# Proposed energy-metabolisms cannot explain the atmospheric chemistry of
Venus

Sean Jordan*1 ,  Oliver Shorttle1,2 and  Paul B. Rimmer2,3,4 1Institute of
Astronomy, University of Cambridge, Madingley Rd, Cambridge CB3 0HA, United
Kingdom  2Department of Earth Sciences, University of Cambridge, Downing St,
Cambridge CB2 3EQ, United Kingdom  3Cavendish Laboratory, University of
Cambridge, JJ Thomson Ave, Cambridge CB3 0HE, United Kingdom  4MRC Laboratory
of Molecular Biology, Francis Crick Ave, Cambridge CB2 0QH, United Kingdom
*corresponding email<EMAIL_ADDRESS>
###### Abstract.

Life in the clouds of Venus, if present in sufficiently high abundance, must
be affecting the atmospheric chemistry. It has been proposed that abundant
Venusian life could obtain energy from its environment using three possible
sulfur energy-metabolisms. These metabolisms raise the possibility of Venus’s
enigmatic cloud-layer SO2-depletion being caused by life. We here couple each
proposed energy-metabolism to a photochemical-kinetics code and self-
consistently predict the composition of Venus’s atmosphere under the scenario
that life produces the observed SO2-depletion. Using this photo-bio-chemical
kinetics code, we show that all three metabolisms can produce SO2-depletions,
but do so by violating other observational constraints on Venus’s atmospheric
chemistry. We calculate the maximum possible biomass density of sulfur-
metabolising life in the clouds, before violating observational constraints,
to be $\sim 10^{-5}\,-\,10^{-3}\,{\rm mg\,m^{-3}}$. The methods employed are
equally applicable to aerial biospheres on Venus-like exoplanets, planets that
are optimally poised for atmospheric characterisation in the near future.

## Introduction

Sulfur dioxide, SO2, is observed to be depleted in the cloud layer of Venus by
orders of magnitude compared to its below-cloud abundance [1]. Gas-phase
sulfur chemistry at the cloud top is dominated by the photochemical formation
of sulfuric acid, where one photon of light, one H2O molecule, and up to two
SO2 molecules, are consumed in the production of one H2SO4 molecule, which
then condenses and forms the bulk of the mass loading in the clouds. According
to this reaction scheme, the maximum depletion of SO2 that could possibly be
achieved is within a factor of $2$ of the H2O mixing ratio, however this is
inconsistent with the observations of $\sim 150\,{\rm ppm}$ below-cloud SO2
and $\sim 10\,{\rm ppb}$ above-cloud SO2, yet only $\sim 30\,{\rm ppm}$ below-
cloud H2O – a factor of five lower than the SO2 mixing ratio. Thus, Venus’s
observed SO2 depletion represents a key gap in our understanding of the
planet’s atmosphere [2, 3, 4], for which no full-atmosphere model can self-
consistently predict via exclusively gas-phase chemistry [5]. This has led to
the conclusion that there must be some currently unknown chemical pathway
responsible for the depletion of SO2 in the middle atmosphere. One abiotic
solution has been proposed which hypothesises that the delivery of hydroxide
salts to the cloud layer in the form of mineral dust can enable reduction of
SO2 within the cloud droplets to reproduce the observed trend [6]. It is not
yet known whether this in-droplet mineral chemistry is occurring in the clouds
of Venus, or whether the delivery of mineral dust to the cloud droplets is
sufficient for this mechanism to produce the observed SO2 depletion. This
raises the possibility of an alternative hypothesis: the observed depletion of
SO2 in the clouds of Venus is due to the bio-chemistry of sulfur-metabolising
life in an extant aerial biosphere harboured within the cloud layer.

While the present day surface of Venus is uninhabitable, the lower cloud layer
lies in the region of the atmosphere where pressure-temperature conditions are
suitable for life (47-57 km altitude). This has led to suggestions of
microbial life being a possible explanation for the existence of large and
irregularly shaped (‘mode 3’) aerosol particles in the lower clouds and being
the identity of the unknown UV-absorber [7, 8, 9]. It has recently been
suggested that the metabolic activity of microbial life in the clouds could
also be the origin of the reduced chemical species PH3 and NH3, which are
biotically produced on Earth and have been tentatively observed in the
Venusian atmosphere at abundances many orders of magnitude greater than those
predicted given their short photochemical lifetimes [10, 11, 12, 13]. A
plausible life cycle for a strictly aerial biosphere in the clouds has been
investigated [11], and a suite of mission concepts for a future Venus Life
Finder Mission have been recently presented to the community [14]. It it
therefore timely to provide novel and quantitative tests of the potential
metabolisms of a hypothetical aerial biosphere to better understand the
requirements and limits of life in the clouds, if it exists. Given that the
terrestrial biosphere has had an inexorable effect on the Earth’s atmospheric
chemistry throughout geological time, investigating Venusian biochemistries
self-consistently with the atmospheric chemistry is crucially important to
assessing the viability of the aerial life-hypotheses.

It has historically been proposed that hypothetical Venusian life could have
originated at the planet’s surface or been seeded from Earth, if Venus was
ever habitable in its past. This surface-dwelling life would have subsequently
migrated to the temperate region of the cloud layer as Venus underwent runaway
greenhouse warming, residing in the clouds today as an aerial biosphere [7,
15, 8]. Runaway greenhouse warming is instigated by an increase in the surface
temperature of a planet which causes an increase in the evaporation of surface
water to the atmosphere. Since H2O is a strong greenhouse gas, the increase in
atmospheric H2O further exacerbates the warming effect, creating a positive
climate feedback. Atmospheric H2O is eventually lost via photo-dissociation
followed by hydrogen loss to space, and the record of the past water content
is left imprinted in the atmospheric deuterium to hydrogen ratio (D/H). The
enhanced D/H ratio measured in the atmosphere of Venus provides evidence of
past water loss [16], however it is unclear whether this water was ever
present as liquid at the surface or always as gas in a hot steam atmosphere.
If liquid water was present at the surface of Venus in its past then the
planet’s slow rotation rate would have enabled a stabilising cloud-climate
feedback mechanism to be established [17], maintaining habitable surface
conditions despite the increasing Solar luminosity over time. This phase of
habitability could have lasted for up to $\sim 900$ Myrs if Venus has always
been in a stagnant lid tectonic regime [18], or if early Venus had Earth-like
plate tectonics then this habitable period could have lasted for up to $4$
Gyrs, until runaway greenhouse was instigated by voluminous magmatism [17]. In
contrast, other modelling has suggested that, due to its proximity to the Sun,
Venus could never have cooled sufficiently for liquid water to condense at the
surface following its magma ocean phase, and thus a stabilising cloud feedback
could never have been established [19]. In this case, life in the clouds would
likely require an aerial origin of life scenario within cloud droplets,
utilising very different biochemistry to terran life.

Regardless of the proposed origins or adaptations of hypothetical life on
Venus, the requirement of life to obtain energy and materials from its
environment is universal. A stable biosphere must consume metabolic feedstock
molecules and release metabolic products to liberate energy regardless of the
particulars of the biochemical machinery. A Venusian aerial biosphere has been
suggested to be dependent on a sulfur-based energy-metabolism[20, 21] for
three reasons: 1) the chemical energy liberated via currently proposed sulfur-
based metabolisms [20, 21] is sufficient to meet the energy requirements of a
cell [22]; 2) the relatively high abundance of sulfur species (e.g., SO2, OCS,
H2S) in the Venusian atmosphere means these are the abundant feedstock
molecules that could support a global biosphere; and, 3) terrestrial
acidophillic organisms provide a precedent for life in an acid environment
which uses sulfur-metabolism as a primary source of energy (e.g.,
acidithiobacillus ferooxidans). Further support for sulfur-metabolising life
in Venus’s clouds comes from an extensive analysis into the spectral
absorptivities of proteins and cofactors used by terrestrial sulfur-
metabolising microbes. Significant overlap was found between the spectral
appearance of these molecules and the UV features of Venus’s spectrum,
currently attributed to the unknown absorber [9].

Three sulfur-based energy-metabolisms for putative Venusian microbes have been
proposed by Schulze-Makuch (2004) and Schulze-Makuch & Irwin (2006) [20, 21],
each of which utilise the relatively abundant sulfur species in the atmosphere
and can, in principle, induce the depletion of SO2 as a result of their
respective biochemical effects on the atmospheric chemistry. One proposed
metabolsim, ‘A’, is a primitive anoxygenic photosystem used by microbes on
Earth, oxidising hydrogen sulfide (H2S) to elemental sulfur (S) meanwhile
fixing carbon from atmospheric CO2 into organics [20]:

(1) $\displaystyle 2\,\ce{H2S}+\ce{CO2}$
$\displaystyle\ce{->[light]}(\ce{CH2O})+\ce{H2O}+\ce{S2},$

where (CH2O) represents carbon locked in organic molecules (e.g., glucose). We
couple this photosystem to reaction 2, where the organics produced by the
anabolism of reaction 1 are consumed in catabolism to liberate chemical
energy:

(2) $\displaystyle\ce{SO2}+(\ce{CH2O})$
$\displaystyle\rightarrow\ce{CO2}+\ce{H2O}+\ce{S}.$

We have constructed reaction 2 as a hypothetical variant of terrestrial
oxygenic respiration with O2 replaced by SO2. The widespread use of O2 for
respiration by life on Earth is believed to be an adaptation of anoxygenic
life following oxygenation of the Earth’s atmosphere to contain abundant
O2[23]. In contrast, O2 is not observed in the upper atmosphere of Venus
leading to an inferred upper limit on its abundance of $<2.8\,{\rm ppm}$ [4].
One measurement of O2 within the cloud layer suggests an abundance of $\sim
70\,{\rm ppm}$[24] however this detection remains controversial, and is
discrepant with the results of modelling[6] and the subsequent non-detection
of O2[25]. Taking the widely accepted view that there is not substantial O2 in
the atmosphere of Venus, we assume that hypothetical Venusian life would have
evolved to utilise the abundant SO2 in the atmosphere in an analogous fashion
to terrestrial oxygenic life’s respiration of atmospheric O2. The net
metabolic reaction for metabolism A is therefore:

(3) $\displaystyle\text{Metabolism A (net):}\quad 2\,\ce{H2S}+\ce{SO2}$
$\displaystyle\ce{->[light]}2\,\ce{H2O}+\ce{S2}+\ce{S}.$

Reaction 3 is expressed in it’s simplest stoichiometric ratio. In reality life
is more likely to be be performing higher multiples of reaction 3,
constructing and burning longer chain organic molecules, and releasing longer
sulfur allotropes, rather than S and S2. We use this representation of the
metabolism for modelling simplicity when coupling biochemistry to the
atmospheric chemistry, and this does not impact any of the results that we
will present.

Metabolism B is a chemoautotrophic pathway proposed in the literature for
hypothetical Venusian life, that exploits the redox disequilibrium between CO,
H2 and SO2 in the atmosphere [21]:

(4) $\displaystyle\text{Metabolism B:}\quad\ce{H2}+2\,\ce{CO}+\ce{SO2}$
$\displaystyle\rightarrow 2\,\ce{CO2}+\ce{H2S}.$

Metabolism C is also a chemoautotrophic pathway proposed in the literature for
hypothetical Venusian life, and exploits the redox disequilibrium between CO
and SO2 in the atmosphere [21]:

(5) $\displaystyle\text{Metabolism C:}\quad 3\,\ce{CO}+\ce{SO2}$
$\displaystyle\rightarrow\ce{OCS}+2\,\ce{CO2}.$

Both of the proposed chemotrophic metabolisms, B and C, could yield $\sim
240\,{\rm kJmol^{-1}}$ of energy to a putative Venusian microbe, which would
be a sufficient source of chemical energy for life to thrive [22], provided
the reactant species are in high enough abundance.

We present a framework for the coupling of each proposed energy-metabolisms to
a photochemical-kinetic network for modelling planetary atmospheres in 1D [26,
6], and self-consistently solve for the atmospheric composition of Venus
required to reproduce the observed SO2-profile biochemically. We incorporate
each sulfur-based metabolism into our reaction network in turn, impose that
the reaction rate can only be non-zero between $47-57\,{\rm km}$ altitude
(where ambient atmospheric temperatures are between $0-100\,{\rm{}^{\circ}C}$
and aerosols are large enough to accommodate microbial colonies [11]), and
investigate the parameter space of metabolic activity versus availability of
metabolic inputs. While plausible metabolic pathways and survival strategies
for an extant Venusian biosphere have been suggested in previous studies [15,
20, 21, 9, 11] none have been investigated self-consistently with the wider
atmospheric chemistry. By coupling proposed metabolic requirements with self-
consistent models of atmospheric chemistry, we here produce a rigorous test of
whether the proposed metabolisms could support a Venusian aerial biosphere.
The results of our investigation show that, while proposed metabolic
incorporation of gas-phase species can reproduce the sulfur-depletion, in
every case the resultant atmospheric chemistry violates separate observational
constraints on the lower atmosphere of Venus.

## Results

### Predicting SO2-depletion biochemically.

Each of the three metabolisms are capable of reproducing the observed
SO2-depletion biochemically in the middle atmosphere of Venus when coupled to
our photochemical-kinetics network. For each metabolism, Figure 1 shows the
depletion of the metabolic reagent species, above $47\,{\rm km}$ altitude, as
a percentage of the metabolism-free reference abundance. The figure describes
where each metabolic pathway is rate-limited (within the white space) or
reagent-limited (within the darkest coloured contour for a given reagent
species), with transitional regions between each limiting regime. The
resultant SO2-depletion is shown with red contours in every panel. We take a
surface boundary condition for the initial chemical composition that is
consistent with past lower-atmosphere [27, 28] and full-atmosphere [5, 6]
models as our fiducial initial atmospheric composition. This surface condition
corresponds to species that have either been observed or are thermochemically
predicted to be present at the base of the Venus atmosphere [6]. For the
fiducial atmospheric composition, each metabolism is either rate-limited or
reagent limited: metabolism A is limited by the H2S abundance; metabolism B by
the H2 abundance; and metabolism C by the CO abundance. Therefore, none of the
three metabolisms initially provide a significant destruction pathway for SO2.

To fit the SO2 observations, each metabolism must be in the SO2-limiting
regime, and thus the abundances of the other limiting metabolic inputs must be
increased from the fiducial (and observationally constrained) surface boundary
conditions. Figure 1 demonstrates how, for each metabolic pathway, the net
metabolic rate and atmospheric composition can be tuned to successfully
reproduce the observed SO2 profile (white dotted line inside the SO2-limiting
region).

Figure 1. Depletion contours of metabolic inputs. Contour plot showing
depletion of metabolic reagents within the cloud layer for each metabolic
pathway, (a), (b), and (c). Depletion is shown as a function of net metabolic
rate (y-axis) and initial surface mixing ratio of the limiting metabolic
reagent species (x-axis). The depletion of each species is calculated with
respect to the species’ column abundance in a reference model run without the
metabolism present. The predicted or observed surface mixing ratios of
metabolic reagent species and their 1$\sigma$ errors are indicated with grey
errorbars [6], and the metabolic models reproducing SO2-depletion are
indicated with a white dotted line.

### Comparing the resultant atmospheric chemistry with observational
constraints.

In order to reproduce the observed SO2-depletion biochemically, the predicted
abundance of metabolic input species for each of the three metabolisms
violates a separate observational constraint in the lower atmosphere of Venus.
In figure 2 we plot the mixing ratios of SO2, H2S and CO as functions of
altitude, for the model atmosphere that best fits the SO2 observations for
each metabolic pathway. Alongside, we show the fiducial atmosphere, predicted
from thermochemistry, photochemistry and chemical kinetics, which does not
correctly predict the SO2 observations, and the abiotic model atmosphere of
Rimmer et al., (2021) which successfully predicts the SO2 observations by
hypothesising aqueous droplet chemistry [6]. All three proposed metabolisms
fit the SO2-depletion with a similar accuracy to the droplet chemistry
hypothesis [6], with metabolism A providing the closest fit of any hypothesis
to the middle-atmosphere observations (Figure 2). However, this comes at the
cost of the other metabolic reagent species not then fitting their below-cloud
observational constraints well.

Figure 2. Atmospheric profiles resulting from biochemical SO2-depletion.
Atmospheric profiles of (a) SO2, (b) H2S and (c) CO mixing ratios as a
function of altitude for the Rimmer et al., (2021) models (with and without
droplet chemistry) [6], compared with the three proposed metabolic pathways
[20, 21] under conditions that reproduce the SO2 observations. Observational
data and their 1$\sigma$ errors are shown in black except for the observation
of H2S below $20\,{\rm km}$ altitude which is shown in grey, as this is
possibly a spurious detection [29].

Metabolisms B and C each violate well-constrained lower-atmosphere
observations of CO, requiring $300\,{\rm ppm}$ and $400\,{\rm ppm}$ of CO at
the surface respectively, approximately $\sim 15\times$ and $\sim 20\times$
greater than observations indicate. Below-cloud CO has been measured in situ
by gas chromatography instruments on board the Venera 12 and Pioneer Venus
descent probes [30, 31], by remote detection via IR spectra in a $2.3\,{\rm\mu
m}$ atmospheric transparency window by the Venus Express orbiter [32, 33], and
via remote Earth-based detection [34, 35]. All of the data generally agree on
deep atmosphere CO mixing ratios between $17-45\,{\rm ppm}$ and therefore a CO
mixing ratio as high as that required by metabolisms B and C would be
unexpected.

Metabolism A requires $300\,{\rm ppm}$ of H2S at the surface in order for life
to be responsible for the observed SO2-depletion. H2S has been measured at an
abundance of $3\,{\rm ppm}$ below the cloud layer via a mass spectrometer
instrument on-board the Pioneer Venus descent probe [29], however this
reported value is one to two orders of magnitude larger than the currently
expected surface mixing ratio of $\sim 10-100\,{\rm ppb}$ from thermochemical
modelling [27, 6]. The H2S abundance otherwise only has an inferred upper
limit at the cloud top derived from non-detection by Earth-based platforms
[36]. From Figure 2 we find that, even when the H2S mixing ratio is on the
order of $100{\text{'s}\,\text{ppm}}$ in the cloud layer, H2S is so
efficiently photolysed by the irradiating Solar flux that the profile always
remains consistent with the inferred upper limit at the cloud top (with the H
atoms liberated forming H2, and the sulfur atoms liberated mainly forming
sulfur allotropes, S_x for $x=1,...,8$). Given the relative lack of
observational constraints on the deep atmosphere H2S abundance, it remains a
remote possibility that H2S exists with the required surface mixing ratio for
SO2-depletion via metabolism A. However, assuming a volcanic source of SO2 and
H2S in the Venus atmosphere [37], this would require that the molar ratio of
volcanically degassed H2S/SO2 is $\sim 2$. This is in disagreement with the
oxidising nature of the Venus atmosphere, the observation of oxidised sulfur-
containing minerals at the Venera 13, Venera 14, and Vega 2 landing sites [38,
39, 37], and the inference that the volcanic H2S abundance should be lower
than that of SO2 due to the very low water content in the atmosphere [37].

## Discussion

Our results demonstrate that the three sulfur-based metabolic pathways
proposed for Venusian aerial life are capable of reproducing the observed
SO2-depletion in the cloud layer of Venus, but in each case they require a
source of chemical reducing power roughly equal in abundance to below-cloud
SO2: the atmosphere otherwise does not possess enough reducing power for life
to exploit to generate the observed SO2-depletion.

It is possible that our understanding of the below-cloud and in-cloud
atmosphere is not correct. There is a consensus picture of Venus’s atmosphere,
however this picture still has several mysteries that models are not able to
solve, and certain observational data are discrepant with this picture.
Discrepant measurements include in-situ and ground-based detections of H2,
H2O, H2S, and O2, at higher abundances than expected[24, 31, 29, 41], although
these data are not consistent with each other without introducing spatial and
temporal variability or significant systematic error. If the discrepant
measurements were not made in error, then they must somehow be reconciled with
the known atmospheric chemistry and thus reveal potential uncertainties in the
consensus picture. However, even if these outliers represented the global
atmosphere of Venus, they would still be inconsistent with these metabolic
pathways explaining the sulfur-depletion in Venus’s atmosphere. If, as seems
likely, the standard description of the lower-atmosphere composition of Venus
is broadly correct [6], then we must conclude that either the SO2-depletion is
due to a different metabolic pathway, or SO2-depletion on Venus is not related
to life.

If any diagnostic feature of a biosphere utilising the proposed sulfur energy-
metabolisms is being imprinted in the atmospheric chemistry of Venus, then the
enigmatic cloud-layer SO2-depletion is likely to be it. This is because the
observed SO2-depletion does not yet have a proven abiotic solution, even
though the rest of the atmospheric sulfur-chemistry can be explained by
abiotic models to within at least an order of magnitude accuracy [6]. It may
be that the products of these metabolisms could in principle provide good
biosignatures, but only in the case that their presence or abundances are not
explained by abiotic sources and sinks. This criteria for metabolic products
to be diagnostic biosignatures is not met by the three suggested metabolisms
that we investigate here: the presence of CO2, H2O, H2S, OCS, and sulfur
allotropes in the atmosphere of Venus is already consistent with an abiotic
source from volcanic degassing [37]; the enhanced abundance of OCS and H2S
produced via metabolisms B and C remain undetectable above the cloud layer due
to efficient photochemical destruction by the Solar flux; and any H2O produced
via metabolism A would very likely be retained by the hypothetical
microorganism as part of its survival strategy rather than being released back
to the atmosphere, given the environmental stress posed by the relative
desiccation of Venus’s atmosphere compared to the Earth’s atmosphere [42].
Since there are limited ways to utilize SO2 as a metabolic input, and since
all other reactants would seem to exist in even more limited abundances than
those investigated in the study (CO, OCS, H2S, H2), this strongly suggests
there is no unknown sulfur-metabolism responsible for the observed
SO2-depletion.

Alternatively, a Venusian biosphere may exist and not be influencing the
atmospheric chemistry in a diagnostic and observable way, in which case we can
now place upper limits on the potential biomass density in the cloud layer
before observational constraints are explicitly violated for the three
proposed metabolisms. In order to calculate this limiting biomass density, an
alternative removal mechanism for cloud-layer SO2 must be incorporated into
the model to self-consistently predict the SO2-depletion. One alternative
biologically-driven mechanism for removing SO2 has been suggested by Bains et
al., ($2021$)[43], which hypothesises that life in the clouds, fixing nitrogen
from N2 and using atmospheric H2O, can produce NH3 that reacts with SO2 to
form ammonium sulfites and sulfates, however this has not yet been
demonstrated with fully self-consistent chemical modelling. An abiotic route
to removing SO2 in the cloud layer has been suggested by Rimmer et al.,
($2021$) by incorporating aqueous droplet chemistry. The droplet chemistry
hypothesis posits that an additional source of hydrogen could be delivered to
the clouds as hydroxide salts in mineral dust. This would enable aqueous
reactions within the cloud droplets to provide a destruction pathway for
aqueous SO2 dissolved in the droplets from the gas-phase. A dust flux to the
clouds of $\sim\,16\,{\rm Gt/yr}$ containing $5\,{\rm wt.\%}$ salt would be
sufficient to explain the observed SO2-depletion. For more details see Rimmer
et al., ($2021$) [6]. We here repeat our metabolic analysis, now coupling each
metabolism in turn to the abiotic model that incorporates droplet chemistry in
order to reproduce the observed SO2-depletion self-consistently [6], and
investigate how far we can increase the metabolic activity and abundance of
metabolic input species within observational limits.

The first observational constraints to be exceeded as the biomass density is
increased are the constraints on the abundances of metabolic inputs below the
clouds. Figure 3 shows the three metabolic models with their metabolic input
abundances increased as far they can be while remaining just within an
observed upper-limit, or within observational error. For metabolism B this
corresponds to a $\sim 10\,{\rm ppm}$ upper limit on H2 below the cloud layer
[40, 31], and for metabolism C the upper end of the estimated error on the
largest measured CO mixing ratio below the clouds ($\sim 60\,{\rm ppm}$). For
metabolism A, assuming that the in situ observation of $3\,{\rm ppm}$ at the
surface is correct, the maximum below-cloud H2S within observational
uncertainty corresponds to $\sim 5\,{\rm ppm}$ H2S [29].

Figure 3. Atmospheric profiles resulting from the maximum limit of metabolic
activity within observational constraints. Atmospheric profiles of (a) H2, (b)
H2S and (c) CO mixing ratios as a function of altitude for the fiducial model
and droplet-chemistry (abiotic) model [6], compared with the three proposed
metabolic models at the limit of violating an observational constraint in the
atmosphere. Observational data and their 1$\sigma$ errors are shown in black
except for the observation of H2S below $20\,{\rm km}$ altitude [29], and the
observation of H2 at $50\,{\rm km}$ altitude [41], which are shown in grey, as
these are possibly spurious detections.

We estimate the corresponding biomass density in the cloud layer for the three
limiting metabolic models using a simple energetic argument. The reactions of
metabolisms B and C are chemotrophic and therefore the $\Delta G_{r}$
liberated by each metabolic reaction is already known. If we make an
assumption about the wavelength of light absorbed by the photosystem of
metabolism A then an effective $\Delta G_{r}$ liberated by the net metabolic
reaction can be assumed, give or take some efficiency factor and some
numerical factor for the ratio of photons required per metabolic input
molecule consumed in the synthesis of long-chain organic molecules (these
numerical factors would act to reduce the Gibbs free energy per mole for
metabolism A, thus our estimate here is a relevant upper limit). The average
volumetric rate of the net metabolic reaction thus provides us with an upper
limit on the net energy released via the metabolic activity. We can therefore
make an estimate of the maximum biomass density as a function of cell power,
under the assumption that cells are $\sim 0.5\,{\rm\mu m}$ in radius [9] and
the density of cellular material is of the order $\sim 1\,{\rm g\,cm^{-3}}$.
The results are shown in figure 4, and further details of this calculation can
be found in the Methods section.

Figure 4. Maximum biomass density as a function of cell power requirement in
the Venusian atmosphere. Biomass density in the cloud layer for metabolisms A,
B, and C, as a function of cell power. The red dashed line indicates the
estimated minimum limit on cell power for terrestrial microorganisms [44] and
the red hatch region is the excluded region of the parameter space. The blue
shaded region indicates the range of usual cell maintenance power
requirements, measured in a laboratory setting, for terrestrial microorganisms
[22].

The possible biomass range in our model is critically dependent on the power
requirements of a single cell. We can first obtain an absolute upper bound on
the possible biomass density in the Venusian cloud layer for each proposed
energy-metabolism, by adopting that $P_{\rm cell}\approx P_{\rm cell,\,min}$.
The red dashed line in Figure 4 shows the estimated minimum cell-power,
$P_{\rm cell,\,min}\sim 10^{-21}\,{\rm W}$ for terrestrial microbial life
[44]. The biomass density estimates corresponding to this $P_{\rm cell,\,min}$
are $2\times 10^{5}\,{\rm mg\,m^{-3}}$ for metabolism A, $3\times 10^{6}\,{\rm
mg\,m^{-3}}$ for metabolism B, and $5\times 10^{6}\,{\rm mg\,m^{-3}}$ for
metabolism C. These estimates are inconsistent with the mass loading estimate
of $\sim 0.1-100\,{\rm mg\,m^{-3}}$ for the Venusian lower-cloud droplets
themselves [9], which is problematic as these droplets would host the
microbial life in the hypothesised life-cycle [11]. This treatment must
therefore be a large overestimate of the maximum possible biomass because of
the minimum power assumption we have made: Life in Venus’s clouds would
inevitably have far greater power requirements than the minimum estimated
power requirements of terrestrial life.

More realistic values for the power requirements of terrestrial microorganisms
are those of cell maintenance power – the power per cell required to prevent
population decay – estimated in a laboratory setting to be between
$0.019-4700\times 10^{-15}\,{\rm J\,s^{-1}\,cell^{-1}}$ (indicated with a blue
shaded region in Figure 4) [22]. Using these values, we calculate a range of
biomass estimates between $\sim 3\times 10^{-5}-8\,{\rm mg\,m^{-3}}$ for
metabolism A, $\sim 6\times 10^{-4}-160\,{\rm mg\,m^{-3}}$ for metabolism B,
or $\sim 1\times 10^{-3}-270\,{\rm mg\,m^{-3}}$ for metabolism C. The majority
of values in these ranges of maximum biomass density would be consistent with
the mass loading in the Venusian cloud layer, and the values at the upper end
of the ranges are comparable to the past estimate of $14\,{\rm mg\,m^{-3}}$
[9], and to the estimated biomass density of the Earth’s own aerial biosphere,
$44\,{\rm mg\,m^{-3}}$ [9].

We can instead express this result in terms of the maximum cell concentration
within the aerosol substrate by considering the number density of cells in the
cloud layer and the aerosol mass loading of the Venusian lower-cloud droplets
(see the Methods section for details). For our maximum biomass density ranging
between order $10^{-5}-10^{2}\,{\rm mg\,m^{-3}}$, and aerosol mass loading
estimates ranging from $0.1-100\,{\rm mg\,m^{-3}}$[9], we obtain an estimated
range of values for the maximum possible cell concentration between
$10^{5}-10^{15}\,{\rm cells\,mL^{-1}}$. Comparing to the measured values of
cell concentrations of ${10^{3}-10^{5}\,{\rm cells\,mL^{-1}}}$ from hyperarid
environments on Earth such as the Atacama desert[45], we find that the lower
end of the maximum cell concentration range for a Venusian aerial biosphere
intersects the uppermost value measured in hyperarid terran environments, and
otherwise all other values of maximum cell concentration in the range exceed
cell concentrations observed in hyperarid terran environments. A Venusian
biosphere more abundant than a terrestrial biosphere, even in a hyperarid
environment, would be unexpected given extreme stresses in the Venusian
environment: to loss of biomass out of the cloud layer; and, to desiccation in
the sulfuric acid of the cloud droplets. We note, however, that hyperarid
terran environments consist of dry soils whereas the hyperarid environment of
the Venusian cloud layer consists of liquid sulfuric acid and thus there is no
true analog environment on Earth that can be used as a close comparator to
that of Venus.

The vast chemical challenges that the acidic environment of Venus would pose
to an aerial biosphere would likely result in maintenance power requirements
that are larger than any encountered on Earth, which would favour biomass
density and cell concentration values below even the lower limit of our
estimated biomass ranges (i.e., $\lesssim 3\times 10^{-5},\,6\times 10^{-4}$,
and $1\times 10^{-3}\,{\rm mg\,m^{-3}}$ for metabolism A, B, and C,
respectively). We note also that these estimates have not factored in
limitations on biomass from metabolic processes that are not related to
energy-capture but are still necessary for life: fixation of nutrients from
the atmosphere for growth, for example the energetically costly process of
nitrogen fixation from atmospheric N2, and the incorporation and availability
of metals for use in enzymes to catalyze reactions. This may pose a problem
for the life hypothesis proposed by Bains et al., ($2021$), which relies on
life fixing nitrogen from atmospheric N2 in a non-energy-capture metabolism to
explain the SO2-depletion biochemically[43]. Given these additional challenges
life faces, we emphasise again that the analysis we present here is the most
optimistic scenario possible for the hypothetical life that we have
considered, in which the only limit imposed on a hypothetical biosphere is the
availability of metabolic inputs required to capture chemical energy. These
arguments suggest a very high cell power requirement should be favoured for
hypothetical Venusian aerial life, with the consequence that any such sulfur-
metabolising biosphere would then have to be extremely low in biomass density,
at least four orders of magnitude lower than the terrestrial aerial biosphere,
to remain consistent with atmospheric observations.

We can compare the above estimated biomass densities to the abundance of
matter required to produce the unexplained spectral properties observed at UV
wavelengths in the cloud layer, which are attributed to an ‘unknown UV-
absorber’. It has been suggested that the unknown UV-absorber may be explained
by the presence of sulfur-metabolising microorganisms harboured in the clouds,
with significant overlap found between the spectral appearance of various
proteins and photosynthetic pigments used by some sulfur-metabolising
microorganisms on Earth [9]. Alternatively, one abiotic contender to explain
the UV-absorber is suggested to be a $1\,\%$ solution of FeCl3 dissolved
within $75\,\%$ H2SO4 cloud droplets [46, 47]. Performing a comparison of our
biomass density estimates to this abiotic suggestion, $1\,\%$ of the mass-
loading of the lower clouds is equal to $10^{-3}-1\,{\rm mg\,m^{-3}}$ [9]. If
the maximum biomass density we have calculated above is indeed close to or
less than the lower end of our estimated range of values (i.e., large cell
power requirements with respect to known terrestrial values), then, assuming
the biotic matter can produce similar strength spectral properties as the
suggested FeCl3 molecules, only metabolism C could possibly intersect the
lower limit of the absorber’s required mass-loading range ($10^{-3}\,{\rm
mg\,m^{-3}}$) and metabolisms A and B fall short by at least one order of
magnitude. However, if the biological pigments can produce much stronger
absorption than simple salts then the potential biomass density we have
constrained in the Venusian cloud layer may possibly still be a viable
explanation to the unknown UV-absorber. Future studies would be needed to
compare the standard absorbance of such proposed biological pigments to the
observed spectral properties of the cloud aerosol.

One final consideration that could be made is that of the ecological
stoichiometry of a combined ecosystem, featuring a combination of metabolisms
in a steady state where the fluxes of metabolic products and inputs consumed
are equally balanced. For any significant biosphere an ecosystem is needed and
nutrient recycling is critical. Given that the terrestrial biosphere would
collapse within a short time period in the absence of nutrient recycling
between different species in a diverse ecosystem, it is also unlikely that one
species of organism could exist by itself in the cloud layer of Venus, with
the speculative exception perhaps of an autotrophic species using
photosynthesis and only maintaining a very low biomass, such as that described
by metabolism A. We have demonstrated that each of the metabolic hypotheses
investigated here can be individually ruled out on the basis of violating
lower atmosphere constraints on the required metabolic input abundances, and
we will now demonstrate how our results also show that a multi-metabolism
ecosystem recycling the required chemical species exclusively within the cloud
layer cannot explain the sulfur depletion in Venus’s clouds.

If we were to take, for example, the net effect of metabolism A and B together
then the flux of H2S consumed by life with metabolism A, could be produced at
the same rate by life utilising metabolism B. This results in a net reaction:

(6) $\displaystyle 2\,\ce{H2}+3\,\ce{SO2}+4\,\ce{CO}$
$\displaystyle\rightarrow 2\,\ce{H2O}+4\,\ce{CO2}+\ce{S2}+\ce{S}$

Even if arguments could be posed for the efficient production of CO from the
abundant CO2 in the atmosphere via photolysis or perhaps another metabolism in
a hypothetical ecosystem, the reaction inevitably still requires input of H2
to the atmosphere, inconsistent with observations. It has been suggested that
H2 could be replenished in the atmosphere via hydration reactions of ferrous
iron oxide minerals at the surface [48]. Even if there were sufficient H2O in
the atmosphere to enable this, the resulting below-cloud H2 mixing ratio would
violate the observational upper limit of $<10\,{\rm ppm}$ [31]. Production of
H2 via hydration of ferrous iron oxide mineral dust within the cloud layer
exclusively would not be possible because this mineral dust would be dissolved
in the sulfuric acid cloud droplets and thus any hydrogen in the reaction
would exist as aqueous H+ ions. Mineral dust therefore cannot provide a source
of H2 localised within the cloud layer as a potential metabolic input and H2
could only be replenished by surface-atmosphere processes. The observational
upper limit on H2 abundance below the clouds therefore poses a strict upper
limit on the productivity of any biosphere recycling nutrients in ecosystem
interactions. In the absence of ecosystem interactions, the requirement for
metabolic inputs to be replenished by surface-atmosphere processes extends to
all chemical species and therefore below-cloud observations of any limiting
input species poses a strict upper limit on the productivity of the biosphere.

Ultimately there simply is not enough reducing power in the lower atmosphere
of Venus for the suggested forms of life to exploit in order to deplete SO2
and thus be of sufficient biomass to influence the global atmospheric
chemistry. We have demonstrated this result for life using each of the energy-
capture metabolisms proposed in the literature so far, however our results
extend generally to: a) chemotrophs exploiting energy gradients from gaseous
atmospheric species; or b) phototrophs utilising gaseous atmospheric species
as electron donors. The exception to this conclusion is phototrophic life that
captures light energy not for CO2 reduction but to support general cellular
demands, in which case the potential biomass density in the clouds cannot be
constrained via the availability of metabolic inputs in the atmosphere, and
future studies are needed to constrain this potential biomass via the
available light flux. There may also still plausibly be a low mass biosphere
present, the only observable effect of which is to release relatively small
amounts of trace metabolic gases. The recent detections of phosphine gas in
the atmosphere of Venus may be consistent with this, in the absence of any
abiotic phosphine production pathway [10, 49, 50, 51, 12].

Upcoming missions to Venus may be able to determine whether or not the cloud
layer hosts a high mass biosphere using a currently unidentified metabolism, a
low mass biosphere producing only trace metabolic products, or no biosphere at
all. However, the prospect remains that other, possibly wetter, Venus-like
exoplanets could host a habitable niche in the temperate region of their
atmospheres. Aerial biospheres in general therefore have significant
implications on the number and observability of potentially habitable planets
beyond the Solar System. The hypothetical ‘habitable zone’ around a star has
resulted in the search for Earth-sized exoplanets within the range of orbital
distances for which an Earth-like atmosphere could permit condensation of
liquid water at the surface [52]. Interior to the habitable zone is the region
where surface temperatures are too high for water to condense at the surface,
and this region is more likely to be populated by Venus-like exoplanets [53].
For Venus-like exoplanets with permanent cloud cover in the temperate region
of the atmosphere, the possibility of an aerial biosphere extends the inner
edge of the habitable zone much closer to the host star, where orbital periods
are shorter and transiting planetary signals have enhanced signal to noise
ratios, compared to planets within the surface-water habitable zone. The
presence of a biosphere at high altitude also means that biologically induced
features in the atmospheric chemistry will be imprinted in the observationally
accessible region of the atmosphere, compared to surface life obscured by
thick cloud cover. It has been demonstrated that the metabolic inputs H2S
(metabolism A) and SO2 (metabolisms A, B, and C), and metabolic products OCS
(metabolism C) and H2S (metabolism B), will not be depleted photochemically in
the observable region of a Venus-like planet’s atmosphere around cooler host
stars, in the same way they are in Venus’s atmosphere[54]. This means that
with high precision retrievals of the upper atmosphere, the aerial biosphere
hypothesis may be possible to test on Venus-like exoplanets. The framework
that we present in this paper, self-consistently coupling metabolic reaction
networks with a photochemical-kinetics network for the atmosphere of Venus, is
equally applicable to simulating the effect of an aerial biosphere on the
atmospheric chemistry of potentially habitable Venus-like exoplanets. If the
new and exotic biome of a strictly aerial biosphere is possible in principle,
then given the sheer number and diversity of exoplanets, we might expect that
the first detected sign of life beyond the Solar System will originate from an
aerial biosphere harboured on a world interior to the surface-water habitable
zone.

## Methods

### The model.

We model the atmosphere of Venus using a photochemical-diffusion model [6].
The model is composed of a $1{\rm D}$ Lagrangian solver, Argo, and a network
of reactions, Stand2020, that treats H/C/N/O/S/Cl chemistry accurately within
a temperature range of $100-30,000\,{\rm K}$ [26, 55]. The model has been
shown to reproduce the known atmosphere of Venus within approximately an order
of magnitude of observations given the inclusion of a scheme of aqueous
chemistry that aids the gas-phase depletion of SO2 and H2O [6]. The chemical
network, Stand2020, is a list of reactants, products, and rate constants, for
every chemical reaction considered in the atmosphere. For thermochemical
reactions the reaction rates are temperature dependent, and for photochemical
reactions the reaction rates are dependent on the photon flux as a function of
wavelength incident at a given layer of the atmosphere. These chemical
reactions determine the chemical production and loss rates of each species. At
each altitude step in the atmosphere the reactions are solved by Argo as a set
of time-dependent, coupled, non-linear differential equations:

(7) $\dfrac{dn_{\rm X}}{dt}=P_{\rm X}-L_{\rm X}n_{\rm
X}-\dfrac{\partial\Phi_{\rm X}}{\partial z},$

where, at a given height $z\,{\rm(cm)}$ and time $t\,{\rm(s)}$, $n_{\ce{X}}$
(cm-3) is the number density of species $\ce{X}$, $P_{\rm X}$ (cm-3 s-1) is
the rate of production of species X, $L_{\rm X}$ (s-1) is the rate constant
for loss of species X, and $\partial\Phi_{\rm X}/\partial z$ (cm-3 s-1)
describes the divergence of the vertical diffusion flux, encapsulating both
eddy- and molecular-diffusion.

Argo follows a parcel of gas as it rises from the surface to the top of the
atmosphere and back down again. An initial condition for the chemical
composition is input at the base of the atmosphere, listed in table 1. At
every altitude step on the journey upwards Argo solves equation 7 for all
species in the atmosphere, based on the pressure, temperature, and species’
abundances at that altitude in the atmosphere. The time interval over which
Argo solves for is prescribed by the eddy diffusion profile which parametrises
vertical transport through the atmosphere. The pressure-temperature profile
and eddy-diffusion profile for modern Venus are each taken from past
photochemical-kinetics models of the lower [27] and middle [56] atmosphere. At
the top of the atmosphere, Argo takes the incident stellar spectrum and, at
each step on the journey downwards, includes photochemical reactions driven by
the stellar flux irradiating that layer of the atmosphere. Argo iterates this
procedure until a convergence criteria is met and thus a global solution is
found.

CO_2 |  N_2 |  SO_2 |  H_2O |  CO |  OCS |  HCl |  H_2 |  H_2S
---|---|---|---|---|---|---|---|---
0.96 | 0.03 | 150 ppm | 30 ppm | 20 ppm* | 5 ppm | 500 ppb | 3 ppb* | 10 ppb*
Table 1. Initial surface abundances for the fiducial atmosphere [6]. *The
initial surface abundance of species marked with an asterisk are varied for
some of the models.

We test the effect of each of the net metabolic reactions, 3, 4 and 5 in turn,
by coupling them to our atmospheric network and solving for the atmospheric
composition of Venus. For each proposed metabolism the individual net
metabolic reaction is added to the Stand2020 reaction network with an
effective rate constant that is unconstrained. This rate is not synonymous
with the biochemical rate of the metabolic reaction itself, rather it is an
‘effective’ net volumetric rate of all microbial activity in our one
dimensional and diurnally averaged model. In this sense, the effective rate of
the reaction relates to the biomass density in the model. The effective
reaction is constrained to occur within the temperate region of the cloud
layer by prescribing a condition that depends on atmospheric temperature. For
a given effective rate, $k$, the reaction constant in the network depends on
temperature (and thus altitude / pressure) as:

(8) ${\rm Rate}=\begin{cases}k,&\text{if $273<T\,{\rm(K)}<373$}.\\\
0,&\text{otherwise}.\end{cases}$

With each metabolism coupled to the Stand2020 network in turn, we iterate Argo
over a grid of models to explore the parameter space of net metabolic rate
versus reagent abundance. The SO2 abundance remains fixed at it’s fiducial
value. For metabolism A we gradually increase the H2S abundance from $10\,{\rm
ppb}$ up to $500\,{\rm ppm}$. For metabolism B we gradually increase the H2
abundance from $3\,{\rm ppb}$ up to $10\,{\rm ppm}$, and then increase both
the H2 and CO abundances in tandem in the ratio 2:1 respectively, from
$10\,{\rm ppm}$ up to $300\,{\rm ppm}$ for H2 and from $20\,{\rm ppm}$ up to
$600\,{\rm ppm}$ for CO. For metabolism C we gradually increase the CO
abundance from $20\,{\rm ppm}$ up to $650\,{\rm ppm}$.

### Biomass density calculation.

For the model atmospheres that incorporate the metabolic activity of a
putative Venusian biosphere at the maximum possible limit before transgressing
observational constraints, we can estimate the maximum biomass density
averaged over the volume of the cloud layer, under the assumption that the
availability of metabolic inputs is the limiting factor on the productivity of
the biosphere (in practise other factors, such as availability of metals for
catalytic enzymes, will likely limit the productivity of the biosphere). For
volumetric reaction rate $R$ ($\rm s^{-1}\,m^{-3}$) of the metabolic reaction
in the model, we convert to the rate of energy liberated metabolically per
unit volume in the cloud layer, $\epsilon$ ($\rm W\,m^{-3}$):

(9) $\epsilon=\frac{R\,\Delta G_{\rm r}}{N_{\rm A}},$

where $\Delta G_{r}$ ($\rm J\,mol^{-1}$) is the Gibbs free energy release from
the metabolic reaction and $N_{\rm A}$ is Avogadro’s constant. The average
cell number density in the cloud layer, $n_{\rm cell}$ ($\rm cells\,m^{-3}$),
is then given by:

(10) $n_{\rm cell}=\frac{\epsilon}{P_{\rm cell}},$

and the average biomass density:

(11) $m_{\rm bio}=\frac{\epsilon\,M_{\rm cell}}{P_{\rm cell}},$

where $P_{\rm cell}$ ($\rm W\,cell^{-1}$) represents the power requirement of
a single microbe of mass $M_{\rm cell}$ (mg). This allows for a comparison to
be made to the measured mass loading of aerosol in the atmosphere, and to the
estimated mass loading of microbes in the Earth’s (transiently) aerial
biosphere. We can further make comparison to measured cell concentrations in
analogue terrestrial environments that are most similar to the hyperarid
environment of the Venusian cloud layer. To do so, we estimate the volume of
aerosol per unit volume of the atmosphere, $v_{\rm aero}$ ($\rm mL\,m^{-3}$),
within the cloud layer, from the reported aerosol mass loading $m_{\rm aero}$
($\rm mg\,m^{-3}$) and an assumed density of aerosol material, $\rho_{\rm
aero}$ ($\rm g\,mL^{-1}$):

(12) $v_{\rm aero}=\frac{m_{\rm aero}}{\rho_{\rm aero}}.$

The cell concentration within the aerosol substrate, $n_{\rm cell,\,aero}$
($\rm cells\,mL^{-1}$), is then given by:

(13) $n_{\rm cell,\,aero}=\frac{n_{\rm cell}}{v_{\rm aero}}.$

Combining equations (9 \- 13) and evaluating when the volumetric metabolic
rate and metabolic input abundances are at the limit of transgressing
observational data, gives us our final set of equations used to evaluate the
range of maximum possible biomass densities and maximum possible cell
concentrations referenced in the main text:

(14) $\displaystyle m_{\rm bio,\,max}$ $\displaystyle=\frac{R\,\Delta G_{\rm
r}\,M_{\rm cell}}{N_{\rm A}\,P_{\rm cell}},$ (15) $\displaystyle n_{\rm
cell,\,aero,\,max}$ $\displaystyle=\frac{R\,\Delta G_{\rm r}\,\rho_{\rm
aero}}{N_{\rm A}\,m_{\rm aero}\,P_{\rm cell}}.$

The analysis here refers primarily to the chemotrophic metabolisms where
$\Delta G_{r}\sim 240\,{\rm kJ\,mol^{-1}}$ is known a priori. Photosynthetic
life would obtain energy from sunlight rather than a redox disequilibrium in
the atmosphere, however the phototrophic metabolism can be treated with the
same analysis if an assumption is made about the wavelength of light absorbed
in reaction 1. For example, the energy delivered in sunlight at a wavelength
of $500\,{\rm nm}$ is approximately $\sim 240\,{\rm kJ\,mol^{-1}}$ (i.e. per
${\rm N_{A}}$ of photons absorbed where ${\rm N_{A}}$ is Avogadro’s constant)
and so the above analysis can be repeated exactly for the phototroph
(metabolism A) give or take some efficiency factor, and some numerical factor
for the different possible wavelengths of light being utilised (the exception
to this energetic analysis would be phototrophic life that captures light
energy for cell maintenance without any redox chemistry occuring, in which
case the potential biomass cannot be constrained by the rate of consumption of
metabolic input species). Recent radiative transfer modelling has shown that,
below $\sim 59\,{\rm km}$ altitude, while biologically-damaging UV radiation
is screened by the cloud layer, photosynthetically active light at longer
wavelengths ($>400\,{\rm nm}$) continues to penetrate [57]. We therefore
assume $\Delta G_{r}\sim 240\,{\rm kJ\,mol^{-1}}$ for each of the three net
metabolic reactions. While our choice of $500\,{\rm nm}$ for the wavelength
utilised in the photosynthetic metabolism is for simplicity in calculating the
biomass, this value sits well within the range of common terrestrial
photosynthetic wavelengths ($\sim 400-1000\,{\rm nm}$) including the
wavelengths utilised by anoxygenic phototrophs in deep water columns [58]. We
further assume a typical value for cell density of a microbe to be of the
order $\rho_{\rm cell}\sim 1\,{\rm g\,cm^{-3}}$ and typical size of $r_{\rm
cell}\sim 0.5\,{\rm\mu m}$ in order to evaluate $M_{\rm cell}$. We can then
investigate how the choice of cell power requirement, $P_{\rm cell}$, affects
our estimates for the biomass density, in figure 4.

The required introduction of the cell power requirement term provides the
largest uncertainty to this energetic analysis, varying over at least 10
orders of magnitude for terrestrial microbial life alone. It is for this
reason why we must evaluate a range of possible estimates for $m_{\rm
bio,\,max}$ and $n_{\rm cell,\,aero,\,max}$. Assuming that the stresses posed
by the acidic and hyperarid Venusian environment pose greater survival
challenges to life compared to conditions encountered on Earth, we can
significantly constrain our estimated range of values of $m_{\rm bio}$ and
$n_{\rm cell,\,aero}$ at the limiting metabolic reaction rate, by imposing
that $P_{\rm cell}\gtrsim P_{\rm max,\,\bigoplus}$:

(16) $\displaystyle m_{\rm bio,\,max}$ $\displaystyle\lesssim\frac{R\,\Delta
G_{\rm r}\,M_{\rm cell}}{N_{\rm A}\,P_{\rm max,\,\bigoplus}},$ (17)
$\displaystyle n_{\rm cell,\,aero,\,max}$
$\displaystyle\lesssim\frac{R\,\Delta G_{\rm r}\,\rho_{\rm aero}}{N_{\rm
A}\,m_{\rm aero}\,P_{\rm max,\,\bigoplus}},$

Taking $P_{\rm max,\,\bigoplus}\,=\,4700\times 10^{-15}\,{\rm
J\,s^{-1}\,cell^{-1}}$ we evaluate limits $m_{\rm bio,\,max}\lesssim 3\times
10^{-5}\,{\rm mg\,m^{-3}}$ for metabolism A, $m_{\rm bio,\,max}\lesssim
6\times 10^{-4}\,{\rm mg\,m^{-3}}$ for metabolism B, and $m_{\rm
bio,\,max}\lesssim 1\times 10^{-3}\,{\rm mg\,m^{-3}}$ for metabolism C, as
quoted in the abstract.

## Data Availability

The data generated in this study have been deposited in the Harvard online
database under accession code (https://doi.org/10.7910/DVN/AKUTME).

## Code Availability

The methods underlying the code and the chemical network are all publicly
available in the set of papers [26, 59, 55, 6]. The ARGO software may be made
available upon reasonable request from the authors.

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## Acknowledgements

S.J. thanks the Science and Technology Facilities Council (STFC) for the PhD
studentship (grant reference ST/V50659X/1). P.B.R. thanks the Simons
Foundation for funding (SCOL awards 599634).

## Author Contributions

S.J., O.S., and P.B.R. conceived of the study and contributed extensively to
the manuscript. S.J. implemented the simulations and analysed the output data.

## Competing Interests

The authors declare no competing interests.


#  Rectification of Confined Soft Vesicles Containing Active Particles

M. C. Gandikota and A. Cacciuto<EMAIL_ADDRESS>Department of Chemistry,
Columbia University
3000 Broadway, New York, NY 10027

###### Abstract

One of the most promising features of active systems is that they can extract
energy from their environment and convert it to mechanical work. Self
propelled particles enable rectification when in contact with rigid
boundaries. They can rectify their own motion when confined in asymmetric
channels and that of microgears. In this paper, we study the shape
fluctuations of two dimensional flexible vesicles containing active Brownian
particles. We show how these fluctuations not only are capable of easily
squeezing a vesicle through narrow openings, but are also responsible for its
rectification when placed within asymmetric confining channels (ratchetaxis).
We detail the conditions under which this process can be optimized, and sort
out the complex interplay between elastic and active forces responsible for
the directed motion of the vesicle across these channels.

##  I Introduction

It is well known that the second law of thermodynamics prohibits the
extraction of work from any cyclic process performed on a system in thermal
equilibrium with a single thermal bath Leighton and Sands (1965). This exploit
can instead be achieved in systems that are subject to active fluctuations.
Seminal experiments on micro-gears placed in a suspension of bacteria have
demonstrated how sustained and rectified rotational motion of a gear can be
achieved Angelani  _et al._ (2009); Di Leonardo  _et al._ (2010); Sokolov  _et
al._ (2010). Further work demonstrated how motion rectification can occur when
E. coli is confined in a chamber divided by funnel shaped walls Galajda  _et
al._ (2007); Wan  _et al._ (2008); Reichhardt and Reichhardt (2017) and when
active components such as active colloids or chains are subject to asymmetric
potentials Angelani  _et al._ (2011); Yariv and Schnitzer (2014); Wu  _et al._
(2015) or when confined in asymmetric periodic channels Ghosh  _et al._
(2013); Ao  _et al._ (2014); Lu  _et al._ (2017).

Motion rectification breaks both space and time symmetries and is thus not
permitted in equilibrium systems except as transients. In contrast,
rectification can be achieved in thermal systems using time-dependent
spatially asymmetric potentials Reimann (2002); Hänggi and Marchesoni (2009).
Time-dependent potentials are, however, unnecessary for active particles to
achieve rectification of motion given that active systems are inherently out
of equilibrium at the single particle level Ramaswamy (2010); Marchetti  _et
al._ (2013); Zöttl and Stark (2016).

While previous studies on active rectification have mostly focused on active
components directly interacting with hard passive objects, be that a gear or a
micro-channel, here we show how rectification can also develop as a result of
the interplay between active and elastic forces. We report the rectification
of soft vesicles enclosing active Brownian particles when confined in an
asymmetric periodic channel.

Experimentally, the vesicle can be realized by enclosing active particles
inside lipid membranes, and it has been shown that large non-equilibrium shape
deformations capable of reshaping the vesicle can develop under certain
conditions Vutukuri  _et al._ (2020). Experiments on macroscopic soft circular
scaffolds that enclose small centimeter size actuators have also been shown to
be capable of squeezing through tight spaces Boudet  _et al._ (2021). Overall,
the shape and motility aspects of these vesicles have been studied in detail
by employing simple models such as a closed flexible boundary enclosing
spherical or rod shaped active particles Paoluzzi  _et al._ (2016); Wang  _et
al._ (2019); Quillen  _et al._ (2020). However, to the best of our knowledge,
rectification aspects of vesicles undergoing active fluctuations have not been
paid any attention. Yet this is an important problem especially given the
experimental demonstration of directional motion of fibroblasts and epithelial
cancerous cells when confined within asymmetric periodic channels Le Maout
_et al._ (2020). While cells are complicated entities involving sub-structures
such as the cytoskeleton, here, we show that rectification can be achieved by
far simpler deformable entities such as a two dimensional soft vesicle that
encloses spherical active particles.

##  II Model

We model the two dimensional vesicle as a flexible closed boundary consisting
of $N$ self-avoiding passive spheres of diameter $\sigma$ connected with stiff
harmonic springs. The active fluctuations of the vesicle are due to $N_{A}$
active Brownian particles (ABP) of spherical shape and diameter $\sigma$
enclosed within its perimeter. These particles are activated using a self-
propelling force of constant magnitude $v_{p}$ and follow these equations of
motion:

$\begin{split}\frac{d\boldsymbol{r}_{i}(t)}{dt}&=\frac{1}{\gamma}\boldsymbol{f}(\\{r_{ij}\\})+v_{p}\,\boldsymbol{\hat{n}}_{i}(t)\,\delta_{t_{i},1}+\sqrt{2D}\,\boldsymbol{\xi}(t),\\\
\frac{d\boldsymbol{{\hat{n}}}_{i}(t)}{dt}&=\sqrt{2D_{r}}\,\boldsymbol{\xi}_{r}(t)\times\boldsymbol{\hat{n}}_{i}(t),\end{split}$
(1)

where $i$ is the particle index, $\boldsymbol{\hat{n}}$ is the axis of
propulsion, and $t_{i}$ is a binary index, which is set to zero (one) for
passive (active) particles. The Kronecker delta function $\delta_{t_{i},1}$
activates the self-propulsion term for the active particles. The translational
diffusion coefficient $D$ is related to the temperature $T$ and the
translational friction $\gamma$ via the Stokes-Einstein relation $D=k_{\rm
B}T\gamma^{-1}$. Likewise, the rotational diffusion coefficient, $D_{r}=k_{\rm
B}T\gamma_{r}^{-1}$, with $D_{r}=3D\sigma^{-2}$. The solvent-induced Gaussian
white-noise terms for both the translational $\boldsymbol{\xi}$ and rotational
$\boldsymbol{\xi}_{r}$ motions are characterized by
$\langle\boldsymbol{\xi}(t)\rangle=0$ and
$\langle\xi_{m}(t)\xi_{n}(t^{\prime})\rangle=\delta_{mn}\delta(t-t^{\prime})$.
$\boldsymbol{f}(\\{r_{ij}\\})$ indicates the excluded volume forces between
particles and the stretching forces between neighboring monomers of the
vesicle. Excluded volume forces between any two particles are enforced via a
Weeks-Chandler-Andersen (WCA) potential
$U_{w}=4\varepsilon\left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}+\frac{1}{4}\right].$
The bonds between neighboring monomers obey the harmonic potential
$U_{s}=k_{s}(r_{i,i+1}-\sigma)^{2}$ where $r_{i,i+1}$ is the distance between
consecutive monomers along the chain. The spring constant $k_{s}$ is set to
$2500k_{\rm{B}}T/\sigma^{2}$.

Figure 1: Probability distribution of local curvatures $\kappa$ for different
numbers of ABPs. The vertical dashed lines are guides to the eye indicating
$\kappa=0$ (flat region) and $\kappa=r^{-1}_{0}$ (the curvature of the exact
circle described by the vesicle). The $*$ sign on the x-axis is the pore
curvature in the circular cavities. The inset is a typical snapshot from our
simulations with $\phi=0.072$. The flat distribution of curvatures of the
passive ring ($N_{A}=0$) is shown as a black dotted line.

We performed our simulations using the numerical package LAMMPS Plimpton
(1995) and set $\sigma$, $\tau=\sigma^{2}D^{-1}$ and $k_{\rm B}T$ as the units
of length, time and energy respectively. All simulations were run with a time
step of $\Delta t=10^{-5}\tau$. We collected statistics for up to $10^{10}$
time steps. The strength of the active forces is reported in terms of the
Péclet number defined as $Pe=v_{p}\sigma/D$. The packing fraction of the ABPs
$\phi=N_{A}/(\pi r_{0}^{2})$ where $r_{0}$ is the radius of the exact circular
shape of the soft vesicle. We used a soft vesicle composed of $N=64$ colloidal
particles unless otherwise stated.

##  III Shape fluctuations and escape from a circular cavity

We first study the effect of the active particles on the shape of a soft
vesicle in open boundary conditions.

For an inextensible soft vesicle, the shape statistics of the vesicle is fully
determined by the distribution of the local curvatures. We fitted osculating
circles to every three consecutive monomers and determined the local
curvatures by taking the inverse of their radii. The probability distribution
of the local curvatures $\rho_{k}$ for varying number of active particles,
$\phi$, at $Pe=100$ is shown in Fig. 1. For large $\phi$, the particles
distribute homogeneously along the vesicle perimeter and push on it. The
imposed tension on the vesicle suppresses the shape fluctuations Vutukuri  _et
al._ (2020); Iyer  _et al._ (2022) resulting in a rather sharp curvature
distribution centered around $\kappa=r_{0}^{-1}=0.098\sigma^{-1}$. For small
$\phi$, particles create local kinks on the vesicle and stretch flat the
region of the vesicle connecting any two kinks. This results in a broad
distribution of curvatures that peaks at $k/r_{0}^{-1}\simeq 0$ and extends
all the way to $\kappa/r_{0}^{-1}=10$. For intermediate values of $\phi$, the
distribution presents a peak near $\kappa/r_{0}^{-1}=0$ and develops a
shoulder (rather than a distinct second peak observed for run-and-tumble
particles Paoluzzi  _et al._ (2016)) for larger values of $\kappa$, suggesting
the presence of moderately curved regions generated by particle clustering on
the surface that are then connected by flat regions (see a typical snapshot in
Fig. 1.) Curvature-mediated particle clustering is a well known phenomenon for
ABP near a curved surface Fily  _et al._ (2014), and plays an important role
in understanding the active shape of these flexible boundaries.

Figure 2: Average escape time of a soft vesicle from a circular cavity as a
function of number of ABPs, $N_{A}$ for different values of $Pe$. The inset is
a typical snapshot from our simulations for $\phi=0.072$ at $Pe=100$.

The presence of these large curvature fluctuations is of crucial importance
when considering the ability of vesicles to squeeze through narrow openings,
as they can be exploited to drive the vesicle out of a confinement cavity with
a narrow opening. We study this phenomenon by considering a circular hard wall
constructed by freezing in space a number of WCA particles of diameter
$\sigma$ as shown in the schematic of Fig. 2. The wall has an opening of size
$7.9\sigma$ which allows the passage of the vesicle only if the local
curvature of the vesicle at this pore is greater than
$1/7.9\sigma=0.25\sigma^{-1}$. This curvature is marked with a “$\ast$” on the
$\kappa/r_{0}^{-1}$ axis of Fig. 1, and indicates how vesicles containing
small numbers of active particles will be more likely to generate large enough
shape fluctuations to drive the vesicle out of the cavity.

Figure 3: Establishing a preferential direction of motility using asymmetric
walls. The vesicle escapes to the left for all finite $\gamma$ in the wall
geometry specified in the schematic. The solid curve is Eq. (3). Here,
$\phi=0.03$ and $Pe=150$.

The average escape time $\left<t_{e}\right>$ is calculated over a hundred runs
for a range of $Pe$ and $\phi$ and is shown in Fig 2. Interestingly, the
escape time has a distinctly non-monotonic dependence on $N_{A}$. This can be
understood by considering that while small number of active particles lead to
large curvature fluctuations of the vesicle, larger values of $\phi$ result in
overall larger driving forces and cooperative pushing on the vesicle (see
snapshot in Fig. 2). However, if the vesicle is filled with too many active
particles, curvature fluctuations become rare, and the vesicle escape is
hindered. We find that $\phi\in[0.012,0.024]$ provides the best compromise
between these two tendencies for the range of Peclét numbers considered in
this study.

##  IV Rectification in asymmetric periodic channels

We have now seen the interplay of the high-curvature segments of the vesicle
and the directed motility of active particles which squeeze the vesicle
through a pore. However, a vesicle containing active particles in free space
does not have a preferential direction of motion. The mean square displacement
(MSD) of the center of mass of this system can be calculated to be (see
Appendix A) Paoluzzi  _et al._ (2016); Hagen  _et al._ (2009),

$\left<\bm{r}^{2}(t)\right>=\frac{4D\;t}{N+N_{A}}+\frac{2\;Pe^{2}\sigma^{2}N_{A}}{9\;(N+N_{A})^{2}}\left(e^{-D_{r}t}+D_{r}t-1\right).$
(2)

The vesicle is diffusive in the limit of $t\gg\tau$ with the long time
diffusion constant attaining its maximum for $N=N_{A}$ (see Appendix VII.1).
The MSD of a vesicle containing active particles can be considered to be the
MSD of a single active particle with re-scaled diffusion constant and Péclet
number while keeping the rotational diffusion constant unchanged.

For the vesicle to achieve directional motion, we need to break the spatial
symmetry in the system Reimann (2002). For instance, we can constrict the
vesicle between two vertical walls as shown in the schematic of Fig. 3, where
the active particles are equally distributed between the left and right
regions. With this setup, the vesicle has equal probability of escaping
towards the left or the right side of the space, i.e. $P^{L}=P^{R}=0.5$. The
probabilities are computed by calculating the number of times the vesicle
translocates to the left (or right) divided by the total number of
translocation events. A simple way of breaking the spatial symmetry is to add
a second set of walls tilted of an angle $\gamma$ with respect to the vertical
axis as depicted in Fig. 3. Crucially, as shown in Fig. 3, for any $\gamma>0$,
$P^{L}>0.5$ and it monotonically increases with $\gamma$. For large $Pe$ and
$N_{A}$, the vesicle conforms to the shape of the walls with most of the
active particles located along its perimeter with their axis of propulsion
parallel to the surface normal. In this limit one can estimate the relative
forces pushing the vesicle on either side. While all particles on the left
hand side contribute to push the vesicle their way by applying an homogeneous
internal pressure to the left lobe, particles along the slanted walls of the
right hand side generate no contribution to the overall pressure on the right
lobe. In this limit, $P^{L}(\gamma)$ can be approximated to be (see Appendix
B),

$P^{L}(\gamma)=\frac{1}{2(1-\gamma/\pi)},$ (3)

and is shown as the solid curve in Fig. 3.

Figure 4: The position of the vesicle’s center of mass denoted by $x$ as a
function of time $t$. The vesicle displays leftward rectification when
confined within a periodic asymmetric channel The inset is a snapshot from
simulations. Here, $\phi=0.072$ and $Pe=150$.

We next consider a channel constructed with periodic cavities incorporating
such a sloped geometry as shown in Fig. 4. The cavity is characterized by a
width $x_{0}=30\sigma$, a pore size $d_{0}=7.6\sigma$ and
$\gamma=\text{atan}(0.6)$.

Figure 5:  a) The leftward escape rate of vesicle $k^{L}$ is non-monotonic
with respect to $N_{A}$. Inset: The probability of leftward transitions
consistently increases with the number of ABPs. Here, $Pe=150$. b) The
leftward escape rate monotonically increases with $Pe$ while the probability
of leftward transitions monotonically decreases (inset). Here, $\phi=0.024$.

The preferential leftward motion of the center of mass of the vesicle, $x$, is
readily observed and highlighted in Fig. 4. The rectification of motion as
measured by $P^{L}$ consistently increases with $\phi$ as seen in the inset of
Fig. 5(a). However, the escape rate from the narrow (left) side of the cavity
in the periodic system, $k^{L}$, is non monotonic as seen in Fig. 5(a). The
escape rate has its maximum at $\phi\approx 0.024$ and decreases with further
increase in $\phi$ consistent with the minimum of the average escape time in
Fig. 2. The escape rate monotonically increases with $Pe$ while the
rectification of motion monotonically decreases as can be seen in Fig. 5(b)
and its inset. Clearly, for $Pe=0$, that is when the vesicle is passive, we do
not expect any rectification and expect $P^{L}=0.5$. However, measuring
$P^{L}$ for $Pe<50$ is a challenge as each translocation event becomes
extremely slow making it very hard to collect sufficient statistics.

We have already identified one mechanism that explains why the left side of
the cavity is the more likely escape route for the vesicle. There is a
secondary driving force that becomes important for intermediate to large
number of particles and results from the collective tendency of active
particles to cluster around highly curved regions. To understand how this
comes about, we considered a non-periodic, isolated triangular cavity as
depicted in Fig. 6. Within this cavity, the vesicle faces no asymmetry
immediately upon exiting the cavity. If we factor the escape probabilities
into the probability of docking, $P_{d}$, and that of translocating, $P_{t}$,
then the ratio between $P_{L}$ and $P_{R}$ can be written as

$\frac{P^{L}}{P^{R}}=\left(\frac{P^{L}_{d}}{P^{R}_{d}}\right)\left(\frac{P^{L}_{t}}{P^{R}_{t}}\right).$
(4)

Our simulations on this system indicate that the docking probability ratio
$P^{L}_{d}/P^{R}_{d}$ increases with $N_{A}$ whereas the transition
probability ratio $P^{L}_{t}/P^{R}_{t}$ remains roughly independent of
$N_{A}$. See Fig. 6. Crucially, in this cage, we observe a cross-over behavior
where for small values of $\phi$, the vesicle rectifies its motion to the
right, whereas for larger values of $\phi$, leftward rectification is
achieved. This result suggests that for small number of particles, when the
vesicle can undergo large curvature fluctuations, the slanted side of the cage
effectively acts as a barrier against the vesicle passage by reducing the
amount of space available to it. However, when $\phi$ becomes sufficiently
large, and large shape fluctuations become rare, the slanted side of the
cavity can promote the formation of high curvature regions (as the vesicle
conforms to the shape of the walls) which in turn drive the accumulation of
active particles that can cooperatively push the vesicle through the left side
pore. Clearly for large $\phi$ the docking probability decreases for both
sides of the cavity, however, the decrease is more pronounced on the right
side where such cooperative behavior is harder to achieve. Since, as was
already established above (see Eq. 3), the probability of translocation is
more likely on the slanted side of the cavity. For about $\phi>0.036$ the
overall rectification direction switches from right to left.

Figure 6: Relative docking and translocation probabilities for a vesicle
confined inside an isolated triangular cavity for different values of $\phi$.
The inset is a snapshot of a vesicle in this cavity. The lines connecting the
data points are a guide to the eye.

##  V Relevance of vesicle size for rectification

Lastly, we consider the behavior of a vesicle placed in a periodic asymmetric
channel with a single active particle in its interior. Individual active
particles are known to rectify their motion in asymmetric channels when the
persistence length of the particles is larger than the cavity size (Knudsen
regime) Ghosh  _et al._ (2013). Operating in the Knudsen regime for the
isolated particle ($Pe=100$), we find that while a particle in the absence of
the vesicle readily rectifies its motion, when we place it inside the vesicle
the response of the vesicle is crucially dependent on the vesicle size.
Vesicles with a diameter $2r_{0}$ smaller than the pore size $d_{0}$ do not
rectify their motion. However, for diameters $2r_{0}>d_{0}$, vesicles rectify
to the left (see table 1).

One could expect that placing an active particle within a small vesicle
($2r_{0}<d_{0}$) could move it away from the Knudsen regime because the
effective Péclet number of the particle-vesicle system decreases with vesicle
size according to $Pe\rightarrow Pe/(1+N)$ (see Eq. 2). However, the
rectification of larger vesicles under the same conditions is counter
intuitive. In fact, if anything, one should expect the particle-vesicle system
to further reduce its Pećlet number upon increasing the size of the vesicle
($N$). The emergence of $d_{0}$ as a relevant lengthscale for the problem is
an important result because it shows that the rectification of the vesicle is
not a simple consequence of the rectification of the individual particles, but
emerges from a complex interplay between active and elastic forces.

Table 1:  Probability that a vesicle containing a single active particle with $Pe=100$ escapes from the left side of an asymmetric channel for different ratios of vesicle size, $2r_{0}$ to pore size, $d_{0}$ $2r_{0}/d_{0}$ | 0.27 | 0.55 | 1.09 | 1.64 | 2.18
---|---|---|---|---|---
$P^{L}$ | 0.50 | 0.50 | 0.50 | 0.53 | 0.62

This can be qualitatively understood by considering that when a particle
squeezes part of a vesicle through the right opening of the cavity, it can
only keep on transporting the vesicle to the next cell if its axis of
propulsion forms an angle, $|\phi|$, with the x axis that is smaller than that
formed by the normal to the slanted side of the cavity. This constraint does
not apply when a particle is attempting to transport a vesicle through the
left opening. In this case, a particle can complete the translocation process
independently of its orientations as it faces a flat vertical wall, rather
than a slanted wall on its way back. When that happens, the particle can still
complete the translocation by dragging part of the vesicle vertically as it
slides along the flat wall for any value of $|\phi|>0$.

##  VI Conclusions

To conclude, we studied the interplay of elastic and active forces in an
active vesicle. We measured the shape fluctuations and motility of a soft two-
dimensional vesicle containing active Brownian particles in free space, in
confined cavities and inside an asymmetric periodic channel. When
unconstrained, we observe large positive curvature fluctuations which are
suppressed upon increasing the density of active particles inside the vesicle.
When confined within a spherical cavity containing a small opening, the
average escape time of the vesicle has a non-monotonic dependence on the
number of active particles, suggesting an optimal particle density for
enhanced vesicle translocation. When placed inside an asymmetric periodic
channel the motion of the vesicle rectifies as long as its diameter is larger
than the size of the cavity’s smallest constriction.

Our simulations clearly show how vesicle rectification emerges from the
complex coupling between the active and elastic forces of our model and is not
a mere mere result of active particles pushing and sliding against the
triangular asymmetric cavity Ghosh  _et al._ (2013). In fact, for low particle
packing fraction, where it is easy to induce large curvature fluctuations on
the vesicle, rectification is determined by the number of ways particles can
push the vesicle across the asymmetric cavity once the translocation process
is initiated. At moderate to large packing fractions, high curvature
fluctuations, necessary for a translocation event to occur, can only develop
as a result of the collective behavior of the particles which tend to more
likely cluster near regions with the largest curvature.

It is quite remarkable that this model, which could be considered as what is
possibly the simplest active model of a cell, can generate a sufficient level
of complexity to produce behavior, such as ratchetaxis of a living cell when
placed in asymmetric confining channels Le Maout  _et al._ (2020).

## Acknowledgements

A.C. acknowledges financial support from the National Science Foundation under
Grant No. DMR-2003444. The authors acknowledge Albert Chen for a careful
reading of the manuscript.

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##  VII Appendix

###  VII.1 Mean Squared Displacement of soft vesicle containing active
particles

Figure 7: a) Mean squared displacement of center of mass of a vesicle
containing active particles. With no active particles, the motion remains
diffusive at all times. With a finite number of active particles enclosed, the
vesicle has a ballistic regime at $t\approx\tau$ and is diffusive for
$t\gg\tau$. The solid black lines are the closed form expression from Eq. 8.
b) The long time diffusion constant (Eq. 10) is non-monotonic and attains
maximal value when the number of active particles $N_{A}$ in the vesicle
equals the number of colloids forming the flexible boundary of the vesicle
$N$. Here, $N=64$.

In this section, we evaluate the closed form expression for the mean squared
displacement of the center of mass of the soft vesicle. This calculation
follows the procedure adopted for run-and-tumble active particles within a
vesicle Paoluzzi  _et al._ (2016). The center of mass of the vesicle is
defined as,

$\bm{r}_{\rm
cm}=\frac{1}{N+N_{A}}\left(\sum_{i=1}^{N}\bm{r}_{i}+\sum_{j=1}^{N_{A}}\bm{r}_{j}\right),$
(5)

where $\bm{r}_{i},\bm{r}_{j}$ denote the positions of the particles in the
flexible boundary and the enclosed active colloids respectively. The velocity
of the center of mass $\bm{v}=d\bm{r}/dt$ can be written explicitly in terms
of the constituent particle velocities as,

$\bm{v}_{\rm
cm}=\frac{1}{N+N_{A}}\left(\sum_{i=1}^{N}\bm{v}_{i}+\sum_{j=1}^{N_{A}}\bm{v}_{j}\right).$
(6)

The dynamics of the particle velocities is encoded in the Brownian equations
of motion in Eq. 1 of the main text. The total force on the center of mass is
the sum of all forces on all particles and given the symmetry of the problem
the interaction forces between colloids add to zero Paoluzzi  _et al._ (2016).
We integrate the equations of motion of the colloidal particles to find the
position of the center of mass,

$\begin{split}\bm{r}_{\rm
cm}(t)=&\frac{1}{N+N_{A}}\;\bigg{(}\sqrt{2D}\;\sum_{i=1}^{N}\int_{0}^{t}\bm{\xi}_{i}(t^{\prime})\;dt^{\prime}\\\
+&\sqrt{2D}\;\sum_{j=1}^{N_{A}}\int_{0}^{t}\bm{\xi}_{j}(t^{\prime})\;dt^{\prime}\\\
+&v_{p}\;\sum_{j=1}^{N_{A}}\int_{0}^{t}\bm{n}_{j}(t^{\prime})\;dt^{\prime}\bigg{)}.\end{split}$
(7)

The mean squared displacement of the vesicle is,

$\begin{split}\left<\bm{r}_{\rm
cm}^{2}(t)\right>=&\frac{1}{(N+N_{A})^{2}}\times\\\
&\Bigg{(}2D\;\sum_{i=1}^{N}\int_{0}^{t}\int_{0}^{t}\left<\bm{\xi}_{i}(t^{\prime})\cdot\bm{\xi}_{i}(t^{\prime\prime})\right>dt^{\prime}dt^{\prime\prime}\\\
&+2D\;\sum_{i=j}^{N_{A}}\int_{0}^{t}\int_{0}^{t}\left<\bm{\xi}_{j}(t^{\prime})\cdot\bm{\xi}_{j}(t^{\prime\prime})\right>dt^{\prime}dt^{\prime\prime}\\\
&+v_{p}^{2}\;\sum_{j=1}^{N_{A}}\int_{0}^{t}\int_{0}^{t}\left<\bm{n}_{j}(t^{\prime})\cdot\bm{n}_{j}(t^{\prime})\right>\;dt^{\prime}dt^{\prime\prime}\Bigg{)}.\end{split}$
(8)

Given that noise on the colloidal particles is not correlated with each other
and that the system is confined to two dimensions, we have
$\left<\bm{\xi}_{i}(t^{\prime})\cdot\bm{\xi}_{i}(t^{\prime\prime})\right>=2\;\delta(t^{\prime}-t^{\prime\prime})$.
The first two sets of integrals in Eq. 8 can be done employing the Dirac delta
function. The third set of integrals involving the time-correlation of the
orientation unit vector is evaluated following Hagen et al. Hagen  _et al._
(2009). The ensemble of active particles considered in Hagen  _et al._ (2009)
has a fixed initial direction of self-propulsion, which contributes direction
specific terms to the mean squared displacement. However, since we numerically
calculate the mean squared displacement of the vesicle along a trajectory,
these terms average out to zero. Following the evaluation of the integrals in
Eq. 8, we are able to write the mean squared displacement of the center of
mass of the vesicle confined to two dimensions as,

$\left<\bm{r}_{\rm
cm}^{2}(t)\right>=\frac{4D\;t}{N+N_{A}}+\frac{2\;Pe^{2}\sigma^{2}N_{A}}{9\;(N+N_{A})^{2}}\left(e^{-D_{r}\;t}+D_{r}\;t-1\right).$
(9)

The numerical results are tested against this expression in Fig. 7(a). The
long time diffusion constant $D_{\infty}$ defined as,

$D_{\infty}=\lim_{t\to\infty}\frac{1}{4t}\left<\bm{r}_{\rm
cm}^{2}(t)\right>=\frac{D}{N+N_{A}}+\frac{Pe^{2}\sigma^{2}N_{A}D_{r}}{18(N+N_{A})^{2}},$
(10)

is non-monotonic as a function of $N_{A}$ (See Fig. 7b). The number of active
particles $N_{A}^{0}$ needed to attain the maximal $D_{\infty}$ is calculated
by setting the derivative of $D_{\infty}$ with respect to $N_{A}$ to be zero.
This condition can be simplified to,

$\frac{N_{A}^{0}}{N+N_{A}^{0}}=\frac{1}{2}-\frac{3}{Pe^{2}}.$ (11)

For $Pe\rightarrow\infty$, $N_{A}^{0}=N$.

###  VII.2 Preferential motility of a vesicle stuck between sloped walls

Figure 8: Geometries used to calculate $P^{L}(\gamma)$. We ignore the finite
pore width. In (b), we assume the length of the ring in contact with the wall
$p$ to be the same as the arc length in (a).

In this section, we estimate the probability of leftward transitions
$P^{L}(\gamma)$ (Eq. 3 of main text) for the system in the inset of Fig. 3 of
main text. We approximate the soft vesicle as two rings of equal radius $R$
whose perimeters are $2\pi R$ each. We also ignore the finite pore width and
set it to zero to allow for a simple calculation. See Fig. 8(a). The angle
$\gamma$ quantifies the slope of the slanted wall with respect to the $y-$
axis. The wall bisects the right side ring at two points. The angle bisected
by the arc between these two points is $\alpha$, and the arc length is
$p=R\alpha$. We assume that for $N_{A}\gg 1$, the ring conforms to the wall as
in Fig. 8(b), where the portion of the ring in contact with the wall is taken
to be $p$. The length of curved portion of the right ring is $q=2\pi
R-2p=2R(\pi-\alpha)$. From the geometry of the system, we have
$\alpha=\pi-2\theta$ and $\theta=\pi/2-\gamma$. In the high $Pe$ limit, all
particles are expected to be at the vesicle boundary. The number of active
particles at the vesicle boundary is proportional to the length of the
vesicle, and the force on the ring scales with the number of active particles.
The probability of leftward transition is then,

$P^{L}=\frac{2\pi R}{2\pi R+q}.$

While all the active particles on the left ring push the vesicle leftwards,
only the active particles on the free curved portion of the right ring push
the vesicle rightwards. Using the geometric relations discussed above, $P^{L}$
as a function of the angle $\gamma$ can be found to be Eq. 3 of the main text.