In a recent preprint, Chretien et al suggest that mitochondria are 10 degrees warmer than the surrounding cytosol. Ten degrees across the membrane of a small organelle struck me as an unreasonably large gradient and I had difficulties imagining how such a gradient could be maintained. And indeed, the quick calculation below shows that such a gradient can't possibly be real.
To get a rough idea of the heating power required to maintain a gradient, lets assume a steady state scenario and ignore the fact that mitochondria are curved (which would add constant of order one here and there but doesn't affect the huge discrepancy we will find below). The steady state power \(P\) is then approximately $$ P = \kappa \frac{\Delta T}{w}A $$ where \(\kappa\) is the heat conductance, \(\Delta T = 10^{\circ}K\) is the temperature difference, \(w = 100nm\) is the distance over which temperature changes, and \(A=1 \mu m^2\) is the approximate area of a mitochondrion. A more difficult number to estimate is the heat conductance of the mitochondrial membrane. This paper by Park et al estimates the whole cell cytosolic heat conductance to be \(\kappa = 0.5 W/m/K\). Müller and Müller-Plathe report a roughly similar number for lipid bilayers. Overall, watery environments tend to have a heat conductance of around \(1W/K/m\). Plugging these numbers into the above equation, we find $$ P = \frac{0.5 W}{mK}\frac{10K}{10^{-7}m} 10^{-12}m = 0.5\times 10^{-4}W $$ That is a lot of power for a single cell. Previous studies estimate the typical power consumption of a cell to be around \(10^{-11} - 10^{-10}\)W, i.e., we are off by a factor of at least a million (and hence no need to worry factors of \(2\pi\)).
Let's extrapolate this to the entire body: We have about \(10^{13}\) cells in our body each of which contains dozens of mitochondria. This would correspond to a total power of \(10^{10}\)W or roughly the power of 10 nuclear power stations. Of course only a small fraction of cells operate at full throttle, but nevertheless the gradients reported by Chretien et al are implausible. The authors don't discuss the physical requirements and constraints (no mention of heat conductance in the article).
This rough order of magnitude calculation echoes an earlier critique by Baffou et al who have argued much more generally that intra-cellular temperature differences can't exceed \(10^{-5}K\) and that many reports of large intra-cellular temperature gradients are probably artefacts. Thanks a lot to Anthonie Muller for pointing me to these references in the comment section on the preprint.