Bioreactor growth rates

My home bioreactor took 48 hours to get going where the instructions said 24-36 hours was typical, but it got going. About two weeks after startup the active cell culture had matured and I decided to put it into production. The instructions indicated that 1.5 hours at room temperature would be an adequate first stage reaction period. Four hours into it the first reaction stage wasn't finished, so I put the lot into the fridge and went to bed; clearly the time estimates were not representative.

I knew putting it in the fridge would slow the reaction down to the point where I could pick it up again the next day, because it's a bioreactor and they're sensitive to temperature - specifically, the reproduction rate of the cell culture slows down dramatically when cooled.

That's when I realized that my bioreactor had been reacting more slowly than the instructions suggested was normal every step of the way.

This is also when I remembered that "room temperature" is a fairly wide range, and I tend to keep my home toward the low end of that range when it's cool outside, as it is for all but about 3 months of summer. I suspect that the time estimates given are more likely for the middle of the temperature range.

Clearly, this calls for some math. If I want to take this reaction to completion, I will need to allow the correct amount of time for each reaction stage.

A batch bioreactor in its growth phase grows at a rate that varies with temperature according to the Arrhenius equation:

$\mu'_R = Ae^{-E_a/RT}$

Unfortunately, I haven't yet found the growth rate $$\mu'_R$$, the activation energy $$E_a$$, or the constant $$A$$, for the particular strains of microorganisms I have in my bioreactor. It probably doesn't help that I don't know what they are, exactly, other than being a mixture of Saccharomyces and lactobaccilus. Note that those are genus-level names, which is a step broader than species. There are lots of options for what exactly they could be, and there's a good chance I have more than one species from each genus.

Fortunately, I'm just comparing the same thing at different temperatures, and so I can cancel out and otherwise ignore most of the numbers that I don't know.

After checking a few different sets of instructions, most of which said nothing more specific than "room temperature" or "a warm place", I found that about 25C, or 298K, seems to be what is assumed for the reaction temperature, which is pretty warm for "room temperature" but describes "a warm place" quite well. My house is about 20C, or 293K, most of the year.

As far as the actual difference in rate, all we really need to look at is how much the $$e^{-E_a/RT}$$ part changes with temperature. At $$T_1$$ of 293K and $$T_2$$ of 298K, we have these equations:

$\mu_1 = Ae^{-E_a/RT_1}$ $\mu_2 = Ae^{-E_a/RT_2}$

All I'm looking for is the rate ratio, $$\mu_1 \over \mu_2$$, so that I can use the ratio calculate how much more time to allow for each reaction stage. I don't care what either of the numbers actually are. I will divide the first equation by the second to get:

${\mu_1 \over \mu_2} = {Ae^{-E_a/RT_1} \over Ae^{-E_a/RT_2}}$

Simplifying somewhat:

${\mu_1 \over \mu_2} = e^{E_a/RT_2 - E_a/RT_1}$

I got rid of the $$A$$ by cancelling, but couldn't do that to the $$E_a/R$$. $$R$$ is a constant, and according to one of my textbooks, $$E_a$$ ranges from 42,000 to 84,000 J/mol for cell growth. So, I'll calculate it twice, to get the edges of the expected range.

Plugging in the numbers, for 42,000J/mol I get $$\mu_1/\mu_2 = 0.75$$, and for 84,000J/mol I get $$\mu_1/\mu_2 = 0.56$$. Or, my bioreactor growth rate is 56% - 75% of the expected growth rate due to the lower temperature. This matches my observations; the instructions indicate that the bioreactor volume can double in 6 hours due to CO2 gas production and collapse before 12 hours is up as gas production stops and it deflates under its own weight, but I have found it's rarely even starting the collapse when I check on it after 12 hours.