Mitocondrial pyruvate metabolism

Mitocondrial pyruvate metabolism(MPM) is a pseudo reaction that represents the total ATP production from one unit of pyruvate in the mitochondrian.

Chemical reaction

$Pyruvate + 13ADP + 13Pi \leftrightarrow 13ATP$

Rate equation

• Chemical reactions proceed to equilibrium within closed systems. For a simple reaction $A \rightarrow B$ it is defined as $K_{eq} = \frac{[B]_{eq}}{[A]_{eq}}$ where forward and reverse rates are equal.
• Equilibrium is not reached in open system due to influx and outflux. Mass action ratio $\tau$ for $A \rightarrow B$ reaction is defined as $\tau = \frac{[B]_{ob}}{[A]_{ob}}$ where subscript ob represents observable at a given point.
• Deviation from equilibrium is measured with Disequilibrium constant $\rho$ as $\rho = \frac{\tau}{K_{eq}}$
• Given the simple uni molecular reaction $A \leftrightarrow B$ the mass action equation can be modified as

$v=K_1A-K_2B$
$v=K_1A \left(1-\frac{K_2B}{K_1A} \right)$

Considering $K_{eq} = \frac{K_1}{K_2}$ and $\tau = \frac{P_1P_2 \ldots}{S_1S_2 \ldots}$ we have,

$v=K_1A \left(1-\frac{\tau}{K_{eq}} \right)$

• The generalized arbitrary mass action ratio gives us

$v = K_1A^{n_1}B^{n_2} \ldots \left(1-\frac{\tau}{K_{eq}} \right)$ Or,
$v = K_1A^{n_1}B^{n_2} \ldots (1 - \rho)$

For eg. for the reaction $2A + B \leftrightarrow C + 2D$, the rate law would be $K_1A^2B \left(1-\rho\right)$

• This equation demonstrates how a rate expression can be divided into parts that include both kinetics and thermodynamic properties .

• Given the net rate of reaction $v = v_f \left( 1 - \frac{v_r}{v_f} \right)$, we have

$v = v_f \left(1 - \rho \right)$

In this model

• The rate law is defined as $v = K_1[Pyruvate][ADP]^{13}[Pi]^{13}\left(1-\frac{\frac{[ATP]^{13}}{[Pyruvate][ADP]^{13}[Pi]^{13}}}{K_{eq}}\right)$
  
• The overall standard free-energy change for Pyruvate metabolism is $\Delta G^o{'}= -50.3 Kj/Mol$. Note The overall standard free energy is calculated by adding the standard free energy for all the reactions in TCA cycle.

Calculating $K_{eq}$ value from these free energy gives $\Delta G' = - 50.3\ kJ.mol^{-1}$, $Keq = exp(-\frac{\Delta G'}{RT}) = exp(\frac{50300}{8.31*298.15}) \approx 656010875.924588$. In order to ensure that the uncertainty does not affect the model equilibrium a small uncertainty of 5% can be considered for transporter. In our model we have applied this approach. So $K_{eq} = 654904512.15 \pm 32800543.79$.

• The Flux of pyruvate consumed by mitochondria measured for AS_30D is $v = 1.8$ .
• The initial concentrations for substrates and products are $ATP=8.7 \pm 3 (5)$, $ADP = 2.7 \pm 1.3$, $Pyruvate = 8.5 \pm 3.6$ and $Pi = 7.5$.
• Considering the stoichiometry of the model the value of $K_1$ is $5.1344e-017$. If we ingnore the stiochiometric constants while calculating $K_1$, the value calculated from the above mentioned values are $0.01045$
• To calculate the uncertainty of $K_1$ we first looked at the uncertainty on the substrate and product concentration. The maximum uncertainty reported for these values are $\approx 50%$. In our model we considered this $50%$ uncertainty in its mean value giving value of $0.01045 \pm 0.0052$

• Alternative As this is a psuedo-reaction, the above mentioned formula fails to reach steady state. So a constant flux mentioned in Hernandez et. al. can be used. The maximum relative precent of error for other constant fluxes are reported to be 31%. We consider the same relative percent error for this reaction because of it being a psuedo-reaction.

Parameter values

Parameter	Value	Organism	Remarks
$K_1$	$7.78E^{-015}$ In our model we used $0.01045 \pm 0.0052$
$K_{eq}$	$654904512.15$

Alternative

Parameter	Value	Organism	Remarks
$V$	$1.0^{-04}$		Constant Flux

Parameters with uncertainty

Parameter	Value	Organism	Remarks
$K_1$	$7.78E^{-015} \pm 3.89E^{-015}$
$K_{eq}$	$654904512.15 \pm 32745226$

Alternative

Parameter	Value	Organism	Remarks
$V$	$1.0^{-04}\pm 3.1^{-05}$		Constant Flux