## The stumbling block of geometry

Recall from Part 2 how respiration works. Redox reactions generate a proton gradient across a membrane, which is then used to power the synthesis of ATP. An intact membrane is necessary for energy generation. Eukaryotic cells use the inner mitochondrial membrane to generate ATP, while bacteria, which do not have organelles, must use their external cell membrane.

The limitation for bacteria is geometric. For simplicity, imagine a bacterium shaped like a cube, then double its dimensions. A cube has six sides, so if our cubic bacterium had dimensions of one thousandth of a millimetre each way (1 ^m), doubling its size would quadruple the surface area, from 6 ^m2 (1 X 1 X 6) to 24^m2 (2 X 2 X 6) ^m2. The volume of the cube, however, depends on its length multiplied by its breadth by its depth, and this rises eightfold, from 1 ^m3 (1 X 1 X 1) to 8 ^m3 (2 X 2 X 2). When the cube has dimensions of 1 ^m each way, the surface area to volume ratio is 6/1 = 6; with dimensions of 2 ^m each way, the surface area to volume ratio is 24/8 = 3. The cubic bacterium now has half as much surface area in relation to its volume. The same thing happens if we double the dimensions of the cube again. The surface area to volume ratio now falls to 96/64 = 1.5. Because the respiratory efficiency of bacteria depends on the ratio of surface area (the external membrane used for generating energy) to volume (the mass of the cell using up the available energy) this means that as bacteria become larger their respiratory efficiency declines hyperbolically (or more technically, with mass to the power of 2/3, as we'll see in the next Part).

This decline in respiratory efficiency is coupled to a related problem in absorbing nutrients: the falling surface area to volume ratio restricts the rate at which food can be absorbed relative to the requirement. These problems can be mitigated to some extent by altering the shape of the cell (for example, a rod has a larger surface-area-to-volume ratio than a sphere) or by folding the membrane into sheets or villi (as in our own intestinal wall, which is subject to the same need to maximize absorption). Presumably, however, there comes a point when complex shapes are selected against, simply because they are too fragile, or too difficult to replicate with any accuracy. As any spatially challenged plasticine modeller knows, an imperfect sphere is much the most robust and replicable shape. We aren't alone: most bacteria are spherical (cocci) or rod-like (bacilli) in shape.

In terms of energy, a bacterial cell with double the 'normal' dimensions will produce half as much ATP per unit volume, while being obliged to divert more energy towards replicating the cellular constituents, such as proteins, lipids, and carbohydrates, that make up the extra cell volume. Smaller variants, with smaller genomes, will almost invariably be favoured by selection. It is therefore hardly surprising that only a handful of bacteria have achieved a size comparable with eukaryotes, and these exceptions merely prove the rule. For example, the giant sulfur bacterium Thiomargarita namibiensis (the 'sulfur pearl of Namibia'), discovered in the late 1990s, is eukaryotic in size: 100 to 300 microns in diameter (0.1 to 0.3 mm). Although this caused some excitement, it is actually composed almost entirely of a large vacuole. This vacuole accumulates raw materials for respiration, which are continually washed up and swept away by the upwelling currents off the Namibian coast. Their giant size is a shamâ€”they amount to no more than a thin layer covering the surface of a spherical vacuole, like the rubber skin of a water-filled balloon.

Geometry is not the only stumbling block for bacteria. Think again about proton pumping. To generate energy, bacteria need to pump protons across their external cell membrane, into the space outside the cell. This space is known as the periplasm, because it is itself bounded by the cell wall.1 The cell

1 Technically the periplasm refers to the space between the inner and outer cell membranes of Gram-negative bacteria. These are named after the way in which they are coloured by a particular stain known as the Gram stain. Bacteria that are coloured by this stain are called Gram-positive; bacteria that are not stained are called Gram-negative. This odd behaviour wall presumably helps to keep protons from dissipating altogether. Peter Mitchell himself observed that bacteria acidify their medium during active respiration, and presumably more protons are free to disperse if the cell wall is lost. Such considerations may help to explain why bacteria that lose their cell wall become fragile: they not only lose their structural support but also lose the outer boundary to their periplasmic space (of course they retain the inner boundary, the cell membrane itself). Without this outer boundary, the proton gradient is more likely to dissipate, at least to some extentâ€”some protons appear to be 'tethered' to the membrane by electrostatic forces. Any dispersal of proton gradient is likely to disrupt chemiosmotic energy production: energy is not produced efficiently. As energy production runs down, all other aspects of a cell's housekeeping are forced to run down too. Fragility is the least of what we would expect; it's more surprising that the denuded cells can survive at all.