12 h

Meeq R S

ratio 14/86

(2'R)-(P)-cis-B (unstable) 60°C, 24 h quantitative

60°C, 24 h quantitative

S

S Meeq

Meax

hν, 365 nm 10°C, 12 h

R S

ratio 89/11

(2'R)-(P)-trans-D (unstable)

R S (2'R)-(M)-cis-C (stable)

Scheme 1. General concept of molecular motors. R denotes the stereochemistry of the stereogenic center at the 2-position of the rotor. P and M denote helicity of the entire molecule (23). Cis and trans denote whether the naphthalene part of rotor is at the identical (cis) or opposite side (trans) as the R-substituent of the stator.

a proper approach to quantify the rotational dynamics and efficiency of synthetic rotational motors is lacking. This is in contrast to the situation for biological molecular motors, generally defined as Brownian motors (28–34) or steppers (35–38), which have been the inspiration of numerous studies. Here, we report characterization of the rotation dynamics of a synthetic molecular motor (Scheme 1) upon simultaneous input of photons and heat. Next, all empirically determined reaction constants were inserted in a mathematical model based on the theory of Markov processes. Finally, we show how the model can be used to optimize rotational behavior (i.e., efficiency) of existing molecular motors. Results and Discussion Empirical Results. Several experiments were necessary to fully characterize the rotation dynamics of the molecular motor system as depicted in Scheme 1 (R ⫽ OCH3). In a first experiment, 100% of the stable isomer A (17) was simultaneously illuminated (␭ ⫽ 365 Author contributions: E.M.G., S.J.v.d.M., and B.L.F. designed research; E.M.G., S.J.v.d.M., and M.M. performed research; B.L.F. contributed new reagents/analytic tools; E.M.G., S.J.v.d.M., M.M., and B.L.F. analyzed data; and E.M.G., S.J.v.d.M., M.M., and B.L.F. wrote the paper. The authors declare no conflict of interest. This article is a PNAS Direct Submission. 1Present address: University of Edinburgh, School of Chemistry, The King’s Buildings, West

Mains Road, Edinburgh EH9 3JJ, United Kingdom. 2To

whom correspondence should be addressed. E-mail: [email protected].

This article contains supporting information online at www.pnas.org/cgi/content/full/ 0903710106/DCSupplemental.

PNAS 兩 October 6, 2009 兩 vol. 106 兩 no. 40 兩 16919 –16924

CHEMISTRY

We deal with the issue of quantifying and optimizing the rotation dynamics of synthetic molecular motors. For this purpose, the continuous four-stage rotation behavior of a typical light-activated molecular motor was measured in detail. All reaction constants were determined empirically. Next, we developed a Markov model that describes the full motor dynamics mathematically. We derived expressions for a set of characteristic quantities, i.e., the average rate of quarter rotations or ‘‘velocity,’’ V, the spread in the average number of quarter rotations, D, and the dimensionless Pe´clet number, Pe ⴝ V/D. Furthermore, we determined the rate of full, four-step rotations (⍀eff), from which we derived another dimensionless quantity, the ‘‘rotational excess,’’ r.e. This quantity, defined as the relative difference between total forward (⍀ⴙ) and backward (⍀ⴚ) full rotations, is a good measure of the unidirectionality of the rotation process. Our model provides a pragmatic tool to optimize motor performance. We demonstrate this by calculating V, D, Pe, ⍀eff, and r.e. for different rates of thermal versus photochemical energy input. We find that for a given light intensity, an optimal temperature range exists in which the motor exhibits excellent efficiency and unidirectional behavior, above or below which motor performance decreases.

1.0

[(M)-trans-A] [(P)-cis-B] [(M)-cis-C] [(P)-trans-D]

0.9 0.8

fraction

0.7 0.6

pAA A stable

0.5 0.4

pDA

0.3 0.2

pCA+

pAB B unstable

pBA pCA-

0.1

pBB

pDB+ pDBpCB

0.0 0.0

0.5

1.0

1.5

2.0

time/104 s

2.5

3.0

3.5

4.0

Fig. 1. Development of four motor isomers (M)-trans-A, (P)-cis-B, (M)-cis-C, and (P)-trans-D starting from [(M)-trans-A]0 ⫽ 1.00. Symbols: empirical data. Lines: modeling with matrix P0 (see Eqs. 1 and 2).

nm) and heated (55.5 °C) to induce rotation of the motor molecules. See supporting information (SI) for details. The development of concentrations of all four isomers in course of time was monitored by 1H NMR spectroscopy (Fig. 1). Care was taken to ensure that light intensity and temperature were identical and constant throughout both experiments. When pure A was irradiated and heated, a relatively fast cis-trans isomerization into B was observed. The fraction of A decreased from 1.00 to 0.10 in 1.3 ⫻ 104 s, passing through a shallow minimum. The fraction of B reached a maximum of 0.59 at t ⫽ 6.6 ⫻ 103 s and then slowly decreased to a final value of 0.43 at equilibrium. The third isomer in the cycle (C) was formed next, and its fraction reached a final value of 0.17. The fourth isomer (D) was not observed during the first 1.2 ⫻ 103 s of the experiment and from that moment, fraction D slowly increased to a final magnitude of 0.28. The order of formation of isomers is in line with the unidirectional rotation behavior of this system. The observation that the third isomer (C) initially developed faster than the fourth (D), although its final fraction value at equilibrium is significantly lower (0.17 versus 0.28), is especially notable. A similar result was observed when the other stable isomer, C, was taken as starting material (experiment 2). In final equilibrium, equal fraction values of A, B, C, and D were expected for both experiments ([A]eq ⫽ 0.12, [B]eq ⫽ 0.43, [C]eq ⫽ 0.17, [D]eq ⫽ 0.28). This was indeed the case, within the measurement error of the NMR method (⬇3%). It demonstrates that we were able to keep experimental input parameters, most notably the illumination intensity, constant in both experiments. From Fig. 1, we determined a first set of reaction rates, namely kAB (27.4 ⫻ 10⫺5 s⫺1), kAC (1.4 ⫻ 10⫺5 s⫺1), and kAD (not observed). For this, a first-order exponential fit of the decay of A, and the growth of B, C, and D during the first 1.2 ⫻ 103 s of experiment 1 (Fig. 1) was made. Multiple reaction steps for low time scales can be neglected. Likewise, kCD, kCA, and kCB were determined from the first 1.2 ⫻ 103 s of experiment 2. Minor cross-reactions were observed, i.e., 5% of the reaction product of A became C directly, and 7% of reacting C became A. To determine reaction rates kBA, kBC, kBD, kDA, kDB, and kDC, additional experiments were carried out with pure isomers B and D. The detailed results of all experiments, including the values of all 12 relevant reaction constants, are given in the SI. It is assumed that the temperature does not have an influence on the conversion rates governed by photon input and vice versa. Modeling the Evolution of the Molecular Fractions. We proceed by deriving a Markov model for molecular rotation dynamics. In Scheme 2, a schematic overview of all mathematically possible transitions (denoted by arrows) in our motor system is depicted. 16920 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0903710106

pAD

pDD

D unstable

pBC

pAC- pAC+

pBD- pBD+

pDC

C stable

pCD pCC Scheme 2. Schematic representation of four-stage rotational behavior of a molecular motor. For example, pAB describes the conversion probability from isomer A to B, during a time interval ⌬t. The scheme allows for so-called ‘‘self-transitions,’’ such as pBB, which represents the probability that a molecule of type B does not react at all during ⌬t.

Cross-conversions of A into C and B into D and vice versa are split in positive (⫹, clockwise) and negative (⫺, counterclockwise) rotational movements because these conversions can take place in either direction. To characterize the system, we note that motor rotation is a stochastic process. The dynamics of a given molecule will be a random walk through Scheme 2, which is an example of a Markov process. The basic property of a Markov process is that every next step is dependent only on the present state of the system, i.e., it does not depend on its history. This assumption is certainly valid for synthetic molecular motors such as shown in Scheme 1. A quarter rotation, once the motor is excited, happens on the scale of tens of picoseconds (27, 39–42), whereas the interval between two jumps (i.e., two quarter rotations) typically ranges from seconds to hours. Due to the huge ratio of both time scales (⬎1010), we can safely conclude that in between two quarter rotations, a molecular motor will have lost its memory. There is extensive literature on Markov processes (43, 44). Their dynamics are controlled by probabilities, which are, in our case, related to the motor’s reaction constants. The fact that we typically have 1019 molecules in a container, fully justifies a stochastic approach. To model reaction dynamics, we first defined a time-dependent row vector X(t) ⫽ [A(t), B(t), C(t), D(t)]. Its elements equal the normalized concentration, or fraction, of the four isomers of the molecular motor, at a time t (see Fig. 1). Next, a 4 ⫻ 4 matrix P0 was determined from Scheme 2. This matrix governs how X changes in a small time interval ⌬t by the collection of all reactions, i.e.: X共t ⫹ ⌬t兲 ⫽ X共t兲 P0

[1]

The elements of P0 are related to the probabilities pKL in Scheme 2 (with K,L 僆 {A, B, C, D}). They are given by pKL ⫽ k KL 䡠 ⌬t pKK ⫽ 1 ⫺

共for K ⫽ L兲



p KL

[2a] [2b]

L⫽K

Geertsema et al.

W 䡠 P0 ⫽ W

[3]

Thus, W is a left-side eigenvector of P0 with eigenvalue 1. We calculated W from Eq. 3 and the reaction constants; W ⫽ [wA ⫽ 0.13, wB ⫽ 0.43, wC ⫽ 0.17, wD ⫽ 0.27]. The correspondence between model and empirical data are very accurate, given the considerable standard error of NMR (⬇3%) and the experimental challenge of keeping the input of photons and heat constant during the experiments. This result establishes that the matrix P0 and the empirically determined reaction constants adequately describe the fraction development of experiments 1 and 2, once again justifying our approach. We note that matrix P0 is easily modified to be applied to rotational motors with any number of cyclic stages as well as nonrotational (i.e., linear) motors (45). Rotation Dynamics. The Markov process described by P0 is appro-

priate for studying the development of the fractions of isomers A, B, C, and D in time. However, it does not allow one to determine all dynamic parameters. It is essential to realize that in dynamic equilibrium, the molecules keep on rotating (like in experiments 1 and 2), although the fraction values of isomers A, B, C, and D do not change anymore. To calculate the full rotation dynamics in equilibrium, a more extended Markov process and matrix, P, closely related to P0, is introduced. For this, we consider all possible conversion pairs or jump types (the arrows in Scheme 2) as the unknowns in our problem, e.g., AA or BC. The proper state space S, thus reads: S ⫽ {AA, AB, AC⫹, AC⫺, AD, BB, BC, BD⫹, BD⫺, BA, CC, CD, CA⫹, CA⫺, CB, DD, DA, DB⫹, DB⫺, DC}. Because S has jump types as elements, it allows us to find the dynamic properties in equilibrium. Within S, the signs ⫹ and ⫺ for cross-conversions (A 7 C and B 7 D) refer to a clockwise and counterclockwise jump, respectively. We assume equal probabilities for both paths: pA,C⫺ ⫽ pA,C⫹ ⫽ 1⁄2kA,C⌬t, with equivalent expressions for the other cross-probabilities, BD, DB, and CA. Thus, four extra conversion probabilities are introduced into matrix P compared with the initial matrix P0. The elements of the resulting 20 ⫻ 20 matrix P, incorporating all conversion probabilities, are formally given by (with KL, MN 僆 S): PKL,MN ⫽ p MN

if

L⫽M

[4a]

PKL,MN ⫽ 0

if

L⫽M

[4b]

During an interval ⌬t, a fixed fraction of molecules will be making a certain conversion. The equilibrium fraction making a jump KL 僆 S in a time interval ⌬t is formally denoted by WKL. Because only molecules in state K can make the transition KL, WKL is proportional to wK (the equilibrium isomeric fraction calculated from P0, see Eq. 3) and to the transition probability pKL: WKL ⫽ w K 䡠 p KL

[5]

The next ingredient to determine rotation dynamics is the jump observable (␾). It is a function that assigns how many quarter rotations a molecule is making during a jump. For example; ␾ (AA) ⫽ 0, ␾ (AB) ⫽ 1, ␾ (AC⫹) ⫽ 2, ␾ (AC⫺) ⫽ ⫺2, ␾ (AD) ⫽ ⫺1. A motor that performed a random walk through Scheme 2 of n steps, i.e., for a period t ⫽ n䡠⌬t, will have reached ⌽t quarter Geertsema et al.

rotations at time t. Thus, ⌽t, the so-called net jump counter, is the sum of all jumps:

冘 n

⌽t ⫽

␾i

[6]

i⫽1

Because each molecule performs its own, specific random walk, each molecule has its own net jump count (⌽t), after a time t. In equilibrium, the collection of net jump counts (⌽t) of all molecules is distributed according to a normal distribution, due to the central limit theorem (43, 44). Hence, for each isomer K, the distribution of net jump counts is given by:

␮t共K兲共⌽兲 ⫽

c K共⌽兲

冑2 ␲ Dt

e

⫺ 共 ⌽⫺Vt 兲 2 2Dt

[7]

Here, ␮(K) t (⌽) denotes the fraction of molecules of isomer species K, that have ⌽t ⫽ ⌽ at a time t. Eq. 7 represents a drifting Gaussian distribution, which is a general solution for biased diffusion processes (46–48). The expression includes a prefactor cK(⌽), which depends on the isomer involved. See SI for a more elaborate consideration of the prefactor. Furthermore, it contains a rotation velocity or ‘‘drift,’’ V, and a broadening term 2Dt, of which the quantity D acts as a diffusion constant. The quantities V and D, measured in quarter rotations, are the same for all isomers because they are global properties of the Markov process. It is straightforward to find an expression for V (44, 45). It equals the sum of all jumps per time interval ⌬t: V⫽

1 ⌬t



␾共KL兲 䡠 W KL ⫽

KL僆S



␾ 共KL兲 䡠 w K 䡠 k KL

[8]

KL僆S

Here, we used Eqs. 5 and 2 to find that V can be calculated from the equilibrium fractions wA, wB, wC, and wD (Eq. 3) and the reaction constants. In our case, V has units of quarter rotations per second. The broadening D ⫽ D0 ⫹ Dm of the net jump counts is given by a more involved formula. Its first term, D0, exhibits a standard form to determine broadening: 1 ⌬t

D0 ⫽



关␾共KL兲 ⫺ V⌬t兴 2 䡠 W KL

[9]

KL僆S

For the second term we find (43, 44) Dm ⫽

2 ⌬t



共␾共KL兲 ⫺ V⌬t兲 䡠 W KL

KL僆S0

冋冘 冘 ⬁

k⫽1 MN僆S

共P k兲 KL,MN共 ␾ 共MN兲 ⫺ V⌬t兲



[10]

where S0 is a subset of S, excluding self-conversions. Because numerical values for all parameters in Eqs. 8-10 can be inferred from our empirical data, one can calculate V and D ⫽ D0 ⫹ Dm by using Eqs. 2, 4, and 5. Having obtained expressions for V and D, we calculated the full rotation dynamics (see Eq. 7). We assumed the conditions of experiment 1, i.e., we started with 100% A and made use of the empirical reaction rates. For our specific motor at 55.5 °C, we found: V ⫽ 1.1 ⫻ 10⫺4 s⫺1 and D ⫽ 2.0 ⫻ 10⫺4 s⫺1. In Fig. 2, we plotted ␮(K) t (⌽), the fraction of molecules that performed a given number of quarter rotations, ⌽, for all isomers together. This graph can equivalently be described as a plot of the probability density of reaching a certain number of quarter rotations at a time t. We depicted two curves, one at t ⫽ 5 ⫻ 105 s (red) and another for t ⫽ PNAS 兩 October 6, 2009 兩 vol. 106 兩 no. 40 兩 16921

CHEMISTRY

The matrix P0 was determined for our specific molecular system, using Eqs. 2 a and b and the 12 empirical reaction constants. Making use of standard linear algebra after letting ⌬t 3 0, we solved Eq. 1 for the initial conditions in experiments 1 and 2 (see SI). The calculated isomeric fractions for experiment 1 are visualized as solid lines in Fig. 1. The motor system finally reached a dynamic equilibrium. This implies that X converged to a vector W, according to: X(t 3 ⬁) 3 W ⫽ [wA, wB, wC, wD]. The elements wK of W form the equilibrium fractions of the four isomers. From Eq. 1, it follows directly that

0.10

A

B

0.05

D

Molecular Fraction

0.08 A

0.00 108

0.06

C

A 112

116

0.04 0.02 0.00

0

40

80

120

160

B

Number of Quarter Rotations

Fig. 2. Fraction of molecules, ␮t(K)(⌽), that has performed a certain number of quarter rotations, ⌽. We use Eqs. 7-10, with the empirically determined reaction constants (40), to find V ⫽ 1.1 ⫻ 10⫺4 s⫺1; D ⫽ 2.0 ⫻ 10⫺4 s⫺1. Two full curves are shown: at t ⫽ 5 ⫻ 105 s (red, left) and at t ⫽ 106 s (black), after starting with 100% isomer A at t ⫽ 0 (experiment 1). The curves are spiked, drifting Gaussians. The spikes are related to the equilibrium concentrations of the four isomers. Because, e.g., wB ⬎ wA, more motors will have made 4n ⫹ 1 quarter rotations, than 4n quarter rotations (n 僆 Z). This is illustrated in the Inset, which zooms in on the curve at t ⫽ 106 s.

106 s (black). These distributions have the shape of spiked Gaussians, with well-defined, global V and D values. The spiky appearance is due to the prefactor cK(⌽) which differs for each isomer as cK(⌽) ⬀ wK. Therefore, the Gaussians in Fig. 2 are modulated periodically, with a period of four quarter rotations. For example, isomer B, with wB ⫽ 0.43, is responsible for the large spikes. The local minima, on the other hand, are due to the isomers A and C, with their lower equilibrium concentrations wA and wC. This is illustrated in Fig. 2 Inset, which zooms in on the curve at t ⫽ 106 s. We stress that the full dynamics of the equilibrium motor system is captured in Eq. 7-10. As an independent check, we also numerically calculated ␮(K) t (⌽) from the four-variable differential equation that describes the evolution of A, B, C and D. For large times, like in Fig. 2, the simulations are indistinguishable from the curves calculated with Eqs. 7-10. Optimizing Motor Dynamics. Interestingly, it is possible to adjust motor performance by tuning the two types of energy supplied to the system: heat and light. Reaction constants kBC, kCB, kAD, and kDA describe the thermally activated jumps from the unstable to the stable forms. Hence, they follow the relation k ⫽ (kBTexp(⫺⌬G‡/RT))/h. On the other hand, the rates kAB, kAC, kBA, kBD, kCA, kCD, kDB, and kDC scale with the illumination intensity. Obviously changes in the reaction rates result in different rotation dynamics, via Eqs. 2 and 4. This triggers the question if temperature and light intensity can be tuned in such a manner that the motors rotate optimally. To substantiate on this interesting issue, we calculated V (see Fig. 3A) and D (see Fig. 3B) in the temperature range 250 ⬍ T ⬍ 370 K, for the specific motor used in our experiments. We considered two different illumination intensities. The first (solid curves) equals the value in our experiments whereas for the second (dashed) we assume a light input that is 10 times smaller. The temperature dependent rates were determined by using empirical values: ⌬G‡ (B 3 C) ⫽ 25.6 kcal mol⫺1, ⌬G‡ (C 3 B) ⫽ 30.3 kcal mol⫺1 and ⌬G‡ (D 3 A) ⫽ 25.3 kcal mol⫺1, ⌬G‡ (A 3 D) ⫽ 30.0 kcal mol⫺1. As can be seen from Fig. 3A, V increases both with increasing T and increasing light input (black vs. red curves). At higher temperatures, V reaches saturation. The saturation value depends on the light input (it differs a factor 10 for the two curves), because the light-induced steps are rate limiting in this temperature region. At lower T, however, the thermal steps become rate limiting, so that V decreases dramatically. Because the light input is not dominant any longer, the two curves approach each other. As for the temperature dependence of D, we observe two S-shaped curves. At higher T, saturation similar to the one for V is seen. At lower T, however, D also 16922 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0903710106

C

Fig. 3. Values of V (A), D (B), and Pe (C) versus temperature, calculated for the molecular motor studied in experiments 1 and 2. We incorporated the thermally activated behavior of the rates kBC, kCB, kAD, and kDA. The black, solid curves have the same light input as in experiments 1 and 2. For the red, dashed curves, the light input is assumed 10 times smaller.

reaches a plateau value, unlike V. The reason for the latter is that D is not limited by the ever slower thermal steps, because it takes all possible jumps into account (see Eqs. 9 and 10). Hence, the light-induced steps also determine D at low T, as confirmed by the factor of 10 between the curves with high and low light input. Pe´clet Number. A measure giving insight in motor performance is formed by the dimensionless Pe´clet number Pe. It is defined as (taking a quarter rotation as a unit step) (35, 38):

Pe ⫽ V/D

[11]

Thus Pe, which originated in fluid dynamics, quantifies the ratio of directed motion and diffusive spreading. For molecular motors, Pe is optimal if all forward rate constants are matched (i.e., equal) and the backward rate constants are small. Indeed, if the forward rates have very dissimilar values, ‘‘traffic jams’’ (or pile up of a certain isomer) will limit the rotation efficiency so that Pe is small. The spikes in Fig. 2 are the signature of such ‘‘piling up.’’ Nevertheless, even for optimized synthetic molecular motors, under continuous irradiation, Pe is limited to unity, i.e., Pe ⱕ 1. This is a direct consequence of the stochastic nature of the rotation process. It illustrates the fundamental difference between synthetic molecular engines and daily-life macroscopic engines. The latter have Pe ⬎⬎ 1 because their movement (work) is tightly coupled to distinct quantities of supplied energy. As shown in Fig. 3C, Pe reaches a maximum value of 0.75 at ⬇340 K for the motor and light intensity used in experiments 1 and 2. This fact can be understood as follows. At low T, the rates of the thermal steps are much smaller than the light-induced rates, i.e., the forward rates are not matched at all, and Pe is small. Upon increasing T, the temperaturedependent rates increase until values close to the light-sensitive rates are reached. At that point, Pe arrives at a maximum. As T increases further, the thermal rates start to dominate the light-induced rates, and Pe Geertsema et al.

decreases again. Reasoning along these lines, it becomes clear why for a lower light input, the maximum in Pe shifts to the left in Fig. 3C. In this case, optimal matching to the light-induced rates is already achieved at lower T. We conclude that it is possible to optimize Pe once the system has been characterized properly. Full Rotation Speed (⍀) and Rotational Excess (r.e.). The Pe forms an excellent figure of merit for molecular motor performance. However, to quantify unidirectionality, which is an essential property of rotational molecular motors, a further calculation is needed. Unidirectionality is a measure of the number of successful full (fourstep) forward rotations versus the number of backward rotations. If only forward rotations take place, a motor is considered 100% unidirectional. Interestingly, for perfect unidirectionality, it is not necessary that the forward reaction constants (k’s) are ‘‘matched,’’ in contrast to the case of Pe. Hence, a motor that is 100% unidirectional, may have a low Pe due to dissimilar forward rates. This emphasizes that we need to quantify both Pe and an independent quantity for unidirectionality, the rotational excess (r.e.). Note that a motor with Pe ⫽ 1 must also be unidirectional, so that r.e. ⫽ 1. The definition of r.e. is inspired by similar quantities in chemistry, such as enantiomeric and diastereomeric excess that describe selectivity in asymmetric catalysis (49) and chiral resolution (50). We define:

⍀⫹ ⫺ ⍀⫺ ⍀⫹ ⫹ ⍀⫺

[12]

Here, ⍀⫹ denotes the number of full forward rotations per unit time. The definition of ⍀⫹ is much more stringent than that of V. We define a full forward rotation as a rotation from A to A (or B to B, etc.) in the forward direction that contains no backward jumps in between. A similar definition holds for ⍀⫺, which describes the rate of full backward rotations. If the motor is fully unidirectional, we have ⍀⫺ ⫽ 0, and r.e. ⫽ 100%. If there are as many forward as backward rotations, the motor does not rotate on average and r.e. ⫽ 0%. A calculation of ⍀⫹ requires knowledge about the exact sequence of partial rotations. As soon as a backward jump takes place, a rotation should be disregarded. To take this into account, we adapt our Markov approach. Intuitively, it works in the following way (a mathematically adequate description can be found in SI): First, we define five mathematical ‘‘graphs’’ similar to Scheme 2. Four of these are used as ‘‘counters’’ for full forward rotations: GA, GB, GC, and GD. The fifth graph, G⫺, acts as a pool of molecules that rotated backwards in their last attempt. To intuitively explain the function of ‘‘subgraphs’’ GA, GB, GC, and GD as well as of G⫺, let us suppose that a molecule has just jumped backwards from state B to state A. Therefore, it is by definition part of graph G⫺. Next, the molecule rotates forward by a quarter turn to become B. At that point, it is transferred to graph GA, which follows molecules that started as isomer A and are ‘‘trying’’ to make a full forward rotation. Indeed, if the molecule keeps rotating forward only, without going backwards at all, it stays within GA. Upon completion of a full turn, we add one full rotation (⫹1) to our counter. If, on the other hand, a backward rotation takes place before the full rotation is completed, the complete rotation attempt is disregarded. This is done by transferring this molecular state to G⫺ again. From there, another attempt can be made for a full rotation, etc. By properly connecting the subgraphs GA, GB, GC, GD, and G⫺, a complete graph is formed that describes the Markov problem fully. From this extended graph, a 20 ⫻ 20 matrix P⫹ was derived, which serves as the basis for our calculations of ⍀⫹. The method we use to actually compute ⍀⫹ is related to the way we calculate V above. An analogous procedure is used for ⍀⫺. In Fig. 4A, we show the rate of full forward rotations (⍀⫹, black) and full backward rotations (⍀⫺, blue), as well as their difference (⍀eff ⫽ ⍀⫹ ⫺ ⍀⫺, red), calculated versus temperature. Again, we assumed two light intensities: the photon input of experiments 1 and 2 (solid lines) and 1/10 of that (dashed lines). Remarkably, a clear Geertsema et al.

B

Fig. 4. Full rotation rates and rotational excess versus temperature. (A) Rate of forward full rotations (⍀⫹, black) and backward full rotations (⍀⫺, blue) as well as their difference (⍀eff, red) versus temperature. Solid lines represent the light intensity used in experiments 1 and 2; dashed lines are for 1/10 of that light input. (B) Rotational excess (r.e.) versus temperature, for the same two light intensities. (Inset) (100% ⫺ r.e.) plotted semilogarithmically vs. temperature. Values of r.e. ⬎99.8% are reached.

optimum in ⍀eff is found in both cases. The maximum value of ⍀⫹ scales with the light input: It is 10 times higher for a 10-times-higher intensity (1.1 ⫻ 10⫺4 and 1.1 ⫻ 10⫺5 full rotations per second, respectively). Moreover, the temperature at which the maximum occurs also depends on light input. It is located at ⬇370 K for the standard light intensity and at ⬇350 K for 1/10 the intensity. Finally, for high temperatures (T ⬎ 450 K), ⍀eff is ⬇100 times smaller for the lower light input than for the higher intensity. We explain these interesting results in the following way. At temperatures below 310 K, the thermal rates are much slower than the photochemical rates (k⌬ ⬍⬍ kh␯). The thermal rates are therefore rate determining and inhibit relatively fast full rotations to take place. For this reason, ⍀eff is basically the same for both light inputs. When the temperature rises, the thermal rates increase and ⍀eff increases correspondingly. A maximum is reached when the thermal rates and light-induced rates match, such that each forward quarter rotation can be followed by another step forward. However, as the temperature increases further, the thermal rates become dominant. Moreover, the probability of a backward thermal step becomes larger than the probability of a light-induced forward jump. At that point, ⍀eff will start decreasing due to our stringent demands: only uninterrupted rotations are included. For a lower light input, this point is reached for a lower temperature, explaining the shift of the maximum in Fig. 4A. Once again, this shows the importance of an accurate balance between photon and heat input to achieve the highest rate of total rotations (i.e., motor efficiency). In Fig. 4B, the r.e. of the motor studied is plotted versus temperature, for the two light intensities considered. At temperatures ⬍275 K, r.e. is close to 0%. In the region at ⬇300 K, a steep rise to ⬎99% is observed, after which, r.e. flattens off to a value ⬎99.8% for higher temperatures. Once more, the curves for different light intensities are shifted over the temperature axis. These observations are rationalized by the fact that the thermal PNAS 兩 October 6, 2009 兩 vol. 106 兩 no. 40 兩 16923

CHEMISTRY

r.e. ⫽ 100%

A

reaction rates have dropped to practically zero in the region ⬍275 K, i.e., much below the photochemical rates. Therefore, full rotations do not take place through thermal steps from states B to C and D to A, but primarily through photochemical reactions between states A and C and B and D. Because these conversions take place in both directions, backward and forward, in a 50-to-50 ratio, r.e. is 0%. When the temperature is increased, the thermal conversions that take place quantitatively (⬎⬎99%) in the forward direction, become competitive with the photochemical rates, and r.e. increases consequently to almost 100%. For the lower light intensity, this point is reached at lower temperatures, explaining the shift in Fig. 4B). We conclude that for temperatures that are high enough, these motors are almost perfectly unidirectional. Optimal Conditions for the Motor Studied. Having calculated V, D, Pe,

⍀eff, and r.e., we can finally investigate the question at which conditions the motor studied rotates optimally. We assume the light intensity in experiments 1 and 2. Fig. 4 reveals that the highest motor efficiency is achieved between 345 and 370 K, because at these temperatures, ⍀eff is near its maximum and r.e. is ⬎99%. Furthermore, Fig. 3 shows that the Pe is near its maximum, whereas V has practically reached its saturation value. Hence, the motors are both very efficient in their rotations (high Pe), as well as almost perfectly unidirectional (r.e. ⬇ 1). Remarkably, it is not useful to raise the temperature further. Motor rotation becomes neither faster nor more efficient at higher temperatures. This, at first sight, counterintuitive result, forms one of the main conclusions of our Markov study.

rectional four-stage rotating motor was completely characterized by a temperature and photon-cycled experimental routine based on 1H NMR spectroscopy. All its relevant reaction rates were obtained empirically. By using Markov theory, the probability density of observing a certain number of quarter rotations at time t was calculated. Spiked, drifting Gaussians were obtained. Next, expressions were found for the quarter rotation velocity V and rotational spread D. This allowed us to determine the Pe, (V/D), 0 ⱕ Pe ⱕ 1, for rotational motors, which is a measure for motor performance. Furthermore, we defined the quantity r.e., 0 ⱕ r.e. ⱕ 1, as the relative ratio of the rate of full forward rotations (⍀⫹) versus the rate of full backward rotations (⍀⫺). The effective rate of full rotations is given by ⍀eff ⫽ ⍀⫹ ⫺ ⍀⫺. We demonstrated that V, D, Pe, ⍀eff, and r.e. can be optimized by tuning the temperature and light input, thus achieving maximum efficiency and unidirectionality. Remarkably, there is an optimal temperature range for motor rotation at a given light intensity. Increasing the temperature beyond that range reduces motor performance. Our work presents a pragmatic and powerful combination of theory and experiment for optimization and control of molecular motor performance. This provides a pivotal step forward toward possible nanoscientific applications. Matlab Routines Matlab routines to calculate Figs. 3 and 4 are included in SI.

Summary We applied the theory of Markov processes to quantify the general reaction dynamics of synthetic molecular motors. A specific unidi-

ACKNOWLEDGMENTS. We thank Dr. Martijn Wubs for fruitful discussion on the manuscript. The work was supported by The Netherlands Organization for Scientific Research.

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