Back in the bad old days, a home brewer was happy just to have a reliable yeast culture to pitch into his/her wort. The average home brewer today is no longer content with having access to yeast cultures that get the job done without leaving a trail of metabolic trash that is a mile wide. He/she wants to be able to compute and hit the exact number of cells needed to ferment a given batch of wort. The cold hard truth is that this level of precision is neither obtainable, nor is it necessary in a home brewery. A yeast culture is like a nuclear weapon in that a brewer only needs to be within a reasonable distance of his/her target in order to complete the task at hand. In this blog entry, we will cover how the yeast biomass in a starter or fermentation grows, and why home brewers are placing emphasis on precision where it is not needed.
Brewing yeast is a mystery to many home brewers. It is easily the most technically complex part of the brewing process, after all, brewers make wort, yeast makes beer. Fermentation is little more than controlled spoilage. In the case of beer, the spoilage microorganism is the yeast culture that we pitch. We want the pitched yeast culture to “own” the wort, and we want to ensure that it owns the wort quickly enough to prevent any wild microflora (yeast, bacteria, mold) that may have hitched a ride on airborne particulate matter from gaining a foothold in our fermentation. We accomplish this task by pitching a large number of active yeast cells while practicing good brewery hygiene.
Yeast cells go through three distinct phases during fermentation before entering a state known as quiescence. The phase that we will be discussing in this blog entry is known as the exponential phase (also known as the logarithmic phase). The exponential phase is where the yeast biomass grows. This phase is called the exponential phase because the cell count grows exponentially at a rate of 2n, where n is the number of minutes that have elapsed since the culture entered the exponential phase divided by the replication period in minutes (computer scientists who are reading this blog entry will recognize this growth pattern as O(2n), or binary exponential growth). The reason why the yeast biomass grows at a rate of 2n is because each mother cell buds a daughter cell during a replication period.
With the above said, let’s examine the basic formula for approximating the cell count at a given number of minutes into the exponential phase. yeast_cell_count_at_time_t = initial_yeast_cell_count * 2n, where n equals the number of minutes that have elapsed since the beginning of the exponential phase (time_t) divided by the replication period (replication_period) Let’s apply the formula shown above with time_t equal to 90 minutes and replication_period equal to 90 minutes.
time_t = 90 minutes into the exponential phase
initial_cell_count = 200 billion
replication_period = 90 minutes
n = 90 / 90 = 1 replication period
yeast_cell_count_at_time_t = 200 billion * 2^1 = 400 billion cells, where the symbol "^" denotes raised to the power of
After ninety minutes of exponential growth, the culture has doubled in size.
Let’s extend time_t to six hours, which equals three hundred and sixty minutes.
n = 360 / 90 = 4
yeast_cell_count_at_time_t = 200,000,000,000 (200 billion) * 2^4 = 3,200,000,000,000 (3.2 trillion) cells
After four replication periods, the cell count is now sixteen times larger than it was when it was pitched. Herein, lies the explosive power of exponential growth.
We can determine the number of replication periods necessary to reach a target cell count given an initial cell count by re-writing the equation to solve for n. We will refer to the variable n as the number_of_replication_periods and the variable cell_count_at_time_t as target_cell_count in our re-written equation.
number_of_replication_periods = log (target_cell_count / initial_cell_count) / log(2)
Let’s set target_cell_count to 3.2 trillion and initial_cell_count to 200 billion to verify that the formula produces the number 4 for the number of replication periods.
number_of_replication_periods = log (3,200,000,000,000 / 200,000,000,000) / log(2) = 4
With that said, the yeast calculator that we used for our latest recipe determined that we needed to pitch 200 billion cells. Our culture only contains 150 billion cells. How much impact will underpitching by 50 billion cells make in the amount of time necessary to reach maximum cell density for 5 gallons, which is approximately 3.8 trillion cells?
number_of_replication_periods = log (3,800,000,000,000 / 200,000,000,000) / log(2) = ~4.25
number_of_replication_periods = log (3,800,000,000,000 / 150,000,000,000) / log(2) = ~4.66
As long as there is enough oxygen in solution to support cellular health, the difference in exponential growth time between pitching 150 billion cells and 200 billion cells is 4.66 – 4.25 = 0.41 * 90 = ~37 minutes.
Okay, let’s pitch half of the number of cells that our yeast calculator computed.
number_of_replication_periods = log (3,800,000,000,000 / 100,000,000,000) / log(2) = ~5.25
Once again, as long as there is enough oxygen in solution to support cellular health, the difference in exponential growth time between pitching 100 billion cells and 200 billion cells is 5.25 – 4.25 = 1.0 * 90 = 90 minutes.
As one can clearly see, underpitching by as much as 50% only lengthens the exponential growth phase by 90 minutes. The key is to ensure that there is adequate dissolved oxygen to support cellular health when underpitching, as there is almost always enough carbon (sugar is carbon bound to water; hence, the term carbohydrate). While brewing species within the Saccharomyces genus do not respire in brewer’s wort due to being Crabtree positive, they do use oxygen for ergosterol and unsaturated fatty acid (UFA) biosynthesis by shunting oxygen and a small amount of carbon to the respirative metabolic pathway during the lag phase. These compounds are used by cells to maintain their plasma membranes. Plasma membrane health determines how well a yeast cell can take in nutrients and expel waste through its cell wall.
If yeast cultures are like nuclear weapons, why do we have pitching guidelines? Well, most pitching guidelines are for slurry, not laboratory grown yeast. Slurry is a mixture of various age cells that have been through one or more fermentations. These cells have been subjected to ethanol and brewery-related environmental stress. Most starters are grown from laboratory-cultured yeast. Laboratory-prepared growth media and environmental conditions are designed to maximize biomass growth while minimizing stress.
With that said, there is a significant challenge that places a lower bound on our pitched cell count; namely, sanitation. No real-world brewery is sterile. Airborne microflora has an opportunity to contaminate a culture or a medium every time it is exposed to air. Residual surface contamination from less than adequate cleaning and/or sanitation increases the chance of wild microflora gaining a foothold in a fermentation. Bacteria are the biggest threat because they too grow exponentially, but their replication period is one third that of yeast; hence, the bacteria cell count increases by a factor of eight every time the yeast cell count doubles. If we normalize the bacteria growth model to that of the yeast growth model, we end up with the equation shown below.
bacteria_cell_count_at_time_t = initial_bacteria_cell_count * 8n, where n equals the number of minutes that have elapsed since the beginning of the exponential phase divided by the yeast replication period
Since the bacteria cell count doubles in one third of the amount of time that it takes the yeast cell count to double, the bacteria cell count grows at a rate of 2^3 every time the yeast cell count grows at a rate of 2^1.
To give readers an idea of how this difference allows a tiny number of bacteria cells to overtake a larger number of yeast cells, let’s calculate the 2n and 8n multipliers out to 16 yeast eplication periods (n = 1 to 16).
Cell count multiplier for the 2n growth pattern = 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536
Cell count multiplier for the 8n growth pattern = 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, 134217728, 1073741824, 8589934592, 68719476736, 549755813888, 4398046511104, 35184372088832, 281474976710656
After sixteen yeast replication periods, the yeast cell count can be as much as 65,536 times larger than it was when we started whereas the bacteria cell count can be as much as a whopping 281,474,976,710,656 times larger than when we started, that is, if there is sufficient carbon, oxygen, and space to support that much growth. Herein, lies the reason why we need to pitch a large number of cells. We want our yeast culture to rapidly shutdown the replication of competitors by dominating a batch of wort. In modern vernacular, we want the culture to “own” the wort to the extent that nothing else stands a chance of tainting our controlled spoilage process. However, that feat can be easily accomplished with modern commercial yeast cultures without having to worry about pitching a precise number of yeast cells.
In closing, hopefully readers have gained an understanding of the explosive nature of exponential growth. Exponential growth can basically erase a difference in cell counts that is less then a factor of two, and make up to a factor of four difference in cell counts insignificant, which is why yeast cultures are like nuclear weapons. There are even times when we want to purposely underpitch, but that is a topic for a future blog entry.