# Review on Rehydrating Dry Yeasts and Viability Stains

Eureka and welcome to a new science post. I would like to discuss the results from a paper about rehydrating yeasts and viability staining. Since dry yeasts are widely used in the homebrewing scene and even on industrial scale, lots of discussions are about the effects of rehydrating the dry yeasts before use. Some rehydrate the active dry yeast (ADY) in some water and others just sprinkle the yeast in the beer. Some prefer to use warm water and others prefer colder temperatures.

During the drying process, the water flows out of the yeast cells rather rapidly and leads to a collapse of the cytoskeleton (Rodriguez-Porrata et al (2008)). During the first minutes of the rehydration process, the cell’s membrane are not functionally active yet and lead to membrane leakage. In this process, molecules from within the cell flow out of the cell. During the rehydration process, the cellular membranes get repaired and thus stop the membrane leaking. If the cells cannot stop the leakage, it is going go die.Jenkins et al studied the effects of rehydration conditions on yeast viability and came to some remarkable results. The authors studied three different yeast strains (LAL1 a lager strain, LAL2 Nottingham strain and LAL4 a Munich strain), rehydrated the cells at different temperatures and for a different amount of time. In addition, they measured the viabilities at certain time points during the rehydration process using four different techniques: Slide cultures, methylene blue, MgANS (8-anilino-1-naphthalene-sulfonic acid hemi-magnesium salt hydrate) and Oxonol staining. For the slide cultures, a small volume of yeast suspension was added to a small amount of agar on a slide and the arising microcolonies from the yeast cells within the agar were counted after 18 h. Thus a staining independent method to asses viability.

For this test, Jenkins et al used 1 g of ADY and added 10 times the weight of water (water temperatures 25°C and 30°C). Time point A was taken immediately after adding the ADY to the water. Time point B was taken after leaving the ADY rehydrate for 15 min. The yeast-water mixture was then mixed and samples were taken after additional 15 min each (time points C1 to C4).

Lets look what they found out. In case of the Lager strain, the viabilities at the different time points are shown in Fig 1. I would like to leave the different temperatures aside since its effect is strain dependent and not important for my main message here. What they could observe is a lower viability at time point A compared to the other time points. This effect seems to be independent of the water temperature (not shown). Further on to notice are the different viability values one obtained using the four techniques. For the Lager strain, measuring the viability using methylene blue lead to lower values compared to the other two staining techniques.

Fig 1: Viabilities of Lager strain at different time points at 25°C by Jenkins et al (2011)

The lower viabilities at time point A could also be observed for the Nottingham (Fig 2) and the Munich yeast (Fig 3). Again, the lower viabilities seem not to depend on the water temperature of the water used for the rehydration process (not shown).

Fig 2: Viabilities of Nottingham strain at different time points at 25°C by Jenkins et al (2011)

Fig 3: Viabilities of Munich strain at different time points at 25°C by Jenkins et al (2011)

Putting this observations together. Measuring the yeast viability using staining methods such as methylene blue within the first minutes of rehydration seems to lead to significant lower values compared to values obtained from later time points. What is the reason for this you may ask? This is where it gets interesting.

The first question one has to address is how yeast viabilities can increase in the first place (as observed in the figures from time point A to B and C1 in case of the staining). Please note that the viability measured with slide cultures in case of the Munich strain are highly similar to later time points (Fig 3). Viability, in a biological sense, can only increase by the formation of new, viable cells. One other way would be for dead yeasts to get alive again which is not very likely in my opinion (though I don’t have any proof for my statement). Due to the chosen time points, it is very unlikely for the yeasts to undergo divisions and thus increasing the viabilities again from time point A to B and C1. There has to be a different effect.

The answer to the question lies within the methods. As previously mentioned, dried yeasts don’t have active membranes. During the rehydration process, the membranes get repaired again (Rodriguez-Porrata et al). Since most of the viability stainings, such as methylene blue rely on active membranes (the dye should only be taken up by dead cells with inactive membranes), dry yeasts behave much like dead cells within the first time points and thus leading to lower viability values. With time, the membranes get active again, and the dye is probably exported from the cells and thus increasing the viability. This is a very nice example of a method’s limitations.

However, there seems to be an effect in the slide culture method as well. This might be due to osmotic pressure issues when the dry yeasts get in contact with a highly osmotic agar which enhances the leaking of the cells.

Assessing viability measurements using methylene blue on dry yeasts is a valid method but has to be done with caution. Especially if one wants to assess whether dry yeasts have a higher viability if added to water first or pitched to wort directly. Consider the right time points for such experiments.

I hope this was interesting to read and might give you a better insight into viability stains and its limitations.

# Bibliography

Jenkins DM, Powell CD, Fischborn T, Smart A (2011) Rehydration of Active Dry Brewing Yeast and its Effect on Cell Viability, J. Inst. Brew. 117(3), 377-382 (http://onlinelibrary.wiley.com/doi/10.1002/j.2050-0416.2011.tb00482.x/abstract)

Rodriguez-Porrata B, Novo M, Guillamon J, Rozès N, Mas R, Cordero Otero R (2008) Vitality enhancement of the rehydrated active dry wine yeast. International Journal of Food Microbiology. 126, 116-122
(http://www.sciencedirect.com/science/article/pii/S0168160508002560)

# BBA/EBY Brett Experiment Update 1

Hello fellow BBA/EBY experiment collaborators. This is the first update concerning the BBA/EBY Brett Experiment. I would like to begin by thanking Jeff for the good collaboration so far and all the other people who are willing to take the risk testing some of my strains. Additional thanks to Luke and Ryan for their contributions about the evaluation sheets. Cheers to all that. And thanks for all the people offering to send me some unique dregs and yeasts as well. I would like to proceed with some numbers:

• 32 collaborators are officially in for the experiment (One subscriber, George Peterson did not write me any email yet)
• 308 samples will be sent out for the experiment so far (might further increase)
• From the 32 collaborators, 1 is not from the US (and it isn’t me)
• 7 collaborators will test the entire 20 strains (awesome!)

I would like to give you further information about the experiment today and cover some other questions I got asked so far.

# You signed up but would like to test more/less strains?

• Write me an email with your request. We will find a solution

# Updates concerning the recipe/process:

• Fermentation temperatures and fermentation times. We left this one open so people can pick what they feel most comfortable with. Please feel free to play with fermentation temperatures and times if you like. One request though. Please remember the numbers such as fermentation temperatures and times for the evaluation later on
• Please try to measure the terminal gravities before bottling. This is necessary to get the attenuation levels for the different strains
• Split batch sizes. Take whatever fermentation volume you feel most comfortable with. Jeff and I will both brew a 10 gal (40 L) batch and split the batch into 0.5 gal shares to test all the 20 strains. If someone would like to brew more than that, please adjust the pitching rate
• Pitching rates. Expect to get about 1.2 million yeast cells per tube (see picture below) and go from there.Pitching the 1 mL liquid culture into a 200 mL DME unstirred starter should give you roughly 25 billion (+/- 3 billion) yeast cells after 10 to 14 days. Corresponding to roughly 12 mL of yeast slurry. This should be enough yeast to pitch 0.5 gal directly. If you need bigger cell counts, use the http://yeastcalc.com/ and use 25 billion cells as initial cell count if you have done a 200 mL starter first. As far as I can tell from my Brett starters, the starter volumes are quite comparable to normal yeast but Brett need more time. So don’t expect the Bretts to eat through a starter within 24 h. To evaluate how much yeast you got after each starter, try to estimate the yeast sediment volume (in mL) and multiply it by two to get to the cells in billion. For example, 12 mL of yeast slurry are equal to 24 billion yeast cells
• When should I bottle the beers? Bottle the beers as soon as you think they are ready. It is hard to make any predictions here since I have no idea how the strains perform. Some strains might be done fermenting after seven days, others might need more time. I for my part will give the primary fermentation about two weeks and then evaluate which beers to bottle

# Yeast shipping

• I am currently stepping up all the 20 strains to have enough viable yeast for the shipping. This will take another few days for sure before I can prepare the first tubes. I plan to send out the yeasts on Monday, the 26th of August. According to my post office, you should get the yeasts after three to seven days
• The yeasts will be sent out without any cooling pads as Bretts should be fine with ambient temperatures. Further on, I will supply the yeasts with fresh media to give them something to do during the shipping. As long as the parcels are not exposed to sun light for several hours, they should be fine. If someone expects problems with this approach, please let me know

# I got the yeasts, what’s next to do?

• I would advise you to prepare small starters (200 mL at max) some days after the 26th of August to be prepared for the yeasts. For a 200 mL starter, add 20 g of dry malt extract to 200 mL of water and sterilize it using a pressure cooker if possible. Mason jars could be well suited for the starters. Any smaller volumes works as well. Just don’t go beyond a starter volume of 10 mL.
1. Wash your hands thoroughly with soap first
2. Flick at the bottom of the tubes to get the yeast pellet on the bottom of the tube back into suspension. Prevent any vivid shaking of the tubes
3. Sanitize the tubes if possible with alcohol (Vodka etc) or a sanitizer
4. Remove the Parafilm wrap from the top of the tube. There might be some pressure forcing some gas out of the tubes or even lift the lid. So be prepared. I would press down the lid with my thumb (or any other finger you like) while removing the parafilm and then gently open the lid
5. Open the tube, avoid touching any inner parts of the tube lid and pour the entire content of the tube into your yeast starter
6. Shake your starter a bit and leave it as it is
• If you planned to do the batch for the experiment later on, please prepare the starters as well. The Bretts can be stored in the starters for weeks to months and it would be better for the yeasts to let them recover from the shipping procedure
• If this in no option, store the tubes at a cool place. Don’t freeze them! They might not survive that
• Expect to wait at least 10 days before you see signs of activity in the starters. I know this might be hard for some of you but please don’t write to me complaining about dead yeasts before the 10 days mark
• If you encounter any problems during the starter steps (no signs of fermentation visible after 10 days, contaminations etc) please let me know what problem you encountered (contacteurekabrewing@gmail.com). I will send you some fresh yeasts immediately

# What about the evaluation process?

If there is anything left unanswered, please let me know. That’s it for the first update. Thanks for your help and cheers, Sam

# Collaboraters for BBA/EBY Brett Experiment wanted

Hello fellow yeast ranchers and Brett aficionados. Are you interested to be part of a large-scale Brettanomyces fermentation project? If yes, please read on. If not just ignore this post.

All started with the latest update of my yeast database which now consists of close to 100 isolated yeast and bacteria strains. After the last database update in mid June 2013, Bikes, Beer & Adventures (aka Jeffrey Crane) contacted me and asked for the strains to give me feedback in return. Since I haven’t actually brewed with any of the strains yet and kind of postponed the testing part for later on, we both agreed on an experimental setup to test all the different Brettanomyces strains in a single Brett-beer experiment. Jeffrey and I came up with a nice experiment to test some of the Brettanomyces strains I currently have.

The basic idea for this experiment is to brew a rather simple beer, split the batch and ferment the split parts with individual Brettanomyces strains only. Then evaluate the outcome of the beers, evaluate the individual strains and return the results to me (or make them otherwise available to me like posting on a blog etc). The recipe for this experiment, developed by Jeffrey, is mentioned below.

Recipe:

## BBA/EBY-Brettanomyces Experiment

Numbers: Volume [L] 18 (5 gal)
Original gravity 12.5°P (1.050)
Terminal gravity Will depend on strain used
Color Around 10 EBC
IBU 25 IBU
ABV Will depend on strain used
Grains: Pilsner malt (4 EBC) 2.5 kg (5.5 lbs)
Munich malt (14.5 EBC) 0.5 kg (1.0 lbs)
Wheat malt (4 EBC) 1 kg (2.0 lbs)
Acid malt (5 EBC) 0.17 kg (6 oz)
Hops: Styrian Goldings (4.5% AA) Add 28.3 g (1 oz) and boil for 60 min
Saazer (4.5% AA) Add 28.3 g (1 oz) and boil for 10 min
Yeast: EBY Various strains in split-batch-mode
Water: Mash: 10.4 L (2.75 gal), sparge: 19.8 L (5.2 gal) @78°C (172°F)
Rest: Mash in @67°C (152°F), 60 min @67°C (152°F), sparge at 74°C (165.2°F)
Boil: Boil for 60 min
Fermentation: Oxygenate your wort as normal
Primary 1-31 days at around 20°C (68°F)
Secondary 1-31 days
Maturation: Carbonation (CO2 vol) 2
Maturation time N/A

# How does the experiment work?

As a participant of the BBA/EBY Brettanomyces Experiment you will receive a small amount of the Brettanomyces strain(s) from me, propagate them to have enough yeast to pitch, follow the recipe and procedure mentioned above, split the batch and let the shares ferment with the Brett strains. Then evaluate the beers and give me feedback about the individual strains. After the experiment, you are free to keep the individual strains and use them as you like. One request though, please only subscribe to the experiment if you follow the recipe, procedure mentioned above. There will be time for individual tests later on.

# What about the individual Brettanomyces strains?

The strains are all isolated from different sour beers by myself and haven’t been tested on full-scale yet. If everything works as planned the following 20 strains will be sent out:

EBY001 (B. girardin I), EBY002 (B. dreifonteinii I), EBY005 (B. cantillon I), EBY007 (B. italiana I), EBY008-013 (B. cantillon II to VII), EBY014-015 (B. lostfontain I and II), EBY016-017 (B. lambeek I and II), EBY019 (B. cucurbita I), EBY020 (B. jurassienne), EBY021 (B. bruery I), EBY035 (B. cucurbita II), EBY038 (B. cantillon VIII) and EBY048 (B. italiana II).

For more information about the individual strains check out my page with all the strains I currently have (go to EBY program).

• I will either send you the entire set of Brettanomyces (around 20 strains) or a subset of them. Please notice that I can only send out 8 complete sets of Brettanomyces (8 samples of each strain)
• Yeasts will be sent to you in 1.5 mL Eppendorf tubes (1 mL of liquid culture) and I only charge the shipping costs
• A 200 mL unstirred dry malt extract starter (20 g of DME to 200 mL of water) should be enough to get you enough yeast to ferment 2 L of wort (0.52 gal). I recommend to start the yeast starters three weeks before the brew day. If possible, sterilize your yeast starters using a pressure cooker as it might take more than a week before any signs of fermentation are visible. Brettanomyces grow really slow. If the yeast starter is not sterile, any contamination can easily out compete the slow-growing yeasts
• If you send me some interesting yeast/dregs in return, I will send you the Brettanomyces strains for free
• I will send out the yeasts somewhere between mid to end of August 2013

# How can I participate?

Please only participate if you follow the recipe, procedure mentioned above. The whole purpose of this experiment is to get as much information about the strains as possible.

Please further notice that I haven’t tested most of the Brettanoymces strains on full-scale yet. I therefore cannot make any predictions about the fermentation performance nor the aroma, flavor profile of the out coming beers. I wouldn’t be surprised if there are strains not suitable for fermenting beers.

• First, go to http://doodle.com/hc2tgynku7khf8s8 and choose the yeast strain(s) you want to test. Please notice that I only send out eight samples of each yeast strains. I kindly ask you to only choose the yeast strains you are actually going to test
• If you are interested in the strains but don’t want to participate in this experiment, please subscribe to https://groups.google.com/forum/#!forum/eureka-brewing-yeast to get email alerts of future strain releases
• Second, write me an email to contacteurekabrewing@gmail.com with your shipping address and your name you used in the doodle list
• Third, the world-wide shipping costs (yes, I ship the stuff around the globe) for all the 20 yeast strains should be less than 20 USD (a rough estimate). However, I will charge you only the costs I actually paid at my post office in the end. To make my life easier, I would like to deposit all the parcels at my post office (by mid to end of August 2013) and then send you the quotes I had to pay afterwards. I therefore can’t tell you the exact shipping cost before the actual shipping
• And finally, payments can be done using Paypal (please add another 1 USD to cover the fees) or directly to my Swiss bank account

I would be really happy to get as much collaborators as possible. So please spread the word. If there are more collaborators out there than yeast strains, or the available yeasts are gone within an instant, I might even think about another re-release of yeast strains.

If anything is not clear feel free to comment below or write me an email to contacteurekabrewing@gmail.com.

I am really looking forward to this experiment which will hopefully be fun, exciting and a real adventure. Cheers, Sam

# #68 Dark Berliner Sour

Eureka, its time for another recipe. Actually this one is not as straight forward as you might expect from my previous recipes and might be hard to reproduce. The idea for this beer came up during the lautering process of my #67 Koschei Imperial Stout batch. Brewed 51 L (13.5 gal) of Imperial Stout and could not throw away the second runnings which still had a gravity of 12°P (1.048). I therefore used the runnings as a base for this recipe and went from there.

I collected the runnings up to a total volume of 10 L (2.6 gal) and added a package of Wyeast’s Lactobacillus delbrueckii to the unboiled mash as the mash reached a temperature of 40°C (104°F) and let the mash sit at 40°C for a three days until the sourness was at a good level. I then let the wort cool down to around 20°C (68°F) and added the unboiled, per-soured wort on top of a Wyeast’s 3191 Berliner Weisse cake. I left the beer on the Berliner Blend for nearly two months and kegged the soured Stout into a small keg. Kegged the beer on the 9th of December 2012 and left the keg at a relatively warm place to mature. I then forgot about this beer for a while…

This changed in late Spring of 2013. I re-discovered this particular keg in my cellar during an inventory and was quite excited to try a first sip of this beer. The ABV for this one is around 5 %. For a style, it should be something like a dark Berliner Weisse. Since Weisse originates from white in German, it would not much sense to call it a Dark Berliner Weisse. I therefore simply call it a Dark Berliner Sour. Or maybe there is already a suitable beer style for this kind of beer. Let me know if there is a matching beer style for my beer.

Aroma: Smells like a cold brewed coffee gone sour with a touch of lemon, dark chocolate and bonfire smoke. Can even detect a hint of gingerbread. Impressive aroma profile and really interesting aroma combination.

Appearance: Deep black color, clear with lots of bubbles rising to the top. Not very long-lasting off-white head

Flavor: Hint of dark chocolate, very subtle roast character and a nice level of sourness. Even some red grape and wood character like in certain wines. Tobacco is there as well. The sourness and the flavors from the roasted barley really go along really nice.

Mouthfeel: Light body, average carbonation level, very dry but not too thin, silky and lightly sour finish. Detectable astringency. Very refreshing. Leaves a smoky impression on the tongue like you get after smoking a cigar.

Overall Impression: Quite impressed how this one turned out. Despite the roast character, this beer has a lot of flavors common in red wines. Not only that, it reminds me of Jolly Pumpkin’s Madrugada Obscura. With the exception that this one is not really funky. I am further surprised how the sourness plays with the roasty, astringency characters. The play along really nicely.

I expected to get a huge mess of a beer. Simply because the grist of the Imperial Stout was not really destined to turn into a sour beer. Nevertheless, the beer turned out to be way more complex than expected and is a very interesting one. Maybe not the kind I would drink for a whole evening but I have others that drink this one in pints. I might brew another batch of this one in the future and mature it in my Whisky barrel to see how this one turns out after some time in a wooden barrel. Cheers and thanks for reading

# A glimpse into copper sulfate agar

Eureka, I would like to publish some preliminary results from my latest plating experiments. I am still interested in isolating Brettanomyces from different sources and still play around with different agar media to see what their impact is on the entire isolation process. The latest experiment I performed was a large scale bromocresol screening on different Saccharomyces yeasts to see whether bromocresol can be used to differentiate between Saccharomyces and Brettanoymces. My insight from this experiment: bromocresol green as a tool to differentiate between Brettanomyces (known to grow as white colonies) and Saccharomyces might only work within a small time frame. In addition, some Saccharomyces strains grew as white colonies in presence of bromocresol green (possible false positive strains).

Yet another approach is to add copper sulfate to the agar media to inhibit the growth of domesticated yeasts [Yakobson, 2010, Taylor et al, 1984]. Wild yeasts therefore should be able to grow in presence of copper sulfate. I wanted to give this agar a go to see if it can be used to differentiate between domesticated Saccharomyces strains and wild yeasts (Brettanoymces in my case). I started by adding 0.6 g copper sulfate to 1 L of Sabouraud agar and streaked some strains on the plates. As controls, plain Sabouraud agar plates were used to test the viability of the strains (not all plates shown).

Fig 1: Saccharomyces yeasts on Sabouraud agar (1056 = Wyeast American Ale, 1084 = Wyeast Irish Ale, PtPtince = EBY049, Y05 = EBY050)

The four domesticated Saccharomyces strains plated on plain Sabouraud agar showed a nice growth phenotype (Fig 1). Streaking the same strains on copper sulfate containing Sabouraud agar revealed that only one strain (WY1084 Irish Ale) was impaired in its growth (Fig 2). All the remaining Saccharomyces strains grew as normal. From this observation one can already conclude that the addition of copper sulfate to the agar media impaired only 25% of the domesticated Saccharomyces strains tested.

Fig 2: Saccharomyces yeasts on CuSo4-Sabouraud agar (1056 = Wyeast American Ale, 1084 = Wyeast Irish Ale, PtPtince = EBY049, Y05 = EBY050)

Plating Brettanomyces and isolated Saccharomyces strains on copper agar media revealed a growth phenotype for all tested Brettanomyces strains (Fig 3, 4). Only the Saccharomyces isolate (B04 green in Fig 3) and the bacteria strain (I10 in Fig 4) did not grow on copper sulfate agar. Since B04 green was isolated from a Gueuze, it can be argued that this particular strain might be a non-domesticated Saccharomyces strain. On the other hand, it might be a domesticated yeast strain concluding from the lacking growth on copper sulfate. Including the previous observation that only a small part of domesticated Saccharomyces strains were impaired in their growth makes it even harder to allocate the isolated yeast strain to domesticated or non-domesticated Saccharomyces.

Fig 3: Different yeasts on CuSO4-Sabouraud agar (B04 = EBY004 Brettanomyces, B04green = EBY041 Saccharomyces, B05 = EBY005 Brettanomyces, B02 = EBY002 Brettanomyces)

Fig 4: Different yeasts/bacteria on CuSO4-Sabouraud agar (B01 = EBY001 Brettanomyces, I10 = EBY024 Bacteria, I05 = EBY009 Brettanomyces, I11 = EBY013 Brettanomyces)

This small-scale experiment revealed that a copper sulfate addition to Sabouraud agar media does not impair most of the domesticated Saccharomyces strains tested. All the Brettanomyces strains tested in this experiment grew in presence of copper sulfate.

It seems to me that copper sulfate used at a concentration of 0.6 g per liter of Sabouraud agar media was not useful to differentiate non-domesticated from domesticated Saccharomyces yeasts. Simply because it could not inhibit the growth of most of the domesticated yeasts tested. As an outlook, one might increase the concentration of copper sulfate to levels where it impairs most of the domesticated Saccharomyces strains. Then test the Brettanomyces under the same conditions and see if they still grow or not. Maybe even change the Sabouraud agar to MYGP like published by Taylor et al. It is not clear to me yet if I even further investigate the use of copper sulfate.

# A glimpse into yeast growth kinetic models

Eureka, science post! Some math, model building and biology: I would like to start talking about yeast growth kinetic models today. In general, growth kinetics describes how different conditions (substrate, oxygen amount, metabolism products, inoculation rate, growth inhibitors etc) influence the growth of a microorganism in a time dependent manner. At the end one can construct mathematical models to describe the growth behaviour. In most cases one begins with experiments with fixed conditions and the growth of the organism is observed over time. A next experiment can be conducted with the change of one condition and the growth over time is observed once again. And so forth.

All these kinetic models can be very powerful tools. Not only can one improve for example the efficiency of yeast propagation but also investigate the effect of different parameters such as the inoculation rate of yeast, the substrate concentration, oxygen- or ethanol concentration on yeast growth. One application for growth kinetics could be to calculate yeast propagation like done in other yeast growth calculators for homebrewers.

This post is a general introduction about yeast kinetics and I would like to show you how one can get from experimental data to a simple growth kinetic model. Future posts will go into more detail and I would like to give a general introduction first.

# Exponential growth of microorganism

One of the most basic models in mathematical biology is the exponential growth equation for microorganisms. This equation can be used to calculate the amount of microorganisms or biomass (X) after a certain amount of time (t). Biomass can be the physical mass of the microorganisms, the cell concentration or any other measurement related to specify how many microorganisms there are (like optical density). I will stick to the physical mass of microorganisms as biomass in this post. In the exponential growth equation model no inhibition or substrate limitation is included. Substrate by the way is a term for any food source for the microorganisms such as glucose, maltose etc.

$X_t = X_{t0} \cdot e^{\mu_{max} \cdot t - t_0}$

• Xt is the biomass concentration at the time point t [g L-1]
• X0 is the biomass concentration at the time point zero [g L-1]
• µmax is the maximum specific growth rate [h-1]
• t Time [h]
• t0 Time where the experiment begins. Normally zero [h]

Lets make an example. Lets say you have a 1 L yeast starter (with indefinite amount of substrate) and add 10 g of yeast (X0) at the beginning (t0). You might ask yourself how many yeast cells you have after waiting for 90 min. The only thing you might not know is the specific growth rate (µmax). These coefficients can be looked up and for yeast µmax is somewhere around 0.5 h-1. My solution for this question would be 21.2 g L-1. So roughly double the amount.

You can even calculate the doubling time (tD) after changing the previous equation (doubling time specifies the time needed to double the initial amount of microorganisms):

$ln(\frac{X_t}{X_{t0}}) = \mu_{max} \cdot t$

By the way, the equation above will be important later on to get µmax. Moving on, at the time of doubling (tD), Xt equals 2 times Xt0:

$ln(\frac{2 X_{t0}}{X_{t0}}) = \mu_{max} \cdot t_D$

$ln(2) = \mu_{max} \cdot t_D$

$t_D = \frac{ln(2)}{\mu_{max}}$

Common doubling times for Saccharomyces cerevisiae are around 90 min [1]. This gives you a µmax of roughly 0.5 h-1.

In the exponential growth equation model substrate limitations are not included and therefore makes it not that useful to fully describe the growth behaviour if you want to investigate the effect of the substrate concentration itself. Let go to the next model.

# Monod equation for biomass

A very widely used model to describe the growth of microorganisms is the Monod model. The whole model is based on the Michaelis-Menten equation widely used in enzyme kinetics. I don’t want to go into further details here how one came up with these equations. In the end, the Monod model is again a simplified version of reality (like every model is). The first equation for the Monod model is:

$\mu = \mu_{max} \frac{S}{K_S + S}$

• µ is known as the specific growth rate [h-1]
• µmax is the maximum specific growth rate [h-1]
• S is the substrate concentration [g L-1]
• KS is the substrate saturation constant [g L-1]

We already know µmax, S is just the substrate concentration. For example the amount of dry malt extract you use for your yeast starter. Or glucose or whatever you are interested in. Just keep in mind that µmax depends on the used substrate. The definition of µ is:

$\mu = \frac{dX}{dt} \cdot \frac{1}{X}$

• µ is known as the specific growth rate [h-1]
• X is the biomass concentration [g L-1]
• dX is the change of the biomass concentration [g L-1]
• dt is the change of time when dX happens [h]

and depends on the change of biomass within an indefinite amount of time (dt). You could therefore write the Monod equation like:

$\frac{dX}{dt}= \mu_{max} \frac{S}{K_S + S} \cdot X$

and you already have your first differential equation for your model to describe the change of biomass in a time dependent manner.

This leaves KS. This is similar to the Michaelis constant Km in the Michaelis-Menten equation. In the case of the Michaelis-Menten equation, Km describes the substrate concentration at which the enzyme reaction is equal to half of the maximum speed. In the case of Monod, KS describes the substrate concentration (therefore S as index) where you have half of µmax. This is all so far for the first part of Monod.

# Monod equation for substrate

In the previous section, I introduced the basic Monod equation which can be used to describe the change of biomass over time. Next we like to include the change of substrate. In case of yeast one might be interested to investigate the behaviour of dry malt extract. For that we have to introduce another coefficient:

$Y_{X/S} = \frac{\frac{dX}{dt}} {\frac{dS}{dt}}$

• YX/S Biomass on substrate yield coefficient [gX gS-1]
• dX is the change of the biomass concentration [g L-1]
• dS is the change of the substrate concentration [g L-1]
• dt is the change of time when dX and dS happen [h]

YX/S simply defines how much biomass you can get from substrate (thus the index X and S). For example how much biomass you get from one gram of substrate. Looking at the previous equations, you can already see how to get a differential equation for the substrate:

$\frac{dS}{dt} = - \frac{\mu_{max} \frac{S}{K_S + S} \cdot X}{Y_{X/S}}$

Per definition, the equation is multiplied by minus one because the slope of dS over dt is negative (substrate is metabolized and therefore only decreases).

This leaves us to look at product kinetics. In case of yeast, one might be interested to describe the production of ethanol as a product. This is very similar to the substrate shown above. In this case you need another yield factor:

$Y_{P/X} = \frac{\frac{dP}{dt}}{\frac{dX}{dt}}$

• YP/X Product on biomass yield coefficient [gP gX-1]
• dP is the change of the product concentration [g L-1]
• dX is the change of the biomass concentration [g L-1]
• dt is the change of time when dX and dP happen [h]

In this case the product depends on the biomass and not on the substrate. Once again one can write down the differential equation for the product formation:

$\frac{dP}{dt} = \mu_{max}\frac{S}{K_S + S} \cdot X \cdot Y_{P/X}$

Are you still reading? Yes? You must either be very interested and/or be a math geek like me… I would like to stop with the introduction here and discuss other things such as inhibition etc in a future post. I now would like to share some information how you could/can get the coefficients for the equations above from empirical data.

# Coefficient determination from experimental data

All the data below is from a S. cerevisiae batch cultivation I cultivated during my undergraduate studies. I don’t want to get much into detail here but some information to understand the values you get afterwards. The batch cultivation was done in a 16 L reactor (4.2 gal) under sterile conditions and glucose as the only carbon source (substrate). All the values below therefore are for glucose only.

During the cultivation (8 h), every 30 min the optical density (OD) and glucose concentration was measured and every 60 min the dry mass was determined. The optical density is another measurement method to get an idea about the yeast concentration. The glucose was measured enzymatically and the dry mass was determined by filtration, drying of the filter paper and finally determining the mass of yeast on the filter paper.

# µmax determination based on dry mass

As previously mentioned, the following equation can be used to get the µmax value by plotting the logarithmic ratio of the biomass against the time (t). This should give you a linear function with the slope µmax as shown in Fig 1:

$ln(\frac{X_t}{X_{t0}}) = \mu_{max} \cdot t$

Fig 1: Log dry mass ratio against time. µmax determined from slope of linear fit function

One can easily see that the amount of yeast grew steadily up to the time point of six hours where the yeast concentration stayed the same (Fig 1). After six-hour the whole glucose was already metabolized (not shown) and no further yeast growth could be observed (Fig 1). The linear fit function therefore only makes sense between time point zero and six hours. The slope of the linear fit function was 0.43 h-1. Please remember, this µmax value is for the dry mass and is close to known values [2].

Lets quickly calculate the doubling time for this µmax. This give you a doubling time of roughly 97 min. Not that far away from the 90 min stated in source [1].

# µmax determination based on optical density (OD)

The same can be done for the OD (Fig 2).

Fig 2: Log OD ratio against time. µmax determined from slope of linear fit function

Once again after five to six hours, the yeast growth came to a stop (Fig 2). This time the slope of the linear fit function was 0.53 h-1. A nice example that not every µmax is the same. One has to be very careful how the specific µmax was determined. Either based on the optical density, dry mass weight or even the cell count.

# Biomass on substrate yield coefficient (YX/S)

To determine the yield coefficient, one can plot the measured yeast dry mass or OD against the measured glucose values and use the linear slope to determine the yield factors (Fig 3).

Fig 3: Determine the biomass to glucose yield factor

The yield factor based on the dry mass was -0.5 g g-1, and -0.1 g g-1 for the OD. Known yield factors for yeast based on glucose and dry mass are between -0.3 and – 0.5 g g-1 [3].

# Building growth models

Now its time to put all the equations and values into a model. The only coefficient not known so far is KS. This value is hard to determine and the easiest way to get this value is by iterative approaches. You simply use the model to get KS. I use Berkley Madonna for this purpose. What you have to do is input all the different differential equations, the measured values and run an iterative algorithm to let the model function approximate the measured values. If you do this the right way you might get graphs like below (Fig 4).

Fig 4: Yeast kinetic model 1, explanation in text below

In this case the substrate concentration (S) and the yeast concentration (X) are plotted against time. In addition, the black dots correspond to the measured glucose concentrations. I included the values I used for the different coefficients in the graph as well. Unfortunately, one cannot fit the measured glucose concentration curve with the numerical values of the determined coefficients. I therefore had to use slightly different values for the yield coefficient (in this case 0.3 g g-1) and 0.51 h-1 for µmax. The differences between the numerical values of the parameters might be due to measuring inaccuracies for the glucose concentration and/or yeast concentration. In the end, one can determine KS to be around 0.031 g L-1.

You now can use the model to investigate how the different conditions affect the growth of the yeast. For instance different substrate conditions or inoculation rates. All this can be useful to understand the behaviour of yeast growth under different conditions. However, one has to keep in mind that all this is based on a simplified model and it does not have to represent reality. Model building is sometimes hard work because iteration processes might get stuck and lead to wrong results (such as negative substrate concentrations).

# Summary

Growth kinetics models can be used to describe the growth of microorganisms. Because biological systems can hardly be approximated by simple fit functions, more sophisticated methods need to be applied. Such as the models described in this post.

# Outlook

This was just a basic introduction about growth kinetics and models. Future posts will go into more details covering additions to the basic Monod model. The next post concerning growth kinetics will be about Brettanomyces growth and is a nice organism to introduce inhibition effects.

I would like to do some small-scale experiments with yeast propagation and determine the individual model-parameters in the future. The resulting models therefore might be useful to approximate yeast starters under different conditions. Just be patient, I am currently really busy with my real scientific work. Thanks for reading and comment if you like. Cheers!

# Bibliography

[1] : http://dbb.urmc.rochester.edu/labs/sherman_f/yeast/4.html, “An Introduction to the Genetics and Molecular Biology of the Yeast Saccharomyces cerevisiae” (2012)
[2] : http://www.atcc.org (2012)
[3] : B. Sonnleitner, Lecture slides Bioprozesstechnik 1, ZHAW Wädenswil, 2010