Cellular Metabolism

Metabolic Pathways & Reaction Balancing

Metabolism is an intricate network of chemical reactions catalyzed (or not) by various enzymes, resulting in a balanced and dynamic network of autocatalytic processes. Sometimes this web of reactions can be observed in the countless so-called metabolic maps available, printed or on the internet. However, metabolic pathways are classified for better understanding into subsets of reactions relevant to a certain group of nutrients or biological compounds, such as carbohydrates, lipids and nucleic acids, for example.
And in a broader view of these subsets, a further classification results in several sequential and interdependent biochemical reactions, the so-called metabolic pathways or routes. These metabolic pathways summarize catabolic or anabolic processes involving substrates, products, enzymes, cofactors and coenzymes, as illustrated in glycolysis, gluconeogenesis, glycogenolysis, citric acid cycle, respiratory chain, pentose pathway, fatty acid beta-oxidation, among others.
Thus, a metabolic pathway can be considered as a linear combination of enzymatically catalyzed reactions (e.g. anaerobic glycolysis). In addition, one can also consider biochemical reactions as biochemical equations, and therefore as a linear system of biochemical equations with resolution by linear algebra. Thus, one can use matrix relations to solve the stoichiometric balance (mass and charge), obtaining the final net reaction from a set of known reactions. In short, biochemical equations as matrix equations (Alberty 1991).

Matrix operation

By applying linear algebra, it is possible to obtain the final balance of sequential reactions by the following relation, also seen in the Biothermodynamics chapter:

\[ A = \begin{pmatrix} a_{11} & a_{12} & a_{13} & ... & a_{1n} \\ a_{21} & a_{22} & a_{23} & ... & a_{2n} \\ a_{31} & a_{32} & a_{33} & ... & a_{3n} \\ ... & ... & ... & ... & a_{jn}\\ \end{pmatrix} ,\ b = \begin{pmatrix} b_1 \\ b_2 \\ b_3 \\ ... \\ b_n \\ \end{pmatrix} ,\ y = \begin{pmatrix} y_1 \\ y_2 \\ y_3 \\ ... \\ y_n \\ \end{pmatrix} \tag{1}\]

In this case, A represents the stoichiometric matrix of reactions and compounds, b the metabolic pathway vector, and y the balanced net reaction vector. The pathway vector indicates the number of times each reaction must occur to produce the final net reaction. Matrix A is arranged in such a way as to present each reaction in each column and each reactant in each row, filling it with the stoichiometric coefficients of reactants and products. For these, it is necessary to present a positive sign for products (they are formed) and a negative sign for reactants (they are consumed).
The matrix operation to obtain the net reaction involves only a cross product, such as:

\[ A * b = y \tag{2}\]

Obtaining the metabolic pathway vector for aerobic glycolysis:

Assuming that we know the summarized net reaction for a set of metabolic reactions, such as aerobic glycolysis (in turn including anaerobic glycolysis, the citric acid cycle, and oxidative phosphorylation), we can easily obtain the pathway vector. From this calculated pathway vector, we can obtain the final net reaction of the more detailed set of reactions. In practice, this detailing involves the participation of oxidation-reduction coenzymes, ADP, ATP, and Pi.

For example, consider the reactions below, related to a glycolysis synthesis (Alberty 1996):

\[ glucose \rightleftarrows 2\,pyruvate \\ pyruvate + CoA \rightleftarrows acetyl \, CoA + CO_2 acetyl \, CoA + 3 H_2O \rightleftarrows 2 CO_2+CoA \frac{1}{2} O_2 \rightleftarrows H_2O \tag{3}\]

\[ acetyl \, CoA + 3 H_2O \rightleftarrows 2 CO_2+CoA \frac{1}{2} O_2 \rightleftarrows H_2O \\ \]

The sequential reactions above have as a net resulting reaction:

$$ glucose + 6O_2 CO_2 + 6 H_2O

$$ {#eq-eqGlicoSimpFin}

To obtain the route vector, it is necessary to elaborate the matrix relation, such that:

\[ A = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 2 & -1 & 0 & 0 \\ 0 & 0 & -3 & 1 \\ 0 & -1 & 1 & 0 \\ 0 & 1 & -1 & 0 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & -1/2 \\ \end{pmatrix} \tag{4}\]

\[ y = \begin{pmatrix} -1 \\ 0 \\ 6 \\ 0 \\ 0 \\ 6 \\ -6 \\ \end{pmatrix} \tag{5}\]

Thus, the solution for the route vector is:

\[ b = A^{-1} * y \tag{6}\]

It turns out that the linear system for the set of equations of the glycolytic pathway is overestimated (there are more equations than unknowns), not allowing the solution by the solve function directly, as illustrated in the Biothermodynamics chapter. In this case, the vector b can be obtained by least squares solution:

\[ b = (A^{T}*A)^{-1}*A^T * y \tag{7}\]

Solving the vector of pathways in R:

# Matrix solution for the vector of metabolic pathways

A <- matrix(c(-1, 0, 0, 0, 2, -1, 0, 0, 0, 0, -3, 1, 0, -1, 1, 0, 0, 1,
-1, 0, 0, 1, 2, 0, 0, 0, 0, -1 / 2), nrow = 7,
byrow = TRUE) # matrix A of reaction stoichiometry
rownames(A) <- list("glucose", "pyruvate", "H2O", "CoA", "acetyl CoA"
, "CO2", "O2") # reactant labels
A # matrix A of reactions

           [,1] [,2] [,3] [,4]
glucose      -1    0    0  0.0
pyruvate      2   -1    0  0.0
H2O           0    0   -3  1.0
CoA           0   -1    1  0.0
acetyl CoA    0    1   -1  0.0
CO2           0    1    2  0.0
O2            0    0    0 -0.5

y <- c(-1, 0, 6, 0, 0, 6, -6) # vector y of routes

solve(t(A) %*% A) %*% t(A) %*% y

     [,1]
[1,]    1
[2,]    2
[3,]    2
[4,]   12

Obtaining the balance of ATP, ADP, Pi and redox coenzymes in glycolysis

With the obtained route vector ({1,2,2,12}), it is now possible to apply it to a more extensive set of glycolysis equations, to this time solve the balancing of reactions involving oxidation-reduction coenzymes, ATP, ADP, and Pi:

\[ glucose+2Pi+2ADP+2NAD^+ \rightleftarrows 2 \,pyruvate+2ATP+2NADH+2H_2O \, , \\ glycolysis \, anaerobic, \] \[ pyruvate+CoA+NAD^+ \rightleftarrows acetyl-CoA+CO_2+NADH \\ \]

\[ acetyl-CoA + 4NAD^++ADP+Pi+2H_2O \rightleftarrows2CO_2+4NADH+ATP+CoA, \, \\ \, pyruvate \, dehydrogenase \] \[ NADH+ \frac{1}{2}O_2+3Pi+3ADP \rightleftarrows NAD^++4H_2O+3ATP \, ,\\ oxidation \, phosphorylative \tag{8}\]

It is worth mentioning a simplification by replacing GTP and GDP with ATP and ADP (Krebs cycle), since they are interconvertible in metabolism. For simplification, the interconversion of FAD and NAD in the net reaction is also assumed (Alberty 1996) :

\[ FADH + NAD^+ \rightleftarrows FAD + NADH \tag{9}\]

With the reactions present in Equation 8 and the route vector obtained previously, the new stoichiometric matrix can be constructed and the least squares solution (Equation 7) can be applied to obtain the net reaction of glycolysis. And to do this, simply apply the Equation 2) cross product:

A <- matrix(c(-1, 0, 0, 0, -2, 0, -1, -3, -2, 0, -1, -3, -2, -1,
-4, 1, 2, -1, 0, 0, 2, 0, 1, 3, 2, 1, 4, -1, 2, 0,
-2, 4, 0, -1, 1, 0, 0, 1, -1, 0, 0, 1, 2, 0, 0, 0,
0, -1 / 2), nrow = 12, byrow = TRUE)
# matrix A of reaction stoichiometry
rownames(A) <- list("glucose", "Pi", "ADP", "NAD", "pyruvate", "ATP",
"NADH", "H2O", "CoA", "acetyl CoA", "CO2", "O2")
# reagent labels
A # reaction matrix A

           [,1] [,2] [,3] [,4]
glucose      -1    0    0  0.0
Pi           -2    0   -1 -3.0
ADP          -2    0   -1 -3.0
NAD          -2   -1   -4  1.0
pyruvate      2   -1    0  0.0
ATP           2    0    1  3.0
NADH          2    1    4 -1.0
H2O           2    0   -2  4.0
CoA           0   -1    1  0.0
acetyl CoA    0    1   -1  0.0
CO2           0    1    2  0.0
O2            0    0    0 -0.5

y <- c(1, 2, 2, 12) # vector y of routes

A %*% y

           [,1]
glucose      -1
Pi          -40
ADP         -40
NAD           0
pyruvate      0
ATP          40
NADH          0
H2O          46
CoA           0
acetyl CoA    0
CO2           6
O2           -6

# solve(t(A)%*%A)%*%t(A)%*%y

From this result, the final net reaction of the glycolytic pathway can be assembled as:

\[ glucose + 6O_2 + 40 ADP + 40 Pi \rightleftarrows 6CO_2+ 40ATP \tag{10}\]

The example above illustrates the use of matrix algebra to solve problems of balancing biochemical reactions. By its nature, it is a comprehensive and algorithmic method (therefore, programmable), although it is not the only one. Other proposed solutions for balancing and conservation of mass and charge include direct inspection by screening and error based on mnemonic rules, balancing by half-equation method, and the oxidation number method, not discussed here.

References

Alberty, Robert A. 1991. “Chemical Equations Are Actually Matrix Equations.” Journal of Chemical Education 68 (12): 984.

———. 1996. “Calculation of Biochemical Net Reactions and Pathways by Using Matrix Operations.” Biophysical Journal 71 (1): 507–15.