Using Catalysis to Drive Chemistry Away from Equilibrium: Relating Kinetic Asymmetry, Power Strokes, and the Curtin-Hammett Principle in Brownian Ratchets

Abstract

Chemically fueled autonomous molecular machines are catalysis-driven systems governed by Brownian information ratchet mechanisms. One fundamental principle behind their operation is kinetic asymmetry, which quantifies the directionality of molecular motors. However, it is difficult for synthetic chemists to apply this concept to molecular design because kinetic asymmetry is usually introduced in abstract mathematical terms involving experimentally inaccessible parameters. Furthermore, two seemingly contradictory mechanisms have been proposed for chemically driven autonomous molecular machines: Brownian ratchet and power stroke mechanisms. This Perspective addresses both these issues, providing accessible and experimentally useful design principles for catalysis-driven molecular machinery. We relate kinetic asymmetry to the Curtin-Hammett principle using a synthetic rotary motor and a kinesin walker as illustrative examples. Our approach describes these molecular motors in terms of the Brownian ratchet mechanism but pinpoints both chemical gating and power strokes as tunable design elements that can affect kinetic asymmetry. We explain why this approach to kinetic asymmetry is consistent with previous ones and outline conditions where power strokes can be useful design elements. Finally, we discuss the role of information, a concept used with different meanings in the literature. We hope that this Perspective will be accessible to a broad range of chemists, clarifying the parameters that can be usefully controlled in the design and synthesis of molecular machines and related systems. It may also aid a more comprehensive and interdisciplinary understanding of biomolecular machinery.

Figure 1

Definitions of key concepts as used in this Perspective. Terms in bold are explicitly defined. Specific examples, when provided, always refer to the anhydride formation reaction in the single-bond rotary motor fueled by carbodiimide hydration (Figure 2).

Figure 2

Carbodiimide-fueled single-bond rotary motor. (a) Chemomechanical cycle for motor molecule 1. The horizontal and vertical directions show mechanical (conformational) and chemical transitions, respectively. (S) and (R) indicate the axial stereochemistry of the conformations of the motor around the C–N bond. Subscripts “d” and “a” in the species labels show the chemical states, which are the diacid (d) and the anhydride (a) forms of the motor. In the rate constants of the chemical processes (vertical transitions), superscripts “S” and “R” indicate whether a reaction involves the motor in the S or R conformation, respectively. Subscripts “f” and “h” indicate the anhydride formation and hydrolysis reactions, respectively. In the rate constants of the mechanical processes (horizontal transitions), superscripts “d” and “a” indicate whether a reaction involves the motor in the diacid or in the anhydride form, respectively. The subscript “s” stands for “stepping”. In all of the rate constants, + and – show the forward and backward reactions, respectively. Processes shown in red and blue indicate the kinetic resolution in the anhydride formation and hydrolysis reactions, respectively. Dashed arrows indicate the kinetically unfavored processes of the two competing chemical reactions. The anhydride formation reaction occurs faster from (S)-1_d than from (R)-1_d because the chiral carbodiimide fuel forms a more stable transition state (i.e., with lower activation energy) with rotamer (S)-1_d. The hydrolysis occurs faster from (R)-1_a than from (S)-1_a because the chiral nucleophilic catalyst forms a more stable transition state with conformer (R)-1_a. Consequently, under continuous fuel-to-waste turnover, the motor rotates in the clockwise direction, as indicated by the curved red and blue arrows at the corners, with F_C–H = 2.4 (see the text). (b) Kinetic resolution in the anhydride formation reaction. The anhydride formation reaction from (S)-1_d occurs faster than that from (R)-1_d because the activation energy for the former (ΔG_f^⧧,S) is smaller than the latter (ΔG_f^⧧,R). ΔG_d^° = μ_(R)-1d^° – μ_(S)-1d^° is the difference of the standard chemical potentials of (R)-1_d and (S)-1_d, which is zero in this motor. ΔΔG_f^⧧ = ΔG_f^⧧,R – ΔG_f^⧧,S + ΔG_d^° is the free energy difference of the two transition states for competing reaction paths. (c) Kinetic resolution in the hydrolysis step. The hydrolysis reaction from (R)-1_a occurs faster than that from (S)-1_a because the activation energy for the former (ΔG_h^⧧,R) is smaller than the latter (ΔG_h^⧧,S). ΔG_a^° = μ_(S)-1a^° – μ_(R)-1a^° is the difference of the standard chemical potentials of (S)-1_a and (R)-1_a, which is zero in this motor. ΔΔG_h^⧧ = ΔG_h^⧧,S – ΔG_h^⧧,R + ΔG_a^° is the free energy difference of the two transition states for competing reaction paths.

Figure 3

Three equivalent expressions for the ratcheting constant (K_r). The equations show how K_r can be related to the Curtin–Hammett asymmetry factor (F_C–H), an experimentally accessible quantity for determining the directionality of Brownian ratchets. See SI section 2 for the derivation.

Figure 4

Changes in the free energy profiles for three different Brownian ratchet scenarios. Dotted and solid lines indicate the free energy profiles before and after the modification, respectively. Dashed boxes in (b) and (c) highlight the differences with respect to (a). Downward arrows indicate where free energies are varied. As a consequence of free energy variations, the values of the rate constants and equilibrium constants explicitly shown in the figure are affected. The red and blue energy profiles refer to the anhydride formation and hydrolysis reactions, respectively. (a) Case 1: Hammond’s postulate. The free energy of (S)-1_d and the transition state for the corresponding anhydride formation reaction vary. Consequently, the rate constants k_–f^S and k_–h^S and the equilibrium constant K_s^d are affected. Such a change could be implemented by, for example, the use of a chiral countercation. To realize the change shown here, the transition state of the rate-determining step must come early in the reaction sequence so that the transition state is similar to the reactants. (b) Case 2: change without affecting the transition states’ free energies—only the free energy of (S)-1_d varies. Consequently, the rate constants k_+f^S and k_–h^S and the equilibrium constant K_s^d are affected. Such a change could be implemented by, for example, the use of a chiral countercation. To realize the change shown here, the transition state of the rate-determining step must come late in the reaction sequence so that the transition state is similar to the products. (c) Case 3: opposite power strokes (chemical processes are unaffected). The free energies of (S)-1_d, (S)-1_a, and the transition state between them varies. Consequently, the equilibrium constants K_s^a and K_s^d are affected. Such a change might be implemented by, for example, the use of a chiral solvent.

Figure 5

Kinesin walker. (a) Chemical reactions involving kinesin’s heads. Different states of kinesin are distinguished by the type of nucleotide attached to each head (T, ATP; D, ADP; E, empty; DP, ADP and inorganic phosphate). The heads are considered to be chemically identical but distinguishable (orange and purple in the cartoons) and are always attached to the microtuble except when they bind ADP. ADP unbinds with forward rate constant k_–D^F/B and backward rate constant k_+D^F/B. ATP is hydrolyzed with forward rate constant k_+h^F/B and backward rate constant k_–h^F/B, consuming water and eventually releasing inorganic phosphate P_i. The whole process is modeled as a single coarse-grained reaction. One empty head binds ATP with forward rate constant k_+T^F/B and backward rate constant k_–T^F/B. Superscripts F and B indicate whether the reaction happens at the front (right) or back (left) head. (b) Chemomechanical cycle for kinesin. The horizontal and vertical directions show mechanical (conformational) and chemical transitions, respectively. The rate constants k_+s and k_–s indicate forward and backward stepping, respectively. Processes shown in blue indicate the kinetic resolution in the ADP release. Under continuous fuel-to-waste turnover, kinesin moves left to right, corresponding to clockwise cycling as indicated by the curved blue arrows at the corners, with F_C–H = 1.25 × 10⁶ (see the text). (c) Kinetic resolution in the ADP release. Due to the equivalence of the two heads, the same free energy profile applies to both the top and bottom parts of the chemomechanical cycle. ΔG_D^⧧,B and ΔG_D^⧧,F are the activation energies for ADP release from D:T and T:D, respectively. ΔG_s^° is the difference in the standard chemical potentials of D:T and T:D. ΔΔG_D^⧧ = ΔG_D^⧧,B – ΔG_D^⧧,F + ΔG_s^° is the free energy difference in the two transition states for competing reaction paths. Since ΔΔG_D^⧧ is nonzero, one product (T:E) forms preferentially over the other (E:T), which leads to directional movement of kinesin.