Proton gradients and pH oscillations emerge from heat flow at the microscale

Proton gradients are essential for biological systems. They not only drive the synthesis of ATP, but initiate molecule degradation and recycling inside lysosomes. However, the high mobility and permeability of protons through membranes make pH gradients very hard to sustain in vitro. Here we report that heat flow across a water-filled chamber forms and sustains stable pH gradients. Charged molecules accumulate by convection and thermophoresis better than uncharged species. In a dissociation reaction, this imbalances the reaction equilibrium and creates a difference in pH. In solutions of amino acids, phosphate, or nucleotides, we achieve pH differences of up to 2 pH units. The same mechanism cycles biomolecules by convection in the created proton gradient. This implements a feedback between biomolecules and a cyclic variation of the pH. The finding provides a mechanism to create a self-sustained proton gradient to drive biochemical reactions.

T he establishment of ion gradients holds an important place for the sustainment of all current life forms. Regions of different pH are essential for various cellular processes such as the synthesis of DNA, RNA, and proteins as well as their degradation and recycling [1][2][3][4] . These processes are accommodated in specialized compartments separated by biological membranes and important in lysosomes, mitochondria, and cell vacuoles. To form complex reaction networks, the variation of pH is essential to create different reaction conditions 5,6 . In the process of chemiosmosis, for example, the channeled movement of ions across a membrane drives the phosphorylation of ADP to ATP, life's most commonly used energy currency 7 . Since ancient microbes, such as the last universal common ancestor (LUCA), are assumed to be chemiosmotic 8 , it has been argued toward the significance of proton gradients at the origin of life [9][10][11] . In addition, the synthesis of precursor molecules for early molecular evolution requires differing pH values for the synthesis of purine and pyrimidine ribonucleotides 12 , aminonitriles 13 , amino acids 14 , and phosphoenol pyruvate 15 .
For living systems, the creation of pH differences is tedious and requires complex protein-mediated transport mechanisms. On prebiotic earth, natural pH gradients would form by laminar mixing of fluids with different pH values, e.g., at the interface between disequilibria at submarine hydrothermal vents and the Hadean ocean [16][17][18] . However, these systems rely on a continuous influx of mass and energy. Molecules would experience this gradient only for a short time and are washed out into a pH-equilibrated reservoir, keeping the mechanism transient. A repetitive pH oscillation inside a closed system would be desirable and keep molecules at one location without being lost by a flow or by diffusion.
Here we found a mechanism to form stable pH gradients and continuous cycling of pH in a closed system. Thermal energy, the form of energy necessary for the mechanism, triggers an accumulation mechanism for individual ionic species. The heat flow forms a temperature gradient, spanning across sub-millimetersized water-filled compartments. This is likely a common setting on early earth, found for example in geothermally heated porous rocks such as hydrothermal vents or cooling volcanic sites 19,20 . This elementary setting has previously been shown to concentrate dilute nucleotides 21 , accumulate lipids to facilitate the formation of vesicles 22 , shift polymerization reactions toward longer DNA/ RNA strands 23 , and selectively replicate longer polymers 24 . This study now extends these characteristics to include the formation of a stable pH gradient by thermally separating dissolved buffer molecules of different charge states. This locally shifts the buffer equilibrium, yielding pH differences of up to two units and provides another aspect for a promising long-term microhabitat for the onset of molecular evolution.
In the following experiments, we demonstrate the formation of proton gradients in various buffer solutions. These include prebiotically plausible solutions of phosphate buffers, amino acids, and nucleotides [25][26][27][28][29] . In a temperature gradient, the thermophoretic movement of each ionic species is different. As a result, the concentration of ionic species is locally shifted, prompting the formation of proton gradients (Fig. 1). As indicated by the modeling, the formation of proton gradients is not affected by smaller concentrations of large biomolecules or vesicles. Such entities are shuttled inside the already established pH gradient by laminar convection, prompting them to move regularly between regions of varying pH. Thereby, each molecule undergoes individual pH oscillations with respect to their molecular properties, mostly their diffusion coefficient and Soret coefficient. The pH oscillations enable chemical reaction pathways that rely on pH fluctuations. The comparably fast shuttling of vesicles could trigger proton gradients across protocellular membranes without the need for active proton pumps.

Results
Thermally driven accumulation. The formation of pH gradients is facilitated by a thermally driven accumulation mechanism inside a water-filled compartment. These flow chambers are subjected to a spatially confined horizontal temperature gradient. Based on the second law of thermodynamics, energy fluxes are necessary to maintain states of low entropies, in this case accumulated regions of molecules. The accumulation mechanism relies on the superposition of two effects: (i) thermal convection drives molecules up-and downwards in the flow chamber and (ii) thermophoresis-the movement of molecules in a temperature gradient (Fig. 2a). In combination, the two effects result in an efficient net transport of molecules, as small as single ions, to the bottom of the flow chamber and a simultaneous depletion at the top. The flux j i for an ionic species i inside the flow chamber is composed of diffusion, thermophoresis, and convection. It can be described by Where v denotes the convection flow, c i , D i , and S Ti the concentration of the ionic species i, diffusion coefficient, and Soret coefficient, respectively 23 . The Soret coefficient S Ti = D Ti D i −1 is defined as the quotient of the thermal and collective diffusion coefficient. Using Debye's approach, an analytical solution for the accumulation mechanism in a flow chamber at equilibrium is given by 30,31 where c/c 0 denotes the relative concentration of a species at the bottom of the chamber, c 0 the initial concentration, the temperature difference ΔT, the aspect ratio of the chamber r = h w −1 with the height h and width w, and the experimental prefactor κ i ranging between 0 and 0.42 (see Eqs. 10 and 11). The experimental prefactor κ i is calculated for each ionic species and depends on experimental parameters such as the temperature difference ΔT, the chamber width w, and the diffusion coefficient of the species D i .  Fig. 1 Formation of pH gradients by chemical or thermal energy. a Modern cells run on an elaborate protein system to maintain proton gradients across a membrane. Chemical energy is used to pump protons against their concentration gradient which is then harnessed by chemiosmosis. b Heat fluxes across confined solutions, a common geological setting, induce a movement of ions that results in a stable pH gradient of up to 2 pH units. The charge selective thermophoretic accumulation of buffer molecules such as amino acids, phosphates, or nucleotides keeps the water selfionization persistently out of equilibrium. The formation of a pH gradient is achieved in a closed system and biomolecules are repetitively exposed to differences in pH For a given buffer reaction in solution, each buffer subspecies accumulates differently. Its respective accumulation efficiency, as shown in Eq. (2), strongly depends on molecular properties which define its diffusion and Soret coefficients. As a result, different ratios in concentrations of proton acceptor and donor species are present along the height of the flow chamber, which then form a pH gradient (Fig. 2b, c). Chamber widths above 150 µm for Earth's gravity field are optimal for the accumulation of fast diffusing, small molecules to achieve a maximum pH gradient. These pores exhibit high convection velocities, which are necessary to counterbalance the fast diffusion of the molecules. As a result, the accumulation mechanism concentrates small molecules at the bottom of the pore quickly. Longer cannot compensate for the convection on the considered experimental and simulation timescales. Until they reach their steady state of thermophoretic accumulation after days or even weeks, they shuttle through the thermal habitat and the zones of different pH. The Soret coefficient of the phosphate buffer species are estimated with respect to their net charge (see Supplementary Table 2) 32 .
Phosphate-mediated formation of stable pH gradients. At a pH of 7, phosphate buffer consists mostly of two ionic species, hydrogen phosphate (HPO 4 2− ) and dihydrogen phosphate (H 2 PO 4 − ). Both ionic species obtain different vertical accumulation profiles along the flow chamber, as the dominant contribution to the thermophoretic force approximately scales with the charge of the molecules squared (Fig. 3a). Based on their net charge, HPO 4 2− accumulates tenfold stronger compared to H 2 PO 4 − , locally shifting the acid-base equilibrium in the buffer solution. The acid-base reaction in general and for the phosphate buffer solution in particular is defined by where HX and H 2 PO 4 − denote the proton donors and X − and HPO 4 2− the proton acceptors. High concentrations of HPO 4 2− at the bottom of the chamber drive a protonation reaction that locally reduces the hydronium ion concentration (Fig. 3b). Simultaneously, a deprotonation reaction of highly concentrated H 2 PO 4 − releases hydronium ions at the top of the chamber. The acid-base reaction locally shifts the pH along the chamber. The pH of a solution is derived by the Henderson-Hasselbalch where c acceptor and c donor denote local proton acceptor and proton donor concentrations, respectively. Given sufficiently high buffer concentrations, the pH is only affected by the ratio of proton acceptors to donors, and not by their total concentrations. The pH at the bottom of the chamber pH bottom is therefore derived by calculating the corresponding proton acceptor/donor concentration at the bottom, given by Eq.
with Δ κ i S T i ð Þ¼κ acceptor S T acceptor À κ donor S T donor and r ¼ hw À1 the aspect ratio of the chamber. By assuming κ acceptor ≈κ donor and 1/ln(10) ≈ 0.43, pH bottom is given by The maximum pH difference ΔpH inside the flow chamber, arising between the lowest and highest regions of the chamber, is given by with ΔS T = S T acceptor −S T donor . The steepness of the pH gradient is affected by the temperature difference ΔT, Soret coefficient S T , and height h and width w of the chamber ( Fig. 3c and Supplementary Fig. 4). An elongated flow chamber can compensate for lower temperature differences, achieving similar or higher pH gradients (Fig. 3d). The theoretical model is applicable for shallow proton gradients and is in good agreement with experimental data such as the formation of pH gradients in solutions of phosphate. Here the model predicts a proton gradient of ΔpH = 0.40, 0.90, and 1.2, which describes comparably well the experimental values of ΔpH = 0.34, 0.69, and 0.93 for a given temperature difference of ΔT = 8, 12, and 14 K, respectively (Fig. 3c). The pH model (Eq. 8), however, neglects the build-up of an electrical field that leads to an electrophoretic movement of ions, inhibiting the formation of a pH gradient. As a result, the  . The primary requirement is an acid-base reaction and therefore buffer solutions of HEPES, TRIS, single nucleotides formed stable pH gradients (Fig. 4a). The fluorescent pH indicator SNARF-1 could also act as a buffer, however, due to the low concentrations of 50 µM, only shallow pH gradients emerged as shown in the control measurement. Solutions of TRIS and phosphate achieve the highest pH gradients of ΔpH = 0.47 and 0.79 at concentrations of 200 mM after 60 min. Amino acids have the capability to react with both bases and acids. Prebiotically plausible amino acids, such as alanine, arginine, glycine, proline, and serine, at concentrations of 200 mM formed proton gradients ranging between ΔpH = 0.1 and 1.1 within 60 min (Fig. 4b). Finite-element simulations describe the formation of the pH gradient in detail by assuming a difference in Soret coefficient ΔS T of 0.022, 0.019, 0.016, 0.012, and 0.003 K −1 for the acceptor and donor species of alanine, glycine, arginine, serine, and proline, respectively. These Soret coefficients are in the range for unevenly charged ionic species 32 . Our models indicate that the thermophoretic properties of each buffer species are the major source for differences in the proton gradients. However, the thermophoretic properties of small molecules are rather tedious to measure and not all available, making it difficult to give a detailed prediction for individual buffers 33 . Mixtures of various buffer or amino acids are expected to form an averaged pH gradient, derived from Eq. 2 and seen on the example for phosphate/arginine mixtures (see Supplementary Fig. 8). This pH gradient, however, is shifted by the exact ionic composition of the solution (see Eqs. [12][13][14]. Interestingly, short chains of amino acid monomers, shown for tetra-glycine and penta-alanine, form an inverted proton gradient of ΔpH = −0.33 and −0.06,  (Fig. 4c). Here the accumulation of proton donors is much stronger compared to proton acceptors, resulting in a deprotonation reaction of proton donors that releases hydronium ions at the bottom of the chamber. The inversion of the pH gradient is explained by a negative difference in the Soret coefficient ΔS T = S T acceptor −S T donor , which is the result of additional non-ionic contributions to thermophoresis. The non-ionic contribution S T NI of the Soret coefficient S T (see Eq. 12) scales linearly with the polymer length, as a result of local, moleculesolvent interaction around the tube-like polymer 32,34,35 . In the case of longer peptides, the contribution of the Soret coefficient which depends on the protonated state S T CM becomes comparatively smaller. As a result, the difference in Soret coefficient ΔS T between the proton acceptor and donor decreases or is even inverted for longer peptides.
Convectively driven pH oscillations. Molecules and particles accumulate and undergo laminar convection. The molecular motion inside the chamber is described by convection, Brownian motion, and thermophoresis (Fig. 5a, see Supplementary Movie 2). As a result, molecules shuttle in the arising pH gradient and are subjected to subsequent pH oscillations. These pH oscillations strongly depend on the molecular properties. Larger macromolecules are more prone to shuttle in the convective flow and exhibit faster pH cycle times compared to smaller molecules, which tend to solely accumulate at the bottom of the chamber (Fig. 5b). For example, 10-mer strands of RNA complete pH oscillations of at least ΔpH = 0.5, 1.0, and 1.5 in 11, 38, and 73 min, respectively (Fig. 5c). Longer DNA/RNA strands such as 200 mers undergo faster pH oscillations of at least ΔpH = 0.5, 1.0, and 1.5 in 7, 14, and 26 min, respectively.
The accumulation mechanism not only defines the magnitude of pH oscillations for a species, but also affects the probability for a species to remain in a specific pH region (Fig. 5d). The flow chamber efficiently accumulates DNA/RNA strands in the region of 3-10 nt, while smaller and larger molecules accumulate significantly weaker within the considered timescales. A 10-mer oligonucleotide inside an already formed pH gradient ranging from 7.7 to 5.6 has probabilities to stay in pH regions pH > 7.2 and pH > 6.7 of 0.81 and 0.98, respectively. This probability decreases for shorter and longer oligonucleotides. The accumulation kinetics hereby strongly depends on the flow chamber's width; smaller chambers achieve larger accumulation for longer molecules (see Supplementary Fig. 13). The inversion of the pH gradients, facilitated by longer peptides, allows for a microscopic chemical feedback from polymerization back to the pH gradient in these chambers. Now, long molecules that are concentrated at the bottom of the chamber remain at low pH values for prolonged periods if the pH gradient is inverted.

Discussion
Our experimental and theoretical findings show that heat fluxes across confined solutions form and sustain stable pH gradients. The mechanism solely requires a dissolved acid-base reaction buffer that is given by prebiotically plausible biomolecules such as phosphate, amino acids, or RNA. Phosphate for example is assumed to play an important role in prebiotic chemistry because it functions as a nucleophilic catalyst, pH buffer, and chemical buffer 29 . The formation of pH gradients is hereby achieved in a closed system that allows molecules to undergo repetitive pH oscillations without being washed away into an equilibrated reservoir.
The dissipation of free thermal energy, the sole form of energy necessary, is a geologically abundant scenario on early Earth. Although the formation of pH gradients depends on the thermal gradient that drives the accumulation mechanism, shallow thermal gradients can be compensated by longer flow chambers 36 . The geometry of the flow chamber is realized to mimic the simplest geometrical setting with dimensions of 8.0 × 0.17 mm. However, the shape of the flow chamber is found to have a minor impact on the accumulation mechanism 21 .
The thermally driven formation of pH gradients can be incorporated into a microfluidic device that forms tunable pH gradients and subjects molecules to pH oscillations defined in range and frequency. This device potentially allows for size selection and certain replication mechanisms to be functional 24 .
Interestingly, the existing pH gradient can simultaneously be utilized by actively driven biomolecules, not only based on Brownian motion. The accumulation mechanism thereby subjects molecules to convectively driven pH cycles, resulting in a b c d  permanent pH oscillation. Importantly, the oscillatory behavior applies for each individual species of the solution and strongly depends on its molecular properties. Hereby, specific molecular species harness the pH gradient by undergoing extensive pH oscillations, while others strongly accumulate by remaining in a limited pH region. This enables the formation of biomolecules that rely on fluctuations in pH during their assembly. The yield of pH-dependent multicomponent reaction systems, such as a threecomponent purine/pyrimidine reaction system, could be shifted along the height of the chamber. Here, purine ribonucleotides precursors would be synthesized at the top of the chamber at pH 5 while pyrimidine precursors would be synthesized at the bottom at pH 7 12 . An exciting prospect of this work is the periodical formation of proton gradients across vesicles. The leakiness of the first membranes 37 , made of fatty acids or phosphorylated isoprenoids 38,39 , ensures a short-term pH equilibration with its surrounding. Based on the convective flow, vesicles are transported into different pH regions while preserving the equilibrated pH value at the inside, forming pH gradients across the membrane. These pH gradients could be harnessed by primordial forms of the ATP synthase to drive an energy storage system 11,40 .
Our results conclusively show in both theory and experiment how heat flows form stable pH gradients. Proton gradients hold an important place for the sustainment of all current life forms and have also been argued to play a central role in powering the origin of life 9,10 . The simplicity of this mechanism opens various possibilities for rebuilding biological systems bottom up-to either understand them from a mechanistic perspective or to elucidate how cellular life might have originated.
The mechanism that forms pH differences here also drives the accumulation of dilute biomolecules 21,24 , overcoming the concentration problem of the origin of life. The combination of such an accumulation mechanism that simultaneously subjects biomolecules to pH oscillations prompts interesting future experiments. In the origin of life, the abundance of geological environments that can form stable pH gradients in a closed system could have opened vast chemical possibilities for the formation and sustainment of the first replication cycles of prebiotic biomolecules.

Methods
Microfluidic flow chamber. The temperature gradient is applied across a 3Dprinted flow chamber, in between a silica wafer and a transparent sapphire window (Sappro, Germany, see Supplementary Fig. 1). Hereby, a water bath (CF41 Kryo-Kompakt-Thermostat, Julabo, Germany) cools down the silica wafer while simultaneously heating up the sapphire with temperature PID-controlled heater cartridges (12 V, 6/20 mm, RepRap, France, see Supplementary Fig. 2). These heater cartridges are inserted into copper blocks that are thermally connected to the sapphire window. The silica wafer (Si-Mat-Silicon Materials e.K., Germany) has a diameter of 100 mm, a thickness of 525 ± 25 µm, a 100 nm SiO 2 coating, and a < 100 > orientation. The boundaries of the flow chamber are fabricated by a 3D printer using polylactic acid (PLA) material (Filamentworld, Germany). The flow chamber is positioned using a translation stage (Thorlabs, Germany). For in-and outflow of the chamber, capillaries with a height and width of 50 × 500 µm (vitrotubes 5003, CMScientific, UK) connected to a BTP E-60 tubing (0.76 × 1.22 mm, Instech Laboratories, Inc., USA) are used. A high precision syringe pump (neMESYS, Cetoni, Germany) and 250 µL microsyringes (Hamilton, USA) ensure an accurate loading of the flow chamber at flow rates ranging from 1000 to 1 nL s −1 .
Finite-element simulations. The finite-element simulations consist of three consecutive calculations of a two-dimensional flow chamber. The flow chamber is mimicked by using a rectangularly shaped compartment with a horizontal aligned temperature gradient.
The temperature profile of the flow chamber is calculated using partial differential equations for transient heat transfer. Thereby, a temperature of T max and T min is applied to the left and right vertical boundaries, respectively.
The convective flow profile inside the flow chamber is derived by numerically solving the incompressible Navier-Stokes equation. The micrometer-sized width and a moderate temperature gradient lead to a Rayleigh number of Ra L = 2. As a result, the heat flow as well as the temperature profile almost entirely corresponds to heat transfer by conduction. The heat transfer caused by convection are negligible. Finite-element simulation of the heat transfer and fluid flow were preformed consecutively to decrease the computational time, however, yielded the same results compared to a coupled simulations of heat transfer/fluid.
Molecular concentrations for each buffer species are derived by simultaneously solving acid-base reaction kinetics, electrostatics, and molecular flux equations. Following Debye's approach, the simulation neglects perturbations at the upper and lower reversal points by assuming, where v denotes the flow velocity. Explicit reaction rates and kinetics are listed below for both cases, solutions of phosphate buffer and amino acids (see Supplementary Tables 1-9).
pH detection/calibration method. A spatially resolved pH is derived using the ratiometric dye SNARF-1. Following a modified Henderson-Hasselbalch equation (see Eq. 9) 41 , the fluorescence intensity ratio R = F λ1 /F λ2 provides a measure for the pH. The wavelengths λ 1 and λ 2 denote the emission wavelength of SNARF-1 at λ 1 = 580 nm and λ 2 = 640 nm. The fluorescent images are recorded simultaneously using a beam-splitting device and background correction.
The modified Henderson-Hasselbalch equation comprises of four parameters (Ra, Rb, a, and b), derived from the ratio-to-pH calibration curve (see Supplementary Fig. 3). The parameters a and b denote for the pK a of SNARF-1 and the weighting of the spectra parameters while Ra and Rb account for the maximum and minimum fluorescence rations. Temperature-dependent shifts of the pH calibration curve are corrected by linearly fitted parameters a, b, Ra, and Rb (see Supplementary Fig. 3 and Supplementary Table 10). If not mentioned separately, shown measurements were conducted in 50 µM SNARF-1, a buffer/amino acid concentration of 200 mM and a temperature gradient of ΔT = 14 K.
Experimental prefactor κ. The experimental prefactor for the accumulation ratio is defined by 31 with q defined as where ΔT denotes the temperature gradient, α the volume expansion of the solvent, g standard gravity, w the width of the chamber, ρ 0 the density of the solvent, η the viscosity of the solvent, and D the diffusion coefficient of the species. The accumulation prefactor strongly affects the accumulation efficiency. Therefore, the width of the flow chamber is specifically designed for certain molecules.
Thermophoresis. The thermophoretic effect moves molecules along a temperature gradient, mostly from the warm to the cold side. However, thermophoresis is still subject to active research. The Soret coefficient gives a measure for a molecules thermophoretic movement. It can thereby be described by four contributions, the capacitor S T CM , Seebeck effect S T EL 42, 43 , non-ionic S T NI , and the temperature contribution 32 .
Recently, it has been suggested that thermophoresis on charged molecules is dominated by Seebeck effects and ion shielding 32 . The Seebeck effect solely depends on the ionic composition of the solution. Here each ion in solution is subjected to thermophoresis and generates a global electric field. Within the electric field, molecules are subjected to electrophoresis. The electrical field is given by where E denotes the electric field, Z i , c i , and S Ti denotes the charge number, concentration, and Soret coefficient of an ionic species. The Seebeck contribution can be approximated by where µ DNA denotes the electric mobility. As shown in Supplementary Fig. 7, the Seebeck contribution strongly affects the formation of pH gradients.
Charge of amino acids and peptides. The net charge of a peptide sequence or single amino acid depends on the pH and is given by 44 Z ¼ X i N i 10 pKai 10 pH þ 10 pKai À X j N j 10 pH 10 pH þ 10 pKaj ð15Þ where Z denotes the net charge of the peptide sequence, N i the number of histidine, lysine, and arginine residues, and the N terminus with their respective pK ai values, and N j the number of cysteine and the C terminus with their respective pK aj values.
The net charge of an amino acid changes with respect to the alpha-amino group (pK ≈ 9), the carboxylic acid group (pK ≈ 2) and in some cases the side chain. The initial pH is adjusted to be situated in the detection range of SNARF-1 (pH 5-9), therefore solely the dissociation of the alpha-amino group and side chain are detectable (see Supplementary Fig. 8).
Random walk model. Random walk simulations investigate the pH cycling statistics of RNA/DNA strands. These molecules are placed inside a 0.17 × 8.0 mm flow chamber filled with 200 mM of phosphate solution. In theory and experiment, a temperature difference of ΔT = 30 K forms a pH gradient of up to ΔpH = 2.1 (see Figs. 2c and 5a). The particle movement (N = 1000) inside the already established pH gradient is simulated for a given time-step of Δt ¼ 1 ms for a total simulation time of 3 h. The displacement Δsðx; yÞ of these particles is given by Where Δt denotes the time-step, vðx; yÞ the convective flow profile, and T the local temperatures. Finite element simulations calculate the convective flow profile (see chapter finite-element simulations) while the Brownian motion is implemented by a randomly directed movement βðtÞ. Diffusion and Soret coefficients for RNA/ DNA strands in the range of 1-200 nt are given by D ¼ 470 n À0:53 μm 2 s À1 and S T ¼ 0:1 n 0:5 K À1 , where n denotes the length of the strand. The experimental scaling law of the Soret coefficient S T is adapted from Mast et al. 23 .
Data availability. The data that support the findings of this study are available from the corresponding author on reasonable request.