Energy landscape design principle for optimal energy harnessing by catalytic molecular machines

Under temperature oscillation, cyclic molecular machines such as catalysts and enzymes could harness energy from the oscillatory bath and use it to drive other processes. Using an alternative geometrical approach, under fast temperature oscillation, we derive a general design principle for obtaining the optimal catalytic energy landscape that can harness energy from a temperature-oscillatory bath and use it to invert a spontaneous reaction. By driving the reaction against the spontaneous direction, the catalysts convert low free-energy product molecules to high free-energy reactant molecules. The design principle, derived for arbitrary cyclic catalysts, is expressed as a simple quadratic objective function that only depends on the reaction activation energies, and is independent of the temperature protocol. Since the reaction activation energies are directly accessible by experimental measurements, the objective function can be directly used to guide the search for optimal energy-harvesting catalysts.

Figure 1

A general model of a catalyzed reaction, where the catalyst undergoes a cyclic pathway consisting of $N$ intermediate states and $2N$ transitions. $A$ and $B$ represent the sets of reactants and products, which can join/leave the catalytic loop at arbitrary locations. We choose a convention that the forward reaction goes clockwise. The transition from $j$ to $i$ and its rate $R_{ij}$ is illustrated by the energy landscape

Figure 2

The $2N$-dimensional design space of $\mathbf{r}=(R_{21},R_{12},...,R_{ji},R_{ij},...,R_{1N},R_{N1})$. The yellow $\beta$ locus crossing three points $AOB$ is the set of $\mathbf{r}\left(\beta \right)$ corresponding to a given energy landscape at all possible inverse temperature $\beta$'s. The blue surface represents a $(2N-1)$ manifold defined by $\tilde{\mathcal{A}}\left(\mathbf{r}\right)=\text{const}$ that contains point $O$. The gradient of $\tilde{\mathcal{A}}\left(\mathbf{r}\right)$ at point $O$ is shown as the blue normal vector $\nabla \tilde{\mathcal{A}}$. When we consider temperature oscillation between ${\beta }_{1,2}={\beta }_0\pm \mathrm{\Delta }\beta$, their corresponding rate matrices are points $A$ and $B$. The rate matrix at ${\beta }_0$ is represented by point $O$. The effective rate matrix ${\widehat{R}}^{*}$ corresponds to the midpoint $D$ between $A$ and $B$. At infinitesimal temperature amplitude, the vector $\overrightarrow{OC}=\overrightarrow{OB}-\overrightarrow{AO}$ becomes $d^2\mathbf{r}/d{\beta }^2$ in Eq. (11).

Figure 3

Reaction rate (average probability current at periodic steady state) vs temperature oscillation frequency $f={\tau }^{-1}$ for both the optimal energy landscapes for reaction inversion and enhancement, obtained for $\mathcal{A}=-\mathrm{\Delta }G=1, {\alpha }_{\text{max}}=11, {\beta }_0=0.9$, and $\mathrm{\Delta }\beta =0.3$.

Figure 4

Objective function $\mathcal{C}$ and the scaled affinity change $2\mathrm{\Delta }\tilde{\mathcal{A}}/\mathrm{\Delta }{\beta }^2$ for randomly generated energy landscapes under the same set restriction with the optimization problem ($\mathcal{A}=-\mathrm{\Delta }G=1, {\alpha }_{\text{max}}=11, {\beta }_0=0.9$). The red and black crosses correspond to the optimal energy landscapes for reaction inversion (minimizing $\mathcal{C}$) and enhancement (maximizing $\mathcal{C}$).