Note how calcium is ejected from the cell in the lower southwest portion of the figure. This is why Chapter 10 of A World from Dust talks about calcium signals that are instigated by opening up calcium doors in the membrane and letting the calcium flow in. A NET begins with such a calcium influx, like many other signals. Calcium floods the cell and uses its unusual charge to aggregate and reshape proteins, reconfiguring the cell in myriad ways, resulting in the formation of the NET.
But then something remarkable happens: calcium's chemistry builds a NET. The cell just turns itself inside out, and a NET forms. The reason why this works is because the NET is made from DNA, and when DNA is ejected into the high calcium concentrations outside the cell, the sticky calcium binds phosphate in DNA, crosslinking it into a dense net that engulfs and immobilizes the germs. I'm reminded of Spiderman on the Electric Company (see 2:21):
So because of the imbalance of calcium outside the cell, all it takes to catch a germ is for one cell to play Spiderman, turn itself inside out, and spill its DNA. Calcium automatically solidifies DNA and the immune system uses a fundamental aspect of biochemistry to make an automatic net. Long ago, calcium was ejected to avoid cross-linking DNA. Now, DNA ejected into the calcium-rich exterior of a cell automatically makes a net.
I'd expect a system like this to evolve wherever water-based life uses a periodic table like ours.