Examples

In order to demonstrate the application of the graphical simplification rules, we consider two examples. First, we reduce the model of the enzymatic reaction with competitive inhibition from Example 7.2 and Figure 7.2. Then, we consider the more complex PTS model from Figure 7.3.

Example 7.9 (Enzymatic reaction). In the network from Example 7.2 (p. 140) and Figure 7.2, we assume that the rate-limiting step is the actual conversion of the products A and B into the reactant C in reaction 5. Accordingly, as the first step towards a reduced model, we set R₁ = R₂ = R₃ = R₄ = R₆ = 0 (Figure 7.11 left). After applying the simplification rule from Figure 7.9 to reactions 1, 3 and 6 we get in step 2 the network in Figure 7.11 (right). By applying rules from Figure 7.5 this can be simplified in step 3 to Figure 7.12 (left). In step 4 we apply again the rule from Figure 7.9 and get Figure 7.12 (right). In step 5, a translation of the stoichiometric coefficients (Figure 7.6) finally leads to the reduced model in Figure 7.13. The figure shows that the inhibitor I and the enzyme E do not participate in the overall reaction, but they influence the reaction rate via ‘side effects’ of their thermokinetic potentials on the capacities C˜_A, C˜_B and C˜_C. The reduced concentrations ˜˜c_A,c˜˜_B, ˜˜c_E and c˜_I represent the overall concentrations of A, B, E and I, respectively. This result is consistent with the discussion in Section 6.7 (p. 134).

CEB

CEA

C_EI

R5

CEAB

CB

C_A CI

CE

CC

R5

CEAB

C˜I

C˜_A

C˜B

C˜E

CC

˜

c_E =c_E+c_EI+c_EA+c_EB, C˜_E =C_E + ˜ξ_IC_EI+ ˜ξ_AC_EA+ ˜ξ_BC_EB,

˜

c_I =c_I+c_EI, C˜_I =C_I+ ˜ξ_EC_EI,

˜

c_A=c_A+c_EA, C˜_A=C_A+ ˜ξ_EC_EA,

˜

c_B=c_B+c_EB, C˜_B=C_B+ ˜ξ_EC_EB.

Figure 7.11.: Reduction of the enzymatic reaction model – step 1 and 2.

7. Graphical Representation of TK Models

R5

C˜E

CC

C˜I

C˜A

C˜B

CEAB

R5

˜˜ CE

CC

C˜I

˜˜ CA

˜˜ CB

˜˜

c_E = ˜c_E+ ˜c_EAB C˜˜_E = ˜C_E+ ˜ξ_Aξ˜_BC_EAB,

=c_E+c_EI+c_EA+c_EB+c_EAB, =C_E+ ˜ξ_IC_EI+ ˜ξ_AC_EA+ ˜ξ_BC_EB+ ˜ξ_Aξ˜_BC_EAB,

˜˜

c_A= ˜c_A+c_EAB C˜˜_A= ˜C_A+ξ˜˜_Eξ˜_BC_EAB,

=c_A+c_EA+c_EAB, =C_A+ξ˜˜_EC_EA+ξ˜˜_Eξ˜_BC_EAB,

˜˜

c_B= ˜c_B+c_EAB C˜˜_B= ˜C_B+ξ˜˜_Eξ˜_AC_EAB

=c_B+c_EB+c_EAB, =C_B+ξ˜˜_EC_EB+ξ˜˜_Eξ˜_A.

Figure 7.12.: Reduction of the enzymatic reaction model – step 3 and 4.

˜˜ C_E C˜_I

˜˜ CB

R˜₅

CC

˜˜ C_A

R˜₅=ξ˜˜_E⁻¹R₅

Figure 7.13.: Reduced model of the enzymatic reaction.

7. Graphical Representation of TK Models

Example 7.10 (PTS). In Example 5.20 (p. 94) the thermokinetic resistances and capacities for the PTS model from Rohwer et al. [87] were derived. The graphical representation of this model is given in Figure 7.3.

The model contains 10 resistances and 13 capacities. To reduce the model we approximate the 6 lowest resistances by zero: R₁ = R₂ = R₅ = R₆ = R₉ = R₁₀ = 0. This leads to the approximated but unreduced model in Figure 7.14 with4resistances but still 13capacities. By applying the rules for the reduction of the kinetic submodel (Figure 7.8), we get the reduced model in Figure 7.15. This model contains only 7 capacities but is not suited for a direct simulation because it contains stoichiors that are directly linked without intermediate capacities or resistances. This means that instead of the removed capacitiesC_EI, C_EI·^P, C_{HP r}·^P, C_IIA·^P, C_IICB and C_IICB·^P, the network in Figure 7.15 contains capacities of value zero. We can apply the rules for the reduction of partially serial reactions from Figure 7.7 to resolve this problem partially. In order to avoid crossing lines, the remaining elements need to be rearranged and one gets Figure 7.16.

By applying the rules for the equivalence of stoichior elements (Figure 7.5), we can completely resolve this problem in the two steps shown in Figure 7.17 and Figure 7.18. This network could be simulated as a TK model. However, the negative stoichiometric coefficients are uncommon and we can get rid of them by a translation of the stoichiometric coefficients (Figure 7.6) and get the reduced model in Figure 7.19. The model contains a conservation relation for each of the four proteinsEI, HP r,IIA and IICB. Only the conservation relations for EI and IICB in Figure 7.19 have the form required for using the simplification rule in Figure 7.10. After reduction of these two conservation relations we get the reduced model in Figure 7.20.

Figure 7.21 shows simulation results of the original and the reduced model. The respective curves are qualitatively similar but partly deviate quantitatively. The fluxes in the reduced model are overestimated in steady state because some of the resistances that add to the overall resistance of the PTS are set to zero. Thus, by increasing the remaining resistances we can get a much better fit of original and reduced model. In a real modeling work, the parameters of the detailed model are not known, but the reduced parameters are estimated using the measurement data. Then, the adapted resistance values can be directly identified from the measurement values.

7. Graphical Representation of TK Models

CEI·^P CHP r CIIA·^P CIICB

ξP EP

ξP yr

R3R4 R7R8

ξGlc

CEI CIIA

ξGlc·^P CEI·^P·^{HP r}

CP yr·^P·^EI CHP r·^P·^IIA CIIA·^P·^IICB CIICB·^P·^Glc

CHP r·^P CIICB·^P

Figure 7.14.: Approximated but unreduced PTS model.

ξP EP

ξP yr

R3R4 R7R8

ξGlc

ξGlc·^P C˜IICB·P·Glc

C˜HP r·P·IIA

C˜P yr·P·EI CEI·P·HP r CIIA·P·IICB

C˜HP r

C˜IIA

C˜_{P yr}·^P·^EI =C_{P yr}·^P·^EI+ξ_{P EP}⁻¹ C_EI+ξ_{P yr}⁻¹ C_EI·^P, ˜c_{P yr}·^P·^EI =c_{P yr}·^P·^EI+c_EI+c_EI·^P, C˜_{HP r}·^P·^IIA=C_{HP r}·^P·^IIA+ ˜ξ_IIA⁻¹ C_{HP r}·^P + ˜ξ_{HP r}⁻¹ C_IIA·^P, c˜_{HP r}·^P·^IIA =c_{HP r}·^P·^IIA+c_{HP r}·^P +c_IIA·^P, C˜_IICB·^P·^Glc =C_IICB·^P·^Glc+ξ⁻¹_Glc·^P C_IICB+ξ⁻¹_GlcC_IICB·^P, ˜c_IICB·^P·^Glc =c_IICB·^P·^Glc+c_IICB+c_IICB·^P,

C˜_IIA=C_IIA−ξ_{HP r}·^P·^IIAξ˜_IIA⁻² C_{HP r}·^P, ˜c_IIA =c_IIA−c_{HP r}·^P, C˜_{HP r} =C_{HP r}−ξ_{HP r}·^P·^IIAξ˜_{HP r}⁻² C_IIA·^P, c˜_{HP r} =c_Hpr−c_IIA·^P.

Figure 7.15.: Reduction of the PTS model – step 1.

7. Graphical Representation of TK Models

C˜HP r·P·IIA

CEI·P·HP r C˜P yr·P·EI C˜IICB·P·Glc CIIA·P·IICB

R˜3

R˜4

ξP yr R˜7

R˜8

ξGlc·^P ξGlc

ξP EP C˜IIA

C˜HP r

R˜₃=ξ_{P yr}R₃, R˜₄=ξ_{P EP}R₄, R˜₇=ξ_Glc·^PR₇, R˜₈=ξ_GlcR₈.

Figure 7.16.: Reduction of the PTS model – step 2.

C˜HP r·P·IIA

CEI·P·HP r C˜P yr·P·EI C˜IICB·P·Glc CIIA·P·IICB

R˜3

R˜4

ξP yr R˜7

R˜8

ξGlc·^P ξGlc

ξP EP −1

−1

C˜IIA

C˜HP r

Figure 7.17.: Reduction of the PTS model – step 3.

7. Graphical Representation of TK Models

C˜HP r·^P·^IIA

CEI·^P·^{HP r} C˜P yr·^P·^EI C˜IICB·^P·^Glc CIIA·^P·^IICB R˜3

R˜4

ξP yr R˜7

R˜8

ξGlc·P

C˜IIA ξGlc

ξP EP −1

−1 C˜HP r

Figure 7.18.: Reduction of the PTS model – step 4.

R˜3

ξP yr

ξGlc

ξGlc·P

CIIA·^P·^IICB C˜HP r·^P·^IIA C˜IICB·^P·^Glc

C˜P yr·^P·^EI CEI·^P·^{HP r}

R˜8

ˆ˜ R4

ξP EP

ˆ˜ R7

C˜IIA

C˜HP r

ˆ˜

R₄= ˜ξ_IIAR˜₄= ˜ξ_IIAξ_{P EP}R₄, Rˆ˜₇= ˜ξ_{HP r}R˜₇= ˜ξ_{HP r}ξ_Glc·^P R₇.

Figure 7.19.: Reduction of the PTS model – step 6.

7. Graphical Representation of TK Models

C˜HP r·^P·^IIA

ξP yr

ξP EP

ξGlc

ξGlc·P

˜˜ R8

˜˜

CIICB·^P·^Glc

˜˜

CP yr·^P·^EI

˜˜ R3

˜ˆ R˜4

˜ˆ R˜7

C˜HP r

C˜IIA

ξ˜_{P yr}·^P·^EI = c_EI,tot C_EI,tot

˜˜

ξ_{P yr}·^P·^EI, ξ˜_IICB·^P·^Glc= c_IICB,tot C_IICB,tot

˜˜

ξ_IICB·^P·^Glc,

˜

c_{P yr}·^P·^EI = ˜˜c_{P yr}·^P·^EI =c_{P yr}·^P·^EI+c_EI+c_EI·^P,

˜

c_IICB·^P·^Glc= ˜˜c_IICB·^P·^Glc =c_IICB·^P·^Glc+c_IICB+c_IICB·^P,

˜˜

C_{P yr}·^P·^EI = c_EI,tot C_EI,tot

C˜_{P yr}·^P·^EI = c_EI,tot

C_EI,tot(C_{P yr}·^P·^EI+ξ_{P EP}⁻¹ C_EI+ξ⁻¹_{P yr}C_EI·^P),

˜˜

C_IICB·^P·^Glc= c_IICB,tot C_IICB,tot

C˜_IICB·^P·^Glc = c_IICB,tot

C_IICB,tot(C_IICB·^P·^Glc+ξ_Glc⁻¹·^PC_IICB+ξ_Glc⁻¹C_IICB·^P),

˜˜

R₃= C_EI,tot c_EI,tot

R˜₃= C_EI,tot

c_EI,tot ξ_{P yr}R₃, R˜ˆ˜₄= C_EI,tot c_EI,tot

ˆ˜

R₄= C_EI,tot c_EI,tot

ξ˜_IIAξ_{P EP}R₄,

˜ˆ

R˜₇= C_IICB,tot c_IICB,tot

ˆ˜

R₇= C_IICB,tot c_IICB,tot

ξ˜_{HP r}ξ_Glc·^P R₇, R˜˜₈= C_IICB,tot c_IICB,tot

R˜₈= C_IICB,tot

c_IICB,tot ξ_GlcR₈, with

c_EI,tot= ˜c_{P yr}·^P·^EI+c_EI·^P·^{HP r} =c_EI+c_EI·^P +c_{P yr}·^P·^EI+c_EI·^P·^{HP r},

c_IICB,tot= ˜c_IICB·^P·^Glc+c_IIA·^P·^IICB=c_IICB+c_IICB·^P +c_IICB·^P·^Glc+c_IIA·^P·^IICB, C_EI,tot= ˜C_{P yr}·^P·^EIξ˜˜_{P yr}·^P·^EI+C_EI·^P·^{HP r}

= (C_{P yr}·^P·^EI+ξ⁻¹_{P EP}C_EI+ξ_{P yr}⁻¹ C_EI·^P)ξ˜˜_{P yr}·^P·^EI+C_EI·^P·^{HP r}, C_IICB,tot= ˜C_IICB·^P·^Glcξ˜˜_IICB·^P·^Glc+C_IIA·^P·^IICB

= (C_IICB·^P·^Glc+ξ⁻¹_Glc·^P C_IICB+ξ⁻¹_GlcC_IICB·^P)ξ˜˜_IICB·^P·^Glc+C_IIA·^P·^IICB.

Figure 7.20.: Reduced PTS model.

7. Graphical Representation of TK Models J₃ [mM min⁻¹]

t [10⁻¹min]

0.5 200

100

00 1.0

300

J₄ [mM min⁻¹]

t[10⁻¹min]

0.5 200

100

00 1.0

300

J₇ [mM min⁻¹]

t [10⁻¹min]

0 0.5

0 1.0

40 20

J₈ [mM min⁻¹]

t[10⁻¹min]

0 0.5

0 1.0

20 40

cIIA [µM]

t [10⁻¹min]

0 0.5

0 1.0

10 5

cHP r [µM]

t[10⁻¹min]

0 0.5

0 1.0

10 20

c_{HP r}·^P·^IIA [µM]

t [10⁻¹min]

0 0.5

0 1.0

30 20 10

c_{P yr}·^P·^EI [µM]

t[10⁻¹min]

0 0.5

0 1.0

4 2

Figure 7.21.: Simulation results with the original (solid) and the reduced (dashed) PTS model.

The dotted line shows the reduced model with corrected resistances: R⁰₃ = 1.3R₃, R⁰₄ = 1.3R₄, R₇⁰ = 1.3R₇ and R⁰₈ = 1.3R₈. The clamped concentrations are c_{P EP} = 2800µM, c_{P yr} = 900µM, c_Glc = 500µM, c_Glc·^P = 50µM. The initial conditions of the detailed model are c_EI,0 = 5µM, c_{HP r,0} = 50µM, c_IIA,0 = 40µM, c_IICB,0 = 15µM, c_EI·^P·^{HP r,0} = c_EI·^P,0 = c_{HP r}·^P·^IIA,0 = c_{HP r}·^P,0 = cIIA·^P·^IICB,0 = cIIA·^P,0 = cIICB·^P·^Glc,0 = cIICB·^P,0 = cP yr·^P·^EI,0 = 0.01µM. The initial conditions of the reduced model were computed numerically from the initial conditions of the original model.

7. Graphical Representation of TK Models

Im Dokument Thermokinetic modeling and model reduction of reaction networks (Seite 146-154)