I naively thought that mash-in was only producting sugars via the alpha and beta amylase, but it seems (from Briggs) that total nitrogen (TN; protein), soluble nitrogen (orprotein) and free amino nitrogen (FAN) are also produced.
Why are they important in beer? Is it correlated to the amount of fermentable sugar or is it something completely different?
Its completely different than the sugar part of the equation. Its not something to worry about as the mash provides all the necessary nitrogen under normal conditions.
Nitrogen levels are purely a concern for yeast performance.
Grain contains protein. Protein contains nitrogen.
The amount of nitrogen-containing substances in malt does not correlate with the produced fermentable suger. Yeast needs (like most creatures) amino acids for propagation. If no amino acids are available, yeast can synthesize them from inorganic compounds, e.g. ammonium (which is usually added as nutrition salt for cider fermentation [apple juice has not enough amino acids or other nitrogen sources]). Malt contains naturally proteins. To produce amino acids as yeast nutrition, these proteins must be degraded. Modern malt houses produce malt with already degraded proteins, therefore usually no protein rest during mashing has to be made. A small amount of non-degraded proteins are needed for the foam, but too much makes the beer hazy. Hence, for brewing purpose, barley with a lower protein content than for baking is used. A protein rest (around 55 °C) should usually be made for adjunct-rich mashes.
