The picture above shows a simple five-celled organism. It has a thick outer membrane and thinner membranes between the cells. As in many organisms, a substance introduced in one cell can move across the cell boundaries into other cells. We are interested in modeling this phenomenon, called diffusion.
We start with a simple model of diffusion. We think of time as being broken into short discrete units. Suppose that in each unit of time each molecule of a particular substance has a 25% chance of crossing either of the boundaries of that cell into an adjacent cell. For the middle three cells, this means that with probability 50% the molecule remains in the cell; with probability 25% it crosses into the cell on its right; and with probability 25% it crosses into the cell on its left. For the two end cells, this means that with probability 75% each molecule remains in the cell and with probability 25% it crosses into the one adjacent cell.
We model this situation using two different kinds of models.
With this model we simulate a number of different molecules. Each molecule starts in a particular cell. For each molecule, in each unit of time we pick a random number between zero and one. If the molecule is in one of the middle three cells then if the random number is between zero and 0.25 the molecule crosses into the cell on its left; if the random number is between 0.25 and 0.50 the molecule crosses into the cell on its right; and if the random number is between 0.50 and one the molecule remains in its current cell. If the molecule is in one of the two end cells then if the random number is between zero and 0.25 the cell crosses into the adjacent cell and if the random number is between 0.25 and one it remains in its current cell.
Thus, in each unit of time some of the molecules will remain in their current cell and others will move into an adjacent cell.
This model is based on the ideas that we developed in the preceding section on Markov Chains. We begin with an initial probability vector that tells us the fraction of all the molecules that are initially located in each cell. For example, the initial probability vector (0, 0, 1, 0, 0) would represent a situation in which the substance was concentrated entirely in the middle cell. We can also think of this vector as describing the probability that a molecule chosen at random starts out in each cell.
A table or matrix describes the probability that in each unit of time a particular molecule moves from one cell to another.
From cell
1 2 3 4 5
To cell
1 0.75 0.25 0.00 0.00 0.00
2 0.25 0.50 0.25 0.00 0.00
3 0.00 0.25 0.50 0.25 0.00
4 0.00 0.00 0.25 0.50 0.25
5 0.00 0.00 0.00 0.25 0.75
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Using this information we can predict the probability that a molecule chosen at random is found in each cell at each time in the future as we did in the preceding section. With a large number of molecules this is close to the fraction of the molecules that are found in each cell at that time. We have just touched upon an interesting idea at the foundation of probability. Click here for more discussion of this idea.
Click here to open a new window with a Java applet. Arrange your windows so that they overlap and you can move back-and-forth between the two windows by clicking on the inactive window to make it active.
This applet examines both models above. It starts out with all the molecules in the middle cell. Notice that the red graph, on the left, has a full middle column and the other columns are empty. The blue graph, on the right, also has full middle column and its other columns are empty. Each column represents the number of molecules of the diffusing substance in one cell. The red graph, on the left, will demonstrate the Markov Chain model and the blue graph, on the right, will demonstrate the simulation model. Click on the green bar at the far right to watch the two demonstrations.
Now open the laboratory for this section for your computer algebra system by clicking on the appropriate icon in the navigation bar. Use your CAS to answer the questions below.
