To review the concept to exponential growth.
To introduce a differential equation.
To gain mathematical intuition for a function using limit case analysis.
Limit Case Analysis
Hi everybody! Welcome back to Synthetic Biology One. Today we’re going talk about exponential growth in bacteria. Our goal is to develop some intuition for the power of exponential growth. We also want to introduce some equations to show how we will represent math in this course. Ready? Let’s get started.
You have probably heard it before. Bacteria are capable of exponential growth. If a single bacterium is left alone in a test tube, a bacterial culture will rapidly grow until all the nutrients are depleted. Under ideal conditions, a single bacterium can divide once per hour or even faster. At that rate, we expect 4 bacteria in 2 hours. 16 bacteria in 4 hours. 1000 bacteria in 10 hours and 16 million bacteria after 1 day.
In synthetic biology, we use bacteria as a tool. But unlike other tools, bacteria can build themselves. This means bacteria-based biotechnology can reach a large scale quickly. The power of exponential growth is one reason synthetic biology can have a large impact in a short time. So how can we understand exponential growth? Let’s start with a simple mathematical model.
Here we have a differential equation. It is almost the simplest equation we could write. But since this is the first time we have introduced math let’s walk through it piece by piece. This equation has 3 variables. C represents the number of cells. t represents time. r represents the specific growth rate of our cells. In the example I mentioned above, this division rate is once per hour.
Because this equation describes a physical process, each term has physical units. The derivative on the left tells us the number of new cells being created per hour. It has units of cells per hour. The term C, on the right, represents the total number of cells. It is expressed simply as a number. Finally r represents the growth rate of the cells: one division per hour. It has units of per hour, or hours to the minus 1.
This equation gives rise to these two curves. Here, I have plotted the number of new cells produced per hour. Notice how rapidly it increases from just a few cells to more cells than we can show on this graph. This extremely rapid increase is classic exponential growth. Here, I have plotted the total number of cells. Notice that the curves have almost the same shape. This makes sense. During cell division, each cell produces a new cell. Therefore, the creation of new bacteria is exactly proportional to the number of existing bacteria.
The differential equation for exponential growth has an exact solution, shown here. The total number of cells, C, is equal to the starting number of cells, C zero, and an exponential term including the growth rate and time. Now, we could plug these equations into a computer and calculate the exact number of cells that we expect to produce, given a number of starting cells and a length of time. But you and me, we aren’t computers. I want to find something human in this equation. I want to ask: how does this equation make me feel?
To get an intuition for this model, I’m going to perform some simple thought experiments to see if the model matches my view of reality. To do those experiments, I’ll do what’s called Limit Case Analysis. We are going to push the parameters of this model to the limit, using the largest and smallest values that we can possibly imagine, and see if the model behaves reasonably.
Let’s try it for the parameter C, which represents the starting number of cells in our system. What is the smallest number of cells you can possibly imagine? One cell? What about zero cells. How does our model behave when we start with zero cells? Well in this case it doesn’t matter what the growth rate is. Anything times zero is zero. So no new cells are created, and the population is still zero at any future time. Does this make sense? Of course it does. If you start with nothing, you get nothing. So our model is good.
Now let’s go the other way. What happens when the number of cells becomes very large? Well, nothing happens. As the number of cells continues to increase, the growth rate continues to increase forever. How does that make us feel? Well it makes me feel bad because it is impossible. Exponential growth can never continue forever. If bacteria were really allowed to replicate every hour for more than a few days, pretty soon the universe would contain nothing but bacteria. If our model is allowing impossibly high growth values, it may be a sign that we have forgotten an important constraint on the system.
Checking a model for extremely high or extremely low values using limit case analysis is a useful sanity check for when you build your own models. When we return to the bacterial growth model, we will try to fix this problem by adding new constraints to the model that account for the fact that bacteria can not grow exponentially forever. Until then, happy modeling!