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

• Nature Communications 8, Article number: 1897 (2017)
• doi:10.1038/s41467-017-02065-3
Accepted:
Published online:

## Abstract

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.

## Introduction

The 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 recycling1,2,3,4. 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 conditions5, 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 currency7. Since ancient microbes, such as the last universal common ancestor (LUCA), are assumed to be chemiosmotic8, it has been argued toward the significance of proton gradients at the origin of life9,10,11. In addition, the synthesis of precursor molecules for early molecular evolution requires differing pH values for the synthesis of purine and pyrimidine ribonucleotides12, aminonitriles13, amino acids14, and phosphoenol pyruvate15.

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 ocean16,17,18. 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-millimeter-sized 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 sites19, 20. This elementary setting has previously been shown to concentrate dilute nucleotides21, accumulate lipids to facilitate the formation of vesicles22, shift polymerization reactions toward longer DNA/RNA strands23, and selectively replicate longer polymers24. 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 nucleotides25,26,27,28,29. 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

$j i =- D i ∇ c i - S T i D i ∇T c i +v c i$
(1)

Where v denotes the convection flow, c i , D i , and STi the concentration of the ionic species i, diffusion coefficient, and Soret coefficient, respectively23. The Soret coefficient STi  = DTi 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 by30, 31

$c c 0 =exp κ i S T i Δ T r$
(2)

where c/c0 denotes the relative concentration of a species at the bottom of the chamber, c0 the initial concentration, the temperature difference ∆T, 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 ∆T, the chamber width w, and the diffusion coefficient of the species D i .

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 (HPO42−) and dihydrogen phosphate (H2PO4). 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, HPO42− accumulates tenfold stronger compared to H2PO4, 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

$HX+ H 2 O⇌ X - + H 3 O +$
(3)
$H 2 PO 4 - + H 2 O⇌ HPO 4 2- + H 3 O +$
(4)

where HX and H2PO4 denote the proton donors and Xand HPO42− the proton acceptors. High concentrations of HPO42− 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 H2PO4 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 equation

$pH=p K a +log c acceptor c donor$
(5)

where cacceptor and cdonor 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 pHbottom is therefore derived by calculating the corresponding proton acceptor/donor concentration at the bottom, given by Eq. (2) at equilibrium.

$pH bottom = p K a +log c 0 acceptor c 0 donor exp Δ κ i S T i Δ T r = p K a +log c 0 acceptor c 0 donor + Δ ( κ i S T i ) Δ T r ln ( 10 )$
(6)

with $Δ κ i S T i = κ acceptor S T acceptor - κ donor S T donor$ and $r=h w - 1$ the aspect ratio of the chamber. By assuming κacceptorκdonor and 1/ln(10) ≈ 0.43, pHbottom is given by

$pH bottom ≈ pH init +0.43κ S T acceptor - S T donor ΔTr$
(7)

The maximum pH difference ΔpH inside the flow chamber, arising between the lowest and highest regions of the chamber, is given by

$ΔpH≈0.86κ Δ S T ΔTr$
(8)

with ∆ST = ST acceptorST donor. The steepness of the pH gradient is affected by the temperature difference ∆T, Soret coefficient ST, 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 model is not applicable for steep proton gradients, e.g., formed by large differences in temperature or by large differences in Soret coefficients of proton donors/acceptors. In addition, above formula is subjected to the limitations of the Henderson–Hasselbalch equation, i.e., limited to acid/base concentration in the millimolar regime and pH values of about 7. Better descriptions of the experimental results are obtained using time-dependent finite-element simulations (Fig. 3c), including explicit reaction kinetics for the self-ionization of water, acid–base reaction kinetics, and electrostatics (see Supplementary Tables 19). Simulation results are derived using the time-dependent finite-element model with a commercial solver.

### 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 buffer29. 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 chambers36. 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 mechanism21.

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 functional24.

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 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 three-component 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 712.

An exciting prospect of this work is the periodical formation of proton gradients across vesicles. The leakiness of the first membranes37, made of fatty acids or phosphorylated isoprenoids38, 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 system11, 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 life9, 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 biomolecules21, 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 3D-printed 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 SiO2 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.

### Ratiometric pH detection

The pH in the solution is derived using the ratiometric dye SNARF-1 (5-[and-6]-Carboxy SNARF-1, Invitrogen AG, CA) at concentrations of 50 µM (see Supplementary Fig. 3 and Supplementary Table 10). The starting pH of each solution was adjusted close to the pK of the SNARF-1 dye to ensure maximum accuracy of the pH measurement. For recording the fluorescence images at λ1 = 580 nm and at λ2 = 640 nm simultaneously, a self-built fluorescence microscope was used with a CMOS Stingray 145-b camera (AVT, Germany), a x2 objective (LD, NA 0.055, Mitutoyo, Japan) and an Optosplit 2 (Cairn Research, UK). The ratiometric dye is excited using a 470 nm LED (M470L2-C4, Thorlabs, Germany) in combination with a ratiometric filterset (BrighLine HC 482/35, HCBS506, Brightline HC 580/23, H606LP, BrightLineXF 643/20, AHF Analysentechnik AG, Germany). Depicted error bars are given from duplicate measurements (n = 2), and in particular cases from triplicates.

### 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 Tmax and Tmin 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 RaL = 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 19).

### 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 = Fλ1/Fλ2 provides a measure for the pH. The wavelengths λ1 and λ2 denote the emission wavelength of SNARF-1 at λ1 = 580 nm and λ2 = 640 nm. The fluorescent images are recorded simultaneously using a beam-splitting device and background correction.

$pH=a+blog R - R a R b - R$
(9)

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 pKa 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 by31

$κ= q ∕ 120 1 + q 2 ∕ 10080$
(10)

with q defined as

$q= Δ T α g ρ 0 w 3 6 η D$
(11)

where ΔT denotes the temperature gradient, α the volume expansion of the solvent, g standard gravity, w the width of the chamber, ρ0 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 STCM, Seebeck effect STEL42, 43, non-ionic STNI, and the temperature contribution32.

$S T = S T CM + S T EL + S T NI +1∕T$
(12)

Recently, it has been suggested that thermophoresis on charged molecules is dominated by Seebeck effects and ion shielding32. 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

$E= k B T ∇ T e ∑ i Z i c i S T i ∑ j Z j 2 c j$
(13)

where E denotes the electric field, Z i , c i , and STi denotes the charge number, concentration, and Soret coefficient of an ionic species. The Seebeck contribution can be approximated by

$S T EL =- k B T μ DNA e D DNA ∑ i Z i c i S T i ∑ j Z j 2 c j$
(14)

where µDNA 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 by44

$Z= ∑ i N i 1 0 p K a i 1 0 pH + 1 0 p K a i - ∑ j N j 1 0 pH 1 0 pH + 1 0 p K a j$
(15)

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 $Δt=1ms$ for a total simulation time of 3 h. The displacement $Δs ( x , y )$ of these particles is given by

$Δs x , y = 4 ⋅ D ⋅ Δ t ⋅η t +Δt⋅ v x , y + D ⋅ S T ⋅ ∇ T x , y$
(16)

Where $Δt$ denotes the time-step, $v ( x , y )$ 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 $η ( t )$. Diffusion and Soret coefficients for RNA/DNA strands in the range of 1–200 nt are given by $D=470× n - 0.53 μm 2 s - 1$ and $S T =0.1× n 0.5 K - 1$, where n denotes the length of the strand. The experimental scaling law of the Soret coefficient ST 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.

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## Acknowledgements

We thank Matthias Morasch, Victor Sojo, and Dieter Braun for discussions and valuable comments on the manuscript. Financial support from the NanoSystems Initiative Munich, the Simons Collaboration on the Origin of Life, the Ludwig-Maximilians-Universität Munich Initiative Functional Nanosystems, and the SFB 1032 Project A4 is acknowledged.

## Author information

### Affiliations

1. #### Systems Biophysics, Physics Department, Center for Nanoscience, Ludwig-Maximilians-Universität München, 80799, Munich, Germany

• Lorenz M. R. Keil
• , Friederike M. Möller
• , Michael Kieß
• , Patrick W. Kudella
•  & Christof B. Mast

### Contributions

L.K., M.K., P.K. and C.B.M. performed the experiments. L.K. and F.M. performed the simulations. L.K. and C.B.M. conceived and designed the experiments, analyzed the data, and wrote the paper.

### Competing interests

The authors declare no competing financial interests.

### Corresponding author

Correspondence to Christof B. Mast.