Tensile forces and shape entropy explain observed crista structure in mitochondria

Abstract

We present a model from which the observed morphology of the inner mitochondrial membrane can be inferred as minimizing the system's free energy. In addition to the usual energetic terms for bending, surface area, and pressure difference, our free energy includes terms for tension that we hypothesize to be exerted by proteins and for an entropic contribution due to many dimensions worth of shapes available at a given energy. We also present measurements of the structural features of mitochondria in HeLa cells and mouse embryonic fibroblasts using three-dimensional electron tomography. Such tomograms reveal that the inner membrane self-assembles into a complex structure that contains both tubular and flat lamellar crista components. This structure, which contains one matrix compartment, is believed to be essential to the proper functioning of mitochondria as the powerhouse of the cell. Interpreting the measurements in terms of the model, we find that tensile forces of ∼20 pN would stabilize a stress-induced coexistence of tubular and flat lamellar cristae phases. The model also predicts a pressure difference of -0.036 ± 0.004 atm (pressure higher in the matrix) and a surface tension equal to 0.09 ± 0.04 pN/nm.

Figure 1

Electron tomography of the three mitochondrial volumes. These three volumes were used for measurements that were inserted in the free energy model to obtain thermodynamic parameters in each mitochondrion. (Left) Representative sections of constant z from the tomogram volumes. (Right) Three-dimensional models from volume segmentation and rendering. The successive views rotate about a vertical axis showing different parts of the three-dimensional models. The outer membrane is shown in translucent blue, the inner membrane in white and cristae in various colors. (a and b) Normal mitochondria from HeLa cells where dual-axis tomography enabled volume reconstruction of the mitochondria. (c) Normal mitochondrion from mouse embryonic fibroblast cells where dual-axis serial tomography on four successive sections enabled full volume reconstruction of the mitochondrion. Scale bar, 250 nm.

Figure 2

The simplified geometry of a crista membrane that is used to set up the equations of the free energy model. There are three sections of easily characterized geometry: 1), a flat lamellar section at the center, with radius R and thickness 2r; 2), N tubular membranes, with radius r and length L; and 3), a semicylindrical membrane with radius r that wraps around the edge of the lamellae.

Figure 3

Comparison of predicted (shaded) and measured (solid) values of N, R, and L for the second mitochondria. The predicted values are obtained from the model for Δp = −0.034 atm, with σ = 0.058 pN/nm.

Figure 4

Tensile force f as a function of the size of a crista ℓ as predicted by our model for Δp = −0.033 atm and σ = 0.058 pN/nm. The constancy of f for larger ℓ values is a crucial part of our argument that we can view the tube-lamella coexistence as a phase equilibrium. Calculations are performed for N = 12, 15, and 18. The curves are parameterized by the tube length L. The upper branches correspond to smaller values of L than we have observed. (Inset) Tube radius r as a function of ℓ. Note that at low ℓ, r decreases, which causes f to increase.