Wavy membranes and the growth rate of a planar …Wavy membranes and the growth rate of a planar chemical garden: Enhanced diffusion and bioenergetics Yang Dinga, Bruno Batistab, Oliver

Wavy membranes and the growth rate of a planarchemical garden: Enhanced diffusion and bioenergeticsYang Dinga, Bruno Batistab, Oliver Steinbockb, Julyan H. E. Cartwrightc,d, and Silvana S. S. Cardosoa,1

aDepartment of Chemical Engineering and Biotechnology, University of Cambridge, Cambridge CB2 3RA, United Kingdom; bDepartment of Chemistry andBiochemistry, Florida State University, Tallahassee, FL 32306-4390; cInstituto Andaluz de Ciencias de la Tierra, Consejo Superior de InvestigacionesCientíficas–Universidad de Granada, E-18100 Armilla, Granada, Spain; and dInstituto Carlos I de Física Teórica y Computacional, Universidad de Granada,E-18071 Granada, Spain

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved June 29, 2016 (received for review May 16, 2016)

To model ion transport across protocell membranes in Hadeanhydrothermal vents, we consider both theoretically and experi-mentally the planar growth of a precipitate membrane formed atthe interface between two parallel fluid streams in a 2D micro-fluidic reactor. The growth rate of the precipitate is found to beproportional to the square root of time, which is characteristic ofdiffusive transport. However, the dependence of the growth rateon the concentrations of hydroxide and metal ions is approxi-mately linear and quadratic, respectively. We show that such adifference in ionic transport dynamics arises from the enhancedtransport of metal ions across a thin gel layer present at thesurface of the precipitate. The fluctuations in transverse velocity inthis wavy porous gel layer allow an enhanced transport of thecation, so that the effective diffusivity is about one order ofmagnitude higher than that expected from molecular diffusionalone. Our theoretical predictions are in excellent agreement withour laboratory measurements of the growth of a manganesehydroxide membrane in a microfluidic channel, and this enhancedtransport is thought to have been needed to account for thebioenergetics of the first single-celled organisms.

chemical gardens | hydrothermal vents | origin of life | chemobrionics

In hydrothermal vents, precipitation membranes grow at theboundary between seawater and the mineral-rich liquid flowing out

of the vent. The same sort of semipermeable precipitation mem-brane forms in a laboratory setting in chemical gardens, as well as inother chemobrionic systems (1–3) (Fig. 1). Such membranes areincreasingly viewed as having played a vital role in the emergenceof life on Earth over 4 billion years ago (4, 5). Both acidic “blacksmoker” (Fig. 1A) and alkaline “white smoker” (Fig. 1B) ventsdisplay a complex internal structure of precipitation membranes. Itis from the geochemistry of the alkaline white smoker vents such asthose of the Lost City that, it is thought, the biochemistry of life maymost plausibly have arisen (6, 7) and much experimental work isbeing carried out along these lines (8, 9). It has been suggested thatLUCA, the Last Universal Common Ancestor, i.e., the progenitorof all present life on Earth, needed to possess a “leaky membrane”to function in bioenergetic terms in the hydrothermal vent envi-ronment (10). Here, we demonstrate that flow over a growing, wavymembrane enhances diffusive transport of solute toward and acrossits surface, owing to the tortuous paths of the fluid. Such wavymembranes would be the best candidates for the origin of life, froma point of view of exchange of chemicals between the membraneand the surrounding environment. The irregular surface would en-sure that even when the external flow is parallel to the membrane,there would be a strong transverse exchange, much faster thanmolecular diffusion. Our finding of enhanced diffusion thus helps toexplain the “leakiness” present in protocells before LUCA that theymay have inherited from chemical gardens.Reactive interfaces play in general an important role in many

chemical (11, 12), biological, and engineering processes in in-stances ranging from heterogeneous catalysis (13) to proteinactivity in biological membranes (14, 15) and to environmental

flows (16). For liquid–liquid interfaces, most chemical researchfocuses on immiscible liquids such as certain polymer melts (17,18) and emulsions because sharp interfaces between miscibleliquids are difficult to generate and swiftly decay owing to dif-fusion and other transport phenomena. This limitation is elimi-nated or at least weakened if the two liquids produce a gel-like orsolid reaction product that effectively compartmentalizes thesystem. Such resulting reaction products trace the macroscopicshape of the original solution interface but can reveal interestingcomplexity at microscopic length scales that result from the un-usually steep concentrations during their formation. Further-more, the separating membrane or wall enforces its own rules onthe evolution of the reaction systems as it controls transmembranetransport according to reactant-specific permeabilities. Prime ex-amples of such phenomena are a class termed chemobrionic sys-tems, which include the so-called chemical gardens (1).Classical chemical gardens (19–21) consist of hollow precipi-

tate tubes that form when a metal salt seed is placed into aque-ous solutions of silicate, carbonate, phosphate, borate, and someother anions. The thin, hollow tubes typically consist of an outerlayer of silica (in silicate systems) and an inner layer rich in metalhydroxides or oxides. Numerous chemical modifications havebeen reported including tube-forming reactions involving poly-oxometalates (22) and organic reactants (23). The seed particlemay be replaced by injected salt solutions (24). Furthermore, onecan reduce the dimensionality of the system by the injection of saltsolutions from a point-like source into a thin, horizontal layer ofsilicate solution contained within a Hele–Shaw cell (25–27). Thisapproach was taken further by studying the formation of pre-cipitate membranes in a microfluidic device at the planar interfacebetween two parallel, cocurrent reactive fluid streams (28). The

Significance

In hydrothermal vents on the ocean floor, precipitation mem-branes grow at the boundary between seawater and mineral-rich liquid flowing out of the vent. Such membranes areincreasingly viewed as having played a vital role in the emer-gence of life on Earth, but their bioenergetics is unclear. Here,we present a laboratory and theoretical study that quantifiesionic transport across an analog membrane. We demonstratethat flow over a growing, wavy-membrane topography en-hances diffusive transport across its surface. This enhanceddiffusion helps to explain the “leakiness” present in early pro-tocells from chemical gardens. More generally, the work is ofinterest in fluid-flow control via surface topography, and the op-posite: predesigned flow perturbations to shape membrane for-mation, in biology, chemistry, and physics.

Author contributions: O.S., J.H.E.C., and S.S.S.C. designed research; Y.D., B.B., O.S., J.H.E.C.,and S.S.S.C. performed research; and O.S., J.H.E.C., and S.S.S.C. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. Email: [email protected].

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product wall in these experiments is analogous to the tube wall in3D chemical gardens but provides excellent control over the ex-perimental parameters and also allows a direct view of the wallcross-section. As found in an earlier study that measured the dryweight of tubes formed over different reaction times (29), the wallthickness increases as the square root of time, thus suggesting adiffusion-controlled mechanism. In addition, the wall thickeningoccurs strictly in the direction of slightly acidic, metal salt solu-tion and not toward the alkaline silicate solution. Whereas thisobservation was originally made for silicate systems, the resultsfrom microfluidic growth experiments confirmed this finding forprecipitation reactions between sodium hydroxide and variousmetal salt solutions including magnesium, manganese, iron, co-balt, and copper (28). Here we present experimental measure-ments on the growth of precipitate membranes in a microfluidicsystem together with an analytical model that captures key as-pects of the growth dynamics. Our results show that the smallvariations in the membrane thickness and the presence of aporous gel layer create a fluctuating component of the flow ve-locity which, compared with pure diffusion, yields a faster growthof the precipitate.

Experimental MethodsOur experiments use a Y-shaped microfluidic device that has been describedin detail in ref. 28 (Fig. 2A). It is constructed from pairs of metacrylate plates(5 × 4 cm2) sandwiching a precut parafilm membrane (approximate thick-ness, 130 μm). The cutouts are prepared by a computer-controlled cuttingtool (Silhouette Portrait) and define two inflow channels that combine toform the reaction channel. The latter is 2 cm long and 3 mm wide, yielding atotal volume of about 8 μL. The reactant solutions, NaOH and MnCl2 withconcentrations in the range 0–0.5 M, are delivered by a syringe pump (KDScientific 200) through tubing and barb fittings glued to holes on the topplate. Before the experiment, the device channels are pumped full of water.Once the reactant solutions reach the device, the pump rate is set to a constantvalue of 2 mL/h per syringe. We monitor the formation of the precipitate usinga charge-coupled device camera (COHU 2122) and subsequently analyze thevideo frames using in-house Mathematica scripts. All experiments are carriedout at room temperature.

Experimental ResultsThe parallel flow of the two solutions along the reaction channelinduces the formation of a nearly linear precipitate membrane(Fig. 2A). The membrane width wðtÞ is easily monitored andincreases only in the direction of the Mn2+ solution, while themembrane interface with NaOH solution remains stationary(Fig. 2B). Fig. 2C graphs the square of this width as a function oftime for four different concentrations of MnCl2. The data arewell described by the relation w2 =Det, where De is an effectivediffusion coefficient. Deviations from this simple dependenceare observed only for high reactant concentrations and long re-action times but, even in the case of [MnCl2] = 0.5 M, the square-root dependence holds well for the first 2 h of the experiment.The surface of the membrane facing the Mn2+ solution shows

a thin layer that is distinctly different from the opaque main partof the precipitate membrane. Based on our optical micrographs,this layer is translucent and appears somewhat granular (Fig.3A). We interpret it as having a gel-like structure with large poresthat allow for some fluid flow. The image sequence in Fig. 3Ashows that the layer is present throughout all stages of the growthprocess. No comparable feature is observed on the hydroxide sideof the precipitate membrane. The image sequence also reveals thatthe manganese side of the earlier membrane is not perfectly planar.Furthermore, these small variations in the membrane width decayover time, ultimately yielding very straight boundaries between thesolid precipitate, its gel-like zone, and the manganese solution.Quantitative details of these two features are illustrated in Fig. 3

B andC. The data in Fig. 3B show the width of the gel-like layer as afunction of the elapsed reaction time revealing an essentially con-stant value of ∼ 40± 5 μm for t> 30 min. During the first 30 min,the width increases smoothly suggesting the decay of slow transients.

A B C

500 m 100 m10 cm

Fig. 1. Wavy membranes in (A and B) hydrothermal vents and (C) achemical garden. (A) Longitudinal section of the Olivia chimney from theEast Pacific Rise. Metals such as copper, zinc, and iron dissolved in the ex-pelled hot fluids precipitate to form the metallic center of the chimney ofthis acidic black smoker (2) [Image courtesy of C. E. J. de Ronde, GNS Science(Lower Hutt, New Zealand).] (B) A section through an alkaline white smokerhydrothermal vent wall of the Lost City (3), showing its microporous structure.(Reproduced from ref. 3, with permission from AAAS, science.sciencemag.org/content/307/5714/1428.) (C) Scanning electron micrograph of the wavy, in-terior surface of a silica-zinc-(hydr)oxide tube formed by injection of a zincsulfate solution into a host solution of sodium silicate.

w(t)Mn2+OH-

A

C D

B

Fig. 2. (A) Y-shaped microfluidic device used for the production of a pre-cipitate membrane (brown line) at the interface of NaOH and MnCl2 solu-tions. (B) Optical micrograph of the membrane (black region). The membraneaverage thickness is wðtÞ, where t denotes the elapsed reaction time. (C)Evolution of the squared membrane width for four different concentrationsof Mn2+ (0.1, 0.2, 0.25, and 0.5 M) and [OH−] = 0.5 M. (D) Evolution of thesquared membrane width for four different concentrations of OH− (0.1, 0.3,0.4, and 0.5 M) and [Mn2+] = 0.5 M. The straight lines in C and D are linearregressions that yield concentration-dependent slopes De.

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The double-logarithmic graph in Fig. 3C shows the evolutionof the amplitude of the width variations as detected from im-ages including those in Fig. 3A. We approximate the data set fort> 30 min by a power law δðμmÞ= 115=

ffiffiffiffiffiffiffiffiffiffiffiffiffitðminÞp

, represented asthe continuous line in Fig. 3C; this approximation lies within theerror bars of experimental uncertainty and allows us to obtain ananalytical solution in the modeling below.

Model and Comparison with ExperimentsNow consider the parallel flow of two aqueous solutions, onecontaining anion An− at concentration cA0 and the other cationBm+ at concentration cB0. The reaction

jνAjAðaqÞ+ jνBjBðaqÞ→ jνCjCðsÞ [1]

occurs at the interface between the two streams to form aprecipitate layer of product C, whose thickness wðtÞ grows withtime. A gel layer of dissolved precipitate CðaqÞ develops at thesolid precipitate surface (Fig. 4), growing rapidly to a steady stateof thickness l � wðtÞ. A balance of the two chemical speciesacross the precipitate and gel layers requires that

∂cA∂t

=DA∂2cA∂x2

, [2a]

∂cB∂t

=DB∂2cB∂x2

. [2b]

Here, cA and cB are the concentration of A in the pore fluid in theprecipitate and the concentration of B in the gel, respectively; DA

is the diffusivity of A in the porous precipitate and DB is thediffusivity of B in the gel layer. The transverse spatial position isdenoted by x. We assume the precipitate is impermeable to themetal ion and that the reaction is almost instantaneous, so thatreaction and precipitation occur only at the plane of contact oftwo ions. The initial conditions at t= 0 are that cA = cA0 for x< 0and cB = cB0 for x> 0. At the boundaries of the precipitate and gellayers, x= 0 and x=wðtÞ+ l, these concentrations remain constantwith time owing to the relatively fast flow in the injected streams,so that cA = cA0 and cB = cB0, respectively. The boundary condi-tions at the reaction plane x=wðtÞ express conservation of totalmass and the stoichiometric constraint

ρsSdwdt

=dmA

dt+dmB

dt= S

�−DAMA

dcAdx

+DBMBdcBdx

�, [3a]

−jνBjDAdcAdx

= jνAjDBdcBdx

. [3b]

Here, S is the surface area of the reaction plane, ρs is the bulkdensity of the solid precipitate, m denotes mass in the precipi-tate, andM the molar mass. Owing to the instantaneous reaction,we also impose that at least one of the concentrations at theprecipitation plane, cAs or cBs, is zero. This condition then sepa-rates two regimes, one in which transport is limited by the supplyof A (cAs = 0) and the other in which the growth is limited by thetransport of B (cBs = 0) to the reaction site.For sufficiently long times, such that the gel layer has

attained constant thickness and the concentration profile of Btherein is linear, the integration of Eq. 2 with the initial andboundary conditions leads to the following analytical relationsfor the effective diffusivity De of A and B in the system and theconcentration of the ion in stoichiometric excess at the re-action plane:ffiffiffiffiffiffiffiffiffiDe

4DA

r· exp

�De

4DA

�· erf

ffiffiffiffiffiffiffiffiffiDe

4DA

r=�MA +MB

jνBjjνAj

�cA0 − cAsffiffiffi

πp

ρs, [4a]

De =�DB

ffiffit

p

l

�MA

jνAjjνBj+MB

�2ðcB0 − cBsÞ

ρs

�2. [4b]

In Fig. 5 we show that almost all of the measurements for Delie, within experimental error, on the lines predicted by Eq. 4afor DA = 9.8× 10−10 m2/s and DB(m2/s) = 5.8× 10−7=

ffiffiffiffiffiffiffitðsÞp

. Thediffusivity of A in the precipitate was estimated from its moleculardiffusivity in solution DA∞= 5.3× 10−9 m2/s, the measured po-rosity of the precipitate ϕ= 0.52, and its tortuosity τ= 2.8, asDA =ϕDA∞=τ; the relatively high value for the tortuosity is

A

B

C

Fig. 3. (A) Micrograph sequence of the precipitate surface facing the Mn2+

solution and the accompanying gel-like layer. Field of view: 480×180 μm2.Time between frames: 25 min. (B) Width of the gel-like layer as a function ofthe elapsed reaction time. (C) Double-logarithmic plot of the amplitude ofthe width variations of the gel-like layer as a function of the elapsed re-action time. The linear regression line δðμmÞ= 115=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitðminÞp

is shown.

Fig. 4. Sketch of the concentration profiles of A and B in the precipitateand gel layers.

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consistent with a precipitate composed of channels at an angleof ∼ 20° to the membrane surface (30). For the transport of Bin the gel layer, we need to consider hydrodynamic effects. Theexperiments show that the precipitate has a rough surface withprotrusions into the gel. The amplitude of the protrusions δdecays with time, from ∼20 to 1 μm (Fig. 3), following the ap-proximate dependence δðμmÞ= 115=

ffiffiffiffiffiffiffiffiffiffiffiffiffitðminÞp

. The stream of Bis injected with velocity v∼ 0.0015 m/s and drags the fluid in thegel layer over the protrusions. Evidence of this motion is seenby tracking the paths of a few bubbles in the gel layer and in theouter stream (Fig. 6). The bubbles move with longitudinalspeeds of ∼8.5 and 19 μm/s in the gel and injected stream,respectively. The bubble speed in the stream is much slower thanthe average fluid speed because, owing to buoyancy, they lie close tothe upper surface of the top plate of the microchannel. Importantly,our observations show that the bubble paths are tortuous in the gellayer, indicating that the fluid there is moving to and fro themembrane surface (Fig. 6B). The bubble transverse speed inthe irregular part of its path is ∼2.4 μm/s; this suggests that thefluid is moving to and fro the membrane with a horizontalspeed u∼ 2.4× 10−4 m/s. The forced flow over the wavy to-pography and within the porous structure of the gel is re-sponsible for this transverse mixing. We therefore expecttransverse dispersive transport of B on the order of DB ∼ avδ(31), where the constant a should be smaller than 1. Com-parison between the theory and the experiments suggests thata∼ 0.06, so that DB(m2/s) = 5.8× 10−7=

ffiffiffiffiffiffiffitðsÞp

. Thus, at a time of200 min, DB ∼ 5.3× 10−9 m2/s, i.e., the dispersive transport is thenapproximately an order of magnitude larger than the moleculardiffusivity of the manganese cation in water, DB∞ = 7.0× 10−10 m2/s.

This prediction is in agreement with all of the experimentalobservations in Fig. 5.We can identify two regimes of behavior in Eq. 4a, according

to the limiting driving force for the precipitate growth, either thetransport of A or B to the reaction plane. When transport of B islimiting, De grows quadratically with driving concentration dif-ference across the gel layer; this behavior is a consequence of theflow of fluid in the porous gel layer over the wavy-membranetopography, which induces the transverse localized mixing. Incontrast, when the transport of A determines the precipitation,the diffusive transport grows linearly with the concentrationdifference across the precipitate layer. The growth of the pre-cipitate is maximized for the stoichiometric concentration ratiocA0=cB0 ∼ νA=νB.Our experimental method combined with theoretical analyses

will also allow interesting studies of the concentration thresholdfor continuous membrane formation. An earlier study (32)reported this value as 0.1 mol/L for the case of iron sulfide tubes.For lower reactant concentrations, only rising plumes of indi-vidual colloidal particles were observed. This threshold phe-nomenon should also exist in our experimental setting which isideally suited for systematic experiments due to the ease of ob-servation and the system’s simple geometry. Other targets forfuture studies include the investigation of shear rates on the wallgrowth and the associated waviness. There are also potentialtechnological applications associated with enhanced diffusionacross wavy membranes. One can envisage both fluid flow con-trol via surface topography, and the opposite: Predesigned flowperturbations could be used to shape the membrane. We anticipatethat the work presented here will provide valuable starting pointsfor such efforts.More broadly, we have shown that a wavy membrane and

neighboring gel layer is the essential factor driving enhanceddiffusion. Both laboratory chemical gardens and hydrothermalvents in the oceans are examples of chemobrionic systems thatpossess such membranes. In the case of hydrothermal vents this

[OH - ] (M)

0 0.1 0.2 0.3 0.4 0.5 0.6

De

(10-1

1m

2/s

)

0.5

1

1.5

2

2.5

3

3.5TheoryExperiment

[Mn 2+ ] (M)

0 0.1 0.2 0.3 0.4 0.5 0.6

De

(10-1

1m

2/s

)

0.5

1

1.5

2

2.5

3

3.5TheoryExperiment

A

B

Fig. 5. Variation of the effective diffusivity with the concentrations of Aand B. Experimental data are shown with the prediction of Eq. 4a forDA = 9.8× 10−10 m2/s and DB(m

2/s) = 5.8× 10−7=ffiffiffiffiffiffiffiffitðsÞp

. Regimes of growthcontrolled by the transport of either the anion (blue) or the cation (red) tothe reaction surface are shown.

A

B

Fig. 6. Visualization of fluid flow near the precipitate surface. (A) Examplesof microbubble pathlines in the gel layer (red) and in the manganese stream(blue) during a period of 10 s. (B) Bubble motion in the gel layer relative tothe precipitate surface. The bubble trajectory was measured after ∼3 h ofprecipitate growth for [MnCl2] = [NaOH] = 0.5 M.

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attribute takes on a great significance because the diffusion ofmaterial across such a membrane must have been fundamentalto the origin of life. Indeed, there is an apparent paradox at theheart of the earliest protobiochemistry. A membrane had both toprotect the complex chemistry from its environment yet at thesame time to allow the passage of materials; to be at once per-meable to some ions and so allow mixing, but at the same time tomaintain steep pH gradients (7, 33, 34). Waviness provides aplausible mechanism to achieve enhanced transport across rel-atively thick abiotic hydrothermal vent membranes at the verybeginnings of the climb in complexity toward life. Prebioticchemistry needed to sustain disequilibrium with its surroundingsto possess pH and reduction potential by maintaining protonand redox gradients (34). The evolution of ion-tight biologicalmembranes was contingent on the prior invention of active ionpumping, or chemiosmosis (7, 35, 36). Long before the emer-gence of chemiosmosis with its active mechanism of ion trans-port, waviness constitutes a natural passive mechanism for theenhancement of transport across a semipermeable membranethat in our view forms part of the answer to the puzzle of the

origin of membrane bioenergetics (33) and thereby the passagefrom passive osmosis in a hydrothermal vent vesicle to activechemiosmosis in a protocell. There is a vast distance between achemical garden membrane and LUCA, but the fact that LUCAstill needed a leaky membrane even at that much later stage inthat climb in complexity (7) demonstrates what a vital aspect thision transport mechanism constituted for protolife. “Prebioticchemistry has been approached from the intellectual tradition ofsynthetic chemistry, and the apotheosis is the ‘one-pot synthesis.’But cells are not simply a pot of chemicals; they have a structure inspace” (7); waviness is how hydrothermal vent pores––microfluidicreactor vessels that are marvels of natural chemical engineering––optimize their fluid mechanics (37).

ACKNOWLEDGMENTS. S.S.S.C. acknowledges the financial support of the UKLeverhulme Trust Project RPG-2015-002. J.H.E.C. acknowledges the financialsupport of the Spanish Ministerio de Ciencia e Innovación (Project FIS2013-48444-C2-2-P). O.S. acknowledges support by the US National Science Founda-tion under Grant 1609495. B.B. thanks the Brazilian National Council forScientific and Technological Development for a Postdoctoral Fellowship.

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