Modeling the Interplay of Oscillatory Synchronization and Aggregation via Cell-Cell Adhesion

TL;DR

A linear stability analysis of the incoherent, spatially uniform state of the ‘clocks’ lets us classify possibly emerging patterns depending on model parameters, and determines a range of possible far-from equilibrium patterns when baseline adhesion strength is zero.

Abstract

We present a model of systems of cells with intracellular oscillators (`clocks'). This is motivated by examples from developmental biology and from the behavior of organisms on the threshold to multicellularity. Cells undergo random motion and adhere to each other. The adhesion strength between neighbors depends on their clock phases in addition to a constant baseline strength. The oscillators are linked via Kuramoto-type local interactions. The model is an advection-diffusion partial differential equation with nonlocal advection terms. We demonstrate that synchronized states correspond to Dirac-delta measure solutions of a weak version of the equation. To analyze the complex interplay of aggregation and synchronization, we then perform a linear stability analysis of the incoherent, spatially uniform state. This lets us classify possibly emerging patterns depending on model parameters. Combining these results with numerical simulations, we determine a range of possible far-from equilibrium patterns when baseline adhesion strength is zero: There is aggregation into separate synchronized clusters with or without global synchrony; global synchronization without aggregation; or unexpectedly a ``phase wave" pattern characterized by spatial gradients of clock phases. A 2D Lattice-Gas Cellular Automaton model confirms and illustrates these results.

Figures (9)

• Figure 1: Graphs of the functions $F_2(k^*)$ (red, solid) and $G_2(k^*)$ (blue, dashed); see Proposition \ref{['linearization']}.
• Figure 2: Graphical representations of the six pattern onsets listed in Table \ref{['tabonsets']} in one spatial dimension. For each pattern onset, the top panel shows a heat map of the linearized patterns $\Delta R(x,\theta)$ in $x-\theta-$ space, the bottom panel shows the corresponding pattern in the total spatial density $\Delta R_{\rm{tot}}(x)=\int_0^{2\pi}\Delta R(x,\theta)\,d\theta$. For presentation purposes, we scaled densities to take values between $-1$ and $+1$. Here the length of the spatial interval is $L=10$. For AI, AGS and ALS, we chose $k_1=\frac{3\pi}{5}$. Additionally for ULS, $k_2=\frac{3\pi}{5}$ and for ALS, $k_2=\frac{\pi}{5}$.
• Figure 3: Graphical representation of the result in Proposition \ref{['propjk']}. Note that the images are slightly different in one and two spatial dimensions, although the overall appearance is very similar.
• Figure 4: Phase diagrams of onsets of patterning in $J^*K^*-$space. We show the case of $d=1$ spatial dimension; the case $d=2$ is very similar (Figure \ref{['figjk']}). See Table \ref{['tabonsets']} for the classification of the different pattern onsets. Results are based on Propositions \ref{['propj0']} and \ref{['propjk']}. Left diagram: Case $J_0^*<J^1_{\rm{crit}}.$ In this case, there is no spatial aggregation and pattern onsets are nonconstant in clock space only. The three possibilities are uniform incoherent (UI), uniform locally synchronized (ULS) and uniform globally synchronized (UGS). Right diagram: Case $J_0^*>J^1_{\rm{crit}}.$ In this case, spatial aggregation occurs simultaneously with patterning in clock space only. The three possibilities are aggregated incoherent (AI), aggregated locally synchronized (ALS) and aggregated globally synchronized (AGS). For the reader's convenience, we include the example densities patterns in $x\theta$-space from Figure \ref{['figonsets']}.
• Figure 5: Phase diagram of far-from-equilibrium patterning in one spatial dimension for Equation (\ref{['diffeq']}) in $J^*K^*-$space (with $J^*=0$). This is based on the onset diagram in Fig \ref{['figjk']} (left panel). The horizontal line on the $K^*>0$-half plane corresponds to the critical value $J^d_{\rm{crit}}$ in $d=1$ dimension. For example numerical computation for the parameters indicated by Roman numerals, see Figure \ref{['fig_example_comp']}. Specifically, $(J^*,K^*)$ values are: I $(-1,-1)$, II $(0.4,-1)$, III $(1,-1)$, IV $(-1,1)$, V $(0.05,1)$, VI $(0.4,1)$, VII $(1,1)$.

Theorems & Definitions (15)

• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Proposition 4.1
• Proposition 4.2
• proof
• Proposition 4.3
• proof
• Proposition A.1