Emergence of the mitochondrial reticulum from fission and fusion dynamics

Abstract

Mitochondria form a dynamic tubular reticulum within eukaryotic cells. Currently, quantitative understanding of its morphological characteristics is largely absent, despite major progress in deciphering the molecular fission and fusion machineries shaping its structure. Here we address the principles of formation and the large-scale organization of the cell-wide network of mitochondria. On the basis of experimentally determined structural features we establish the tip-to-tip and tip-to-side fission and fusion events as dominant reactions in the motility of this organelle. Subsequently, we introduce a graph-based model of the chondriome able to encompass its inherent variability in a single framework. Using both mean-field deterministic and explicit stochastic mathematical methods we establish a relationship between the chondriome structural network characteristics and underlying kinetic rate parameters. The computational analysis indicates that mitochondrial networks exhibit a percolation threshold. Intrinsic morphological instability of the mitochondrial reticulum resulting from its vicinity to the percolation transition is proposed as a novel mechanism that can be utilized by cells for optimizing their functional competence via dynamic remodeling of the chondriome. The detailed size distribution of the network components predicted by the dynamic graph representation introduces a relationship between chondriome characteristics and cell function. It forms a basis for understanding the architecture of mitochondria as a cell-wide but inhomogeneous organelle. Analysis of the reticulum adaptive configuration offers a direct clarification for its impact on numerous physiological processes strongly dependent on mitochondrial dynamics and organization, such as efficiency of cellular metabolism, tissue differentiation and aging.

Figure 1. Large-scale structure of mitochondria.

(A) Confocal micrograph (cross-section through cellular body) visualizing the mitochondrial network (scale bar is 1.3 µm) in a HeLa cell. (B) Result of thresholding and skeletonization of the reticulum in (A). Upper rectangle: decomposition of the reticulum into a set of interconnected linear segments. Lower inset: positions of detected degree k = 1 (magenta) and 3 (blue) network nodes. (C) Graph representation of the mitochondrial reticulum using three node types: k = 1 (magenta), 2 (green) and 3 (blue). The reticulum can be represented as a set of linear segments consisting of one or more edges (black rods) liking the nodes. (D) Two types of elementary network transformations comprising the reticulum fission/fusion reactions. (E) Segment types present in the network: (1-1) separate open-end segments, (2-2) separate loop segments, (1-3) surface and (3-3) internal segments of branched clusters.

Figure 2. Steady state solutions ( Eq. 3 ) of differential-algebraic model of the mitochondrial reticulum ( Eq. 2 ).

Numbers x_k of network nodes of degrees k = 1,2,3 are plotted as a function of relative intensities of fusion and fission for a mitochondrial reticulum of size L = 3·10⁴ approximately corresponding to a HeLa cell line. A unique reticulum configuration characterized by the number of mitochondrial tips x ₁ (red), bulk nodes x ₂ (green), and branching sites x ₃ (blue) corresponds to each value of the ratio between fusion and fusion rates for tip-to-tip (c ₁) and tip-to-side (c ₂) processes. Schematic drawings of representative network configurations are shown in the insets, along with their approximate location in the parameter space. Unlike the fragmented or hyperfused networks resulting from disproportionate fusion and fission activities, in the physiologically relevant parameter range the reticulum configuration is heterogeneous and especially sensitive to changes in c ₁ and c ₂.

Figure 3. Chondriome segment lengths (L = 3·10⁴).

(A) Examples of segment length distributions for different values of tip-to-tip and tip-to-side fusion/fission rates c ₁ and c ₂ respectively (stars: c ₁ = 0.1; open circles: c ₁ = 10; red color: c ₂ = 5.0·10⁻⁵; blue color: c ₂ = 5.0·10⁻⁶) in the agent-based representation. The same data plotted using semi-log scaling (inset) highlight deviations from the geometrical decay in short segments due to influence of loop structures. (B) Distributions for loops (light gray diamond), open segments (gray square) and total (dark gray star) for the simplified network (c ₂ = 0) as calculated by the stochastic algorithm, compared to the exact analytical results: Dotted and dashed lines are contributions from loops and open segments, respectively, according to Eq. 5 (after normalization). (C) Markers - mean values differentiated by segment types as determined by the agent-based simulations: (1-1) - plus; (1-3) - diamond; (2-2) - cross; (3-3) - circle. Solid line: from the deterministic model. Dotted and dashed lines: Limiting values at c ₂ = 0 (Eq. 5) for sizes of loops (2-2) and open-ended (1-1) segments respectively.

Figure 4. Sizes of disconnected mitochondrial clusters.

(A) Distribution of cluster sizes in the vicinity of percolation transition (c ₁ = 0.1, c ₂ = 2.2·10⁻⁵, L = 3·10⁴) from agent-based simulation (stars) superimposed with n(j), Eq. 7 (black solid line). Contributions from 1-segment (blue diamonds, r = 1), 3-segment (green circles, r = 3) and 5-segment (red crosses, r = 5) clusters are shown separately together with corresponding fits, Eq. 6 (colored dashed lines). (B) Typical configurations of clusters for r = 1, 2 and 3 segments (for clarity, bulk nodes are not shown). (C) Scattered plots of cluster size j vs. number of segments per cluster for different c ₂ (legend) and c ₁ = 10, L = 3·10⁴. Note the elevated spread of cluster sizes in the vicinity of the critical point (yellow, green). The scatter slopes for various c ₂ reflect change in the mean segment length.

Figure 5. Percolation transition in the mitochondrial network for c ₁ = 0.1, L = 3·10⁴.

(Upper part) Fractional size of the largest cluster in the system (blue stars) and the corresponding average number of segments per cluster in non-giant clusters (pink circles). The position of the critical point (arrow) corresponds to approximately 30% of largest cluster size fraction. For comparison, the experimentally determined largest cluster size fraction was 35±20% (dashed line). (Bottom part) Qualitative differences between subcritical, critical and supercritical regimes of the reticulum (c ₂ = 1.7·10⁻⁶, 1.9·10⁻⁵ and 5.0·10⁻⁴ respectively, which corresponds to the ratios of end nodes (blue) to branching nodes (yellow) x ₁/x ₃≈31, 2.7 and 0.14 respectively): Although numbers of network edges (overall mitochondrial mass) are the same for the three configurations, the cell-wide connectivity is only possible above the critical value of tip-to-side fusion/fission ratio c ₂≈2.0·10⁻⁵ (for clarity, bulk nodes are not shown and segments (green) are sketched of equal length).