Stochastically pumped adaptation and directional motion of molecular machines

Abstract

Recent developments in synthetic molecular motors and pumps have sprung from a remarkable confluence of experiment and theory. Synthetic accomplishments have facilitated the ability to design and create molecules, many of them featuring mechanically bonded components, to carry out specific functions in their environment—walking along a polymeric track, unidirectional circling of one ring about another, synthesizing stereoisomers according to an external protocol, or pumping rings onto a long rod-like molecule to form and maintain high-energy, complex, nonequilibrium structures from simpler antecedents. Progress in the theory of nanoscale stochastic thermodynamics, specifically the generalization and extension of the principle of microscopic reversibility to the single-molecule regime, has enhanced the understanding of the design requirements for achieving strong unidirectional motion and high efficiency of these synthetic molecular machines for harnessing energy from external fluctuations to carry out mechanical and/or chemical functions in their environment. A key insight is that the interaction between the fluctuations and the transition state energies plays a central role in determining the steady-state concentrations. Kinetic asymmetry, a requirement for stochastic adaptation, occurs when there is an imbalance in the effect of the fluctuations on the forward and reverse rate constants. Because of strong viscosity, the motions of the machine can be viewed as mechanical equilibrium processes where mechanical resonances are simply impossible but where the probability distributions for the state occupancies and trajectories are very different from those that would be expected at thermodynamic equilibrium.

The molecular machines necessary for life must carry out their function in the fluctuating environment of a biological cell. Nowhere is the dynamic aspect of biology at the molecular level more evident than in the bilayer membrane surrounding most cells and organelles, a small portion of which is shown schematically in Fig. 1. Three transmembrane proteins are included in the depiction: (i) an ion channel that provides a conduit for conduction of ions across the membrane; (ii) a molecule that facilitates the transport of some substance, S, across the membrane; and (iii), an Na⁺K⁺ ATPase that uses energy from ATP hydrolysis to maintain ion gradients across the membrane. The transporter acts as a catalyst to reduce the barrier for transport of S. In and of itself, the transporter cannot drive the substance S from low to high chemical potential. The transporter, however, is near a possible source of energy, the ion channel. The ion channel provides a path for conduction of ions from high to low electrochemical potential, thus dissipating energy. Every time the ion channel opens or closes, the local electric field across the membrane changes, and because the flow of ions down the electrochemical gradient when the channel is open dissipates energy, the resulting electrical noise is a nonequilibrium fluctuation. These fluctuations, when coupled to conformational transitions of the transporter, can provide energy to drive transport of S from low to high chemical potential, even if S itself is uncharged (1, 2). The mechanism by which this occurs is known as electroconformational coupling (3–5). The ion gradient is formed and maintained by an active ion pump, the Na⁺, K⁺ ATPase. The sensitivity of many membrane proteins to the membrane potential has provided a means for studying their kinetics by examining the frequency response of transport processes to applied oscillating electric fields or other perturbation (6–12). The theoretical investigations also show another, perhaps more fundamentally important, nonlinear effect (13). The zero-average applied fluctuating fields cause a shift in the average state occupancies of many membrane proteins, thereby creating a nonequilibrium steady state. The fluctuations also can provide energy to power the membrane protein for accomplishing work on the environment by pumping a substance from low to high chemical potential. These results motivate investigation of two very general questions—how do external fluctuations drive chemical fluxes and ultimately lead to the formation of nonequilibrium steady states and to adaptation of the system to the fluctuating environment and how can these effects be used to design molecular machines that harness energy from external fluctuations to drive directional motion and to maintain themselves in a nonequilibrium steady state. It is on these broad and overarching questions and on what synthetic molecular machines can teach us about biomolecular machines that we focus in the present perspective article.

Fig. 1.

Schematic illustration of a transporter molecule in a membrane that facilitates the flow of some substance S from high chemical potential to low. Also shown are an ion channel and a Na⁺K⁺ ATPase. Nonequilibrium fluctuations can allow the transporter to act as an active molecular machine that drives the transport process uphill. The theory describing this effect was originally termed electroconformational coupling.

Thermodynamics of Interaction Between a Molecule and Its Environment

Molecules interact with their environment through intensive thermodynamic parameters $F_i$, the so-called generalized thermodynamic forces that include temperature (T), pressure (p), membrane potential ($\mathrm{\Delta }\mathit{\psi }$), applied mechanical force ($\overrightarrow{F}$), chemical potentials of ligands (${\mathit{\mu }}_L$), etc. The amplitude of the effect of the generalized forces depends on the canonically conjugate extensive thermodynamic parameters (known as generalized displacements), ${\mathit{\theta }}_{\mathit{i}}$, such as entropy (S), molecular volume (V), displacement charge (q), extension ($\ell$), and number of ligand molecules bound (${\mathit{n}}_L)$, respectively. The generalized forces influence the relative free energies of different states of the macromolecule according to the relation (14) $\mathit{dG}={\displaystyle {\sum }_{\mathit{i}}\mathit{\theta }\mathit{d}{\mathit{F}}_{\mathit{i}}}-\mathcal{A}\hspace{0.17em}\mathit{d\xi }$, where $\mathcal{A}$ is the affinity and $\mathit{\xi }$ is the extent of reaction, as introduced by de Donder. The extent of reaction is zero when there is only substrate and unity when there is only product. In a nonfluctuating environment, $\mathcal{A}$ = 0 when $\mathit{dG}/\mathit{d}\mathit{\xi }=0$, and therefore, the extent of reaction connects Gibb’s free energy with the concept of chemical affinity. The change in internal energy due to the response of the system is given by $\mathit{dU}={\displaystyle {\sum }_{\mathit{i}}{\mathit{F}}_{\mathit{i}}\mathit{d}{\mathit{\theta }}_{\mathit{i}}}$. The extensive parameters are defined in terms of the change in free energy due to a change in the corresponding intensive parameter, with all other intensive parameters held constant, ${\mathit{\theta }}_{\mathit{i}}\equiv {\partial \mathit{G}/\partial {\mathit{F}}_{\mathit{i}}|}_{{\mathit{F}}_{\mathit{j}\ne \mathit{i}}}$.

Robertson and Astumian (15, 16) generalized the electroconformational coupling theory to describe a Michaelis–Menten enzyme (Fig. 2) that catalyzes the reaction $\text{S}\begin{array}{c}{\text{K}}_{\text{eq}}\\ \rightleftharpoons \end{array}\text{P}$ by the mechanism $\text{S}+\text{E}\rightleftharpoons {\text{E}}_{\text{L}}\rightleftharpoons \text{E}+\text{P}$ in the presence of an externally driven rapid fluctuant (Fig. 2). The steady-state ratio between S and P is given by the relation

$${\frac{\left[\text{P}\right]}{\left[\text{S}\right]}|}_{\text{ss}}={\text{K}}_{\text{eq}}\langle e^{\mathrm{\Delta }\theta \left[\delta F\left(t^{\ast }\right)-\delta F\left(t^{\ast \ast }\right)\right]}\rangle ,$$ [1]

where here and elsewhere in this paper, energies are given in units of the thermal energy $k_{\text{B}}T$. The ${\text{K}}_{\text{eq}}$ is the standard equilibrium constant between substrate and product. The factor $\mathrm{\Delta }\theta ={\theta }_E-{\theta }_{E_L}$ is the change in the extensive molecular parameter (generalized displacement) due to binding that characterizes the response of the catalyst to the fluctuating generalized thermodynamic force $\delta F\left(t\right)$. The quantity $\langle e^{\hspace{0.17em}\mathrm{\Delta }\theta \left[\delta F\left(t^{\ast }\right)-\delta F\left(t^{\ast \ast }\right)\right]}\rangle$ is the weighted average of the exponential of the “excess” work, $\mathrm{\Delta }\theta \left[\delta F\left(t^{\ast }\right)-\delta F\left(t^{\ast \ast }\right)\right]$, exchanged between the source of the fluctuations, the catalyst, and the environment in the forward conversion $\text{S}\to \text{P}$ in a path characterized by the value of the force, $\delta F\left(t^{\ast }\right)$, at which substrate binds and the force, $\delta F\left(t^{\ast \ast }\right)$, at which the product dissociates. This excess work can be positive, negative, or zero for any one path, and depending on the interaction with the energy barriers (16), the net effect of the fluctuations can favor either substrate or product relative to the equilibrium case.

Fig. 2.

Illustration of the effect of environmental fluctuations on the energy levels for a Michaelis–Menten enzyme. We set the free energy of the unbound enzyme state $\text{E}$ as the quantity against which all other energies are measured, including the free energy, $\mathrm{\Delta }G$, of the bound state when $\delta F=0$ as well as the reference chemical potential ${\mu }^0$ and the difference in chemical potentials of P and S: $\ln \left({\text{K}}_{\text{eq}}\left[\text{S}\right]/\left[\text{P}\right]\right)=\mathrm{\Delta }\mu$.

The deceptively simple result, Eq. 1, is quite general and represents an extension of the standard equilibrium law of chemistry to systems where external fluctuations act to continuously feed energy into the molecular machines—enzymes—that catalyze chemical reactions. Time-dependent fluctuations $\delta F\left(t\right)$, whether arising from an electric field generator plugged into an electric wall socket (7), gradients near a thermal vent (17), or external modulation of pH or redox potential (18), provide a source of energy that can power a molecular machine to drive formation of a nonequilibrium steady state. To better reflect the generality of the theory behind Eq. 1, recent usage has shifted to describe the mechanism by which macromolecular catalysts harvest energy from a fluctuating environment as stochastic conformational pumping (19, 20) rather than electroconformational coupling. The stochastic pumping theory provides a general mechanism for how molecular systems adapt to their fluctuating environment. The interaction between the external fluctuations and the transition state energies plays a central role in determining the effect of fluctuations on a chemical system (16) and in particular, in determining the steady-state concentrations. This is very different from in equilibrium, where the concentration ratios are determined solely by the free energies of the states themselves.

Importance of Kinetic Asymmetry

Consider the simple elementary reaction, $A\begin{array}{c}{k}_{AB}\\ \rightleftharpoons \\ k_{BA}\end{array}B$, characterized by the free energies, $G_A^0$ and $G_B^0$, of states A and B, respectively, as well as by the transition state energy, $G_{AB}^{‡,0}$, all evaluated at some reference condition. The rate constants are $k_{AB}^0=\mathbb{A}{\mathit{e}}^{G_{AB}^{‡}-G_A^0}$ and $k_{BA}^0=\mathbb{A}{\mathit{e}}^{G_{AB}^{‡}-G_B^0}$, where $\mathbb{A}$ is a frequency factor. The equilibrium ratio between the concentrations of A and B is ${\left[B\right]/\left[A\right]|}_{\text{eq},0}=k_{AB}^0/k_{BA}^0={\mathit{e}}^{G_A^0-G_B^0}$. If one of the intensive thermodynamic parameters is different from the reference value by an amount $\mathit{\delta }\mathit{F}$, the free energies of the states are $G_A=G_A^0+\mathit{\delta }\mathit{F}{\mathit{\theta }}_A$ and $G_B=G_B^0+\mathit{\delta }\mathit{F}{\mathit{\theta }}_B$, and the activation energy is $G^{‡}=G^{‡,0}+\mathit{\delta }\mathit{F}{\mathit{\theta }}^{‡}$. The rate constants can be expressed as $k_{AB}=k_{AB}^0{\mathit{e}}^{\mathit{\delta }\mathit{F}({\mathit{\theta }}_A-{\mathit{\theta }}^{‡})}$ and $k_{BA}=k_{BA}^0{\mathit{e}}^{\mathit{\delta }\mathit{F}({\mathit{\theta }}_B-{\mathit{\theta }}^{‡})}$, and the equilibrium ratio is

$${\frac{\left[B\right]}{\left[A\right]}|}_{\text{eq},\mathit{\delta }\mathit{F}}\hspace{0.17em}=\hspace{0.17em}\underset{\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{p}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}}{\underbrace{\frac{k_{AB}^0{e}^{\mathit{\delta }\mathit{F}\left({\mathit{\theta }}_{\mathit{A}}-{\mathit{\theta }}^{‡}\right)}}{k_{BA}^0{e}^{\mathit{F}\left({\mathit{\theta }}_{\mathit{B}}-{\mathit{\theta }}^{‡}\right)}}={\mathit{e}}^{{\mathit{G}}_{\mathit{A}}^0-{\mathit{G}}_{\mathit{B}}^0}{\mathit{e}}^{\mathit{\delta }\mathit{F}\left({\mathit{\theta }}_{\mathit{A}}-{\mathit{\theta }}_{\mathit{B}}\right)}}}.$$ [2]

This ratio is independent of the properties of the transition state (i.e., does not depend on $\mathbb{A},{\mathit{\theta }}^{‡},\text{or}\hspace{0.17em}{G}_{AB}^{‡,0}$). The second equality in Eq. 2 is known as microscopic reversibility and holds at every instant. If $\mathit{\delta }\mathit{F}$ is constant or fluctuates very slowly, the concentrations are at every instant given by the equilibrium value, and the concentration ratio is independent of the properties of the transition state and specifically does not depend on ${\mathit{\theta }}^{‡}$. If, however, the intensive parameter $\mathit{F}$ is caused to fluctuate rapidly, the system never reaches thermodynamic equilibrium, because there is a continual flow of energy as As are converted into Bs and vice versa in the always changing energy landscape. The difference between low-frequency and high-frequency fluctuations in the system $A\begin{array}{c}{k}_{AB}\\ \rightleftharpoons \\ k_{BA}\end{array}B$ from a kinetic perspective can easily be understood by considering the difference between the average of the ratio of rate constants $\overline{{\mathit{k}}_{AB}/{\mathit{k}}_{BA}}$ (low frequency) and the ratio of the averages of the rate constants $\left(\overline{{\mathit{k}}_{AB}}\right)/\hspace{0.17em}\left(\overline{{\mathit{k}}_{BA}}\right)$ (high frequency) (16). The system does attain an apparent “steady state” in the presence of high-frequency fluctuations, where the observed ratio of the concentrations does not change.

The condition for steady state is $\left[A\right]\overline{{\mathit{k}}_{\mathit{AB}}^0{\mathit{e}}^{\mathit{F}\left(\mathit{t}\right)({\mathit{\theta }}_{\mathit{A}}-{\mathit{\theta }}^{‡})}}\hspace{0.17em}=\left[B\right]\overline{{\mathit{k}}_{\mathit{BA}}^0{\mathit{e}}^{\mathit{F}\left(\mathit{t}\right)({\mathit{\theta }}_{\mathit{B}}-{\mathit{\theta }}^{‡})}}$, where the overbar denotes a time average. Using microscopic reversibility, Eq. 2, this relation becomes

$${\frac{\left[\mathit{B}\right]}{\left[\mathit{A}\right]}|}_{\text{ss},\mathit{\delta }\mathit{F}}={\mathit{e}}^{{\mathit{G}}_{\mathit{A}}^0-{\mathit{G}}_{\mathit{B}}^0}\frac{{\displaystyle {\int }_{\mathit{t}}^{\mathit{t}+\mathit{T}}{e}^{\mathit{\delta }\mathit{F}\left(\mathit{t}\right)\left({\mathit{\theta }}_{\mathit{B}}-{\mathit{\theta }}^{‡}\right)}{\mathit{e}}^{\mathit{\delta }\mathit{F}\left(t\right)\left({\mathit{\theta }}_{\mathit{A}}-{\mathit{\theta }}_{\mathit{B}}\right)}\mathit{dt}}}{{\displaystyle {\int }_{\mathit{t}}^{\mathit{t}+\mathit{T}}{e}^{\mathit{\delta }\mathit{F}\left(\mathit{t}\right)\left({\mathit{\theta }}_{\mathit{B}}-{\mathit{\theta }}^{‡}\right)}\mathit{dt}}}\equiv {\mathit{e}}^{{\mathit{G}}_{\mathit{A}}^0-{\mathit{G}}_{\mathit{B}}^0}\langle {\mathit{e}}^{\mathit{\delta }\mathit{F}\left(\mathit{t}\right)\left({\mathit{\theta }}_{\mathit{A}}-{\mathit{\theta }}_{\mathit{B}}\right)}\rangle .$$ [3]

The mathematical form is that of an ensemble average of the exponent of the excess work done in the transition, where the integration is carried out over all possible values of $\delta F$ at the moment of transition from A to B weighted by the rate constant for transition from B to A at that value of $\delta F$. The expression is simple to evaluate in three cases. If ${\theta }^{‡}={\theta }_B$, the ratio is $e^{G_A^0-G_B^0}\overline{e^{\delta F\left(t\right)({\theta }_A-{\theta }_B)}}$; if ${\theta }^{‡}={\theta }_A$, the ratio is $e^{G_A^0-G_B^0}{\left[\overline{e^{\delta F\left(t\right)({\theta }_A-{\theta }_B)}}\right]}^{-1}$, and if ${\theta }^{‡}={\theta }_B+{\theta }_A/2$, the ratio simplifies to $e^{G_A^0-G_B^0}$. The ensemble average $\langle e^{\delta F\left(t\right)\left({\theta }_A-{\theta }_B\right)}\rangle$ that determines the steady-state ratio is governed by the interaction between the transition state and the external fluctuation and is not a state function. We can conveniently parametrize the kinetic asymmetry in terms of an apportionment factor $z$, where ${\theta }^{‡}=z\hspace{0.17em}{\theta }_A+(1-z){\theta }_B$. The rate constants can then be written $k_{AB}=k_{AB}^0{e}^{z\mathrm{\Delta }\theta \delta F}$ and $k_{BA}=k_{BA}^0{e}^{(1-z)\mathrm{\Delta }\theta \delta F}$, where $\mathrm{\Delta }\theta =\left({\theta }_{{\rm A}}-{\theta }_B\right)$. The factor $z$ is usually a number between zero and one. It is possible for the $z$ to depend on $F$, as is often the case in biological systems (21), but we will not explicitly discuss this effect in this paper. The kinetic asymmetry can be used by evolution to great effect in designing enzymes—macromolecular catalysts—to harvest energy from external fluctuation to do chemical, mechanical, and/or electrical work on their environment.

Stochastically Pumping an Enzyme (15, 16)

Consider the Michaelis–Menten enzyme mechanism shown in Fig. 2 with fluctuations of a single intensive parameter $\mathrm{\delta }\mathit{F}\left(\mathrm{t}\right)$ as discussed in the previous section. The rate constants for the binding and dissociation of S and P are ${\mathit{k}}_{+\mathit{i}}\left(\mathit{t}\right)={\mathit{k}}_{+\mathit{i}}^0{\mathit{e}}^{-{\mathit{z}}_{\mathit{i}}\mathrm{\Delta }\mathit{\theta }\mathit{\delta }\mathit{F}\left(\mathit{t}\right)}$ and ${\mathit{k}}_{-\mathit{i}}\left(\mathit{t}\right)={\mathit{k}}_{-\mathit{i}}^0{\mathit{e}}^{(1-{\mathit{z}}_{\mathit{i}})\mathrm{\Delta }\mathit{\theta }\mathit{\delta }\mathit{F}\left(\mathit{t}\right)}$. We focus on a situation where ${\mathit{\theta }}_{\text{S}}={\mathit{\theta }}_{\text{P}}$ and hence, where ${\mathit{k}}_{+1}\left(\mathit{t}\right){\mathit{k}}_{-2}\left(\mathit{t}\right)/{\mathit{k}}_{+2}\left(\mathit{t}\right){\mathit{k}}_{-1}\left(\mathit{t}\right)={\mathrm{K}}_{\text{eq}}$ at every instant. The free energy released/absorbed for each turnover at fixed conditions is the difference in chemical potential of substrate and product, $\mathrm{\Delta }\mathit{\mu }=\left[\mathrm{S}\right]/\left[\mathrm{P}\right]\ln \left({\mathrm{K}}_{\text{eq}}\right)$. When the binding of S and release of P occur at different times in the presence of the external fluctuations, an “excess work” ${\mathcal{W}}_{\mathit{S}}=\mathrm{\Delta }\mathit{\theta }\left[\mathit{\delta }\mathit{F}\left({\mathit{t}}^{\ast }\right)-\mathit{\delta }\mathit{F}\left({\mathit{t}}^{\ast \ast }\right)\right]$ is exchanged between the enzyme and the environment. The ratio of the forward and reverse rate constants for such a trajectory is ${\mathit{k}}_{+1}\left({\mathit{t}}^{\ast }\right){\mathit{k}}_{-2}\left({\mathit{t}}^{\ast \ast }\right)/{\mathit{k}}_{+2}\left({\mathit{t}}^{\ast \ast }\right){\mathit{k}}_{-1}\left(t^{\ast }\right)={\mathrm{K}}_{\text{eq}}{\mathit{e}}^{-\mathrm{\Delta }\mathit{\theta }\left[\mathit{\delta }\mathit{F}\left(t^{\ast }\right)-\mathit{\delta }\mathit{F}\left({\mathit{t}}^{\ast \ast }\right)\right]}$. The probability for a conversion of $\text{S}\to \text{P}$, in which S binds at time $\left(t^{\ast }\right)$ and P dissociates at time $\left(t^{\ast \ast }\right)$, is the product of the rates divided by a normalization factor $\mathit{Z}$, $\mathit{\pi }\left(\mathrm{S}\stackrel{\mathit{S}}{\to }\mathrm{P}\right)=\left[\mathrm{S}\right]{\mathit{k}}_{+1}\left(t^{\ast }\right){\mathit{k}}_{-2}\left(t^{\ast \ast }\right)/\mathit{Z}$, and the probability for the microscopic reverse process (22) is the product of the rates, $\mathit{\pi }\left(\mathrm{P}\stackrel{{\mathit{S}}^{†}}{\to }\mathrm{S}\right)=\left[\mathrm{P}\right]{\mathit{k}}_{-1}\left(t^{\ast }\right){\mathit{k}}_{+2}\left(t^{\ast \ast }\right)/\mathit{Z}$. Using the requirement that these two probabilities, averaged over all possible values of $\mathit{\delta }\mathit{F}\left(t^{\ast }\right)$ and $\mathit{F}\left(t^{\ast \ast }\right)$, must equal one another at steady state, we immediately obtain Eq. 1. A numerical example illustrating the effect of fluctuations is shown (see Fig. 5), where the chemical reaction starts at equilibrium and is driven to a nonequilibrium steady state by the enzyme that serves as a conduit to allow energy to flow from the fluctuations to the chemical reaction. Consider this result more deeply (16). We start with the chemical reaction at equilibrium where $\left[\text{P}\right]/(\left[\text{S}\right]+\left[\text{P}\right])={\mathrm{K}}_{\text{eq}}/(1+{\mathrm{K}}_{\text{eq}})$ and with concentrations in the millimolar range. External fluctuations $\mathrm{\delta }\mathit{F}\left(\mathit{t}\right)$ introduce latent energy, but with ${\mathit{\theta }}_{\text{S}}={\mathit{\theta }}_{\text{P}}$, there is no mechanism for the chemical reaction to absorb this energy; therefore, the reaction remains in equilibrium. When a tiny amount of an enzyme that catalyzes the reaction and that has conformational transitions with $\mathrm{\Delta }\mathit{\theta }\ne 0$ is added, the reaction is driven away from equilibrium until reaching a steady state, where $\left[\text{P}\right]/\left[\text{S}\right]+\left[\text{P}\right]={\mathrm{K}}_{\text{eq}}\langle {\mathit{e}}^{{\mathcal{W}}_S}\rangle /(1+{\mathrm{K}}_{\text{eq}}\langle {\mathit{e}}^{{\mathcal{W}}_S}\rangle )$. The ${\mathcal{W}}_{\mathit{S}}=-\mathrm{\Delta }\mathit{\theta }\left[\mathit{F}\left(t^{\ast }\right)-\mathit{\delta }\mathit{F}\left(t^{\ast \ast }\right)\right]$ is the excess work exchanged with the environment in the conversion $\text{S}\stackrel{S}{\to }\text{P}$ in the path $\mathit{S}$ specified by binding S when $\mathit{\delta }\mathit{F}=\mathit{F}\left(t^{\ast }\right)$ and dissociation of P when $\mathit{\delta }\mathit{F}=\mathit{\delta }\mathit{F}\left(t^{\ast \ast }\right)$. The ensemble average

$$\frac{{\displaystyle {\sum }_{\mathit{S}}\hspace{0.17em}\mathit{\pi }\left(\mathrm{P}\stackrel{{\mathit{S}}^{†}}{\to }\mathrm{S}\right){\mathit{e}}^{\hspace{0.17em}{\mathcal{W}}_{\mathit{S}}}}}{{\displaystyle {\sum }_{\mathit{S}}\hspace{0.17em}\mathit{\pi }\left(\mathrm{P}\stackrel{{\mathit{S}}^{†}}{\to }\mathrm{S}\right)}}\equiv \langle {\mathit{e}}^{\hspace{0.17em}{\mathcal{W}}_{\mathit{S}}}\rangle$$ [4]

can be greater than or less than one depending on the kinetic asymmetry (16). The path ${\mathit{S}}^{†}$ is the microscopic reverse of $\mathit{S}$. The excess work and the path probabilities for any specific trajectory $\mathit{S}$ obey the relations ${\mathcal{W}}_{{\mathit{S}}^{†}}=-{\mathcal{W}}_{\mathit{S}}$ and $\mathit{\pi }\left(\mathrm{S}\stackrel{\mathit{S}}{\to }\mathrm{P}\right)=\mathit{\pi }\left(\mathrm{P}\stackrel{{\mathit{S}}^{†}}{\to }\mathrm{S}\right){\mathit{e}}^{\hspace{0.17em}{\mathcal{W}}_{\mathit{S}}}\hspace{0.17em}{\mathit{e}}^{{\mathit{\mu }}_S-{\mathit{\mu }}_P}$, respectively.

Fig. 5.

Comparison of two molecular switches that differ only by the spacer [(A) blue (electrostatic) or (B) green (steric) ball] that determines the energy barrier between the two recognition sites. (C) The redox potential of the solution controls the redox state of the mobile ring. The behaviors at equilibrium or in the presence of slow fluctuations of the redox potential are the same, but the responses to very fast fluctuations of the redox potential, however, are very different.

Sinusoidal oscillations were used in the numerical example shown in Fig. 3, but random fluctuations, including fluctuations with random lifetimes, random amplitudes, random lifetimes and amplitudes, and even pseudowhite noise can drive the formation of a nonequilibrium steady state (23), an effect that was confirmed experimentally for several cases (24). The key requirement is that the fluctuation be autonomous (i.e., externally generated), leading to a situation in which the probability for a transition is strongly correlated with the energy released in the transition, a correlation that requires kinetic asymmetry.

Fig. 3.

Numerical calculation of the effect of a sinusoidal forcing $\delta F\left(t\right)=\cos \left(2\pi ft\right)$ with $f=400/s$ on the enzyme shown in Fig. 2. All four rate constants were set to 1/s when $\delta F=0$, and $\mathrm{\Delta }\theta$ was taken to be one. The kinetic apportionment constants were chosen as indicated on the graph.

A simple two-state stochastic pump can, with fluctuations that take on only specific values, be mapped onto a Markov kinetic process described in terms of transitions between distinct kinetic states (23, 25). For mechanisms with more than two enzyme states and hence, with more than one relaxation time, the effect of the fluctuation can be a nonmonotonic function of frequency (15, 16). This nonmonotonic behavior results from a dispersive windowing between relaxation times of the system and is not a resonance. In water, the viscous drag and thermal noise totally dominate any inertial forces that drive the physical motions of molecules (26). Mechanical resonances in this regime are simply impossible, and even chemical resonances (27) require very specific kinetic features not present for most catalysts. Analogizing the effect of external oscillations or fluctuations on chemical systems with the shattering of a wine glass by an opera singers voice or with the collapse of a bridge is incorrect and seriously misleading.

The theoretical description of stochastic pumping provides several important physical insights. First, external fluctuations can be harnessed by many catalysts to drive a chemical reaction away from equilibrium to a nonequilibrium steady state, thereby storing energy. The necessary requirements are quite ubiquitous among macromolecules. Second, kinetic asymmetry is essential for energy coupling and adaptation. Dissipation alone is not sufficient. Third, the simplest version of a stochastic pump is an energy well surrounded by two energy barriers, where the energy well and one (or both) of the energy barriers can be driven by external fluctuations. As we will see, this structure can be rather simply implemented in construction of a synthetic molecular machine.

To understand how the molecular machines of life (28, 29) function, we can follow Richard Feynman’s implicit suggestion written on his last blackboard at Cal Tech, “I do not understand that which I cannot create,” and turn to synthetic chemistry, a science that creates the object of its study as emphasized by the great French chemist Marcellin Berthelot who observed “Le Chimie cree son objet.”

Tomar, Portugal 2004

In 2004, David Leigh organized a conference in Tomar, Portugal on “Smart Materials,” matter that responds and adapts to its environment (30). Much of the meeting centered on switches and logic gates, but there was also vigorous discussion about how to achieve directional motion. The meeting in Tomar came on the heals of publication of a remarkable paper by Leigh et al. (31) showing how directional cyclic motion in a mechanically interlocked ring system—a catenane—could be induced by time-dependent modulation of the chemical composition of the medium. Theoretical ideas about directional motion discussed at the meeting focused on the overdamped environment of molecules in solution where viscous drag dominates inertial forces [low Reynold’s number regime (26)], on the characteristics of motion arising from omnipresent thermal noise (32), and on the constraints imposed by the energetic requirements of microscopic reversibility (33). Attention centered on the extension of a diffusion-based model of Richard Feynman known as a Brownian ratchet to the description molecular systems. In the simplest incarnation of Feynman’s ratchet, the height of a saw-tooth barrier fluctuates (34, 35), giving rise to directed motion if the saw-tooth potential is asymmetric. This simple model had captured the imagination of scientists working in a very wide range of disciplines from quantum dots (36) to the ribosome (37) and synthetic molecular machines (38, 39) and has led to many experimental tests in numerous different contexts (40–42). A key idea in these systems with significant thermal noise is that input energy can be used to prevent backward (undesired) motion, leaving behind the desired forward motion. This idea has been loosely described as a Brownian motor (43, 44) principle. Based on this principle, a model was proposed for a catenane-based rotor, in which one ring could be viewed as moving on a potential energy surface defined by the orther ring that was attached to a surface and where rotational motion was generated by an oscillating applied field (45). The connection between directional motion of mechanically interlocked motion and the Feynman–Smoluchowski ratchet was made explicit in an experimental paper by Hernandez et al. (46) describing a reversible synthetic rotor. The give and take between experiment and theory at the meeting in Tomar, at a Nobel Symposium in Sweden (2005), at a Solvay Conseil in Brussels (2007), and at a Foresight Institute Symposium in Los Angeles (2013) led to the nucleus of an idea for implementing a ratchetting stochastic pump based on mechanically bonded (47) rotaxanes and catenanes. The flexibility afforded by the ability to incorporate recognition sites and chemical barriers with different properties that can be controlled by the chemical makeup of the milieu of the molecule would play a crucial role.

There and Back Again, a Rotaxanes Tale

Consider a simple molecular switch constructed of mechanically bonded molecules (48), such as shown in Fig. 4, where a long polymer is threaded through a ring molecule, a cyclobisparaquat tetracation (CBPQT⁺⁴), such that the ring is trapped—mechanically bonded—on the polymeric thread by two bulky diisopropyl end groups. Since the thread with bulky end groups is often somewhat rigid it is described as having a dumbell (DB) structure. In a major breakthrough (49), the polymeric thread was made asymmetric by incorporation of two different moieties, a benzidine unit and a biphenyl residue, that served as recognition sites for the CBPQT⁺⁴ ring.

Fig. 4.

Langevin simulation of the motion of a CBPQT⁺⁴ ring shuttling between two recognition sites of a molecular switch. For a barrier height of $\sim$10 k_BT, the lifetimes are of order 1–10 ms, while the barrier interaction time is around $\sim$10 ps.

Under basic conditions, the ring spends most (84%) of its time associated with the benzidine unit, with only 16% of its time associated with the biphenyl residue. Acidification of the medium, however, results in protonation of the benzidine after which, due to Coulombic repulsion, the CBPQT⁺⁴ ring spends most of the time associated with the biphenyl residue. A subsequent design (50) (see Fig. 6) settled on a similar structure but with different recognition sites. One recognition site (see green in Fig. 6) is a tetrathiafulvene (TTF) that interacts more strongly with the CBPQT⁺⁴ than does a 1, 5 dioxynaphthalene (DNP) (see red in Fig. 6). When the TTF is oxidized to TTF⁺² dication, Coulombic repulsion weakens the interaction with the CBPQT⁺⁴, so that the CBPQT⁺⁴ is more stable when encircling the DNP group. The average location of the ring (i.e., on the TTF or on the DNP site) depends on whether the TTF is reduced or oxidized and hence, on the redox potential in the solution. However, whether under oxidizing or reducing conditions, the equilibrium between the two recognition sites is a dynamic process, with the ring continually shuttling back and forth between the two sites.

Fig. 6.

Design for a synthetic molecular pump. Reduction recruits a ring from the bulk to bind to the active site, and oxidation results in the ring undergoing thermally activated transition to the collecting chain. Repetition of several cycles of reduction and oxidation brings several rings onto the collecting chain to form a highly nonequilibrium structure.

When driven by an external oscillation or fluctuation of the redox potential, the ensemble of molecules cannot be in thermodynamic equilibrium, and the average state probabilities (concentrations) will not be Boltzmann distributed according to their energies. The transitions between states, however, remain simple equilibrium motions—the rings do not experience violent kicks or judo throws and are most assuredly not “beaten into shape” by the oxidation/reduction of the TTF recognition site. The physical process of transition of a CBPQT⁺⁴ ring from the oxidized TTF⁺² site to the DNP group is precisely the same as it would be had the system as a whole been in equilibrium for a long time. At the single-molecule level, we can apply the results of equilibrium stochastic thermodynamics even in the presence of external driving.

Beyond Boltzmann

The equilibrium character of the motion is evident in the movement of the CBPQT⁺⁴ ring as modeled by a Langevin equation. The potential energy profile can be calculated by density functional theory (51), and because of the viscous environment, no inertial term is needed. In Fig. 4, we see a realization of this motion for the case that TTF is not oxidized. Two back-and-forth transits between the DNP and TTF sites are shown. These transitions can be broken into “waiting times,” in which the ring undergoes Brownian motion near one interaction site, and transitions in which the ring makes a “hop” from one site to the other. This transition time, also known as the barrier interaction time, is very much shorter than the lifetime on one or the other recognition sites. We might imagine that, if the barrier were made steeper (e.g., by incorporating either a steric or an electrostatic barrier between the two recognition sites), the barrier interaction time would be longer, but in fact, the barrier interaction time is shorter when the barrier is steeper. The waiting time, however, increases exponentially with the height of the barrier. Although the decrease in barrier interaction time on increasing the barrier height may seem counterintuitive to anyone with experience walking in the mountains, the result is an easily derived corollary of the principle of microscopic reversibility (22). The most probable trajectory from any point a to a different point b is the microscopic reverse of the most probable trajectory from point b to point a. We can focus on what is known as the “last touch first touch” (LTFT) time—that time between when a ring last touches point a and first touches point b—and prove (22) the remarkable relation ${\tau }_{\text{LTFT}}(\text{a}\to \text{b})\hspace{0.17em}=\hspace{0.17em}{\tau }_{\text{LTFT}}(\text{b}\to \text{a})$ that holds for any points a and b for either reduced or oxidized form. Furthermore, the statistical properties of the ensembles of forward and microscopic reverse trajectories are tied by symmetry relations from which the very general equality for the ratio of the conditional probabilities for a forward and backward transition at any time is derived (22, 52):

$$\frac{\pi \left(\text{a}\to \text{b},\hspace{0.17em}t\right)}{\pi \left(\text{b}\to \text{a},\hspace{0.17em}t\right)}=e^{\left(G_{\text{a}}^0-G_{\text{b}}^0\right)}{e}^{\left[\delta F\left(t\right)\left({\theta }_{\text{a}}-{\theta }_{\text{b}}\right)\right]}.$$ [5]

The simplicity of Eq. 5 results from a separation of timescales, where it is recognized that the barrier interaction time is so short that any hop from one state to another is accomplished at what is effectively a constant, albeit not the reference, value of $F$. The net transition probabilities can be integrated to obtain the average of the ratio of the probabilities for the forward and backward transitions:

$$\frac{{\displaystyle \int p\left(\text{a}\right)\pi \left(\text{a}\to \text{b},\hspace{0.17em}t\right)dt}}{{\displaystyle \int p\left(\text{b}\right)\pi \left(\text{b}\to \text{a},\hspace{0.17em}t\right)dt}}=\hspace{0.17em}\frac{{\displaystyle \int p\left(\text{a}\right)\hspace{0.17em}\pi \left(\text{b}\to \text{a},\hspace{0.17em}t\right)\hspace{0.17em}{e}^{\left(G_{\text{a}}^0-G_{\text{b}}^0\right)}{e}^{\left[\delta F\left(t\right)\left({\theta }_{\text{a}}-{\theta }_{\text{b}}\right)\right]}dt}}{{\displaystyle \int p\left(\text{b}\right)\pi \left(\text{b}\to \text{a},\hspace{0.17em}t\right)dt}},$$ [6]

where Eq. 5 was used to write $\pi \left(\text{a}\to \text{b},\hspace{0.17em}t\right)$ in terms of $\pi \left(\text{b}\to \text{a},\hspace{0.17em}t\right)$. Setting the ratio to unity, we arrive at the steady-state condition

$${\frac{p\left(\text{b}\right)}{p\left(\text{a}\right)}|}_{\text{ss}}=e^{\left(G_{\text{a}}^0-G_{\text{b}}^0\right)}\langle e^{\left[\delta F\left(t\right)\left({\theta }_{\text{a}}-{\theta }_{\text{b}}\right)\right]}\rangle ,$$ [7]

where the angle brackets indicate the average of some quantity characteristic of a forward transition over all values of $\delta F$ weighted by the probability of the microscopic reverse transition at that value of $\delta F$. The averaging over trajectories to arrive at Eq. 7 is just the same as that used in Eq. 3 for a Markov-state chemical kinetic model. Eq. 7 is not a “nonequilibrium” relation but rather, is a relation that is derived based on recognition that, even in the presence of nonequilibrium fluctuations, of external driving, or of chemical potential differences of a substrate and product, the transitions of an individual molecular machine are equilibrium processes (22, 33).

Eq. 5 can be derived most easily by use of the Onsager–Machlup thermodynamic action theory (22). The impression of many scientists is that the Onsager–Machlup theory is valid only very near equilibrium in the linear regime. This characterization, while technically accurate, is very misleading. Onsager and Machlup (53) describe explicitly the requirements for application of their theory: “The essential physical assumption about the irreversible processes is that they are linear; i.e., that the fluxes depend linearly on the forces that cause them.” The equilibria required are mechanical equilibrium—there must be no acceleration—and thermal equilibrium, where the velocities equilibrate rapidly in comparison with the position so that the temperature is well-defined. In the context of a physical process, such as the motion of a ring molecule from one recognition site to another in a mechanically bonded molecule, this amounts only to the low Reynold’s number condition (26) that the viscous drag force is much greater than the inertial force. Even if some external pulling force is applied, the resulting motion is still strongly overdamped. It is almost impossible to change the applied force so rapidly as to bring a single macromolecule out of mechanical equilibrium (54). The conditions necessary for application of the stochastic thermodynamic theory of Onsager and Machlup (53) are, at least in molecular science, ubiquitous, even when the system is far from thermodynamic equilibrium.

Kinetic Asymmetry in a Molecular Switch

We can better appreciate how kinetic asymmetry arises by considering two specific molecular switches shown in Fig. 5 A and B. (i) One of these switches has an energy barrier set by a charged group and hence, determined by electrostatic interactions, and (ii) the other has an energy barrier set by an uncharged group and hence, determined by steric interactions. The switching is triggered by changing the redox potential of the solution and thereby, changing the oxidation state of the mobile ring as shown in Fig. 5C. If this is carried out slowly so that the ensemble of switches remains in thermodynamic equilibrium (i.e., with probability distributions given by a Boltzmann distribution according to the instantaneous redox potential of the solution), there is no difference in the response of the two types of switches. In the limit that switching between the reduced and oxidized forms of the mobile ring occurs rapidly in comparison with the relaxation time of the system, however, the concentrations of rings on the two recognition sites, red and blue in Fig. 5, attain an approximately constant apparent steady state. The fluctuations favor, relative to the equilibrium case at the midpoint redox potential, the red site in the molecule with the electrostatic barrier and the blue site in the molecule with the steric barrier (55). For fast fluctuation between a redox potential, where the mobile ring is almost certainly oxidized, and a redox potential, where the mobile ring is almost certainly reduced, at steady state, the ring in Fig. 5A will reside almost certainly on the blue recognition site, while the ring in Fig. 5B will reside almost certainly on the red recognition site. The ability to influence how absorption of energy from an externally driven fluctuation by selection of the barriers (transition states) can be used to great effect in designing molecular machines.

Synthetic Molecular Pump

With a fundamental picture of single-molecule thermodynamics and kinetics in hand, we can understand how the stochastic pump mechanism serves to provide a design principle for a molecular assembler, where external fluctuations facilitate the creation and maintenance of a nonequilibrium structure that would, in thermodynamic equilibrium, exist in only minute amounts. The first steps toward this goal were aimed toward achieving directed motion in a pseudorotaxane, where a CBPQT ring would thread onto the DB from one end and leave by the other end (51). This goal was accomplished by use of two different types of barriers—an isopropyl phenyl (IPP) group that hinders threading of a CBPQT by steric hindrance and a pyridinium group that slows threading by Coulombic repulsion—to provide the necessary kinetic asymmetry. The kinetic barriers were placed on either end of a long polymer, with a DNP recognition site in the middle. The supramolecular moiety formed when the CBPQT ring encircles the DNP is a pseudorotaxane, because while significant, the energy barriers presented by the IPP and pyridinium (PY⁺) groups for dissociation are by no means insuperable. Under oxidizing conditions, CBPQT⁺⁴ binds from the bulk, predominately over the IPP group. The Coulombic repulsion between the PY⁺ and the tetracation essentially precludes passage of CBPQT⁺⁴ over PY⁺.

When the ring is reduced, the interaction with the DNP is significantly decreased, and therefore, the ring most likely dissociates. Furthermore, the Coulombic repulsion between electrostatic barrier and the ring is not as strong in the reduced state, and passage of CBPQT^+·2 over the PY⁺ is more likely than passage over the IPP group. Thus, in an externally enforced cycle of oxidation and reduction, the ring will most likely bind to DNP over the IPP group and dissociate from DNP over the PY⁺ (i.e., the motion of the ring relative to the rod enforced by the external fluctuations is directional). Directional threading was also accomplished by Arduini et al. (56) using a Calix[6]arene wheel compound instead of a CBPQT ring and with alkyl chains with viologen recognition sites as the molecular thread.

In the next step toward a synthetic molecular pump, Cheng et al. (57) reasoned that substitution of the IPP “speedbump” with the much bulkier diisopropylphenyl group (a “stopper”) could allow the ring to be trapped in a high-energy state. The diisopropylphenyl group provides an almost insurmountable barrier for CBPQT in either the reduced or oxidized form. Thus, the reduced ring CBPQT⁺² moves on and off the dumbbell relatively rapidly over the electrostatic partial barrier on the end opposite the stopper, but movement of CBPQT⁺⁴ on and off the DB is very slow (i.e., after oxidation, the ring is kinetically trapped). However, the interaction between CBPQT^+·2 and DNP is strong, while the interaction between CBPQT⁺⁴ and DNP is relatively weak. To accomplish the goal of moving something from a low chemical potential to a high chemical potential, the interaction strengths should be reversed, so that the ring binds while reduced and then remains associated with the DB in a kinetically trapped high-energy nonequilibrium state when oxidized. This goal was accomplished by substituting a 4, 4′-bipyridinium (BIPY) group for the DNP recognition site, a substitution that has the additional benefit that the BIPY as well as the CBPQT can be reduced and oxidized. The interaction energy difference between BIPY⁺⊂CBPQT⁺² and BIPY⁺²⊂CBPQT⁺⁴ based on Coulomb’s law alone is obviously very large: $\left(\mathrm{\Delta }{G}^{\ast }-\mathrm{\Delta }G\right)\approx 30\hspace{0.17em}\text{kCal/mol}$ (i.e., about 50 kT!). With this setup, the CBPQT⁺² ring binds to the BIPY⁺ and then, when oxidized, remains metastably associated with the DB in a high-energy state, encircling the DB rod around the polyoligomethylene chain for which it has little or no affinity.

The final step (18) was to incorporate a partial steric barrier (an IPP group) between the recognition site and the polyoligomethylene chain on which the CBPQT⁺⁴ would reside when oxidized, thereby separating two compartments, the bulk and the collecting chain, from one another, with a stochastic pump comprising an electrostatic barrier, a recognition site, and a steric speed bump between them as shown in Fig. 6. External driving of the redox potential of the solution provides the energy to pump and maintain a CBPQT ring onto the collecting chain, despite the unfavorable $\mathrm{\Delta }{G}_{\text{out}}$. The steady-state occupancy of the collecting chain with external fluctuations of the redox potential is (18) $\left[\text{CBPQT}\right]\hspace{0.17em}{e}^{-\mathrm{\Delta }{G}_{\text{out}}}\langle e^{W_{\mathcal{S}}}\rangle$. We have, as a very good approximation, $\langle e^{W_S}\rangle \approx e^{\left(\mathrm{\Delta }G-\mathrm{\Delta }{G}^{\ast }\right)}$ (where $\mathrm{\Delta }G>0$ and $\mathrm{\Delta }{G}^{\ast }<0$). The preference for this path is so strong that repeating the cycling between reducing and oxidizing conditions can pump several rings onto the collecting chain, forming a highly nonequilibrium structure with almost unit probability. The structure in the absence of pumping would make up fewer than 1 in 10 million molecules irrespective of the redox potential of the solution or the redox states of the components of the machine.

Prospective and Conclusions

Understanding how macromolecules, including synthetic and biological molecular machines, absorb external energy to drive directional motion, to undergo nonequilibrium self-assembly, and to adapt to their environment is one of most important challenges in contemporary chemistry. Surprisingly, this ability is a ubiquitous property (58) and is not some highly specialized function, and it requires, in addition to energy input, only kinetic asymmetry (16) as a necessary structural feature. The mechanism is stochastic pumping (16, 20), and the governing equation can be written in terms of a stability relation that augments the standard equilibrium constant by a factor, $\langle e^{{\mathcal{W}}_S}\rangle$, that reflects the kinetic asymmetry of the system and is greater than or less than one depending on the correlations between the kinetic factors $\mathit{\pi }\left(\mathrm{P}\stackrel{{\mathit{S}}^{†}}{\to }\mathrm{S}\right)$ and the thermodynamic factors ${\mathit{e}}^{{\mathcal{W}}_{\mathit{S}}}$ (see Eq. 4). If the paths for which ${\mathcal{W}}_{\mathit{S}}$ is positive are more probable than those for which ${\mathcal{W}}_{\mathit{S}}$ is negative, the product P is kinetically stabilized by the fluctuations and vice versa. In short, molecular systems adapt to form structures in which the average energy dissipated in making the favored structure is greater than the average energy dissipated in disassembling the favored structure. In the viscous, noisy environment (26, 32), transitions of single-molecular machines are mechanical equilibrium processes (54), although energy input maintains the distribution among the possible states of the machine away from thermodynamic equilibrium. Viscosity dominates inertia, and mechanical resonances are simply impossible in this regime.

The clear prediction of the stochastic pump model (16) that kinetic asymmetry is an essential component of how molecules adjust to their environment has been challenged in a very recent proposal known as “dissipative adaptation” (59, 60) that asserts that molecules adapt to maximize the dissipation of energy and that makes no mention of kinetic asymmetry. Despite much attention given to the dissipative adaptation model, particularly in the lay literature, the basic claim of the dissipative adaptation model is wrong (55). We can see this in the context of a very simple model, $A^{\text{'}}\rightleftharpoons A\rightleftharpoons I\rightleftharpoons B\rightleftharpoons B^{\text{'}}$, where $\left|{\theta }_{A^{\text{'}}}-{\theta }_A\right|\gg \left|{\theta }_{B^{\text{'}}}-{\theta }_B\right|>0$ but where $z_A=1/2$ and $z_B=1$. In the presence of rapid fluctuations of the conjugate intensive parameter $\delta F\left(t\right)$, the macroscopic state $\mathbf{A}=\left(A,A^{\text{'}}\right)$ dissipates much more energy than the macroscopic state $\mathbf{B}=\left(B,B^{\text{'}}\right)$. In contrast, the macroscopic state $\mathbf{B}$, unlike state $\mathbf{A}$, is kinetically asymmetric. The result from solving the kinetic equations with $\delta F\left(t\right)$ influencing the kinetic parameters is unequivocal—state B, not $\mathbf{A}$, is favored at the nonequilibrium steady state relative to the equilibrium case (55). This result is not consistent with the prediction of the dissipative adaptation model but is consistent with the prediction of the much earlier stochastic pump theory (16, 23).

The almost ubiquitous ability of macromolecules to absorb energy from external fluctuations to maintain a nonequilibrium steady state opens a window for understanding how simple matter becomes complex (61, 62) in the presence of environmental fluctuations. The assembly of the rings onto the collecting chain in the synthetic molecular pump shown in Fig. 6 is an example where kinetic asymmetry has been exploited in the design, the general features of which are highly reminiscent of a biological membrane transporter (63) (Fig. 1). A complex, nonequilibrium structure with several rings on the collecting chain can be assembled by several cycles of externally enforced modulation of the redox potential or in principle, due to cyclic motion in the presence of a combined thermal and redox potential spatial gradient (64). If the assembled CBPQT rings have a catalytic activity not expressed by rings in the bulk, this selection could be further enhanced in a replication network (65). The importance of kinetics and self-replication for understanding molecular adaptation has also been emphasized by Pross (66) and Pascal and Pross (67). Eq. 1 shows one example of kinetic stabilization, but another, termed dynamic kinetic stabilization, involving autocatalysis (i.e., self-replication) has been proposed and discussed by Pascal and Pross (68). Both types of kinetic stabilization emphasize the importance of kinetics in determining the steady-state behavior of molecular system, an aspect that is often overlooked.

Although the push toward application of synthetic molecular machines (69–71) remains in its infancy (72), there have already been important insights provided to the fundamental aspects of the principles by which molecules use energy input from the environment to carry out mechanical and chemical tasks (73), including self-assembly of complex structures from simpler starting materials. A very important step has recently been taken by the research group of David Leigh. These chemists designed a synthetic molecular machine, where one ring of a catenane was caused to rotate about another ring by energy released in a catalyzed reaction (74). The kinetic analysis (74, 75) grounded in the principle of microscopic reversibility reveals an important fact—catalyzed chemical reactions drive processes by an information ratchet (76, 77) mechanism rather than the energy ratchet models by which external fluctuation-driven machines work (41). Blackmond (78) has termed kinetic models of the appearance of homochirality in which microscopic reversibility is ignored as “when pigs fly chemistry.” Kinetic models for free energy transduction without microscopic reversibility similarly require porcine aviation. The recent reappreciation of the importance of microscopic reversibility for transitions of single-molecular machines (22, 33, 78) is perhaps the most significant development in stochastic and nanothermodynamics over the last several decades.

It can only be expected that, using mechanically bonded molecules (48) as a springboard and armed with the organizing principle of microscopic reversibility (33), the continued collaboration between scientists in many different disciplines will lead to a rapidly evolving understanding of the fundamental principles of actively-driven self-organization of complex matter.