Molecular Machines

Once filaments were known to slide, the motors divided into kinds, and the division is not cosmetic.

A team motor detaches after every working stroke. Muscle myosin spends most of its cycle unattached, so a single head accomplishes nothing: force is produced because a thick filament carries hundreds of heads whose cycles are uncorrelated, and at any instant a few per cent of them are pulling. The design suits a situation where the load is large and the distance short, and it is why muscle can be enormously strong and cannot move a single cargo.

A processive motor must never let go. Kinesin, dragging a vesicle along a microtubule for tens of micrometres, has two heads that alternate: the rear head detaches only after the front one has bound, so the molecule is always attached by at least one point and walks hand over hand for a hundred steps or more. That requires the two heads to communicate, which is the field's main unsolved mechanical problem.

A third design abandons the stroke entirely. Howard Berg established in 1973 that a bacterium's flagellum is not a whip but a propeller, turned by a rotary motor embedded in the membrane, driven by protons flowing down their gradient rather than by ATP, spinning at a few hundred revolutions per second and reversing in about a millisecond. The reason rotation rather than reciprocation is the right answer is hydrodynamic: at the Reynolds number of a swimming bacterium, around $10^{-5}$, the equations are time-reversible, so any stroke that merely runs backwards on the return brings the organism back where it started. A corkscrew does not have a return stroke. The constraint belongs to fluid dynamics and the solution to this chapter.

ATP synthase is rotary too, and runs the same machinery in reverse: instead of consuming ATP to turn, it is turned by a proton gradient and makes ATP, which is how almost all of the ATP in the biosphere is produced.

The arithmetic of this field is conducted in piconewtons and nanometres, and it is worth establishing the three numbers that bound everything.

Thermal energy. At body temperature,

$$k_B T = (1.38\times10^{-23})(310) = 4.3\times10^{-21}\ \mathrm{J} = 4.3\ \mathrm{pN\,nm}.$$

One ATP. Hydrolysis under cellular conditions releases about 50 kJ/mol, so per molecule

$$\frac{50{,}000}{6.022\times10^{23}} = 8.3\times10^{-20}\ \mathrm{J} = 83\ \mathrm{pN\,nm},$$

which is about 20 $k_BT$. This is the entire budget for one step of one motor.

One kinesin step. The optical-trap measurements give a step of 8.2 nm — the spacing of tubulin subunits along a microtubule — and a stall force of about 5 to 6 pN. The work done against that load is

$$W = (6\ \mathrm{pN})(8.2\ \mathrm{nm}) = 49\ \mathrm{pN\,nm}.$$

So the efficiency is

$$\frac{49}{83} = 59\%.$$

A protein converting chemical energy to mechanical work at nearly 60% efficiency, which is better than a car engine and close to the thermodynamic constraints of the cycle.

The same calculation for ATP synthase, approached from the other direction, agrees in a way that is worth noticing. The motor turns through 120° for each ATP, three per revolution, against a measured torque of about 40 pN·nm. The work per revolution is $2\pi \times 40 = 251$ pN·nm, so per ATP

$$\frac{251}{3} = 84\ \mathrm{pN\,nm},$$

which is the full energy of one ATP. The enzyme runs at essentially 100% efficiency, and since it normally runs in reverse — using a proton gradient to make ATP — that is what it has to do.

Now the complication that makes this field distinctive. A working stroke is about 50 pN·nm and thermal energy is 4.3 pN·nm, so the ratio is only about twelve. The probability of a thermal fluctuation large enough to undo a step goes as $e^{-12} \approx 6\times10^{-6}$, so backward steps are rare but real, and they are observed. More importantly, during the step itself the motor is being struck by water molecules at every moment; it is not pushing through a vacuum. Estimate the viscous drag on a 5-nm domain moving 8 nm in a millisecond, at $6\pi\eta a v$ with water's viscosity: the force required is of order $10^{-4}$ pN, four orders of magnitude below what the motor generates. Viscosity is not the obstacle. Randomness is.

The resolution, formalised by Frank Jülicher, Armand Ajdari and Jacques Prost in 1997, is that these machines do not fight thermal motion but exploit it. Model the motor as a particle diffusing in an energy landscape whose shape switches as the chemical cycle proceeds. The movement is supplied by thermal collisions, free of charge; what the ATP pays for is biasing the landscape so that a fluctuation in the forward direction is captured and one in the backward direction is not. At this scale a ratchet is the right design, and the stochastic stepping, the occasional backward step and the high efficiency are all consequences of it rather than imperfections. The connection is to the physics of systems held away from equilibrium, which is why single-molecule experiments became the test bed for non-equilibrium statistical mechanics.

The measurements behind those numbers were made possible by an instrument from another thread entirely. An optical trap — a laser focused tightly enough that its intensity gradient pulls a transparent bead towards the focus — exerts and measures forces of exactly a few piconewtons, with position resolution of a nanometre. Arthur Ashkin's invention, described under lasers and photonics, turned out to be matched to biology's scale by coincidence.

Karel Svoboda and Steven Block used one in 1993 to record a single kinesin molecule. The position trace is a staircase: flat dwells punctuated by abrupt 8-nanometre advances, with the dwell times lengthening as ATP is diluted, which shows one ATP per step. Increasing the trap stiffness loads the motor until it stalls, giving the force. This is mechanics on one molecule, and it made the quantities in the previous section measurable rather than inferred.

Ivan Rayment's structure of the myosin head, the same year, explained how a motor amplifies. The chemical event at the nucleotide site is a rearrangement of a few ångströms. It is relayed through a converter domain to a long helix acting as a lever arm, whose far end sweeps about 10 nm. Protein engineers then shortened and lengthened that helix and found the step size changed in proportion — about as direct a confirmation of a mechanical hypothesis as molecular biology affords.

The most startling observation came in 1997, and needed no interpretation at all. Kazuhiko Kinosita's group fixed single F1-ATPase molecules to a slide, attached a fluorescent actin filament to the central shaft of each, and added ATP. Under the microscope the filaments turned — anticlockwise, in discrete 120° steps, three per revolution, exactly as Paul Boyer's kinetic model had predicted and John Walker's structure had implied. A single protein molecule, visibly rotating.

What remains unresolved is the coordination itself. The gating between kinesin's two heads, and the far more elaborate cycle of dynein, involve states lasting microseconds in a molecule that must be under load for the mechanism to work at all — so structures catch the wrong moments and single-molecule traces report position rather than chemistry. The physics of these machines, meanwhile, has become a test bed for non-equilibrium statistical mechanics, since a single motor is a system in which work and heat fluctuate by as much as their averages, and the fluctuation relations that describe that regime were verified on exactly this kind of apparatus.