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# Wet-dry cycles away from equilibrium catalyse chemical reactions

Ivar Svalheim Haugerud, Pranay Jaiswal, Christoph A. Weber  Faculty of
Mathematics, Natural Sciences, and Materials Engineering: Institute of
Physics, University of Augsburg, Universitätsstraße 1, 86159 Augsburg, Germany

###### Abstract

Recent experimental studies suggest that wet-dry cycles and coexisting phases
can each strongly alter chemical processes. The mechanisms of why and to which
degree chemical processes are altered when subject to evaporation and
condensation are unclear. To close this gap, we developed a theoretical
framework for non-dilute chemical reactions subject to non-equilibrium
conditions of evaporation or condensation. We find that evaporation and
condensation can each accelerate the initial rate of chemical processes by
more than an order of magnitude, depending on the substrate-solvent
interaction. When maintaining the system out of equilibrium by wet-dry cycles,
the cycle frequency controls the chemical turnover. Strikingly, there is
resonance behavior in the cycle frequency where the turnover is maximal. This
resonance behavior enables wet-dry cycles to select specific chemical
reactions suggesting a potential mechanism for chemical evolution in prebiotic
soups at early Earth.

The presence of catalysts severely alters the speed of chemical reactions. In
contrast to reactants, they are not converted during the turnover of a
substrate to a product. Catalysts affect the activation energy $\Delta E$
without changing the thermodynamic free energies of the substrate and product
state Masel  (see Fig. 1). The use of catalysts ensures economic efficiency of
most applications in chemical and biomolecular engineering Singh and Tandon
(2014); Werner  _et al._ (2010); Blaser and Studer (1999). In living systems,
catalytic activity is mediated by enzymes. These enzymes are specialized
proteins involved in almost all metabolic processes in the cell Alberty
(2003); Srere (1987); Underkofler  _et al._ (1958). Enzymes do not solely
maximize the speed of chemical reactions; they are optimal concerning reaction
speed and specificity Eigen and Hammes (1963); Eigen (1968), i.e., they
catalyze only specific reactions in a mixture of extremely many reacting
components. However, this optimal catalytic property for life had to develop
during evolution Kauffman (2011); Oparin (1965); Orgel (1994). In other words,
there were likely no enzymes at the molecular origin of life. It remains one
of the mysteries of the molecular origin of life how functional products such
as self-replicating RNA strands could have formed despite activation barriers
significantly larger than $k_{B}T$ Hay  _et al._ (2010); Claeyssens  _et al._
(2006).

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Figure 1: Wet-dry cycles in chemically reacting systems: (a) Wet-dry cycles
can enhance chemical reaction rates by effectively reducing the energetic
barrier $\Delta E=E_{\text{a}}-\mu_{A}$ for a reaction to occur by increasing
the chemical potential of the substrate $\mu_{A}$, where
$E_{\text{a}}=-k_{B}T\log{\left(k_{c}/k_{0}\right)}$ is the activation energy,
$k_{B}T\log{k_{0}}$ is some reference energy and $k_{c}$ is the reaction rate
coefficient. In the case of catalysts, chemical reactions are sped up due to
lowering the barrier by increasing $k_{c}$. (b) Wet-dry cycles lead to non-
dilute conditions, which affect chemical reactions and may induce liquid-
liquid phase separation. The cyclic behavior is included through an
oscillating reservoir (c) with periods of evaporation and condensation.

An alternative perspective on how to speed up chemical reactions is through
non-equilibrium conditions Mast  _et al._ (2013); Morasch  _et al._ (2019);
Baaske  _et al._ (2007); Bartolucci  _et al._ (2022); Ianeselli  _et al._
(2022); Matreux  _et al._ (2021). In fact, non-equilibrium applies to many
processes in chemical engineering, almost all processes in living cells and
prebiotic soup at early Earth. The reason is that these are all open systems
that can exchange matter and entropy with their environment, termed dissipate
systems Nicolis (1986); England (2015); Brogliato  _et al._ (2020). Continuous
dissipation is often achieved by cycles of the system’s control parameters,
such as temperature or chemical potentials. Recently studied cases are wet-dry
cycles Maguire  _et al._ (2021); Fares  _et al._ (2020); Becker  _et al._
(2018); Tekin  _et al._ (2022); Dass  _et al._ (2023).

The ability of wet-dry cycles to affect the synthesis of compounds that are
also relevant for the molecular origin of life is progressively backed by
experiments (Forsythe  _et al._ , 2015; Mamajanov  _et al._ , 2014; Becker
_et al._ , 2018). Wet-dry cycles have been applied to catalyze the formation
of polyacids (Weber, 1989), esters (Forsythe  _et al._ , 2015; Mamajanov  _et
al._ , 2014), oligopeptides (Lahav  _et al._ , 1978; Rodriguez-Garcia  _et
al._ , 2015; Fox and Harada, 1958), nucleosides (Becker  _et al._ , 2018),
polynucleotides (Rajamani  _et al._ , 2008; Da Silva  _et al._ , 2015),
polyethersulfone Fang  _et al._ (1994), phosphorylation (Maguire  _et al._ ,
2021) and lipid structures (Rajamani  _et al._ , 2008). Up to now, the
mechanisms of how wet-dry cycles affect chemical processes, particularly in
comparison to catalysts, remain elusive.

Unraveling such mechanisms is complex since reducing the solvent amount
through evaporation generally leads to non-dilute conditions. As a result,
non-linear thermodynamic contributions to chemical activities emerge in
addition to altered kinetic activation barriers. Additionally, non-dilute
systems can induce phase transitions, such as liquid-liquid phase separation
Weber  _et al._ (2019); Kato  _et al._ (2021); Hyman  _et al._ (2014), gel
formation Blankschtein  _et al._ (1985); Semenov and Rubinstein (1998); Deviri
and Safran (2020); Bartolucci  _et al._ (2023), and solid precipitation Lewis
(2010); Agarwal and Peters (2014). Recent experiments show that coacervate
formation can be triggered by wet-dry cycles Fares  _et al._ (2020). To
interpret the existing experimental studies on the effects of wet-dry cycles
on the synthesis of chemical buildings and to decipher the underlying physio-
chemical mechanisms, a theoretical framework is lacking.

This work proposes a kinetic theory based on thermodynamic principles for a
non-dilute chemically reacting mixture coupled to a cyclic wet-dry reservoir
that drives the system out of equilibrium; see Fig. 1b,c. We show to what
extent and under which conditions both evaporation and condensation of solvent
can speed up or slow down rates of chemical reactions. In contrast to
catalysts that solely affect activation barriers, we find that solvent
evaporation and condensation can significantly alter the chemical potentials
of substrate and product once the system is driven toward non-dilute
conditions, illustrated in Fig. 1. We also study how wet-dry cycles allow for
continuous chemical activity by keeping the system out of equilibrium. For
this case, a resonance cycling frequency exists for each
evaporation/condensation rate at which the chemical turnover is maximal. This
resonance behavior provides a selection pressure for specific chemical
reaction networks suggesting wet-dry cycles as potential catalyst supplements
to kick-start the molecular evolution of life Carrea  _et al._ (2005); Eijsink
_et al._ (2005); Schulz  _et al._ (2022).

## Theory for wet-dry cycles with chemical reactions

We consider a mixture of volume V composed of $M$ components, each of particle
number $N_{i}$ ($i=1,...,M$) in the Gibbs ($T$-$P$-$N_{i}$) ensemble with the
Gibbs free energy $G(T,P,N_{i})$. Evaporation and condensation of molecule $i$
is governed by the difference between the bulk chemical potential
$\mu_{i}=\left({\partial G}/{\partial N_{i}}\right)_{T,p,N_{j\neq i}}$, and
the reservoir chemical potential $\mu_{i}^{\text{r}}$. In all our studies, the
chemical potential of the reservoir $\mu_{i}^{\text{r}}$ is a control
parameter that can be constant or vary in time in the case of wet-dry cycles.
The evaporation and condensation flux through the surface area $A$ is given as

$\displaystyle\frac{\dot{N_{i}}}{A}$
$\displaystyle=k_{\text{e},i}\left[\exp{\frac{\mu_{i}^{\text{r}}}{k_{\text{B}}T}}-\exp{\frac{\mu_{i}}{k_{\text{B}}T}}\right]\,,$
(1)

where $k_{\text{B}}T$ denotes the thermal energy and $k_{\text{e},i}$ is the
kinetic evaporation/condensation coefficient for component $i$. Equilibrium is
reached when the chemical potentials between the reservoir $r$ and the bulk
are equal, $\mu_{i}=\mu_{i}^{\text{r}}$, implying no exchange between the bulk
and the reservoir, $\dot{N_{i}}=0$.

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Figure 2: Evaporation/condensation phase diagrams: (a)
Evaporation/condensation equilibrium for different reservoirs are displayed
together with the constraint line of constant $\phi_{B}/\phi_{A}$. The binodal
encloses the blue-shaded area of phase coexistence, where the system phase
separates into two phases connected by straight black tie lines. (b) For four
sets of interaction strengths, creating four qualitatively different phase
diagrams (i-iv), the orange-shaded region displays the initial states
resulting in a phase-separated equilibrium. (c) The constraint of constant
$\phi_{B}/\phi_{A}$ at phase-separated equilibria are qualitatively different
for the different types of diagrams when varying the reservoir through
$\mu_{C}^{r}$.

To calculate the chemical potential, we consider the mean field Gibbs free
energy $G$ per volume $V$:

$\displaystyle\frac{G(T,p,\\{\phi_{k}\\})}{V}=k_{\text{B}}T$
$\displaystyle\sum_{i=0}^{M}\left[\frac{\phi_{i}}{\nu_{i}}\log{\phi_{i}}+\frac{\omega_{i}\phi_{i}}{\nu_{i}}\right]$
$\displaystyle+\frac{1}{2}$
$\displaystyle\sum_{i=0}^{M}\sum_{j=0}^{M}\chi_{ij}\phi_{i}\phi_{j}+p\,,$ (2)

where $\chi_{ij}$ is the interaction parameter characterizing the interaction
between component $i$ and $j$. Moreover, $\omega_{i}$ and $\nu_{i}$ denote the
internal energy and molecular volume of component $i$, respectively. The
volume fraction of a component is defined by the volume it occupies relative
to the total system volume, $\phi_{i}={\nu_{i}N_{i}}/{V}$,
$i\in\\{1,2,...,M\\}$. The solvent volume fraction $\phi_{M}$ is fixed through
incompressibility

$\phi_{M}=1-\sum_{i=1}^{M-1}\phi_{i}\,.$ (3)

The volume change due to evaporation and condensation is equal to the volume
of particles removed from or added to the system per unit of time

$\dot{V}=\sum_{i}\nu_{i}\dot{N_{i}}\,.$ (4)

Moreover, the volume change alters the volume fractions of all components,
indirectly affecting their chemical potentials. This change in chemical
potentials affects the chemical reactions governed by

$r_{i\rightharpoonup j}-r_{i\leftharpoondown
j}=k_{c,ij}\left[\exp{\frac{\mu_{i}}{k_{\text{B}}T}}-\exp{\frac{\mu_{j}}{k_{\text{B}}T}}\right]\,.$
(5)

Evaporation/condensation, therefore, has the ability to enhance chemical
reaction rates. The time evolution of the non-solvent volume fraction of a
homogeneous system follows

$\partial_{t}\phi_{i}=\sum_{j\neq i}^{M}r_{i\leftharpoondown
j}-r_{i\rightharpoonup
j}-\phi_{i}\frac{\dot{V}}{V}+\frac{\nu_{i}\dot{N_{i}}}{V}\,,$ (6)

where the last term is the particle change due to evaporation/condensation.

If the interaction strength between two components surpasses a critical value,
the system will phase separate into two phases (denoted I and $\textup{II})$
with different composition $\phi_{i}^{\textup{I}/\textup{II}}$, but same
chemical potential $\mu_{i}^{\textup{I}}=\mu_{i}^{\textup{II}}$. For a phase-
separated system, the chemical reactions concomitantly occur in both phases,
while the fluxes related to evaporation/condensation are considered to occur
at a surface area enclosing the dilute phase. As the two phases are at phase
equilibrium, the choice of which phase is in contact with the reservoir
becomes irrelevant. We consider that evaporation/condensation of
macromolecules is negligible compared to the solvent Loche  _et al._ (2022),
such that $k_{e,i}=k_{e}\delta_{iC}$. When only the solvent is
evaporating/condensing, and no chemical reactions occur, the ratio of each
non-solvent volume fraction, $\phi_{i}/\phi_{j}$, is conserved. The kinetic
algorithm to evolve the system according to Eq. (6) for homogeneous and phase-
separated systems builds on a method proposed by Bauermann and Laha Bauermann
_et al._ (2022a); see Appendix A for a discussion on the limitations of the
theory.

## Results

### Thermodynamics of evaporation/condensation mixtures without chemical
reactions

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Figure 3:  Evaporation/condensation and chemically reacting phase diagrams:
(a) The intersection of the equilibrium lines of evaporation/condensation and
the chemical reaction gives the thermodynamic equilibrium point all initial
states converge towards. An example of the change in volume fraction is
displayed as a black arrow, which is the sum of a chemical (red) and
evaporation/condensation component (green). (b) The set of all such points
produces a flow field in the phase diagram, all converging at the
thermodynamic equilibrium point, displayed for $k_{e}/k_{c}=4$. (c) Phase-
separated thermodynamic equilibrium states are only achievable for a single
reservoir value for each interaction strength. The phase coexistence line
separates solvent-poor and solvent-rich states until the critical point (red
cross).

Equilibrium states composed of coexisting phases can be induced through
evaporation/condensation if the equilibria are compatible. Without chemical
reactions, such a compatible equilibrium exists when both evaporation-
condensation equilibrium and phase equilibrium are fulfilled concomitantly.
This case corresponds in Fig. 2 to the intersection between the binodal and
the mononodal line set by the reservoir chemical potential
$\mu_{C}^{\text{r}}$ (orange dashed line). A cone-shaped domain of compatible
equilibria is spanned by two distinct conservation lines of constant
$\phi_{B}/\phi_{A}$ (orange domains in Fig. 2). The number and area of such
compatible equilibrium domains depend on the interactions among the molecules.
The area of compatible domains increases the more $A$ and $B$ interact
differently with the solvent (case i in contrast to ii and iii), or when $A$
and $B$ phase-separate independently of the solvent (case iv). Both trends
arise as the energy density depends strongly on the solvent amount for
constant $\phi_{B}/\phi_{A}$.

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Figure 4: Chemical speed-up through evaporation/condensation: The pathway
through the phase diagram during evaporation (a-c) and condensation (d-f)
towards the same equilibrium (red dot) for different $k_{e}/k_{c}$ are shown
in (a) and (d), respectively. Their respective product over time is displayed
in (b) and (e), where a significant speed up through the different pathways is
observed. How this speed-up depends on the interaction strengths for
$k_{e}/k_{c}=10$ is displayed in (c) and (f). The top row (a-c) shows the
results for evaporation, and the bottom row from condensation (d-f).

The equilibrium states are homogeneous when evaporation-condensation and phase
equilibrium are incompatible. Such incompatible states correspond to the
complement of the orange shaded domains in Fig. 2. The area of incompatible
equilibria increases when the non-solvent components interact similarly with
the solvent or when there is no phase separation between them, as discussed
for compatible equilibria. All equilibria are incompatible for some values of
the reservoir chemical potential (green and yellow lines in Fig. 2).

The phase behavior of the evaporating/condensing mixtures (i-iv) without
chemical reactions can be summarized by the thermodynamic phase diagram
spanned by the reservoir chemical potential $\mu_{C}^{\text{r}}$ and the
conserved variable $\phi_{B}/\phi_{A}$. The different interactions (i-iv) lead
to very different shapes where two phases can coexist with very different
values of the conserved variable $\phi_{B}/\phi_{A}$, as shown in Fig. 2. For
case iv, where the coexisting phases have a similar solvent volume fraction,
the non-solvent components phase separates independently of the solvent. As a
result, each phase has a large difference in $\phi_{B}/\phi_{A}$, creating a
significant phase-separated domain.

### Thermodynamics of evaporation/condensation mixtures with chemical
reactions

To illustrate the interplay between evaporation/condensation and chemical
reaction, we consider a simple reaction of the form: $A\rightleftharpoons B$,
where $k_{c}$ is the reaction rate coefficient
($k_{c,ij}=k_{c}\delta_{iA}\delta_{jB}$ in Eq. (5)). In the presence of
chemical reactions, $\phi_{B}/\phi_{A}$ is not conserved, and compatible
equilibria are no longer achievable. Thermodynamic equilibrium, therefore,
always corresponds to a homogeneous state where the conditions of chemical
equilibrium ($\mu_{A}=\mu_{B}$) and evaporation-condensation equilibrium
($\mu_{C}=\mu_{C}^{r}$) are fulfilled; see the intersection between the red
and green line in Fig. 3. Any initial state will evolve following a flow field
in the phase diagram toward the fixed point of thermodynamic equilibrium; see
Fig. 3. Each point in this flow diagram has two independent directions that
characterize the rates of change in the average volume fractions, which
correspond to the chemical turnover flux of constant $\phi_{A}+\phi_{B}$ and
evaporation-condensation flux of constant $\phi_{B}/\phi_{A}$; see vectors
shown in Fig. 3. Such fluxes, visualized by the length of each vector, are set
by $k_{e}$ and $k_{c}$. Gibbs’ phase rule (discussed in appendix C) makes the
domain of coexisting states inaccessible at thermodynamic equilibrium, such
that small changes in reservoir chemical potential $\mu_{C}^{\text{r}}$ close
to the binodal line pronounces significant changes in the equilibrium
compositions. This behavior is shown in Fig. 3, where solvent-rich and
solvent-poor equilibrium states are separated by the phase coexistence line
ending up at the critical point (red cross) in Fig. 3.

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Figure 5: Resonance behavior in chemical reactions with an oscillating
reservoir: The top row (a-c) shows the conversion from $A$ to $B$ on the path
towards a limit cycle, while the bottom row (d-f) shows equivalent quantities
during limit cycles. The change in the phase-diagram pathway with cycling
frequency $\Omega$ is displayed in (a) and (d). Both cases exhibit a resonance
behavior between the cycling frequency of the reservoir and the chemical
turnover, quantified by the conversion half-time (b,c) and periodic chemical
turnover (Eq. (8)) (e,f). Both values increase with $k_{e}/k_{c}$ at the
resonance frequency.

### Speed-up and slow-down of chemical reactions through evaporation and
condensation

Evaporation and condensation can speed up chemical reactions by driving the
system along different phase diagram trajectories. For unidirectional chemical
reactions, where $\omega_{\text{product}}\ll\omega_{\text{substrate}}$, these
trajectories are characterized by solvent-poor or solvent-rich conditions
during the relaxation to thermodynamic equilibrium (Fig. 4). These different
solvent conditions have a dramatic impact on the half-time $\tau_{1/2}$ to
turn over $50\%$ of $A$ to $B$ (Fig. 4). For the depicted molecular
interactions, i.e., $A$ favors the solvent while $B$ does not, a quick removal
of the solvent (large $k_{e}/k_{c}$) causes an increase in the chemical
potential of $A$. This increase in the substrate’s chemical potential,
visualized in Fig. 1, speeds up the turnover from $A$ to $B$.

The threshold between slow-down and speed-up of chemical reactions corresponds
to no interactions between the components, as seen in Fig. 4 and 4. The
leading term to quantify the effect is the substrate-solvent interaction,
which enters exponentially as $\chi_{AC}\phi_{C}$ in the reaction rate in
equation (12). The sign of $\chi$ determines whether evaporation or
condensation aids the chemical turnover, while the magnitude determines the
speed-up. In other words, reactions of solvophilic substrates are sped up
through evaporation, and solvophobic substrates under condensation. Though
other interactions also affect the speed-up, they are less relevant. All
statements about interaction strengths are qualitatively reversed once we
switch from evaporation (Figs. 4(a-c)) to condensation (Figs. 4(d-f)).

### Chemical reactions subject to wet-dry cycles

Now we discuss the effects on chemical reactions when the system is subject to
cycles of the reservoir chemical potential

$\mu_{C}^{r}(t)=\langle\mu_{C}^{r}\rangle+\mu_{C,r}^{\text{amp}}\sin{\left(\Omega
t\right)}\,.$ (7)

Here, $\Omega$ denotes the cycling frequency, $\langle\mu_{C}^{r}\rangle$ the
average reservoir value and $\mu_{C,r}^{\text{amp}}$ the oscillation
amplitude. We now discuss two cases of cycles: (i) the path to the limit cycle
and (ii) the limit cycle. For case (i), we are interested in the chemical
turnover for a unidirectional chemical reaction, as before, and how it is
affected by cycles. After the initial phase, the trajectory becomes periodic.
In case (ii), we are interested in the chemical turnover rate during such a
limit cycle.

(i) We find that wet-dry cycles can speed up chemical turnover relative to no
cycling, with a resonance peak at a specific cycling frequency. Different
cycling frequencies $\Omega$ result in the different pathways displayed in
Fig. 5, with different conversion half-times. Oscillating reservoirs decrease
the conversion half-time relative to no oscillations,
$\tau_{1/2}(\Omega)/\tau_{1/2}(0)$, by up to a factor of $5$ for the
parameters used in Fig. 5. The resonance cycling frequency
$\Omega_{\text{res}}$ is stable, changing less than a decade when varying
$k_{e}/k_{c}$ over four decades, displayed in Fig. 5. Slow chemical reactions
have the largest speed-up due to wet-dry cycles.

(ii) With repeated cycling, the system will eventually enter a limit cycle,
where the path in the phase diagram is a closed loop (Fig. 5). To quantify the
chemical activity per period, we define the relative chemical turnover per
cycle,

$\mathcal{R}=\frac{\Omega}{2\pi}\int_{0}^{2\pi/\Omega}\mathop{}\\!\mathrm{d}t\lvert
r^{\rightharpoonup}-r^{\leftharpoondown}\rvert.$ (8)

This quantity is at its minimal $\mathcal{R}=0$ if no chemical reactions take
place during the cycle, and maximal $\mathcal{R}=\Omega/(2\pi)$ if all
molecules undergo both the forward and backward reaction per cycle.

Different cycling frequencies $\Omega/k_{c}$ produce different limit-cycles in
the phase diagram, as shown in Fig. 5. Small $\Omega/k_{c}$ allows the system
to remain close to chemical equilibrium, while large values do not allow the
system to respond to the change in the reservoir. In between, there exists a
frequency where the system has time to react to the reservoir and be driven
away from the chemical equilibrium line, maximizing $\mathcal{R}$. For larger
values of $k_{e}/k_{c}$, the ability of the system to respond to the reservoir
increases, making $\mathcal{R}$ larger as well. With increasing $k_{e}/k_{c}$,
the system is able to follow the reservoir equilibrium, saturating
$\mathcal{R}$.

The resonance behavior provides a selection mechanism for specific chemical
reactions. This selection is achieved by the selective speed-up of specific
chemical reactions, while other reactions do not benefit from the wet-dry
cycles. The selection conditions depend on the value of the evaporation rate
coefficient $k_{\text{e}}$, which, however, does not change much for water for
typical temperature and pressure oscillation on Earth Hisatake  _et al._
(1993); Heinen and Vrabec (2019); Xiong and Yuen (1991). A specific resonance
frequency, and thus the selection of specific chemical reactions, can be
realized by varying the system’s geometry, i.e., surface-to-volume ratio
$A/V_{0}$ of the system (Fig. 6). Increasing the surface area to volume ratio
from a puddle to a thin film increases the resonance frequency, selecting for
faster chemical processes.

(a)

Figure 6: Resonance behavior determined by system’s geometry: The resonance
frequency of the periodic chemical turnover in Fig. 5 can be reached by
varying the geometry of the mixture through its surface-to-volume ratio.
Different reaction time-scales lead to different resonance frequencies, making
the resonance behavior act as a selection mechanism. For this figure, the
evaporation speed is $5$ cm/day.

## Conclusion

In this work, we develop a thermodynamic description for evaporation and
condensation of non-dilute reacting mixtures and study the effect on chemical
reactions. We find that kinetics of evaporation and condensation can
significantly speed up chemical reactions. This speed-up stems from
evaporation and condensation driving the system away from equilibrium. For
wet-dry cycles, the turnover from substrate to product can be sped up even
further. We found that the cycling frequency strongly affects the rate of
chemical reactions per cycle. A key finding of our work is the existence of a
resonance cycling frequency where turnover is maximal. This resonance behavior
can act as a selection mechanism to only speed up specific reactions. The
mechanism for how evaporation and condensation for constant and cycling
reservoirs affect chemical reactions relies on effectively lowering the free
energy barrier $\Delta E=E_{\text{a}}-\mu_{A}$ for a reaction to occur (Fig.
1). By reducing the amount of solvent through evaporation, a solvophilic
component will interact less with the solvent and more with the other
components, making it less energetically favorable to be solvophilic.
Likewise, solvophobic substrates interacting more with the solvent makes it
less energetically favorable to be solvophobic. In both cases, the substrate
chemical potential $\mu_{A}$ increases, thereby reducing the free energy
barrier $\Delta E$. For realistic thermodynamic and kinetic parameters of
chemical compounds in aqueous solutions Clary (1990); Kok and Rudin (1982);
Russell  _et al._ (2015); Zwicker  _et al._ (2017); Hisatake  _et al._ (1993);
Xiong and Yuen (1991), speed-ups of chemical reactions by more than a factor
10 are possible through wet-dry cycles. This pronounced effect on chemical
reactions suggests a strong contender as a naturally occurring process
analogous to enzymes. However, while enzymes speed up reactions by lowering
the kinetic contributions of the energetic barrier between the substrate and
the product, evaporation/condensation increases the chemical potential of the
substrate $\mu_{A}$ and thereby decrease the energetic barrier (Fig. 1).

Our findings suggest that wet-dry cycles could have played a vital role in
speeding up prebiotic chemistry on the early Earth, where biological enzymes
were absent. During early Earth settings, systems were likely exposed to
repeated wet-dry cycles. The reservoir oscillations can originate from weather
changes such as temperature, humidity, or pressure (Mulkidjanian  _et al._ ,
2012; Damer and Deamer, 2015), or other phenomena such as salt deliquescence
(Campbell  _et al._ , 2019), or freeze-thaw cycles (Vlassov  _et al._ , 2004;
Le Vay  _et al._ , 2021; Mutschler  _et al._ , 2015). Alternatively,
oscillations in the solvent can be found in foams Tekin  _et al._ (2022), or
porous rock containing trapped gas-bubbles Ianeselli  _et al._ (2022); Matreux
_et al._ (2021).

An important implication of our work is that the resonance behavior in the
frequency of wet-dry cycles provides a selection mechanism for chemical
reactions. Since the system’s geometry determines the resonance frequency,
different chemical reactions are selected in a puddle versus a mesoscopic
aqueous droplet. This suggests that pores of different sizes subject to wet-
dry cycles can provide a setting for chemical selection and evolution at the
molecular origin of life.

## Acknowledgement

We thank Evan Spruijt and Iris Smokers for discussions on the experimentally
observed effects of wet-dry cycles on chemical reactions and acknowledge their
feedback on the manuscript. We also thank Sudarshana Laha for stimulating
discussions and Hidde Vuijk, Samuel Gomez, and Gaetano Granatelli for feedback
on the manuscript. Figure 1 b,c was created using BioRender bio . C. Weber
acknowledges the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation program (Fuelled Life, Grant Number
949021) for financial support.

##  Appendix A Limitations of theory

Table 1: The interaction strengths $\chi_{ij}$, internal energies $\omega_{i}$, and reservoir chemical potential values $\mu_{C}^{r}$ used for all figures is specified here. For all figures in this work, we have chosen $\nu_{i}=1$, $k_{c}=1$, and $\omega_{C}=0$.  Fig. | $\chi_{AC}/k_{B}T\quad$ | $\chi_{BC}/k_{B}T\quad$ | $\chi_{AB}/k_{B}T\quad$ | $\omega_{A}/k_{B}T\quad$ | $\omega_{B}/k_{B}T\quad$ | $\mu_{C}^{r}/k_{B}T$
---|---|---|---|---|---|---
2 | 3.0 | 0.6 | -0.6 | -0.2 | 0.1 | (-0.2, -0.5, -0.9)
2 i | 3.0 | 0.6 | -0.6 | -0.2 | 0.1 | -0.2
2 ii | 2.84 | 2.04 | -0.6 | -0.2 | 0.1 | -0.15
2 iii | 1.72 | 1.72 | -2.19 | -0.1 | 0.4 | -0.2
2 iv | -0.60 | 0.60 | 3.0 | -0.2 | 0.1 | -1.40
3 | 3.0 | 0.6 | -0.6 | -0.2 | 0.1 | -0.5
4 a,b | 3 | -3 | 2 | -10 | 0.0 | -10
4 d,e | 3.5 | 2 | -2 | -10 | 0.0 | 2.77
5 a,b,c | 3 | -0.6 | 0.6 | 0 | -6.0 | $\langle-0.3\rangle$
5 d,e,f | 3 | -0.6 | 0.6 | 0.4 | 1.8 | $\langle-0.538\rangle$

We consider the case where phase equilibrium holds at each time during the
chemical kinetics. As a result, each thermodynamic quantity is homogeneous in
each phase. This simplified case is satisfied if diffusion rates of reacting
components are fast compared to chemical and evaporation rates:

$k_{e},\,\,k_{c},\,\,\Omega\ll\frac{D_{\text{min}}}{h^{2}}\,,$ (9)

where $h$ is the height of our system and $D_{\text{min}}$ is the smallest
diffusion coefficient out of all the components. The height becomes the
relevant length scale, as gradients in the horizontal direction are smaller
than in the vertical direction.

Evaporation can strongly decrease the diffusion coefficient, potentially
violating the condition (9) as reacting compounds can become more crowded.
However, removing the solvent also decreases the system’s height $h$,
potentially maintaining the validity of the condition above throughout the
evaporation process.

The validity of this approximation must be understood through kinetic
measurements of diffusion coefficients and chemical reaction rates. For a
diffusion coefficient of $100\,\mu\text{m}^{2}/\text{s}$ and a reaction time
scale of minutes, this is satisfied for system sizes of $0.1$mm. Though the
rate coefficients should decrease with less solvent fraction, the already
measured experimental speed-up Forsythe  _et al._ (2015); Mamajanov  _et al._
(2014); Becker  _et al._ (2018); Maguire  _et al._ (2021) shows that the
change in the kinetic coefficient does not overshadow the energetic benefit
possible through evaporation/condensation. Furthermore, gradients in the
system can be incorporated using the effective droplet model (Bauermann  _et
al._ , 2022b), which might lead to new behavior for slow diffusion.

We have considered parameters consistent with the approximate assumption of
phase equilibrium during the wet-dry cycles and the reaction kinetics. The
parameters used are listed in table 1.

##  Appendix B Unidirectional chemical reactions and analytic scaling relation

Unidirectional chemical reactions are achieved by choosing a large difference
in the internal energies between the product and substrate

$\omega_{\text{product}}\ll\omega_{\text{substrate}}.$ (10)

In turn, the chemical reaction becomes dominated by the direction from
substrate to product, where the backward pathway is exponentially damped by
the factor
$(\omega_{\text{product}}-\omega_{\text{substrate}})/k_{\text{B}}T$. The
chemical reaction in equation (5) can, under these conditions, be
approximately written as

$r^{A\rightharpoonup B}-r^{A\leftharpoondown
B}\approx\tilde{k}_{c}\phi_{A}\exp{\frac{\tilde{\mu}_{A}}{k_{\text{B}}T}}\,,$
(11)

when $A$ is acting as the substrate and $B$ as the product. We have defined
$\tilde{k_{c}}\equiv k_{c}\exp{\omega_{A}/k_{\text{B}}T}$, and
$\tilde{\mu}_{A}$ represents the non-entropic contribution to the chemical
potential of component $A$, defined as

$\frac{\tilde{\mu}_{A}}{\nu_{A}}=\phi_{B}\chi_{AB}+\phi_{C}\chi_{AC}+p-\Gamma+k_{\text{B}}T/\nu_{A}\,.$
(12)

To speed up the chemical reaction, this quantity should be maximized. During
evaporation, we reduce $\phi_{C}$, increasing $\phi_{A}$ and $\phi_{B}$. A
negative value of $\chi_{AC}$ increases the substrate’s internal energy. While
for condensation, $\phi_{C}$ is increased relative to $\phi_{A}$, such that
the chemical potential is increased for positive values of $\chi_{AC}$. Though
the interaction between $\chi_{AB}$ might appear as important as $\chi_{AC}$,
the volume fraction of the product will generally vary less during
evaporation/condensation. However, it will become important towards the later
part of the turnover process. Importantly,
$\Gamma=\sum_{i=1}^{M}\sum_{j=1}^{M}\chi_{ij}\phi_{i}\phi_{j}/2+k_{B}T\sum_{i=1}^{M}\phi_{i}/\nu_{i}$
also depends on the volume fractions and will vary during
evaporation/condensation. The changes in $\Gamma$ through
evaporation/condensation will still contribute less to the overall speed-up as
its interaction terms contribute only as second order in the $\phi$’s.

The reaction coefficients $k_{c,ij}$ can generally depend on the volume
fractions and differ in each phase. However, for simplicity, we treat it as
constant and equal in the two phases.

##  Appendix C Gibbs’ phase rule

For a system of $\mathcal{C}$ components, $\mathcal{N}$ external thermodynamic
parameters, and $\mathcal{P}$ number of phases, the number of degrees of
freedom $\mathcal{F}$ can be calculated from

$\mathcal{F}=\mathcal{C}+\mathcal{N}-\mathcal{P}.$ (13)

The number of degrees of freedom gives the number of independent intensive
variables uniquely defining the thermodynamic equilibrium. The number of
components $\mathcal{C}$ is calculated as the number of components needed to
specify the composition of all phases (incompressibility reduces $\mathcal{C}$
by one) minus the number of constraints between the concentrations (reduced by
the number of chemical reactions). Furthermore, the evaporation/condensation
equilibrium can be regarded as an external intensive variable defining the
system. Such that the set $\\{T,p,\mu_{C}^{r}\\}$ or alternatively
$\\{T,\mu_{C}^{r}\\}$, gives

$\mathcal{F}=\begin{cases}5-\mathcal{P}&\mbox{for}\,\quad\\{T,p,\mu_{C}^{r}\\}\\\
4-\mathcal{P}&\mbox{for}\,\quad\\{T,\mu_{C}^{r}\\}\end{cases}$ (14)

Therefore, two-phase coexistence in a system is defined as a volume in
$\\{T,p,\mu_{C}^{r}\\}$-space and a plane in $\\{T,\mu_{C}^{r}\\}$-space,
while the homogeneous system has an additional degree of freedom. For a fixed
reservoir and a single chemical reaction, the degree of freedom becomes
$\mathcal{F}=2-\mathcal{P}$, such that two-phase coexistence is only
achievable for a single point in the phase diagram. This is consistent with
the phase diagram in Fig. 3.

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