Life's Ratchet: How Molecular Machines Extract Order from Chaos

by Hoffmann, Peter M.

  Helmholtz and Mayer had already realized that any violation of the first or second law by living organisms would mean that a perpetual-motion machine was possible—a machine that could either make energy out of nothing or operate at 100 percent efficiency. Both are impossible. The impossibility of a perpetuum mobile provided the main argument against the existence of a vital force. The argument went as follows: If there were a mysterious force that did not come from physical energy conversions, but was somehow inherent in life itself, such a force could add additional energy to a living system—energy that was not supplied as food. Helmholtz’s obituary in the Proceedings of the Royal Society made a similar argument: “Helmholtz was led to the discussion of this subject [the conservation of energy] by reflections on the nature of the ‘vital force.’” He had convinced himself that if it were true that living organisms could restrain or liberate the action of chemical or physical forces, perpetual motion would be realized.” A life force could create energy out of nothing—a prospect that was clearly absurd. Even in the eighteenth century, most physicists firmly believed that the creation of a perpetual-motion machine was—in principle—impossible.

  The impossibility of a vital force can be directly related to the impossibility of realizing a Maxwellian demon. This is the argument: Why did eighteenth-century biologists postulate a vital force? Because they were looking for something that could explain the “directed activity” seen in living organisms. It was a way to explain “purpose,” to explain the apparent “intelligence” that seemed to operate in humble cells and microbes. This directed activity had to ultimately stem from the directed activity of the molecules that made up the organism. Who could direct this activity? Only a molecular Maxwell demon, who, by using his intelligence, injects purpose into the otherwise senseless motions of molecules.

  But a Maxwell demon is impossible, because a perpetual-motion machine is impossible. The connection between information and entropy, made clear by the thought experiment of the demon, shows that intelligence, purpose, or vital forces can play no role at the molecular scale if the statistical second law of thermodynamics is supposed to hold. Before Maxwell’s demon was conceived, Helmholtz and Mayer had already realized that vital forces had no place in the play of molecules, not even molecules that live in our cells.

  Feynman’s Ratchet

  Our discussion leaves us empty-handed: To make life work, we need something like a Maxwell demon—something that can create directed activity out of chaos. Yet, a Maxwell demon is impossible. The existence of such a being or object would lead to the creation of perpetual-motion machines and violate the second law. The fact that the second law can sometimes be seemingly violated does not help much, either, because it can only be violated by single molecules, at random, and not repeatedly.

  Strictly speaking, the second law is a statistical law, that is, the result of averaging over long times or many molecules. It is therefore not truly violated by rare events that seem to run counter to it. On average, nothing, not even molecules, violates the second law. And you cannot make a functioning machine out of a molecule that only works occasionally and usually moves in the wrong direction.

  Marian Smoluchowski devised another simplified version of Maxwell’s demon, which is very relevant to the mystery of molecular machines. Looking at Maxwell’s demon, he wondered if it would be possible to devise a tiny machine that could extract work from the random motions of molecules in a uniform-temperature environment. There is energy contained in the random thermal motion of atoms, the molecular storm. But how could we harness this energy? Smoluchowski’s machine consisted of a wire with a ratchet attached at the other end. Years later, physicist and Nobel laureate Richard Feynman read Smoluchowski’s work and devised a particularly fruitful incarnation of Smoluchowski’s machine: Feynman’s ratchet.

  What kind of molecular device could channel random molecular motion into oriented activity? Such a device would need to allow certain directions of motion, while rejecting others. A ratchet, that is, a wheel with asymmetric teeth blocked by a spring-loaded pawl, could do the job (Figure 5.2). Old-fashioned watches have ratchets in their windup mechanisms, as do pulleys. As the son of a watchmaker, I have seen tiny ratchets in windup wristwatches many times. The ratchet allows us to wind up our watch but not let it unwind. It allows easy motion in one direction, but blocks motion in the opposite direction. Maybe, nature has made molecular-size ratchets that allow favorable pushes from the molecular storm in one direction, while rejecting unfavorable pushes from the opposite direction. This surely would be a nice way to harvest energy from random motion.

  FIGURE 5.2. Ratchet and pawl—a simple device designed to block motion in one direction (clockwise in this case), but allow it in the other direction (counterclockwise).

  I am sure you already sense that there is something fishy about this suggestion. After all, every other attempt to make machines, doors, or demons that could extract useful work from the molecular storm has failed. But surely, this one looks promising: You just need to adjust the spring of the pawl to the right stiffness, and the wheel should block backward motion, while easily sliding in the forward direction. What could go wrong with this idea?

  The Ratchet Fails

  Alas, Feynman showed that this hypothetical machine was also impossible. To get a molecular-size ratchet and pawl to work, the pawl needs to be spring-loaded so that it moves up and down, notch after notch. To allow the ratchet to rotate, the pawl must be pushed up to the height of one of the teeth. The backward step is restricted because a much larger force is needed to push the pawl up the steep edge of the tooth rather than the gentle incline on the other side. The energy or work is the same in either case, because work equals force times distance. Pushing the pawl up the gentle incline takes less force, but more distance, while pushing it up the steep edge takes more force and less distance.

  FIGURE 5.3. A molecular-size ratchet and pawl would need to have a weak enough spring to allow collisions with molecules to turn it. But such a weak spring would also allow the water molecules to randomly open the pawl, allowing the ratchet to slip backward.

  For the ratchet-and-pawl machine to extract energy from the molecular storm, it has to be easy to push the pawl over one of the teeth of the ratchet. The pawl spring must be very weak to allow the ratchet to move at all. Otherwise, a few water molecules hitting the ratchet would not be strong enough to force the pawl over one of the teeth. Just like the ratchet wheel, the pawl is continuously bombarded by water molecules. Its weak spring allows the pawl to bounce up and down randomly, opening from time to time, allowing the ratchet to slip backward, as shown in Figure 5.3. Worse, because the spring is most relaxed when the pawl is at the lowest point between two teeth, the pawl spends most of its time touching the steep edge of one of the teeth. When an unfavorable hit pushes the ratchet backward just as the pawl has opened, it does not need to go far to end up on the incline of the next tooth, and the spring will push the pawl down the incline—rotating the ratchet backward! Feynman calculated the probabilities of the ratchet’s moving forward and backward and found them to always be the same. The ratchet will move, bobbing back and forth, but it will not make any net headway.

  Any simple device, when stuck in an isolated, uniform-temperature bath can only move randomly—no matter how ingeniously designed. The impossibility of making simple, passive machines that can extract oriented work from random thermal energy, be it Smoluchowski’s trap door or Feynman’s ratchet, is a powerful illustration of the second law of thermodynamics. Work cannot be repeatedly extracted from an isolated reservoir at uniform temperature. If it were possible to make machines that could do this, our energy problems would be solved: Such machines would convert heat in our environment back into ordered mechanical energy. Imagine placing such a contraption into your backyard. It would make the air in the backyard colder and turn the extracted heat into electricity. That would be wonderful, but alas, nobody has been able to violate the second law of thermodynam
ics. At least nobody larger than ten nanometers. And even at ten nanometers, it only happens randomly and rarely.

  So, we seem to have hit a snag: We know our cells are full of tiny machines, and we know they must be molecular in size—but we have not yet explained how they work. Does life have the mysterious power to defy the second law of thermodynamics—a law that rules supreme in the inanimate universe? Or are we missing a crucial ingredient that allows life’s molecular machinery to use the molecular storm without violating this all-powerful physical law?

  6

  The Mystery of Life

  What is the characteristic feature of life? When is a piece of matter said to be alive? When it goes on “doing something,” moving, exchanging material with its environment, and so forth, and that for much longer period than we would expect an inanimate piece of matter to “keep going” under similar circumstances.

  —ERWIN SCHRÖDINGER, WHAT IS LIFE?

  For molecules, moving deterministically is like trying to walk in a hurricane: the forces propelling a particle along the desired path are puny in comparison to the random forces exerted by the environment. Yet cells thrive. They ferry materials, they pump ions, they build proteins, they move from here to there. They make order out of anarchy.

  —R. DEAN ASTUMIAN, “MAKING MOLECULES INTO MACHINES”

  MOLECULAR MACHINES READ AND TRANSLATE DNA; MAKE new machines; operate the processes that makes cells reproduce, transport nutrients, and expel wastes; and help the cell change shape and move about. These tiny machines are the basis of life. But how do they work?

  So far, our attempts to explain how molecular machines can do useful work in the midst of the molecular storm have been foiled by the second law of thermodynamics. No machine, however ingeniously designed, can directly convert the random thermal energy into “oriented, coherent activity,” in the words of Jacques Monod. What are we missing?

  Thou Shalt Not Violate the Second Law

  In a September 2000 spoof in the satirical newspaper The Onion, concerned voters demanded that their legislators repeal the second law of thermodynamics: “‘Why can’t disorder decrease over time instead of everything decaying?’ . . . ‘Is that too much to ask? This is our children’s future we’re talking about!’” If the second law were a dogma imposed by nefarious physicists, Congress might tell us to get rid of it. But we are not that powerful (conspiracy theories about evil scientists notwithstanding). The second law is an inescapable (macroscopic) consequence of the randomizing power of the inescapable (microscopic) molecular storm.

  As explained by physicists Dean Astumian and Peter Hänggi in their 2002 Physics Today article on Brownian motors, a typical molecular motor uses about 100 to 1,000 ATP molecules per second. This translates into performing work at a rate of only 10−16 watts (1 watt = 1 joule/second). It would take 1021 molecular motors to generate as much power as a typical car engine—an unimaginably large number. Yet, this number of molecular machines barely fills a teaspoon—a teaspoon that could generate 130 horsepower! The power density of molecular machines (the power generated per volume the machine occupies) is very large, about 108 watts per cubic meter. A car engine has a power density that is a thousand times smaller. These little machines are extremely efficient: As mentioned earlier, our entire bodies, which are based on these machines, operate on a power “budget” of just 100 watts, the same as a (large) incandescent light bulb.

  If that were not astonishing enough, consider the world these machines inhabit. At the nanoscale, nothing can escape the molecular storm. As Astumian and Hänggi point out, every molecular machine in our bodies is hit by a fast-moving water molecule about every 10−13 seconds. Each collision delivers on average 4.3 × 10−21 joules of energy (the energy is determined by the product of Boltzmann’s constant and body temperature measured in degree Kelvin). This translates into an average power input of more than 10−8 watts. Remember that a molecular machine generates only about 10−16 watts in power. Thus, the power input from the random pounding of water molecules is a hundred million times larger than the power output of our machine! To put this into perspective, compare this to a car in a windstorm. How powerful would a storm need to be to transfer a hundred million times more power to a car than the car’s engine can generate? Taking the air resistance of a car into account, the storm would need a wind speed of an astounding seventy thousand miles per hour to have the same effect that the molecular storm has on our hapless molecular machines! I doubt even a Hummer would get very far against a storm like that.

  A Molecular Hummer

  An important difference between a macroscopic storm and our molecular storm is that the molecular storm has no preferred direction. In other words, every collision with a water molecule comes from a random direction. Storms of our everyday experience blow in a more or less constant direction and can perform useful work (for example, move a sailboat or a windmill). At the nanoscale, not only is the molecular storm an overwhelming force, but it is also completely random. It is difficult to imagine how a molecular storm can serve as an energy source for useful work.

  What to do? We can think of two possibilities. We could build an incredibly robust machine, a kind of molecular super-Hummer, which could make small, measured steps while resisting the thermal chaos surrounding it. Alternatively, we could make a “floppy” machine that works according to the motto “If you can’t beat ’em, join ’em.” That is, make a machine that could “harvest” favorable pushes from the random hits it receives. Since the latter was our approach in Chapter 5 and has so far failed miserably, it may be tempting to go with the first option. But the super-Hummer option would clearly not work, either. Such a molecular machine would need to be made exceedingly “stiff” to withstand the molecular storm. If it were that stiff, how could it move at all? As soon as it wanted to move, it would have to weaken its springs. But if the springs were weakened, the machine would be pushed around in the storm just as before. Moreover, where does motion come from in the first place? As we have seen, the only motion existing at the molecular scale is the molecular storm. How could a molecular machine move, while resisting this motion?

  Let’s stick with the notion that our machine must somehow tame the molecular storm, regardless of whether the machine is robust or floppy. To understand how molecular machines tame the chaos of thermal motion, we first need to learn what these machines actually look like. Obviously, they are not truly the ratchets or car engines we used as convenient analogies. To describe molecular machines, we need to learn a little bit of biochemistry.

  Squeeze to Fit

  Molecular machines must change their shapes to create locomotion. How is this possible? A good starting point is to look at chemical reactions: In chemical reactions, molecules must change shape to combine in novel ways. These changes in shape are driven by a reduction in free energy.

  If this were all there were to driving chemical reactions, they would happen in a snap. Yet, many chemical reactions take time. Your car does not turn into a pile of rust overnight, and your milk does not curdle as soon as you open the carton. The speed of chemical reactions is explained by the concept of activation energy. During any chemical reaction, there is an awkward moment when molecules no longer have their original shapes, but neither do they have their final shapes. They form an intermediate state between their initial state (reactants) and their final state (products). This intermediate state, called the transition state, tends to be uncomfortable for the molecules.

  Figure 6.1 illustrates the transition state with the closing of a cardboard box. One way to close a cardboard box is to overlap each of the four flaps with a neighboring flap. When I try to do this, there is always a tricky moment when I am trying to push down all four flaps at once—one always pops out. But once I’ve succeeded, the box is perfectly happy and shut. Thus we have two comfortable (low-energy) states. Initial state: box (and flaps) open; final state: box closed. To get from one state to the other, we have to cross an intermediate uncomfortable, high
-energy state (shown as a question mark in the figure).

  FIGURE 6.1. Folding a cardboard box to close it. Both the open and the closed box are stable states, but to get from one to the other, we need to push all four flaps together at once, an often quite unstable and awkward state of the system.

  The same holds true for chemical reactions. As reactants turn into products, chemical bonds become strained for a moment before the molecules relax into their final shapes. This strained state has higher energy than either the original or the final state of the transformation. In Figure 6.2, two molecules are not quite compatible, but are “trying” to combine and form a new compound molecule with lower energy. The two molecules can lower their energy if they combine, but first they need to obtain enough energy to pass over the transition state. In the cardboard box example, I needed to push the flaps down with enough force so they snapped into place. In other words, I needed to supply sufficient energy to get past the awkward arrangement of the four flaps pushing against each other. In chemistry, the energy required to cross over the transition state is called the activation energy. It is the minimum amount of energy required to activate a chemical reaction.