- To correct our model of exponential growth by including a realistic constraint.
To continue using limit case analysis to explore new mathematical models.
To describe the phases of bacterial growth.
- Exponential Phase
- Logistic Phase
- Stationary Phase
Hi everybody! Welcome back to Synthetic Biology One. Today we are going to try, for the second time, to model growth in bacteria. If you remember, the last time we tried to do this, we were sad because the model that we came up with didn’t work. We did get the essence of exponential growth, bacteria doubling over and over again, but we forgot to include a way to make them stop. In our model, bacteria could just grow and grow forever until they reached impossible population sizes. So let’s try it again, but this time be a little less nuts.
Here is that original equation we were using for exponential growth. And here is the new equation for logistic growth. Logistic in the sense of logistics, like managing limited resources. Let’s walk through and look at the differences. We still have C, representing the number of cells. We still have r, the specific growth rate. You could imagine, for example, one division per hour. The physical unit of r is per hour, or hours-1. We have also introduced a new term K. K is called the carrying capacity. It represents the absolute upper limit on the number of bacteria that can live in our test tube. It is just what we need to keep our model sane.
Let’s look closer at this term in the parentheses (1 – C/K). Imagine what will happen when C, the number of cells, equals K, the maximum number of cells allowed. Well in that case, C/K = 1, which means that 1 – C/K = 0. Zero times anything is zero, so the solution is zero growth. This expression, the logistic term, gets smaller and smaller until eventually it forces the entire growth rate to be zero.
Let’s try some more limit case analysis to see how this model makes us feel. Does it behave nicely, or does it go crazy for some parameters? How about K. What does this model look like when K gets very small? Well, for small values of K, growth stops quickly. Whenever C equals K, growth stops. And if C somehow becomes larger than K, growth becomes negative, meaning that the number of cells decreases. Totally reasonable.
Now what about for large values of K? Well, if K is much much larger than C, then C/K will be almost zero. That means 1 – C/K will me almost 1. Well 1 times anything is itself, so the logistic term essentially does not affect the growth rate. In this case, the logistic growth model looks very much like the exponential model. The cell production rate equals r C. How does this make us feel? Well it makes sense to me. If K is very very large, meaning the bacteria have lots of space and nutrients, then growth really is basically exponential, at least for a while.
Here we have a side-by-side comparison of logistic growth and exponential growth. In this plot, I am showing the new cell production rate as a function of the number of existing cells. You can see that, for exponential growth, new cell production just keeps increasing forever. However, in the logistic model, new cell production reaches a maximum and then returns to zero once the test tube becomes too crowded for new growth. Notice how the two models look similar in the beginning, when there are very few cells relative to the carrying capacity. In this case, growth really is exponential.
Like last time, this differential equation is simple enough to have an exact solution. Like last time, we can put it in a computer to calculate the number of cells over time. When we look at a logistic growth curve, we can think of 3 phases of bacterial growth. Here, in the exponential phase, the cells have unlimited resources and the growth is almost exponential. Here, in the log phase, cells continue to grow but more slowly. Finally in the stationary phase, resources are exhausted and growth is stopped.
The logistic model is very close to reality, and matches well with the growth of real bacteria in real test tubes. To me, it is super amazing that we can capture all of that complexity of billions of bacteria growing together with just a simple model of r, C and K.
Until next time – happy growing!