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Simplified model

Simplifying assumptions

[PK] and [LDH] are high, so their reactions are not rate limiting.

[ATP] and [Glu] are high enough to saturate GS.

Steady-state GS model

GS-Mg-ADP-Glu-P + NH₄⁺ → GS-Mg-ADP-(Glu-NH₃)-P → GS-Mg-ADP-Pi-Gln

Math

(see Orsi and Tipton, 1979)

$$Vt = [P] + K_m \ln\frac{[S]_0}{[S]_0-[P]} $$ $$Vt = ([S]_0-[S]) + K_m \ln\frac{[S]_0}{[S]}$$

Solving for $$[S]$$ gives us:

$$K_m W\left(\frac{[S]_0}{K_m} e^{\frac{[S]_0}{K_m} - \frac{Vt}{K_m}}\right)$$

This is identical to equation (13) in Goličnik, 2012.

The gory details

GS Model

from https://dx.doi.org/10.1021/bi00169a007

GS-Mn + ATP → GS-Mn-ATP + Glu → GS-Mn-ATP-Glu → GS-Mn-ADP-Glu-P + NH₄⁺ → GS-Mn-ADP-(Glu-NH₃)-P → GS-Mn-ADP-Pi-Gln

(We are actually using Mg, not Mn.)

LDH model

from https://core.ac.uk/download/pdf/81950158.pdf

LDH + NAD⁺ ⇌ LDH-NAD⁺ + lactate ⇌ LDH-NAD⁺-lactate ⇌ LDH-NADH-pyruvate ⇌ LDH-NADH + pyruvate ⇌ LDH + NADH

PK model

from https://pmc.ncbi.nlm.nih.gov/articles/PMC1184192

      PK-PEP                             PK-ATP + Pyr
    ↗↙       ↘                         ↗              ↘
 PK            PK-PEP-ADP → PK-Pyr-ATP                  PK + ATP + Pyr
    ↘        ↗↙                        ↘              ↗
      PK-ADP                             PK-Pyr + ATP

Combined model


                                                                     LDH-NAD⁺-lactate ⇌ LDH-NAD⁺ + lactate ⇌ LDH + NAD⁺
                                                                            ⇵ (rxn measured by absorbance change)
                                                                     LDH-NADH-Pyruvate
                                                                            ⇵
      PK-PEP                             PK-ATP + Pyr              Pyr + LDH-NADH ⇌ LDH + NADH
    ↗↙       ↘                         ↗              ↘             ⇵
 PK            PK-PEP-ADP → PK-Pyr-ATP                  PK + ATP + Pyr
    ↘        ↗↙                        ↘              ↗       ⇵
      PK-ADP                             PK-Pyr + ATP        ATP + GS-Mg
                                                                 ⇵
                                                             GS-Mg-ATP + Glu ⇌ GS-Mg-ATP-Glu ⇌ GS-Mg-ADP-Glu-P + NH₄⁺
                                                                                                        ⇵
                                                                                               GS-Mg-ADP-(Glu-NH₃)-P
                                                                                                        ⇵
                                                                                                 GS-Mg-ADP-Pi-Gln