Wikipedia:Reference desk/Archives/Mathematics/2013 November 14

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November 14

Basic logarithms

I work in a lab and culture cells. They go from about 10% to 70% confluent (coverage of container) in about 48 hours. I want to calculate the doubling time (time from 10% to 20% to 40% etc) and I'm pretty sure that I need to use logarithms but can't remember how. I got an A in A level Maths 10 years ago so clearly that was a wasted effort.

I remember Log(base)A = C where b = A^C and I'm pretty sure base is 2 because I'm interested in the rate of doubling. — Preceding unsigned comment added by 129.215.47.59 (talk) 18:45, 14 November 2013 (UTC)

I believe the Exponential_growth article is what you are looking for. OldTimeNESter (talk) 19:28, 14 November 2013 (UTC)

I believe you are assuming exponential growth, which is reasonable for cells at small times, but not so true for large times (if they are filling up 70% of the container, resource restrictions might apply, and thus a logistic model would be more suitable).

If you want to stick to the exponential growth then C(t), the concentration at time t, is given by C(t) = C(0) exp(kt), where C(0) is the concentration at time 0 and k is a constant specifying the growth rate. Your data is C(0) = 10 and C(48) = 70. From there you can solve for k to find the growth rate. Once you have k, you can solve 2C(0) = C(0) exp(kt) for t, and the result will be the doubling time.

Now, concerning the logarithms, I belive you might have the definition wrong. Log_b (A) = C means that b^C = A (b is called the base). It's easy to remember: the log_b of A is what power you need to raise b to in order to get A. Using this, and remembering that log_e (exp(x)) = x (that is, log and exp "cancel each other") you can solve the equations from the previous paragraph. Good Luck. - Syats (talk) 19:33, 14 November 2013 (UTC)

You say you want to focus on the doubling time of the covered fraction. But maybe you should focus on the half-life H of the uncovered portion q. This suggests modeling $q=2^{-t/H}$ where t is elapsed time. Then $\log _{2}q=(-t)\cdot (1/H)$ and you could see if some value of H makes this fit the data on various (q, t) combinations well. Duoduoduo (talk) 11:22, 15 November 2013 (UTC)

Yes logistic growth is what you need for the full problem, unless you can convincingly argue that resources are not limiting at ~70% coverage. In one dimension, logistic regression is not that hard, and some software packages (e.g. the free and popular R_(programming_language)) will do it for you. You can still do an exponential fit to the early phase of growth, but (naively) it would be safest to only use data points up to ~50%, where you can be more sure that the organisms are not competing too strongly. The main point is, remember that "doubling time" can only make sense and be used in a limited time frame, otherwise we'd all be knee-deep in your lab cultures by now ;) SemanticMantis (talk) 16:35, 15 November 2013 (UTC)