Determining rates in an ODE

Question

I am building a model with a system of ODEs for the interaction of bacteria with human cells.

I want to determine the rate parameters in the model and in the literature I found that the maximum number of bacteria that can get attached to a single cell is 20 and this maximum number will be reached within 4 hours. I want to know how I can use these information to come up with the rates that I want.

In the model I have a rate with which unattached bacteria gets attached. As it takes 4 hours for 20 bacteria to attach, can I take the rate with which a bacterium moves to the attached state to be 4/20?

Then I have another rate, namely the rate with which cells move into the infected cell stage. This happens when a cell can no longer attach any more bacteria (having reached its maximum capacity for attachment) is reached. So, can I take it to be 1/4 hours?

My model is somewhat similar to the model found here (the rate with which cells become infected in my model is modelled similarly to the rate $\omega$; $P$ is the bacteria, $E_A$ is the uninfected cells, $E_U$ is the infected cells):

Answer 1

In the model I have a rate in which unattached bacteria gets attached. As it takes 4 hours for 20 bacteria to attach, can I take the rate that a bacteria moves to attached state be 4/20.

That strongly depends on how the attachment works. Consider the following mechanisms of attachment:

1. There is a big reservoir of mostly randomly moving bacteria. Whenever a bacterium is in the vicinity of a cell that is not yet covered with bacteria, it will seek out a free space and become attached to the cell. In this case, your model would make sense as the 4 hours in question are determined only by how long it statistically takes for 20 bacteria to come within the vicinity of the cell. A differential equation that models the number of bacteria $n$ attached to a single cell would be:

   $$\frac{\mathrm{d} n}{\mathrm{d} t} = \frac{20}{4\,\mathrm{h}} = 5 \,{\mathrm{h}}^{-1}$$

2. There is a big reservoir of randomly moving bacteria. When a bacterium collides directly with a cell, it “sticks”. Twenty bacteria fully cover a cell such that there is no way another bacterium can directly collide with the cell (it can only hit other bacteria). In this case, your model is not accurate, as the more bacteria are already attached to a cell, the less likely it is for another one to hit a free spot and become attached.

   Instead you would the number of bacteria $n$ attached to a single cell to behave roughly exponentially over time, for instance: $n=20·(1-\exp(-ρt))$. The corresponding differential equation could be something like $$\frac{\mathrm{d} n}{\mathrm{d} t} = ρ·(20-n).$$ (Note that $ρ=\frac{1}{\mathrm{h}}$ roughly matches your observations.)

The model you quote suggest something along the lines of option 2. However, things are a bit different:

• There is no infinite reservoir of bacteria.
• It’s a population model and not a model for a single cell.
• There are other things going on which I can only understand with studying the paper.