# Modeling bacterial growth with differential equations

I hope this is the right place for this question. I am working on building a growth model for bacteria for a risk assessment, and would like to move the growth model past static temperature conditions. The primary growth model, using the modified Gompertz equation, is:

$$\log_{10} R = A_g \cdot \exp(-\exp\{\mu\cdot \exp(1)\cdot(l-t)/A_g +1\}),$$

where $R$ is the relative population. There are secondary models for the three parameters $A_g$, $\mu$ and $l$ (i.e., the max number of bacteria at the stationary phase; the growth rate; and the lag phase, respectively). For example, for mu the equation is:

$$\sqrt{\mu}=0.0421\cdot(T-12.0570),$$ where $T$ is temperature.

All of the secondary models are dependent upon temperature, since bacteria obviously favor warmer conditions and grow faster, up to a certain point.

My trouble is that one paper specifying a different growth model (logistic) was able to model bacterial growth under ANY temperature regime you can think of by deriving a differential equation for their primary model. I have little experience in differential equations and sadly, I am actually just not even sure which part of my model would need to be integrated. Since I am interested in how the changing parameters (due to changing temperature) lead to growth in the end, I would think I would need to integrate with respect to $T$? But I am unsure if this means solving the differential equations of the second-order parameters (because they are the only ones that directly have $T$ in them) or if that means doing something else to the primary equation?

For reference, the paper I am using most often is:

Corradini, M. G., Amézquita, A., Normand, M. D., & Peleg, M. (2006). Modeling and predicting non-isothermal microbial growth using general purpose software. International journal of food microbiology, 106(2), 223-228.

(Open source PDF available through Google Scholar)

• There is the Gompertz equation, and the Gompertz differential equation (I'm relying on wikipedia for this). The former is the solution of the latter. Perhaps you can take your formulation and from it create a "modified Gompertz differential equation"? – Jeff Snider Jan 30 '14 at 21:04
• This might be a good route, if I understood more. Basically I think I am looking for a conceptual explanation of what needs to happen, rather than mechanistic? I've taken a fair amount of calculus but it was a while ago, and I'm getting stuck in the weeds on this one. For instance: I know that if I wanted to calculate overall growth for a simple varying temperature, I could just modify the second-order parameters, stick them into the primary model, and see much growth happened for each sequential time period. But I want to do that for something consistently decreasing, and not sure (cont) – HFBrowning Jan 30 '14 at 21:19
• what I need to integrate/or differentiate with respect to. Plus the first option is an excessive amount of work. Here's a pic of what's "easy" to me, and then what I actually want to do. So I think it's a very easy question for math people probably. [IMG]i60.tinypic.com/24y4wsy.png[/IMG] – HFBrowning Jan 30 '14 at 21:22
• In general when you have a complex differential equation and you're more interested in the result than the math, solving it is best done by a numerical package. There are a lot of different packages to choose from. The first thing you need is to formulate your problem as a differential equation, where your equation tells you the rate of change rather than the current population size. – Jeff Snider Jan 30 '14 at 21:26
• Jeff - exactly! Your last sentence is what I am struggling with but maybe didn't articulate well. – HFBrowning Jan 30 '14 at 21:28

For some suitable constants $(a,b,c)$ determined by $(A_g,\mu,l)$, one has $$\ln\ln R=a-\exp(b-ct)$$ hence $$\frac{R'}{R\ln R}=c\exp(b-ct)=c(a-\ln\ln R)$$ that is, $$R'=cR\ln R(a-\ln\ln R)$$