# A state model of sodium channels

I am studying by myself Human Physiology. I have encountered the following question:

In the following given model of sodium channel with 3 states open closed blocked (which I assume means inactivated), the rates of going from state to stated are given in the picture below.

a. Write matrix $Q$ for the system and find it's eigenvalues. b. What does these values represent?

If anyone has an idea what is the required matrix I will be grateful. Thanks!

• Interesting question, looks like homework but i won't give you a minus. Can you describe full description, i mean, what these numbers mean. Suppose it is charge need to be possessed to change the state, if so, i can do that, it looks like simple matrix model if we add some probabilities of generating charge. I also suppose that you need to get eigenvalues just to find stable system state and that's good. – dshulgin Apr 26 '16 at 11:51
• I think Q is supposed to be an adjacency matrix that represents the graph in your diagram. Take a look here: en.wikipedia.org/wiki/Adjacency_matrix and here: cs.elte.hu/~lovasz/eigenvals-x.pdf – Justas Apr 26 '16 at 22:49
• This is a homework question. You should at least make some attempt at the answer. – WYSIWYG Apr 27 '16 at 8:37
• Thank you all. It is from HW I found in the net. So, no other information valid for me. I usually, of course, make a lot of self effort. But, sometimes, like in this case, even though you know you have the right tools, you just don't know where to start, especially, if not all the definitions are given... So, Thanks for your help and comments! – user135172 Apr 27 '16 at 14:25

## 1 Answer

There is no standard notation called Q (matrix).

However in this case I think the matrix that they are referring to is the state transition matrix (similar to the adjacency matrix as mentioned by Justas in the comments, but with rates instead of just the connections). Basically you have three states (lets call them A, B and C) and there is a rate for transition from one state to the other. You can also represent these rates in the form of transition probabilities.

You just represent this as a 3×3 matrix. Put zero where there are no such transitions (for example there is no transition from A to A or A to C):

$$\begin{array}{|l|c|c|c|} \hline & A & B & C \\\hline A & 0 & 10 & 0 \\\hline B & 100 & 0 & 50 \\\hline C & 5 & 0 & 0 \\\hline \end{array}$$

You can then calculate the eigenvalue for this matrix. How to calculate eigenvalues and what is their significance is off-topic in this site. You can easily find out how to calculate eigenvalues. Their significance is something that is not that easy to understand but you can read more about that and can perhaps ask a precise question in Mathematics Stack Exchange. Basically, they tell you how the system proceeds in different directions (denoted by eigenvectors).

• About the eigen-values, I know that if I view this matrix as a correlation matrix, then the eigenvalues are the variance values in the 3 element basis of eigen-vectors. Can I apply this character here? Anyway, Thank you!! – user135172 Apr 27 '16 at 14:22
• @user135172 just that this is not the correlation matrix. However, the concept of eigenvalues and eigenvectors is similar. Just like any linear algebra problem. If a matrix is a function applied to a vector (which may denote state of the system), then the eigenvalues somewhat denote the amplification factor of the matrix (along different directions). – WYSIWYG Apr 27 '16 at 18:19