Estimating yeast growth

Recently there has been a lot of focus on yeast growth calculators for starters. But most of the various calculators out there base their data on work published in Chris White and Jamil’s yeast book. Unfortunately the yeast growth example given in that book was only for a non agitated starter. When a starter is constantly stirred all the yeast is kept in suspension in a homogeneous nutrient environment. That is not true for non agitated starters where yeast will sediment and only evolution of CO2 will cause agitation. As a result stirred and still starters are expected to show different growth behavior that cannot be simply approximated by adding a constant scaling factor to yeast growth.

Jamil’s pitching rate calculator supports stirred starter fermentation but the growth curve used for that mode is a simple scaling of the growth curve for non agitated starters. Jamil never published how he arrived at the model used in his calculator. As a result I have to draw conclusions based on what I can observe when I run data points through his calculator.

Before I get into comparing growth curves, I need to explain how I look at yeast growth. The primary factor in yeast growth is the available sugar. Within a practical range of wort concentrations the actual concentration of wort will have little impact on yeast growth. I’m still going to test this in a controlled experiment but observations from yeast propagation in my brewing seem to support that. Because of that I focus not not how many Million yeast cells are in a ml of wort, but rather how many Billion yeast cells get to share a gram of extract (sugars, proteins, minerals, etc) that is dissolved in that wort.

As for yeast growth, I care about how many new cells of yeast can be grown from that gram of extract. In their yeast book Chris and Jamil use a yield factor defined as new yeast growth in Million/ml per degree Plato drop of apparent extract. This takes into account that different worts can have different attenuation levels and with it varying amounts of fermentable sugars. While this is correct it requires knowledge of the wort’s fermentability and most brewers don’t know the fermentability of their starter wort. Furthermore the uncertainty of starter wort fermentability is likely +/- 5 % and this is well within the imprecision that one can expect from yeast calculators. I expect yeast growth calculators to have an error of +/- 15% or more. Because of that I feel confident in using Billion cells growth per gram of extract (B/g) as the yeast growth metric that should be tracked. Assuming a starter attenuation of 75%, the conversion factor between  “yield factor” and specific growth (B/g) is 13.3:

1 B/g = 13.3 M/(ml*P)

Now that I established how I plot yeast growth I can show some charts.

This one compared Mr Malty data for simple starters and stirred starters with the simple starter data from the Yeast book as well as a 2.7x scaled version of the simple starter data:

It is apparent that the growth curve for simple starter matches the data from the book, which makes sense. What surprises me, however, is that the stir plate data is not just scaling the yield for a given innoculation rate, but it also scales the inoculation rate. This means that the yeast growth calculator has different optimal innoculation rates for simple and stirred starters. That is something I don’t follow and my data contradicts that. More on that in a moment.

The following is a chart with my data. I have been using WY2042 since it is a low flocculant lager strain. Most of that has been presented in a previous blog post (Yeast growth experiments – some early results). What’s new is the non agitated data points and a few data points for using yeast that was stored in the fridge for 5 days before being used.

When the data is potted as growth in billion cells per gram over initial billion cells per gram, the data makes more sense. As the amount of extract available to each cell of the starting population approaches the amount of extract needed to grow a new cell, the growth per extract starts to fall. This is because the initial population is in a resting state and when sugar becomes available all vital yeast cells start to consume the sugar. A cell will not start budding unless it consumed all the resources needed to grow a new cell. If this was an ideal culture, where every cell consumes sugar at the same rate and needs the same amount of sugar to grow a daughter cell, cell growth would stop once there are more cells per gram of extract that it takes to grow the same amount of cells. This is because none of the cells would have access to the amount of nutrients needed to grow a new cell. But this is not such an ideal culture and some cells will consume nutrients faster than others and will be able to grow daughters while others can’t.

I also expect that yeast growth of a large population of older cells is reduced compared to young cells since the old cells will need to fill their reserves before they can start accumulating nutrients for growth. I don’t have enough data on that yet to quantify this effect.

This effect of dropping yeast yield is not as pronounced for a non agitated culture. This is because in such a culture not all yeast cells have the same access to wort nutrients due to sedimentation. Yet another reason why one cannot estimate the yeast growth characteristics in a stirred starter from a non agitated starter.

Based on my observations and knowledge of yeast growth so far, I think the following yeast growth model should be used for calculating expected yeast growth in stirred starters:

If (initial cells < 1.4 Billion/gram extract)
  yeast growth is 1.4 Billion / gram extract
If (initial cells between 1.4 and 3.5 Billion / gram extract)
  yeast growth is 2.33 - 0.67 * Billion initial cells per gram extract
else
  no yeast growth

For non agitated starters a growth rate of 0.4 Billion per gram should be assumed over the full initial cell density range (0 – 3.5 B/g). I don’t think that starters with more than 3.5 Billion cells/g  are practical. I expect the growth rate in starters to be affected by the shape of the vessel, the volume of the wort and more. The 0.4 Billion new cells per gram proposed here is a rough guess based on the data points I have so far.

Update:

The proposed model for stirred starters has been implemented in:

 

38 thoughts on “Estimating yeast growth

  1. Your resulting plot seems to fall into the classical “stationary —> decline” phase end of the normal culture curve as far as I can tell. That is once nutrient supply vs demand reaches equilibrium and starts to decrease, a decline in population and proliferation is observed. So apparently at 3.5 B/g that equilibrium is reached for this strain.
    Would be interesting to see these values for ale strains. I imagine they’d be lower due to higher energy demands of S. cerevisiae…

  2. I’m not sure if that’s the case here. I think what you talking about is the population over time plot that can be found in many publications on yeast growth. These charts here plot new growth for an existing population of varying size. The 3.5 B/g point is a rough guess at this point. As I look more closely at the mechanics of yeast growth I may be able to make more sense of what’s happening for large population densities. In an experimental setting it is difficult to measure low growth due to the relatively large error in cell counting.

    I do plan to examine other strains as well. For now I’m focusing my efforts on this one to gain more understanding of the principles that are involved. But from general brewery propagations I see that other strains also tend to show a growth of 1.2 – 1.6 B/g when the start out with less than 1 B/g cell density.

  3. Population over time is a reflection of both birth and death. Here you just look at birth without counting death but accounting for nutrient content so it is essentially looking at the same thing, and plotting the same thing, from a slightly different angle.
    Speaking of errors in cell counts. I’m almost finished with an article about cell counts and sizes and I site your work in there for a few points. Would you like to review it and give me some of your thoughts before I post it? If so, where could I email you?

  4. Dimitri, these plots don’t plot anything over time. The time span involved is the time it takes for the yeast to grow, which is generally 1 to 2 days. During that time there is negligible death and since I’m not doing viability staining for cell counts I wouldn’t know which cells are dead and which aren’t. The few times I did check viability I found 95%+ at start end end of the experiment. You’re welcome to send your article to my e-mail and I’m glad to take a look.

  5. Well population over time doesn’t really plot over time as well. Not when you really look into it. It’s more of a reflection of cell birth/death vs nutrient/waste and time just a function that stretches as far as the latter ratio allows.
    What you’re looking, as far as I can understand, is proliferative capacity of initial population over nutrient available. So not exactly the same, but in that general vicinity. Actually what you can derive from this is an approximation of cell density in a given culture once you work out an equation that fits this plot. Metabolic activity assay cell count estimation essentially (which I touch in the article).
    Very cool.
    I’ll email you once it’s done.

  6. Hello Braukaiser, thank you for sharing your yeast growth experiments. I somewhat struggle to understand why you observe higher growth per extract values at lower initial cell per extract values than the calculators. If I recall correctly you basically used 10°P starter wort for the first data point and let it ferment, right? For the next experiment, did you add another volume of wort directly or did you decant the starter wort off the yeast sediment first?

    Yet another thing. It seems to me that higher initial cells per g lead to a decreased growth per extract. Higher initial cells per g mean a thicker yeast starter, right? How can you be sure to not have a vitamin, growth factor etc. depletion in such a thick yeast starter and therefore have a decreased yeast growth which leads to a lower growth per extract value?

    Please forgive me for asking all these questions. I really want to understand your results here since I am looking into yeast kinetics as well and my models kind of predict other results. Thank you for all your great experiments and viele Grüsse aus der Schweiz, Gruss Samuel

    • Samuel,

      What I seem to be observing in stirred starters is that I see a constant growth per extract up to some cell density per extract. Comparison with other calculators is difficult since no data exists on how the curve for stirred starters was obtained. There are inherent differences between still and stirred starters that make it impossible to simply predict stirred starter behavior from still starters.

      In the experiments I allowed the starter to complete fermentation and decanted the spent starter beer from the yeast before I added fresh wort.

      At this point I don’t exactly know what the limiting nutrient is and I plan to investigate this. But I don’t think that this matters for the results of my experiments. As there are more cells per exact each cell will get a lower share of the limiting nutrient. Once that share of limiting nutrient becomes less than what’s actually needed to grow a new cell, you will see a drop in growth since all cells will consume the nutrient but not all will end up accumulating enough to grow a new cell.

      Your comments and questions are welcome, especially if you are also researching this topic. You can contact me here or though e-mail (kai at braukaiser.com)

  7. Kai, thank you for your explanations. Your observation of constant growth per extract up to higher cell densities now explains why one obtains a linear correlation between the starter volume and the final cell concentration with the brewer’s friend calculator.

    I agree that you cannot simply apply results obtained by non agitated starters to agitated ones. I will have a look at my models how the different yeast calculators get from the non-agitated starter kinetics to the agitated ones. At the moments it looks like most of the calculators simply multiply the non-agitated values by a certain factor to get to the agitated starters…

    Concerning the nutrients, that’s what I was talking about. For me the drop at initial cells per gram > 1.4 might be due to a lack of a limiting factor. One way of testing this would be to do two parallel starter experiments with an initial cells per extract of about 3.0 in a 10°P wort. Baker’s yeast might be suitable here. Then simply add some additional yeast nutrients or dead yeast cells to one of the two starters and determine the growth per extract in both starters. If there is an observable difference in yeast growth, maybe a nutrient depletions might be a cause of the growth drop.

    Another lacking factor might be oxygen…

    I will have to check my models again and will get back to you. Thanks again for your explanations. Cheers Samuel

    • Samuel,

      I don’t think that the drop in growth per extract for very high inoculation rates can be attributed only to the limitation of a given nutrient. A nutrient limitation is also what limits the growth at lower inoculation rates and in that lower range I don’t see as much dependency on the inoculation rate. Granted, there is lots of noise in that data and more experiments are needed. In particular, I’m planning experiments with high inoculation rates and yeast cultures of various ages to see how much of the extract is needed to replenish depleted reserves and how much can go to proliferation.

      I would be very interested details of your models.

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  12. Quick question: when you calculate cells per gram extract, is that the original extract in the starter (i.e. 100g in a 1L starter), or is it the extract drop during growth (i.e. less than 100g)?

    From how your work is implemented at yeastcalc and brewer’s friend, it appears to be the former, but I wanted to double check.

    Thanks,
    Andrew

    • Correct. I consider the initial amount of extract in the wort. I know that different fermentabilities affect the amount of sugar available to the yeast. But I consider these variations small compared to the general variations in yeast growth.

  13. Since there is 100 g of extract in 1000 ml starter with dry extract, do you think there is also 100 g of extract in 1000 ml mashed starter?

    What about fermentability in mashed starters, is it near to extract starters (assuming we mash at lower temperature range, eg. 145F), or there is significant difference in fermentability between these two?

    • if the starter has about 10 plato OG, then there are about 100g extract in 1000 ml. as for the fermentability, I don’t think it matters much since there is more uncertainty in the yeast growth than in the fermentability of the wort.

  14. If the initial inoculation rate is very low for example .075bn / gm would there be an issue with yeast health. I have read about too low an inoculation rate causing severe cell scarring. Would yeast grown in the starter impact on the beer they are fermenting?

    • At this point I don’t buy the argument about the excessive scarring. in any culture there are about 50% cells with no scars, 25% with one scar, 13% with two scars and do forth. thus the bulk of the population will have a low scar count regardless of the initial pitch rate. but there might be other reasons for not going to low with the initial pitch rate.

  15. Thanks Kai.
    You have mentioned there might be reasons for not going too low with the initial pitch rate. Would you advise against it based on the what you have experienced?
    If so, do you have a minimum number that you feel comfortable with?

    • I would say that if you start from a vial or smack pack you should always be able to get away with just one propagation step. This only applies to 5 to 10 gal batches. When propagating from slant it is advisable to step up the culture due to the increased risk of contamination if the starting population is too small.

      In my yeast propagation practice, the 3rd step has an innoculation rate of about 0.1 B/g and the 4th and final step tends to be around 0.5 B/g. This seems to work well. Maybe 0.1 B/g is a reasonable lower bound. That’s what you get if you have a growth of 1 B/g and step up by 10x. In practice the innoculation rate will be higher since you can expect more than 1 B/g growth and brewers don’t generally use propagation steps smaller than 10x.

  16. Kai,thank you for providing this valuable information. I was wondering whether Chris White and/or Jamil has weighed in on this new information yet, and whether they have changed their assumptions as a result? Thanks.

    • I contacted Jamil a while back but his response was that I have to work with more different strains and that I should check my process. I have not pushed him further on this. He can weigh in when I present this data at the NHC.

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  18. Hi Kai. You are probably bombarded with email after your NHC talk but I wanted to chime in with the discussion. You state that the limited factor of yeast growth is nutrient access between a stirred and non-stirred starter. I wanted to ask your opinion on oxygen access and crabtree effect. non-stirred starters will go into fermentation mode, and stirred starters will have some oxidative respiration going on, how much, I am not sure. That will influence growth. That being said, a 10P wort is too high (if I am not mistaken 9 grams per liter is the cut-off for Sacch.) for oxidative growth because of the crabtree effect. This will complicate your results. I agree with Sam that you have shown an effect at higher inoculation rates, and that the most obvious effect would be nutrient depletion. Regarding the bud scars, there are studies where they report a shift in average scars per yeast cell during aging of the culture due to the fact that younger cells grow slower due to overcrowding/nutrient depletion. Best, Jasper

    • Jasper,

      I talked about Crabtree effect in my NHC presentation and plan to blog about it as well. It is my understanding that yeast will use oxygen for aerobic respiration if it is available. However the aerobic capacity is limited and any excess glucose that is taken up by the yeast will be fermented. I think that this is happening in stirred starters with access to air: some aerobic and some anaerobic metabolism concurrently.

      I think there is some truth to the fact that fresh daughter cells and mother cells grow differently. I came across a source that said that daughter cells need to consume more nutrients to bud on their own compared to mother cells which need to consume less nutrients to bud again. That, however, should shift the balance to more cells with no scars since it takes less time for a mother cell to grow a new bud than it takes for a daughter cell to have a bud of its own and become a mother cell. This is actually something I could investigate with the yeast growth model I have.

  19. This looks great – are you able to comment on practical limitations??

    As an example, if I had an old (say 25% viable) smackpack and used that in a 2 liter starter, I would end up 236 billions cells, inoculation rate of 16.6 million/ml and growth rate of 8.44. This appears to be more than enough to ferment out a decent sized beer.

    This seems too good to be true, do you have any recommendations around the upper / lower limits for inoculation / growth rates that may help me with deciding whether or not steps are needed??

    • I know its a late reply. But I’m just now going through a back-log of unapproved comments.

      You are basing the growth rate only on the amount of cells you start out with. That’s not how I like to approach this. A 2 l starter can easily give you 200-300B new cells and that’w why you generally don’t need multiple steps when starting with a smack-pack or a vial of yeast.

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  21. kai, using the braukaiser stirplate it looks like i could only use a tenth of a wl vial instead of a whole one if i up the statrer size from 1 to 1.5. would that be correct? thanks!

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  23. Hi Kai,

    Just curious as to how you reached the following assumption:

    Assuming a starter attenuation of 75%, the conversion factor between ”yield factor” and specific growth (B/g) is 13.3:
    1 B/g = 13.3 M/(ml*P)

    I have been developing a yeast calculator and have been trying to prove this conversion factor. However based on my calculations the conversion factor I come to is:

    1 B/g = 15.97 M/(ml*P)

    Thanks,

    Glen Bentley

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