Fig. 2. Distinct lateral and vertical mass-transport modes at different bulk heights.

A Phase diagram for bulk height and E:D ratio showing three types of mass-transport modes, identified by linear stability analysis, that exist in overlapping regimes: lateral instability (green), local membrane-to-membrane instability ("m-to-m'', blue) and local membrane-to-bulk instability ("m-to-b'', magenta). See Supplementary Fig. 3 for representative dispersion relations in the various regimes. Green dot-dashed line marking the transition from chaotic to coherent patterns⁵. A representative example of a chaotic pattern is shown in Supplementary Movie 4 for the parameter combination marked by the red star. Black dots mark the parameters used for the simulations shown in Fig. 1d. Red line: parameter range for adiabatic sweeps shown in Supplementary Fig. 6 demonstrating hysteresis as a signature of multistability. Panels B–D illustrate the mass-transport modes at different bulk heights. The top row shows laterally isolated compartments to illustrate local vertical oscillations due to vertical bulk gradients. The bottom row illustrates the interplay of lateral and vertical mass-transport modes in a laterally extended system. B For low bulk height, the bulk height is too small for significant vertical concentration gradients to form. Hence, a laterally isolated compartment does not exhibit any instabilities (top). In a laterally extended system (bottom), exchange of mass can drive a lateral instability (green arrows); see Supplementary Movie 5. The cartoon of an E. coli cell illustrates that this instability also underlies pattern formation in vivo; see Supplementary Movies 19 and 20. C For bulk heights above H_c, vertical concentration gradients become significant enough to enable vertical membrane-to-membrane oscillations (blue arrows); see Supplementary Movie 6. These oscillations do not require lateral exchange of mass, i.e., they occur in a laterally isolated bulk column (top). The cartoon of an E. coli cell illustrates that the membrane-to-membrane oscillations in a laterally isolated compartment can also be pictured as equivalent to in vivo pole-to-pole oscillations. The laterally extended in vitro system constitutes a continuum of such oscillators (bottom). D For bulk heights larger than the penetration depth of vertical gradients, the top and bottom membrane effectively decouple (see Supplementary Movies 7 and 8). In this regime, which corresponds to the classical in vitro regime, the bulk in-between the membranes acts as an effective protein reservoir that facilitates membrane-to-bulk oscillations.