At its core, what you're seeing in all of these steps are molecular interactions - neurons fire to the rate they do because they build up sodium, potassium, or calcium ions; different chemical signaling chains are what prompt the transcription of given genes; charge affinities between the amino acids and with their environment are what create the shapes of proteins and give them their properties and capabilities. Effectively, each atom - each electron on each atom - is affecting these interactions, and that's what's driving the whole system.
At the molecular level, the properties of a molecule are (effectively) determined by their structure and charge distribution - the atomic composition of the molecule, where are the electrons likely to congregate, which bonds are stronger or weaker, and where are atoms likely to be able to be added or removed. These affect how the molecule reacts with other molecules, and each reaction that changes either the structure or the charge distribution changes how the molecule will react going forward.
So, the computation model is effectively a physical/structural one - how do these structures meet, compare, and combine, played out over trillions of interactions and interaction chains.
(I'm consciously ignoring the quantum side, because A) I don't understand it well enough and B) the structure/charge lens seems to be basically sufficient.)
It's worth taking a look at some of the cell chemical pathways to see how some of this plays out - take a look at things like the Krebs cycle, which are basically steps of interactions in which a molecule or several are modified step by step in a series of splits, joins, adds, and subtractions that allow for the next step.
Part of what makes this tricky is that, while you can "zoom out" and focus on larger systems like neurons or genomes, the molecular interaction model shows up all over the place - neurons fire to the degree they do because of charge accumulation, DNA & RNA transcription are strongly affected by Weird Molecular Interactions, protein folding is at least partially a product of charge affinities, enzymes work as they do because of structure and charges. This is why a lot of these problems are enormously computationally difficult - it's a physical system, not a logical one - there's no way to isolate any layer from any other layer.
(I deeply welcome corrections on any of the above, by the way - I've spent time reading on all of this, but I'm not a professional by any stretch. The above is the model I've acquired over time, not reality.)