Thermodynamics of Brownian Ratchets

1. First law of thermodynamics: You can’t win. 2. Second law of thermodynamics: You can’t even break even. 3. Third law of thermodynamics: You can’t stay out of the game.

Source: Voet, Donald; Voet, Judith G. (2010-11-22). Biochemistry, 4th Edition (BIOCHEMISTRY (VOET)) (Page 52). John Wiley & Sons, Inc. 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( Thermodynamics

• First: the change in the internal energy of a closed system is equal to the amount of heat supplied to the system, minus the amount of work done by the system on its surroundings. à The law of conservation of energy can be stated: The energy of an isolated system is constant.

• 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, which implies that it is impossible for any procedure to bring a system to the absolute zero of temperature in a finite number of steps

Source: Wikipedia, http://en.wikipedia.org/wiki/Laws_of_thermodynamics 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.( The(Ratchet(As(An(Engine( Ratchet,(pawl(and(spring.( Feyman’s Ratchet

Feynman’s (Brownian) ratchet: what is it and what was the point of explaining it?

1. Rod 2. Paddle wheel: the fins on the left 3. A pawl: that yellow thing that forces the ratchet not to go back in the other direction 4. A ratchet: that saw toothed gear on the right 5. A spring/bearing (m) low friction, to allow rotation. (From the fun explanation @: http://gravityandlevity.wordpress.com/2010/12/07/feynmans-ratchet-and-the-perpetual- motion-gambling-scheme/) 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  − −  ΩL=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 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  − −  ΩL=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

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

Escherichia coli ATP synthase H. Wang and G. Oster (Nature 396:279-282 1998)

2 Feyman’s Ratchet

Feynman’s (Brownian) ratchet: what is it and what was the point of explaining it?

In one sentence:

The first moral is the Second Law itself: it is impossible to extract directed motion from a random process (a single heat reservoir). Anyone who claims they can do so is either mistaken or a charlatan. (From the fun explanation @: http://gravityandlevity.wordpress.com/2010/12/07/feynmans-ratchet-and-the-perpetual- motion-gambling-scheme/) In other words….

Well, the Brownian ratchet was a ‘thought’ experiment, along with Maxwell’s demon. It was used historically to attempt to explain away the 1) potential existence of a molecular machine that can perform…. 2) perpetual motion, without violating the laws of thermodynamics, namely the conservation of energy and

of molecular machines.

In other words, a fraud J as Feynman found…. and if anyone tries to convince you otherwise … just say:

But, by decree of thermodynamics, Feynman’s ratchet cannot work as a heat engine. It plainly violates the Second Law, which says that useful work can only be obtained by the flow of energy from high to low temperature. This device purports to get energy from a single temperature reservoir: that of the air around it.

Where does it go wrong? (hint: Brownian motion also affects the pawl)

(From the fun explanation @: http://gravityandlevity.wordpress.com/2010/12/07/feynmans-ratchet-and-the-perpetual- motion-gambling-scheme/) Reformulated

The goal of these thought experiments was…

To prove the nonexistence/existence of a perpetual motion machine of the second kind

“a machine which spontaneously converts thermal energy into mechanical work.”

When the thermal energy is equivalent to the work done, this does not violate the law of conservation of energy. However it does violate the more subtle second law of thermodynamics (see also entropy).

The signature of a perpetual motion machine of the second kind is that there is only one heat reservoir involved, which is being spontaneously cooled without involving a transfer of heat to a cooler reservoir.

This conversion of heat into useful work, without any side effect, is impossible, according to the second law of thermodynamics.

From Wikipedia: http://en.wikipedia.org/wiki/Perpetual_Motion_Machine#Classification Rephrased

1. Fallacy: “Oh, you forgot to take into account friction,” they’ll say, and then they’ll give you a short lecture on the First Law of thermodynamics. “Energy is neither created nor destroyed,” they’ll say.

1. The truth, however, is that most perpetual motion machines that you are likely to encounter do not violate energy conservation. Rather, the tricky and persistent scientific “scams” violate the much more nebulous Second Law of Thermodynamics, which says (in one of its formulations):

– It is impossible for a device to receive heat from a single reservoir and do a net amount of work.

(From the fun explanation @: http://gravityandlevity.wordpress.com/2010/12/07/feynmans-ratchet-and-the-perpetual- motion-gambling-scheme/) What are you really trying to say…

Although at first sight the Brownian ratchet seems to extract useful work from Brownian motion, Feynman demonstrated that if the entire device is at the same temperature, the ratchet will not rotate continuously in one direction but will move randomly back and forth, and therefore will not produce any useful work.

A simple way to visualize how the machine might fail is to remember that the pawl itself will undergo Brownian motion. The pawl therefore will intermittently fail by allowing the ratchet to slip backward or not allowing it to slip forward.

è Feynman demonstrated that if T1 = T2, then pawl failure rate = rate of forward ratcheting, è So that no net motion results over long enough periods

From Wikipedia: http://en.wikipedia.org/wiki/Brownian_ratchet 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." If you really wanted the thing to work… • You could make T2 -> T1, such that the temperature is taken from the temperature gradient….

• BUT YOU JUST DID THE OPPOSITE OF WHAT YOU SAID YOU WANTED TO DO!! – You are using TWO thermal reservoirs! There’s also some heat wasted…. – Wait a minute…this looks like a mini heat engine à “a system that performs the conversion of heat or thermal energy to mechanical work by bringing a working substance from a high temperature state to a lower temperature state” – …..which complies with the second law of thermodynamics…which says what? • It is impossible for a device to receive heat from a single reservoir and do a net amount of work. So how does this help us…

This means we have to basically introduce this directed energy somehow, in lay terms (because I’m not a physicist, I like lay terms).

Which is exactly what scientists did recently à almost 40 years after Feynman’s famous talk! 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 Wait! What was all that business with the demon?

OK, this is another thought experiment (too much thinking here) created by physicist James Maxwell.

Why on earth would he do that? Well, he was trying to say that the second law of thermodynamics wasn’t ABSOLUTELY certain à it only has statistical certainty.

So, again, it tries to violate this second law (this is getting personal).

From Wikipedia: http://en.wikipedia.org/wiki/Maxwell's_demon Another fun explanation at http://alienryderflex.com/maxwells_demon.shtml What?

Imagine that you have two compartments: one with hot and the other cold water.

If both compartments were to be mixed together, you’d reach an equilibrium, right? OK, but what if you had this little pesky hypothetical being (let’s call it a demon) that can open and allow molecules through to reverse this mixing/equilibrium process, but without expending energy and separating hot from cold by only allowing molecules hotter-than-average through to one side through a ‘door’.

Check out the new experiment that ‘converts information to pure energy’ http://www.livescience.com/8944-maxwell-demon-converts-information-energy.html Right, the violation again.

In other words, going from a disordered (higher entropy) to an ordered system (lower entropy) without expending energy.

This violates the second law of thermodynamics…which says what? à entropy should not decrease in an isolated system à It is impossible for a device to receive heat from a single reservoir and do a net amount of work. à it is impossible to extract directed motion from a random process (a single heat reservoir).

à It is impossible to extract an amount of heat QH from a hot reservoir and use it all to do work W . Some amount of heat QC must be exhausted to a cold reservoir. This precludes a perfect heat engine.

Check out the new experiment that ‘converts information to pure energy’ http://www.livescience.com/8944-maxwell-demon-converts-information-energy.html 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, Maxwells demon.[14,15,20]

Angew. Chem. Int. Ed. 2007, 46, 72 – 191 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.angewandte.org 75 Angewandte Molecular Devices Chemie

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 Feynmans 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. Feynmans 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 characteristics device (Figure of 4b) this consists size regime of a miniature axle with vanes attached to one end, surrounded by direction. Exactly analogous to Smoluchowskis 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 Feynmans 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 ideas 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 characteristics 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), progressively 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 flashing 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 potential 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 flashing 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 potential 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 particles interaction with it—are applied globally It appears to us that information-ratchet mechanisms of and independent of the particles 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 Maxwells 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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Figure 9. A rocking ratchet.[39f] a) The Brownian particles start out in energy minima on the potential-energy surface with the energy barriers

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 particles interaction with it—are applied globally It appears to us that information-ratchet mechanisms of and independent of the particles 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 Maxwells 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