Brownian(Mo*on(
Reem(Mokhtar( What(is(a(Ratchet?(
A(device(that(allows(a(sha9(to(turn(only(one(way(
hLp://www.hpcgears.com/products/ratchets_pawls.htm(
Feynman,(R.(P.,(Leighton,(R.(B.,(&(Sands,(M.(L.((1965).(The$Feynman$Lectures$on$Physics:$ Mechanics,$radia7on,$and$heat((Vol.(1).(AddisonKWesley.( Thermodynamics(
2nd(law,(by(Sadi(Carnot(in(1824.( ( • Zeroth:(If(two(systems(are(in(thermal(equilibrium(with( a(third(system,(they(must(be(in(thermal(equilibrium( with(each(other.(( • First:(Heat(and(work(are(forms(of(energy(transfer.( • Second:(The(entropy(of(any(isolated(system(not(in( thermal(equilibrium(almost(always(increases.( • Third:(The(entropy(of(a(system(approaches(a(constant( value(as(the(temperature(approaches(zero.( ( hLp://en.wikipedia.org/wiki/Laws_of_thermodynamics( Why(is(there(a(maximum(amount(of(work( that(can(be(extracted(from(a(heat(engine?( • Why(is(there(a(maximum(amount(of(work(that( can(be(extracted(from(a(heat(engine?( • Carnot’s(theorem:(heat(cannot(be(converted( to(work(cyclically,(if(everything(is(at(the(same( temperature(!(Let’s(try(to(negate(that.( The(Ratchet(As(An(Engine( Ratchet,(pawl(and(spring.( The(Ratchet(As(An(Engine( Forward rotation Backward rotation
ε energy to lift the pawl ε energy to lift the pawl ε Lθ work done on load θ ε + Lθ energy to rotate wheel θ Lθ work provided by load by one tooth f 1 L / − −(ε+ θ ) τ1 ε + Lθ energy given to vane L fB = Z e Boltzmann factor for work provided by vane b −1 −ε /τ2 f fB = Z e Boltzmann factor for ν fB ratching rate with ν T2 T2 attempt frequency tooth slip ν f b slip rate with f fL power delivered B ν B θ attempt frequency ν
ε energy provided to ratchet
T1 T1
Equilibrium and reversibility Ratchet Brownian motor
b f ε+ Lθ ε ratching rate = slip rate f f τ1 − − B = B f b Leq θε= −1 τ1 τ2 Angular velocity of ratchet: Ω =θν (fB − fB )= θν e − e τ 2 Reversible process by increasing the ε ε load infinitesimally from equilibrium − − ΩL=0 →θν e τ1 − e τ 2 Leq. This forces a rotation leading to Without load: heating of reservoir 1 with τ2 ε Lθ dq1=ε+Leq ? and cooling of − − Ω(L)τ1=τ 2=τ→θν e τ e τ −1 reservoir 2 as dq2=−ε: Equal temperatures: L dq1 ε + eq θ τ1 = = τ1 − dq 2 ε τ 2 τ Ω 1 rectification dq dq 12dS dS 0 +=+=12 L ττ12 τ isentropic process 2
Escherichia coli ATP synthase H. Wang and G. Oster (Nature 396:279-282 1998)
2 Forward rotation Backward rotation
ε energy to lift the pawl ε energy to lift the pawl ε Lθ work done on load θ ε + Lθ energy to rotate wheel θ Lθ work provided by load by one tooth f 1 L / − −(ε+ θ ) τ1 ε + Lθ energy given to vane L fB = Z e Boltzmann factor for work provided by vane b −1 −ε /τ2 f fB = Z e Boltzmann factor for ν fB ratching rate with ν T2 T2 attempt frequency tooth slip ν f b slip rate with f fL power delivered B ν B θ attempt frequency ν
ε energy provided to ratchet
T1 T1
Equilibrium and reversibility Ratchet Brownian motor b f ε+ Lθ ε ratching rate = slip rate f f τ1 − − B = B f b Leq θε= −1 τ1 τ2 Angular velocity of ratchet: Ω =θν (fB − fB )= θν e − e τ 2 Reversible process by increasing the ε ε load infinitesimally from equilibrium − − ΩL=0 →θν e τ1 − e τ 2 Leq. This forces a rotation leading to Without load: heating of reservoir 1 with τ2 ε Lθ dq1=ε+Leq ? and cooling of − − Ω(L)τ1=τ 2=τ→θν e τ e τ −1 reservoir 2 as dq2=−ε: Equal temperatures: L dq1 ε + eq θ τ1 = = τ1 − dq 2 ε τ 2 τ Ω 1 rectification dq dq 12dS dS 0 +=+=12 L ττ12 τ isentropic process 2
Escherichia coli ATP synthase H. Wang and G. Oster (Nature 396:279-282 1998)
2 Forward rotation Backward rotation
energy to lift the pawl ε ε energy to lift the pawl ε Lθ work done on load θ ε + Lθ energy to rotate wheel θ Lθ work provided by load by one tooth f 1 L / − −(ε+ θ ) τ1 ε + Lθ energy given to vane L fB = Z e Boltzmann factor for work provided by vane b −1 −ε /τ2 f fB = Z e Boltzmann factor for ν fB ratching rate with ν T2 T2 attempt frequency tooth slip ν f b slip rate with f fL power delivered B ν B θ attempt frequency ν
ε energy provided to ratchet
T1 T1
Equilibrium and reversibility Ratchet Brownian motor
b f ε+ Lθ ε ratching rate = slip rate f f τ1 − − B = B f b Leq θε= −1 τ1 τ2 Angular velocity of ratchet: Ω =θν (fB − fB )= θν e − e τ 2 Reversible process by increasing the ε ε load infinitesimally from equilibrium − − ΩL=0 →θν e τ1 − e τ 2 Leq. This forces a rotation leading to Without load: heating of reservoir 1 with τ2 ε Lθ dq1=ε+Leq ? and cooling of − − Ω(L)τ1=τ 2=τ→θν e τ e τ −1 reservoir 2 as dq2=−ε: Equal temperatures: L dq1 ε + eq θ τ1 = = τ1 − dq 2 ε τ 2 τ Ω 1 rectification dq dq 12dS dS 0 +=+=12 L ττ12 τ isentropic process 2
Escherichia coli ATP synthase H. Wang and G. Oster (Nature 396:279-282 1998)
2 Forward rotation Backward rotation energy to lift the pawl ε ε energy to lift the pawl ε Lθ work done on load θ ε + Lθ energy to rotate wheel θ Lθ work provided by load by one tooth f 1 L / − −(ε+ θ ) τ1 ε + Lθ energy given to vane L fB = Z e Boltzmann factor for work provided by vane b −1 −ε /τ2 f fB = Z e Boltzmann factor for ν f ratching rate with ν T2 T2 B tooth slip attempt frequency ν f b slip rate with f fL power delivered B ν B θ attempt frequency ν
ε energy provided to ratchet
T1 T1
Equilibrium and reversibility Ratchet Brownian motor b f ε+ Lθ ε ratching rate = slip rate f f τ1 − − B = B f b Leq θε= −1 τ1 τ2 Angular velocity of ratchet: Ω =θν (fB − fB )= θν e − e τ 2 Reversible process by increasing the ε ε load infinitesimally from equilibrium − − ΩL=0 →θν e τ1 − e τ 2 Leq. This forces a rotation leading to Without load: heating of reservoir 1 with τ2 ε Lθ dq1=ε+Leq ? and cooling of − − Ω(L)τ1=τ 2=τ→θν e τ e τ −1 reservoir 2 as dq2=−ε: Equal temperatures: L dq1 ε + eq θ τ1 = = τ1 − dq 2 ε τ 2 τ Ω 1 rectification dq dq 12dS dS 0 +=+=12 L ττ12 τ isentropic process 2
Escherichia coli ATP synthase H. Wang and G. Oster (Nature 396:279-282 1998)
2 ATP synthase, H. Wang and G. Oster (Nature 396, 279, 1998) Myosin
Muscle myosin is a dimerof two identical motor heads that are anchored to the thick filament (top) by a coiled-coil (gray rod extending to the upper right). The helicalactin filament is shown at the bottom (gray). Myosin's catalytic core is blue and its mechanical elements (converter, lever arm helix and surrounding light chains) are colored yellow or red. In the beginning of the movie, the myosin heads are in the prestroke ADP-Pi state (yellow) and the catalytic cores bind weakly to actin . Once a head docks properly onto an actinsubunit (green), phosphate (Pi) is released from the active site. Phosphate release increases the affinity of the myosin head for actin and swings the converter/lever arm to the poststroke, ADP state (transition from yellow to red). The swing of the lever arm moves theactin filament by ~100 Å the exact distance may vary from cycle to cycle depending upon the initial prestroke binding configuration of the myosin onactin . After completing the stroke, ADP dissociates and ATP binds to the empty active site, which causes the catalytic core to detach from actin. The lever arm thenrecocks back to its prestroke state (transition from red to yellow). The surface features of the myosin head and the actin filament were rendered from X-ray crystal structures by Graham Johnson (fiVth media: www.fiVth.com) using the programs MolView, Strata Studio Pro and Cinema 4D. PDB files used were ADP-AlF4-smooth muscle myosin (prestroke, yellow: #1BR2) and nucleotide-free chicken skeletal myosin (poststroke, red: #2MYS). Transitions between myosin crystal structure states were performed by computer coordinated extrapolations between the knownprestroke and poststrokepositions.
http://www.sciencemag.org/feature/data/1049155.shl
Kinesin Molecular gears
The two heads of the kinesindimer work in a coordinated manner to move processivelyalong the microtubule. The catalytic core (blue) is bound to a tubulinheterodimer (green, b-subunit; white, a-subunit) along a microtubule protofilament (the cylindrical microtubule is composed of 13protofilament tracks). In solution, both kinesinheads contain ADP in the active site (ADP release is rate-limiting in the absence of microtubules). The chaotic motion of the kinesinmolecule reflects Brownian motion. One kinesinhead makes an initial weak binding interaction with the microtubule and then rearranges to engage in a tight binding interaction. Only one kinesinhead can readily make this tight interaction with the microtubule, due to restrai nts imposed by the coiled -coil and pre-stroke conformation of the neck linker in the bound head. Microtubule binding releases ADP from the attached head. ATP then rapidly enters the empty nucleotide bind ing site, which triggers the neck linker to zipper onto the catalyti c core (red to yellow transition). This action throws the detached head forward and allows it to reach the next tubulinbinding site, thereby creating a 2-head -bound intermediate in which the neck linkers in the trailing and leading heads are pointing forward (post-stroke; yellow) and backwards (pre-stroke; red) respectively. The trailing head hydrolyzes the ATP (yellow flash of ADP -Pi), and reverts to a weak microtubule binding state (indicated by the bouncing motion) and releases phosphate (fading Pi). Phosphate release also causes the unzippering of the neck linker (yellow to red transition). The exact timing of the strong-to-weak microtubule binding transition and the phosphate release step are not well-defined from current experimental data. During the time when the trailing head execut es the previously described actions, the leading head releases ADP, binds ATP, and zippers its neck linker onto the catalytic core. This neck linker motion throws the trailing head forward by 160 Å to the vicinity of new tubulinbinding site. After a random diffusional search, the new lead head docks tightly onto the binding site which comp letes the 80 Å step of the motor. The movie shows two such 80 Å steps of the kinesinmotor. The surface features of the kinesin motor domains and the microtubule protofilament were rendered from X -ray and EM crystallographic structures by Graham Johnson (fiVth media: www.fiVth.com) using the programs MolView, Strata Studio Pro and Cinema 4D. PDB files used were human conventional kinesin (prestroke, red: #1BG2) and rat conventional kinesin(poststroke, yellow: #2KIN). In human conventional kinesin, the neck linker is mobile and its located in the prestrokestate is estimated from cryo - electron microscopy data. Transitions between states were performed by performing computer-coordinated extrapolations between the prestrokeand poststroke positions. The durations of the events in this sequence were optimized for clarity and do not necessarily reflect the precise timing of events in the ATPasecycle. http://chem.iupui.edu/Research/Robertson/Robertson.html#Gears
Diffusion in asymmetric potentials Driven Brownian ratchets
electrostatic potential
chemical potential
R. Dean Astumian, Science 1997 276: 917-922.
3 ATP synthase, H. Wang and G. Oster (Nature 396, 279, 1998) Myosin
Muscle myosin is a dimerof two identical motor heads that are anchored to the thick filament (top) by a coiled-coil (gray rod extending to the upper right). The helicalactin filament is shown at the bottom (gray). Myosin's catalytic core is blue and its mechanical elements (converter, lever arm helix and surrounding light chains) are colored yellow or red. In the beginning of the movie, the myosin heads are in the prestroke ADP-Pi state (yellow) and the catalytic cores bind weakly to actin . Once a head docks properly onto an actinsubunit (green), phosphate (Pi) is released from the active site. Phosphate release increases the affinity of the myosin head for actin and swings the converter/lever arm to the poststroke, ADP state (transition from yellow to red). The swing of the lever arm moves theactin filament by ~100 Å the exact distance may vary from cycle to cycle depending upon the initial prestroke binding configuration of the myosin onactin . After completing the stroke, ADP dissociates and ATP binds to the empty active site, which causes the catalytic core to detach from actin. The lever arm thenrecocks back to its prestroke state (transition from red to yellow). The surface features of the myosin head and the actin filament were rendered from X-ray crystal structures by Graham Johnson (fiVth media: www.fiVth.com) using the programs MolView, Strata Studio Pro and Cinema 4D. PDB files used were ADP-AlF4-smooth muscle myosin (prestroke, yellow: #1BR2) and nucleotide-free chicken skeletal myosin (poststroke, red: #2MYS). Transitions between myosin crystal structure states were performed by computer coordinated extrapolations between the knownprestroke and poststrokepositions.
http://www.sciencemag.org/feature/data/1049155.shl
Kinesin Molecular gears
The two heads of the kinesindimer work in a coordinated manner to move processivelyalong the microtubule. The catalytic core (blue) is bound to a tubulinheterodimer (green, b-subunit; white, a-subunit) along a microtubule protofilament (the cylindrical microtubule is composed of 13protofilament tracks). In solution, both kinesinheads contain ADP in the active site (ADP release is rate-limiting in the absence of microtubules). The chaotic motion of the kinesinmolecule reflects Brownian motion. One kinesinhead makes an initial weak binding interaction with the microtubule and then rearranges to engage in a tight binding interaction. Only one kinesinhead can readily make this tight interaction with the microtubule, due to restrai nts imposed by the coiled -coil and pre-stroke conformation of the neck linker in the bound head. Microtubule binding releases ADP from the attached head. ATP then rapidly enters the empty nucleotide bind ing site, which triggers the neck linker to zipper onto the catalyti c core (red to yellow transition). This action throws the detached head forward and allows it to reach the next tubulinbinding site, thereby creating a 2-head -bound intermediate in which the neck linkers in the trailing and leading heads are pointing forward (post-stroke; yellow) and backwards (pre-stroke; red) respectively. The trailing head hydrolyzes the ATP (yellow flash of ADP -Pi), and reverts to a weak microtubule binding state (indicated by the bouncing motion) and releases phosphate (fading Pi). Phosphate release also causes the unzippering of the neck linker (yellow to red transition). The exact timing of the strong-to-weak microtubule binding transition and the phosphate release step are not well-defined from current experimental data. During the time when the trailing head execut es the previously described actions, the leading head releases ADP, binds ATP, and zippers its neck linker onto the catalytic core. This neck linker motion throws the trailing head forward by 160 Å to the vicinity of new tubulinbinding site. After a random diffusional search, the new lead head docks tightly onto the binding site which comp letes the 80 Å step of the motor. The movie shows two such 80 Å steps of the kinesinmotor. The surface features of the kinesin motor domains and the microtubule protofilament were rendered from X -ray and EM crystallographic structures by Graham Johnson (fiVth media: www.fiVth.com) using the programs MolView, Strata Studio Pro and Cinema 4D. PDB files used were human conventional kinesin (prestroke, red: #1BG2) and rat conventional kinesin(poststroke, yellow: #2KIN). In human conventional kinesin, the neck linker is mobile and its located in the prestrokestate is estimated from cryo - electron microscopy data. Transitions between states were performed by performing computer-coordinated extrapolations between the prestrokeand poststroke positions. The durations of the events in this sequence were optimized for clarity and do not necessarily reflect the precise timing of events in the ATPasecycle. http://chem.iupui.edu/Research/Robertson/Robertson.html#Gears
Diffusion in asymmetric potentials Driven Brownian ratchets
electrostatic potential
chemical potential
R. Dean Astumian, Science 1997 276: 917-922.
3 ATP synthase, H. Wang and G. Oster (Nature 396, 279, 1998) Myosin
Muscle myosin is a dimerof two identical motor heads that are anchored to the thick filament (top) by a coiled-coil (gray rod extending to the upper right). The helicalactin filament is shown at the bottom (gray). Myosin's catalytic core is blue and its mechanical elements (converter, lever arm helix and surrounding light chains) are colored yellow or red. In the beginning of the movie, the myosin heads are in the prestroke ADP-Pi state (yellow) and the catalytic cores bind weakly to actin . Once a head docks properly onto an actinsubunit (green), phosphate (Pi) is released from the active site. Phosphate release increases the affinity of the myosin head for actin and swings the converter/lever arm to the poststroke, ADP state (transition from yellow to red). The swing of the lever arm moves theactin filament by ~100 Å the exact distance may vary from cycle to cycle depending upon the initial prestroke binding configuration of the myosin onactin . After completing the stroke, ADP dissociates and ATP binds to the empty active site, which causes the catalytic core to detach from actin. The lever arm thenrecocks back to its prestroke state (transition from red to yellow). The surface features of the myosin head and the actin filament were rendered from X-ray crystal structures by Graham Johnson (fiVth media: www.fiVth.com) using the programs MolView, Strata Studio Pro and Cinema 4D. PDB files used were ADP-AlF4-smooth muscle myosin (prestroke, yellow: #1BR2) and nucleotide-free chicken skeletal myosin (poststroke, red: #2MYS). Transitions between myosin crystal structure states were performed by computer coordinated extrapolations between the knownprestroke and poststrokepositions.
http://www.sciencemag.org/feature/data/1049155.shl
Kinesin Molecular gears
The two heads of the kinesindimer work in a coordinated manner to move processivelyalong the microtubule. The catalytic core (blue) is bound to a tubulinheterodimer (green, b-subunit; white, a-subunit) along a microtubule protofilament (the cylindrical microtubule is composed of 13protofilament tracks). In solution, both kinesinheads contain ADP in the active site (ADP release is rate-limiting in the absence of microtubules). The chaotic motion of the kinesinmolecule reflects Brownian motion. One kinesinhead makes an initial weak binding interaction with the microtubule and then rearranges to engage in a tight binding interaction. Only one kinesinhead can readily make this tight interaction with the microtubule, due to restrai nts imposed by the coiled -coil and pre-stroke conformation of the neck linker in the bound head. Microtubule binding releases ADP from the attached head. ATP then rapidly enters the empty nucleotide bind ing site, which triggers the neck linker to zipper onto the catalyti c core (red to yellow transition). This action throws the detached head forward and allows it to reach the next tubulinbinding site, thereby creating a 2-head -bound intermediate in which the neck linkers in the trailing and leading heads are pointing forward (post-stroke; yellow) and backwards (pre-stroke; red) respectively. The trailing head hydrolyzes the ATP (yellow flash of ADP -Pi), and reverts to a weak microtubule binding state (indicated by the bouncing motion) and releases phosphate (fading Pi). Phosphate release also causes the unzippering of the neck linker (yellow to red transition). The exact timing of the strong-to-weak microtubule binding transition and the phosphate release step are not well-defined from current experimental data. During the time when the trailing head execut es the previously described actions, the leading head releases ADP, binds ATP, and zippers its neck linker onto the catalytic core. This neck linker motion throws the trailing head forward by 160 Å to the vicinity of new tubulinbinding site. After a random diffusional search, the new lead head docks tightly onto the binding site which comp letes the 80 Å step of the motor. The movie shows two such 80 Å steps of the kinesinmotor. The surface features of the kinesin motor domains and the microtubule protofilament were rendered from X -ray and EM crystallographic structures by Graham Johnson (fiVth media: www.fiVth.com) using the programs MolView, Strata Studio Pro and Cinema 4D. PDB files used were human conventional kinesin (prestroke, red: #1BG2) and rat conventional kinesin(poststroke, yellow: #2KIN). In human conventional kinesin, the neck linker is mobile and its located in the prestrokestate is estimated from cryo - electron microscopy data. Transitions between states were performed by performing computer-coordinated extrapolations between the prestrokeand poststroke positions. The durations of the events in this sequence were optimized for clarity and do not necessarily reflect the precise timing of events in the ATPasecycle. http://chem.iupui.edu/Research/Robertson/Robertson.html#Gears
Diffusion in asymmetric potentials Driven Brownian ratchets
electrostatic potential
chemical potential
R. Dean Astumian, Science 1997 276: 917-922.
3 ATP synthase, H. Wang and G. Oster (Nature 396, 279, 1998) Myosin
Muscle myosin is a dimerof two identical motor heads that are anchored to the thick filament (top) by a coiled-coil (gray rod extending to the upper right). The helicalactin filament is shown at the bottom (gray). Myosin's catalytic core is blue and its mechanical elements (converter, lever arm helix and surrounding light chains) are colored yellow or red. In the beginning of the movie, the myosin heads are in the prestroke ADP-Pi state (yellow) and the catalytic cores bind weakly to actin . Once a head docks properly onto an actinsubunit (green), phosphate (Pi) is released from the active site. Phosphate release increases the affinity of the myosin head for actin and swings the converter/lever arm to the poststroke, ADP state (transition from yellow to red). The swing of the lever arm moves theactin filament by ~100 Å the exact distance may vary from cycle to cycle depending upon the initial prestroke binding configuration of the myosin onactin . After completing the stroke, ADP dissociates and ATP binds to the empty active site, which causes the catalytic core to detach from actin. The lever arm thenrecocks back to its prestroke state (transition from red to yellow). The surface features of the myosin head and the actin filament were rendered from X-ray crystal structures by Graham Johnson (fiVth media: www.fiVth.com) using the programs MolView, Strata Studio Pro and Cinema 4D. PDB files used were ADP-AlF4-smooth muscle myosin (prestroke, yellow: #1BR2) and nucleotide-free chicken skeletal myosin (poststroke, red: #2MYS). Transitions between myosin crystal structure states were performed by computer coordinated extrapolations between the knownprestroke and poststrokepositions.
http://www.sciencemag.org/feature/data/1049155.shl
Kinesin Molecular gears
The two heads of the kinesindimer work in a coordinated manner to move processivelyalong the microtubule. The catalytic core (blue) is bound to a tubulinheterodimer (green, b-subunit; white, a-subunit) along a microtubule protofilament (the cylindrical microtubule is composed of 13protofilament tracks). In solution, both kinesinheads contain ADP in the active site (ADP release is rate-limiting in the absence of microtubules). The chaotic motion of the kinesinmolecule reflects Brownian motion. One kinesinhead makes an initial weak binding interaction with the microtubule and then rearranges to engage in a tight binding interaction. Only one kinesinhead can readily make this tight interaction with the microtubule, due to restrai nts imposed by the coiled -coil and pre-stroke conformation of the neck linker in the bound head. Microtubule binding releases ADP from the attached head. ATP then rapidly enters the empty nucleotide bind ing site, which triggers the neck linker to zipper onto the catalyti c core (red to yellow transition). This action throws the detached head forward and allows it to reach the next tubulinbinding site, thereby creating a 2-head -bound intermediate in which the neck linkers in the trailing and leading heads are pointing forward (post-stroke; yellow) and backwards (pre-stroke; red) respectively. The trailing head hydrolyzes the ATP (yellow flash of ADP -Pi), and reverts to a weak microtubule binding state (indicated by the bouncing motion) and releases phosphate (fading Pi). Phosphate release also causes the unzippering of the neck linker (yellow to red transition). The exact timing of the strong-to-weak microtubule binding transition and the phosphate release step are not well-defined from current experimental data. During the time when the trailing head execut es the previously described actions, the leading head releases ADP, binds ATP, and zippers its neck linker onto the catalytic core. This neck linker motion throws the trailing head forward by 160 Å to the vicinity of new tubulinbinding site. After a random diffusional search, the new lead head docks tightly onto the binding site which comp letes the 80 Å step of the motor. The movie shows two such 80 Å steps of the kinesinmotor. The surface features of the kinesin motor domains and the microtubule protofilament were rendered from X -ray and EM crystallographic structures by Graham Johnson (fiVth media: www.fiVth.com) using the programs MolView, Strata Studio Pro and Cinema 4D. PDB files used were human conventional kinesin (prestroke, red: #1BG2) and rat conventional kinesin(poststroke, yellow: #2KIN). In human conventional kinesin, the neck linker is mobile and its located in the prestrokestate is estimated from cryo - electron microscopy data. Transitions between states were performed by performing computer-coordinated extrapolations between the prestrokeand poststroke positions. The durations of the events in this sequence were optimized for clarity and do not necessarily reflect the precise timing of events in the ATPasecycle. http://chem.iupui.edu/Research/Robertson/Robertson.html#Gears
Diffusion in asymmetric potentials Driven Brownian ratchets
electrostatic potential
chemical potential
R. Dean Astumian, Science 1997 276: 917-922.
3 DNA transport by a micromachined Brownian ratchet device Geometrical Brownian ratchet I
Joel S. Bader et al., PNAS 96. 13165 (1999)
A. van Oudenaarden and S. G. Boxer, Science 1999; 285: 1046 -1048.
Geometrical Brownian ratchet II Unidirectional molecular rotation
T. Ross Kelly et al., Nature 401(1999)150
Chemically driven rotation Light driven rotation
N. Koumura et al ., Nature 401(1999)152
4 DNA transport by a micromachined Brownian ratchet device Geometrical Brownian ratchet I
Joel S. Bader et al., PNAS 96. 13165 (1999)
A. van Oudenaarden and S. G. Boxer, Science 1999; 285: 1046 -1048.
Geometrical Brownian ratchet II Unidirectional molecular rotation
T. Ross Kelly et al., Nature 401(1999)150
Chemically driven rotation Light driven rotation
N. Koumura et al ., Nature 401(1999)152
4 DNA transport by a micromachined Brownian ratchet device Geometrical Brownian ratchet I
Joel S. Bader et al., PNAS 96. 13165 (1999)
A. van Oudenaarden and S. G. Boxer, Science 1999; 285: 1046 -1048.
Geometrical Brownian ratchet II Unidirectional molecular rotation
T. Ross Kelly et al., Nature 401(1999)150
Chemically driven rotation Light driven rotation
N. Koumura et al ., Nature 401(1999)152
4 DNA transport by a micromachined Brownian ratchet device Geometrical Brownian ratchet I
Joel S. Bader et al., PNAS 96. 13165 (1999)
A. van Oudenaarden and S. G. Boxer, Science 1999; 285: 1046 -1048.
Geometrical Brownian ratchet II Unidirectional molecular rotation
T. Ross Kelly et al., Nature 401(1999)150
Chemically driven rotation Light driven rotation
N. Koumura et al ., Nature 401(1999)152
4 DNA transport by a micromachined Brownian ratchet device Geometrical Brownian ratchet I
Joel S. Bader et al., PNAS 96. 13165 (1999)
A. van Oudenaarden and S. G. Boxer, Science 1999; 285: 1046 -1048.
Geometrical Brownian ratchet II Unidirectional molecular rotation
T. Ross Kelly et al., Nature 401(1999)150
Chemically driven rotation Light driven rotation
N. Koumura et al ., Nature 401(1999)152
4 DNA transport by a micromachined Brownian ratchet device Geometrical Brownian ratchet I
Joel S. Bader et al., PNAS 96. 13165 (1999)
A. van Oudenaarden and S. G. Boxer, Science 1999; 285: 1046 -1048.
Geometrical Brownian ratchet II Unidirectional molecular rotation
T. Ross Kelly et al., Nature 401(1999)150
Chemically driven rotation Light driven rotation
N. Koumura et al ., Nature 401(1999)152
4 Maxwell’s demon Quantum demon? (ask Milena Grifoni)
W. Smoluchowski (1941): No automatic, permanently effective perpetual motion machine van violate the second law by taking advantage of statistical fluctuations (Feynman: the demon is getting hot). Such device might perhaps function if operated by intelligent beings.
W.H. Zurek, Nature 341(1989)119: The second law is safe from intelligent beings as long as their abilities to process information are subject to the same laws as these of universal Turing machines.
Fluctuations of µm-sized trapped colloidal particles Noise ratchet
G.M. Wang et al., Phys. Rev. Lett. 89(2002)050601
F opt vopt
physics.okstate.edu/ ackerson/vackerson/
1 t vFs ds Σt = ∫ opt ⋅ opt ( ) τ 0 Entropy production
5 Maxwell’s demon Quantum demon? (ask Milena Grifoni)
W. Smoluchowski (1941): No automatic, permanently effective perpetual motion machine van violate the second law by taking advantage of statistical fluctuations (Feynman: the demon is getting hot). Such device might perhaps function if operated by intelligent beings.
W.H. Zurek, Nature 341(1989)119: The second law is safe from intelligent beings as long as their abilities to process information are subject to the same laws as these of universal Turing machines.
Fluctuations of µm-sized trapped colloidal particles Noise ratchet
G.M. Wang et al., Phys. Rev. Lett. 89(2002)050601
F opt vopt
physics.okstate.edu/ ackerson/vackerson/
1 t vFs ds Σt = ∫ opt ⋅ opt ( ) τ 0 Entropy production
5 Maxwell’s demon Quantum demon? (ask Milena Grifoni)
W. Smoluchowski (1941): No automatic, permanently effective perpetual motion machine van violate the second law by taking advantage of statistical fluctuations (Feynman: the demon is getting hot). Such device might perhaps function if operated by intelligent beings.
W.H. Zurek, Nature 341(1989)119: The second law is safe from intelligent beings as long as their abilities to process information are subject to the same laws as these of universal Turing machines.
Fluctuations of µm-sized trapped colloidal particles Noise ratchet
G.M. Wang et al., Phys. Rev. Lett. 89(2002)050601
F opt vopt
physics.okstate.edu/ ackerson/vackerson/
1 t vFs ds Σt = ∫ opt ⋅ opt ( ) τ 0 Entropy production
5 The Feynman Thermal Ratchet"
Pforward~exp(-Δε/kT1)" Pbackward~exp(-Δε/kT2)" " 2 2 τrel ≈ Cl /(4π κ)" works only if T1>T2 !!" motor protein conformational change: µs" wrong " decay of temperature gradient over 10 nm: ns" model" Brownian Ratchet" (A.F. Huxley 57)"
Cargo" Net " Thermal motion" transport"
Motor"
Track"
perpetuum mobile? Not if ATP is used to switch the off-rate." Three-bead assay " with ncd" Myosin: averaged power strokes" (Veigel et al. Nature 99, 398, 530)" Myosin Power Stroke"
ADP" Mechano-chemical cycle:" +Pi"
attach" M*ATP"
myosin" working stroke" actin" M*ADP*Pi"
ADP" detach" M" recovery stroke"
Pi" Conformational Change of Single ncd Molecule" 10 2 µM ATP
5
0
MT displacement [nm] -5 0 400 800 1200 1600 time [ms]
ADP*Pi" ADP" release"
DeCastro, Fondecave, Clarke, Schmidt, Stewart, Nature Cell Biology (2000), 2:724"
Stepping and Stalling of a Single Kinesin Molecule "
300 7 250 6 240 200 5 224 force[pN] [nm] 150 4 208 3 192 100 2 176 50 time [s] displacement [nm] 1 160 ~ 6 pN stall force" 6 6.5 7 7.5 8 8.5 9 0 0
-50 -1 0 10 20 30 40 50 60 ~ 8 nm steps" time [s]
Svoboda, Schmidt, Schnapp, Block, Nature (1993), 365: 721" Randomness Δt=const." parameter" clockwork "
r = 0!
k" k" k" k" k" 1 -> 2 -> 3 -> 4 -> 5 -> 6"
Poisson process" r = 1! k 1 -> 2" Randomness parameter for single kinesin" " (Visscher,Schnitzer, Block ( 99)," Nature 400, 184)" Thermal Motion of a Trapped/Tethered Particle
200 150 Time series:" 100 50 0 2 2 kBT -50 var( x) = x − x = -100 κ -150 Displacement [nm] Displacement -200 0 0.05 0.1 0.15 0.2 0.25 0.3 Time [s] Spectrum:" 100 k T /Hz] B 2 S( f ) = 2 2 2 [nm S(f) ≈ S π γ ( f + f ) 10 0 c S(f)
1 f c density,
0.1 κ 4γkBT slope = -2 f = , S = spectral c 2πγ 0 κ 2 0.01
Power 1 10 100 103 Frequency, f [Hz] k T trapped bead attached to motor: var(x) = B κ trap +κ motor Efficiency, Invertability and$ Processivity of Molecular Motors"
F. Jülicher, Institut Curie, Paris"
A. Parmeggiani" L. Peliti (Naples)" A. Ajdari (Paris)" J. Prost (Paris)"
http://www.curie.fr/~julicher" Mechano-chemical coupling" 4" M-ATP" 3" M-ADP-P" 2" M-ADP" x" 1" M"
∂t Pi + ∂ x Ji = −∑ωijPi + ∑ω jiPj j ≠ i j ≠i
Ji = −µ(k BT∂ xPi + Pi∂ xWi − fext Pi ) l mean velocity" v dx J = ∫ ∑ i 0 i Example: weakly bound state"
weakly bound"
strongly bound"
ATP"
ADP-P" ADP" Example: identical shifted states"
U"
ω12 ω21
a" l"
U = 20 7.4 10 −4 kg / s k T ξ = ⋅ (Δµ− ΔW) / kT B a ω12 = ωe = 0.1 l l =16nm
ω21 = ω
chemical free energy of hydrolysis:"
Δµ = µ ATP − µ ADP − µP 0 0 µi ≈ µi + kT ln(Ci / Ci ) Dissipation rates"
l motion" H = W − f x Q˙ dx J H 0 i i ext within a state:" i = ∫ i ∂ x i ≥ 0 + kBT ln(Pi )
chemical transitions:" l Q˙ = dx (ω P − ω P )(H − H + Δµ) ≥ 0 α ∫ 12 1 21 2 1 2 0
total internal dissipation of the motor:" ˙ ˙ ˙ Q = Q1 + Q2 + Qα ≥ 0 Efficiency of energy transduction"
Energy conservation:" fext force" v velocity" ˙ Δµ chemical " r chemical " Q = rΔµ + fextv energy" rate"
mechanical work" chemical work" total internal dissipation"
f v η = − ext rΔµ
A. Parmeggiani, F. Jülicher, A. Ajdari and J. Prost, PRE 60, 2127 (1999) " Angewandte Molecular Devices Chemie
there would be little viscosity to slow it down. These effects always cancel each other out, and as long as a temperature can be defined for an object it will undergo Brownian motion appropriate to that temperature (which determines the kinetic energy of the particle) and only the mean-free path between collisions is affected by the concentration of particles. In the absence of any other molecules, heat would still be transmitted from the hot walls of the container to the particle by electromagnetic radiation, with the random emission and absorption of the photons producing the Brownian motion. In fact, even temperature is not a particularly effective modulator of Brownian motion since the velocity of the particles depends on the square root of the temperature. So to reduce random thermal fluctuations to 10% of the amount present at room temperature, one would have to drop the temperature from 300 K to 3 K.[12,13] It seems sensible, therefore, to try to utilize Brownian motion when designing molecular machines rather than make structures that have to fight against it. Indeed, the question of how to (and whether it is even possible to) harness the inherent random motion present at small length scales to generate motion and do work at larger length scales has vexed scientists for some considerable time.
Figure 1. The fundamental difference between a “switch” and a 1.2.1. The Brownian Motion “Thought Machines” “motor” at the molecular level. Both translational and rotary switches influence a system as a function of the switch state. They switch The laws of thermodynamics govern how systems gain, between two or more, often equilibrium, states. Motors, however, process, and release energy and are central to the use of influence a system as a function of the trajectory of their components particle motion to do work at any scale. The zeroth law of or a substrate. Motors function repetitively and progressively on a thermodynamics tells us about the nature of equilibrium, the system; the affect of a switch is undone by resetting the machine. first law is concerned with the total energy of a system, while a) Rotary switch. b) Rotary motor. c) Translational switch. d) Transla- tional motor or pump. the third law sets the limits against which absolute measure- ments can be made. Whenever energy changes hands, Carter, N. J., & Cross,however, R. A. (body (2005). to body Mechanics or form to form) of the it is kinesin the second step. Nature, 435(7040), law308–312. of thermodynamics" which comes into play. This law provides the link between the fundamentally reversible laws ard motion of tiny particles within translucent pollen grains of physics and the clearly irreversible nature of the universe in suspended in water.[7] An explanation of the phenomenon— which we exist. Furthermore, it is the second law of now known as Brownian motion or movement—was provided thermodynamics, with its often counterintuitive consequen- by Einstein in one[8] of his three celebrated papers of 1905 and ces, that governs many of the important design aspects of how was proven experimentally[9] by Perrin over the next to harness Brownian motion to make molecular-level decade.[10] Scientists have been fascinated by the implications machines. Indeed, the design of tiny machines capable of of the stochastic nature of molecular-level motion ever since. doing work was the subject of several celebrated historical The random thermal fluctuations experienced by molecules “thought-machines” intended to test the very nature of the dominate mechanical behavior in the molecular world. Even second law of thermodynamics.[14–18] the most efficient nanoscale machines are swamped by its effect. A typical motor protein consumes ATP fuel at a rate of 1.2.1.1. Maxwell’s Demon 100–1000 molecules every second, which corresponds to a 16 maximum possible power output in the region of 10À to Scottish physicist James Clerk Maxwell played a major 17 [11] 10À W per molecule. When compared with the random role (along with Ludwig Boltzmann) in developing the kinetic 8 environmental buffeting of approximately 10À W experi- theory of gases, which established the relationship between enced by molecules in solution at room temperature, it seems heat and particle motion and gave birth to the concept of remarkable that any form of controlled motion is possible.[12] statistical mechanics. In doing so, Maxwell realized the When designing molecular machines it is important to profundity of the statistical nature of the second law of remember that the presence of Brownian motion is a thermodynamics which had recently been formulated[19] by consequence of scale, not of the nature of the surroundings. Rudolf Clausius and William Thomson (later Lord Kelvin). In It cannot be avoided by putting a molecular-level structure in an attempt to illustrate this feature, Maxwell devised the a near-vacuum, for example. Although there would be few thought experiment which has come to be known as random collisions to set such a Brownian particle in motion, Maxwell s demon.[14,15,20]
Angew. Chem. Int. Ed. 2007, 46, 72 – 191 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.angewandte.org 75 Reviews D. A. Leigh, F. Zerbetto, and E. R. Kay
Maxwell envisaged a gas enclosed within a container, to to describe all manner of hypothetical constructs designed to and from which no heat or matter could flow. The second law overcome the second law of thermodynamics by continually of thermodynamics requires that no gradient of heat or extracting energy to do work from the thermal bath.[14] pressure can spontaneously arise in such a system, as that Maxwell noted that the demon principle could be would constitute a reduction in entropy. Maxwell imagined demonstrated in a number of different ways; a “pressure” the system separated into two sections by a partition demon, for example, (Figure 2b) could sort particles so that (Figure 2). Having just proven that the molecules in a gas at more end up in one end of a vessel than the other, which required different information to operate from the original tem- perature demon (the direction of approach of a particle, not its speed).
1.2.1.2. Szilard’s Engine
Both Maxwell and Thomson appreciated that the operation of these systems for separating Brownian particles appeared to rely on the demon s “intelli- Figure 2. a) Maxwell’s “temperature demon” in which a gas at uniform temperature is sorted into “hot” gence” as an animate being, but and “cold” molecules.[15] Particles with energy higher than the average are represented by red dots while did not try to quantify it. Leo blue dots represent particles with energies lower than the average. All mechanical operations carried out Szilard made the first attempt to by the demon involve no work—that is, the door is frictionless and it is opened and closed infinitely mathematically relate the slowly. The depiction of the demon outside the vessel is arbitrary and was not explicitly specified by demon s intelligence to the ther- Maxwell. b) A Maxwellian “pressure demon” in which a pressure gradient is established by the door being opened only when a particle in the left compartment approaches it.[15c] modynamics of the process by considering the performance of a machine based on the “pressure demon”, namely, Szilard s engine a particular temperature have energies statistically distrib- (Figure 3).[17] Szilard realized that the operations which the uted about a mean, MaxwellCarter, postulated N. J., & a tinyCross, “being” R. A. able(2005). to Mechanicsdemon carriesof the kinesin out can step. be reduced to a simple computational detect the velocities of individualNature, 435 molecules(7040), 308–312. and open" and process. In particular, he recognized the process requires close a hole in the partition so as to allow molecules moving measurement of the approach of the particle to gain faster than the average to move in one direction (R L in information about its direction and speed, which must be ! Figure 2) and molecules moving slower than the average to remembered and then acted upon.[24] move in the other (L R in Figure 2). All the time, the ! number of particles in each half and the total amount of 1.2.1.3. Smoluchowski’s Trapdoor energy in the system remains the same. The result of the demon s endeavors being successful would be that one end of The concept of a purely mechanical Brownian motion the system (the “fast” end, L) would become hot and the machine which does not require any intelligent being to other (the “slow” end, R) cold; thus a temperature gradient is operate it was first explored by Marian von Smoluchowski set up without doing any work, contrary to the second law of who imagined the Maxwell system as two gas-containing thermodynamics. compartments with a spring-loaded trapdoor in place of the After its publication in “Theory of Heat”,[15b] Thomson demon-operated hole (Figure 4a).[16] If the spring is weak expanded upon Maxwell s idea in a paper[21] read before the enough, it might appear that the door could be opened by Royal Society of Edinburgh on 16 February 1874, and collisions with gas molecules moving in one direction (L R ! published a few weeks later in Nature,[22] introducing the in Figure 4a) but not the other, and so allow transport of term “demon” for Maxwell s being.[23] In using this word, molecules preferentially in one direction thereby creating a Thomson apparently did not intend to suggest a malicious pressure gradient between the two compartments. Smolu- imp, but rather something more in keeping with the ancient chowski recognized (although could not prove) that if the Greek roots of the word (more usually daemon) as a door had no way of dissipating the energy it gains from supernatural being of a nature between gods and humans. Brownian collisions it would be subject to the same amount of Indeed, neither Maxwell nor Thomson saw the demon as a random thermal motion as the rest of the system and would threat to the second law of thermodynamics, but rather an not then function as the desired one-way valve. illustration of its limitations—an exposition of its statistical nature. This, however, has not been the view of many 1.2.1.4. Feynman’s Ratchet and Pawl subsequent investigators, who have perceived the demon as an attempt to construct a perpetual motion machine driven by Richard Feynman revisited these ideas in his celebrated thermal fluctuations. The term “Maxwell s demon” has come discussion of a miniature ratchet and pawl—a construct
76 www.angewandte.org 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Angew. Chem. Int. Ed. 2007, 46, 72 – 191 Angewandte Molecular Devices Chemie
Figure 4. a) Smoluchowski’s trapdoor: an “automatic” pressure demon (the directionally discriminating behavior is carried out by a wholly mechanical device, a trapdoor which is intended to open when hit from one direction but not the other).[16] Like the pressure demon shown in Figure 2b, Smoluchowski’s trapdoor aims to transport Figure 3. Szilard’s engine which utilizes a “pressure demon”.[17] particles selectively from the left compartment to the right. However, a) Initially a single Brownian particle occupies a cylinder with a piston in the absence of a mechanism whereby the trapdoor can dissipate at either end. A frictionless partition is put in place to divide the energy it will be at thermal equilibrium with its surroundings. This container into two compartments (a b). b) The demon then detects means it must spend much of its time open, unable to influence the ! the particle and determines in which compartment it resides. c) Using transport of particles. Rarely, it will be closed when a particle this information, the demon is able to move the opposite piston into approaches from the right and will open on collision with a particle position without meeting any resistance from the particle. d) The coming from the left, thus doing its job as intended. Such events are partition is removed, allowing the “gas” to expand against the piston, balanced, however, by the door snapping shut on a particle from the doing work against any attached load (e). To replenish the energy used right, pushing it into the left chamber. Overall, the probability of a by the piston and maintain a constant temperature, heat must flow particle moving from left to right is equal to that for moving right to into the system. To complete the thermodynamic cycle and reset the left and so the trapdoor cannot accomplish its intended function [18] machine, the demon’s memory of where the particle was must be adiabatically. b) Feynman’s ratchet and pawl. It might appear that erased (f a). To fully justify the application of a thermodynamic Brownian motion of the gas molecules on the paddle wheel in the ! concept such as entropy to a single-particle model, a population of right-hand compartment can do work by exploiting the asymmetry of Szilard devices is required. The average for the ensemble over each of the teeth on the cog of the ratchet in the left-hand compartment. these devices can then be considered to represent the state of the While the spring holds the pawl between the teeth of the cog, it does system, which is comparable to the time average of a single multi- indeed turn selectively in the desired direction. However, when the particle system at equilibrium, in a fashion similar to the statistical pawl is disengaged, the cog wheel need only randomly rotate a tiny mechanics derivation of thermodynamic quantities. amount in the other direction to move back one tooth whereas it must Carter, N. J., & Cross,rotate R. randomly A. (2005). a long way Mechanics to move to the next of tooth the forward. kinesin If the step. paddle wheel and ratchet are at the same temperature (that is, T1 =T2) these rates cancel out. However, if T T then the system will Nature, 435(7040), 308–312." 1 ¼6 2 designed to illustrate how the irreversible second law of directionally rotate, driven solely by the Brownian motion of the gas thermodynamics arises from the fundamentally reversible molecules. Part (b) reprinted with permission from Ref. [18]. laws of motion.[18] Feynman s device (Figure 4b) consists of a miniature axle with vanes attached to one end, surrounded by a gas at temperature T1. At the other end of the axle is a difficult (and temperature gradients cannot be maintained ratchet and pawl system, held at temperature T2. The question over molecular-scale distances, see Section 1.3), what this posed by the system is whether the random oscillations hypothetical construct provides is the first example of a produced by gas molecules bombarding the vanes can be plausible mechanism for a molecular motor—whereby the rectified by the ratchet and pawl so as to get net motion in one random thermal fluctuations characteristic of this size regime direction. Exactly analogous to Smoluchowski s trapdoor, are not fought against but instead are harnessed and rectified.
Feynman showed that if T1 = T2 then the ratchet and pawl The key ingredient is the external addition of energy to the cannot extract energy from the thermal bath to do work. system, not to generate motion but rather to continually or Feynman s major contribution from the perspective of cyclically drive the system away from equilibrium, thereby molecular machines, however, was to take the analysis one maintaining a thermally activated relaxation process that stage further: if such a system cannot use thermal energy to do directionally biases Brownian motion towards equilibrium.[26] work, what is required for it to do so? Feynman showed that This profound idea is the key to the design of molecular-level when the ratchet and pawl are cooler than the vanes (that is, systems that work through controling Brownian motion and is
T1 > T2) the system does indeed rectify thermal motions and expanded upon in Section 1.4. can do work (Feynman suggested lifting a hypothetical flea attached by a thread to the ratchet).[25] Of course, the machine 1.2.2. Machines That Operate at Low Reynolds Number now does not threaten the second law of thermodynamics as dissipation of heat into the gas reservoir of the ratchet and Whilst rectifying Brownian motion may provide the key to pawl occurs, so the temperature difference must be main- powering molecular-level machines, it tells us nothing about tained by some external means. Although insulating a how that power can be used to perform tasks at the nanoscale molecular-sized system from the outside environment is and what tiny mechanical machines can and cannot be
Angew. Chem. Int. Ed. 2007, 46, 72 – 191 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.angewandte.org 77 Angewandte Molecular Devices Chemie
Figure 4. a) Smoluchowski’s trapdoor: an “automatic” pressure demon (the directionally discriminating behavior is carried out by a wholly mechanical device, a trapdoor which is intended to open when Angewandte Molecular Devices hit from one direction but not the other).[16] Like the pressure demon Chemie shown in Figure 2b, Smoluchowski’s trapdoor aims to transport Figure 3. Szilard’s engine which utilizes a “pressure demon”.[17] particles selectively from the left compartment to the right. However, a) Initially a single Brownian particle occupies a cylinder with a piston in the absence of a mechanism whereby the trapdoor can dissipate at either end. A frictionless partition is put in place to divide the energy it will be at thermal equilibrium with its surroundings. This container into two compartments (a b). b) The demon then detects means it must spend much of its time open, unable to influence the ! the particle and determines in which compartment it resides. c) Using transport of particles. Rarely, it will be closed when a particle this information, the demon is able to move the opposite piston into approaches from the right and will open on collision with a particle position without meeting any resistance from the particle. d) The coming from the left, thus doing its job as intended. Such events are partition is removed, allowing the “gas” to expand against the piston, balanced, however, by the door snapping shut on a particle from the doing work against any attached load (e). To replenish the energy used right, pushing it into the left chamber. Overall, the probability of a by the piston and maintain a constant temperature, heat must flow particle moving from left to right is equal to that for moving right to Figure 4. a) Smoluchowski’s trapdoor: an “automatic” pressure into the system. To complete the thermodynamic cycle and reset the left and so the trapdoor cannot accomplish its intended function demon (the directionally discriminating behavior is carried out by a [18] wholly mechanical device, a trapdoor which is intended to open when machine, the demon’s memory of where the particle was must be adiabatically. b) Feynman’s ratchet and pawl. It might appear that hit from one direction but not the other).[16] Like the pressure demon erased (f a). To fully justify the application of a thermodynamic Brownian motion of the gas molecules on the paddle wheel in the shown in Figure 2b, Smoluchowski’s trapdoor aims to transport ! right-hand compartmentFigure 3. Szilard’s can engine do work which byutilizes exploiting a “pressure the demon”. asymmetry[17] of particles selectively from the left compartment to the right. However, concept such as entropy to a single-particle model, a population of in the absence of a mechanism whereby the trapdoor can dissipate the teeth on thea) Initially cog of a single the ratchet Brownian in particle the left-handoccupies a cylinder compartment. with a piston Szilard devices is required. The average for the ensemble over each of at either end. A frictionless partition is put in place to divide the energy it will be at thermal equilibrium with its surroundings. This these devices can then be considered to represent the state of the While the springcontainer holds into the two pawl compartments between (a theb). teeth b) The demon of the then cog, detects it does means it must spend much of its time open, unable to influence the ! indeed turn selectivelythe particle and in the determines desired in which direction. compartment However, it resides. when c) Using the transport of particles. Rarely, it will be closed when a particle system, which is comparable to the time average of a single multi- approaches from the right and will open on collision with a particle pawl is disengaged,this information, the cog the wheel demon need is able only to move randomly the opposite rotate piston a into tiny particle system at equilibrium, in a fashion similar to the statistical position without meeting any resistance from the particle. d) The coming from the left, thus doing its job as intended. Such events are mechanics derivation of thermodynamic quantities. amount in thepartition other isdirection removed, allowing to move the back “gas” to one expand tooth against whereas the piston, it mustbalanced, however, by the door snapping shut on a particle from the rotate randomlydoing a long work against way to any move attached to load the (e). next To replenish tooth forward. the energy Ifused the right, pushing it into the left chamber. Overall, the probability of a by the piston and maintain a constant temperature, heat must flow particle moving from left to right is equal to that for moving right to paddle wheel andinto the ratchet system. are To complete at the thesame thermodynamic temperature cycle (thatand reset is, theT1 =T2left) and so the trapdoor cannot accomplish its intended function [18] these rates cancelmachine, out. the However, demon’s memory if T1 ofT where2 then the particlethe system was must will be adiabatically. b) Feynman’s ratchet and pawl. It might appear that designed to illustrate how the irreversible second law of erased (f a). To fully justify the¼6 application of a thermodynamic Brownian motion of the gas molecules on the paddle wheel in the directionally rotate, driven! solely by the Brownian motion of the gas thermodynamics arises from the fundamentally reversible concept such as entropy to a single-particle model, a population of right-hand compartment can do work by exploiting the asymmetry of molecules. Part (b) reprinted with permission from Ref. [18]. the teeth on the cog of the ratchet in the left-hand compartment. [18] Szilard devices is required. The average for the ensemble over each of laws of motion. Feynman s device (Figure 4b) consists of a these devices can then be considered to represent the state of the While the spring holds the pawl between the teeth of the cog, it does miniature axle with vanes attached to one end, surrounded by system, which is comparable to the time average of a single multi- indeed turn selectively in the desired direction. However, when the particle system at equilibrium, in aCarter, fashion similar N. to J., the statistical& Cross, pawlR. isA. disengaged, (2005). the cogMechanics wheel need only randomlyof the rotate kinesin a tiny step. a gas at temperature T1. At the other end of the axle is a difficult (andmechanics temperature derivation of thermodynamic gradients quantities. cannot be maintainedamount in the other direction to move back one tooth whereas it must ratchet and pawl system, held at temperature T . The question over molecular-scale distances, seeNature Section, 435 1.3),(7040), what 308–312. thisrotate randomly" a long way to move to the next tooth forward. If the 2 paddle wheel and ratchet are at the same temperature (that is, T1 =T2) these rates cancel out. However, if T T then the system will posed by the system is whether the random oscillations hypothetical construct provides is the first example of a 1 ¼6 2 designed to illustrate how the irreversible second law of directionally rotate, driven solely by the Brownian motion of the gas produced by gas molecules bombarding the vanes can be plausible mechanismthermodynamics for arises a molecular from the fundamentally motor—whereby reversible themolecules. Part (b) reprinted with permission from Ref. [18]. rectified by the ratchet and pawl so as to get net motion in one random thermallaws of fluctuations motion.[18] Feynman characteristic s device (Figure of 4b) this consists size regime of a miniature axle with vanes attached to one end, surrounded by direction. Exactly analogous to Smoluchowski s trapdoor, are not fought against but instead are harnessed and rectified. a gas at temperature T1. At the other end of the axle is a difficult (and temperature gradients cannot be maintained Feynman showed that if T1 = T2 then the ratchet and pawl The key ingredientratchet and is pawl the system, external held at addition temperature ofT2. energy The question to theover molecular-scale distances, see Section 1.3), what this cannot extract energy from the thermal bath to do work. system, notposed to generate by the system motion is whether but rather the random to continually oscillations orhypothetical construct provides is the first example of a produced by gas molecules bombarding the vanes can be plausible mechanism for a molecular motor—whereby the Feynman s major contribution from the perspective of cyclically driverectified the by system the ratchet away and pawl from so as to equilibrium, get net motion in thereby one random thermal fluctuations characteristic of this size regime molecular machines, however, was to take the analysis one maintainingdirection. a thermally Exactly activatedanalogous to relaxation Smoluchowski processs trapdoor, thatare not fought against but instead are harnessed and rectified. T T [26] stage further: if such a system cannot use thermal energy to do directionallyFeynman biases showed Brownian that if motion1 = 2 then towards the ratchet equilibrium. and pawl The key ingredient is the external addition of energy to the cannot extract energy from the thermal bath to do work. system, not to generate motion but rather to continually or work, what is required for it to do so? Feynman showed that This profoundFeynman idea s is major the key contribution to the design from the of molecular-level perspective of cyclically drive the system away from equilibrium, thereby when the ratchet and pawl are cooler than the vanes (that is, systems thatmolecular work through machines, controling however, was Brownian to take the motion analysis one and ismaintaining a thermally activated relaxation process that [26] T > T ) the system does indeed rectify thermal motions and expanded uponstage further:in Section if such 1.4. a system cannot use thermal energy to do directionally biases Brownian motion towards equilibrium. 1 2 work, what is required for it to do so? Feynman showed that This profound idea is the key to the design of molecular-level can do work (Feynman suggested lifting a hypothetical flea when the ratchet and pawl are cooler than the vanes (that is, systems that work through controling Brownian motion and is [25] attached by a thread to the ratchet). Of course, the machine 1.2.2. MachinesT1 > ThatT2) the Operate system does at indeed Low Reynolds rectify thermal Number motions and expanded upon in Section 1.4. now does not threaten the second law of thermodynamics as can do work (Feynman suggested lifting a hypothetical flea attached by a thread to the ratchet).[25] Of course, the machine 1.2.2. Machines That Operate at Low Reynolds Number dissipation of heat into the gas reservoir of the ratchet and Whilst rectifyingnow does not Brownian threaten the motion second law may of thermodynamics provide the key as to pawl occurs, so the temperature difference must be main- powering molecular-leveldissipation of heat machines, into the gas reservoir it tells of us the nothing ratchet and about Whilst rectifying Brownian motion may provide the key to tained by some external means. Although insulating a how that powerpawl can occurs, be so used the temperature to perform difference tasks at must the be nanoscale main- powering molecular-level machines, it tells us nothing about tained by some external means. Although insulating a how that power can be used to perform tasks at the nanoscale molecular-sized system from the outside environment is and what tinymolecular-sized mechanical system machines from the outside can and environment cannot is beand what tiny mechanical machines can and cannot be
Angew. Chem. Int. Ed. 2007, 46, 72 – 191 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.angewandte.org 77 Angew. Chem. Int. Ed. 2007, 46, 72 – 191 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.angewandte.org 77 Reviews D. A. Leigh, F. Zerbetto, and E. R. Kay potential does not have to be regular. As long as the two sets of energy minima and maxima are repetitively switched in the same order, the particle will tend to be transported from left to right in Figure 7, even though occasionally it may move over a barrier in the wrong direction. The basic pulsating ratchet requirements can also be realized in another way. A potential such as that shown in Figure 7a can be given a directional drift velocity. Such systems are often termed “traveling potential” ratchets. This principle is essentially the same as macroscopic devices such as Archimedes screw, yet in the presence of thermal fluctuations these systems exhibit Brownian ratchet charac- teristics and are closely related to tilting the potential in one direction (as discussed in Section 1.4.2.2). Clearly, however, an asymmetric potential is not absolutely required, nor in fact are thermal fluctuations; imagine, for example, a particle “surfing” on a traveling wave on the surface of a liquid. This category of mechanism is at the boundary between fluctua- Figure 8. A temperature (or diffusion) ratchet.[39f] a) The Brownian tion-driven transport and transport as a result of potential particles start out in energy minima on the potential-energy surface @ gradients, with the precise location on this continuum with the energy barriers kBT1. b) The temperature is increased so that the height of the barriers is ! k T and effectively free diffusion is dependent on the importance of thermal fluctuations and B 2 allowed to occur for a short time period (much less than required to spatial asymmetry under the conditions chosen. In terms of reach global equilibrium). c) The temperature is lowered to T1 once chemical systems, the traveling-potential mechanism has most more, and the asymmetry of the potential means that each particle is relevance for the field-driven processes discussed in Section 5 statistically more likely to be captured by the adjacent well to the right and the self-propulsion mechanisms of Section 6, while in rather than the well to the left. d) Relaxation to the local energy Section 7 we shall discuss some situations in which the minimum (during which heat is emitted) leads to the average position balance between random fluctuations and external direc- of the particles moving to the right. Repeating steps (b)–(d), progres- sively moves the Brownian particles further and further to the right. tional driving force is shifted strongly towards the latter. The Note the similarities between this mechanism and that of the on–off traveling motion does not have to be continuous, but rather ratchet shown in Figure 6. may take place in discrete steps. Furthermore, arranging a continuously traveling potential to “jump” distances which Carter, N. J., & Cross, R. A. (2005). Mechanics of the kinesin step. are multiples of its period, at random or regular intervals, can therefore, it seems that applyingNature temperature, 435(7040), variations 308–312. to " be used to escape from the inherent directionality of the any process which involves an asymmetric potential-energy traveling-wave scheme and it has been shown that the ratchet surface could result in a ratchet effect.[47] In chemical terms, dynamics are essentially unaffected. In the limiting situation, this means that the rotation of a chiral group around a this can be reduced to dichotomous switching between two covalent bond can, at least in principle, be made unidirec- spatially shifted potentials which are otherwise identical, tional in such a manner. This concept is explored further in which is very similar to a rocking ratchet (see Section 1.4.2.2) relation to a molecular ratchet system in Section 2.1.2. and is also relevant to the unidirectional motors discussed in An unbiased driving force can also be achieved by Section 2.2. applying a directional force in a periodic manner so that, over time, the bias averages to zero, thus generating a 1.4.2.2. Tilting Ratchets[39f ] “rocking” ratchet. The simplest form of this is shown in Figure 9. Periodic application (to the left and then right) of a In this category, the underlying intrinsic potential remains driving force that allows the particle to surmount the barriers unchanged and the detailed balance is broken by application (for example, by applying an external field if the particle is of an unbiased driving force to the Brownian particle. Perhaps charged) results in transport. Motion over the steep barrier is the most apparent unbiased driving force is heat, and ratchet again most likely as it involves the shortest distance. Such a mechanisms based on periodic or stochastic temperature mechanism is physically equivalent to tilting the ratchet variations are generally termed “temperature” or “diffusion” potential in one direction then the other. Of course, if the ratchets. In its simplest form (Figure 8) this mechanism is very driving force is strong enough and is constantly applied in one similar to the on–off ratchet. Initially the thermal energy is direction or the other, thermal fluctuations are not necessary, low so that the particles cannot readily cross the energy which would then correspond to a power stroke. barriers. A sudden increase in temperature can be applied so Analogues of a rocking ratchet in which the applied that kB T is much greater that the potential amplitude causing driving force is a form of stochastic noise are known as the particles to diffuse as if over a virtually flat potential- “fluctuating force” ratchets (in certain cases, also “correla- energy surface (Figure 8b). Returning to the original, lower, tion” ratchets). Finally, a tilting ratchet can be achieved in a temperature (Figure 8c) is equivalent to turning the potential symmetric potential if the perturbation itself results in spatial back on in Figure 6c and more particles will have moved to asymmetry (becoming very similar to the travelling-potential the next well to the right than to the left. Under this scheme, models in Section 1.4.2.1). This may be rather clear for the
82 www.angewandte.org 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Angew. Chem. Int. Ed. 2007, 46, 72 – 191 Angewandte Molecular Devices Chemie into two basic types—pulsating ratchets and tilting ratchets— number of biological processes[42b] as well as being the basis [39f] Angewandte and are the subject of a recent major review byMolecular Reimann, Devices for a [2]catenane rotary motor (see Section 4.6.3), is illus- Chemie and information ratchets, which are much less common in the trated in Figure 7. It consists in physical terms of an physics literature, but have been discussedinto by two Parrondo, basic types—pulsatingasymmetric ratchets potential-energy and tilting ratchets— surfacenumber (comprising of biological a processes periodic[42b] as well as being the basis [39h,42b,45] Astumian, and others. Both energyand ratchets are the subject and of a recent major review by Reimann,[39f] for a [2]catenane rotary motor (see Section 4.6.3), is illus- information ratchets bias the movement ofand a information Brownian ratchets, which are much less common in the trated in Figure 7. It consists in physical terms of an substrate.[46] However, we will also show (Sectionsphysics 1.4.3 literature, and but have been discussed by Parrondo, asymmetric potential-energy surface (comprising a periodic Astumian, and others.[39h,42b,45] Both energy ratchets and 4.4) that they offer clues for how to go beyond ainformation simple switch ratchets bias the movement of a Brownian with a chemical machine to enable tasks tosubstrate. be performed[46] However, we will also show (Sections 1.4.3 and through the non-equilibrium control of conformational4.4) that they offerand clues for how to go beyond a simple switch co-conformational changes within molecular structures.with a chemical machine to enable tasks to be performed through the non-equilibrium control of conformational and co-conformational changes within molecular structures. 1.4.2.1. Pulsating Ratchets[39f] 1.4.2.1. Pulsating Ratchets[39f] Pulsating ratchets are a general category of energy ratchet Pulsating ratchets are a general category of energy ratchet in which potential-energy minima and maximain are which varied potential-energy in a minima and maxima are varied in a periodic or stochastic fashion, independent ofperiodic the position or stochastic of fashion, independent of the position of the particle on the potential-energy surface.the In its particle simplest on the potential-energy surface. In its simplest form this can be considered as an asymmetricform this sawtooth can be considered as an asymmetric sawtooth potential being repetitively turned on and off faster than potential being repetitively turned on and offBrownian faster particles than can diffuse over more than a small fraction Brownian particles can diffuse over more than aof small the fraction potential energy surface (an “on–off” ratchet, of the potential energy surface (an “on–off”Figure 6). ratchet, The result is net directional transport of the Figure 6). The result is net directional transportparticles across of the the surface (left to right in Figure 6). More general than the special case of an on–off ratchet, particles across the surface (left to right in Figureany 6).asymmetric periodic potential may be regularly or More general than the special case of an on–offstochastically ratchet, varied to give a ratchet effect (such mechanisms any asymmetric periodic potential may beare regularly generally termed or “fluctuating potential” ratchets). As stochastically varied to give a ratchet effect (suchwith mechanisms the simple on–off ratchet, most commonly encountered examples involve switching between two different potentials are generally termed “fluctuating potential”and ratchets). are therefore As often termed “flashing” ratchets. A classic with the simple on–off ratchet, most commonlyexample, encountered which has particular relevance for explaining a examples involve switching between two different potentials and are therefore often termed “flashing” ratchets. A classic example, which has particular relevance for explaining a
Figure 7. Another example of a pulsating ratchet mechanism—a flash- ing ratchet.[42b] For details of its operation, see the text.
series of two different minima and two different maxima) along which a Brownian particle is directionally transported by sequentially raising and lowering each set of minima and An example of a pulsating ratchet mechanism—an on–off Figure 6. Figure 7. Another example of a pulsatingmaxima. ratchet The mechanism—a particle starts flash- in a green or orange well ratchet.[39f] a) The Brownian particles start out in energy minima on the ing ratchet.[42b] For details of its operation,(Figure see 7a the or text. c). Raising that energy minimum while lowering potential-energy surface with the energy barriers @ kBT. b) The poten- tial is turned off so that free Brownian motion powered diffusion is those in adjacent wells provides the impetus for the particle to allowed to occur for a short time period (much less than required to change position by Brownian motion (Figure 7b 7c or 7d ! ! reach global equilibrium). c)series On turning of thetwo potential different back on minima again, and7e). By two simultaneously different maxima) (or beforehand) changing the relative the asymmetry of the potentialalong means which that the a particles Brownian have a greater particleheights is directionally of the energy transported barrier to the next energy well, the probability of being trapped in the adjacent well to the right rather kinetics of the Brownian motion in each direction are than the adjacent well to theby left. sequentially Note this step involves raising raising and the lowering each set of minima and An example of a pulsating ratchet mechanism—an on–off different and the particle is transported from left to right. Figure 6. energy of the particles. d) Relaxationmaxima. to the The local energyparticle minima starts in a green or orange well ratchet.[39f] a) The Brownian particles start out in energy(during minima which on heat the is emitted) leads to the average position of the Note that the position of the particle does not influence the particles moving to the right.(Figure Repeating 7a steps or (b)–(d)c). Raising progressively that energysequence minimum in which while (or lowering when, or if) the energy minima and potential-energy surface with the energy barriers @ kBT. b) The poten- moves the Brownian particles further and further to the right. maxima are changed. Furthermore, the switching of the tial is turned off so that free Brownian motion powered diffusion is those in adjacent wells provides the impetus for the particle to allowed to occur for a short time period (muchCarter, less than N. J., required & Cross, to R. A. (2005).change Mechanics position of the by kinesin Brownian step. motion (Figure 7b 7c or 7d Angew. Chem. Int. Ed. 2007, 46, 72 – 191 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim! ! www.angewandte.org 81 reach global equilibrium). c) On turning the potentialNature, back435(7040), on again, 308–312." 7e). By simultaneously (or beforehand) changing the relative the asymmetry of the potential means that the particles have a greater heights of the energy barrier to the next energy well, the probability of being trapped in the adjacent well to the right rather kinetics of the Brownian motion in each direction are than the adjacent well to the left. Note this step involves raising the different and the particle is transported from left to right. energy of the particles. d) Relaxation to the local energy minima (during which heat is emitted) leads to the average position of the Note that the position of the particle does not influence the particles moving to the right. Repeating steps (b)–(d) progressively sequence in which (or when, or if) the energy minima and moves the Brownian particles further and further to the right. maxima are changed. Furthermore, the switching of the
Angew. Chem. Int. Ed. 2007, 46, 72 – 191 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.angewandte.org 81 Angewandte Molecular Devices Chemie into two basic types—pulsating ratchets and tilting ratchets— number of biological processes[42b] as well as being the basis and are the subject of a recent major review by Reimann,[39f] for a [2]catenane rotary motor (see Section 4.6.3), is illus- and information ratchets, which are much less common in the trated in Figure 7. It consists in physical terms of an physics literature, but have been discussed by Parrondo, asymmetric potential-energy surface (comprising a periodic Astumian, and others.[39h,42b,45] Both energy ratchets and information ratchets bias the movement of a Brownian substrate.[46] However, we will also show (Sections 1.4.3 and 4.4) that they offer clues for how to go beyond a simple switch with a chemical machine to enable tasks to be performed through the non-equilibrium control of conformational and co-conformational changes within molecular structures.
1.4.2.1. Pulsating Ratchets[39f]
Pulsating ratchets are a general category of energy ratchet in which potential-energy minima and maxima are varied in a periodic or stochastic fashion, independent of the position of the particle on the potential-energy surface. In its simplest form this can be considered as an asymmetric sawtooth potential being repetitively turned on and off faster than Brownian particles can diffuse over more than a small fraction of the potential energy surface (an “on–off” ratchet, Figure 6). The result is net directional transport of the particles across the surface (left to right in Figure 6). More general than the special case of an on–off ratchet, any asymmetric periodic potential may be regularly or stochastically varied to give a ratchet effect (such mechanisms are generally termed “fluctuating potential” ratchets). As with the simple on–off ratchet, most commonly encountered examples involve switching between two different potentials and are therefore often termed “flashing” ratchets. A classic example, which has particular relevance for explaining a
Figure 7. Another example of a pulsating ratchet mechanism—a flash- ing ratchet.[42b] For details of its operation, see the text. Carter, N. J., & Cross, R. A. (2005). Mechanics of the kinesin step. series of two different minima and two differentNature maxima), 435(7040), 308–312." along which a Brownian particle is directionally transported by sequentially raising and lowering each set of minima and An example of a pulsating ratchet mechanism—an on–off Figure 6. maxima. The particle starts in a green or orange well ratchet.[39f] a) The Brownian particles start out in energy minima on the (Figure 7a or c). Raising that energy minimum while lowering potential-energy surface with the energy barriers @ kBT. b) The poten- tial is turned off so that free Brownian motion powered diffusion is those in adjacent wells provides the impetus for the particle to allowed to occur for a short time period (much less than required to change position by Brownian motion (Figure 7b 7c or 7d ! ! reach global equilibrium). c) On turning the potential back on again, 7e). By simultaneously (or beforehand) changing the relative the asymmetry of the potential means that the particles have a greater heights of the energy barrier to the next energy well, the probability of being trapped in the adjacent well to the right rather kinetics of the Brownian motion in each direction are than the adjacent well to the left. Note this step involves raising the different and the particle is transported from left to right. energy of the particles. d) Relaxation to the local energy minima (during which heat is emitted) leads to the average position of the Note that the position of the particle does not influence the particles moving to the right. Repeating steps (b)–(d) progressively sequence in which (or when, or if) the energy minima and moves the Brownian particles further and further to the right. maxima are changed. Furthermore, the switching of the
Angew. Chem. Int. Ed. 2007, 46, 72 – 191 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.angewandte.org 81 Angewandte Molecular Devices Chemie
Figure 9. A rocking ratchet.[39f] a) The Brownian particles start out in energy minima on the potential-energy surface with the energy barriers
@ kBT. b) A directional force is applied to the left. c) An equal and opposite directional force is applied to the right. d) Removal of the force and relaxation to the local energy minimum leads to the average position of the particles moving to the right. Repeating steps (b)–(d) progressively moves the Brownian particles further and further to the right.
Carter, N. J., & Cross, R. A. (2005).Figure Mechanics 10. A type of information of the kinesin ratchet mechanism step. for transport of a Brownian particle along a potential-energy surface.[39h,42b,45] Dotted periodic force cases (imagine applying the electric field Nature, 435(7040), 308–312." arrows indicate the transfer of information that signals the position of discussed above for longer in one direction than the other), the particle. If the signal is distance-dependent—say, energy transfer but is less so for stochastic driving forces. In general, these from an excited state which causes lowering of an energy barrier— mechanisms are known as “asymmetrically tilting” ratchets. then the asymmetry in the particle’s position between two barriers provides the “information” which transports the particle directionally 1.4.2.3. Information Ratchets[39h,42b,45] along the potential energy surface.
In the pulsating and tilting types of energy ratchet mechanisms, perturbations of the potential-energy surface— or of the particle s interaction with it—are applied globally It appears to us that information-ratchet mechanisms of and independent of the particle s position, while the perio- relevance to chemical systems can arise in at least three ways: dicity of the potential is unchanged. Information ratchets 1) a localized change to the intrinsic potential-energy surface (Figure 10) transport a Brownian particle by changing the depending on the position of the particle (Figure 10); 2) a effective kinetic barriers to Brownian motion depending on position-dependent change in the state of the particle which the position of the particle on the surface. In other words, the alters its interaction with the potential-energy surface at that heights of the maxima on the potential-energy surface change point; or 3) switching between two different intrinsic periodic according to the location of the particle (this requires potentials according to the position of the particle.[39h,49] An information to be transferred from the particle to the surface) example of the first of these types, in which the system whereas the potential-energy minima do not necessarily need responds to the “information” from the particle by lowering to change at all. This switching does not require raising the the energy barrier to the right-hand side (and only to the potential energy of the particle at any stage, rather the motion right-hand side) of the particle, is shown in Figure 10. can be powered with energy taken entirely from the thermal The particle starts in one of the identical-minima energy bath by using information about the position of the particle. wells (Figure 10a). The position of the particle lowers the This is directly analogous to the mechanism required of kinetic barrier for passage to the adjacent right-hand well and Maxwell s pressure demon (Figure 2b, Section 1.2.1.1), but it moves there by Brownian motion (10b 10c). At this point ! does not break the second law of thermodynamics as the it can sample two energy wells by Brownian motion, and a required information transfer (actually, information era- random reinstatement of the barrier has a 50% chance of sure[48]) has an intrinsic energy cost that has to be met returning the particle to its starting position and a 50% externally. chance of trapping it in the newly accessed well to the right
Angew. Chem. Int. Ed. 2007, 46, 72 – 191 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.angewandte.org 83 Angewandte Molecular Devices Chemie
Figure 9. A rocking ratchet.[39f] a) The Brownian particles start out in energy minima on the potential-energy surface with the energy barriers
@ kBT. b) A directional force is applied to the left. c) An equal and opposite directional force is applied to the right. d) Removal of the force and relaxation to the local energy minimum leads to the average position of the particles moving to the right. Repeating steps (b)–(d) progressively moves the Brownian particles further and further to the right.
Figure 10. A type of information ratchet mechanism for transport of a Brownian particle along a potential-energy surface.[39h,42b,45] Dotted periodic force cases (imagine applying the electric field arrows indicate the transfer of information that signals the position of discussed above for longer in one direction than the other), the particle. If the signal is distance-dependent—say, energy transfer but is less so for stochastic driving forces. In general, these from an excited state which causes lowering of an energy barrier— mechanisms are known as “asymmetrically tilting” ratchets. then the asymmetry in the particle’s position between two barriers provides the “information” which transports the particle directionally 1.4.2.3. Information Ratchets[39h,42b,45] along the potential energy surface. Carter, N. J., & Cross, R. A. (2005). Mechanics of the kinesin step. Nature, 435(7040), 308–312." In the pulsating and tilting types of energy ratchet mechanisms, perturbations of the potential-energy surface— or of the particle s interaction with it—are applied globally It appears to us that information-ratchet mechanisms of and independent of the particle s position, while the perio- relevance to chemical systems can arise in at least three ways: dicity of the potential is unchanged. Information ratchets 1) a localized change to the intrinsic potential-energy surface (Figure 10) transport a Brownian particle by changing the depending on the position of the particle (Figure 10); 2) a effective kinetic barriers to Brownian motion depending on position-dependent change in the state of the particle which the position of the particle on the surface. In other words, the alters its interaction with the potential-energy surface at that heights of the maxima on the potential-energy surface change point; or 3) switching between two different intrinsic periodic according to the location of the particle (this requires potentials according to the position of the particle.[39h,49] An information to be transferred from the particle to the surface) example of the first of these types, in which the system whereas the potential-energy minima do not necessarily need responds to the “information” from the particle by lowering to change at all. This switching does not require raising the the energy barrier to the right-hand side (and only to the potential energy of the particle at any stage, rather the motion right-hand side) of the particle, is shown in Figure 10. can be powered with energy taken entirely from the thermal The particle starts in one of the identical-minima energy bath by using information about the position of the particle. wells (Figure 10a). The position of the particle lowers the This is directly analogous to the mechanism required of kinetic barrier for passage to the adjacent right-hand well and Maxwell s pressure demon (Figure 2b, Section 1.2.1.1), but it moves there by Brownian motion (10b 10c). At this point ! does not break the second law of thermodynamics as the it can sample two energy wells by Brownian motion, and a required information transfer (actually, information era- random reinstatement of the barrier has a 50% chance of sure[48]) has an intrinsic energy cost that has to be met returning the particle to its starting position and a 50% externally. chance of trapping it in the newly accessed well to the right
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tures can be created, controlled, ordered, and manipulated by inputs of energy.
These are the principles used to develop the concepts for the compartmentalized molecular machines described in Section 4.4. However, it is interesting to consider that they are equally relevant for the design of interacting chemical reaction cascades and catalytic cycles that operate far from equilibrium. If biology, mathematics, and physics provide the inspira- tion and strategies for controlling molecular-level motion, it is through chemistry that artificial molecular-level machine mechanisms must be designed, constructed, and made to work. The minimum requirements for such systems must be the restriction of the 3D motion of the machine components and/or the substrate as well as a change in their relative positions induced by an input of energy. Ways of achieving control over conformational dynamics involving single bonds and double bonds, as well as over co-conformational dynam- ics in kinetically stable supramolecular and mechanically bonded systems, are discussed in Sections 2–7.
2. Controlling Motion in CovalentlyAngewandte Bonded Molecular Devices Molecular Systems Chemie
2.1. Controlling Conformational Changes 2.1.1. Correlated Motion through Nonbonded Interactions Scheme 1. a) The three residual diastereoisomers of molecular gear tures can be created, controlled, ordered, and manipulated 1.[57b] Correlated disrotation maintains the phase relationship between by inputs of energy. In compounds where two or more bulky planar substitu- the labeled blades (shown here in red) for each isomer. Interconver- sion between isomers (“gear slippage”) requires correlated conrotation ents are geminally or vicinally connected, the congested steric or uncorrelated rotation. b) Molecular gear train 2.[60] c) Vinyl molec- These are the principles used to develop the concepts for environment often demands a helical ground state where all ular gear 3 (shown as the racemic residual diastereomer);[61] a the rings are tilted in the same direction, thus generating a molecular gear 4 with a 2:3 gearing ratio;[62j] and a molecular gear 5 the compartmentalized molecular machines described in structure reminiscent of the macroscopic screw propellers based on revolution around a metallocene and with a 3:4 gearing [63] Section 4.4. However, it is interesting to consider that they found on aeroplanes and boats. The lowest energy rotational ratio. The arrows show correlated motions but are not meant to processes in such systems generally involve correlated motion imply intrinsic directionality. are equally relevant for the design of interacting chemical of the planar “blades” (known as “cogwheeling”), and their reaction cascades and catalytic cycles that operate far from investigation played an important early role in illustrating equilibrium. how motion can be controlled by molecular structure.[53] 1 Inspired by the work of O¯ ki on the high threefold torsional processes by 30–40 kcalmolÀ in these systems, largely If biology, mathematics, and physics provide the inspira- barrier in bridgehead-substituted triptycenes,[54] the research because of the remarkable ease of the disrotary motion ° 1 tion and strategies for controlling molecular-level motion, it is groups of Mislow and Iwamura independently replaced the (DG = ca. 1–2 kcalmolÀ ). This result actually represents through chemistry that artificial molecular-level machine twofold aryl “rotators”[1k] of molecular propellers with 9- stricter selectivity than occurs for the selection rules that triptycyl units, thereby creating so-called “molecular govern correlated torsions under orbital symmetry control mechanisms must be designed, constructed, and made to gears”.[53c,55–57] In these systems, the blades of each triptycyl (see Section 2.1.5). work. The minimum requirements for such systems must be group are tightly intermeshed, so that correlated disrotatory Iwamura and co-workers successfully extended the the restriction of the 3D motion of the machine components motion is strongly preferred. Such “dynamic gearing” mirrors dynamic gearing concept to multiple gear systems in which the operation of a macroscopic bevel gear,[58] with correlated each adjacent pair must disrotate, so that the behavior of the and/or the substrate as well as a change in their relative conrotation or uncorrelated rotation amounting to gear outermost rotators is dependent of the number of linkages in positions induced by an input of energy. Ways of achieving slippage. This threshold mechanism maintains the relation- the chain. In 2 therefore (Scheme 1b), despite the large control over conformational dynamics involving single bonds ship between torsional angles of the two rotators. Changing number of possible conformations and increased flexibility of this relationship can only occur by one of the higher energy the chain, the two outer triptycyl units conrotate and the and double bonds, as well as over co-conformational dynam- rotational processes, so that stereoisomerism arises when at phase relationship of the two chlorine labels is strictly ics in kinetically stable supramolecular and mechanically least one blade of each triptycyl is differentiable (illustrated maintained in both phase isomers.[60] [59] bonded systems, are discussed in Sections 2–7. for 1, Scheme 1a). As the isomers differ in the phase Dynamic gearing has also been realized in vicinally linked relationship between the substituted rings, the term “phase triptycyl rotors, such as 3 (Scheme 1c),[61] and in systems with isomers” was coined for this subset of residual isomerism.[56c] gearing ratios other than 3:3, for example, 2:3 (4),[62] and 3:4 Experimental and theoretical studies confirm that the corre- (5).[63] Although the low-energy dynamic processes observed 2. Controlling Motion in Covalently Bonded lated disrotation is generally favored over other torsional in 5 could not be unequivocally shown to be correlated, this
Molecular Systems Angew. Chem. Int. Ed. 2007, 46, 72 – 191 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.angewandte.org 85
2.1. Controlling Conformational Changes 2.1.1. Correlated Motion through Nonbonded Interactions Scheme 1. a) The three residual diastereoisomers of molecular gear 1.[57b] Correlated disrotation maintains theCarter, phase relationshipN. J., & betweenCross, R. A. (2005). Mechanics of the kinesin step. In compounds where two or more bulky planar substitu- the labeled blades (shown here in red) for each isomer. Interconver- sion between isomers (“gear slippage”)Nature requires, correlated 435(7040), conrotation 308–312." ents are geminally or vicinally connected, the congested steric or uncorrelated rotation. b) Molecular gear train 2.[60] c) Vinyl molec- environment often demands a helical ground state where all ular gear 3 (shown as the racemic residual diastereomer);[61] a the rings are tilted in the same direction, thus generating a molecular gear 4 with a 2:3 gearing ratio;[62j] and a molecular gear 5 structure reminiscent of the macroscopic screw propellers based on revolution around a metallocene and with a 3:4 gearing found on aeroplanes and boats. The lowest energy rotational ratio.[63] The arrows show correlated motions but are not meant to processes in such systems generally involve correlated motion imply intrinsic directionality. of the planar “blades” (known as “cogwheeling”), and their investigation played an important early role in illustrating how motion can be controlled by molecular structure.[53] 1 Inspired by the work of O¯ ki on the high threefold torsional processes by 30–40 kcalmolÀ in these systems, largely barrier in bridgehead-substituted triptycenes,[54] the research because of the remarkable ease of the disrotary motion ° 1 groups of Mislow and Iwamura independently replaced the (DG = ca. 1–2 kcalmolÀ ). This result actually represents twofold aryl “rotators”[1k] of molecular propellers with 9- stricter selectivity than occurs for the selection rules that triptycyl units, thereby creating so-called “molecular govern correlated torsions under orbital symmetry control gears”.[53c,55–57] In these systems, the blades of each triptycyl (see Section 2.1.5). group are tightly intermeshed, so that correlated disrotatory Iwamura and co-workers successfully extended the motion is strongly preferred. Such “dynamic gearing” mirrors dynamic gearing concept to multiple gear systems in which the operation of a macroscopic bevel gear,[58] with correlated each adjacent pair must disrotate, so that the behavior of the conrotation or uncorrelated rotation amounting to gear outermost rotators is dependent of the number of linkages in slippage. This threshold mechanism maintains the relation- the chain. In 2 therefore (Scheme 1b), despite the large ship between torsional angles of the two rotators. Changing number of possible conformations and increased flexibility of this relationship can only occur by one of the higher energy the chain, the two outer triptycyl units conrotate and the rotational processes, so that stereoisomerism arises when at phase relationship of the two chlorine labels is strictly least one blade of each triptycyl is differentiable (illustrated maintained in both phase isomers.[60] for 1, Scheme 1a).[59] As the isomers differ in the phase Dynamic gearing has also been realized in vicinally linked relationship between the substituted rings, the term “phase triptycyl rotors, such as 3 (Scheme 1c),[61] and in systems with isomers” was coined for this subset of residual isomerism.[56c] gearing ratios other than 3:3, for example, 2:3 (4),[62] and 3:4 Experimental and theoretical studies confirm that the corre- (5).[63] Although the low-energy dynamic processes observed lated disrotation is generally favored over other torsional in 5 could not be unequivocally shown to be correlated, this
Angew. Chem. Int. Ed. 2007, 46, 72 – 191 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.angewandte.org 85 Reviews D. A. Leigh, F. Zerbetto, and E. R. Kay system is one of the first molecular structures in which the facile rotation of p fragments in metal- locenes was utilized, thus prompting analogy with a ball bearing.[64] The same comparison can be applied to a more recent series of trinuclear sandwich complexes in which multiple correlated motions have been dem- onstrated.[65] Each silver cation in [Ag3(6)2] has a linear coordination geom- etry, bound to one nitro- gen donor from each tris- monodentate disk-shaped ligand (red)—the result- ing complex is helical, with all the thiazolyl rings tilted in the same direction (Figure 11b). Random helix inversion is a low-energy process, which occurs through a Figure 11. a) Chemical structure of tris-monodentate disk-shaped ligand 6 and hexakis-monodentate ligand [65] nondissociative “flip” 7. b) Schematic representation of complex [Ag3(6)2] arbitrarily shown as its M-helical enantiomer. c) Description of the two correlated rotation processes occurring in [Ag (6)(7)]. The direction of rotation for motion whereby rotation 3 an M P transition through ligand exchange is opposite to that for the potentially subsequent P M’ ! ! of the heteroaromatic transition by the nondissociative flip mechanism. Schematic representations reprinted with permission from rings so that they point Refs. [65a,c]. in the opposite direction Carter, N. J., & Cross, R. A. (2005). Mechanics of the kinesin step. is accompanied by a 1208 Nature, 435(7040), 308–312." relative rotation of the two large disks (illustrated for 2.1.2. Stimuli-Induced Conformational Control around a Single [65a] [Ag3(6)(7)] in Figure 11c). In the heteroleptic analogue Covalent Bond
[Ag3(6)(7)], a similar coordination geometry is adopted so that only every alternate thiazolyl ring in the hexakis- As a first step towards achieving controlled and externally monodentate disk ligand (blue) is bound at any one time. initiated rotation around C C single bonds, Kelly and co- À The “flip” helix reversal, during which the ligand and metal workers combined triptycene structures with a molecular- partners do not change, can clearly still occur. Ligand recognition event.[67] In the resulting “molecular brake” 8 exchange is also rapid, however, and most likely occurs via the trigonal transition state illustrated in Figure 11c. The overall result is a correlated rotation of the heteroaromatic rings together with a 608 relative rotation of the two disks. If a ligand exchange and a flip step occur concurrently, the sense of rotation in each step is opposite, so that overall a 608 relative rotation of the disk-shaped ligands occurs.[65b,c] While all these and other[66] related studies clearly demonstrate the role steric interactions can play, at equilib- rium the submolecular motions are nondirectional even within a partial rotational event. Simply restricting the thermal rotary motion of one unit by a larger blocking group or by the similarly random motion of another unit cannot, in itself, lead to directionality. A molecular machine requires some form of external modulation over the dynamic processes to drive the system away from equilibrium and break detailed balance. Scheme 2. “Molecular brakes” induced by a) metal-ion binding[68] and b) redox chemistry.[70] EDTA = ethylenediaminetetraacetate, mCPBA = meta-chloroperbenzoic acid.
86 www.angewandte.org 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Angew. Chem. Int. Ed. 2007, 46, 72 – 191 Reviews D. A. Leigh, F. Zerbetto, and E. R. Kay system is one of the first molecular structures in which the facile rotation of p fragments in metal- locenes was utilized, thus prompting analogy with a ball bearing.[64] The same comparison can be applied to a more recent series of trinuclear sandwich complexes in which multiple correlated motions have been dem- onstrated.[65] Each silver cation in [Ag3(6)2] has a linear coordination geom- etry, bound to one nitro- gen donor from each tris- monodentate disk-shaped ligand (red)—the result- ing complex is helical, with all the thiazolyl rings tilted in the same direction (Figure 11b). Random helix inversion is a low-energy process, which occurs through a Figure 11. a) Chemical structure of tris-monodentate disk-shaped ligand 6 and hexakis-monodentate ligand [65] nondissociative “flip” 7. b) Schematic representation of complex [Ag3(6)2] arbitrarily shown as its M-helical enantiomer. c) Description of the two correlated rotation processes occurring in [Ag (6)(7)]. The direction of rotation for motion whereby rotation 3 an M P transition through ligand exchange is opposite to that for the potentially subsequent P M’ ! ! of the heteroaromatic transition by the nondissociative flip mechanism. Schematic representations reprinted with permission from rings so that they point Refs. [65a,c]. in the opposite direction is accompanied by a 1208 relative rotation of the two large disks (illustrated for 2.1.2. Stimuli-Induced Conformational Control around a Single [65a] [Ag3(6)(7)] in Figure 11c). In the heteroleptic analogue Covalent Bond
[Ag3(6)(7)], a similar coordination geometry is adopted so that only every alternate thiazolyl ring in the hexakis- As a first step towards achieving controlled and externally monodentate disk ligand (blue) is bound at any one time. initiated rotation around C C single bonds, Kelly and co- À The “flip” helix reversal, during which the ligand and metal workers combined triptycene structures with a molecular- partners do not change, can clearly still occur. Ligand recognition event.[67] In the resulting “molecular brake” 8 exchange is also rapid, however, and most likely occurs via the trigonal transition state illustrated in Figure 11c. The overall result is a correlated rotation of the heteroaromatic rings together with a 608 relative rotation of the two disks. If a ligand exchange and a flip step occur concurrently, the sense of rotation in each step is opposite, so that overall a 608 relative rotation of the disk-shaped ligands occurs.[65b,c] While all these and other[66] related studies clearly demonstrate the role steric interactions can play, at equilib- rium the submolecular motions are nondirectional even within a partial rotational event. Simply restricting the thermal rotary motion of one unit by a larger blocking group or by the similarly random motion of another unit cannot, in itself, lead to directionality. A molecular machine requires some form of external modulation over the dynamic processes to drive the system away from equilibrium and break detailed balance. Scheme 2. “Molecular brakes” induced by a) metal-ion binding[68] and b) redox chemistry.[70] EDTA = ethylenediaminetetraacetate, mCPBA = meta-chloroperbenzoic acid.
86 www.angewandte.org 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Angew. Chem. Int. Ed. 2007, 46, 72 – 191 Carter, N. J., & Cross, R. A. (2005). Mechanics of the kinesin step. Nature, 435(7040), 308–312." Angewandte Molecular Devices Chemie
(Scheme 2a),[68] free rotation of a triptycyl group is halted by rotation in chemical structures does not have to be compli- the conformational change brought about by complexation of cated), thus constituting a temperature ratchet.[75] However, the appended bipyridyl unit with Hg2+ ions, thus effectively proving this experimentally by quantifying the net rotation putting a “stick” in the “spokes”.[69] A similar “braking effect” appears to be nontrivial. As a different solution, Kelly et al. has been demonstrated by using redox chemistry and N- proposed a modified version of the ratchet structure (11a, arylindolinone 9 (Scheme 2b); both oxidized forms of the Scheme 3), in which a chemical reaction is used as the source sulfur side chain exhibit markedly increased barriers to of energy.[76] If the amino group is ignored, all three energy rotation around the N aryl bond (see À also 45, Scheme 21).[70, 71] Kelly et al. next extended their investigation of restricted Brownian rotary motion to a molecular realiza- tion (10) of the Feynman adiabatic ratchet and pawl,[72] in which a helicene plays the role of the pawl in attempting to direct the rotation of the attached triptycene “cog wheel” in one direction as a result of the chiral helical structure. Although the calculated energetics for rotation showed an asymmetric poten- tial energy profile (Figure 12b), 1H NMR experiments confirmed that rota- tion occurred with equal frequency in both directions. This result is, of course, in line with the conclusions of the Feynman thought experiment (Sec- tion 1.2.1.4).[73] The rate of a molecular transformation (clockwise and anti- Scheme 3. A chemically powered unidirectional rotor.[76] Priming of the rotor in its initial state with clockwise rotation in 10 included) phosgene (11a 12a) allows a chemical reaction to take place when the helicene rotates far enough up ! depends on the energy of the transition its potential well towards the blocking triptycene arm (12b). This gives a tethered state 13a, for which state (and the temperature), not the rotation over the barrier to 13b is an exergonic process that occurs under thermal activation. Finally, the shape of the energy barrier: state urethane linker can be cleaved to give the original molecule with the components rotated by 1208 with functions such as enthalpy and free respect to each other (11b). energy do not depend on a system s history.[74] Thus, although rotation in 10 follows an asymmetric potential-energy surface, the principle minima for the position of the helicene with respect to the of detailed balance at equilibrium requires that transitions in triptycene “teeth” are identical: the energy profile for 3608 each direction occur at equal rates.[75] rotation would appear as three equal energy minima, The essential element missing from 10 needed to turn the separated by equal barriers. As the helicene oscillates back triptycene directionally is some form of energy input to drive and forth in a trough, however, sometimes it will come close it away from equilibrium and break the detailed balance. In enough to the amine for a chemical reaction to occur (as in principle, this could be achieved rather simply for 10 by a 12b). Priming the system with a chemical “fuel” (phosgene in periodic fluctuation in temperature (causing directional this case to give the isocyanate 12a) results in “ratcheting” of the motion some way up the energy barrier (13a). Continu- ation of the rotation in the same direction, over the energy barrier, can occur under thermal control and is now an exergonic process (giving 13b) before cleavage of the urethane gives the system rotated by 1208 (11b). Although the current version of the system can only carry out one third of a full rotation, it demonstrates the principles required for a fully operating and cyclable rotary system under chemical control and represents a major advance in the experimental realization of molecular-level machines. Exemplified by the motor proteins from nature, contin- ually operating (autonomous) chemically powered molecular motors may be classified as a subset of catalysts—catalyzing Figure 12. a) Kelly’s molecular realization (10) of Feynman’s adiabatic ratchet and pawl which does not rotate directionally at equilibrium.[72] the conversion of high-energy “fuel” molecules into lower b) Schematic representation of the calculated enthalpy changes for energy products while undergoing a conformational change rotation around the single degree of internal rotational freedom in 10. but returning to the starting state on completion of each cycle.
Angew. Chem. Int. Ed. 2007, 46, 72 – 191 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.angewandte.org 87
Carter, N. J., & Cross, R. A. (2005). Mechanics of the kinesin step. Nature, 435(7040), 308–312." Angewandte Molecular Devices Chemie
(Scheme 2a),[68] free rotation of a triptycyl group is halted by rotation in chemical structures does not have to be compli- the conformational change brought about by complexation of cated), thus constituting a temperature ratchet.[75] However, the appended bipyridyl unit with Hg2+ ions, thus effectively proving this experimentally by quantifying the net rotation putting a “stick” in the “spokes”.[69] A similar “braking effect” appears to be nontrivial. As a different solution, Kelly et al. has been demonstrated by using redox chemistry and N- proposed a modified version of the ratchet structure (11a, arylindolinone 9 (Scheme 2b); both oxidized forms of the Scheme 3), in which a chemical reaction is used as the source sulfur side chain exhibit markedly increased barriers to of energy.[76] If the amino group is ignored, all three energy rotation around the N aryl bond (see À also 45, Scheme 21).[70, 71] Kelly et al. next extended their investigation of restricted Brownian rotary motion to a molecular realiza- tion (10) of the Feynman adiabatic ratchet and pawl,[72] in which a helicene plays the role of the pawl in attempting to direct the rotation of the attached triptycene “cog wheel” in one direction as a result of the chiral helical structure. Although the calculated energetics for rotation showed an asymmetric poten- tial energy profile (Figure 12b), 1H NMR experiments confirmed that rota- tion occurred with equal frequency in both directions. This result is, of course, in line with the conclusions of the Feynman thought experiment (Sec- tion 1.2.1.4).[73] The rate of a molecular transformation (clockwise and anti- Scheme 3. A chemically powered unidirectional rotor.[76] Priming of the rotor in its initial state with clockwise rotation in 10 included) phosgene (11a 12a) allows a chemical reaction to take place when the helicene rotates far enough up ! depends on the energy of the transition its potential well towards the blocking triptycene arm (12b). This gives a tethered state 13a, for which state (and the temperature), not the rotation over the barrier to 13b is an exergonic process that occurs under thermal activation. Finally, the shape of the energy barrier: state urethane linker can be cleaved to give the original molecule with the components rotated by 1208 with functions such as enthalpy and free respect to each other (11b). energy do not depend on a system s history.[74] Thus, although rotation in 10 follows an asymmetric potential-energy surface, the principle Carter,minima N. for J., the & position Cross, ofR. the A. helicene(2005). with Mechanics respect to of the the kinesin step. of detailed balance at equilibrium requires that transitions in Naturetriptycene, 435 “teeth”(7040), are 308–312. identical: the" energy profile for 3608 each direction occur at equal rates.[75] rotation would appear as three equal energy minima, The essential element missing from 10 needed to turn the separated by equal barriers. As the helicene oscillates back triptycene directionally is some form of energy input to drive and forth in a trough, however, sometimes it will come close it away from equilibrium and break the detailed balance. In enough to the amine for a chemical reaction to occur (as in principle, this could be achieved rather simply for 10 by a 12b). Priming the system with a chemical “fuel” (phosgene in periodic fluctuation in temperature (causing directional this case to give the isocyanate 12a) results in “ratcheting” of the motion some way up the energy barrier (13a). Continu- ation of the rotation in the same direction, over the energy barrier, can occur under thermal control and is now an exergonic process (giving 13b) before cleavage of the urethane gives the system rotated by 1208 (11b). Although the current version of the system can only carry out one third of a full rotation, it demonstrates the principles required for a fully operating and cyclable rotary system under chemical control and represents a major advance in the experimental realization of molecular-level machines. Exemplified by the motor proteins from nature, contin- ually operating (autonomous) chemically powered molecular motors may be classified as a subset of catalysts—catalyzing Figure 12. a) Kelly’s molecular realization (10) of Feynman’s adiabatic ratchet and pawl which does not rotate directionally at equilibrium.[72] the conversion of high-energy “fuel” molecules into lower b) Schematic representation of the calculated enthalpy changes for energy products while undergoing a conformational change rotation around the single degree of internal rotational freedom in 10. but returning to the starting state on completion of each cycle.
Angew. Chem. Int. Ed. 2007, 46, 72 – 191 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.angewandte.org 87 Reviews D. A. Leigh, F. Zerbetto, and E. R. Kay
Mock and Ochwat have proposed 14, which undergoes the catalytic cycle illustrated in Scheme 4, as a minimalist model of a molecular motor.[77] In the process of catalyzing the hydration of its ketenimine fuel (red box) to give a
Scheme 5. Proposed system for unidirectional rotation around an aryl– aryl bond based on ring opening/ring closing of a chiral lactone.[79] Although this cannot function at equilibrium, it could if additional
features were added such that k15 16-aS k16-aS 15 and k15 16-aR ! ¼6 ! ! ¼6 Scheme 4. Proposed system for chemically driven directional 1808 k16-aR 15. rotary motion fueled by hydration of a ketenimine (red box).[77] For ! clarity, the catalytic cycle for a single enantiomer of motor 14 is shown. This cycle produces the opposite enantiomer, which undergoes an analogous cycle in which rotation must be biased in the opposite modification is also necessary to achieve directional 3608 direction. A number of potential side reactions (not illustrated) which rotation; since the ring-closing step is the exact reverse of the would divert the system from the catalytic cycle were found to be ring-opening step, both must proceed by the same transition
kinetically insignificant under the reaction conditions employed. state so that the rate in either direction is the same (k15 16-aS = ! k16-aS 15 and k15 16-aR = k16-aR 15). No energy would be con- Carter, N. J., & Cross, R. A. (2005). Mechanics of the kinesin! step. ! ! sumed by the mechanism proposed in Scheme 5 and, of Nature, 435(7040), 308–312." carboxamide waste product (blue box), epimerization of the course, the principle of detailed balance (Section 1.4.1) stereogenic center in the motor occurs—formally rotation demands that unidirectional rotation cannot occur in a around a C O bond (as indicated for (R)-14 in Scheme 4). system at equilibrium. À The cycle operates (Scheme 4) under the specific kinetic Stereoselective ring opening of racemic biaryl lactones
conditions k2,k4 > k1[AcCH=C=NtBu] > k3[H2O]. Kinetic using chiral reagents results in a directional rotation of about analysis based on spectrophotometric measurements was 908, which can be extended to a full half-turn by chemo- used to identify reaction conditions under which this relation selective ring closing to give the lactone by using a hydroxy is satisfied. Over a single catalytic cycle, therefore, the group in the other ortho position. This effect has been molecular chirality should ensure some biasing of the rota- demonstrated by Branchaud and co-workers[80] and, in an tional direction for formal 1808 rotation around the C O independent effort, Feringa and co-workers have successfully À bond. However, the subsequent cycle on the opposite extended this strategy to obtain a full 3608 rotation around a enantiomer will experience the antipodal directing effect so C C single bond.[81] The process involves four intermediates À that the biasing effect is the exact reverse to before.[78] (A–D, Figure 13a), in each of which rotation around the The restriction of rotational motion arising from steric biaryl bond is restricted: by covalent attachment in A and C, interactions between ortho substituents in biphenyl systems and through nonbonded interactions in B and D. Directional forms the basis for another system designed to exhibit rotation to interchange these intermediates requires a ste- directional rotation through ring opening/ring closing of a reoselective bond-breaking reaction in steps (1) and (3) and a chiral lactone (Scheme 5).[79] Ring opening of lactone 15 with regioselective bond-formation reaction in steps (2) and (4). a bulky nucleophile should proceed preferentially through Unlike 15 above, lactones 17 and 19 (Figure 13b) exist as
attack at one of the two diastereotopic faces (k15 16-aS > racemic mixtures as a result of a low barrier for small ! k15 16-aR). Equilibration of the ring-opened diastereomers amplitude rotations around the aryl–aryl bond. Reductive ! should then occur by random oscillations around the aryl–aryl ring opening with high enantioselectivity is, however, achiev- bond, with the bulky nucleophile preferentially avoiding able for either lactone by using a homochiral borolidine passing over the cyclohexanol ring. Ring closing is then catalyst, and the released phenol can subsequently be proposed to be faster from the 16-aR isomer because of the orthogonally protected to give 18a or 20a. The ring-opened proximity of the reacting groups, so that a net 3608 rotation compounds are produced in near-enantiopure form in a will be accomplished by a number of molecules in the system. process which involves directional rotation of 908 around the Experimentally, equilibration of the two ring-opened diaste- biaryl bond, governed by the chirality of the catalyst, and reomers in the current system occurs too fast for determi- powered by consumption of borane. The ortho substitution of nation of the selectivity of the ring-opening reaction. Sim- these species results in a high barrier to axial rotation. ilarly, demonstrating directional selectivity for the ring- Oxidation of the benzylic alcohol (18a 18b or 20a 20b) ! ! closing step will present a significant challenge. A further primes the motor for the next rotational step. Selective
88 www.angewandte.org 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Angew. Chem. Int. Ed. 2007, 46, 72 – 191 Angewandte Molecular Devices Chemie
Figure 13. a) Schematic representation of unidirectional rotation around a single bond, through four states.[81] In states A and C rotation is restricted by a covalent linkage, but the allowed motion results in helix inversion. In states B and D rotation is restricted by nonbonded interactions between the two halves of the system (red and green/blue). These forms are configurationally stable. The rotation relies on stereospecific cleavage of the covalent linkages in steps (1) and (3), then regiospecific formation of covalent linkages in steps (2) and (4). b) Structure and chemical transformations of a unidirectional rotor. Reactions: 1) Stereoselective reduction with (S)-CBS then allyl protection. 2) Chemoselective removal of the PMB group resulting in spontaneous lactonization. 3) Stereoselective reduction with (S)-CBS then protection with a PMB group. 4) Chemoselective removal of the allyl group resulting in spontaneous lactonization. 5)Oxidation to carboxylic acid.
IV III removal of one of the protecting groups on the enantiotopic tion of the metal center ([Ce (21)] [Ce (21)]À) increases ! phenols results in spontaneous lactonizationCarter, N. when J., thermally& Cross, R.the A. rate (2005). of racemization Mechanics by over of 300-fold.the kinesin[82c] This step. effect is driven axial rotation brings theNature two, reactive 435(7040), groups 308–312.thought" to derive from a reduced p–p interaction between the together—again probably a net directional process because two ligands as a consequence of the larger ionic radius of the of the steric hindrance of the ortho substituents (although this metal center in the lower oxidation state. Similarly, complexes is not demonstrated because the chirality is destroyed in this formed around the smaller ZrIV ion show very slow rotational step). Figure 13b illustrates the unidirectional process ach- dynamics at pH 7,[83] yet protonation results in facile race- ieved using the (S)-Corey–Bakshi–Shibata ((S)-CBS) cata- mization.[82a] The effect is retarded by oxidation of the lyst; rotation in the opposite sense can be achieved by porphyrin ligands, probably as a result of electrostatic employing the opposite borolidine enantiomer and swapping repulsion of incoming protons, but also because of the fact the order of phenol protection and deprotection steps. that the highest occupied molecular orbital (HOMO) of the complex is antibonding, so removal of electrons should 2.1.3. Stimuli-Induced Conformational Control in Organometallic Systems
Controlling the facile rotary motion of ligands in metal sandwich or double-decker complexes (introduced in Section 2.1.1) is con- ceptually similar to controlling rotation around covalent single bonds, and stimuli-induced control in such metal complexes has also been demonstrated. In metal bisporphyrinate com- plexes such as [Ce(21)] (Scheme 6) rotary motion corresponds to enantiomerization when the ligands possess D2h symmetry. This situation provides a convenient handle for monitoring the kinetics of rotation.[82] In com- plex [Ce(21)], rotation around the metal center is slow enough to permit isolation of the two Scheme 6. Controlling the rate of relative rotations of porphyrin ligands in cerium bis[tetrakis- enantiomers by chiral HPLC. However, reduc- arylporphyrinate] double-decker complex [Ce(21)].[82c]
Angew. Chem. Int. Ed. 2007, 46, 72 – 191 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.angewandte.org 89 Angewandte Molecular Devices Chemie tion and internal motion around several bonds. Such “induced fit” mechanisms are the basis of many allosteric systems, whereby recognition of one species affects the binding properties or enzymatic activity at a remote site in the same molecule. Allosterism, cooper- ativity, and feedback are central to many functional biological systems[98] and synthetic approaches towards achieving similar effects have been extensively reviewed elsewhere.[84, 99] Here we are interested primarily in the dynamic processes themselves and, therefore, in many cases where the conformational changes are rather small or poorly defined, such systems do not fall under our definition of molecular machines. However, certain examples do exhibit impressive control over the molec- Figure 15. Binding-induced conformational changes in molecular clips. a) Dimethylene- ular shape or conformation and merit discussion here in bridged aromatic clip 27 in which binding to 1,2,4,5-tetracyanobenzene results in a [102] their own right, together with other examples of stimuli- large-amplitude induced-fit conformational change (motion indicated by arrows). b) The three accessible conformations of the diphenylglycouril-based clips illustrated induced conformational changes around several bonds. for simple derivative 28 (methoxy substituents are not shown on the space-filling Some of the first and most elegant examples of models).[103] The syn,anti (sa) form dominates in solution, but only the anti,anti form is synthetic allosteric receptors were introduced by Rebek able to bind aromatic guests—in an induced-fit mechanism. c) Glycouril-based clip 29 et al. (Scheme 9).[99a, 100] In 25, the binding of a metal ion in which an additional binding site (a crown ether) can control the conformation of the such as tungsten to the bipyridyl ligand forces copla- clip molecule, preorganizing the p-electron-rich binding site and resulting in a positive, [104] narity of this unit, thus distorting the crown ether heterotropic allosteric effect. The space-filling representations are reprinted with permission from Ref. [103a]. conformation and diminishing its ability to bind potas-
not principally adopt the anti,anti (aa) binding conformation, only doing so in the presence of a suitable guest.[103] Similar structures can incorporate two conformationally coupled binding sites and exhibit allosteric effects. Glycouril- based clip 29 bears two crown ether substituents and binds potassium ions to form a complex with 1:2 stoichiom- etry. The binding event stabilizes the aa conformation, preorganizing the electron-rich naphthalene units in a pseudoparallel arrangement, thus increasing the binding affinity for Scheme 9. a) Negative heterotropic allosteric receptor 25 binds alkali metal ions, with selectivity simple electron-poor aromatic com- + [100] for K . Chelation of tungsten to the bipyridyl moiety forces this unit to adopt a rigid pounds such as 1,3-dinitrobenzene.[104] conformation in which the 3 and 3’ substituents are brought close together. The resulting conformation of the crown ether does not favor binding through all the oxygen atoms and so In atropoisomeric host molecules, an + induced-fit binding process at elevated affinity for K ions is reduced. In fact [W(CO)3(25)] shows a preference for binding the smaller Na+ ion. b) Positive homotropic allosteric receptor 26.[101] temperatures can be used to dynam- ically select the single optimal binding Carter, N. J., & Cross, R. A. (2005). Mechanics of the kinesin step. conformation for a particular guest, sium ions—a negativeNature heterotropic, 435(7040), allosteric 308–312. effect." The same which can then be “saved” by cooling the system to ambient research group later developed this concept to create positive, temperature and removal of the templating guest.[105] homotropic allosteric receptor 26, in which binding one The addition of an external ligand or guest can also be molecule of Hg(CN)2 preorganizes the remaining binding site used to disrupt a preexisting intramolecular interaction and for a second binding event.[101] These strategies have since cause a change in conformation. This is the case for 4,4’- either directly spawned or indirectly inspired an extraordi- bipyridyl-capped zinc porphyrin 30 in which the pyridyl and narily wide range of increasingly sophisticated synthetic porphyrin planes are switched from perpendicular to mutu- allosteric receptors.[84, 99] ally parallel on addition of pyridine (Scheme10).[106] Binding to molecular tweezer and clip receptors It is well known that the relative stabilities of the two (Figure 15) can result in significant conformational changes different chair forms of six-membered alicyclic rings can be in the host, as favorable interactions with the guest are manipulated by covalent substitutions.[107, 108] However, in maximized. In 27, for example, the distance between the trisaccharide 31 this conformational change can be achieved sidewalls decreases from 14.5 to 6.5 on binding to aromatic by adding and removing metal ions (Scheme 11).[109] In the [102] 4 guests, while diphenylglycouril-based clips such as 28 do absence of metal ions, the C1 conformation is preferred for
Angew. Chem. Int. Ed. 2007, 46, 72 – 191 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.angewandte.org 91
Vol 435|19 May 2005|NATURE ARTICLES
Figure 1 | Example optical trapping records. Three superimposed records Two successive experiments on the same molecule are shown. For automated showing the movement of single kinesin molecules towards stall force, over dwell time calculations and step-averaging, a t-test step finder was applied to time. In record 1 (black), the trap and microtubule remain fixed throughout, the bead position data. The inset shows the t-test profile for the first part of and the kinesin walks with 8-nm steps away from the trap centre (dashed record 1. Steps are defined where the t-value exceeds a preset threshold value line) to stall force and finally detaches. In record 2 (red), on reaching 4 pN (dotted line). The located steps are shown offset just above record 1. In all the microtubule is moved rapidly (upward arrow), pulling the kinesin to records, detachment events (steps larger than 12 nm recognized by the step about 14 pN force. In some instances the kinesin responds with processive finder) are marked with D. Conditions: single kinesin molecules on 560-nm backward steps to stall force (7–8 pN). More commonly at forces above polystyrene beads, 1 mM ATP. The trap stiffnesses for the beads in these 10 pN, the kinesin would detach after a few or no backward steps, the bead records were 0.064, 0.067 and 0.064 pN nm21, respectively. The force scale returning to trap centre. Record 3 (blue) shows the opposite procedure. On represents a trap stiffness of 0.065 pN nm21. Stage movements (arrows) reaching the 4-pN trigger force, the microtubule is quickly moved were typically complete within 200 ms. The data shown are 1-ms boxcar (downward arrows) to apply a large negative (assisting) force to the kinesin. filtered.
To do this we used an automatic algorithmCarter, (see Methods) N. J., to & find Cross,Above R. stall A. force,(2005). backward Mechanics steps predominate. of the Their kinesin dwell times step. steps within a data set of multiple records, andNature to determine, 435 the(7040), dwell depend 308–312. on the ATP" concentration, confirming that backward steps, time (the time interval between the previous step and the current like forward steps, are triggered by ATP binding. It has been step) and amplitude (the distance moved) for each step. Figure 2a reported17 that the mean dwell time for single backward steps is shows the incidence and the amplitude of forward and backward load-independent. We confirm their conclusion and extend it to steps over the range 215 pN (assisting load) to 15 pN (inhibitory include processive backward stepping under super-stall loads of up to load). The inset shows that the ratio of the numberþ of forward steps 15 pN. Figure 2c plots force–velocity curves calculated by dividing to the number of backward steps at any particular load depends the mean step size (forward steps positive, backward steps negative) exponentially on the load. At the stall force of 7.2 pN (ref. 17) this by the mean dwell times shown in Fig. 2b. ratio is 1. The stall force does not seem to depend on the ATP concentration, because the choice between forward and backward Steps are microsecond events with no substeps stepping depends only on load, not on ATP concentration. There are two reports of substeps in the literature7,26, and several Figure 2b plots the variation of dwell time with load for forward current models for the kinesin mechanism predict that the 8-nm step and backward steps. At all loads, the dwell time for both forward and consists of two or more mechanical substeps, with more than one backward steps decreases as the ATP concentration is increased, biochemical kinetic step occurring during an 8-nm physical step. To confirming that ATP binding is required for both forward and search for mechanical substeps we attached kinesin to smaller beads, backward steps17. Under forward (assisting) loads (in the range 22 to give an improved temporal response7, and used a step-averaging to 215 pN in Fig. 2b) only forward steps are observed, and their algorithm to increase spatial resolution. Our algorithm automatically dwell time depends only on the ATP concentration, not on the load, finds steps and then does a global fit to find the best single- indicating that a chemical step (ATP binding) is limiting. Under exponential fit across the full data set of steps (see Methods). The backward load, the dwell time for forward steps depends exponen- origins of these fits are then used to synchronize the raw data records tially on the load, as expected for a particle diffusing over an for the steps so that they can be ensemble averaged. We found that the activation energy barrier in accordance with Kramers theory23–25. resulting average step was a simple monophasic event; no mechanical The load–dwell-time curve for forward steps is exponential above substeps were detectable (Fig. 3) either before or during the major about 3 pN both at 1 mM ATP and at 10 mM ATP, and the exponents event. are similar at high and low ATP concentrations, which is consistent To test the limits of detection in our system we generated synthetic with a single, common, rate-limiting mechanical step under high data by adding substeps of defined amplitude and duration to real, backward load at both high and low ATP. At any particular backward recorded noise. The simulations (Supplementary Information) show load, dwell times are exponentially distributed (Supplementary that a bead position substep lasting more than 30 ms would be Fig. S2). necessary to increase the averaged bead rise time detectably beyond
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Vol 435|19 May 2005|NATURE ARTICLES
Vol 435|19 May 2005|NATURE ARTICLES
Figure 3 | Average time course of forward and backward steps. a, b, The same data sets are shown on two timescales: fine (a) and coarse (b). Step positions were determined automatically with a t-test step finder, followed by a least-squares exponential fit for step position refinement. The averaged forward steps (circles) and backward steps (squares) for two bead sizes are shown: the fit time constant for 500-nm beads (black) was faster than that for 800-nm beads (red). For 500-nm beads, forward steps (n 1,693) had time constant 15.3 ms, amplitude 7.39 nm and average force 5.0¼ pN; for backward steps (n 316) these were 19.4 ms, 7.34 nm and 6.1 pN, respectively. For 800-nm¼ beads, forward steps (n 565) had time constant 35.9 ms, amplitude 7.6 nm and force 4.7 pN; for backward¼ steps (n 68) these were 37.3 ms, 7.8 nm and 5.6 pN, respectively. All records used¼ for the step averaging were recorded at 1 mM ATP. Figure 3 | Average time course of forward and backward steps. a, b, The Figure 4 | Model. Before ATP binding, the motor is parked (state 0): the same data sets are shown on two timescales: fine (a) and coarse (b). Step probability of a forward step decreases exponentially with increasing holdfast head remains stably bound to the microtubule and the ADP- positions were determined automatically with a t-test step finder, followed load, whereas the probability of a backward step is constant. At stall, containing tethered head cannot access its new site. Straining this state either by a least-squares exponential fit for step position refinement. The averaged the probability of forward stepping is equal to that of backward forwards or backwards does not induce stepping. ATP (T) binding to the forward steps (circles) and backward steps (squares) for two bead sizes are stepping. Above stall, our model predicts that efficiency goes negative holdfast head sanctions ADP (D) release from the tethered head. Forward shown: the fit time constant for 500-nm beads (black) was faster than that steps ( 1, 2, 3, 0) occur when the tethered head binds in front of the as backward steps begin to predominate. Note, however, that þ þ þ for 800-nm beads (red). For 500-nm beads, forward steps (n 1,693) had holdfast head, backward steps (21, 22, 23, 0) when it binds behind. The although we know that backward stepping requires ATP binding, time constant 15.3 ms, amplitude 7.39 nm and average force 5.0¼ pN; for choice between forward and backward stepping depends on the applied load there is at present no evidence that backward stepping is coupled to backward steps (n 316) these were 19.4 ms, 7.34 nm and 6.1 pN, (the spring). In the figure, the central state 0 represents stall, in which the ATP turnover. backward load applied by the trap (represented by the stretched spring) is respectively. For 800-nm¼ beads, forward steps (n 565) had time constant There are previous reports of substeps6,7, and several current such that the tethered head has an equal probability of stepping forwards or 35.9 ms, amplitude 7.6 nm and force 4.7 pN; for backward¼ steps (n 68) models predict substeps30–32. Our data rule out substeps that take backwards. To make a forward step, the motor needs to make a diffusional these were 37.3 ms, 7.8 nm and 5.6 pN, respectively. All records used¼ for the longer than 30 ms. This indicates that the appearance of substantial excursion in the progress direction. This excursion can then be locked in by step averaging were recorded at 1 mM ATP. Figure 3 | Average time course of forward and backward steps. a, b,irreversible The ADP release from the lead head. substeps in previous analyses might have been an artefact. The same data sets are shown on two timescales: fine (a) and coarse (b). Step current data gainsay models in which kinesin steps along the 4-nm positions were determined automatically with a t-test step finder, followed tubulinFigure monomer 4 | Model. repeatBefore in microtubules. ATP binding, Similarly, the motor our is dataparked argue (state 0): the by a least-squares exponential fit for step position refinement.Carter, The averaged N. J., & Cross, R. A. (2005). Mechanics of the kinesin step. probability of a forward step decreases exponentially with increasing againstholdfast models head that remains predict stably substeps bound arising to the from microtubule rapidly equilibrat- and the ADP-cases the motor head diffuses into a highly strained state, locks on to forward steps (circles) and backward steps (squares) for two bead sizes are load, whereas the probability of a backward step is constant. At stall, containing tethered head30 cannot access its new site. StrainingNature this33 state, the435 either track(7040), and then 308–312. detaches at a" strain-dependent rate. For myosin, ingshown: mechanical the fit time substates constant. The for data 500-nm support beads one-stroke (black) was models faster than that the probability of forward stepping is equal to that of backward forwards or backwards does not induce stepping. ATP (T) bindingthis to the original scheme proved inadequate to account for the mecha- andfor specifically 800-nm beads exclude (red). models For 500-nm that forbid beads, backward forward steps. steps Models (n 1,693) in had stepping. Above stall, our model predicts that efficiency goes negative whichholdfast backward head stepping sanctions synthesizes ADP (D) ATP release31 are from unlikely, the tetheredbecause¼ they head. Forwardnical behaviour and was modified to include multiple bound timesteps constant ( 1, 15.32, m3,s, 0) amplitude occur when 7.39 the nm tethered and average head force binds 5.0 in pN; front for of the as backward steps begin to predominate. Note, however, that predict thatþ theþ frequencyþ of backward steps should decrease at high mechanical states linked by strain-dependent conformational backwardholdfast steps head, (n backward316) these steps were (21, 19.422,m2s,3, 7.34 0) nm when and it 6.1binds pN, behind. The 35 ATP concentration,¼ whereas we observed the opposite. changes . For processive kinesins, a model with biased diffusion to although we know that backward stepping requires ATP binding, respectively.choice between For 800-nm forward beads, and backward forward steps stepping (n depends565) had on time the applied constant load Our proposed model for processive kinesins bears¼ striking simi- capture followed by a single, strain-dependent conformational there is at present no evidence that backward stepping is coupled to 35.9(thems, spring). amplitude In the 7.6 nmfigure, and the force central 4.7 pN; state for 0 backward represents steps stall, (n in which68) the larities to the classical Huxley scheme for myosin34, in that in both ¼ change coupled to ADP release seems adequate. It will now be ATP turnover. thesebackward were 37.3 loadms, applied 7.8 nm by and the 5.6 trap pN, (represented respectively. by All the records stretched used spring)for the is 6,7 stepsuch averaging that the were tethered recorded head at has 1 mMan equal ATP. probability of stepping forwards or There are previous reports of substeps , and several current © 2005 Nature Publishing Group 311 models predict substeps30–32. Our data rule out substeps that take backwards. To make a forward step, the motor needs to make a diffusional Figure 4 | Model. Before ATP binding, the motor is parked (state 0): the longer than 30 ms. This indicates that the appearance of substantial excursion in the progress direction. This excursion can then be locked in by probabilityirreversible of ADP a forward release step from decreases the lead head. exponentially with increasing holdfast head remains stably bound to the microtubule and the ADP- substeps in previous analyses might have been an artefact. The load, whereas the probability of a backward step is constant. At stall, containing tethered head cannot access its new site. Straining this state either current data gainsay models in which kinesin steps along the 4-nm the probability of forward stepping is equal to that of backward forwards or backwards does not induce stepping. ATP (T) binding to the tubulin monomer repeat in microtubules. Similarly, our data argue stepping. Above stall, our model predicts that efficiency goes negative holdfast head sanctions ADP (D) release from the tethered head. Forward against models that predict substeps arising from rapidly equilibrat- cases the motor head diffuses into a highly strained state, locks on to steps ( 1, 2, 3, 0) occur when the tethered head binds in front of the as backward steps begin to predominate. Note, however, that þ þ þ 30 33 the track and then detaches at a strain-dependent rate. For myosin, holdfast head, backward steps (21, 22, 23, 0) when it binds behind. The ing mechanical substates . The data support one-stroke models although we know that backward stepping requires ATP binding, this original scheme proved inadequate to account for the mecha- choice between forward and backward stepping depends on the applied load and specifically exclude models that forbid backward steps. Models in there is at present no evidence that backward stepping is coupled to 31 nical behaviour and was modified to include multiple bound (the spring). In the figure, the central state 0 represents stall, in which the which backward stepping synthesizes ATP are unlikely, because they ATP turnover. backward load applied by the trap (represented by the stretched spring) is predict that the frequency of backward steps should decrease at high mechanical states linked by strain-dependent6,7 conformational such that the tethered head has an equal probability of stepping forwards or There are35 previous reports of substeps , and several current ATP concentration, whereas we observed the opposite. modelschanges predict. For substeps processive30–32 kinesins,. Our data a model rule out with substeps biased diffusionthat take to backwards. To make a forward step, the motor needs to make a diffusional Our proposed model for processive kinesins bears striking simi- longercapture than followed 30 ms. This by indicates a single, that strain-dependent the appearance of conformational substantial excursion in the progress direction. This excursion can then be locked in by 34 irreversible ADP release from the lead head. larities to the classical Huxley scheme for myosin , in that in both substepschange in coupled previous to analyses ADP release might seems have adequate. been an artefact. It will now The be current data gainsay models in which kinesin steps along the 4-nm © 2005 Nature Publitubulinshing monomerGroup repeat in microtubules. Similarly, our data argue311 against models that predict substeps arising from rapidly equilibrat- cases the motor head diffuses into a highly strained state, locks on to ing mechanical substates30. The data support one-stroke models33 the track and then detaches at a strain-dependent rate. For myosin, and specifically exclude models that forbid backward steps. Models in this original scheme proved inadequate to account for the mecha- which backward stepping synthesizes ATP31 are unlikely, because they nical behaviour and was modified to include multiple bound predict that the frequency of backward steps should decrease at high mechanical states linked by strain-dependent conformational ATP concentration, whereas we observed the opposite. changes35. For processive kinesins, a model with biased diffusion to Our proposed model for processive kinesins bears striking simi- capture followed by a single, strain-dependent conformational larities to the classical Huxley scheme for myosin34, in that in both change coupled to ADP release seems adequate. It will now be
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ARTICLES NATURE|Vol 435|19 May 2005
that which we observe. Our data therefore rule out substeps of (trap stiffness) to more than 0.65 pN nm21 during force production. ARTICLES duration more than 30 ms. NATURE|Vol 435|19 May 2005All the various links in the system will contribute series compliances, Figure 3 shows that the time constants for the ensemble averaged but we can nevertheless use the measured stiffness at high load to set a forward and backward steps are very similar. Steps were faster if we lower limit of 0.6 pN nm21 on kinesin head stiffness. This low limit that which we observe. Our data therefore rule outused substeps slightly of smaller(trap stiffness) beads, which to more diffuse than 0.65 more pN nm rapidly21 during (Fig. force 3). production.for the head stiffness is consistent with a diffusional mechanism duration more than 30 ms. All the various links in the system will contribute series compliances,(diffusion through 8 nm followed by capture) because at zero load Figure 3 shows that the time constants for the ensembleModel averaged but we can nevertheless use the measured stiffness at high load to setthe a Kramers time25 for a protein of 5 nm radius to diffuse 8 nm is 7 ms. 21 forward and backward steps are very similar. Steps wereThat faster the ifforward we lower directional limit of 0.6 pN bias nm inon kinesin kinesin can head be stiffness. reversed This lowby limitThe Kramers time rises to 8 ms at a 1.7 pN nm21 kinesin head used slightly smaller beads, which diffuse more rapidly (Fig. 3). for the head stiffness is consistent with a diffusional mechanism pulling backwards(diffusion on the through motor 8 nm suggests followed that by capture) the tethered because at head zero loadstiffness, which sets an upper limit on head stiffness for a diffusional Model makes a diffusionalthe Kramers search timefor its25 for next a protein site, the of 5 outcomenm radius to of diffuse which 8 nmcan is 7 mmechanism.s. That the forward directional bias in kinesin can bebe reversed biased by by anThe applied Kramers load. time rises to 8 ms at a 1.7 pN nm21 kinesin head The absence of substeps within the 8-nm kinesin step indicates pulling backwards on the motor suggests that the tetheredSystem head stiffnessstiffness, (trap which stiffness sets plus an upper bead–kinesin–microtubule limit on head stiffness for a diffusionallink that load transfers from one head to the other in a single mechanical makes a diffusional search for its next site, the outcomestiffnesses) of which can in ourmechanism. experiments increases from about 0.06 pN nm21 event. As previously discussed, we envisage that forward steps and be biased by an applied load. The absence of substeps within the 8-nm kinesin step indicatesbackward steps begin from a common intermediate (the parked state; System stiffness (trap stiffness plus bead–kinesin–microtubule link that load transfers from one head to the other in a single mechanical stiffnesses) in our experiments increases from about 0.06 pN nm21 event. As previously discussed, we envisage that forward steps andFig. 4), in which the trailing head lacks nucleotide and acts as a backward steps begin from a common intermediate (the parked state;holdfast to the microtubule, while the ADP-containing tethered head Fig. 4), in which the trailing head lacks nucleotide and acts asis a unbound and unloaded12,27. ATP binding to the holdfast head acts ARTICLES NATURE|Vol 435|19 May 2005 holdfast to the microtubule, while the ADP-containing tethered headas a gate that allows the tethered head to begin a diffusional search for 12,27 is unbound and unloaded . ATP binding to the holdfast head actsits next binding site. Ordinarily the ensuing search pattern is biased as a gate that allows the tethered head to begin a diffusional search for 21 that which we observe. Our data therefore rule out substeps of (trap stiffness) to more than 0.65 pNits nmnext bindingduring force site. production.Ordinarily the ensuing search pattern is biasedtowards the microtubule plus end. This might be achieved by having duration more than 30 ms. All the various links in the systemtowards will contribute the microtubule series compliances, plus end. This might be achieved by havingthe holdfast head guiding the binding direction of its partner. The Figure 3 shows that the time constants for the ensemble averaged but we can nevertheless use the measured stiffness at high load to set a 16 21 the holdfast head guiding the binding direction of its partner. Theneck linker docking scheme proposed by Rice and colleagues has forward and backward steps are very similar. Steps were faster if we lower limit of 0.6 pN nm on kinesinneck head linker stiffness. docking This scheme low limit proposed by Rice and colleagues16 has used slightly smaller beads, which diffuse more rapidly (Fig. 3). for the head stiffness is consistent with a diffusional mechanism this property, but is ruled out in its original form on several this property, but is ruled out in its original form on several 28 (diffusion through 8 nm followed by capture) because at zero load grounds , and particularly because neck linker docking does not grounds28, and particularly because neck linker docking does not Model the Kramers time25 for a protein of 5 nm radius to diffuse 8 nm is 7 ms. have enough energy associated with it to drive stepping at high have enough energy21 associated with it to drive stepping at high That the forward directional bias in kinesin can be reversed by The Kramers time rises to 8 ms at a 1.7 pN nm kinesin head 29 29 loads . Nevertheless, an ATP-dependent parking and unparking of pulling backwards on the motor suggests that the tethered head stiffness, which sets an upper limitloads on head. Nevertheless, stiffness for a an diffusional ATP-dependent parking and unparking of makes a diffusional search for its next site, the outcome of which can mechanism. the tethered head in a forward-biased position remains in our viewthe tethered head in a forward-biased position remains in our view be biased by an applied load. The absence of substeps withinhighly the 8-nm plausible, kinesin albeit step in indicates our model ATP unparks the tethered headhighly plausible, albeit in our model ATP unparks the tethered head System stiffness (trap stiffness plus bead–kinesin–microtubule link that load transfers from one headrather to the other than inparks a single it. Once mechanical the tethered head locates an empty site onrather than parks it. Once the tethered head locates an empty site on 21 stiffnesses) in our experiments increases from about 0.06 pN nm event. As previously discussed, wethe envisage microtubule, that forward strain-dependent, steps and microtubule-activated ADPthe microtubule, strain-dependent, microtubule-activated ADP backward steps begin from a commonrelease intermediate occurs, converting (the parked the state; newly attached head to strong micro- Fig. 4), in which the trailing headtubule lacks binding nucleotide and and transferring acts as a the load stably to it. In the modelrelease occurs, converting the newly attached head to strong micro- holdfast to the microtubule, whilethere the ADP-containing is a transient tetheredtwo-heads-attached head state, but the load is alwaystubule binding and transferring the load stably to it. In the model is unbound and unloaded12,27. ATPborn binding on one to the head holdfast only. Backwards head acts mechanical strain counteracts thethere is a transient two-heads-attached state, but the load is always as a gate that allows the tethered headintrinsic to begin forward a diffusional bias in search the diffusionalfor search, thus increasing theborn on one head only. Backwards mechanical strain counteracts the its next binding site. Ordinarily the ensuing search pattern is biased probability that the tethered head will bind behind the holdfast head.intrinsic forward bias in the diffusional search, thus increasing the towards the microtubule plus end.For This a backwardmight be achieved step, the by newly having bound tethered head is unloaded as it the holdfast head guiding the binding direction of its partner. The probability that the tethered head will bind behind the holdfast head. undergoes microtubule-activated16 ADP release. Consistent with this neck linker docking scheme proposedis our by observation Rice and colleagues that the dwellhas time for backward steps at highFor a backward step, the newly bound tethered head is unloaded as it this property, but is ruled out in its original form on several undergoes microtubule-activated ADP release. Consistent with this grounds28, and particularly becausebackward neck linker loads docking depends does on thenot ATP concentration but not on the have enough energy associated withload. it to drive stepping at high is our observation that the dwell time for backward steps at high loads29. Nevertheless, an ATP-dependent parking and unparking of backward loads depends on the ATP concentration but not on the Comparison with other models the tethered head in a forward-biased position remains in our view load. highly plausible, albeit in our modelCompeting ATP unparks models the tethered for the kinesin head mechanism agree that a decrease rather than parks it. Once the tetheredin efficiency head locates occurs an empty close siteto stall, on due either to futile cycles30 or to the microtubule, strain-dependent,ATP-driven microtubule-activated backward steps ADP24. In our model, tight coupling ofComparison with other models release occurs, converting the newlystepping attached to head ATP to binding strong micro- is retained under all conditions, but theCompeting models for the kinesin mechanism agree that a decrease tubule binding and transferring the load stably to it. In the model in efficiency occurs close to stall, due either to futile cycles30 or to there is a transient two-heads-attached state, but the load is always 24 born on one head only. BackwardsFigure mechanical 2 | Stepping strain behaviour. counteractsa the, Plot of step amplitudes against load. DataATP-driven backward steps . In our model, tight coupling of intrinsic forward bias in the diffusionalfor 1 mM search, ATP: forward thus increasing steps in blue, the backward steps in cyan. Data for 10 mMstepping to ATP binding is retained under all conditions, but the probability that the tethered headATP: will bind forward behind steps the in holdfast red, backward head. steps in orange. Forward and backward For a backward step, the newly boundsteps tethered have equal head mean is unloaded amplitudes. as it Inset, ratio of forward to backward steps undergoes microtubule-activatedplotted ADP release. against Consistent the load; data with for this1 mM ATPare shown in blue, and for 10 mM 20.95 load Figure 2 | Stepping behaviour. a, Plot of step amplitudes against load. Data ATP in red. The fit is: ratio 802e £ , with the load in piconewtons. is our observation that the dwell time for backward steps¼ at high for 1 mM ATP: forward steps in blue, backward steps in cyan. Data for 10 mM backward loads depends on the ATPb, Plot concentration of dwell times but against not on load. the Data were binned with 1-pN intervals, load. and the mean dwell and s.e.m. were calculated for each bin. The colour keyATP: is forward steps in red, backward steps in orange. Forward and backward as in a; filled circles are mean forward step dwells, open circles are mean steps have equal mean amplitudes. Inset, ratio of forward to backward steps Comparison with other models backward step dwells. Mean dwell data for forward steps above 3 pN wereplotted against the load; data for 1 mM ATPare shown in blue, and for 10 mM fitted to single exponentials (solid lines). The fits are dwell 20.95 load Competing models for the kinesin mechanism0.57 load agree that a decrease 0.55 load ¼ ATP in red. The fit is: ratio 802e £ , with the load in piconewtons. 0.0036e £ at 1 mM ATP and dwell 0.0256e £ at 10 mM ATP, in efficiency occurs close to stall, due either to futile cycles30 or to ¼ with the load in piconewtons. Other fits¼ are by linear regression over the b, Plot of dwell times against load. Data were binned with 1-pN intervals, ATP-driven backward steps24. In our model, tight coupling of ranges shown by the solid lines. In the region 5–8 pN the dwell time and the mean dwell and s.e.m. were calculated for each bin. The colour key is stepping to ATP binding is retained under all conditions, but the distribution for backward steps is distorted slightly by the occurrence of as in a; filled circles are mean forward step dwells, open circles are mean forward steps, and vice versa (see Methods). c, Force–velocity curve. backward step dwells. Mean dwell data for forward steps above 3 pN were Velocities were calculated as mean amplitude divided by mean dwell time for Figure 2 | Stepping behaviour. a, Plot of step amplitudes against load. Data fitted to single exponentials (solid lines). The fits are dwell for 1 mM ATP: forward steps in blue, backwardeach bin, steps taking in cyan.the bins Data and for the 10 m meanM dwell times from b. Red data, 10 mM 0.57 load 0.55 load ¼ ATP; blue data, 1 mM ATP. The zero-load data (squares) are calculated from0.0036e £ at 1 mM ATP and dwell 0.0256e £ at 10 mM ATP, ATP: forward steps in red, backward steps in orange. Forward and backward with the load in piconewtons. Other fits¼ are by linear regression over the steps have equal mean amplitudes. Inset,theratio bead of velocities forward toobserved backward with steps the trap turned off. plotted against the load; data for 1 mM ATPare shown in blue, and for 10 mM ranges shown by the solid lines. In the region 5–8 pN the dwell time 20.95 load 310 ATP in red. The fit is: ratio 802e £ , with the load in piconewtons. distribution for backward steps is distorted slightly by the occurrence of © 2005 ¼Nature Publishing Group b, Plot of dwell times against load. Data were binned with 1-pN intervals, forward steps, and vice versa (see Methods). c, Force–velocity curve. Carter, N.and J., the & mean Cross, dwell and R. s.e.m. A. were (2005). calculated for Mechanics each bin. The colour of key isthe kinesin step. as in a; filled circles are mean forward step dwells, open circles are mean Velocities were calculated as mean amplitude divided by mean dwell time for Nature, 435backward(7040), step dwells. 308–312. Mean dwell data" for forward steps above 3 pN were each bin, taking the bins and the mean dwell times from b. Red data, 10 mM fitted to single exponentials (solid lines). The fits are dwell ATP; blue data, 1 mM ATP. The zero-load data (squares) are calculated from 0.57 load 0.55 load ¼ 0.0036e £ at 1 mM ATP and dwell 0.0256e £ at 10 mM ATP, the bead velocities observed with the trap turned off. with the load in piconewtons. Other fits¼ are by linear regression over the ranges shown by the solid lines. In the region 5–8 pN the dwell time distribution for310 backward steps is distorted slightly by the occurrence of © 2005 Nature Publishing Group forward steps, and vice versa (see Methods). c, Force–velocity curve. Velocities were calculated as mean amplitude divided by mean dwell time for each bin, taking the bins and the mean dwell times from b. Red data, 10 mM ATP; blue data, 1 mM ATP. The zero-load data (squares) are calculated from the bead velocities observed with the trap turned off.
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Vol 435|19 May 2005|NATURE ARTICLES
Vol 435|19 May 2005|NATURE ARTICLES
Figure 3 | Average time course of forward and backward steps. a, b, The same data sets are shown on two timescales: fine (a) and coarse (b). Step positions were determined automatically with a t-test step finder, followed by a least-squares exponential fit for step position refinement. The averaged forward steps (circles) and backward steps (squares) for two bead sizes are shown: the fit time constant for 500-nm beads (black) was faster than that for 800-nm beads (red). For 500-nm beads, forward steps (n 1,693) had time constant 15.3 ms, amplitude 7.39 nm and average force 5.0¼ pN; for backward steps (n 316) these were 19.4 ms, 7.34 nm and 6.1 pN, respectively. For 800-nm¼ beads, forward steps (n 565) had time constant 35.9 ms, amplitude 7.6 nm and force 4.7 pN; for backward¼ steps (n 68) these were 37.3 ms, 7.8 nm and 5.6 pN, respectively. All records used¼ for the step averaging were recorded at 1 mM ATP.
Figure 4 | Model. Before ATP binding, the motor is parked (state 0): the probability of a forward step decreases exponentially with increasing holdfast head remains stably bound to the microtubule and the ADP- load, whereas the probability of a backward step is constant. At stall, containing tethered head cannot access its new site. Straining this state either the probability of forward stepping is equal to that of backward forwards or backwards does not induce stepping. ATP (T) binding to the stepping. Above stall, our model predicts that efficiency goes negative holdfast head sanctions ADP (D) release from the tethered head. Forward steps ( 1, 2, 3, 0) occur when the tethered head binds in front of the as backward steps begin to predominate. Note, however, that holdfastþ head,þ backwardþ steps (21, 22, 23, 0) when it binds behind. The although we know that backward stepping requires ATP binding, choice between forward and backward stepping depends on the applied load there is at present no evidence that backward stepping is coupled to (the spring). In the figure, the central state 0 represents stall, in which the ATP turnover. backward load applied by the trap (represented by the stretched spring) is There are previous reports of substeps6,7, and several current such that the tethered head has an equal probability of stepping forwards or models predict substeps30–32. Our data rule out substeps that take backwards. To make a forward step, the motor needs to make a diffusional longer than 30 ms. This indicates that the appearance of substantial excursion in the progress direction. This excursion can then be locked in by irreversible ADP release from the lead head. substeps in previous analyses might have been an artefact. The current data gainsay models in which kinesin steps along the 4-nm tubulin monomer repeat in microtubules. Similarly, our data argue against models that predict substeps arising from rapidly equilibrat- cases the motor head diffuses into a highly strained state, locks on to ing mechanical substates30. The data support one-stroke models33 the track and then detaches at a strain-dependent rate. For myosin, and specifically exclude models that forbid backward steps. Models in this original scheme proved inadequate to account for the mecha- which backward stepping synthesizes ATP31 are unlikely, because they nical behaviour and was modified to include multiple bound predict that the frequency of backward steps should decrease at high mechanical states linked by strain-dependent conformational ATP concentration, whereas we observed the opposite. changes35. For processive kinesins, a model with biased diffusion to Our proposed model for processive kinesins bears striking simi- capture followed by a single, strain-dependent conformational larities to the classical Huxley scheme for myosin34, in that in both change coupled to ADP release seems adequate. It will now be
© 2005 Nature Publishing Group 311
Figure 3 | Average time course of forward and backward steps. a, b, The same data sets are shown on two timescales:Carter, fine (a )N. and J., coarse & Cross, (b). Step R. A. (2005). Mechanics of the kinesin step. positions were determined automaticallyNature with a t-test, 435 step(7040), finder, followed 308–312." by a least-squares exponential fit for step position refinement. The averaged forward steps (circles) and backward steps (squares) for two bead sizes are shown: the fit time constant for 500-nm beads (black) was faster than that for 800-nm beads (red). For 500-nm beads, forward steps (n 1,693) had time constant 15.3 ms, amplitude 7.39 nm and average force 5.0¼ pN; for backward steps (n 316) these were 19.4 ms, 7.34 nm and 6.1 pN, respectively. For 800-nm¼ beads, forward steps (n 565) had time constant 35.9 ms, amplitude 7.6 nm and force 4.7 pN; for backward¼ steps (n 68) these were 37.3 ms, 7.8 nm and 5.6 pN, respectively. All records used¼ for the step averaging were recorded at 1 mM ATP.
Figure 4 | Model. Before ATP binding, the motor is parked (state 0): the probability of a forward step decreases exponentially with increasing holdfast head remains stably bound to the microtubule and the ADP- load, whereas the probability of a backward step is constant. At stall, containing tethered head cannot access its new site. Straining this state either the probability of forward stepping is equal to that of backward forwards or backwards does not induce stepping. ATP (T) binding to the stepping. Above stall, our model predicts that efficiency goes negative holdfast head sanctions ADP (D) release from the tethered head. Forward steps ( 1, 2, 3, 0) occur when the tethered head binds in front of the as backward steps begin to predominate. Note, however, that holdfastþ head,þ backwardþ steps (21, 22, 23, 0) when it binds behind. The although we know that backward stepping requires ATP binding, choice between forward and backward stepping depends on the applied load there is at present no evidence that backward stepping is coupled to (the spring). In the figure, the central state 0 represents stall, in which the ATP turnover. backward load applied by the trap (represented by the stretched spring) is There are previous reports of substeps6,7, and several current such that the tethered head has an equal probability of stepping forwards or models predict substeps30–32. Our data rule out substeps that take backwards. To make a forward step, the motor needs to make a diffusional longer than 30 ms. This indicates that the appearance of substantial excursion in the progress direction. This excursion can then be locked in by irreversible ADP release from the lead head. substeps in previous analyses might have been an artefact. The current data gainsay models in which kinesin steps along the 4-nm tubulin monomer repeat in microtubules. Similarly, our data argue against models that predict substeps arising from rapidly equilibrat- cases the motor head diffuses into a highly strained state, locks on to ing mechanical substates30. The data support one-stroke models33 the track and then detaches at a strain-dependent rate. For myosin, and specifically exclude models that forbid backward steps. Models in this original scheme proved inadequate to account for the mecha- which backward stepping synthesizes ATP31 are unlikely, because they nical behaviour and was modified to include multiple bound predict that the frequency of backward steps should decrease at high mechanical states linked by strain-dependent conformational ATP concentration, whereas we observed the opposite. changes35. For processive kinesins, a model with biased diffusion to Our proposed model for processive kinesins bears striking simi- capture followed by a single, strain-dependent conformational larities to the classical Huxley scheme for myosin34, in that in both change coupled to ADP release seems adequate. It will now be
© 2005 Nature Publishing Group 311 Fig 1: A rocked electron ratchet. The solid lines are an estimation of the confinement potential experienced by electrons as they traverse the experimental ratchet wave-guide shown in Fig 2 (inset). The Fermi distribution of electrons as a function of energy is indicated by the grey regions, where lighter grey corresponds to an occupation probability of less than one. The boldness of the arrows is indicative of the relative strengths of the contributions of high and low energy electrons to the current across the barrier under negative (a) and positive (b) voltages. The dashed lines indicate the spatial distribution of the assumed voltage drop over the barrier, which is scaled with the local potential gradient of the barrier at zero voltage.
Fig. 2: Main: Measured net current as a function of rocking amplitude at a number of temperatures as indicated. Reversals in the to be thicker and smoother (Fig. 1a), suppressingdirection tunnelling,of the net current but asalso a function reducing of rocking wave amplitude,reflection and of electrons, so favouring transmission of electrons withimplicitly high energies. as a function When of temperature, it is tilted are in observed.the other Data direction taken however, from [6]. Inset: A scanning electron microscope image of the the potential deforms to be thinner andratchet sharper device (Fig. (top view). 1b), enhancingThe dark regions tunnelling are etched buttrenches increasing that wave reflection, thus favouring the transmission electricallyof low energy deplete electrons. a two-dimensional In this way, electron the gas two located contributions at the to the AlGaAs/GaAs interface beneath the surface, forming a one- net current, tunnelling through and excitationdimensional over wave-guide. the energy Due barrier, to quantum flow, confinement on average, inside the in opposite directions. By tuning the temperature, rockingwaveguide, voltage an electron or moving Fermi from energy left to right such will that experience one of an these two contributions exceeds the other, the net currentasymmetric direction potential can barrier be chosen.similar to Inthat the shown present in Fig. 1.paper Note thewe point out side gates (marked SG) which are used to tune the height of the that at parameter values where the contributionspotential barrierof the which two is components experienced by of electrons the net moving particle though current are equalthe ratchet. and oppositeThe left point (that contact is, does where not play the a net significant particle role current in the behaviour goes through of the device zero), as a ratchet. a net energy current still exists because the average energy transported in each direction is not the same. In the following we briefly describe the experimental results ofCarter, [6] and N. introduce J., & Cross, a Landauer R. A. model (2005). for Mechanics this experiment of the which kinesin will step. voltage amplitude of V0 ≈ 1 mV. These results may be interpreted by referring to Fig. 1, which illustrates allowthe idea us thatto quantify high energy the electronsheat current and Naturetunnelling generated, 435electrons by (7040),the travel,ratchet. 308–312. on average, "in opposite directions. At low rocking voltage and temperature a positive net electrical current is measured (corresponding to a current of electrons from right to left in Fig. 1), indicating that tunnelling electrons dominate the net current. As either 2the Experimental temperature or quantum voltage is increased, ratchet the energy range of electrons which contribute to transport widens, resulting in a greater contribution from electrons with energies higher than the barrier, leading to a negative Anet scanning current. Whenelectron the microscopetwo contributions (SEM) are equalimage in of magnitude, the experimental the net current quantum undergoes ratchet a sign device reversal. is shown in Fig. 2 (inset). The darker areas are trenches which were defined by shallow wet etching and electron-beam lithography in a two-dimensional electron gas (2DEG) AlGaAs/GaAs heterostructure. This process created3 The Landaueran asymmetric model one-dimensional (1D) wave-guide connecting 2D electron reservoirs. The crucial featureThe Landauer of the equation ratchet expresses is the asymmetric the current flowing point contactthrough ona mesoscopic the right, device which between can be two adjusted reservoirs in width by applyingas a function a voltage of the Fermi to the distribution 2DEG areas of electrons above in and the reservoirs below the and right of the point energy contact dependent that probability serve as side gates (markedthat an electron SG in willthe SEMbe transmitted image). through The side the gatedevice voltage [7]. It tunesmay be the written energy as: of the 1D wave modes, effectively creating an asymmetric2e ∞ energy barrier which is experienced by the electrons as they traverse the wave- I = t(ε,V )[f R (ε,V )− f L (ε,V )]dε , (1) guide. The h left∫ point contact, which is not influenced by the side gates, plays no significant role in −µav determining the behaviour of the device as a ratchet. The dimensions of the device (~1 µm) were much smallerwhere than the length scales for elastic (6 µm) and inelastic (>10 µm) scattering at the temperatures and voltages used in the experiment1 (kBT and eV0 ≤ 1 meV). f L / R (ε,V )= (2) − ε ± eV 2 A low-frequency square-wave1+ exp voltage of amplitude V0 was applied between the two electron reservoirs to k T adiabatically rock the device, andB the resulting net current, averaged over many periods of rocking, was measuredare the Fermi using distributions phase locking in the lefttechniques. and right reservoirs The direction (the upper/lower of the net symbol current in was± in foundall equations to depend upon temperature,corresponds torocking the left amplitude, (L) and right and (R) the reservoirs, applied gate respectively). voltage. Inε isFig. the 2 energy we show of the measurements electrons, for of the net currentconvenience versus chosen the amplitude relative toof thethe averagerocking ofvoltage the chemical for various potentials temperatures on the leftat constant and right side-gate sides, voltage. - Forµav =(smallµL+ µvoltagesR)/2 (Fig. all 1) . three T is thecurves temperature display of parabolic the 2DEG, behaviour, V is the applied as expected bias voltage for aand lowest e=+1.6 order 10 non-linear 19C. Lastly, t( ,V) is the probability that electrons are transmitted across the barrier at a given bias voltage. effect, which atε the lowest two temperatures (0.6 K and 2 K), turns over to reverse direction at a rocking Eqn. (1) assumes that no inelastic scattering occurs inside the device. In addition, we require the applied bias to be much smaller than the Fermi energy. This2 means that the difference between the Fermi distributions will be negligible at low energies, and allows us to use -µav = -0.5(µL+µR) as the lower limit of integration, independent of the voltage sign. Non-linear effects, which form the basis of this particular ratchet effect, have been taken into account by solving the 1D Schrödinger equation to find t(ε,V ) for each positive and negative bias voltage individually. In order to do this, the energy of the lowest mode of
3 Fig. 2: Main: Measured net current as a function of rocking amplitude at a number of temperatures as indicated. Reversals in the direction of the net current as a function of rocking amplitude, and implicitly as a function of temperature, are observed. Data taken from [6]. Inset: A scanning electron microscope image of the ratchet device (top view). The dark regions are etched trenches that electrically deplete a two-dimensional electron gas located at the AlGaAs/GaAs interface beneath the surface, forming a one- dimensional wave-guide. Due to quantum confinement inside the waveguide, an electron moving from left to right will experience an asymmetric potential barrier similar to that shown in Fig. 1. Note the side gates (marked SG) which are used to tune the height of the potential barrier which is experienced by electrons moving though the ratchet. The left point contact does not play a significant role in the behaviour of the device as a ratchet.
voltage amplitude of V0 ≈ 1 mV. These results may be interpreted by referring to Fig. 1, which illustrates the idea that high energy electrons and tunnelling electrons travel, on average, in opposite directions. At low rocking voltage and temperature a positive net electrical current is measured (corresponding to a current of electrons from right to left in Fig. 1), indicating that tunnelling electrons dominate the net current. As either the temperature or voltage is increased, the energy range of electrons which contribute to transport widens, resulting in a greater contribution from electrons with energies higher than the barrier, leading to a negative net current. When the two contributions are equal in magnitude, the net current undergoes a sign reversal.
3 The Landauer model The Landauer equation expresses the current flowing through a mesoscopic device between two reservoirs as a function of the Fermi distribution of electrons in the reservoirs and of the energy dependent probability that an electron will be transmitted through the device [7]. It may be written as: 2e ∞ I = t(ε,V )[f R (ε,V )− f L (ε,V )]dε , (1) h ∫ −µav where 1 f L / R (ε,V )= (2) − ε ± eV 2 1+ exp k BT are the Fermi distributions in the left and rightFig. reservoirs 3: The bold (the curve upper/lower (corresponding symbol to the left in vertical± in all axis) equations is corresponds to the left (L) and right (R) reservoirs,the difference respectively). between the transmissionε is the energy probabilities of the for electrons, +0.5mV for convenience chosen relative to the averageand of –0.5mV the chemical tilting voltages potentials as a function on theof electron left andenergy. right The sides, dotted and dashed curves (corresponding to the right vertical - µav=(µL+µR)/2 (Fig. 1). T is the temperature of the 2DEG, V is the applied bias voltage and e=+1.6 10 19C. Lastly, t(ε,V) is the probability that electronsaxis) areare the transmitted Fermi ‘windows’, across ∆ thef (ε, barrierV0 = 0.5 at mV), a given centred bias on voltage. an equilibrium Fermi energy, µ0 = 11.7meV, for temperatures of 0.3K Eqn. (1) assumes that no inelastic scattering occursand 2K insiderespectively. the device. As the temperature In addition, is changed we require from 0.3Kthe appliedto bias to be much smaller than the Fermi energy.2K, note Thisthat the means integral that of the the product difference of ∆f and between ∆t will make the a Fermi distributions will be negligible at low energies, transitionand allows from us positiveto use -toµ avnegative, = -0.5 (leadingµL+µ Rto) asa reversalthe lower in the limit of integration, independent of the voltage sign. directionNon-linear of the effects, net particle which current form (Eq. 4). the Small basis oscillations of this in particular ∆t ratchet effect, have been taken into account byexist solving for ε > 1 themeV. 1D Schrödinger equation to find t(ε,V ) for each positive and negative bias voltageCarter, individually. N. J., & Cross,In order R.to do A. this, (2005). the energy Mechanics of the lowest of the mode kinesin of step. Nature, 435(7040), 308–312." the experimental wave guide (Fig. 2, inset) was estimated, resulting in the energy barrier shown in Fig. 1 (for more details see [6]). The height of the barrier corresponds to the confinement energy for lowest 3 mode electrons at the narrowest point in the constriction. To obtain the barrier shape at finite voltage, an assumption about the spatial distribution of the voltage drop needs to be made. Arguing that a smooth potential variation can be approximated by a series of infinitesimally small steps, and that a step-like potential change may be assumed to cause a corresponding step-like voltage drop [8], we distribute the voltage drop in proportion to the local derivative of the barrier [9]. It is important to stress that the qualitative quantum behaviour of the ratchet does not depend upon the details of the voltage drop. The present choice, however, has the desirable side-effect that the barrier height remains independent of the sign of the voltage, resulting in the suppression of the classical contribution to the net current. In the present model we include only contributions to transport from the lowest wave mode. The contribution from higher modes is qualitatively similar, and also negligibly small when the Fermi energy is approximately equal to the height of the barrier. The net current is defined as the time average of the current over one period of rocking with a square-wave
voltage of amplitude V0:
net 1 I = [I (V0 )+ I(− V0 )]. (3) 2 This can also be written as: e ∞ I net = ∆t(ε,V )∆f (ε,V )dε , (4) h ∫ 0 0 −µav
where ∆f (ε,V0 ) ≡ f R (ε,V0 )− f L (ε,V0 ), the ‘Fermi window’, is the difference between the Fermi distributions on the right and left of the barrier, and gives the range of electron energies which will contribute to the current. The Fermi window is centred on ε = 0 and has a width which depends upon the
bias voltage and the temperature of the 2DEG (Fig 3). The term ∆t(ε,V0 ) ≡ t(ε,V0 )− t(ε,−V0 ), also shown in Fig. 3, is the difference between the transmission probabilities for an electron with energy ε under positive (Fig. 1b) and negative (Fig. 1a) voltages. Electrons with energies under the barrier height are more likely to flow from right to left when the barrier becomes thinner (under positive voltage, Fig. 1, right), than from left to right when the barrier becomes thicker (under negative voltage, Fig. 1, left). This results in ∆t being negative in this energy range and then positive for energies above the barrier height where above
4 situation is reversed. As ∆f is adjusted (through changing T, V0 or Fermi energy, µ0) to sample the ∆t curve where it is negative rather than positive, the net current will change sign from positive to negative.
4 Energy current The heat change associated with the transfer of one electron to a reservoir with chemical potential µ is given by [10]: ∆Q = ∆U − µ (5) The internal energy, ∆U, associated with the electron is taken with respect to the same global zero as the chemical potential, which is assumed to be unchanged by the electron transfer. The change in heat in the
two reservoirs upon transfer of one electron from the right to the left is then given by ∆QL/R = ε + (µL+µR)/2 - µL/R = ε ± eV/2. Note that the heat removed from one reservoir by an electron crossing the barrier, differs from the heat it adds to the other reservoir by |eV|, as a result of the kinetic energy acquired by the electron in the electric field driving the current. The heat current entering the left and right reservoirs associated with the particle current generated by a voltage V across the device is then obtained from the equation for the electrical current (Eq. 1). This is
done by replacing the electron charge, –e, by a factor of ∆QL/R inside the integral. The heat current can then be written as 2 ∞ qL / R = m (ε ± eV 2)t(ε,V )∆f (ε,V )dε (6) h ∫ −µav The net heat current into the left and right reservoirs over a full cycle of square-wave rocking, net qL / R = (1 2)[qL / R (V0 )+ qL / R (− V0 )], is then:
∞ net 1 qL / R = m [(ε ± eV0 2)t(ε,+V0 )∆f (ε,+V0 )+ (ε m eV0 2)t(ε,−V0 )∆f (ε,−V0 )]dε (7) h ∫ −µav To obtain an intuitive understanding of the action of the ratchet as a heat pump at parameter values where the net particle current goes through zero, it is helpful to rewrite Eq. (7) as:
∞ ∞ net 1 1 eV0 1 1 qL / R = m ε∆t∆fdε + τ∆fdε = m ∆E + Ω (8) h ∫ h 2 ∫ 2 2 −µav −µav
Here τ(ε,V0 ) = (1 2[t(ε,+V0 )+ t(ε,−V0 )]) is the average transmission probability for an electron under net net positive and negative bias voltage. ∆E = qL − qR is the heat pumped from the left to the right sides of the device due to the energy sorting properties of the ratchet, and can be non-zero only for asymmetric
barriers. Ω = (V0 2)[I(+ V0 ) + I(− V0 )] is the ohmic heating, averaged over one cycle of rocking. ∆E can be interpreted as the heat pumping power of the ratchet, averaged over a period of rocking, while net net Ω = qL + qR is the electrical power input, averaged over a period of rocking. We therefore define a coefficient of performance for the ratchet as a heat pump as:
net net ∆E qR − qL χ(T ,V0 ) = = net net (9) Ω qR + qL
Coefficient of Fig. 4: The heat-pumping coefficient of performance of the Performance ratchet model5 shown in Fig. 1, plotted as a function of (%) rocking voltage and temperature. Each point on the surface corresponds to a set of values of rocking voltage, temperature and Fermi energy for which the net particle current goes through zero.
Temperature Bias Voltage (K) (meV) Carter, N. J., & Cross, R. A. (2005). Mechanics of the kinesin step. Nature, 435(7040), 308–312."
It is important to note that ∆E will be trivially non-zero when the net particle current is non-zero because each electron carries heat. This definition of χ therefore only makes sense for parameter values where the net particle current is zero. To evaluate our model potential (Fig. 1) in terms of a heat pump, we have calculated χ for parameter sets where the net current goes through zero (Fig. 4). As required, the heat pumping power goes to zero for small bias and temperature, corresponding to the linear response limit where, by definition, the ratchet cannot work. The positive coefficient of performance indicates that heat is always pumped from left to right for the range of parameters used in the calculation. Small oscillations in ∆t at energies higher than the barrier exist and placing ∆f around these would result in heat being pumped from right to left. The fact that χ is small means that the total heat deposited in each reservoir due to ohmic heating is much larger than the heat pumped from the left to the right sides of the device. This means that, despite the heat pumping action of the ratchet, the internal energy of both reservoirs increases, but one reservoir is heated slightly less than the other. The experimental quantum ratchet may therefore be viewed as a poorly designed refrigerator, where half of the waste heat is deposited inside instead of outside the refrigerated region. For one reservoir to be cooled using the ratchet effect, ∆E would need to be larger than Ω (χ > 1), so that more heat was pumped out of one reservoir than was deposited there as a result of ohmic heating. The low coefficient of performance of the ratchet of [6] as a heat pump is a result of the fact that the potential barrier studied here transmits electrons with a wide range of energies in both rocking directions (all of which contribute to heating), while the ratio ∆t/τ is less than 1%. Thus ohmic heating of each reservoir greatly exceeds the heat pumped from one side to the other. The heat pumping coefficient of performance of the ratchet would be enhanced by designing a potential which only transmitted electrons with energies higher than the equilibrium Fermi energy in one direction, and only transmitted electrons with energies lower than equilibrium Fermi energy in the other direction, so that ∆t/τ ≅1. One way of achieving this may be to employ resonant tunnelling as a means of energy filtering. Resonant tunnelling barriers have in fact been predicted to be able to cool a reservoir when operated in DC mode [11]. An adaptation of this idea to rocked ratchets is currently under investigation.
This work was supported by the Australian Research Council.
6 situation is reversed. As ∆f is adjusted (through changing T, V0 or Fermi energy, µ0) to sample the ∆t curve where it is negative rather than positive, the net current will change sign from positive to negative.
4 Energy current The heat change associated with the transfer of one electron to a reservoir with chemical potential µ is givensituation by [10]: is reversed. As ∆f is adjusted (through changing T, V0 or Fermi energy, µ0) to sample the ∆t curve where it is negative rather than positive, the net current will change sign from positive to negative. ∆Q = ∆U − µ (5) The internal energy, ∆U, associated with the electron is taken with respect to the same global zero as the chemical4 Energy potential, current which is assumed to be unchanged by the electron transfer. The change in heat in the twoThe reservoirs heat change upon associated transfer of with one the electron transfer from of one the electron right to the to a left reservoir is then with given chemical by ∆ potentialQL/R = ε µ+ is (µgivenL+µ Rby)/2 [10]: - µL/R = ε ± eV/2. Note that the heat removed from one reservoir by an electron crossing the barrier, differs from the heat it adds to the other reservoir by |eV|, as a result of the kinetic energy acquired ∆Q = ∆U − µ (5) by the electron in the electric field driving the current. The internal energy, U, associated with the electron is taken with respect to the same global zero as the The heat current entering∆ the left and right reservoirs associated with the particle current generated by a chemical potential, which is assumed to be unchanged by the electron transfer. The change in heat in the voltage V across the device is then obtained from the equation for the electrical current (Eq. 1). This is two reservoirs upon transfer of one electron from the right to the left is then given by ∆QL/R = ε + done by replacing the electron charge, –e, by a factor of ∆QL/R inside the integral. The heat current can ( + )/2 - = eV/2. Note that the heat removed from one reservoir by an electron crossing the thenµL beµ Rwrittenµ L/Ras ε ± barrier, differs from the heat it adds to the other reservoir by |eV|, as a result of the kinetic energy acquired by the electron in 2the∞ electric field driving the current. qL / R = m (ε ± eV 2)t(ε,V )∆f (ε,V )dε (6) h ∫ The heat current entering−µav the left and right reservoirs associated with the particle current generated by a voltage V across the device is then obtained from the equation for the electrical current (Eq. 1). This is The net heat current into the left and right reservoirs over a full cycle of square-wave rocking, donenet by replacing the electron charge, –e, by a factor of ∆QL/R inside the integral. The heat current can qL / R = (1 2)[qL / R (V0 )+ qL / R (− V0 )], is then: then be written as ∞ net 12 ∞ qL / R = m [(ε ± eV0 2)t(ε,+V0 )∆f (ε,+V0 )+ (ε m eV0 2)t(ε,−V0 )∆f (ε,−V0 )]dε (7) qL / R = mh ∫ (ε ± eV 2)t(ε,V )∆f (ε,V )dε (6) h−µav∫ −µav ToThe obtain net anheat intuitive current understanding into the left of andthe actionright ofreservoirs the ratchet over as aa heatfull pumpcycle at of parameter square-wave values rocking, where the netnet particle current goes through zero, it is helpful to rewrite Eq. (7) as: qL / R = (1 2)[qL / R (V0 )+ qL / R (− V0 )], is then: 1 ∞ 1 eV ∞ 1 1 q net = ∞ε∆t∆fdε + 0 τ∆fdε = ∆E + Ω (8) L /netR m 1 ∫ ∫ m qL / R = mh [(ε ± eV0 h2)t(2ε,+V0 )∆f (ε,+V02)+ (ε m2eV0 2)t(ε,−V0 )∆f (ε,−V0 )]dε (7) h−µav∫ −µav −µav
Here τ(ε,V0 ) = (1 2[t(ε,+V0 )+ t(ε,−V0 )]) is the average transmission probability for an electron under To obtain an intuitive understanding of the action of the ratchet as a heat pump at parameter values where net net positivethe net particleand negative current bias goes voltage. through ∆ zero,E = qitL is −helpfulqR is to the rewrite heat Eq.pumped (7) as: from the left to the right sides of the device due to the energy sorting properties of the ratchet, and can be non-zero only for asymmetric 1 ∞ 1 eV ∞ 1 1 barriers. Ωnet= V 2 I + V + I − V 0 is the ohmic heating, averaged over one cycle of rocking. ∆E qL / R (= 0m )[ (ε∆t∆0 fd) ε +( 0 )] τ∆fdε = m ∆E + Ω (8) h ∫ h 2 ∫ 2 2 can be interpreted as− µ theav heat pumping power−µav of the ratchet, averaged over a period of rocking, while Ω = qnet + q net is the electrical power input, averaged over a period of rocking. We therefore define a Here Lτ(ε,V0R ) = (1 2[t(ε,+V0 )+ t(ε,−V0 )]) is the average transmission probability for an electron under coefficientpositive and of negativeperformance bias forvoltage. the ratchet ∆E = asq anet heat− q pumpnet is theas: heat pumped from the left to the right sides of L Carter,R N. J., & Cross, R. A. (2005). Mechanics of the kinesin step. the device due to the∆E energyqnet − sortingqnet properties of the ratchet, and can be non-zero only for asymmetric χ(T ,V ) = = R L (9) barriers. 0 V 2 I Vnet Inet V is Nature the ohmic , heating, 435(7040), averaged over 308–312. one cycle of" rocking. E Ω = ( 0 Ω)[ (+qR0 )++qL(− 0 )] ∆ can be interpreted as the heat pumping power of the ratchet, averaged over a period of rocking, while net net Ω = qL + qR is the electrical power input, averaged over a period of rocking. We therefore define a coefficient of performance for the ratchet as a heat pump as:
net net ∆E qR − qL χ(T ,V0 ) = = net net 5 (9) Ω qR + qL
5