The Physics and Physical Chemistry of Molecular Machines

Abstract

The concept of a "power stroke"-a free-energy releasing conformational change-appears in almost every textbook that deals with the molecular details of muscle, the flagellar rotor, and many other biomolecular machines. Here, it is shown by using the constraints of microscopic reversibility that the power stroke model is incorrect as an explanation of how chemical energy is used by a molecular machine to do mechanical work. Instead, chemically driven molecular machines operating under thermodynamic constraints imposed by the reactant and product concentrations in the bulk function as information ratchets in which the directionality and stopping torque or stopping force are controlled entirely by the gating of the chemical reaction that provides the fuel for the machine. The gating of the chemical free energy occurs through chemical state dependent conformational changes of the molecular machine that, in turn, are capable of generating directional mechanical motions. In strong contrast to this general conclusion for molecular machines driven by catalysis of a chemical reaction, a power stroke may be (and often is) an essential component for a molecular machine driven by external modulation of pH or redox potential or by light. This difference between optical and chemical driving properties arises from the fundamental symmetry difference between the physics of optical processes, governed by the Bose-Einstein relations, and the constraints of microscopic reversibility for thermally activated processes.

Keywords: Brownian motors; energy landscapes; information ratchet; microscopic reversibility; molecular machines.

Figure 1

a) Energy landscape for the F₁ ATPase from the work of Warshel and Mukherjee.^[30] The fundamental periods Δξ and Δθ for which U(ξ, θ)=U(ξ ± iΔξ, θ ± jΔθ) with i,j = any integers, are shown. b) A “ratchet” representation in terms of two 1D energy profiles with transitions between them for the two chemical states D₁E₂T₃ and E₁T₂D₃ from −240° to +200° (the area enclosed in the bright green dashed box in Figure 1a). The remarkable and salient point is that the mechanism shown by the green arrows (clockwise rotation of 120°) seems by common sense to be far more likely than that shown by the red arrows (counterclockwise rotation by 160°), but, in fact, if ε*=ε these two processes are equally likely, and if ε*<ε counterclockwise rotation (red path) is more likely than clockwise rotation (green path). c) A kinetic lattice model describing the potential energy landscape, where green indicates transition over the barrier (saddle point) ε and red indicates transition over the energy barrier (saddle point) ε*. The dashed box illustrates the part of the kinetic lattice corresponding to the region enclosed in the bright green box in Figure 1a. Four different cycles and their microscopic reverses can be identified, ℱ/ℱ_R in which clockwise rotation is coupled to ATP hydrolysis, ℬ/ℬ_R in which counterclockwise rotation is coupled to ATP hydrolysis, 𝒮/𝒮_R (slip) in which rotation occurs uncoupled to chemistry, and 𝒞/𝒞_R (futile cycling) in which chemistry occurs uncoupled to roation.

Figure 2

Plots of the coupled transport terms in Equation (5a) (solid blue line) and Equation (5b) (dashed lines) are shown for q = e⁻⁷ and e^−X_ξ = e⁻⁵ (dashed green curve) and q = e⁻⁵ and e^−X_ξ = e⁻⁷ (dashed orange curve) with ${\tau }^{-1}={\tau }_0^{-1}{e}^{X_{\theta }/2}$.

Figure 3

Plots of Equation (7) (solid curves) with r₀ = e⁻¹³ for three different values of a. The hyperbolic curve J̃_θ = K(1 − X̃_θ)/(K + X̃_θ) proposed by Hill, with K = 0.2 as used by Hill to fit experimental data obtained for the force versus velocity curve of muscle, is shown as the dashed blue curve.

Figure 4

Energy profiles for the a) forward functional (plus end directed) pathway (green curve) and b) the backward (minus end directed) pathway (red or blue curves) for two-headed myosin V walking on actin are shown (adapted from ref. [35]). The computational results suggest that the “power stroke” in the forward pathway is endergonic (ΔG_ps>0), whereas that in the backward pathway is exergonic (ΔG_ps<0). The forward pathway nevertheless is strongly preferred over the backward pathway because the highest activation barrier in the backward path is much higher than the highest barrier in the forward pathway (green compared with red curve), Δε=ε−ε*. Even when one considers that the energetically costly conformational change in one leg of myosin V is compensated completely by the downhill conformational change in the other leg, still the system goes through the blue curve where ε*>ε, although Δε is much lower than that for the red curve. It should be noted that a complete and simultaneous compensation of the conformational changes in the two legs is unlikely to occur and myosin V most likely adopts a much high energy pathway for back stepping (i.e., red curve).

Figure 5

Schematic illustration of how random energy from a hail storm can make it possible for a very small car to drive uphill given an appropriate brake design (adapted from ref. [8b]). There are two possible mechanisms shown, a) an energy ratchet and b) an information ratchet. The car is modeled as a small green sphere in each, where the fire hydrants act as fiduciary markers. a) In the energy ratchet, the car is equipped with a special brake modelled after a mechanical ratchet shown in the upper left hand corner. When the brake is on, the car is forced into the notch of the ratchet just to the rear of the fire hydrant. When the brake is released, the car tends to roll backward but because of the hail the car also jitters back and forth. Owing to the asymmetry of the ratchet teeth it is more likely for the car to initially move forward past the hydrant to its front than backward past the hydrant to its rear, although eventually the car will move downhill if the brake is kept off for a long time. Reapplying the brake, however, at intermediate times when the car is more likely to have moved the short distance forward past the hydrant to the front than the long distance backward past the hydrant to the rear, forces the car on average forward to the next notch to the front. This process can be repeated, resulting in net uphill motion of the car. Note that the energy comes not from the hail itself, but from the effort expended by the driver in applying the brake—that is, from a power stroke. This mechanism does not require the driver to observe the position of the car in determining whether to apply or release the brake, but only to make sure the brake is not kept off for too long. b) An alternate method involves the driver observing the position of the car relative to the hydrants. If the driver releases the brake only when the car is near the hydrant in front, and applies the brake whenever the car is near the hydrant to the rear, the car inexorably moves uphill, even with a very simple brake that simply prevents slippage and where no force needs be exerted when applying the brake. Here, the car moves uphill by virtue of the information obtained to determine when to apply and release the brake.

Figure 6

A ratchet mechanism inspired by a paper on the bacterial flagellar motor^[52a] is formally similar to a mechanical escapement (upper right hand corner). Similar pictures have been given for many other molecular machines, including myosin moving on actin.^[51] Only the solid arrows are shown for the mechanism in the figure by Xing et al.^[52] where k_on and k_off are described as rates for composite conformational transition and proton association from the periplasm or dissociation to the cytoplasm, respectively. The figure is a trompe l’oeil that leads the naive reader to the conclusion that the slope of the potential dictates the direction of motion and other thermodynamic properties. In fact, the slope does not dictate the direction of rotation—the direction of motion is determined by selection between the pathway to the right shown by the solid arrows and the pathway to the left indicated by the dotted arrows, a selection dictated by the θ dependence of the specificities for binding/release of proton to the cytoplasm/periplasm.^[42,48]

Figure 7

a) Illustration of a catalytically driven shift ratchet. The clear implication is that the slope of the potential dictates the direction of motion. For a system in which the flipping between the two potentials is accomplished externally by, for example, an applied electric field, this is in fact the case. b) However, when the flipping between the two potentials is mediated by the binding of substrate and release of product in a catalytic process, the rate constants are constrained by microscopic reversibility and we see that the direction of motion is governed not by the slopes of the potentials but by the θ dependence of the chemical specificities.^[28]

Figure 8

a) Schematic picture of how a chemical process can drive directed mechanical motion. The mechanism illustrated with the solid arrows involves two “power strokes”, downhill “slides” on the slopes of U₁ and U₂. The mechanism with the dotted blue arrows looks impossible from the point of view of macroscopic physics, but when the activation energies for the chemical processes are equal, ε = ε*, the mechanism with dotted blue arrows in which the motor steps left is just as likely as the mechanism involving the solid blue arrows in which the motor steps right. b) and c) Potential energy surfaces for the ratchet mechanism in Figure 7a for the cases ε* < ε and ε < ε*, respectively, where the most probable trajectories are shown by the white dashed and solid curves.

Figure 9

When a neutral density ball falls on a viscous liquid (a) waves propagate outward from the ball (b,c,d) until finally the liquid, with the ball resting on it, is quiescent. If the ball is removed from the surface by some external means (a′), waves once again propagate outward from where the ball had been (b′,c′,d′), until finally the liquid is again quiescent. This backward mechanism is very different than the microscopic reverse mechanism (gray dashed arrows) for removal of the ball from the surface in which waves spontaneously propagate inward toward the ball (d_R,c_R,b_R) until finally the energy of the wave coalesces at the ball, propelling the ball away from the surface. The external removal of the ball corresponds molecularly to photo-dissociation,^[56] external changes to thermodynamic parameters^[57] (electric field strength, pressure, pH, or redox potential), or to computational disapparition.^[58] From the macroscopic or even mesoscopic perspective, the process d_R → c_R → b_R → a_R seems remarkably unlikely, requiring as it does energy to spontaneously concentrate from many degrees of freedom to the single degree of freedom of the ball. Even so, this is the most likely mechanism for thermally activated dissociation as required by the principle of microscopic reversibility.

Figure 10

Illustration of the difference between a) equilibrium, thermally activated, dissociation in which the most probable path for dissociation of CO is the microscopic reverse of that for association of CO and b) photolysis induced dissociation of CO in which the dissociation is not the microscopic reverse of association.

Figure 11

Schematic figure used to illustrate how input energy can be used to maintain a non-equilibrium steady state in which the relative concentrations in states 2 and 0 are not given by a Boltzmann expression.

Figure 12

Triangle reaction with an external load X_θ and: a) no external driving (ND); b) optical driving (OD); and c) chemical driving (CD). The ratio, r, is the probability of a counterclockwise cycle divided by the probability of a clockwise cycle, and is calculated as the ratio of the product of the counterclockwise rates divided by the product of the clockwise rates. In very bright light, the optical transition coefficients obey the simple relation ω_AE ≈ ω_EA and ω_BE ≈ ω_EB. In contrast, the ratio of each forward and reverse rate constant for each pair of processes that are microscopic reverses of one another must be proportional to the exponential of the free-energy difference of the states they connect [see Eq. (2)].

Figure 13

Anatomy of a “power stroke”. In clockwise cycling, the motor moves through the transition B → A. Seemingly, the energy ΔG_AB provides the power for the power stroke and according to Howard^[82] allows the motor to do work against an applied force or applied torque provided ΔG_AB X_θ, with a maximum efficiency^[51] of η_max = ΔG_AB/X_ξ. This analysis is correct for an optically driven machine, but is totally wrong for a chemically driven motor. In the thermodynamic control limit, irrespective of whether ΔG_AB is positive, negative, or zero, a chemically driven motor can do work against an applied force or applies torque provided X_ξ X_θ, and the maximum eddiciency is bounded only by unity.

Figure 14

Kinetic barrier model for an enzyme. Under the influence of an external driving, ψ(t), which changes the energy of the intermediate E_S and the kinetic barriers (transition states) the enzyme can drive the reaction S⇄P away from rather than towards equilibrium.

Figure 15

Kinetic mechanism for an enzyme, describing the effect of externally driven Poisson fluctuations between two states, ±ψ. The fluctuations can drive the reaction S ⇌ P away from equilibrium.^[57]

Figure 16

Illustration of a synthetic molecular pump. The energy profiles for a ring under both reducing (top) and oxidizing (bottom) conditions are shown, where the overall free-energy difference between having a ring on the collection site is the same irrespective of whether the ring is oxidized (CBPQT⁴⁺) or reduced (CBPQT^2(•+)).

Figure 17

Illustration of a molecular demon that selectively shepherds a ring in an intermediate state to assemble rather than disassemble even though the assembled state is higher in energy. The demon can operate either as a blind energy ratchet (Smoluchowski demon), raising and lowering the energies of states and barriers irrespective of the state of the molecule, or as a sighted information ratchet (Maxwell demon), raising and lowering barriers depending on the state of the molecule. The design principles necessary for synthetic implementation of the two types of “demons” are very different.^[10,49]

Figure 18

Driving by a) an external fluctuation that results in an energy ratchet mechanism; and by b) an energy-releasing catalyzed reaction that results in an information ratchet mechanism.

Figure 19

A kinetic lattice model to describe the mechanism of using energy released by catalysis of a chemical reaction to pump a ring from the bulk onto a high-energy collecting site. The green zigzag path denotes the desired mechanism for coupling.