University of Groningen
Autonomy and Chirality in Molecular Motors
Kistemaker, Jozef Cornelis Maria
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A
UTONOMY AND
C
HIRALITY
IN
M
OLECULAR
M
OTORS
Paranimfen: Jochem van Herpt Anouk Lubbe
Cover: Third generation molecular motors pass a resemblance to butterflies. At the top is a Nijenborgh moth, at the bottom is a Waterman projection.
The research described in this thesis was carried out within the Stratingh Institute for Chemistry (University of Groningen, The Netherlands) and it was financially supported by a European Research Council Grant (ERC; advanced grant no. 227897). Printer: Ipskamp Drukkers BV, Enschede
ISBN: 978-94-034-0178-2 (printed version) ISBN: 978-94-034-0177-5 (electronic version) Copyright © J.C.M. Kistemaker, 2017
All rights reserved. No part of this book may be reproduced in any form, by print, photocopying, or otherwise, without prior written permission of the copyright owner
Autonomy and Chirality
in Molecular Motors
Proefschrift
ter verkrijging van de graad van doctor aan de
Rijksuniversiteit Groningen
op gezag van de
rector magnificus prof. dr. E. Sterken
en volgens besluit van het College voor Promoties.
De openbare verdediging zal plaatsvinden op
vrijdag 17 november 2017 om 14.30 uur
door
Jozef Cornelis Maria Kistemaker
geboren op 22 oktober 1983
te Oldenzaal
Promotor
Prof. dr. B.L. Feringa
Beoordelingscommissie
Prof. dr. N.H. Katsonis
Prof. dr. S. Otto
Prof. dr. R.J.M. Nolte
5
TABLE OF CONTENTS
Table of Contents
Chapter 1: Introduction 7
Chapter 2: First Generation Molecular Motors 23
Chapter 3: Second Generation Molecular Motors 49
Chapter 4: Molecular Motors in Viscous Media 93
Chapter 5: Identification of Two Thermal Isomerization Pathways for Bistable Molecular Motors
119
Chapter 6: Third Generation Molecular Motors 143
Chapter 7: Third Generation Molecular Motors – Exploring Their Key Parameters and Limits
167
Chapter 8: Chemically Driven Rotary Motor with Autonomous Directionality
217
Nederlandse Samenvatting 237
7
Chapter 1: Introduction
Herein is reported: What can be defined as a molecular motor? Using examples, the boundaries of the definition are determined and the various fuels and sources of directionality required to drive motors are discussed. The control motors have over their own fuel use and directionality provide a degree of autonomy and determine to which degree an operator is required. The underlying mechanisms which govern unidirectional motion in molecular motors are presented and highlighted by examples from literature.
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An introduction into molecular motors first demands a clear definition of what constitutes a molecular motor. Going by its dictionary definition it is anything, “consisting of molecules” “that imparts motion”.[1] Several biological machines
have been labelled molecular motors, long before their detailed functioning was elucidated.[2–4] After much effort – well rewarded – the rotary motion of ATP
synthase was revealed and due to its behaviour, it much deserves the characterization of molecular motor, all the more since its fuelled rotary motion is reminiscent of both a combustion engine as well as an electric motor (Figure 1.1).[5]
The largest group of motor proteins arguably constitutes the class of cytoskeletal motors such as myosin and kinesin.[6] These biological motors exhibit a walking
motion along filaments upon the consumption of chemical fuel and thereby act as the main cargo-transporters of the cell. More recently, prestin was discovered as an exceptional motor protein, being mechanically responsive to voltage changes, assumed to be aiding in auditory perception.[7] By the aforementioned definition,
rhodopsin can also be regarded as a biological motor, as it undergoes a conformational change upon the absorption of a photon initiating the phototransduction cascade.[8] Furthermore, the cofactor retinal by itself can even
be regarded as a molecular motor, since the Z–E isomerization upon irradiation imparts motion to its structure.[9,10] If one follows such reasoning to its end, any
enzyme that undergoes conformational change upon substrate binding, or any molecule changing its structure by chemical transformation can be included, which suggests a need for a narrower definition of molecular motor.
A more specific definition of motor, is that of prime mover – i.e. an initial source of motive power designed to drive machinery. This amendment to the definition allows for a facile distinction between the first and last examples. The rotary motion of ATP synthase fuelled by an electrochemical gradient is what drives the ATP producing machine and the translational motion of kinesin fuelled by ATP is what drives the cargo-transporting machine. However, an enzyme’s conformational change fuelled by substrate binding and a molecule’s structural change fuelled by a chemical reaction by themselves do not drive a machine. The case of rhodopsin is at the boundary of the definition and requires scrutiny before rhodopsin is labelled a molecular motor. The photoreceptor cell is a vital biological piece of machinery responsible for vision through the phototransduction cascade. However, rhodopsin and rhodopsin bound retinal are not so much driving this cascade as they are triggering it. One might argue that it lacks the repetitive progressive nature such as found in motors of the macroworld (e.g. combustion engines and clockwork motors). Therefore, it might be more fitting to label rhodopsin as a molecular
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INTRODUCTIONtrigger or switch, which sets the phototransduction cascade in proverbial motion instead of driving it.
Scientists and inventors have long strived to construct smaller and smaller motors of their own.[11] For the engineering of ever more complex machinery the approach
has long been one of miniaturization. This has been successful to such an extent, that when Feynman during his famous lecture offered a prize for anyone who was able to make a motor fit into a 1/64th inch cube (~1/4 mL), it was claimed and
awarded within a year by means of miniaturization of a regular electric motor.[12]
His desire, however, was for the motor to have been made bottom up instead of top down, which would eventually become possible through the rise of nanotechnology.[13]
Figure 1.1. Examples of rotary motors. a) Water wheel. b) Wankel internal combustion engine. c)
Electric motor. d) ATP synthase. Reprinted from [14]. e) Molecular motor based on an overcrowded alkene. f) Butyl methyl sulphide on copper surface driven by STM tip. Reprinted in part with permission from [15].
A wide array of nanomotors has since been developed; some operate very similar to conventional electric motors,[16,17] whereas even more operate in ways
conventional macroworld motors never did.[18–20] These nanomotors exhibit rotary,
translational or other motion and while many of them do not actually drive any machinery yet, they are designed to do so, stocking the toolkits of future nanoengineers. Molecular motors – spanning dimensions of about twenty-six nanometres[21] down to one nanometre (Figure 1.1f)[15] – can be considered a
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up of biological molecular motors,[6,22,23] but the scientific advances made in the
last three decades, inspired by Nature’s example, have provided a multitude of synthetic molecular motors.[24–26]
Besides the dimensions of a motor as discussed above, several features can be identified which characterize the properties of any motor – (i) the type of motion produced by the motor; (ii) fuel is to be consumed by the motor and converted into motion; (iii) directionality is to be achieved to avoid random motion; (iv) turnover is required for a functional motor; and (v) autonomy of the motor over fuel consumption, directionality, and itself as a whole determines to which degree an operator is required.
Figure 1.2. Rotational molecular motor using translational movement of a macrocycle along three
different binding sites in a [2]catenane. Reprinted with permission from [27].
Types of motion
Rotational and translational are common types of motion, however, the distinction between these two types is not always clear-cut. The system devised by Leigh et al. shown in Figure 1.2 is based on a catenane in which a small macrocycle binds sequentially to stations A, B, and C situated on a larger macrocycle.[27,28] A
rotaxane is a macrocycle on an axle, and when that macrocycle is able to move – translationally – between stations on the axle it becomes a machine well known as a molecular shuttle.[29] Connecting the ends of the axle of a three station molecular
shuttle yields a catenane such as the one shown in Figure 1.2. The authors report the resulting motion as rotary, which begs the question whether translation in a circle constitutes rotation. A possible analogy is that of an ice speed skater racing on a straight track or around a rink – does the speed skaters motion over the ice change from translational to rotational going from a straight track to an oval rink? It all depends on the reference frame, nevertheless, once the translational motion can be described using an angle it can be considered as rotation, even when the axis of rotation is outside of the body in motion. For situations where the axis of ration
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INTRODUCTIONis inside the body in motion, this axis is usually characterized as the motors axle, as is the case for all examples in Figure 1.1.
Turnover
With motors being designed to be the initial source of motive power to drive machinery, turnover or repetitive motion is essential. Were ATP synthase and kinesin to stop working after a single 120 degrees turn or a single step, one could hardly claim that they were able to drive machinery using their motive power. Hence, a certain amount of repetitive rotations or steps is required of a motor, and for many motors their turnover can be determined. For the motors in Figure 1.1 the operational turnovers depend strongly on the specific design, despite that, the turnovers of some high-speed examples have been summarized in Table 1.1. Table 1.1. Properties of rotary motors presented in Figure 1.1 (‘A’ indicates autonomy of the motor
over its own fuel use or directionality).
Turnover (rpm) Fuel A Directional source A
a 2.0·101 Water current ✓ Water flow ✗
b 7.0·103 Combustion ✗ 2D asymmetry ✓
c 1.0·105 Electric current ✓ Electric flow ✗
d 8.0·103 Proton gradient / ATP ✓ Chirality ✓
e 1.8·108 Light ✓ Chirality ✓
f 1.8·102 Electric current ✓ Chirality ✗
Fuel and autonomy
Any source of energy might qualify as a fuel, to be used by a motor, limited only by the ingenuity of the engineer (Table 1.1). Rotary molecular motors using chemical sources as fuel will be the subject of Chapter 8 whereas in the other chapters, light will serve as the main fuel for rotary motion. An interesting group of motors constitutes those classified as Brownian motors, particularly those which operate using tilting or information ratchet mechanisms.[18,26] It should be noted
that while for these motors their motion is facilitated by Brownian motion, it is not fuelled by it – preserving the second law of thermodynamics.[30] The fuel of these
motors is whatever drives their ratchet mechanisms, such as temperature pulses, force fluctuation or chemicals (vide infra and Chapter 8). Most of the examples presented in Figure 1.1 are autonomous with respect to their fuel use, meaning that when a motor is offered the appropriate fuel it is able to operate by itself. However, numerous motors do not possess such autonomy of which the internal combustion engine is an example. Whether the fuel of an internal combustion engine is the combustible gas mixture or the combustion itself is irrelevant, what matters is that
12
1
simply providing the motor with either does not allow the motor to function. It needs a timed combustion during operation and to initiate operation the motor has to be brought in motion by other means (vide infra). Other examples of non-autonomous motors are several systems derived from the catenane shown in Figure 1.2, which require a specific sequence in which several different chemical fuels have to be supplied. Providing these fuels all at once or out of the specific sequence would prevent the correct operation of the motor, illustrating the motor’s lack of autonomy with respect to its fuel use and the need for an operator. In a recent example this limitation has been addressed,[31] and even though the rotation is slow
and fuel use rather substantial,[32] it theoretically functions as an autonomous
chemically fuelled rotary motor. Directionality and autonomy
Directionality is an essential requirement without which a motor would not be a motor. For example, molecular rotors[33] differ from rotary molecular motors, in
that the motion in the first group is random, while in the latter it has a preference for a specific direction. Unidirectionality in any motor requires the breaking of symmetry in order to achieve a motion in a non-random direction. In some systems this is achieved by an asymmetric unit installed a priori, as in ratchets, diodes, or the stereogenic centre in rotary molecular motors based on overcrowded alkenes. In other systems directionality is introduced by an asymmetric induction of fuel by the operator, for example, the direction of water flow for a waterwheel or the direction in which current is fed to an electromotor. For several systems the directionality is only set at the initial motion and maintained by forces such as momentum. An interesting example of such a system is the well known combustion engine (vide supra), for which in the early days a dangerous hand-crank was required to set the engine in motion, which was soon replaced by an electric starter engine. Common internal combustion engines arguably are not clear examples of rotary motors, but of reciprocating motion, and it is the piston and crankshaft which convert it into rotary motion. This setup is the underlying reason for the engine’s lack of autonomy over its rotational direction, since a so called kick back can readily reverse the rotation. A true rotary internal combustion engine has been invented by Felix Wankel and bears his name (Figure 1.1b).[34] Wankel engines are
built in an asymmetric fashion, and therefore possess a predetermined directionality, however, they still need a starter engine to provide the motor with its initial momentum due to a lack of autonomy of the motor over its fuel use. When it comes to molecular motors, the symmetry breaking process to achieve directionality is often explained by means of ratchet mechanisms.[26]
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INTRODUCTIONRatchet mechanisms
The term ratchet is used in analogy to the process capturing property of real ratchets, which is achieved in molecular motors with the use of asymmetric energy barriers for its motion. This does not necessarily require a potential energy surface (PES) to resemble the saw-tooth pattern of ratchets, since the shape of the PES of a barrier does not alter microscopic reversibility.[30,35] Two mechanisms used by
synthetic molecular motors are the information ratchet and the pulsating ratchet.[36]
The essential property of an information ratchet is the need for a feedback mechanism of the position of the motor on its PES in order to lower energy barriers leading towards the desired direction. Motors for which this feedback mechanism is executed by an operator lack autonomy in both fuel use as well as directionality. Such an example is the chemically driven rotary molecular motor developed by Fletcher et al. depicted in Figure 1.3.[37]
1a O PMBO 1b PMBO CO2H OAllyl O 1c O AllylO O 1d AllylO CO2H OPMB i ii iii iv
Figure 1.3. Reversible rotary molecular motor chemically driven by an information ratchet
mechanism. Left: identified structures in the motor’s rotational cycle; right: reaction coordinate diagram with corresponding labels. Adapted from [37], for further details see Chapter 8.
Using a chiral source in the form of asymmetric catalysis allows for the selective formation of 1b over 1d from 1a (Figure 1.3). Random rotation around the single biaryl bond – the axis of the motor – of 1b to 1d is restricted due to steric effects. Selective deprotection of 1b facilitates reformation of a lactone in 1c, in which the upper half has undergone a 180 degrees rotation with respect to the lower half. Repeating these steps using the appropriate chemical fuels gives state 1a’, completing the rotational cycle. The information ratchet mechanism is evident in motor 1, where a random provision of fuel would yield random rotation and where each step can be reversed using a different chemical fuel if so desired.
An example of a translational molecular motor using an information ratchet mechanism has been developed by Leigh and co-workers (Figure 1.4).[38] Even
though the four-step sequence constitutes a single full displacement, they report that in theory it could be extended to a polymeric backbone along which the
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translating unit can move in either direction. The steps ii and iv (in which the translating unit moves a single leg) represent the selective steps which require information regarding the molecules position on the PES. Applying the conditions of steps ii and iv to 2a or 2e leaves the translating unit mostly in place, even if the backbone were extended to a polymer. This is due to the presence of the stilbene moiety in a trans configuration, which was introduced to lock the translating unit in place and prevent a leg from taking a big step (for example preventing the hydrazone leg of 2a to step to its terminal aldehyde).[39] Random application of
steps ii and iv to 2d should accordingly lead to random translations to 2a and 2e, and on a polymeric backbone similarly for positions 2b and 2f, signifying the crucial presence of an information ratchet mechanism. The switching between different potential energy surfaces for governing directional movement is known as a pulsating ratchet mechanism, and is used by motor 2 when it photoisomerizes its stilbene moiety (steps i and iii). Even though it is not essential for system 2,[39]
it greatly improves its directional yield. There is a large group of molecular motors for which a pulsating ratchet mechanism is the only and essential mechanism for control over directionality (Figure 1.5).
Figure 1.4. Molecular ‘walker’ driven by light and chemical fuel. Left/top: identified structures in the
motor’s translational path; bottom right: reaction coordinate diagram with corresponding labels. i) UV light; ii) base; iii) blue light; iv) acid. Adapted from [38].
Light driven rotary motors
All molecular motors depicted in Figure 1.5 use similar stimuli and pathways to achieve full autonomous unidirectional rotation. Each motor possesses a central double bond which serves as the axle of rotation. Due to the size of the substituents at the double bond, the appending moieties are forced from planarity giving rise to
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INTRODUCTIONhelical chirality. Combined with a form of point chirality at the stereogenic centres provides diastereoselectivity where the helical chirality is dynamic while the point chirality remains fixed. Excitation of state a to b* using light allows for a photochemical E–Z isomerization to take place forming state c (Figure 1.5). During this process the molecules are brought onto a PES with a different shape allowing for a thermal relaxation to take place. After this relaxation the molecules return to a different position on the original PES where another thermal relaxation is possible. These switches between different potential energy surfaces, with subsequent relaxations to new positions, are what characterizes a pulsating ratchet mechanism. The process from state a to c provides the opposite configuration over the double bond (E/Z), as well as an inversion of the helicity (M/P) while retaining point chirality (R/S). This yields an unfavourable diastereoisomer, making it a metastable state. Metastable state c is able to undergo a thermally activated helix inversion to produce stable state d, in which the upper half has rotated 180 degrees with respect to the lower half. A repetition of the conditions of steps i, ii and iii brings the molecular motors through states e* and f to a full 360 degrees rotation in state a’. The clockwise unidirectional rotation of the motors depicted in Figure 1.5 is explained using a pulsating ratchet mechanism which relies on a positional asymmetry between the potential energy surfaces. This asymmetry in the PESes is inextricably linked to the asymmetry of the molecules. The shape of the PES of the excited state favours clockwise rotation in the examples since for counter clockwise rotation an additional barrier would have to be crossed where the upper half clashes with the lower half such as in the helix inversion step iii. The shape of the ground state for step iii again favours clockwise rotation due to the asymmetry of the molecules’ structure where in an equilibrium the other diastereoisomer is favoured over the one initially formed. It is the chirality of these molecular motors that allows for autonomous government over their rotational direction. Note that for all these molecular motors their enantiomers rotate in the opposite direction. C2 symmetric
molecular motors based on overcrowded alkenes with stereogenic centres in the two alpha positions (3 in Figure 1.5 and e in Figure 1.1) were the first synthetic motor systems for which repetitive rotary motion was reported by Feringa and co-workers.[40] Being the first fully functional synthetic molecular motors, the group
of motors represented by 3 was dubbed the first generation when a modified class was introduced by the Feringa group.[41] The second generation of C
1 symmetric
molecular motors represented by 4 (Figure 1.5) feature only a single stereogenic centre and a symmetric lower half. A plethora of structural modifications have allowed for finely tuned and functional molecular motors based on 3 and 4 (Chapter 2 and Chapter 3). Greb and Lehn constructed functional molecular motor 5, based on an ‘overcrowded’ imine.[42] Similarly, Guentner et al. developed a molecular
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motor based on hemithioindigo with point chirality at a sulphur atom instead of a carbon atom and fuelled by visible instead of UV light.[43]
i X X R R R' R' X X R R' X R' R R N HO OH NH2 O MeO OMe S O X X R R R' R' X X R R R' R' X X R R R' R' X X R R' X X R R' X X R R' O MeO OMe S O O MeO OMe S O O MeO OMe S O N R R N R N R HO OH NH2 HO OH H2N HO OH H2N X R' R R X R' R R X R' R R i i i i i ii ii ii ii ii ii iii iii iii iii iii iii 3 4 5 6 7 8
Figure 1.5. Light driven unidirectional rotary molecular motors (all depicted examples rotate
clockwise). Top to bottom: 3, 1st generation molecular motors[40] (Chapter 2); 4, 2nd generation
molecular motors[41] (Chapter 3); 5, imine based motor[42]; 6, hemithioindigo based motor[43]; 7, chiral
dibenzosuberane based motors[44–46]; 8, chiral hydrogen bond based motor[47]; general reaction
coordinate diagram with corresponding labels. i) UV/vis light; ii) thermal relaxation; iii) thermally activated helix inversion.
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INTRODUCTIONInterestingly, Chen and co-workers have reported several overcrowded alkenes based on dibenzosuberane (7) which have all the ingredients for a molecular motor.[44–46] They have shown steps i and ii to function as required as well as the
presence of diastereoselectivity required for step iii, although they do not report on any occurrence of a thermal helix inversion (step iii). It is possible that the barrier of activation for step iii is of such a magnitude for the reported examples of 7 that it prevents their motor function. However, these barriers have been shown to be fully adjustable,[48] which would open up the possibility for overcrowded alkenes
7 to join the ranks of unidirectional molecular motors.
Frutos and co-workers have used computational photochemistry to design molecular motor 8 which uses chiral hydrogen bonds as the source for unidirectionality.[47] Their proposed molecular motor might suffer from several
drawbacks such as a need for hard UV due to a lack of an extended aromatic system as well as stability issues stemming from the presence of an iminium group in conjugation with the overcrowded alkene. However, these practical limitations can potentially be resolved by small structural modifications, providing nanoengineers with another remarkable synthetic molecular motor.
Temperature ratchets
The information and pulsating ratchet mechanisms discussed above do not require their PES to exhibit a saw-tooth pattern for their functioning. Temperature ratchet mechanisms, also known as diffusion mechanisms, do require such a feature to allow for directional motion to take place (Figure 1.6 left).[26,49] The mechanism
utilizes a flat PES in an excited state at which free diffusion takes place in either direction (step ii in Figure 1.6 left) followed by a return to an asymmetric ground state (step iii). With the initial minimum off centre between two barriers a net displacement occurs in the direction of the proximal barrier. Note that even though the ground state PES resembles the saw-tooth pattern of a ratchet, net displacement occurs in the opposite direction of a macroworld ratchet, highlighting the contrast between the underlying mechanisms governing the motion of macro- and nanomachines.
The molecular motors engineered by Tierney et al. (Figure 1.1f)[15] and Perera et
al. (Figure 1.6 right)[50] are both driven by pulses with an STM tip while positioned
on a metal surface and the underlying mechanism for the directional displacement which both motors exhibit is explained using a temperature ratchet mechanism (Figure 1.6 left). The mechanism for unidirectional rotation of these motors can also satisfactorily be explained using a pulsating ratchet mechanism (Figure 1.5 bottom). Both motors are chiral which could provide an asymmetrically placed
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excited state instead of the relatively flat excited PES required for a temperature ratchet. Tierney et al. discuss at length the remarkable influence the asymmetry of the STM tip has on the rotation of the chiral butyl methyl sulphide. And while Perera et al. make no mention of chirality or the strong helical nature of the appending ferrocene moieties, they do present calculations substantiating the presence of the aforementioned asymmetric PES of the excited state. Ultimately it might even be possible for these motors to exploit both mechanisms in unison to achieve unidirectionality.
Figure 1.6. Temperature ratchet mechanism. Left: a population of a molecular motor a is excited
to b using a stimulus (step i), diffusion to c takes place (step ii) after which return to the ground state gives population d with net displacement (size of circles indicate population sizes). Right: molecular system which functions as a rotational motor driven by STM on a gold surface, reprinted in part with permission from [50].
Outline of this thesis
Molecular motors driven by light based on overcrowded alkenes will be the focus of this thesis. Chapter 2 will provide a specific introduction into first generation molecular motors (3, Figure 1.5) and the use of computational chemistry in the study of their thermal behaviour. Chapters 3, 4 and 5 will deal with second generation motors where Chapter 3 will first provide a general introduction followed by studies of several structural modifications, Chapter 4 goes into the effect of viscosity on the motor and in Chapter 5 the limits of the bistability of the second generation molecular motors are explored. The seeming crucial role asymmetry plays in these molecular motors will be put to the test in symmetric third generation molecular motors in Chapter 6 and Chapter 7 examines how structural changes affect the behaviour of third generation molecular motors. Finally, Chapter 8 will switch to chemically driven molecular motors where autonomy over their behaviour is discussed and the design for a novel chemically driven motor is presented.
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INTRODUCTIONReferences
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[33] G. S. Kottas, L. I. Clarke, D. Horinek, J. Michl, Chem. Rev. 2005, 105, 1281–376, doi:10.1021/cr0300993.
[34] J. B. Hege, The Wankel Rotary Engine: A History, McFarland & Company, Inc., Jefferson, NC, USA, 2002.
[35] E. A. Anslyn, D. A. Dougherty, Modern Physical Organic Chemistry, University Science Books, Mill Valley, CA, 2006.
[36] Fluctuating potential ratchets and flashing ratchets are identified as synonomous as well as subclasses of pulsating ratchets.
[37] S. P. Fletcher, F. Dumur, M. M. Pollard, B. L. Feringa, Science 2005, 310, 80–82, doi:10.1126/science.1117090.
[38] M. J. Barrell, A. G. Campaña, M. von Delius, E. M. Geertsema, D. A. Leigh,
Angew. Chem., Int. Ed. 2011, 50, 285–290, doi:10.1002/anie.201004779.
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[40] N. Koumura, R. W. Zijlstra, R. A. van Delden, N. Harada, B. L. Feringa, Nature
1999, 401, 152–5, doi:10.1038/43646.
[41] N. Koumura, E. M. Geertsema, A. Meetsma, B. L. Feringa, J. Am. Chem. Soc.
2000, 122, 12005–12006, doi:10.1021/ja002755b.
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23 Parts have been published as:
J. Chen, J. C. M. Kistemaker, J. Robertus, B. L. Feringa, J. Am. Chem. Soc. 2014,
136, 14924–14932, doi:10.1021/ja507711h.
Chapter 2: First Generation Molecular Motors
Herein is reported: An introduction into first generation molecular motors and the use of molecular modelling in the study thereof. The potential energy surface of the smallest first generation molecular motor known to date is investigated, followed by the design of a molecular motor as a photoswitchable DNA hairpin linker, and lastly a study on the dynamic behaviour of molecular motors of increasing size in viscous media is reported.
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Introduction
Stilbene is well known example of an alkene that is able to interconvert between its trans and cis isomers.[1] The thermal barrier for rotation around the carbon–
carbon double bond is high (>150 kJ·mol−1) and therefore ensures the stability of
the two stereoisomers at room temperature. However, stilbene (1) and its analogues such as dinaphthylethene 3 are able to undergo photochemical cis-trans isomerization.[2,3] Irradiation of the trans-alkenes with UV-light leads to their
excited state in which relaxation brings the switches through conical intersections to their cis isomer. Photoexcitation of the alkenes allows for the reversed
cis-trans isomerization, though it suffers from a significant side-reaction in which the cis-alkenes undergo photocyclization. Oxidation of these intermediates leads to the
aromatic products 2 and 4. Besides stringent oxygen or otherwise oxidant free conditions, there is a structural ways to prevent undesirable photocyclization. A straightforward method involves the obstruction of the oxidation sites on the aromatic positions ortho to the ethylene. This has been achieved for example in dimesitylethene[4] and dianthracenylethene[5] which remain disubstituted alkenes.
An interesting family of alkenes is comprised of tetrasubstituted alkenes in which the regions between the substituents with a cis vicinal relationship are crowded to such an extent that the double bond is forced away from planarity. These compounds are dubbed overcrowded alkenes and the first examples, such as bifluorenylidene and bixanthylidene, date from the nineteenth century.[6,7] A
striking example of an overcrowded alkene is the fixation of the naphthyl moieties in 3 by the addition of a six membered ring (4) connecting the geminal positions on both sides of the double bond (Figure 2.1).[8] The overcrowded nature of the
double bond in 4 forces the substituents from planarity in both stereoisomers (E–Z nomenclature applies to double bonds with more than 2 substituents) and therefore allows alkene 4 to be optically active. Compound 4 allowed for the first instance of an isolation of the enantiomers of both the E and Z isomers of an overcrowded alkene.
Twenty years later Feringa and co-workers studied the properties and behaviour of the unique olefin 4 in greater detail.[9–12] They identified the pathways for thermal
helix inversion (THI) between the P,P and M,M enantiomers of E-4 and Z-4. In an effort to assign the absolute stereochemistry theoretical CD spectra were calculated which even today is not always a simple task (see Chapter 6). Even though an appreciable correlation was observed between the experimental and theoretical results, a chiral derivative 5 was synthesized to unequivocally determine the absolute stereochemistry (Figure 2.2). This chiral overcrowded alkene confirmed the earlier findings and its photochemical switching behaviour was used to obtain
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FIRST GENERATION MOLECULAR MOTORS
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stable (R,R,Z)-(P,P)-5, however, the presence of metastable states was not reported. Two years later Koumura et al. reported an extended study which shows that this overcrowded alkene undergoes unidirectional rotation (Figure 2.2).[13]
Figure 2.1. Photoisomerization behaviour of stilbenes 1, 3 and 4 and the oxidative cyclization of
cis-1 and cis-3.
Overcrowded alkene 5 was obtained by a McMurry dimerization[14] of its ketone
precursor and found only as a single diastereoisomer (R*,P*,R*,P*,E)-5, which will be designated as ‘stable’, and its enantiomers were resolved by chiral HPLC. Stable-(R,P,R,P,E)-5 is found to be C2-symmetrical with the methyl groups at the
stereogenic centres in a pseudo-axial orientation and the six membered rings in a boat conformation anti-folded with respect to each other. Irradiation with UV-light at low temperature allows it to undergo a photochemical E–Z isomerization to form a metastable (MS) species which was identified as MS-(R,M,R,M,Z)-5 in which the six membered rings are again anti-folded with respect to each other. Prolonged irradiation leads to a perceived equilibrium between the forward and backward photo-isomerization dubbed the photostationary state (PSS). It should be noted that the principle of microscopic reversibility does not hold for these photochemical reactions. Therefore, the two photochemical E–Z isomerizations are unidirectional when investigated individually. The methyl groups in MS-(R,M,R,M,Z)-5 have adopted a pseudo-equatorial orientation which introduces steric strain due their proximity to the hydrogens on their neighbouring stereocentres. The molecule retains a C2-symmetry axis though the axis changed to a perpendicular orientation
with respect to the plane through the π orbitals of the central alkene while for E-5 the axis lies in this plane.
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Figure 2.2. Photochemical and thermal isomerizations of the first unidirectional repetitive molecular
rotary motor.
The strain in MS-(R,M,R,M,Z)-5 is released through a thermally activated helix inversion of the aryl moieties. During the thermal helix inversion (THI) an intermediate is identified in which a single ring-flip of the six membered ring has taken place and the aryl moieties have adopted a syn-folded geometry ((R,M,R,P,Z)-5). This intermediate quickly goes on to flip the other ring, by slipping the other aryl moiety past the first to give rise to stable-(R,P,R,P,Z)-5 with the six membered rings again in boat conformations and the methyl groups in a pseudo-axial orientation facing away from the molecule. The energy of activation for this process was not determined, however, for an analogue methylated in the 7 positions (Figure 2.2)[15–17] the Gibbs free energy of activation was determined to
be 91±3 kJ·mol−1. The conversion of MS-Z-5 to stable-Z-5 is quantitative,
indicative of the large energy difference between the two states (>8 kJ·mol−1), and
therefore locks the unidirectional rotation made in the photochemical and subsequent thermal isomerization steps. The orbital overlap over the double bond has increased in Z-5 with respect to E-5 which is characterized by a bathochromic shift of the lowest energy band in the absorption spectrum. This allows for the use of UV-light of a higher wavelength to facilitate an E–Z isomerization of stable-(R,P,R,P,Z)-5 to MS-(R,M,R,M,E)-5. This photochemical E–Z isomerization (PEZ) is performed at room temperature on account of the higher stability of MS-E-5 with respect to MS-Z-5 and prolonged irradiation gives rise to a PSS. In the metastable
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FIRST GENERATION MOLECULAR MOTORS
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(R,M,R,M,E)-5 the six membered rings adopt an anti-folded conformation though in a twist boat conformation and with the methyl groups in a pseudo-equatorial orientation. The major source of strain does not stem from steric hindrance directly related to the methyl groups but the proximity of the individual aryl moieties to the hydrogens at the opposite stereocentres. A THI is observed at elevated temperatures with an energy of activation of 110 kJ·mol−1 or a Gibbs free energy of activation
for the methylated analogue 107±4 kJ·mol−1. During the THI of
MS-(R,M,R,M,E)-5 to stable-(R,P,R,P,E)-MS-(R,M,R,M,E)-5, both possessing C2-symmetry, the motor passes through
the C1-symmetrical MS-(R,M,R,P,E)-5, which was most clearly identified in an
analogue with isopropyl instead of methyl moieties at the stereogenic centre.[17,18]
This second PEZ–THI sequence completes the 360° rotation with the last THI again locking the unidirectional rotations.
This behaviour constitutes full autonomous rotation which means the molecules are able to rotate when presented a fuel (ultraviolet light) without the need for an operator. Furthermore, this molecular motor rotates in an autonomous unidirectional fashion. Here autonomous unidirectionality means that the molecule rotates in a single direction which is determined by the motor itself, and not by a chiral fuel (e.g. circularly polarized light) or by a chiral auxiliary. The inherent difference between the E/Z isomers imparts one metastable state with a much higher thermal barrier compared with the other. While this adds complexity to the system, it also allows for a high degree of selectivity. Overcrowded alkenes with the capacity to rotate autonomously based on the structure of molecule 5 are dubbed first generation molecular motors.
For photochemical molecular switches, the yield for the switching process expressed by the PSS ratio is of great importance in comparing the efficiency of the systems. For molecular motors, however, the PSS ratio is of no importance, but theoretically the rotational efficiency depends solely on the quantum yield of the photochemical step while the rotational rate depends mainly on light intensity.[19,20]
A hypothetical sample of a concentrated solution of motor 5 would ensure full absorption of UV-light used to bring about PEZ. At elevated temperatures, the quantum yield of 5 would then determine the yield of the rotation which will be directly locked by a subsequent THI. Under such conditions the rate would be fully dependent on the light intensity. Increasing the light intensity shifts the dependence to the THI rate, which would be evident from the observation of the presence of the metastable state, which can again be solved by increasing the temperature. Maximum efficiency with respect to yield might be obtained by an optimization of the quantum yield for the PEZ by structural modification. This efficiency of rotation can also be managed by an increase in light intensity; however, maximum
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rate at lower temperatures can only be realized by lowering the barrier for THI. This was accomplished by a ring contraction of the six membered ring to a five membered ring (Figure 2.3).
Figure 2.3. Changes in structures of first generation molecular motors.
Motor 6 was investigated by ter Wiel et al.[21] and a substitution of the naphthalene
moiety for a xylene moiety by Pollard et al.[22] provided motor 7 which simplified
functionalization. Various methods have been developed to introduce bromines and alcohols in the 5 and 6 positions in enantiopure 7 (2,2',4,4',7,7'-hexamethyl-2,2',3,3'-tetrahydro-1,1'-biindenylidene)[23,24] while similar methods also allows for
the introduction of other functional groups instead of the methyl groups in the 7 position[25,26]. The 1st generation motors appended by five membered rings show an
interesting change in behaviour with respect to their six membered analogues. Where the THI barrier is highest for the E configuration in 5, the THI barrier is highest for the Z configuration in 6 and 7. The folding of the six membered rings allows for the aromatic moieties to pass each other in the Z configuration in which they go through a local minimum with the moieties in a syn-folded configuration. The five membered rings do not allow for such folding in the Z configuration which significantly increases the barrier for the helix inversion. The opposite is observed in the E configuration where there is more room in 6 and 7 with respect to 5 for the aryl moieties to pass the opposite stereocentres in their THI’s. The photochemical and thermal behaviour of compounds 5 and 7 has been studied computationally.[27– 30]
Calculated Behaviour of 1
stGeneration Molecular Motors
The experimental behaviour of 7 was studied by Pollard et al. and is summarized in Figure 2.4.[22] It exhibits the same responses as 5 did to irradiation and heat.
Irradiation with UV-light of E-7 led to a PEZ yielding a PSS between
stable-E-7 and MS-Z-7 of varying ratios, strongly depending on substitution pattern and
solvent.[23,31,32] The metastable state required heating to allow it to undergo THI to
stable-Z-7, characterized by a Gibbs free energy of activation of 101 kJ·mol−1,
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FIRST GENERATION MOLECULAR MOTORS
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analogues though slightly lower than their E isomers. At low temperatures
stable-Z-7 is able to form a PSS upon irradiation of which the ratio can be modified by
solvent selection. The photogenerated MS-E-7 readily undergoes THI to complete the unidirectional rotary cycle. The barrier for this step is significantly lower than the previously described thermal isomerizations (Δ‡G° 71 kJ·mol−1).
Throughout this thesis there will be made use of quantum chemistry to study the behaviour of molecular motors. The main method used for these computational investigations is density functional theory (DFT). Next we would like to give a practical introduction into the use of theoretical quantum chemistry on the thermal behaviour of molecular motors. For this purpose we use the first generation molecular motor 7 as an example.
Figure 2.4. Photo-chemical and thermal isomerizations of 7
The goal is to describe the potential energy surface (PES) of the molecular motor, by identifying the global and local minima and their connectivity through transitions states. The procedure used most starts by a molecular mechanics optimization of the overcrowded alkene, which interestingly most often provides the metastable configuration when Chem3D is used on a structure created in ChemDraw. Manual manipulation and some structural insight are required to transform the geometry to the stable configuration. This can also be obtained by molecular dynamics, which especially for flexible compounds, such as 5, provides
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additional insight into the possible minima conformations. These geometries are then submitted to optimization with a semi-empirical method using the Gaussian 09 program.[33] The geometries obtained by the semi-empirical PM3 method agreed
slightly better than the PM6 method with the geometries optimized at the DFT level, though the PM6 method performed the computations faster. Therefore PM6 was used for PES scans while PM3 was used for single geometry optimizations. Investigation of the PES is performed by modifying molecular coordinates such as bond lengths, angles or dihedral angles in small steps in which the particular coordinate is frozen after each increment while all other coordinates are optimized. Performing such a relaxed PES scan by modifying the dihedral of four consecutive atoms, also called torsion angle (such as θ in Figure 2.5), has been successfully used to describe the PES of biaryls, azobenzenes and bifluorenylidenes.[34–38]
However, for overcrowded alkenes with a degree of flexibility, such as molecular motors, this approach is problematic and leads to a poor description of the PES.[39]
This situation is illustrated by the PES depicted in Figure 2.5 for a scan over the θ-dihedral of motor 7. There is a steep drop in energy past a certain point (indicated by the circle) at which the molecule strongly changes its conformation and attempts to find a transition point from this geometry are mostly fruitless. An elegant though laborious solution to this is the inclusion of an additional restrained dihedral angle in the PES scan which allows for the construction of a three dimensional surface.[30]
This is, however, not a practical solution when regarding computation costs.
Figure 2.5. Relaxed PES scan of 7. Scans using semi-empirical PM6 with a single dihedral angle
constrained. θ = dihedral 7a-1-1’-7a’; φ = dihedral 10-7-7’-10’; ξ = dihedral 10-4-4’-10’. IRC’s indicated by dotted line.
The restrained coordinate under investigation is not required to be a sequence of connected atoms, and a careful study of such ‘disconnected’ dihedrals revealed several combinations which are able to describe the thermal helix inversions in a continuous fashion (Figure 2.5, φ- and ξ-dihedrals). Transition state optimizations from the maxima of these scans (indicated by a circle) readily provide optimized transition state geometries each identified by a single imaginary frequency.
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FIRST GENERATION MOLECULAR MOTORS
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Figure 2.6. Optimized geometries, IRC’s and HOMO-LUMO gaps of 7 [DFT B3LYP/6-31G(d,p)]. The minima and transition states obtained from the scans at the semi-empirical level are submitted to optimizations using DFT. The functional used here is B3LYP which has been proven to be very suitable for the description of the behaviour and properties of overcrowded alkenes.[40–43] As basis set 6-31G(d,p) is used, for which
the obtained geometries agree well with those obtained at higher levels. It is noteworthy that the geometries obtained by semi-empirical PM3 or PM6 often agree equally well or even better with those obtained by DFT B3LYP/6-31G(d,p) than those optimized with the use of smaller basis sets such as STO-3G and 3-21G.
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Transition states and minima are confirmed by the absence or presence of a single imaginary frequency in the vibrational analysis, respectively. To ensure the connectivity of two minima by an identified transition state, intrinsic reaction coordinates (IRC) are calculated (performed using the Firefly QC package,[44]
which is partially based on the GAMESS (US)[45] source code, using the
Gonzalez-Schlegel second order method). In an IRC, the steepest descent in mass-weighted coordinates is followed from a TS in both a ‘forward’ and ‘backward’ direction to the first minimum encountered on its path (Figure 2.6).
Table 2.1. Thermochemistry of motor 7 [DFT B3LYP/6-31G(d,p)].
MS-(E)-7 TS-(E)-7‡ S-(E)-7 MS-(Z)-7 TS-(Z)-7‡ S-(Z)-7 ΔG° (kJ·mol−1) 22.3 97.0 13.8 20.6 125.6 0.0
ΔH° (kJ·mol−1) 20.3 89.4 14.2 21.3 121.3 0.0
ΔS° (J·mol−1·K−1) −6.9 −26.0 1.3 2.4 −14.5 0.0
The calculated behaviour agrees well with the experimental observations. Going from S-(Z)-7 to MS-(E)-7 the HOMO-LUMO gap drops corresponding to the experimentally observed bathochromic shift (schematic excited state depicted in Figure 2.6) and the helicity of the overcrowded alkene is inverted. MS-(E)-7 is calculated to be 8.5 kJ·mol−1 higher in energy than its diastereoisomer S-(E)-7 to
which it isomerizes by a THI (TS-(E)-7‡) with a calculated barrier of 73.1 kJ·mol−1
agreeing well with the experimentally determined barrier of 71 kJ·mol−1. It is
important to note that in 1st generation molecular motors, the THI exists out of two
redundant transition states with identical activation energies. This is explained by the C2-symmetry of the local minima, which does not express a preference for
which half of the molecule rotates first in a THI. Therefore, there are two vibrational modes that lead to two transition states with identical barriers. Such an occurrence would allow twice the amount of isomerizations over an amount of time and thus double the rate (k). A doubling of rates results in a lowering of the theoretical Gibbs free energy barrier by 1.69 kJ·mol−1 (individual Δ‡G° of
TS-(E)-7‡ is 74.8, and corrected for the double rate equals to 73.1 kJ·mol−1). The
HOMO-LUMO gap of S-(E)-7 was found to be the largest of all isomers, and of the maximum absorption bands in the experimental UV-vis spectra of all isomers, that of S-(E)-7 was found to be the most hypsochromic. In contrast, the HOMO-LUMO gap of MS-(Z)-7 was calculated to be the smallest of all isomers corresponding to the strongest bathochromically shifted experimental absorption spectrum. The optimized transition state (TS-(Z)-7‡) was confirmed by IRC to connect the metastable species to the stable configuration S-(Z)-7. Evident from the mass-weighted displacement in the IRC, the THI of Z-7 requires a structurally much more
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significant reorganization than that of E-7 does. Additionally, the barrier of TS-(Z)-7‡ is substantially higher, calculated to have a corrected Gibbs free energy of 103 kJ·mol−1 which agrees very well with the experimentally determined barrier of
101 kJ·mol−1. The IRC’s have been projected on the scans in Figure 2.5 to highlight
the functionality of the disconnected dihedral scan and the failure of the torsion angle to describe the motor’s behaviour during the THI.
First Generation Motor as a Hairpin Switch
Nucleic acids and analogues control key biological processes such as replication, transcription and translation and are therefore attractive targets for the incorporation of photoswitches. Letsinger and Wu were the first to incorporate a photoswitchable linker in the phosphate backbone.[46] Irradiation of the A
6
-stilbene-T6 hairpin led to a 10 °C decrease of the melting temperature (Tm). The authors
propose that this change derives from an unusual stability of the trans hairpin due to π-stacking effects, which is lost upon photoswitching to the cis isomer. Unfortunately, π-stacking also leads to quenching and low isomerization rates. The method was further improved by the work of Sugimoto and co-workers[47], who
also considered the conformational effect of photoisomerization. Using molecular dynamics simulations they calculated the backbone-backbone distance in a hairpin to be 13.30 Å and subsequently synthesized an azobenzene linker that is 13.36 Å long in its trans configuration but contracts to 10.50 Å in the cis isomer. Switching forces the hairpin to become much more conformationally strained, resulting in a
Tm difference of 20 °C for AAAAG-azobenzene-CTTTT. The shift in Tm for a
similar, but slightly longer and more flexible linker was less than 2 °C. With these results in mind, we sought to design a new type of photoswitchable backbone linker. Upon photoisomerization and subsequent thermal helix inversion, the distance between the two halves of a molecular motor changes much more than in an azobenzene or stilbene. Therefore, we hope to induce much larger conformational effects while at the same time reduce π-stacking effects.
Figure 2.7. Proposed first generation molecular motors 8 and 9 for hairpin switching
Two first generation molecular motors were proposed as suitable hairpin switches which were, contrary to the azobenzene example, expected to pull the strands apart in the unfavourable configuration (Figure 2.7). The unfavourable configuration
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being E here in which the phosphates connected to the motor will be pulled away from each other with respect to the desired ‘optimal’ orientation in the Z isomer. Before synthetic effort was to be put towards obtaining these compounds, they were investigated computationally in order to identify the most effective design. After initial optimizations at the semi-empirical level, the stable E and Z isomers of 8 were optimized using DFT B3LYP/6-31G(d,p). Subsequently, PES scans were performed with the oxygen-oxygen bond constrained using DFT B3LYP/6-31G(d,p) which eventually afforded the PES depicted in Figure 2.9.
Figure 2.8. Structures and global minimum geometries of stable E and Z isomers of 8.
Figure 2.9. PES of the oxygen-oxygen distance in 8.
During the initial scan maxima were observed along the PES due to the large range of possible conformations around the single bonds shown in red in Z-8 in Figure 2.8. Beyond each maximum, a steep drop was observed, similar to those described
0 10 20 30 40 5 10 15 20 SCF Energy / kJ·mol −1 O–O distance / Å (Z)-8 (E)-8 13.3
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FIRST GENERATION MOLECULAR MOTORS
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for Figure 2.5. Therefore, after each drop an additional scan was performed in the opposite direction and for each O–O distance the lowest energy conformer is shown in the PES in Figure 2.9. The resulting PES is clearly very shallow for both the E and Z isomers of 8. And although the global minimum is slightly off from the 13.3 Å, which is considered the optimal distance between the two oxygen atoms, the energy cost to bring either isomer to this distance is minimal. With that said, actually none of the conformations with a distance ranging 7–19 Å for (Z)-8 and 12–20 Å for (E)-8 requires additional energy with respect to the global minimum. This is caused by the previously discussed conformational freedom around the single bonds shown in red in Z-8 in Figure 2.8. From Figure 2.9 can be concluded that molecular motor 8 only functions as a geometry altering switch when O–O distances less than 11 Å are paramount. For such cases, (Z)-8 would be accommodated in the overall geometry without a significant cost in energy with respect to the molecules global minimum down to 7 Å, while (Z)-8 would require additional energy and therefore destabilize the overall structure.
Figure 2.10. Structures and global minimum geometries of stable E and Z isomers of 9.
A replacement of the appending benzene moieties in 8 for acetylene moieties provides molecular motor 9 in which the rotating single bonds are reduced to only 2, indicated in red in (Z)-9 (Figure 2.10). The reduction in degrees of freedom results in a much deeper PES compared to 8 (Figure 2.11). Global minima are found at O–O distances of 15.0 Å for (Z)-9 and 17.4 Å for (E)-9, which clearly are not the desired distance of 13.3 Å, though it only costs 1.4 kJ·mol−1 for (Z)-9 while it costs
22.9 kJ·mol−1 for (Z)-9 to reorganize to this distance. From the difference between
the two scans (delta shown in purple in Figure 2.11) it is evident that below 15.0 Å and above 17. 6 Å O–O distance, the difference in energy is larger than 10 kJ·mol−1.
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Therefore, the switching between the two stable states of this molecular motor is very suitable to bring about significant geometrical changes. This molecular motor is currently being synthesized and will be connected to a DNA hairpin.[48,49]
Switching experiments will then investigate the influence of the geometrical change in the molecular motor on the melting temperature of the hairpin.
Figure 2.11. PES of the oxygen-oxygen distance in 9.
Molecular Stirrers
A series of first-generation light-driven molecular motors with rigid substituents of varying length was synthesized (6 and 10–13, Figure 2.12), envisioned to act as
molecular stirrers.[50,51]
Figure 2.12. First generation light-driven molecular motors 6 and 10–13 with different substituents
at the 6- and 6’-positions.
The rotary motion of 6 & 10–13 from stable-(E) to stable-(Z) was studied by 1H
NMR and UV–vis absorption spectroscopy in a variety of solvents with different
0 10 20 30 40 50 60 70 80 5 10 15 20 SCF Energy / kJ·mol −1 O–O distance / Å (Z)-9 (E)-9 13.3 Delta
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polarity and viscosity. Their behaviour agreed with that of previously studied first generation molecular motors where a fast photo-equilibrium is reached between stable-(E) and metastable-(Z) by irradiation with UV-light at lower temperatures after which the metastable state isomerizes through a thermal helix inversion to stable-(Z) at higher temperatures (Figure 2.13).
Figure 2.13. Photochemical isomerization and thermal helix inversion steps of 6 and 10–13. Quantitative analyses of the kinetic and thermodynamic parameters show that the rotary speed is affected by the rigidity of the substituents and the length of the rigid substituents and that the differences in speed are governed mainly by entropy effects (Table 2.2). Most pronounced is the effect of solvent viscosity on the rotary motion when long, rigid substituents are present.
Table 2.2. Kinetic parameters and the viscosity dependent factor α of the helix inversion step of 6
& 10–13 in THF at rt Motor t½ (min) Δ ‡G° (kJ·mol−1) Δ ‡S° (J·mol−1·K−1) Δ ‡H° (kJ·mol−1) α 6 74.0 93.2 −31.7 84.0 0.16 10 78.5 93.3 −31.6 84.0 0.20 11 97.8 93.8 −33.1 84.1 0.19 12 124 94.4 −35.1 84.2 0.28 13 215 95.7 −39.3 84.2 0.40
To understand how solvent viscosity and the shape and size of the substituents affect the rotary speed of motors, Doolittle’s free-volume model,[52] derived from
Kramers’ theory,[53] is applied.[54] According to the model, molecular motion in a
liquid medium is only possible when the molecules available free volume (Vf) is at
least as large as its critical volume (V0). The fluidity (η–1) is proportional to the
probability factor [exp(−V0/Vf)] for the translation motion. Therefore, the
free-volume dependence of the viscosity can be expressed as follows
/ 1
where A is a proportionality factor. Gegiou et al. noted that in contrast to translational diffusion, molecular rearrangements (in their case a photochemical
