Computer Simulation of Metabolism
Compounds Receiving Several ^{13}C Atoms From ^{13}CO_{2}
The following figure illustrates a simple Visual Basic program for simulating the ^{13}C labeling kinetics of compounds containing different numbers of carbon atoms (from 1  7), with each compound assumed to be derived from ^{13}CO_{2} via negligible pools of intermediates.
^{13}C labeling kinetics of a 6 carbon compound derived from ^{13}CO_{2} via negligible pools of intermediates
In the figure shown above the labeling patterns for a 6 carbon compound of pool size 2,000 nmol.gfw^{1} (B2) are illustrated. The 6 carbon compound is both synthesized (B3) and utilized (B4) at a rate of 5,000 nmol.h^{1}.gfw^{1} and so the pool size remains constant with respect to time. At time (t) = 0 h, the compound is envisaged to be at natural ^{13}C abundance (i.e. 1 atom %) (B1). The compound is derived from ^{13}CO_{2} supplied at 90 % ^{13}C (A1) during the "pulse" [beginning at t = 0 h] and at natural ^{13}C abundance (i.e. 1 atom %) (A2) during the "chase". The "chase" is initiated at t = 2.0 h. Simulated are the timecourses of relative abundance (% total; lower left graph panel) and absolute levels (nmol.gfw^{1}; lower right graph) of species with 0, 1, 2, 3, 4, 5, and 6 ^{13}C atoms.
Analogous simulations can be viewed for the labeling patterns of compounds with 7, 5, 4, 3, 2, and 1 carbon atom, assuming the same isotope abundances, flux and pool size as illustrated for the 6 carbon compound above.
Simulations can also be viewed for the labeling patterns of a 7 carbon compound assumed to be at 0 % ^{13}C at t = 0 h, derived from ^{13}CO_{2} supplied at either 100, 95, 90, 70, 40, or 20 % ^{13}C during the pulse, and at 0 % ^{13}C during the chase. In all cases the chase is initiated at t = 2.0 h. It can be seen from these figures that the higher the ^{13}C abundance of ^{13}CO_{2} supplied during the "pulse", the higher the probability of forming the fully labeled species.
For a compound containing 1 C atom the absolute amount (nmol.gfw^{1}) of the species that has zero labeled atoms (F0) at t = 0 h is calculated as follows:
F0 = (1  B1) * B2
[where B2 is the pool size (nmol.gfw^{1}), and B1 is the isotope abundance (atom %) of the compound divided by 100]
For a compound containing 1 C atom the absolute amount (nmol.gfw^{1}) of the species that has 1 labeled atom (F1) at t = 0 h is:
Similarly, for a compound containing 2 C atoms:
F0 = (1  B1) * (1  B1) * B2
F1 = (1  B1) * B1 * 2 * B2
F2 = B1 * B1 * B2
For a compound containing 3 C atoms:
F0 = (1  B1) * (1  B1) * (1  B1) * B2
F1 = (1  B1) * (1  B1) * B1 * 3 * B2
F2 = B1 * B1 * (1  B1) * 3 * B2
F3 = B1 * B1 * B1 * B2
For a compound containing 4 C atoms:
F0 = (1  B1) * (1  B1) * (1  B1) * (1  B1) * B2
F1 = (1  B1) * (1  B1) * (1  B1) * B1 * 4 * B2
F2 = B1 * B1 * (1  B1) * (1  B1) * 6 * B2
F3 = B1 * B1 * B1 * (1  B1) * 4 * B2
F4 = B1 * B1 * B1 * B1 * B2
For a compound containing 5 C atoms:
F0 = (1  B1) * (1  B1) * (1  B1) * (1  B1) * (1  B1) * B2
F1 = (1  B1) * (1  B1) * (1  B1) * (1  B1) * B1 * 5 * B2
F2 = B1 * B1 * (1  B1) * (1  B1) * (1  B1) * 10 * B2
F3 = B1 * B1 * B1 * (1  B1) * (1  B1) * 10 * B2
F4 = B1 * B1 * B1 * B1 * (1  B1) * 5 * B2
F5 = B1 * B1 * B1 * B1 * B1 * B2
For a compound containing 6 C atoms:
F0 = (1  B1) * (1  B1) * (1  B1) * (1  B1) * (1  B1) * (1  B1) * B2
F1 = (1  B1) * (1  B1) * (1  B1) * (1  B1) * (1  B1) * B1 * 6 * B2
F2 = (1  B1) * (1  B1) * (1  B1) * (1  B1) * B1 * B1 * 15 * B2
F3 = (1  B1) * (1  B1) * (1  B1) * B1 * B1 * B1 * 20 * B2
F4 = (1  B1) * (1  B1) * B1 * B1 * B1 * B1 * 15 * B2
F5 = (1  B1) * B1 * B1 * B1 * B1 * B1 * 6 * B2
F6 = B1 * B1 * B1 * B1 * B1 * B1 * B2
For a compound containing 7 C atoms:
F0 = (1  B1) * (1  B1) * (1  B1) * (1  B1) * (1  B1) * (1  B1) * (1  B1) * B2
F1 = (1  B1) * (1  B1) * (1  B1) * (1  B1) * (1  B1) * (1  B1) * B1 * 7 * B2
F2 = (1  B1) * (1  B1) * (1  B1) * (1  B1) * (1  B1) * B1 * B1 * 21 * B2
F3 = (1  B1) * (1  B1) * (1  B1) * (1  B1) * B1 * B1 * B1 * 35 * B2
F4 = (1  B1) * (1  B1) * (1  B1) * B1 * B1 * B1 * B1 * 35 * B2
F5 = (1  B1) * (1  B1) * B1 * B1 * B1 * B1 * B1 * 21 * B2
F6 = (1  B1) * B1 * B1 * B1 * B1 * B1 * B1 * 7 * B2
F7 = B1 * B1 * B1 * B1 * B1 * B1 * B1 * B2
Once the pool sizes of the various species are calculated at t = 0 h, as described above, during each iteration of the program (Z = 1/2000^{th} of the total time scale of the simulation) the pool of each species initially expands in proportion to the product of the rate of synthesis (B3) and the appropriate probability of forming that species from ^{13}CO_{2}. Thus, for a compound containing 1 C atom the absolute amount (nmol.gfw^{1}) of the species that has zero labeled atoms (F0) will expand by the following amount during each iteration (Z):
F0 = F0 + (1  A1) * Z * B3
[where B3 is the rate of synthesis (nmol.h^{1}.gfw^{1}), and A1 is the isotope abundance (atom %) of the supplied ^{13}CO_{2} divided by 100]
For a compound containing 1 C atom, the absolute amount (nmol.gfw^{1}) of the species that has 1 labeled atom (F1) will expand by the following amount during each iteration (Z):
The total pool size of the compound becomes B2 = F0 + F1, the probablity of utilizing the material that has zero labeled atoms becomes PUF0 = F0 / B2, and the probablity of utilizing the material that has one labeled atom becomes PUF1 = F1 / B2. Thus, for a compound containing 1 C atom the absolute amount (nmol.gfw^{1}) of the species that has zero labeled atoms (F0) will decline by the following amount during each iteration (Z):
F0 = F0  PUF0 * Z * B4
[where B4 is the rate of utilization (nmol.h^{1}.gfw^{1}), and PUF0 is the fraction of the total pool of B that has 0 labeled atoms]
For a compound containing 1 C atom, the absolute amount (nmol.gfw^{1}) of the species that has 1 labeled atom (F1) will decline by the following amount during each iteration (Z):
Because the small expansions and contractions in total pool size of the compound that occur during each small iteration are exactly balanced (i.e. B3 = B4), the total pool size of the compound remains constant.
Similarly for a compound with 2 carbon atoms the relevant equations describing synthesis and utilization during each iteration are:
B2 = F0 + F1 + F2
F0 = F0 + ((1  A1) * (1  A1)) * Z * B3
F1 = F1 + (A1 * (1  A1) * 2) * Z * B3
F2 = F2 + (A1 * A1) * Z * B3
B2 = F0 + F1 + F2
PUF0 = F0 / B2
PUF1 = F1 / B2
PUF2 = F2 / B2
F0 = F0  PUF0 * Z * B4
F1 = F1  PUF1 * Z * B4
F2 = F2  PUF2 * Z * B4
For a compound containing 3 C atoms:
B2 = F0 + F1 + F2 + F3
F0 = F0 + ((1  A1) * (1  A1) * (1  A1)) * Z * B3
F1 = F1 + (A1 * (1  A1) * (1  A1) * 3) * Z * B3
F2 = F2 + (A1 * A1 * (1  A1) * 3) * Z * B3
F3 = F3 + (A1 * A1 * A1) * Z * B3
B2 = F0 + F1 + F2 + F3
PUF0 = F0 / B2
PUF1 = F1 / B2
PUF2 = F2 / B2
PUF3 = F3 / B2
F0 = F0  PUF0 * Z * B4
F1 = F1  PUF1 * Z * B4
F2 = F2  PUF2 * Z * B4
F3 = F3  PUF3 * Z * B4
For a compound containing 4 C atoms:
B2 = F0 + F1 + F2 + F3 + F4
F0 = F0 + ((1  A1) * (1  A1) * (1  A1) * (1  A1)) * Z * B3
F1 = F1 + (A1 * (1  A1) * (1  A1) * (1  A1) * 4) * Z * B3
F2 = F2 + (A1 * A1 * (1  A1) * (1  A1) * 6) * Z * B3
F3 = F3 + (A1 * A1 * A1 * (1  A1) * 4) * Z * B3
F4 = F4 + (A1 * A1 * A1 * A1) * Z * B3
B2 = F0 + F1 + F2 + F3 + F4
PUF0 = F0 / B2
PUF1 = F1 / B2
PUF2 = F2 / B2
PUF3 = F3 / B2
PUF4 = F4 / B2
F0 = F0  PUF0 * Z * B4
F1 = F1  PUF1 * Z * B4
F2 = F2  PUF2 * Z * B4
F3 = F3  PUF3 * Z * B4
F4 = F4  PUF4 * Z * B4
For a compound containing 5 C atoms:
B2 = F0 + F1 + F2 + F3 + F4 + F5
F0 = F0 + ((1  A1) * (1  A1) * (1  A1) * (1  A1) * (1  A1)) * Z * B3
F1 = F1 + (A1 * (1  A1) * (1  A1) * (1  A1) * (1  A1) * 5) * Z * B3
F2 = F2 + (A1 * A1 * (1  A1) * (1  A1) * (1  A1) * 10) * Z * B3
F3 = F3 + (A1 * A1 * A1 * (1  A1) * (1  A1) * 10) * Z * B3
F4 = F4 + (A1 * A1 * A1 * A1 * (1  A1) * 5) * Z * B3
F5 = F5 + (A1 * A1 * A1 * A1 * A1) * Z * B3
B2 = F0 + F1 + F2 + F3 + F4 + F5
PUF0 = F0 / B2
PUF1 = F1 / B2
PUF2 = F2 / B2
PUF3 = F3 / B2
PUF4 = F4 / B2
PUF5 = F5 / B2
F0 = F0  PUF0 * Z * B4
F1 = F1  PUF1 * Z * B4
F2 = F2  PUF2 * Z * B4
F3 = F3  PUF3 * Z * B4
F4 = F4  PUF4 * Z * B4
F5 = F5  PUF5 * Z * B4
For a compound containing 6 C atoms:
B2 = F0 + F1 + F2 + F3 + F4 + F5 + F6
F0 = F0 + ((1  A1) * (1  A1) * (1  A1) * (1  A1) * (1  A1) * (1  A1)) * Z * B3
F1 = F1 + ((1  A1) * (1  A1) * (1  A1) * (1  A1) * (1  A1) * A1 * 6) * Z * B3
F2 = F2 + ((1  A1) * (1  A1) * (1  A1) * (1  A1) * A1 * A1 * 15) * Z * B3
F3 = F3 + ((1  A1) * (1  A1) * (1  A1) * A1 * A1 * A1 * 20) * Z * B3
F4 = F4 + ((1  A1) * (1  A1) * A1 * A1 * A1 * A1 * 15) * Z * B3
F5 = F5 + ((1  A1) * A1 * A1 * A1 * A1 * A1 * 6) * Z * B3
F6 = F6 + (A1 * A1 * A1 * A1 * A1 * A1) * Z * B3
B2 = F0 + F1 + F2 + F3 + F4 + F5 + F6
PUF0 = F0 / B2
PUF1 = F1 / B2
PUF2 = F2 / B2
PUF3 = F3 / B2
PUF4 = F4 / B2
PUF5 = F5 / B2
PUF6 = F6 / B2
F0 = F0  PUF0 * Z * B4
F1 = F1  PUF1 * Z * B4
F2 = F2  PUF2 * Z * B4
F3 = F3  PUF3 * Z * B4
F4 = F4  PUF4 * Z * B4
F5 = F5  PUF5 * Z * B4
F6 = F6  PUF6 * Z * B4
For a compound containing 7 C atoms:
B2 = F0 + F1 + F2 + F3 + F4 + F5 + F6 + F7
F0 = F0 + ((1  A1) * (1  A1) * (1  A1) * (1  A1) * (1  A1) * (1  A1) * (1  A1)) * Z * B3
F1 = F1 + ((1  A1) * (1  A1) * (1  A1) * (1  A1) * (1  A1) * (1  A1) * A1 * 7) * Z * B3
F2 = F2 + ((1  A1) * (1  A1) * (1  A1) * (1  A1) * (1  A1) * A1 * A1 * 21) * Z * B3
F3 = F3 + ((1  A1) * (1  A1) * (1  A1) * (1  A1) * A1 * A1 * A1 * 35) * Z * B3
F4 = F4 + ((1  A1) * (1  A1) * (1  A1) * A1 * A1 * A1 * A1 * 35) * Z * B3
F5 = F5 + ((1  A1) * (1  A1) * A1 * A1 * A1 * A1 * A1 * 21) * Z * B3
F6 = F6 + ((1  A1) * A1 * A1 * A1 * A1 * A1 * A1) * 7 * Z * B3
F7 = F7 + (A1 * A1 * A1 * A1 * A1 * A1 * A1) * Z * B3
B2 = F0 + F1 + F2 + F3 + F4 + F5 + F6 + F7
PUF0 = F0 / B2
PUF1 = F1 / B2
PUF2 = F2 / B2
PUF3 = F3 / B2
PUF4 = F4 / B2
PUF5 = F5 / B2
PUF6 = F6 / B2
PUF7 = F7 / B2
F0 = F0  PUF0 * Z * B4
F1 = F1  PUF1 * Z * B4
F2 = F2  PUF2 * Z * B4
F3 = F3  PUF3 * Z * B4
F4 = F4  PUF4 * Z * B4
F5 = F5  PUF5 * Z * B4
F6 = F6  PUF6 * Z * B4
F7 = F7  PUF7 * Z * B4
The nmol amounts of the various species (F0 to F7) or their relative abundance in the total pool are then plotted as a function of time. The "chase" is simply initiated by assigning a new value (A2) of isotope abundance of ^{13}CO_{2} to A1 at the specified time.
Download the Visual Basic program illustrated above. To run this program you must have Visual Basic 5.0 (or greater) installed on your computer.
