### #12 How enzymes propel themselves-II

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As we mentioned in previous episode(http://biophys3min.blogspot.tw/2016/08/11-how-enzymes-propel-themselves-i.html), the authors hypothesized that the local structural changes caused by heat of reaction could produce a local pressure to the solvent, and its reaction force would propel the enzyme and increase its diffusion constant.

However, how much could it increase? Are there any deducible relationship that could be experimentally tested? In order to answer these question, we must have some basic knowledge about diffusion theory.

We are going to introduce Fick's law, the macroscopic version of diffusion theory in episode II, discuss Einstein's microscopic diffusion theory in episode III and finally the Langevin's microscopic diffusion theory in episode IV.
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Diffusion is the transportation of molecules caused by their thermal motion. It may sound boring to you but actually quite important. It is relevant to something like "never eat strong smell food in the classroom", "feeling embarrassed if you fart in an elevator", and the gas exchanges in our bodies. A German physician named Fick derived a famous law called Fick's law in 1855. And now we are going to talk about how long it would take for your classmate to start feeling disgusted if you have a smelly fart.
Consider a classroom looks like the above figure and we focus on the x direction. For 2 tiny cube located at x & x-L, the particles contained in these 2 tiny cubes satisfy the following relation:
Now we would like to derive the net flow rate j between these 2 tiny cubes, or how many particles would flow from one cube to the other in unit time per unit area. Assume that it would take time Δt for the particles to move L, and the particles are equally probable to move to its left or right. The net flow rate j could then be expressed as:
For we know the number of particles in a tiny cube could be written in terms of the concentration, or
So the net flow rate could be simplified into
If we define diffusion constant D by
Then we will get
And this is the well-famed Fick's first law. In 3 dimensional world it should be written as j = -Dc. That means, how much foul-smelling molecules would approach your classmates in unit time per unit area is proportional to the concentration gradient of these molecules.

After knowing the flow, we should be able to calculate how concentration changes with time. According to the mass conservation, the time evolution of the number of particles in a tiny cube at location x could be written as
Dividing LHS and RHS with LYZ and we will get:
If we substitute the above results into Fick's first law, and we will get:
And in 3 dimensional world it should be written as
This is Fick's second law. From this equation, the initial concentration profile of foul-smelling molecules and the diffusion constant of these molecules, we could solve the temporal changing of your fart. And that's our brief introduction to the macroscopic diffusion theory.