TL;DR: For a two-step starter, use 35% of the volume in step 1 and 65% of the total volume in step 2. For a three-step starter, use 15%, 30% and 55% of the total volume in steps 1, 2 and 3, respectively.
Long version:
Suppose I have a fixed volume of starter wort and a fixed amount of yeast to add to the starter and I want to maximize the amount of yeast I pitch. (no arguments against the premise, please)
If I want to do a single step starter, the answer is easy enough: I add the yeast to the starter.
If I want to do a multi-step starter, what volumes do I use? I could fiddle around on yeastcalc until I get numbers I like, but there has to be a better way.
If we model the multiplication rate (MR) as a power function of the inoculation rate(IR in million cells per ml) (which seems to fit the white labs growth chart from http://www.yeastcalc.com/growthcharts.html fairly well), it looks like MR=A*IR^-B, where A is approximately 12 and B is approximately .445. This means that if we put x million yeast cells into v ml of starter, we will get a yield of x*A*(x/v)^-B cells out.
We can use this relationship to calculate the optimal volumes as fractions of your starter volume V.
For a 2-step starter, the optimal fractions are (1-B)/(2-B) or roughly 35% for starter one and 1/(2-B) or roughly 65% for starter two.
For a 3-step starter, the optimal fractions are 1+(B-2)/(B^2-3*B+3) or roughly 15% for starter one, (1-B)/(B^2-3*B+3) or roughly 30% for starter two, and 1/(B^2-3*B+3) or roughly 55% for starter three.
Phew, that brainworm has been gnawing at me all evening.
Long version:
Suppose I have a fixed volume of starter wort and a fixed amount of yeast to add to the starter and I want to maximize the amount of yeast I pitch. (no arguments against the premise, please)
If I want to do a single step starter, the answer is easy enough: I add the yeast to the starter.
If I want to do a multi-step starter, what volumes do I use? I could fiddle around on yeastcalc until I get numbers I like, but there has to be a better way.
If we model the multiplication rate (MR) as a power function of the inoculation rate(IR in million cells per ml) (which seems to fit the white labs growth chart from http://www.yeastcalc.com/growthcharts.html fairly well), it looks like MR=A*IR^-B, where A is approximately 12 and B is approximately .445. This means that if we put x million yeast cells into v ml of starter, we will get a yield of x*A*(x/v)^-B cells out.
We can use this relationship to calculate the optimal volumes as fractions of your starter volume V.
For a 2-step starter, the optimal fractions are (1-B)/(2-B) or roughly 35% for starter one and 1/(2-B) or roughly 65% for starter two.
For a 3-step starter, the optimal fractions are 1+(B-2)/(B^2-3*B+3) or roughly 15% for starter one, (1-B)/(B^2-3*B+3) or roughly 30% for starter two, and 1/(B^2-3*B+3) or roughly 55% for starter three.
Phew, that brainworm has been gnawing at me all evening.
