A simple model for conserved intracellular dynamics exhibits multiscale pattern formation, traveling protein domains and arrested coarsening of lipids in the membrane

Abstract

We model the spatiotemporal dynamics of cellular protein concentrations near membranes composed of different lipids using a three-variable continuum model for membrane-bound protein, cytosolic protein, and the local composition of a binary lipid membrane. The model contains two globally conserved quantities: the total protein content and the average fractions of the two lipid species. It combines a conserved reaction-diffusion model for protein dynamics with a Cahn-Hilliard equation for lipid demixing. Linear stability analysis of the homogeneous steady state and direct numerical simulations show that the lipid dynamics undergoes classical phase separation, whereas the protein dynamics exhibits oscillatory phase separation for intermediate total protein contents, associated with a long-wavelength instability and traveling domains. In parameter regions where both instabilities are present, we find multiscale patterns with larger-scale traveling and rotating protein domains coexisting with smaller-scale stationary lipid domains. In this regime, traveling protein domains coexist with arrested coarsening of stationary lipid domains above a critical coupling. We further show that the main instabilities and phase diagram are well captured by an extension of a recently proposed conserved FitzHugh-Nagumo model for non-reciprocal pattern formation. The extended model consists of two non-reciprocally coupled Cahn-Hilliard equations with different interface tensions, reflecting the distinct physical properties of lipids and proteins. This also explains the observed asymmetry between static lipid patterns and traveling protein patterns.

Figures (7)

• Figure 1: Schematic of the model mechanics. A) The cell membrane's two species of lipid acids (black and white head groups) undergo coarsening. The protein subsystem is composed out of a protein in its active and inactive form. The inactive, fast diffusing species in bulk (labeled B, in lilac) binds with the rate $f_a$ to the cell membrane and becomes active and much less mobile (labeled M, in yellow). A dissociation rate $f_b$, linear in the amount of bound protein, recycles the inactive form. B) Both systems are coupled by first, $f_a$ and $f_b$ dependending on the local lipid composition and second, a shift in the chemical potential of the lipid-lipid interaction that depends on the M-concentration, cf. equations 1-3. The resulting protein pattern, a spot with higher M concentration that is stationary in the uncoupled case, travels with a velocity that depends on the coupling strength of the two systems.
• Figure 2: Linear stability analysis of the coupled system with $\alpha=\beta=0.3$. A-C: Dispersion relations from linear stability analysis of well-mixed initial conditions with different, homogeneous concentrations $(p_0, E_0)$. A: The stationary instability, mediated by a change in $p_0$ with $E_0=B_0+M_0=1.8$ remaining constant. It is associated with the emergence of the Cahn-Hilliard pattern, roughly existing for $p_0 \in [-0.15, 0.85]$. The imaginary part remains zero, whereas the fastest growing mode shifts towards larger k values towards the middle of the instability region, indicated by the positive maximum of the real part at $k\approx2.7 \mu m ^{-1}$. B: An oscillatory instability, associated with the emergence of the APS pattern, mediated by changes in the protein concentration while $p_0=-0.25$ remains constant. The associated FGM remains small, approximately $0.45 \mu m^{-1}$. C: In the middle of the instability region at $(p_0, E_0)=(0.3,2.3)$, both pattern are stable which is indicated by two separate, positive maxima in the dispersion relation. Note that the nonzero imaginary part does not occur at the fastest growing modes, but rather in between the two maxima. There, in the region framed by the two kinks in the curve, the real parts of two eigenvalues fall together. D) The phase diagram obtained from the linear stability analysis by classifying the dispersion relations with respect to the number of occurring maxima in the dispersion relation with $Re>0$ and the presence of oscillatory modes. The empty circles represent the membrane ($p_0$) and protein compositions ($E_0=M_0+B_0$) where only the stationary Cahn-Hilliard pattern forms, filled triangles stand for the existence of only the oscillatory APS pattern and both pattern existing simultaneously is indicated by filled, red diamonds. The transitions for changing either the membrane composition $p_0$ (indicated by A) or the total protein concentration (indicated by B), as well as the exemplary case C correspond to the dispersion relations shown in panels A-C. The supplementary material contains videos with the systematic change of the dispersion curves in the transitions shown here (SM2 - SM5) .
• Figure 3: Phase diagram of the coupled system for $(\alpha=\beta=0.3)$ of size $16.8 \mu m$ with periodic boundary conditions in 2D. A: For different initial conditions of the well-mixed system we obtain a phase diagram for different protein content $E_0=(M_0+B_0) \in [1.5,3.1]$ in increments of $0.1$ and different lipid composition $p_0 \in [-0.5,0.9]$ in increments of $0.1$. The membrane ($p$ as a color gradient from dark gray over orange and yellow to white) demixes into small droplets for $-0.1<p<0.8$. The membrane-bound protein $M$ exists in a homogeneous state of either low or high concentration (blue over white to red overlaid). It forms the bigger APS pattern only in a small region of protein content falling from $E_0\approx 2.6$ to $E_0\approx 2$ over the shown range of lipid composition. Where both pattern coexist, the APS pattern exhibits a finite velocity. B: The time series corresponds to three snapshots of the M species for the state framed in red with $(p_0,M_0+B_0)=(0.2,2.2)$. It wanders as a single front from right to left. Note that due to the coupling to the p-subsystem also the smaller pattern of the demixing membrane is visible as spots.
• Figure 4: Protein pattern velocity investigated in 1D. A: The measured velocity of the M-pattern (left in blue to red color scheme) for low coupling $\epsilon=0.1$ is stochastic in nature, seemingly transiently pinned by the p-pattern. For larger couplings ($\epsilon=0.2$, below), the velocity becomes larger and more regular. Also the coarsening p-pattern is influenced by the M-distribution - small deformations in accordance with the coupled M-pattern can be observed. B: M-pattern velocities, determined from the 1D simulations, for different $\alpha$ and $\beta$. C: The measured velocities are symmetric with respect to $\alpha$ and $\beta$ and therefore collapse on a single curve when plotted over the coupling strength $\epsilon=\sqrt{\alpha\beta}$. Note that for small couplings the measured velocity over a longer time possibly underestimates the characteristic pattern velocity as the motion of the APS pattern is highly stochastic and seems to get pinned by the modulations due to the membrane concentration inhomogeneties. From linear stability, following the arguments of brauns2024nonreciprocal, we find good quantitative agreement with the observed front velocity of the M pattern.
• Figure 5: Snapshots of the domain evolution in the demixing Cahn-Hilliard subsystem describing the lipid membrane composition. Shown are the time points at 10s (top), 100s (middle) and 10000s (bottom) for three different coupling strengths. The uncoupled system (left) exhibits large domains (droplets) of the + phase in a background of - phase. With increasing $\epsilon$ (0.2 in the middle and 0.4 on the right), the coarsening is visibly slowed. A video showing the coarsening behavior in comparison is given in the supplementary materials as SM7 and a systematic quantification of the slowing in figure \ref{['fig:coars']}.