# Two state exponential growth model

I want to write an expression to model exponential growth of a microbe that scholastically and irreversibly acquires a mutation at a fixed rate that slows growth.

I can do it iteratively, but am curious whether a simple expression can be written.

I started with:

$$P_{total} = P_{wt}e^{r_{wt}t} + P_{mutant}e^{r_{mutant}t}$$

How do I factor in the mutation rate that converts $$P_{wt}$$ to $$P_{mutant}$$?

• Can you please explain what your variables stand for? – Remi.b Oct 17 '18 at 0:07
• What is the variable you are interested in tracking? Growth rate? Population size? Do you want to model drift and selection in your model to figure out the probability of a given deleterious mutation to fix? – Remi.b Oct 17 '18 at 0:09
• I want to track the number of wild-type and mutant cells over time, given the growth rate of each and the mutation rate. P_total = total # of cells, P_wt = # of wild-type cells, P_mutant = # of mutant cells, r_wt = growth rate of wild-type cells, r_mutant = growth rate of mutant cells, r_mutation = mutation rate t = time, . – J-- Oct 17 '18 at 0:22
• $P_{total} = P_{wt}e^{(r_{wt}-r_{mutation})t} + P_{mutant}e^{(r_{mutant}+r_{mutation})t}$ ? – J-- Oct 17 '18 at 0:24
• What do you mean by scholastically in this context? – Wrzlprmft Oct 17 '18 at 4:34

### In general

Analytic solutions only exist for special cases. Therefore a good way of approaching these problems is to first act like they don’t and be happy if you can see how to derive one. It’s also more straightforward than wild guessing.

• $$P_\text{w}$$ denotes the original population and $$g_\text{w}$$ its growth rate.
• $$P_\text{m}$$ denotes the mutant population and $$g_\text{m}$$ its growth rate.
• $$r$$ denotes the mutation rate.
The differential equations that describe your problem are: $$\dot{P}_\text{w} = g_\text{w} P_\text{w} - rP_\text{w},\\ \dot{P}_\text{m} = g_\text{w} P_\text{m} + rP_\text{w}.$$ This system of differential equations is linear as it can be written in the form $$\dot{\vec{P}} = A·\vec{P}$$ with $$\vec{P} = (P_\text{w},P_\text{m})$$ and: $$A = \pmatrix{g_\text{w}-r & 0 \\ r & g_\text{m}}.$$ Linear systems in turn can be solved by finding the eigenvalues and eigenvectors of the matrix $$A$$, which are:
• $$\vec{v}_1 = (0,\frac{r}{g_\text{w}-g_\text{m}-r})$$ for the eigenvalue $$λ_1 = g_\text{m}$$.
• $$\vec{v}_2 = \left( 1, \frac{r}{g_\text{w}-g_\text{m}-r} \right)$$ for the eigenvalue $$λ_2 = g_\text{w}-r$$.
This means that the solution can be written as: $$\vec{P} = c_1 e^{λ_1 t} \vec{v}_1 + c_2 e^{λ_2 t} \vec{v}_2,$$ where $$c_1$$ and $$c_2$$ are constants determined by the initial condition: $$\vec{P}(0) = c_1 \vec{v}_1 + c_2 \vec{v}_2.$$ If you assume that you have no mutants at the beginning and you initial wild-type population is $$1$$, you have $$c_1=-1$$ and $$c_2=1$$. With this you get: $$P_\text{w} = e^{ (g_\text{w}-r) t},\\ P_\text{m} = \frac{r}{g_\text{w}-g_\text{m}-r} \Big(e^{(g_\text{w}-r) t} -e^{g_\text{m} t} \Big).$$