The Citric Acid Cycle
Maggy
Tomova
Advisor:
Dr. Wyels
California
Lutheran University
CONTENT:
My Story.
What does the Krebs Cycle do?
Some more Info about the Cycle.
System of differential Equations.
Analysing the System.
Surprises.
Summary.
Ideas for further study.
The people who helped me do this project.
References:
My Story
A long time ago, when I was still young and stupid (that was approximately
8 months ago) I was a Biochemistry major. Then I realized what the purpose
of life was  spend lots of time comfortably stretched in the sun doing
cool Math stuff. After lots of hours spent in dark labs handling P32 (for
those of you who were lucky enough never to have to learn what this is,
it is nasty radioactive stuff) I can really appreciate the beauty of Math.
Still sometimes I feel kind of nostalgic and go back to my Biochemistry
books just to make sure I can still understand them. In one of those moments
I decided to do my senior Math project on a Biochemical Pathway.
What Does the Kreb's Cycle do?

Here is a little background on the Citric acid Cycle.
We will look at glucose because it is one of the main sugars our body uses
for fuel and most other sugars get converted into glucose and enter the
same patheway.
During glycolisis, the braking down of the sugar begins.
The energy of the bonds is harvested and stored in the form of ATP which
is just a high energy molecule. The final step in glycolisis yields pyruvate.
Each glucose molecule produces two pyruvates which are then converted to
acetyl CoA and are moved into the mitochondria where the Citric Acid Cycle
takes place. In the cycle, acetyl CoA binds to oxaloacetate to give citrate
and it gets comletely broken down to CO_{2} so more enrgy can be
harvested. At the end of the cycle only an oxaloacetate is left from the
citrate molecule and it is ready to bind to the next Acetyl CoA. 
This is the last piece of Biochemistry
you have to know before you can read the Math, I promise!
The Citric Acid Cycle has involves eight compounds which I labeled
A to H. Each of the eight steps is performed by an enzyme labeled E_{1}
to E_{8}
_{Regulation is another important Biochemical Phenomenon:}

Regulation assures that no energy will be wasted.

Major regulatory molecules are ATP and ADP.

Model will account for:

Step 1 is inhibited by ATP

Step 3 is activated by ADP and inhibited by ATP.

Step 4 is inhibited by succinyl CoA and by NADH.
Now we can finally do some math! To model this very complex process first
we have to make some simplifying assumptions.
They are:

All enzyme concentrations are constant.

All inhibitor and activator concentrations are constant.

There are no intermediates leaving the cycle.
The System of Differential Equations
After some thinking I came up with this system of eight differential
equations to describe the cycle. k is a reaction coefficient,
i is a reaction coefficient involved in inhibition and a is
a reaction coefficient involved in activation. Also I grouped all constants
together to make the system easier to work with.
In the beginning of this project I was hoping to find all the constants
involved but they haven't been experimentally determined. So I renamed
my constants and tried to learn as much as possible about the system using
differential equations. Here is the system replcing the grouped constants
by w1 to w8 and the constants associated with regulation were replaced
by wa, wb and wc.
Finally I converted this equation to a matrix form:
Analysing the System
Probably the easiest and most efficient way to analyse a matrix like
this is to find the eigenvalues. Unfortunately they are the roots of an
eight degree polynomial and that's about all Maple could tell me about
them. So I looked at the determinant and found that Det(W) = w2 w3 w6
w7 w8 (w4 w5 wa  2 w4 w5 w1 + w4 w1 wc w1 wb w5 + w1 wc wb).
Now I knew that wheather or not the determinant is zero depend entirely
on the values of w1, w4, w5, wa, wb and wc and they are all constants in
regulated steps.
Case 1: If the determinant equals zero, there is a
free variable in the equilibrium solution.

The eigenvalues of a randomly generated matrix with a determinant zero
are:
0, 9, 4.5±3.1i, 1.4 ±2.1i, 7.6 ±2.1i

The equilibrium solution of the matrix is:
A(t) = 4*G(t), B(t) = 8/3*G(t), C(t) = 2*G(t), D(t) = 8/5*G(t),
E(t) = 4/3*G(t), F(t) = 8/7*G(t), G(t) = G(t), H(t) = 8*G(t).

Here is sample solution curves generated by Maple:
Problem: If there is a free variable, starting from
different initial conditions gives different equilibrium points:
Case 2. The determinant does not equal zero.
Problem: The only equilibrium solution is the origin.
Determinant of the matrix = 48960
What would happen if we ignore regulation?
It turns out the determinant of the new matrixis always zero regardless
of the values of the constants.
What can we say about the possible eigenvalues for this matrix?
Because I didn't know the values of the parameters, my advisor suggested
I use random number generator to look at the eignevalues that I get from
my matrix. I used Maple and concluded that:

About 1/4 of the eigenvalues are real.

About 7/100 are real and positive.
This was good news for me because positive eigenvalues implies that the
concentrations go to infinity.Fortunately, it seems like positive eigenvalues
are not all that common for my matrix.
Surprises!

The behavior of a multidimensional system with real eigenvalues is markedly
different from a two dimensional system .
The solution curves of randmly generated 8x8 matrices with 8 real eigenvalues
oscillate. In two dimensions the solutions of a matrix with all real values
are either increasing or decreasing.

Repeated eigenvalues are very common for my system.
Many of my random matrices have 8 identical eigenvalues. This might
be because the system describes a cyclic process.
Summary

If the constants involved in the system of equations were
available I would be able to find the equilibrium solution and determine
the behavior of each of the chemical species using numerical methods.

My model suggests that regulation is of crucial importance
to this pathway which we already know from Biochemical observations.

If the determinant of the matrix is not zero, the only solution
of the system is the origin which is not possible from a Biochemical point
of view.

If the determinant is zero, the equilibrium point depends
on the initial conditions which is also impossible.
Ideas for further study.

Reconsider assumptions, particularly constant inhibitors.

Consider the system in the context of the body.

If parameters are known, use simulations.
Special Thanks
Dr. Wyels
Dr. Stanley
Dr. Revie
Dr. Singkofer
Dr. Tong
References:

Reaction Kinetics, Michael J. Pilling and Paul W. Seaking, Oxford Science
Publications, 1995

Mathematical Models in Biology, Leah EdelsteinKeshet, Random House/New
York, 1988

Biochemistry, Lubert Stryer, W.H Freeman and Company/New York, 1995

Citric Acid Cycle: Control and Compartmentation, Lowenstein, J.M. Marcel
Dekker, 1969

http://www.gene.com/ae/AB/GG/

http://incolor.inetnebr.com/mcanaday/Krebs%20Phases.htm