Mitochondrial cristae modeled as an out-of-equilibrium membrane driven by a proton field

As the places where most of the fuel of the cell, namely, ATP, is synthesized, mitochondria are crucial organelles in eukaryotic cells. The shape of the invaginations of the mitochondria inner membrane, known as a crista, has been identified as a signature of the energetic state of the organelle. However, the interplay between the rate of ATP synthesis and the crista shape remains unclear. In this work, we investigate the crista membrane deformations using a pH-dependent Helfrich model, maintained out of equilibrium by a diffusive flux of protons. This model gives rise to shape changes of a cylindrical invagination, in particular to the formation of necks between wider zones under variable, and especially oscillating, proton flux.

Figure 1

Cross section of a mitochondrion in different states of ATP production [2]. (a) State III, for which sugar in excess is available leading to a high rate of ATP production. (b) State IV, for which no sugar is available leading to a vanishing ATP production rate. Mitochondria (left to right) have diameters of 1500 nm and 500 nm.

Figure 2

Schematic representation of a crista. (a) The plots represent the proton concentration on the surface in states III and IV. The tubes represent the shape of the invagination in state III and IV. (b) Deformation fields of the membrane and intrinsic basis of the deformed surface.

Figure 3

Dynamics of the proton field along the cristae. The plots represent the field $h(s,t)$ solution of Eq. (3) for parameter values given in Table 1 and for $t_{\mathrm{IV}}=0\left[T\right]$, $t_{\mathrm{IV}\to \mathrm{III}}=T/4\left[T\right]$, $t_{\mathrm{III}}=T/2\left[T\right]$, $t_{\mathrm{III}\to \mathrm{IV}}=3T/4\left[T\right]$. The concentration $h(s,t)$ is expressed in proton per nm.

Figure 4

(a) Stability diagram of a Helfrich cylinder as a function of $X=C_{00}R$. (b) 1. $G_n\left(s\right)/R$, Normalized Green functions for different values of $X=-0.9,0,0.9$ and for $\delta \sigma =0.15$ and $\delta C_0=0$. 2. Characteristic lengths of oscillation ${\lambda }_0$ and decay ${\lambda }_d$ as a function of $X$. 3. Maximal amplitude of the deformations driven respectively by a variation of $\sigma$, $A_{n\sigma }$ and by a variation of $C_0$, $A_{nC}$. (c) Three-dimensional representations of a cylinder (top) deformed by $h\left(s\right)=\delta \left(s\right)$ for ($X=0.96$, $\delta \sigma =0.2$, $\delta \kappa =0$, $\delta C_0=0$) (middle), and for ($X=0.96$, $\delta \sigma =-0.2$, $\delta \kappa =0$, $\delta C_0=0$) (bottom). All quantities are dimensionless.

Figure 5

Infinite cylinder submitted to a step of proton concentration. (a) Proton concentration $h\left(s\right)$ along the cylinder. (b) Deformation field $u_n\left(s\right)$ induced by a step of proton for $X=0.9$, $\delta \sigma =0.1$, $\delta C_0=-0.1$. (c) Three-dimensional representation of the corresponding shape. All quantities are dimensionless.

Figure 6

Model crista oscillating between states IV and III. A finite membrane tube is submitted to the proton field $h(s,t)$ represented in Fig. 3. (a) Representation of the shape of the cylinder at different times of the period (the same ones as in Fig. 3). (b) Deformation fields $u_n(s,t)$ and $u_s(s,t)$ associated with these shapes (same colors as in Fig. 3). The plots are obtained for values given in Table 1, $X=0.8$, $\delta \sigma =-0.35\sigma$ per $h(s,t)$, $\delta C_0=1.05$ per $h(s,t)$.