3D pattern formation of a protein-membrane suspension

TL;DR

This work shows that MinDE pattern-forming proteins can generate robust 3D patterns on a suspension of submicrometer liposomes, despite complete membrane discontinuity. A coarse-grained 3D reaction-diffusion framework reveals that the physical properties of dispersed membranes, encoded by parameters $\alpha$, $\beta$, and $\gamma$, effectively rescale binding and diffusion rates, enabling pattern formation across length scales spanning hundreds to thousands of liposomes. The pattern type and wavelength are controlled by liposome size $R$, concentration $c$, and inter-liposome spacing $d$, with a linear stability analysis predicting phase boundaries and a scaling relation where $\alpha \propto c$, $\gamma \propto 1/R$, and $\beta \sim \mathcal{N}$ with $\beta = \frac{2}{\pi}\alpha R$. These findings demonstrate the robustness of MinDE self-organization in 3D and suggest tunable, programmable platforms for studying out-of-equilibrium biomaterials and intracellular-like patterning beyond native in vivo contexts.

Abstract

Many essential cellular processes, including cell division and the establishment of cell polarity during embryogenesis, are regulated by pattern-forming proteins. These proteins often need to bind to a substrate, such as the cell membrane, onto which they interact and form two-dimensional (2D) patterns. It is unclear how the membrane's continuity and dimensionality impact pattern formation. Here, we address this gap using the MinDE system, a prototypical example of pattern-forming membrane proteins. We show that when the lipid substrate is fragmented into submicrometer-sized diffusive liposomes, ATP-driven protein-protein interactions generate three-dimensional (3D) spatially extended patterns, despite the complete loss of membrane continuity. Remarkably, these 3D patterns emerge at scales four orders of magnitude larger than the individual liposomes. By systematically varying protein concentration, liposome size, and density, we observed and characterized a variety of 3D dynamical patterns not seen on continuous 2D membranes, including traveling waves, dynamical spirals, and a coexistence phase. Simulations and linear stability analysis of a coarse-grained model revealed that the physical properties of the dispersed membrane effectively rescale both the protein-membrane binding rates and diffusion, two key parameters governing pattern formation and wavelength selection. These findings highlight the robustness of Min's pattern-forming ability, suggesting that protein-membrane suspensions could serve as an adaptable template for studying out-of-equilibrium self-organization in 3D, beyond in vivo contexts.

Figures (5)

• Figure 1: The formation of 3D patterns on a dispersed membrane is a multiscale phenomena. (A) 3D reconstruction of dynamical patterns of MinD proteins from confocal microscopy. (B) Confocal z-slice of MinD (green) and MinE (magenta) forming traveling waves on the dispersed membrane. The largest lipid vesicles onto which proteins are bound are visible using fluorescence microscopy. The mean liposomes' radius is $\rm 15\,nm$. (C) Intensity profile of MinD (green), MinE (magenta), and the lipids (blue) along the direction of propagation of the wave (black arrow). MinE is located at the rear of the waves, following MinD. The lipids are spatially homogeneous. (D) Schematic of the chemical reaction network between MinD, MinE, ATP, and the dispersed membrane. The typical wavelength $\lambda$ of the traveling waves is significantly larger than the size R of individual liposomes and the inter-liposome distance $d$.
• Figure 2: Proteins concentrations regulate pattern selection.(A) Homogeneous steady-state, (B) Traveling waves, (C) Coexistence phase, (D) Spiral phase. The top row shows middle plane z-slices of MinD fluorescence intensity (Widefield microscopy), the middle row shows kymographs (space-time plots) of the pattern, and the bottom row shows the normalized MinD fluorescence intensity along a line. (E) Experimental phase diagram where the concentrations of MinD and MinE are changed. Each data point represents two to five replicates. For all those experiments, the liposomes suspension properties are kept constant: mean radius $R = 15\,\rm nm$ and concentration $c = 0.4\, \rm mM$.
• Figure 3: The properties of the liposomes suspension regulate pattern formation.(A)-(B) Experimental MinD fluorescence images for two liposomes radius ((A): $R = 100\,\rm nm$, (B): $R = 15\,\rm nm$), at increasing lipid concentration $c$ (from left to right). Scale bar: $\rm 200\, \mu m$. (C) Phase diagram in ($R$, $c$), where $R$ is the mean liposome's radius and $c$ the lipid concentration. Colored backgrounds represent regions of predominance of each type of patterns. (D) Phase diagram in ($R$, $d$) where $d$ is the typical distance between two liposomes. The blue region indicates the region where no pattern form. The dashed line indicates the critical inter-liposome distance $d^{*}$ above which no patterns form. For (C) and (D), each experiment was replicated two to five times. For all the experiments reported in this figure, proteins concentrations were fixed to $\mathrm{[MinD]} = 5.7\, \mu \mathrm{M}$ and $\mathrm{[MinE]} = 8.25\, \mu \mathrm{M}$.
• Figure 4: Model predictions depend on membrane properties and protein concentrations.(A) ([MinE],[MinD]) phase diagram with $\beta = 1$, $\gamma = 1$, and $\alpha = 1$$\mu\text{m}{^{-1}}$ shows the dominance of wave states (teal circles) with Homogeneous Stationary States (HSS, blue triangles) occupying either low [MinE] or low [MinD] compositions; insets show two typical wave patterns, ($\dag$) plane wave-dominated and ($\ddag$) a state populated with targets (closed white point) and spirals (open white points). (B) ($\gamma$,$\beta$) phase diagram shows the emergence of various stationary patterns (red, yellow, and orange squares) and mixed-mode states (green squares) for ${\text{[MinE]}} = 800,{\text{[MinD]}}=1000$ and $\alpha = 1$$\mu\text{m}{^{-1}}$. In both (A) and (B), the dashed black lines correspond to linear stability analysis (LSA) prediction. (C) LSA predictions (patterns form in the teal volume) as a function of all three control parameters $\alpha,\beta$ and $\gamma$. The map between these three control parameters and the liposome size $R$ and total lipid concentration $c$, $\{\alpha\left(c\right),\beta\left(R,c\right),\gamma\left(R\right)\}$, is shown as the embedded surface with colors indicating stability. The intersection of the unstable region and this surface is shown in (D), where it is parameterized by $R$ and $c$ in experimental units. For all 2D simulation images, the domain width is $50\, \rm \mu m$; for 3D, $11 \rm\mu m$.
• Figure 5: Wavelength depends on dispersed membrane properties.(A) The wavelength $\lambda$ increases when the inter-liposome distance $d$ increases and decreases when the mean liposomes radius $R$ decreases. Orange and red data points respectively correspond to: $R = 15\,\rm nm$ and $R = 100\,\rm nm$. Each data point is averaged over 2-5 replicates. Error bars are smaller than the marker size. (B) Linear stability analysis-predicted wavelength as a function of $\beta^{-1}$ for $\gamma=\{1,10,50\}$, respectfully light blue, blue, and black.