Petri Dish uses statistical models to calculate rates of diffusion and active transport. This means that, instead of simulating each individual molecule, we simulate collections of molecules using averages. While this introduces a layer of abstraction, we attempt to ground our statistical models in what is happening on the molecular scale.

Diffusion

Imagine a cell with a volume of 1 µm3 (1 µm = 1 micrometer = 10-6 m) and a surface area of 12 µm2. The O2 concentration is 40,000 molecules/µm3 inside of the cell and 100,000 molecules/µm3 outside.

When O2 molecules on the outside of the cell collide with the cell’s membrane, they can either bounce off of the cell membrane or pass through it. The number of O2 molecules passing through the membrane and entering the cell is proportional to the number of collisions, and that is proportional to both the concentration of O2 molecules outside of the cell and the surface area of the cell membrane itself.

The same thing happens on the inside of the cell, except the number of collisions is proportional to the concentration of O2 molecules inside of the cell.

At the start of this simulation, O2 molecules are diffusing across the cell membrane in both directions, but 2.5 times as many molecules are coming in as going out, causing the O2 concentration inside of the cell to increase. This, in turn, causes the rate of diffusion out of the cell to increase until the rate of diffusion out balances the rate of diffusion in, and a state of dynamic equilibrium is reached.

In this simulation, we decided to keep the O2 concentration outside of the cell constant, assuming that there is an infinite reservoir of O2 molecules available. This won’t always be the case. We are also keeping the O2 concentration within the cell uniform. We feel that this is a reasonable first-approximation that will make it easier for players to analyze what is happening since they won’t have to deal with concentration gradients within the cell itself.

Players will still be able to create zones of higher or lower concentrations within the cell by building vesicles and other types of organelles. Here, a player is storing O2 molecules in a vesicle inside of the cell:

Active Transport

Now let’s add some tranport proteins to our simulation. An O2 transport protein works by selectively binding with an O2 molecule and an energy unit. Once it has both, it uses the energy to transport the O2 molecule across the plasma membrane.

The rate of transport depends on how quickly the transport protein can grab onto an O2 molecule and an energy unit, which depends on the concentration of O2 at the input end of the protein and the concentration of energy inside of the cell. If the transport protein is transporting O2 molecules into the cell, then:

And if the transport protein is transporting O2 molecules out of the cell, then:

min() is a mathematical function that returns the smallest number in a set. If there is more than one transport protein, then the rate of transport is multiplied by the number of transport proteins.

In this simulation, the O2 transport proteins are transporting O2 molecules into the cell and the concentration of O2 outside of the cell is less than the concentration of energy inside of the cell, making it the limiting factor.

Once again, there is a net flow of molecules into the cell, causing the O2 concentration inside of the cell to increase. Diffusion and active transport into the cell are both constant because the O2 concentration outside of the cell is constant, but diffusion out of the cell increases with the increasing O2 concentration inside of the cell. The system reaches dynamic equilibrium when the rate of diffusion out balances both the rate of diffusion and the rate of active transport in, which occurs when the concentration of O2 inside of the cell is higher than the concentration outside. This is the active transport pump at work.

What happens when transport proteins are transporting O2 molecules out of the cell?

There is a still net flow of O2 molecules into the cell, but dynamic equilibrium is reached when the concentration of O2 inside of the cell is still lower than the concentration outside. Net flow into the cell continues until the rate of diffusion in balances both the rate of diffusion and the rate of active transport out.