Diffusive Spreading Across Dynamic Mitochondrial Network Architectures

TL;DR

This work presents a unifying framework for the dispersion of material within temporal networks of spatially embedded units that span across a broad connectivity range and provides a quantitative basis for predicting the homogenization of biomolecules through a mitochondrial population.

Abstract

In eukaryotic cells, mitochondria form networks that range from highly fused interconnected structures to fragmented populations of individual organelles that undergo transient interactions. These structures can be described as temporal networks of physical units, whose dynamic topology is determined by fusion, fission, and motion of the mitochondria through intracellular space. The heterogeneity of the mitochondrial population is governed by diffusive transport and inter-unit exchange of proteins, lipids, ions, and RNA within these networks. We present a unifying framework for the dispersion of material within temporal networks of spatially embedded units that span across a broad connectivity range. Specifically, we consider filling of the networks with a locally produced but globally consumed material, demonstrating that the steady-state content is determined by the balance of timescales for spatial encounter between clusters, local fusion, fission, and diffusive transport within a cluster. As the connectivity increases, filling behavior transitions from three-dimensional spread through a `social network' limited by cluster interactions to low-dimensional transport through a largely stationary `physical network' limited by material diffusivity. We extract parameters for mitochondrial networks in three human cell lines, demonstrating that different cells can access both the social and the physical network regimes. These results provide a quantitative basis for predicting the homogenization of biomolecules through a mitochondrial population. Our framework unifies a variety of temporal network structures into an overarching theory for transport through populations of interacting and interconnected units.

Figures (5)

• Figure 1: Dynamic network simulation framework. (a) Schematic of network formation by interacting units which diffuse through 3D space with diffusivity $D_1$. Nearby units can undergo tip-tip and tip-side fusion with rates $k_{u1},k_{u2}$ while connected nodes undergo fission at rate $k_f, \frac{3}{2}k_f$, respectively. Material spreads along connected units with diffusivity $D_p$ and decays over time with rate $k_d$. Red indicates the concentration of material in a given unit. (b) Snapshots of network filling from a single source unit, at steady state. (i) Fragmented network, ($k_{u1}/k_f = 30$) (ii) Network near percolation transition, with kinetic parameters appropriate to mammalian mitochondrial networks holt2024spatiotemporal ($k_{u1}/k_f = 1000$). (iii) Hyperfused network ($k_{u1}/k_f = 3000$). The ratio of tip-tip and tip-side fusion is set to $k_{u1} = 3k_{u2}$, the dimensionless particle diffusivity is $D_p = 4800$, and the dimensionless material decay rate is $k_d=0.32$ throughout.
• Figure 2: Filling of static networks depends on network dimensionality. Two different sets of networks are considered: near-linear structures (blue) with $k_{u2} = 0.01k_{u1}$ and highly-branched structures (green) with $k_{u2} = 10k_{u1}$. The simulations are run to steady state, and the network structures are then frozen. (a) Steady-state material content supplied by a source unit on the static networks is plotted as a function of the diffusive lengthscale, $\lambda_p=\sqrt{D_p/k_d}$. Solid curves show exact solution (details in SI Appendix), averaged over replicate snapshots. Dashed-circle curves show approximation with the linear motif (inset i), applicable at short $\lambda_p$. Dashed-square curves show continuum solution on a fractal domain (inset ii), applicable at long $\lambda_p$. (b) Calculation of the graph dimension for a single instance of each network type (shown in inset). The number of nodes within a given graph distance is plotted against the graph distance on log-log axes, with the slope giving the intrinsic dimension $d$. Averaging over 21 snapshots from 3 independent simulations yields effective dimensionalities $d = 1.2, d=1.9$, respectively.
• Figure 3: Steady-state material content supplied by a source unit in a system of fragmented clusters. Solid lines show mean-field solutions (Eq. \ref{['eq:meanfieldsoln']}) with unit cluster size $\langle n \rangle=1$, for different fusion rates $k_{u}$. Colored dots show explicit simulation results for a simplified system of interacting spheres with uniform size (inset). Dashed black lines show the limiting behavior for fusion-limited and encounter-limited regimes.
• Figure 4: Spreading on simulated networks exhibits a transition between the static network and social network regime. The steady-state material content is plotted as a function of the decay time for different values of the fusion rate constant $k_{u1}$ (solid lines), which tunes between fragmented networks (blue), networks at the percolation transition (light green) and highly fused networks (yellow). The analytic approximation (Eq. \ref{['eq:slowdp']}, dashed curves) encompases both the fractal continuum limit (for large clusters or fast decay) and the social network limit (for small clusters and slow decay). Plateaus at intermediate decay times correspond to filling of an individual cluster before interactions can occur. Insets show example simulation snapshots for the parameters indicated by the gray arrows. Parameters $N_0 = 250, \ell_0 = 0.5, R=5, k_{u2}=\frac{1}{3}k_{u1}, k_f=1$ are used in both (a) and (b), with the particle diffusivity set to $D_p=4800$ in (a) and $D_p=48$ in (b).
• Figure 5: Extracted structures of mammalian mitochondrial networks exemplify both the static network and social network regimes. (a) Representative raw, segmented, and skeletonized images (top to bottom) for mitochondria from SH-SY5Y (human neuroblastoma), IMR90 (human fibroblast), and U2OS (human osteosarcoma) cells. Segmented images show each connected cluster with a different color. Scale bar: $10\mu\text{m}$. (b) Time to fill half of the mitochondrial network with material supplied from a fixed source, as predicted by the analytic model, is plotted as a function of the material diffusivity for each cell type. Solid curves correspond to a 'prototypical cell', where parameters are averaged within each cell type before plugging into the analytic model. Dashed lines indicate results for individual cells. (c) The distribution of cells by mean cluster size fraction (mean cluster length divided by total mitochondrial network length). The dashed vertical line indicates the threshold ($0.29$) to classify networks as disconnected or hyperfused, as determined by k-means clustering with k=2. (d) Network filling half-times as in (b), with cells grouped by the mitochondrial network class (disconnected or hyperfused). Solid curves are obtained by averaging half-times across individual cells within each class with shaded regions indicating the standard error of the mean. Individual cell results are shown as dashed lines.