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# Non-equilibrium Theoretical Framework and Universal Design Principles of
Oscillation-Driven Catalysis

Zhongmin Zhang<EMAIL_ADDRESS>Department of Chemistry, University of North
Carolina, Chapel Hill, NC 27599-3290, U.S.A.  Zhiyue Lu<EMAIL_ADDRESS>Department of Chemistry, University of North Carolina, Chapel Hill, NC
27599-3290, U.S.A.

###### Abstract

Traditional catalysis theory claims that catalysts speed up reactions by
introducing low activation barriers and cannot alter the thermodynamic
spontaneous reaction direction. However, if environments change rapidly,
catalysts can be driven away from stationary states and exhibit anomalous
performance. We present a geometric non-equilibrium theory and a control-
conjugate landscape to describe and explain anomalous catalytic behaviors in
rapidly oscillatory environments. Moreover, we derive a universal design
principle for engineering optimal catalytic energy landscapes to achieve
desired anomalous catalyst behaviors. Applications include but are not limited
to (1) inverting a spontaneous reaction to synthesize high-free-energy
molecules, and (2) dissipatively speeding up reactions without lowering
activation barriers. In both cases, catalysts autonomously harness energy from
non-equilibrium environments to enable such functionalities.

##  I Introduction

Traditional catalysis theory utilizes catalytic reaction pathways with low
activation barriers to speed up both forward and backward reactions. The
thermodynamic driving force of a directed catalyzed reaction could come from
the free energy change of the reaction itself or from an external steady
supply of free energies (e.g., light-harvesting catalysis [1], or electrolysis
[2]). Recently it has been shown that certain catalysts’ performances can be
altered or even enhanced when time-varying (oscillating) environments are
applied as additional driving forces [3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. In
stochastic thermodynamics, similar observations have been made in energy-
pumped enzymes, molecular machines, and flashing ratchets [13, 14, 15, 16, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35]. This
work theoretically justified that, under environmental oscillation, certain
catalysts can demonstrate surprising behavior that can never be achieved in
stationary environments. For example, our previous work [36] shows that a
catalytic molecular machine under oscillatory temperature can invert a
spontaneous reaction, converting low-free-energy products into high-free-
energy reactants by harnessing energy from the oscillatory environment. These
far-from-stationary-state behaviors can not be explained or designed by
traditional catalysis theory.

Here presents a novel universal theoretical framework and a geometric design
principle to explain and predict anomalous catalytic performances under rapid
environmental oscillation. Examples of such performances include enhanced (or
inverted) effective thermodynamic driving force, enhanced (or inverted)
reaction turnover frequency, and even manipulation of catalytic selectivity.
This general theory is applicable to catalysis pumped by arbitrary choices of
environmental parameters, including temperature, electric field, light
intensity, etc. Most importantly, the theory provides a universal energy-
landscape design principle to guide the design of novel catalysts with maximal
desired performance under environment oscillation. With simplifying
approximations, each _design principle_ can be represented by _convenient
design rules_ directly accessible by experimental test and verification.

In contrast to our recent work focusing on temperature oscillation-driven
molecular machines [36], this work provides a universal design principle of
oscillation-driven catalysis with a broader range of applications in
chemistry. By proposing a novel _control-conjugate landscape_ to represent
catalysts and their response to arbitrary environmental parameters, we develop
a geometric framework to explain the anomalous catalytic performance that can
occur under various oscillatory environments (such as electrical voltage,
light intensity, etc.). The performances are not limited to inverting
spontaneous chemical reactions and this work may find more general and
practical chemistry applications in catalysis, electrochemistry,
photochemistry, etc.

This paper is structured as follows. The Theory section starts with a general
kinetic model that captures the cyclic catalytic pathway of an arbitrary
catalyst. Then we introduce the concept of control conjugate landscape to
describe a catalyst’s kinetic properties under different environmental
conditions. By introducing the kinetic parameter space, we offer a geometric
perspective on the anomalous performances exhibited by catalysts in response
to rapidly oscillating environments. From the geometric interpretation, we
obtained a universal design principle for oscillation-driven catalysis. In the
Application section, we demonstrate two types of anomalous catalysis designed
from the general design principle: energy harvesting and enhanced turnover
frequency. In the Conclusion and Discussion section, we briefly discuss how
the general geometric approach can be extended to more complex environmental
parameters and general applications.

##  II Theory

Figure 1: (a) A simple $(N=3)$-step catalytic cycle, spontaneously converting
reactant R to product P. (b) Traditional energy landscapes at two choices of
environmental parameters $\lambda_{a}$ and $\lambda_{b}$. (c) The control-
conjugate landscape for $\lambda$ explicitly illustrates the catalyst’s
response to a control parameter $\lambda$ in a canonical form (Eq. 3). The
control-conjugate activation $\alpha_{i}$ is the height from the solid line to
the dashed line for the $i$-th transition. Here $\alpha_{-1}=\alpha_{-2}=0$
indicates that the transition rates $r_{-1}$ and $r_{-2}$ are independent of
$\lambda$.

###  II.1 Model

Consider a general catalytic reaction pathway comprising $N$ intermediate
reaction steps. For simplicity, it is illustrated in Fig. 1 as a 3-state
catalytic cycle with $N=3$ intermediate steps, converting a reactant R into a
product P per clockwise cycle. At any stationary environment, each
intermediate reaction steps $i$ from $1,2,\cdots N$ defines a forward reaction
rate $r_{i}$ and a backward reaction rate $r_{-i}$. We assume that at any
stationary condition within the range of interest, the reaction spontaneously
goes in the forward (clockwise) direction ($\Delta G<0$). The dynamics of the
system can be described by the master equation

$\derivative{\vec{p}}{t}=\hat{R}\cdot\vec{p}$ (1)

where the $\vec{p}$ is the probability of each intermediate state at a given
time and $\hat{R}$ is the transition rate matrix determined by the energy
landscape and the environmental condition. In our model, the $\hat{R}$ is a
$N\times N$ matrix whose elements are fully determined by the $2N$ reaction
rates of the reaction pathway. Thus the $\hat{R}$ is equivalent to a $2N$-dim
vector ${\mathbf{r}}=(r_{1},r_{-1},r_{2},r_{-2},\cdots,r_{N},r_{-N})$, where
$r_{\pm i}$ represents the forward and backward reaction rates for the $i$-th
reaction.

###  II.2 Control-conjugate landscape

At any fixed environmental condition, the transition rates of each reaction
can be determined by the energy landscape defined over the transition path
(see Fig. 1b). Each reaction step’s reaction rate follows the Arrhenius law

$r_{i}=A_{i}\exp(-E_{a,i}\beta)$ (2)

where $A_{i}$ is prefactor including concentrations, $\beta=1/kT$ is inverse
temperature, and $E_{a,i}$ is the activation energy given by the energy
landscape (Fig. 1b). We assume that the catalytic reaction occurs in a
stationary chemical bath with the concentrations of reactants and products
held constant.

This work focuses on anomalous catalytic behaviors driven by rapidly
oscillating environmental conditions. In general, the oscillatory
environmental variable (e.g.,reactant or product concentrations, electric
field, temperature, light intensity, etc.) can alter the shape of the energy
landscape and thus impact the rate $r_{i}$ through $A_{i}$, $E_{a,i}$, and/or
$\beta$. Thus, each catalyst under time-varying environmental conditions
evolves on an energy “seascape” – an energy landscape whose shape is
controlled by the changing environmental condition.

Inspired by the form of the Arrhenius law, we introduce a canonical form of
reaction rate where we denote each rate’s dependence on $\lambda$ in the
exponential form with an activation-like quantity $\alpha_{i}$:

$r_{i}=r_{0,i}\exp(-\alpha_{i}\lambda)$ (3)

where $r_{0,i}$ is a control-independent factor, $\lambda$ is the _canonical
control parameter_ , and $\alpha_{i}$ is the _control-conjugate activation_.
As illustrated in Fig. 1b, given two environmental conditions $\lambda_{a}$
and $\lambda_{b}$, the activation energy change caused by the change of
$\lambda$ is $\Delta E_{a,i}=RT\alpha_{i}\Delta\lambda$.

The Eq. 3 leads to a novel representation of energy seascape by a _control-
conjugate landscape_ pictured in Fig. 1c. In the traditional energy landscape,
the inverse temperature $\beta$ universally impacts all reaction rates. In
contrast, the conjugate energy landscape description allows selective
environmental control that only impacts a few reaction steps. In other words,
if the transition rate $r_{j}$ is independent of the external control
$\lambda$, then $\alpha_{j}=0$. In a special case where we choose to oscillate
the environmental temperature $T$, then one can define the canonical control
parameter as $\lambda=\beta=1/kT$, and the corresponding control-conjugate
activation $\alpha_{i}$ reduces to the traditional activation energy
$\alpha_{i}=E_{a,i}$.

###  II.3 Effective kinetics under rapid oscillation

In general, at a constant environmental condition $\lambda$, traditional
kinetics theory allows us to obtain the catalyst’s steady-state performance by
solving for the steady-state probability $\vec{p}_{ss}$ from the master
equation $\hat{R}(\lambda)\cdot\vec{p}_{ss}=0$. At such steady-state, the
_catalytic performance_ is dictated by the $\vec{p}_{ss}$ and the
$\hat{R}(\lambda)$. As a result, the stationary catalytic performance can be
expressed by a function of the transition rates for all $N$ reactions at the
given $\lambda$,

$f(\mathbf{r})=f(r_{1},r_{-1},r_{2},r_{-2},\cdots,r_{N},r_{-N})$ (4)

For instance, the performance can be chosen as the turnover frequency (Eq.
12), thermodynamic affinity (Eq. 8), or catalytic selectivity.

For time-varying environments, traditional kinetic theory can only be extended
to describe slowly-varying environments (quasi-static control), where the
system is maintained at slowly varying non-equilibrium steady states (NESS’s)
at all times. However, when the environment rapidly oscillates, the system can
not catch up with the rapidly changing rates. As a result, it can lead to
hard-to-solve yet novel performances which can never be observed at NESS.

To obtain a universal analytic theory for oscillation-driven catalysis, this
work considers high-frequency oscillation whose period $\tau$ is much smaller
than the fastest transition timescale $\tau\ll\min_{i}(r_{i}^{-1})$. Under
rapid oscillation, the effective reaction rates converge to the time-average
reaction rates ${\mathbf{r}}^{\text{eff}}$. In this limit, the oscillatory-
driven catalyst reaches a pseudo-stationary state[37, 38, 36], whose effective
performance can be obtained by plugging in the effective rates into Eq. 4 as
$f({\mathbf{r}}^{\text{eff}})$. We claim that if the effective rates diverge
from any constant-environment rates (see Fig. 2), the catalyst can demonstrate
anomalous performance (e.g., turnover frequency and thermodynamic affinity)
when exposed to rapidly oscillating environments.

For illustrative purposes, in the rest of the paper, we consider a simple
binary oscillatory environment with half a period in $\lambda_{\text{a}}$ and
another half in $\lambda_{\text{b}}$. At the fast oscillation limit, each
reaction step $i$ has $1/2$ chance to occur with one rate $r_{i,\text{a}}$,
and $1/2$ chance to occur with another rate $r_{i,\text{b}}$. Thus, we can
compute the effective rate as

$r_{i}^{\text{eff}}=\frac{r_{i,\text{a}}+r_{i,\text{b}}}{2}$ (5)

as the arithmetic mean of the two instantaneous rates. Then the effective
performance can be expressed by

$f({\mathbf{r}}^{\text{eff}})=f(\frac{{\mathbf{r}}_{\text{a}}+{\mathbf{r}}_{\text{b}}}{2})$
(6)

which may never be achieved by any stationary environment
$f(\mathbf{r}(\lambda))$.

###  II.4 Geometrical design principle

Figure 2: The geometric intuition of oscillation-induced catalysis is
illustrated in the design space comprising $2N$ reaction rates ${\mathbf{r}}$.
(a) Given reaction rates ${\mathbf{r}}$, the corresponding catalytic
performance $f({\mathbf{r}})$ is captured by a set of isosurfaces
$f({\mathbf{r}})=c$ illustrated by light orange surfaces. The gradient of $f$
is the normal vector to the isosurface. A given catalyst’s rates under any
stationary environment $\lambda$ form a 1-dim locus ${\mathbf{r}}(\lambda)$.
When environments oscillates rapidly between $\lambda_{a}$ and $\lambda_{b}$,
the catalyst achieves an effective state
${\mathbf{r}}_{\text{eff}}=({\mathbf{r}}_{a}+{\mathbf{r}}_{b})/2$, which can
achieve a new performance $f({\mathbf{r}}_{\text{eff}})$. (b) For any catalyst
described by the control-conjugate landscape (Eq. 3), the 1-dim locus
${\mathbf{r}}(\lambda)$ appears to be a straight line in the 2N-dim logarithm
scale representation.

Through geometric arguments in a $2N$-dimensional ${\mathbf{r}}$-space
comprising all reaction rates, we provide a universal landscape design
principle for novel catalysts to enhance or reduce an arbitrary desired
property $f$ under rapid oscillation. In this $2N$-dimensional design space, a
catalyst described by a control-conjugate landscape can be represented by a
1-dimensional locus ${\mathbf{r}}(\lambda)$ parameterized by $\lambda$, shown
as a black solid curve in Fig. 2a. By oscillating $\lambda$, the effective
rate ${\mathbf{r}}^{\text{eff}}$ can be found at the middle point of
${\mathbf{r}}_{\text{a}}$ and ${\mathbf{r}}_{\text{b}}$. Geometrically, the
more curved the locus ${\mathbf{r}}(\lambda)$, the larger the deviation of
${\mathbf{r}}^{\text{eff}}$ from any stationary ${\mathbf{r}}(\lambda)$.
Locally, this deviation is captured by the acceleration vector of the locus as
$\ddot{{\mathbf{r}}}=\differential^{2}{\mathbf{r}}/\differential{\lambda}^{2}$.

To design a catalyst that can achieve enhanced (or suppressed) effective
performance $f({\mathbf{r}}^{\text{eff}})$, we use a geometrical indicator
that combines the deviation vector $\ddot{{\mathbf{r}}}$ and the gradient
vector of performance $\nabla_{\mathbf{r}}f$. Specifically, if the
oscillation-induced deviation aligns with (or against) the steepest increment
of desired performance $f$, the catalyst can exhibit enhanced (or suppressed)
performance under rapid oscillation.

Mathematically, the design principle to achieve desired performance
enhancement (or suppression) can be expressed by the maximization (or
minimization) of a universal _design criterion_

$\mathcal{C}^{[f]}\equiv\nabla_{\mathbf{r}}f\cdot\ddot{\mathbf{r}}=\sum_{i=\pm
1}^{\pm N}\partialderivative{f}{r_{i}}\cdot\derivative[2]{r_{i}}{\lambda}$ (7)

which is the dot product between the gradient vector of performance $f$ and
the acceleration vector. This geometry-inspired criterion rigorously
quantifies the oscillation-induced performance deviation under a small
amplitude of $\lambda$. However, as shown by the following geometrical
argument, the criterion applies well even for large-amplitude $\lambda$
oscillations. This is justified because the ${\mathbf{r}}(\lambda)$ locus
bears a well-behaving geometrical shape – at the logarithm-scale
representation of the ${\mathbf{r}}$-space (see Fig. 2b), the locus reduces to
a straight line, with the angle the line defined by the control-conjugate
activations (see Eq. 3). One can verify the result by applying it to design
the catalytic inversion of spontaneous reaction under rapid temperature
oscillation, where the general principle reduces to the result obtained in our
previous work [36].

This geometric analysis not only provides us with a universal design principle
but also reveals the geometrical origin of the anomalous oscillation-induced
performance change. The performance deviation depends on the nonlinearities of
both the rates (Eq. 3) and the performance function (Eq. 4) in the
${\mathbf{r}}$-space.

##  III Applications

The universal design criterion $\mathcal{C}^{[f]}$ can guide the design of
oscillation-driven catalysts with desired performance $f$. For illustrative
purposes, this paper demonstrates two interesting anomalous behavior:
(Application 1) oscillation-induced inversion of spontaneous reaction
direction, and (Application 2) oscillation-enhanced high turnover frequency
without utilizing low activation barriers as required by the traditional
theory. Notice that the design principle applies to any desired interest
performance $f$, not limited to these two examples. In all cases, the
anomalous behavior is sustained by the energy harnessed from the oscillatory
environment.

Figure 3: Demonstrations of application-1 (a,c,e) and application-2 (b,d,f).
Dynamics at two stationary environments ($\lambda_{a}$ and $\lambda_{b}$) are
shown in red and blue; the oscillation-induced effective dynamics are in
green. (a,b) illustrate stationary reaction rates and effective rates (Eq. 5).
(c,d) illustrate two dimensionless stationery free-energy landscapes $G/RT$
and the effective landscape. The tilt of the (effective) landscape, i.e., the
height difference between two ends, which equals $-\tilde{\mathcal{A}}$,
indicates the (effective) spontaneous direction. (e,f) illustrates the
detailed steady-state currents of each transition (thickness of arrows) and
the steady-state probability of intermediate states (size of disks). $J$
represents the net reaction currents. For both applications, we set
$r_{1,a}=2$, and $r_{-1,a}=r_{2,a}=r_{-2,a}=r_{3,a}=r_{-3,a}=1$ for
$\lambda_{a}=0$. The rates for $\lambda_{b}=-1$ can be calculated from
$r_{i,a}$’s and $\alpha_{i}$’s. For both $\lambda_{a}$ and $\lambda_{b}$,
$\Delta G_{\text{rxn}}/RT=-\tilde{\mathcal{A}}=-\ln 2$.

Application 1 – Inverting Thermodynamic Affinity. Here demonstrated is the
optimal catalytic energy landscape for inverting the spontaneous direction of
the reaction ($\tilde{\mathcal{A}}$). Such inversion is particularly
interesting as the catalyst harnesses energy from the external environment and
stores the energy by converting low-free energy products into high-free-energy
reactants. We choose the performance $f({\mathbf{r}})$ to represent the
thermodynamic spontaneity (i.e., dimensionless thermodynamic affinity)

$\tilde{\mathcal{A}}({\mathbf{r}})=\sum_{i=1}^{N}\log\frac{r_{i}}{r_{-i}}$ (8)

which is simply related to the reaction Gibbs free energy change via
$\tilde{\mathcal{A}}=-\beta\Delta G$. To maximally invert the spontaneous
reaction, one needs to find a catalyst with a very negative
$\mathcal{C}^{[\tilde{\mathcal{A}}]}$. In other words, the universal principle
Eq. 7 for designing strong catalytic inversion of a spontaneous reaction
becomes

$\mathcal{C}^{[\tilde{\mathcal{A}}]}=\sum_{i=1}^{N}\alpha_{i}^{2}-\sum_{i=-1}^{-N}\alpha_{i}^{2}$
(9)

which is a simple quadratic function that depends only on $\alpha_{i}$’s, each
step’s activation level conjugates to the control $\lambda$. In conclusion, a
designed catalyst with very negative $\mathcal{C}^{[\tilde{\mathcal{A}}]}$
tends to strongly invert the spontaneous reaction direction under rapid
oscillatory control.

The design principle Eq. 9 can be experimentally verified. Here we illustrate
an optimal catalyst obtained by the principle under a few constraints: To
prevent the control parameter $\lambda$ from directly impacting the
spontaneity $\tilde{\mathcal{A}}$, we require that

$\derivative{\tilde{\mathcal{A}}}{\lambda}=\sum_{i=1}^{N}(\alpha_{i}-\alpha_{-i})=0$
(10)

and thus, any observed inversion of the reaction’s effective direction is
purely a result of the catalytic response to rapid environment oscillation. By
suppressing $\sum_{i=1}^{N}\alpha_{i}^{2}$ and increasing
$\sum_{i=-1}^{-N}\alpha_{i}^{2}$ under the constraint Eq. 10, the optimal
reaction-inverting catalyst for $N=3$ can be designed as

$\displaystyle\begin{split}(\alpha_{1},\alpha_{2},\alpha_{3})&=(\frac{1}{3}\alpha,\frac{1}{3}\alpha,\frac{1}{3}\alpha)\\\
(\alpha_{-1},\alpha_{-2},\alpha_{-3})&=(\alpha,0,0)\end{split}$ (11)

This design is illustrated in Fig. 3a, where the two sets of reaction rates
respectively correspond to environmental conditions $\lambda_{a}=0$ (red) and
$\lambda_{b}=-1$ (blue), and the effective rates due to rapid oscillation are
shown in green colored bars.

Shown in Fig. 3c, the effective reaction energy landscape for this catalyst
has an inverted tilted direction compared to the stationary-condition
landscapes. In other words, the effective affinity has an opposite sign of the
affinity of the stationary environmental condition, leading to the reaction
direction inversion. As illustrated in Fig. 3e ($\alpha=12$), under constant
environment $\lambda_{a}$ (or $\lambda_{b}$), the steady-state current is
$J_{a}=0.833$ (or $J_{b}=0.176$), whereas a rapid oscillation between these
two conditions leads to an average current $J_{\text{eff}}=-0.158$. Thus the
catalyst actively harnesses energy from the environment to drive the reaction
against its spontaneous direction. A special case of catalytic thermal engines
under temperature oscillation is demonstrated in reference [36].

Application 2 – Enhancing reaction TOF (current). Here we illustrate the
design principle for optimal enhancement of reaction current $J$, which is
equivalent to turnover frequency (TOF). The TOF is actively enhanced by energy
harnessed from the environment, and cannot be explained by traditional
catalysis theory, which focuses on lowering activation barriers. In this case,
we choose performance $f({\mathbf{r}})$ as the (pseudo-)stationary current
$J({\mathbf{r}})$, which can be analytically solved for arbitrary catalytic
cycles (see reference [39]). For a $N=3$-step catalytic cycle, the current is

$\displaystyle J({\mathbf{r}})$
$\displaystyle=\frac{u({\mathbf{r}})}{v({\mathbf{r}})}$ (12) $\displaystyle
u({\mathbf{r}})$ $\displaystyle=r_{1}r_{2}r_{3}-r_{-1}r_{-2}r_{-3}$ (13)
$\displaystyle\begin{split}v({\mathbf{r}})&=r_{1}r_{2}+r_{2}r_{3}+r_{3}r_{1}\\\
&\ +r_{-1}r_{-2}+r_{-2}r_{-3}+r_{-3}r_{-1}\\\ &\
+r_{1}r_{-2}+r_{2}r_{-3}+r_{3}r_{-1}\end{split}$ (14)

Plugging into the universal principle, Eq. 7, we find the design criteria for
enhancing TOF as to increase the value of

$\mathcal{C}^{[J]}=J(\frac{u_{2}}{u}-\frac{v_{2}}{v})$ (15)

where

$\displaystyle\begin{split}u_{2}&=(\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2})r_{1}r_{2}r_{3}\\\
&\ -(\alpha_{-1}^{2}+\alpha_{-2}^{2}+\alpha_{-3}^{2})r_{-1}r_{-2}r_{-3}\\\
\end{split}$ (16)
$\displaystyle\begin{split}v_{2}&=(\alpha_{1}^{2}+\alpha_{2}^{2})r_{1}r_{2}+(\alpha_{2}^{2}+\alpha_{3}^{2})r_{2}r_{3}+(\alpha_{3}^{2}+\alpha_{1}^{2})r_{3}r_{1}\\\
&\
+(\alpha_{-1}^{2}+\alpha_{-2}^{2})r_{-1}r_{-2}+(\alpha_{-2}^{2}+\alpha_{-3}^{2})r_{-2}r_{-3}\\\
&\
+(\alpha_{-3}^{2}+\alpha_{-1}^{2})r_{-3}r_{-1}+(\alpha_{1}^{2}+\alpha_{-2}^{2})r_{1}r_{-2}\\\
&\
+(\alpha_{2}^{2}+\alpha_{-3}^{2})r_{2}r_{-3}+(\alpha_{3}^{2}+\alpha_{-1}^{2})r_{3}r_{-1}\\\
\end{split}$ (17)

Here Eq. 15, as a function of $r_{i}$’s as well as $\alpha_{i}$’s can be
directly used as a design principle for the numerical search of the optimal
design of oscillation-enhanced catalyst. Moreover, by observing this equation,
we can further obtain approximated design rules leading to a simple working
example of a prominent current-enhancing catalyst under environment
oscillation. Notice that ${v_{2}}/{v}$ in Eq. 15 is always positive. Thus to
enhance the performance, or achieve a large positive $\mathcal{C}^{[J]}$, we
need to find a catalyst of both large ${u_{2}}/{u}$ and large $J$. To increase
${u_{2}}/{u}$, one can enhance $\sum_{i=1}^{N}\alpha_{i}^{2}$ and suppress
$\sum_{i=-1}^{-N}\alpha_{i}^{2}$. Combining these considerations, we obtain an
intuitive catalytic landscape characterized by

$\displaystyle\begin{split}(\alpha_{1},\alpha_{2},\alpha_{3})&=(\alpha,\alpha,0)\\\
(\alpha_{-1},\alpha_{-2},\alpha_{-3})&=(\frac{2}{3}\alpha,\frac{2}{3}\alpha,\frac{2}{3}\alpha)\end{split}$
(18)

An example of choosing $\alpha=6$ is summarized in Fig. 3b,d,f. The reaction
rates for $\lambda_{a}$, $\lambda_{b}$, and the effective rates are shown in
Fig. 3b. The corresponding dimensionless free energy landscapes are shown in
Fig. 3d. In Fig. 3f, we verify that the effective reaction TOF (current)
reaches $J_{\text{eff}}=5.93$ under rapid oscillation, which is significantly
greater than the stationary current of both conditions ($J_{a}=0.833$, and
$J_{b}=4.05$).

##  IV Conclusion and Discussion

This paper describes a general non-equilibrium theory framework predicting
what catalysts could demonstrate anomalous performance under rapid environment
oscillation and why. The theory generally applies to catalysts with an
arbitrary number of intermediate steps $N$, and proposes a convenient control-
conjugate landscape to describe the catalyst’s response to different
environmental conditions $\lambda$’s. Furthermore, for any oscillatory control
parameter, the theory provides a geometric design principle for engineering
optimal catalysts with a wide range of oscillation-enhanced performances $f$,
such as inverting spontaneous reaction direction or enhancing reaction rate,
both empowered by the dissipation of energy. Immediate applications following
the two examples listed above can be found in oscillation-enhanced redox
reactions and electrolysis, where one can empower a reaction by low-magnitude
AC voltages rather than high-magnitude DC voltages. For instance, traditional
DC electrolysis is subject to the thermodynamic limit (theoretical cell
voltage) plus an extra kinetic limit (overpotential). With novel catalysts,
the reaction could occur under oscillatory voltage, even if it is below the
theoretical cell voltage. In practice, the weak voltage can avoid undesired
side reactions such as electrolysis of the solvent.

Beyond these two applications, by defining other types of performances $f$,
one can achieve more complex and useful performances such as selectivity
manipulation. For instance, consider a complex catalytic network with multiple
loops, where one loop produces desired products while other loops produce by-
products. We can design oscillation-responsiveness to each loop and define the
performance $f$ as selectivity, which is the ratio between the net current of
the desired-product loop and the total net current among all loops. This
allows us to manipulate the catalytic selectivity by adjusting the external
oscillatory stimulation.

The fundamental geometric intuition in the design space to explain
oscillation-induced anomalous performances is general and not restricted by
the simplicity of control parameter $\lambda$ and the control-conjugate
landscape. As represented in the ${\mathbf{r}}$-space, a catalyst under any
stationary environment is described by a point ${\mathbf{r}}(\lambda)$. When
the environment rapidly oscillates, the catalyst’s kinetic rates are
effectively represented by an average point ${\mathbf{r}}_{\text{eff}}$ in the
same ${\mathbf{r}}$-space, allowing for a new performance that can not be
achieved at any stationary condition. Even if the environmental control is
complicated (e.g., when reaction rates cannot be simply represented by a
control-conjugate landscape, or when the environmental control involves multi-
dimensional $\vec{\lambda}$), one can still utilize the geometric intuition to
obtain design principles for desired oscillation-induced performances.

In the future, this work opens up the opportunity to design catalysts beyond
the stationary theory, where catalysts are not simply designed by lowering
activation barriers as required by the traditional theory. Beyond what is
discussed in this paper, one can also find design principles in terms of the
optimal control protocols of $\lambda(t)$. This can potentially allow
catalysts to function as environmental sensors, producing signaling molecules
only when the environment oscillates according to specific temporal patterns.

###### Acknowledgements.

We acknowledge the fund from the National Science Foundation Grant
DMR-2145256. We appreciate helpful discussions and suggestions for the
manuscript from Hong Qian, Zhixin Lu, Yosuke Kanai, Jahan Dawlaty, Ruicheng
Bao, Shiling Liang, and Jim Cahoon.

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