## Main

Spatiotemporal organization of cells generally emerges through redistribution and transport of molecules via motor proteins1, self-assembling cytoskeletal elements2 or self-organizing reaction–diffusion systems3. Coupling of cargo to energy-dissipating NTPases that drive the transport is usually mediated by specific protein–protein interactions. Non-specific coupling of biomolecules, by contrast, is poorly explored in biology and, so far, only a few examples of molecular transport based on purely physical mechanisms have been reported. For example, a study in mouse oocytes showed that actin-coated vesicles generated a pressure gradient that positioned large objects like the nucleoid in the cell centre4,5. In the Caenorhabditis elegans zygote, cortical flows were shown to couple to the PAR reaction–diffusion system via advective transport6. Another recent example comes from in vitro studies of the Escherichia coli Min system7,8.

The Min system, a paradigmatic model for pattern formation in biology, regulates the site of cell division in E. coli9,10,11,12. The core of this reaction–diffusion system consists of only two proteins, the ATPase MinD and the ATPase activating protein MinE, which interact and reversibly bind to the membrane11,13. Despite this simple reaction scheme, the Min system exhibits rich dynamics that have been explored in vivo9,10,11,14, reconstituted in vitro13,15,16 and described by physical theories17,18,19,20,21. In the rod-shaped E. coli, MinDE oscillate from pole to pole10,11,14. In vitro, MinDE proteins form travelling surface waves or quasi-stationary patterns on planar artificial membranes13,15,16 and exhibit oscillations when geometrically confined22,23. These dynamics can provide spatial cues for particular proteins. MinC specifically binds to MinD and thus follows its movements10,22,24,25,26. In turn, MinC constrains the localization of the main divisome protein FtsZ by inhibiting its polymerization27,28.

Besides this well-described patterning by specific interactions with clear physiological evidence, MinDE self-organization has recently shown an intriguing hidden function in vitro: MinDE regulated the localization of unrelated membrane-bound molecules in space and time in the absence of MinC/FtsZ7,8. These results suggested that MinDE oscillations could further enhance cell division by prepositioning membrane proteins to the cell middle7. However, the underlying physics and the broader biological implications have remained unknown.

Here, through a joint experimental and theoretical investigation, we have deciphered the physical mechanism underlying this non-specific transport phenomenon. We quantitatively probed MinDE-dependent transport using a synthetic membrane-bound cargo based on composite DNA nanostructures. We discerned how the effective size (that is, membrane footprint) and diffusion coefficient of the cargo, as well as the type of MinDE patterns, determine the transport that takes place. Exploiting these effects, we revealed that MinDE can even spatially sort different cargo species. Theoretical analysis of these data demonstrated a diffusiophoretic effect: an effective density-dependent inter-particle friction creates cargo transport along the diffusive fluxes of MinD proteins. This type of NTPase-driven diffusiophoresis might represent a generic active transport mechanism in cells that neither requires motor proteins nor specific protein interactions. As such, it might be particularly important for prokaryotes and could have been prevalent in early stages of life on Earth.

## Probing MinDE-induced transport with a synthetic cargo

We set out to understand how MinDE dynamics can spatiotemporally regulate other membrane-bound, yet unrelated molecules (henceforth referred to as non-specific ‘cargo’) into patterns and gradients7,8. Specifically, we asked whether cargo transport can arise from MinDE patterns per se via thermodynamic forces, or whether it requires active processes and non-equilibrium particle fluxes. To experimentally address this question and test possible mechanisms, we set up a highly controllable minimal in vitro platform. To this end, we reconstituted MinDE pattern formation on supported lipid bilayers (SLBs)13. For simplicity, we first chose conditions under which initially chaotic, laterally moving MinDE waves transition into quasi-stationary labyrinth patterns16. To quantitatively assess the interaction between MinDE and cargo molecules on the membrane, we employed a synthetic cargo: a composite object consisting of a DNA origami nanostructure as scaffold and streptavidin molecules that serve as modular building blocks and connectors to the membrane (Fig. 1a). In particular, the origami29 featured 7 dyes on the upper facet for visualization and 42 sites at the bottom facet that could be specifically addressed for the incorporation of biotinylated oligonucleotide handles. These handles in turn bound to streptavidin coupled to biotinylated lipids in the SLB.

When we initiated MinDE self-organization with ATP in the presence of this synthetic cargo, the cargo components—that is, the origami structures and streptavidin—reorganized into patterns from an initially homogeneous state (Fig. 1b,c, Supplementary Fig. 1 and Supplementary Video 1). As expected, the origami and streptavidin co-localized. Accordingly, in the following, origami fluorescence serves as a proxy for cargo localization. For the entire duration of the experiment, cargo always gathered in MinD-depleted regions, forming patterns that were anti-correlated to the MinDE patterns (Fig. 1b,c). Similarly, when we altered the established MinDE/cargo patterns by adding more MinE, the cargo channel reflected the changes in MinDE patterns, moving in an anti-correlated fashion (Supplementary Fig. 2 and Supplementary Video 2). By contrast, when MinE (which stimulates MinD ATP hydrolysis) was omitted, MinD and the cargo molecules remained uniformly distributed (Extended Data Fig. 1). These findings indicate that the spatial heterogeneity of the cargo is not caused by depletion attraction, such as in filament bundling30. For one, depletion attraction should lead to aggregation of large particles (cargo), even in a homogeneous field of smaller particles (MinD)31. Furthermore, depletion attraction would imply a preferred agglomeration of cargo in regions of high MinD density, which we also did not observe (Supplementary Information). Hence, our data demonstrate that cargo transport is active, because it requires the presence of MinDE and ATP, and thus active MinDE self-organization.

## Effective cargo size determines the extent of cargo demixing

Having shown that MinDE redistribute our synthetic cargo, we next exploited the modular nature of our cargo to systematically vary its interaction with MinDE. MinD and lipid-anchored streptavidin both form a monomolecular layer of height ~5 nm on membranes32,33, whereas the lower facet of the origami scaffold lies at ~5–11 nm above the membrane (Supplementary Note 1). MinDE thus move on the membrane below the altitude of the origami scaffold and should mainly interact with the membrane-bound streptavidin. Hence, varying the number of streptavidin (n {1, 2, 5, 15, 28, 42}) bound to the origami scaffold enables fine control over a large dynamic range of the membrane footprint or effective size of the cargo (henceforth cargo-n, Fig. 1). In this way, we modulated the interaction of the cargo-n with MinDE, as well as its diffusion on the membrane.

To quantitatively assess the interaction between MinDE and the respective cargo, we analysed the resulting, final quasi-stationary patterns (Fig. 1d and Extended Data Fig. 2). As a measure for molecule enrichment, we determined the Michelson contrast, (Imax − Imin)/(Imax + Imin), of the fluorescence images on a scale from zero for a homogeneous distribution to one for a binary distribution. Cargo patterns became much sharper with increasing cargo size (Fig. 1e and Extended Data Fig. 2). This increase in the contrast of cargo patterns was accompanied by sharper and also narrower MinDE patterns, as indicated by an increased region of pixels classified as MinD minima (Fig. 1f,g). Thus, MinDE dynamics dictate the localization of cargo on the membrane in a size-dependent manner, and are in turn also impacted by their presence. At first glance, these results could be interpreted as simply being a consequence of steric repulsion between cargo and MinDE. However, although a static gradient of accessible space would indeed induce a gradient of cargo molecules, this effect should be equal for all cargo species, irrespective of their effective size (Supplementary Information). Therefore, even though the diffusion of MinD on the membrane under the given conditions is very slow (D = 0.013 μm2 s−1)25, we conclude that MinD proteins do not simply act as immobile obstacles on the membrane that would bias cargo diffusion via static volume exclusion, a second option for a thermodynamic force.

## Thermodynamic forces cannot explain cargo transport by MinDE

As our experimental data disqualified both depletion attraction and static volume exclusion as possible explanations for cargo redistribution, we wondered whether mobile MinD proteins could effectively repel cargo in a size-dependent manner by imposing local constraints on the entropy of mixing. To test this third (thermodynamic) hypothesis, we formulated a fully quantitative Flory–Huggins theory (FH) without fitting parameters. Specifically, each origami scaffold crosslinks n streptavidin into a passive polymer-like cargo (Fig. 2a), while the remaining free streptavidin (for n < 15; Extended Data Fig. 2) behave independently (Supplementary Information). Given these constraints, we characterized the membrane in terms of local surface densities of cg cargo, cs free streptavidin and cp MinD (Supplementary Table 1), assuming a well-mixed lattice gas on the microscopic scale. On the mesoscopic scale, we then asked ‘what is the equilibrium distribution of passive particles (cargo and free streptavidin) in the presence of a heterogeneous distribution of active MinD proteins?’ To answer this question, we used our FH model to calculate the corresponding chemical potentials μi for each species (Supplementary Information). Furthermore, we assumed that the passive particles adopt a thermal equilibrium state with vanishing chemical potential gradients (μg = μs = 0) in an adiabatic response to the imposed steady-state distribution of active particles (μp ≠ 0). Our theoretical analysis shows that entropic mixing effects can, in principle, create a gradient of passive cargo in response to a gradient of active MinD, where the cargoes’ crosslinked streptavidin experience a weak entropic bias towards MinD-depleted regions (due to volume exclusion effects). However, this effect will be partly compensated by the analogous entropic repulsion between origami scaffolds, which prevents high cargo densities, in stark contrast to our experimental observations (Fig. 2b,c and Extended Data Fig. 3). Consequently, we also rejected entropic mixing in fixed external (chemical potential) gradients as the mechanism underlying MinDE-dependent cargo transport.

## Coupling between particle fluxes explains cargo transport

Taken together, thermodynamic mechanisms alone cannot explain MinDE-induced cargo transport in our experiments, suggesting that the underlying mechanism is genuinely non-equilibrium in nature. Therefore, we generalized our equilibrium FH model to consider the non-equilibrium dynamics of particles. Specifically, MinDE gradients build up due to particle exchange between the bulk solution and membrane (reactive fluxes)16,18,19,21,25, leading to chemical potential gradients μi on the membrane (thermodynamic forces). According to Onsager’s theory34, thermodynamic forces imply particle fluxes ji. Conceptually, diffusive fluxes on the membrane then arise from an effective force balance of thermodynamic and friction forces between the different macromolecules and lipids (with friction coefficients ξi). Furthermore, in a crowded environment such as the membrane surface in our experiments (mean free path of ~50 Å; Supplementary Information), hydrodynamic or also direct interactions between particles can mediate a ‘mesoscopic friction’ that couples their respective fluxes (with coupling constants ζik). Using these arguments, we formulated a phenomenological theory where each membrane-bound species obeys a Maxwell–Stefan (MS)-like effective force-balance equation35,36:

$$\begin{array}{c}{\boldsymbol{\nabla}} {\mu }_{i}+{\xi }_{i}\frac{{{\bf{j}}}_{i}}{{c}_{i}}+\sum _{k}{c}_{k}{\zeta }_{ik}\left(\frac{{{\bf{j}}}_{i}}{{c}_{i}}-\frac{{{\bf{j}}}_{k}}{{c}_{k}}\right)=0\end{array}.$$
(1)

As before, the index i {p, g, s} refers to MinD proteins, cargo with bound streptavidin and free streptavidin, respectively. Although our theory is intrinsically dynamic (Methods), we first analysed the non-equilibrium steady state, ∂tci = 0, reflecting the quasi-stationary MinDE patterns. Then, cargo and streptavidin exhibit Brownian motion and relax to a thermal equilibrium state with vanishing fluxes jg = js = 0. The MinD protein patterns maintain a non-equilibrium steady state and finite protein fluxes jp ≠ 0 via off-equilibrium chemical reactions (ATPase activity). In the absence of mutual friction between the macromolecules (ζik = 0), equation (1) reduces to the FH model (μg = μs = 0), which readily implies weak cargo redistribution in a static gradient of active proteins. As a decisive factor in addition to these entropic demixing effects, the presence of frictional coupling (ζik ≠ 0) between cargo and MinD leads to advective cargo transport by the non-equilibrium protein fluxes (jp ≠ 0) of MinD (Fig. 2a). The additional bias conferred by these protein fluxes leads to cargo redistribution, which is much stronger than by equilibrium thermodynamic forces alone and thus quantitatively explains our experimental data (Fig. 2b,c and Extended Data Fig. 3).

## MinDE impair cargo diffusion via mesoscopic friction

Next, we simplified our theoretical model by neglecting membrane saturation effects (Supplementary Information), reducing the chemical potential of particles with size ai to μi ≈ kBT ln(aici). The force-balance equation, equation (1), then becomes a generalized Fick’s law:

$$\begin{array}{c}{{\bf{j}}}_{{\rm{p}}}\approx -\frac{{k}_{{\rm{B}}}T}{{\xi }_{{\rm{p}}}+{c}_{{\rm{g}}}{\zeta }_{{\rm{pg}}}+{c}_{{\rm{s}}}{\zeta }_{{\rm{ps}}}}{\boldsymbol{\nabla }}{c}_{{\rm{p}}}\equiv -{D}_{{\rm{p}}}\left({c}_{{\rm{g}}},{c}_{{\rm{s}}}\right){\boldsymbol{\nabla }}{c}_{{\rm{p}}}\end{array}$$
(2)

where the density-dependent diffusion coefficient of MinD, Dp(cg,cs), decreases through friction (that is, interactions) with cargo and streptavidin. For cargo with strong coupling ζpg, maintaining the diffusive fluxes that balance reactive protein turnover requires sharper protein gradients, which explains the progressively sharper and narrower MinDE patterns (Fig. 1f,g).

Assuming that the typically small number of free streptavidin (Supplementary Information) does not notably contribute to the dynamics, csζpscgζpg, one obtains a closed expression for the cargo distribution:

$$\begin{array}{c}{c}_{{\rm{g}}}\left({c}_{{\rm{p}}}\right)={c}_{\times }{W}_{0}\left[\frac{{c}_{{\rm{g}}}\left(0\right)}{{c}_{\times }}\exp \left(\frac{{c}_{{\rm{g}}}\left(0\right)-{c}_{{\rm{p}}}}{{c}_{\times }}\right)\right]\end{array},$$
(3)

with typical interaction density c× = ξp/ζpg. Fitting equation (3) to our experimental data (Fig. 3a,b and Extended Data Fig. 3), we demonstrate that the coupling constant ζpg between MinD and cargo-n has a contribution from the origami scaffold and from the n incorporated streptavidin, ζpg = ζpo + ps, explaining why cargo transport increases with the number of streptavidin per cargo (Fig. 1d–g).

To test our theoretical model experimentally, we performed single-particle tracking of cargo-2 and cargo-42, both in the presence and absence of MinD. In the former case, we emulated high MinD membrane densities in the maxima of MinDE patterns by adding 1 µM MinD and ATP, but no MinE. We found that the diffusion coefficient of cargo-2 decreased from 0.65 ± 0.12 μm2 s−1 in the absence of MinD to 0.06 ± 0.02 μm2 s−1 at high MinD density (Fig. 3c). By contrast, the diffusion coefficient of cargo-42, which was already low in the absence of MinD (0.06 ± 0.02 μm2 s−1), hardly changed at high MinD density (0.036 ± 0.011 μm2 s−1) (Fig. 3c). Subsequently, we used our fitted interaction parameters to predict the diffusion coefficient of cargo at high protein densities, based on the experimental values in the absence of proteins (Supplementary Information). Our predictions were in good quantitative agreement with our experimental findings, validating our model. At first, it might seem counterintuitive that MinD affects cargo-42 diffusion less than that of cargo-2, despite stronger frictional coupling. However, even in the absence of MinD, cargo-42 diffuses much slower than cargo-2 due to the friction between its many streptavidin and the membrane, which dominates over the additional friction with MinD. This observation highlighted the parameter interdependence that arises due to streptavidin serving both as building block and connector to the membrane and raised the question whether cargo transport is affected more by its effective size or its diffusion coefficient. To answer this, we employed an alternative membrane attachment strategy for the DNA origami scaffolds via cholesteryl moieties (Extended Data Figs. 4 and 5 and Supplementary Note 2). We found that cargoes with a larger membrane footprint (that is, effective size) and at the same time similar diffusion as cargo-2 were strongly redistributed (Extended Data Fig. 5 and Supplementary Note 2), corroborating that it is indeed the effective cargo size that determines the interaction with MinD. In conclusion, the dependence of the cargo diffusion coefficient on the ambient protein density is a direct experimental proof of friction between MinD and cargo and that the effective cargo size governs the corresponding friction coefficient.

## MinDE spatially sort different cargo species

Can we use our obtained knowledge to selectively position cargo molecules, that is, to sort them according to their properties, along protein gradients? To answer this question, we placed two differently labelled cargo species, cargo-2 and cargo-42, in the same assay (Fig. 4a). We found, as predicted by our model (Fig. 4d), that cargo-42 gathered in MinD-free regions and was framed by cargo-2 (Fig. 4b,c, Supplementary Fig. 3 and Supplementary Video 3). Thus, cargo-42 exhibited a similar behaviour as when present in the assay alone. By contrast, the localization of cargo-2 relative to MinD changed when cargo-42 was also present (Figs. 1d and 4c). The observed spatial separation of cargo species was not an artefact due to fluorescent channel crosstalk, dye selection or quenching (Extended Data Fig. 6). Hence, the clear MinDE-induced spatial sorting of cargo species according to their effective size further refutes thermodynamic models (Fig. 4d), corroborating that MinDE indeed transport molecules via friction.

## MinDE waves direct net cargo transport via diffusiophoresis

Having confirmed diffusiophoretic transport in the context of quasi-stationary MinDE patterns, we turned to conditions where MinDE form travelling surface waves7,8. As in previous experiments with lipid-anchored streptavidin alone7,8, MinDE waves induced anti-correlated patterns of our synthetic cargo-2 that were superimposed with macroscopic gradients across multiple MinDE wavelengths, indicating net transport (Fig. 5a). Consequently, cargo accumulated between opposing wavefronts (Fig. 5a and Supplementary Video 4). Despite their distinct appearance, stationary as well as moving MinDE patterns are maintained by the same kind of non-equilibrium reaction and diffusion fluxes16,18,19,21,25. Thus, we could directly apply our theory to this dynamic setting, without further assumptions (Methods). We found that, in addition to the interaction between MinDE and cargo, macroscopic transport also requires matching timescales between MinDE wave propagation and cargo diffusion: cargo with small mobility cannot keep up with the MinDE wave. In agreement with our simulations (Fig. 5b), the slowly diffusing cargo-42 hardly formed long-ranged gradients (Fig. 5c and Supplementary Fig. 4), despite its strong frictional coupling to MinDE, supporting our prediction that low cargo mobility can limit macroscopic transport. This was further corroborated by the emergence of macroscopic gradients for cholesterol-anchored cargo-chol-2 and cargo-chol-15 (Fig. 5c and Supplementary Fig. 4), which display similarly high coupling to MinDE, but diffuse faster than cargo-42 (Extended Data Fig. 5).

Next, we exploited the geometry-sensitivity of MinDE waves that travel along the longest axis on geometrically patterned planar membranes with chromium barriers37. On these membranes, MinDE waves indeed transported cargo directionally along their wavevector, resulting in reproducible gradients (Fig. 5d,e, Extended Data Fig. 7 and Supplementary Videos 5 and 6). These density gradients are clearly visible on the macroscopic scale, while directed movement of single cargo molecules cannot be discerned from diffusion (Extended Data Fig. 8 and Supplementary Note 3). This observation highlights the difference between ATP-driven diffusiophoresis and translational motor proteins.

Our findings raised the question of whether MinDE-dependent transport via diffusiophoresis could occur in vivo. To our knowledge, no direct observation of this mechanism in E. coli has been reported, which may be attributed to bacterial sizes close to the optical resolution limit or specific interactions concealing such occurrence. Thus, we reconstituted MinDE oscillations together with inert model peripheral membrane proteins (mCh-MTS)7 in the evolutionary distant fission yeast Schizosaccharomyces pombe38 (Fig. 5f, Extended Data Figs. 9 and 10 and Supplementary Videos 7 and 8). We found that in this in vivo model system MinDE formed dynamic patterns with similar wavelength and velocity as in its native host E. coli11,12 (Supplementary Note 4) and, indeed, spatiotemporally regulated mCh-MTS proteins on intracellular membranes (Extended Data Fig. 10).

## Diffusiophoresis drives molecular transport in biology

In conclusion, combining in vitro and in vivo reconstitution experiments with theoretical modelling and analysis, we showed that the prokaryotic MinDE proteins can non-specifically transport and even sort membrane-bound cargo molecules by a diffusiophoretic effect: the diffusive fluxes of MinD and cargo couple via friction in a mechanical rather than thermodynamic fashion (Fig. 6). This process is driven by ATP-consuming MinDE self-organization, which generates a net diffusive flux of MinD towards low densities. This flux establishes an effective frictional force on cargo, driving diffusiophoretic cargo transport towards areas of low MinD density. Hence, the friction between cargo and MinD increases with the effective size of the cargo (for example, with the number of streptavidin building blocks). The bare diffusion coefficient of cargo only becomes relevant in the case of dynamic MinDE waves where cargo diffusion has to ‘keep up’ with the MinDE waves to induce macroscopic gradients.

Similar transport effects have been reported previously in a non-biological context: diffusiophoresis generally refers to particle transport in fluids, induced by concentration gradients of small solutes39,40,41,42,43,44,45,46,47,48,49. Interestingly, diffusiophoresis was also treated in terms of particle fluxes48, suggesting that a flux-centred viewpoint may be appropriate for multicomponent mixtures out of equilibrium. A phenomenologically similar effect has been demonstrated in gases with the MS theory of diffusion35,36,50,51. In a biologically relevant context, theoretical studies have suggested that diffusiophoresis might occur in enzyme chemotaxis52 and alignment53, during ParABS-mediated chromosome segregation in bacteria54 and that metabolism-dependent gradients of small molecules (for example, ATP) could induce transport of large particles55. The here described diffusiophoresis driven by protein patterns that are maintained by NTPase cycles might constitute a novel mechanism of coupling energy dissipation to active spatiotemporal positioning in cell biology. Diffusiophoretic transport is presumably not a special feature of the E. coli MinDE system or reaction–diffusion systems in general, but can potentially be exerted by any active system producing concentration gradients. For example, such a mechanism could be underlying the secretion-induced protein patterning that has been observed in fission yeast56 or be at play for the plethora of intracellular (actin) waves in eukaryotes whose purpose and mode of action has remained elusive57. The mechanism might not even be limited to the membrane as a reaction surface, but potentially extends to other cellular surfaces and even cytosolic gradients. For example, the strong concentration gradients that are built up during liquid–liquid phase-separation processes could potentially similarly impact other molecules58. We also propose that distinct pattern-forming systems sharing the same reaction space should align to minimize friction, even if their constituents are chemically independent. This could potentially link and synchronize pathways to increase their efficacy or provide a rescue mechanism against mutations affecting the chemical coupling via specific interactions (for example, between MinC and FtsZ). That this non-specific means of transport was discovered and described in an in vitro reconstitution assay is not a coincidence, but highlights that the complexity of cells with more sophisticated and stronger specific interactions presumably masks such occurrence. Finally, simple as it is in comparison to eukaryotic, translational motor proteins, this mechanism could be interpreted as an alternative, more rudimentary mode of mechanochemical coupling and as such might be prevalent in prokaryotes and might have been present in early forms of life.

## Methods

### Theoretical description in the presence of travelling waves

We have formulated our theory, equation (1), in terms of particle fluxes. Given these fluxes ji, the spatiotemporal dynamics of each molecular species is determined by the balance equation

$$\begin{array}{c}{\partial }_{t}{c}_{i}+{\boldsymbol{\nabla }}\cdot {{\bf{j}}}_{i}={r}_{i}\left(\ldots \right)\end{array},$$
(4)

where ri signify reaction terms. We considered two species, passive cargo molecules cg that do not engage in reactions, rg = 0, and active MinD proteins cp. Instead of considering specific reaction terms, we assumed that the interplay between diffusive and reactive fluxes of MinDE leads to travelling waves. Then, we took these travelling MinD protein waves with wavelength λ and speed v:

$${c}_{{\rm{p}}}\left(x,t\right)={c}_{{\rm{p}}}\left(2{\rm{\pi }}\frac{x-vt}{\lambda }\right)\equiv {c}_{{\rm{p}}}\left(z\right)$$
(5)

as a given. As specific waveform, we assumed a skewed sinusoidal curve (Fig. 5b):

$${c}_{{\rm{p}}}\left(z\right)=\frac{2}{{\rm{\pi }}\gamma }\arctan \left[\frac{\gamma \cos z}{1-\gamma \sin z}\right],$$
(6)

where the parameter γ [−1…1] determines the ‘skewness’ of the waveform. In principle, we could also choose a different waveform (for example, a sinusoidal curve that is skewed in the opposite direction). Choosing a different waveform does not change the general result of a macroscopic net transport along the wavevector, but it can change the magnitude of the transport because the local gradient magnitude is affected.

Furthermore, for simplicity, we neglected membrane saturation effects (Supplementary Information), so that the chemical potential of a particle with size ai reduces to μi ≈ kBT ln(aici). This is the same assumption as in the derivation of the fitting function for the stationary patterns, equation (3), and hence has the same limitations at high particle densities. Taken together, the spatiotemporal dynamics of the passive cargo species is determined by the following transport equation:

$$\begin{array}{c}{\partial }_{t}{c}_{{\rm{g}}}=-{\boldsymbol{\nabla }}\cdot {{\bf{j}}}_{{\rm{g}}}={\boldsymbol{\nabla }}\cdot \left[{D}_{{\rm{g}}}\frac{(1+{c}_{{\rm{g}}}/{c}_{\times }){\boldsymbol{\nabla }}{c}_{{\rm{g}}}+({c}_{{\rm{g}}}/{c}_{\times }){\boldsymbol{\nabla }}{c}_{{\rm{p}}}}{1+{c}_{{\rm{g}}}/{c}_{\times }+({D}_{{\rm{g}}}/{D}_{{\rm{p}}})({c}_{{\rm{p}}}/{c}_{\times })}\right]\end{array},$$
(7)

where the interaction density is given by c× = ξp/ζpg and the two diffusion coefficients are given by Dg = kBT/ξg and Dp = kBT/ξp. We used FEniCS59 to numerically solve equation (7) in a one-dimensional geometry with no-flux boundary conditions for the cargo.

### Plasmids, strains and proteins

A list of all plasmids and primers is provided in Supplementary Tables 2 and 3. The plasmids pET28a-His-MinD_MinE13, pET28a-His-EGFP-MinD60, pET28a-His-MinE13 and pET28a-MinE-His16 were used for purification of His-MinD, His-EGFP-MinD, His-MinE and MinE-His, respectively, as described in detail previously61. In brief, proteins were expressed in E. coli BL21 (DE3) and then purified via Ni-NTA affinity and size-exclusion chromatography in storage buffer (50 mM HEPES/KOH pH 7.2, 150 mM KCl, 10% glycerol, 0.1 mM EDTA, 0.4 mM tris(2-carboxyethyl)phosphine (TCEP)). Proteins were snap-frozen in liquid nitrogen and stored in small aliquots until further use at −80 °C.

Plasmids for heterologous co-expression of MinD, MinE and model peripheral membrane proteins pREP41X-sfGFP-MinD, pREP42X-MinE, pREP41X_coex_sfGFP-MinD_MinE, pREP42X-mCherry-BsMTS, pREP42X-mCherry and pREP42X-mCherry-BsMTS were constructed analogously to plasmids devised by Terbush and colleagues38. pREP41X and pREP42X, and the leucine and uracil auxotroph S. pombe strain FY61 [h- ura4-D18 leu1-32], were a kind gift from S. Forsburg, University of Southern California62. All constructs based on pREP41X and pREP42X express the respective proteins under the control of the inducible nmt1* promoter. For details see Supplementary Methods.

### Reconstitution in S. pombe

Details of S. pombe culture and transformation are provided in the Supplementary Methods. For image acquisition, 5-ml cultures with selective medium were inoculated with the respective strains from glycerol stocks or plates and grown for 48–72 h at 30 °C. Cover slides were rinsed with ethanol and ddH2O, and further cleaned in a plasma cleaner with oxygen as the process gas. Subsequently, Grace Bio-LABs reusable culture well gaskets (diameter of 9 mm, GBL103240, Sigma-Aldrich) or sticky slides (#81818, ibidi) were attached to the clean cover slides. Poly-l-lysine solution (0.01%, P4707, Sigma-Aldrich) was added to the wells and incubated for more than 30 min. Wells were washed with the respective media once before 10–50 μl of cell suspension was added for microscopy at a constant room temperature of 23 °C.

### DNA origami nanostructures

The elongated DNA origami nanostructure used here was previously designed and described in ref. 29. The 20-helix bundle with hexagonal lattice is based on the M13mp18 7,429-nucleotide-long scaffold plasmid (p7429; Bayou Biolabs) and was modified using CaDNAno63. Staple oligonucleotides, 5′-Cy3B/Cy5-functionalized oligonucleotides, the 5′-cholesteryl-TEG functionalized oligonucleotides (high-purity, salt-free, Eurofins MWG Operon) and 5′-biotin-TEG functionalized oligonucleotides (Sigma-Aldrich) were purchased or diluted in Milli-Q ultrapure water at a concentration of 100 μM. Origami structures with 1–15 anchors were based on the previous design29, which was further modified for functionalization with 42 anchors (Supplementary Fig. 5). The assembly of the origami structure was performed in a one-pot reaction mix as described previously29. In brief, the components were mixed at a final concentration of 20 nM p7429 scaffold plasmid and 200 nM staple oligonucleotides in folding buffer (5 mM Tris-HCl, 1 mM EDTA, 20 mM MgCl2, pH 8.0) and annealed in a thermocycler (Mastercycler) over a 41-h cooling scheme from 65 to 40 °C. Folded nanostructures were purified to remove excess staple strands by centrifugation (14,000g, three cycles for 3 min, one cycle for 5 min) in Amicon Ultra 100 kDa molecular weight cutoff filters (Merck Millipore) using reaction buffer (25 mM Tris-HCl pH 7.5, 150 mM KCl, 5 mM MgCl2). The concentration of folded Cy5-labelled origami structures was estimated by fluorescence intensity measurements using a one-drop measurement unit of a Jasco FP-8500 spectrofluorometer and subsequent comparison with an intensity calibration curve obtained for free Cy5 dye corrected for the multiple labelling of the origami. Cy3B-labelled DNA origami concentration was measured by absorption at 260 nm on a NanoDrop spectrophotometer (ThermoFisher Scientific) and related to Cy5-labelled structures of known concentrations. Cy3B/Cy5-labelled DNA origami structures contained seven Cy3B/Cy5-labelled oligonucleotides attached to extended staples on the upper facet. At the lower facet, the biotin-functionalized origami contained multiple 18-nucleotide (nt) extensions that were hybridized with complementary 5′ biotin-TEG-functionalized oligonucleotides (5′ biotin-TEG-AACCAGACCACCCATAGC) at defined positions. DNA origami that were bound to the membrane via cholesteryl oligonucleotides contained single or multiple 18-nt extensions on the lower facet that can hybridize with the complementary 5′ TEG-cholesteryl-functionalized oligonucleotides (5′ chol-TEG-AACCAGACCACCCATAGC) supplied in the self-organization assay.

### Preparation of SLBs

SLBs were prepared as described in detail in refs. 13,61. In brief, cover slides were rinsed with ddH2O and ethanol, and a plastic chamber was glued on top. Slides were further cleaned by plasma cleaning with oxygen as the process gas (model Zepto, Diener Electronic). Chloroform-dissolved lipids (Avanti Polar Lipids) were dried by a nitrogen stream and subsequently in a desiccator before slow rehydration at a concentration of 4 mg ml−1 in reaction buffer (25 mM Tris-HCl pH 7.5, 150 mM KCl, 5 mM MgCl2). Small unilamellar vesicles were generated by sonication in a bath sonicator and subsequently added to the cleaned reaction chambers at a concentration of 0.5 mg ml−1. After 4 min of incubation on a 37 °C warm heating block, the SLB was washed 10 times with a total of 2 ml of wash buffer (25 mM Tris-HCl pH 7.5, 150 mM KCl) to remove excess vesicles. All mentioned concentrations refer to the final volume of the reaction chamber of 200 µl. To prepare chambers for self-organization experiments with biotin-functionalized origami, the SLB was generated with a lipid composition of 69/30/1 mol% DOPC/DOPG/biotinyl-CAP-PE or with 70/30/0.01 mol% DOPC/DOPG/biotinyl-CAP-PE for single-particle tracking experiments and subsequently incubated with non-labelled or Alexa568-labelled streptavidin (ThermoFisher Scientific) at a final concentration of 1 µg ml−1. After incubation for 5–10 min, unbound streptavidin was removed by washing five times with a total volume of 1 ml of reaction buffer. The buffer was adjusted to a volume of 100 μl and the origami was incubated at a final concentration of 0.1 nM for 10 min, before the buffer was adjusted to the final volume of 200 μl. To prepare chambers for self-organization experiments with cholesteryl-bound origami, the SLB was generated with a lipid composition of 70/30 mol% DOPC/DOPG. The buffer was adjusted to a volume of 100 µl and the 5′ TEG-cholesteryl-functionalized oligonucleotides for binding were added at a final concentration of 10 nM. Subsequently, the origami was added at a final concentration of 0.1 nM and incubated for 10 min, before buffer adjustment to 200 µl. For experiments involving more than one type of DNA origami, DNA origami species were premixed in DNA LoBind tubes (Eppendorf) before addition to the sample chamber at a final concentration of 50 pM for each DNA origami, keeping the overall DNA origami concentration at 0.1 nM. Note that at these experimental conditions, DNA origami does not bind non-specifically to the lipid membrane in the absence of biotin-TEG-anchors/streptavidin or cholesteryl-TEG-anchors, due to the high net negative charge of both the SLB and DNA origami64,65.

### Chromium-patterned cover slides

Chromium-patterned cover slides for patterned SLBs66,67 were generated by photolithography and metal evaporation. Cover slides were first cleaned by rinsing with pure ethanol and ddH2O and subsequently by plasma cleaning with oxygen as the process gas (model Zepto, 20–60 s, 40–50% power and 0.3 mbar). The vapour of bis(trimethylsilyl)amine (HDMS) was deposited on the cover slide for 2 min as adhesion promoter. Subsequently, positive photoresist (AZ ECI 3027, MicroChemicals) was spin-coated onto the cover slide (40 s, 4,000 r.p.m., start/stop acceleration 2,000 r.p.m. s−1) resulting in an ~3-μm-thick layer of photoresist. After pre-baking (90 s, 90 °C) the photoresist was patterned using ultraviolet lithography (μPG101, Heidelberg Instruments) with a 10-mm write head, nominal output power of 35 mW at a wavelength of 375 nm, before passing a 45% attenuation filter. Afterwards, the slides were post baked (60 s, 110 °C) before applying developer (AZ 351B, NaOH-based, diluted 1:4 (vol/vol), MicroChemicals) for 4 min. Finally, slides were rinsed with ddH2O and dried with a nitrogen stream. Chromium was deposited onto these slides by evaporation at 22–33 mA at a rate of ~1 Å s−1 to a final thickness of ~30 nm. After chromium deposition, the photoresist was lifted off in acetone with sonication for 5 min in a sonicator bath. Afterwards, slides were rinsed with isopropanol and dried with a nitrogen stream. SLBs were formed on the chromium-patterned slides as described above.

### Self-organization assay

Self-organization assays were performed essentially as described in detail in ref. 61. In short, they were performed on preformed SLBs in 200 μl of reaction buffer (25 mM Tris-HCl pH 7.5, 150 mM KCl, 5 mM MgCl2) supplemented with 2.5 mM Mg-ATP (stock: 100 mM ATP, in 100 mM MgCl2, adjusted to pH 7.5) and at a constant room temperature of 23 °C. MinD was typically used at 1 μM (0.3 µM EGFP-MinD, 0.7 µM MinD) with either 1.5 μM MinE-His16 to generate quasi-stationary labyrinth patterns or 5 μM His-MinE7,13 to generate regular travelling surface waves. Time series showing the initial development of patterns were acquired by starting the self-organization with Mg-ATP directly before image acquisition. Tile scans used for quantification of quasi-stationary patterns or gradients induced by regular travelling waves on unconstrained and patterned SLBs were acquired 30–60 min after the start of self-organization, in areas of the chamber that had not been imaged previously.

### Single-particle tracking

Single-particle tracking of DNA origami was conducted at a concentration of DNA origami and anchors that can be described as diluted, so that interaction between individual DNA origami was minimized68. DNA origami was diluted in DNA LoBind tubes (Eppendorf) and added to a chamber at a final concentration of 0.1–1 pM (biotin-functionalized origami, SLB with 70/30/0.01 mol% DOPC/DOPG/biotinyl-CAP-PE; cholesteryl functionalized origami, SLB with 70/30 mol% DOPC/DOPG, 0.1 nM 5′ TEG-cholesteryl-functionalized oligonucleotides). Owing to the superior brightness and photostability, single-particle tracking was exclusively performed using Cy3B-labelled DNA origami. To further reduce photobleaching and blinking as well as photopolymerization of MinD, single-particle tracking was performed in the presence of an oxygen scavenger system (3.7 U ml−1 pyranose oxidase, 90 U ml−1 catalase, 0.8% glucose)69 and trolox. Experiments were performed in the absence of any protein or in the presence of 1 µM MinD (30% EGFP-MinD) and 2.5 mM ATP to emulate conditions in the MinD maxima in a simplified fashion. For single-particle tracking of cargo in the presence of MinDE travelling waves, 1 µM MinD (30% EGFP-MinD) and 5 µM His-MinE were used and tracking was performed more than 30 min after starting the reaction with 2.5 mM ATP to obtain regular waves.

### Microscopy

All images, except for single-particle tracking, were taken on a Zeiss LSM780 confocal laser scanning microscope using a Zeiss C-Apochromat ×40/1.20 water-immersion objective, ×20 air objective for chromium-patterned SLBs or ×60/1.4 differential interference contrast (DIC) oil-immersion objective for S. pombe cells (all Carl Zeiss). Longer time series were acquired using the built-in autofocus system. All two- or three-colour images were acquired with alternating illumination for the 488/633 nm and 561 nm laser lines to avoid crosstalk. EGFP-MinD or sfGFP-MinD was excited using a 488 nm argon laser; Cy3B-labelled origami, Alexa568-streptavidin or mCherry in S. pombe were excited using a 561 nm diode-pumped solid-state (DPSS) laser and Cy5-labelled origami using a 633 nm He–Ne laser. Images were typically recorded with a pinhole size of 2.6–4 Airy units for the EGFP and origami channels, 1 Airy unit for the streptavidin channel, 512 × 512-pixel resolution and a pixel dwell time of 1.27 μs. Time series were typically acquired with ~14-s intervals for unconstrained SLBs or 4-s intervals for chromium-patterned SLBs and S. pombe cells. For single-particle tracking of DNA, origami images were acquired on a custom-built total internal reflection fluorescence microscope (TIRFM)70 using a NIKON SR Apo TIRF ×100/1.49 oil-immersion objective, constructed around a Nikon Ti-S microscope body (both Nikon). Two laser lines (490 nm (Cobolt Calypso, 50 mW nominal) and 561 nm (Cobolt Jive, 50 mW nominal), Cobolt AB)) were controlled in power and timing (AOTF, Gooch & Housego TF-525-250) and spatially filtered (kineFLEX-P-3-S-405.640-0.7-FCS-P0, Qioptiq). The beam was further collimated, expanded (10×) and focused on the objective’s back aperture by standard achromatic doublet lenses. The TIRF angle was controlled by precise parallel offset of the excitation beam (Q545, PI). The emission light was notch-filtered to remove residual excitation light, spectrally separated by a dichroic beamsplitter (T555lpxr-UF1, Chroma Technology Cooperation), bandpass-filtered 525/50 and 593/46 (both Chroma), respectively, and repositioned on two halves of the electron-multiplying charge-coupled device (EMCCD) camera (Andor iXon Ultra 897, Andor Technologies). Images were recorded with Andor Solis software (version 4.28, Andor Technologies).

### Image analysis

All images were processed using Fiji (version v1.52p), MATLAB (R2018a, The Math-Works) or Python (Python Software Foundation). Brightness or contrast adjustments of all displayed images were applied homogeneously.

For line plots, the images were smoothed with a Gaussian filter with pixel width of 2 in Fiji.

For the cross-correlation plots, the images were smoothed with a Gaussian kernel of pixel width 1. The theoretical models were formulated as boundary-value problems and solved in a one-dimensional geometry using a finite-difference scheme using SciPy71. Curve fitting was performed with lmfit72.

### Single-particle tracking analysis

Analysis of single-particle tracking was conducted as described previously using previously established code73. In brief, a custom-written MATLAB code was used to detect DNA origami fluorescence in each frame and extract its position. Origami trajectories on the membrane were analysed using jump-distance analysis74,75. The distances between particle locations between subsequent frames were analysed and diffusion coefficients of particle ensembles were obtained by fitting the cumulative histograms. As, usually, some of the origami in the field of view were immobile and did not diffuse, cumulative histograms of obtained jump distances were fitted with two components, where for the second component the upper boundary was set to 0.1 µm2 s−1, and usually resulted in diffusion coefficients of less than 0.01 µm2 s−1.

### Analysis of MinDE-dependent transport

Analysis of fluorescence intensities and contrast was essentially performed as described earlier7. In brief, tile scans were imported into Fiji, where the EGFP-MinD channel was used for segmentation to generate a binary mask of the patterns. The original non-modified images from the two or three spectral channels were analysed based on the binary mask using a custom-written MATLAB code. The average fluorescence intensity in the Alexa568-streptavidin or origami-Cy5 and EGFP-MinD spectral channel was obtained by pooling the means of individual images from one independent experiment. All means from one independent experiment and condition were pooled together. All fluorescence intensity values from one experimental set were normalized to the fluorescence intensity values obtained for the respective origami with one anchor. The Michelson contrast of the resulting cargo patterns was calculated for every individual image as the difference between the average intensity in the MinD minima and MinD maxima divided by the sum of the average intensities in the MinD maxima and minima. The contrast of the MinD patterns was calculated for every individual image as the difference between the average intensity in the MinD maxima and MinD minima divided by the sum of the average intensities in the MinDE minima and maxima.

### Analysis of fluorescence profiles on chromium-patterned SLBs

Time-series from chromium-patterned SLB patches with regular travelling MinDE waves that aligned along the longest axis of the patch were selected and averaged in Fiji. The resulting average intensity of the cargo was plotted along the wavevector of the MinDE waves (that is, the longest axis of the patch) and exported as csv files. Subsequently, using MATLAB, the fluorescence intensity profiles were aligned at the beginning based on the increase of the signal, and the average profile was generated.

### Reporting Summary

Further information on research design is available in the Nature Research Reporting Summary linked to this Article.