Team:NRP-UEA-Norwich/Modeling/Kinetic

Enzyme kinetics

To analyse the efficiency of the branching (GlgB) and debranching (GlgX) enzymes in glycogen production, we generated a set of differential equations to explain glycogen biosynthesis. We used COPASI₁ to simulate the biochemical reactions involved in the production of glycogen in the bacterial cell. We used the glycogen biosynthesis pathway defined by Wilson W.A. et al. ₂ to build our model. The kinetic parameters were found at http://www.metacyc.org/ .

We generated a deterministic model taking into consideration the following assumptions:

•All enzyme concentrations are constant and similar. We fixed the enzyme concentrations to 1x10_-5 mmol/mL.
• Glucose is available and limiting (initial concentration 10 mmol/ml).
• Reactions follow simple reversible or irreversible Michaelis-Menten kinetics.
•There is no additional flux of substrates after the beginning of the simulation.
• The intracellular metabolite concentrations (ATP, AMP, ADP and PPi) are constant. They are all set to 1x10_-5 mmol/mL.
•Only competitive inhibition by the product occurs. For simplicity, other external inhibitors are not considered in the model.
•GlgX and GlgP were considered together in the same as the action of both enzymes together is required to get debranching of glycogen.

GlgB catalyzes two consecutive reactions. First, it cleaves an alpha-1,4 glycosidic linkage in a linear glucan to form a non-reducing-end oligosaccharide chain that is transferred to a C-6 hydroxyl group of the same or another glucan. GlgP removes up to five glucose units from the glycogen outer chain and GlgX only cuts when there are 3-4 glucose residues left at the branching point.

Enzyme	Km (mmol/mL)	Reference
Pgm(G1P)	2.9x10-4	3
Pgm(G6P)	5.6x10-6	3
GlgC(ADPG)	4x10-5	4
GlgC(ADPG)	1.67x10-4	4
AspP (ADPG)	3.2x10-4	7
GlgA	3.5x10-5	5
GlgB(Glucan1-4)	1.42x10-5	6
GlgX (Glycogen)	1x10-6	Not found
GlgP (Glycogen)	1x10-6	Not found

Basal model

All the reactions of the pathway are highly efficient as all of the initial glucose-6-phosphate is converted to glucose-1-phosphate and ADP-glucose, which is the glycosyl donor used to synthesize glycogen. Part of the glucose-1-phosphate is recovered from GlgX debranching activity (see Figure 1).

We set all the reaction rates to 0.01 mmol/(mL*s) except for GlgX-GlgP which was set to 0.002 mmol/(mL*s) to avoid the immediate degradation of all the glycogen being produced. Those were considered our basal conditions. After 50 minutes, all the initial glucose is being converted into glycogen and some glucose-1-phosphate is being released after the start due to the effect of the debranching enzyme. However, it is again used up until the maximum concentration of glycogen is achieved.

Increasing GlgX and decreasing GlgB

By increasing GlgX concentration in the cell, the reaction rate greatly increases (see Figure 2). The increase in the reaction rate produces a release of glucose-1-phosphate to a point where all the glycogen produced is converted immediately to glucose-1-phosphate.

On the other hand, by decreasing GlgB concentration, part of the glucose is converted to linear 1,4 glucan and cannot get branched.

Therefore, the ratio between the concentration of both enzymes will determine the output. If GlgB is high and GlgX is low, all the initial glucose will be converted to glycogen (basal conditions). If GlgB is low and GlgX is high, all the glycogen will be converted back to glucose-1-phosphate.

Differential equations

$$ \begin{array}{ccl} \frac {\mathrm{d}\left( {{\mathrm{[G6P]}} \, } \right) } {\mathrm{d}{t} } \; &=& \; { \, - \, \, \left(\frac {\frac {{\mathrm{Vf}}_{\mathrm{(Pgm)}} \, \cdot \, {\mathrm{[G6P]}} } {{\mathrm{Kms}}_{\mathrm{(Pgm)}} } \, - \, \frac {{\mathrm{Vr}}_{\mathrm{(Pgm)}} \, \cdot \, {\mathrm{[G1P]}} } {{\mathrm{Kmp}}_{\mathrm{(Pgm)}} } } { {{1} \, + \, \frac{\mathrm{[G6P]}}{{\mathrm{Kms}}_{\mathrm{(Pgm)}} } } \, + \, \frac{\mathrm{[G1P]}}{{\mathrm{Kmp}}_{\mathrm{(Pgm)}} } } \right) } \\ && \\ \frac {\mathrm{d}\left( {{\mathrm{[G1P]}} \, } \right) } {\mathrm{d}{t} } \; &=& \; { \, + \, \, \left(\frac {\frac {{\mathrm{Vf}}_{\mathrm{(Pgm)}} \, \cdot \, {\mathrm{[G6P]}} } {{\mathrm{Kms}}_{\mathrm{(Pgm)}} } \, - \, \frac {{\mathrm{Vr}}_{\mathrm{(Pgm)}} \, \cdot \, {\mathrm{[G1P]}} } {{\mathrm{Kmp}}_{\mathrm{(Pgm)}} } } { {{1} \, + \, \frac{\mathrm{[G6P]}}{{\mathrm{Kms}}_{\mathrm{(Pgm)}} } } \, + \, \frac{\mathrm{[G1P]}}{{\mathrm{Kmp}}_{\mathrm{(Pgm)}} } } \right) } \\ && \\ \; && \; { \, - \, \, \left(\frac { {{\mathrm{Vmax}}_{\mathrm{(GlgC)}} \, \cdot \, {\mathrm{[G1P]}} } \, \cdot \, {\mathrm{[ATP]}} } { { { {{\mathrm{KmB}}_{\mathrm{(GlgC)}} \, \cdot \, {\mathrm{KmA}}_{\mathrm{(GlgC)}} } \, + \, {{\mathrm{KmB}}_{\mathrm{(GlgC)}} \, \cdot \, {\mathrm{[G1P]}} } } \, + \, {{\mathrm{KmA}}_{\mathrm{(GlgC)}} \, \cdot \, {\mathrm{[ATP]}} } } \, + \, {{\mathrm{[G1P]}} \, \cdot \, {\mathrm{[ATP]}} } } \right) } \\ && \\ \; && \; { \, + \, \, \left(\frac {{V}_{\mathrm{("GlgX-GlgP")}} \, {\mathrm{[Glycogen]}} } {{\mathrm{Km}}_{\mathrm{("GlgX-GlgP")}} \, + \, {\mathrm{[Glycogen]}} } \right) } \\ && \\ \; && \; { \, + \, \, \left(\frac {{V}_{\mathrm{(AspP)}} \, \cdot \, {\mathrm{[ADPG]}} } {{\mathrm{Km}}_{\mathrm{(AspP)}} \, + \, {\mathrm{[ADPG]}} } \right) } \\ && \\ \frac {\mathrm{d}\left( {{\mathrm{[ADPG]}} \, } \right) } {\mathrm{d}{t} } \; &=& \; { \, - \, \, \left(\frac {{V}_{\mathrm{(GlgA)}} \, \cdot \, {\mathrm{[ADPG]}} } {{\mathrm{Km}}_{\mathrm{(GlgA)}} \, + \, {\mathrm{[ADPG]}} } \right) } \\ && \\ \; && \; { \, + \, \, \left(\frac { {{\mathrm{Vmax}}_{\mathrm{(GlgC)}} \, \cdot \, {\mathrm{[G1P]}} } \, \cdot \, {\mathrm{[ATP]}} } { { { {{\mathrm{KmB}}_{\mathrm{(GlgC)}} \, \cdot \, {\mathrm{KmA}}_{\mathrm{(GlgC)}} } \, + \, {{\mathrm{KmB}}_{\mathrm{(GlgC)}} \, \cdot \, {\mathrm{[G1P]}} } } \, + \, {{\mathrm{KmA}}_{\mathrm{(GlgC)}} \, \cdot \, {\mathrm{[ATP]}} } } \, + \, {{\mathrm{[G1P]}} \, \cdot \, {\mathrm{[ATP]}} } } \right) } \\ && \\ \; && \; { \, - \, \, \left(\frac {{V}_{\mathrm{(AspP)}} \, \cdot \, {\mathrm{[ADPG]}} } {{\mathrm{Km}}_{\mathrm{(AspP)}} \, + \, {\mathrm{[ADPG]}} } \right) } \\ && \\ \frac {\mathrm{d}\left( {{\mathrm{[1,4glucan]}} \, } \right) } {\mathrm{d}{t} } \; &=& \; { \, + \, \left(\frac {{V}_{\mathrm{(GlgA)}} \, \cdot \, {\mathrm{[ADPG]}} } {{\mathrm{Km}}_{\mathrm{(GlgA)}} \, + \, {\mathrm{[ADPG]}} } \right) } \\ && \\ \; && \; { \, - \, \left(\frac {{V}_{\mathrm{(GlgB)}} \, \cdot \, {\mathrm{[1,4glucan]}} } {{\mathrm{Km}}_{\mathrm{(GlgB)}} \, + \, {\mathrm{[1,4glucan]}} } \right) } \\ && \\ \frac {\mathrm{d}\left( {{\mathrm{[Glycogen]}} \, } \right) } {\mathrm{d}{t} } \; &=& \; { \, + \, \, \left(\frac {{V}_{\mathrm{(GlgB)}} \, \cdot \, {\mathrm{[1,4glucan]}} } {{\mathrm{Km}}_{\mathrm{(GlgB)}} \, + \, {\mathrm{[1,4glucan]}} } \right) } \\ && \\ \; && \; { \, - \, \, \left(\frac {{V}_{\mathrm{("GlgX-GlgP")}} \,\cdot \, {\mathrm{[Glycogen]}} } {{\mathrm{Km}}_{\mathrm{("GlgX-GlgP")}} \, + \, {\mathrm{[Glycogen]}} } \right) } \\ && \\ \end{array} $$

References

1. Hoops S., Sahle S., Gauges R., Lee C., Pahle J., Simus N., Singhal M., Xu L., Mendes P., and Kummer U., 2006, COPASI–A Complex Pathway Simulator , Bioinformatics, 22, p. 3067–74.

2. Wilson W.A., Roach P.J., Montero M., Baroja-Fernández E., Muñoz F.J., Eydallin G., Viale A.M., and Pozueta-Romero J., 2010, Regulation of glycogen metabolism in yeast and bacteria , FEMS Microbiology Reviews, 34, p. 952–85

3. Brautaset T., Petersen S.B., and Valla S., 2002, In vitro determined kinetic properties of mutant phosphoglucomutases and their effects on sugar catabolism in Escherichia coli , Metabolic Engineering, 2, p. 104–14

4. Figueroa C.M., Esper M.C., Bertolo A., Demonte A.M., Aleanzi M., Iglesias A.A., and Ballicora M.A., 2011, Understanding the allosteric trigger for the fructose-1,6-bisphosphate regulation of the ADP-glucose pyrophosphorylase from Escherichia coli , Biochimie, 93, p. 1816–23.

5. Fox J., Kawaguchi K., Greenberg E., and Preiss J., 1976, Biosynthesis of bacterial glycogen. purification and properties of the Escherichia coli B ADPglucose:1,4-alpha-d-glucan 4-alpha-glucosyltransferase , Biochemistry, 15, p. 849–57

6. Mikkelsen R., Binderup K., and Preiss J., 2001, Tyrosine residue 300 is important for activity and stability of branching enzyme from Escherichia coli , Archives of Biochemistry Biophysics, 385, 372–7