U-5R

Derivation of differential equations describing evolution of spin concentrations

Strategy:

I will develop kinetic matrices for  five cases to have from one to five isomers of R. I will explicitly use all rate constants of similar transitions to be able to simplify in the next steps by setting them to one value.

NOTE: The order of species in this model will be: R, RL, R*, R**, ...  to simplify expansion of the model. It is different from U-R model in the existing IDAP code!

Hint: Accurate extraction of K matrix: Workflow for accurate extraction of the K matrix

Definitions of transitions and strategy

Reaction, partial conversion, and net rates

Expression in terms of spin (monomer) concentrations

Derivation of K matrix for U-R mechanism

Derivation for K matrix of U-2R mechanism

Derivation of K matrix for U-3R mechanism

Derivation of K matrix for U-4R mechanism

Derivation of K matrix for U-5R mechanism

Conclusions

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clean up workspace

reset()

Write properly balanced reactions equations for all transitions in the mechanism:

Binding reaction, transition A:

R+L<=>RL

Constants: k_1_A (forward), k_2_A (reverse).

Isomerization of R to R-starred species: transitions B

R <=> R*

Constants: k_1_B_s1 (forward), k_2_B_s1 (reverse).

R <=> R**

Constants: k_1_B_s2 (forward), k_2_B_s2 (reverse).

R <=> R***

Constants: k_1_B_s3 (forward), k_2_B_s3 (reverse).

R <=> R****

Constants: k_1_B_s4 (forward), k_2_B_s4 (reverse).

R <=> R*****

Constants: k_1_B_s5 (forward), k_2_B_s5 (reverse).

Interconversion of R-starred isomers: transitions C

-- R* --

R* <=> R**

Constants: k_1_C_s12 (forward), k_2_C_s12 (reverse).

R* <=> R***

Constants: k_1_C_s13 (forward), k_2_C_s13 (reverse).

R* <=> R****

Constants: k_1_C_s14 (forward), k_2_C_s14 (reverse).

R* <=> R*****

Constants: k_1_C_s15 (forward), k_2_C_s15 (reverse).

-- R** --

R** <=> R***

Constants: k_1_C_s23 (forward), k_2_C_s23 (reverse).

R** <=> R****

Constants: k_1_C_s24 (forward), k_2_C_s24 (reverse).

R** <=> R*****

Constants: k_1_C_s25 (forward), k_2_C_s25 (reverse).

-- R*** --

R*** <=> R****

Constants: k_1_C_s34 (forward), k_2_C_s34 (reverse).

R*** <=> R*****

Constants: k_1_C_s35 (forward), k_2_C_s35 (reverse).

-- R**** --

R****<=> R*****

Constants: k_1_C_s45 (forward), k_2_C_s45 (reverse).

Write reaction rates

Here, we distinguish reaction rate (elementary reaction acts per unit time; denote as "Rate_reaction-label") and conversion rates (number of moles of specific species consumed/produced per unit time, dc/dt). Conversion rates, dc/dt, for species are related to reaction rates, Rate, through molecularity coefficients.

We also distinguish here partial conversion rates from net (overall) conversion rates. The net conversion rate is actual rate of change in measured concentration of the species due to all transitions this species is involved with (denote at Rate_reaction-label_N). Partial conversion rate is the conversion rate of the species along a specific branch of the reaction mechanism (denote 'dC-component-dt-reaction-label'). Summation of the partial conversion rates of the species gives the net conversion rate.

NOTE: In this mechanism all transition involve only one molecules of species of each kind, therefore all  partial conversion rates are equal to reaction rates. This is reflected by setting 'molecularity' to 1 for all transitions. The molecularity sign also indicates whether the species is created or destroyed in this transition.

Strategy:

We need equations for the net convertsion rates for each species. For this purpose, we write partial conversion rates originating from every individual (forward or reverse) process. To obtain the partial conversion rate for a process, we use the reaction rate equation times molecularity of the process in terms of this particular species.

Workflow:

make a list of transitions for every species (creating and destroying it) and write forward and reverse equations for each.

Back to Contents

In the following subsections, I am developing equations to account for net rate of change of every particular species.

R derivation

RL derivation

R* derivation

R** derivation

R*** derivation

R**** derivation

R***** derivation

Species: R

Equations group: 1

R+L<=>RL

Constants: k_1_A (forward), k_2_A (reverse).

Equations subgroup: A

a forward reaction rate

eq1_A_1a:= Rate_1_A = k_1_A*R*L

a partial conversion rate of R in this transition

molecularity:=-1:
eq1_A_1b:= dcRdt_1_A = molecularity*Rate_1_A

The final form

eq1_A_1c:= eq1_A_1b | eq1_A_1a

a reverse reaction rate for the transition

eq1_A_2a:= Rate_2_A = k_2_A*RL

a partial conversion rate of R in this transition

molecularity:=1:
eq1_A_2b:= dcRdt_2_A = molecularity*Rate_2_A

The final form

eq1_A_2c:= eq1_A_2b | eq1_A_2a

R <=> R*

Constants: k_1_B_s_1 (forward), k_2_B_s_1 (reverse).

Equations subgroup: Bs1

a forward reaction rate for the transition

eq1_Bs1_1a:= Rate_1_B_s_1 = k_1_B_s_1*R

a partial conversion rate of R in this transition

molecularity:=-1:
eq1_Bs1_1b:= dcRdt_1_B_s_1 = molecularity*Rate_1_B_s_1

The final form

eq1_Bs1_1c:= eq1_Bs1_1b | eq1_Bs1_1a

a reverse reaction rate for the transition

eq1_Bs1_2a:= Rate_2_B_s_1 = k_2_B_s_1*R_s_1

a partial conversion rate of R in this transition

molecularity:=1:
eq1_Bs1_2b:= dcRdt_2_B_s_1 = molecularity*Rate_2_B_s_1

The final form

eq1_Bs1_2c:= eq1_Bs1_2b | eq1_Bs1_2a

R <=> R**

Constants: k_1_B_s2 (forward), k_2_B_s2 (reverse).

Equations subgroup: Bs2

a forward reaction rate for the transition

eq1_Bs2_1a:= Rate_1_B_s_2 = k_1_B_s_2*R

a partial conversion rate of R in this transition

molecularity:=-1:
eq1_Bs2_1b:= dcRdt_1_B_s_2 = molecularity*Rate_1_B_s_2

The final form

eq1_Bs2_1c:= eq1_Bs2_1b | eq1_Bs2_1a

a reverse reaction rate for the transition

eq1_Bs2_2a:= Rate_2_B_s_2 = k_2_B_s_2*R_s_2

a partial conversion rate of R in this transition

molecularity:=1:
eq1_Bs2_2b:= dcRdt_2_B_s_2 = molecularity*Rate_2_B_s_2

The final form

eq1_Bs2_2c:= eq1_Bs2_2b | eq1_Bs2_2a

R <=> R***

Constants: k_1_B_s3 (forward), k_2_B_s3 (reverse).

Equations subgroup: Bs3

a forward reaction rate for the transition

eq1_Bs3_1a:= Rate_1_B_s_3 = k_1_B_s_3*R

a partial conversion rate of R in this transition

molecularity:=-1:
eq1_Bs3_1b:= dcRdt_1_B_s_3 = molecularity*Rate_1_B_s_3

The final form

eq1_Bs3_1c:= eq1_Bs3_1b | eq1_Bs3_1a

a reverse reaction rate for the transition

eq1_Bs3_2a:= Rate_2_B_s_3 = k_2_B_s_3*R_s_3

a partial conversion rate of R in this transition

molecularity:=1:
eq1_Bs3_2b:= dcRdt_2_B_s_3 = molecularity*Rate_2_B_s_3

The final form

eq1_Bs3_2c:= eq1_Bs3_2b | eq1_Bs3_2a

R <=> R****

Constants: k_1_B_s4 (forward), k_2_B_s4 (reverse).

Equations subgroup: Bs4

a forward reaction rate for the transition

eq1_Bs4_1a:= Rate_1_B_s_4 = k_1_B_s_4*R

a partial conversion rate of R in this transition

molecularity:=-1:
eq1_Bs4_1b:= dcRdt_1_B_s_4 = molecularity*Rate_1_B_s_4

The final form

eq1_Bs4_1c:= eq1_Bs4_1b | eq1_Bs4_1a

a reverse reaction rate for the transition

eq1_Bs4_2a:= Rate_2_B_s_4 = k_2_B_s_4*R_s_4

a partial conversion rate of R in this transition

molecularity:=1:
eq1_Bs4_2b:= dcRdt_2_B_s_4 = molecularity*Rate_2_B_s_4

The final form

eq1_Bs4_2c:= eq1_Bs4_2b | eq1_Bs4_2a

R <=> R*****

Constants: k_1_B_s5 (forward), k_2_B_s5 (reverse).

Equations subgroup: Bs5

a forward reaction rate for the transition

eq1_Bs5_1a:= Rate_1_B_s_5 = k_1_B_s_5*R

a partial conversion rate of R in this transition

molecularity:=-1:
eq1_Bs5_1b:= dcRdt_1_B_s_5 = molecularity*Rate_1_B_s_5

The final form

eq1_Bs5_1c:= eq1_Bs5_1b | eq1_Bs5_1a

a reverse reaction rate for the transition

eq1_Bs5_2a:= Rate_2_B_s_5 = k_2_B_s_5*R_s_5

a partial conversion rate of R in this transition

molecularity:=1:
eq1_Bs5_2b:= dcRdt_2_B_s_5 = molecularity*Rate_2_B_s_5

The final form

eq1_Bs5_2c:= eq1_Bs5_2b | eq1_Bs5_2a

Summary of partial conversion rates for the species

eq1_A_1c; eq1_A_2c;

eq1_Bs1_1c;  eq1_Bs1_2c;
eq1_Bs2_1c;  eq1_Bs2_2c;
eq1_Bs3_1c;  eq1_Bs3_2c;
eq1_Bs4_1c;  eq1_Bs4_2c;
eq1_Bs5_1c;  eq1_Bs5_2c;

Net conversion rate for the species

I will create equations for all five versions of the mechanism.

U-R-RL

dcRdt_N = dcRdt_1_A + dcRdt_2_A + dcRdt_1_B_s_1 + dcRdt_2_B_s_1

Substitute

eq1_R_N__U_R:= % | eq1_A_1c | eq1_A_2c | \
eq1_Bs1_1c |   eq1_Bs1_2c | \
eq1_Bs2_1c |   eq1_Bs2_2c | \
eq1_Bs3_1c |   eq1_Bs3_2c | \
eq1_Bs4_1c |   eq1_Bs4_2c | \
eq1_Bs5_1c |   eq1_Bs5_2c;

U-2R-RL

dcRdt_N = dcRdt_1_A + dcRdt_2_A + dcRdt_1_B_s_1 + dcRdt_2_B_s_1  + dcRdt_1_B_s_2 + dcRdt_2_B_s_2

Substitute

eq1_R_N__U_2R:= % | eq1_A_1c | eq1_A_2c | \
eq1_Bs1_1c |   eq1_Bs1_2c | \
eq1_Bs2_1c |   eq1_Bs2_2c | \
eq1_Bs3_1c |   eq1_Bs3_2c | \
eq1_Bs4_1c |   eq1_Bs4_2c | \
eq1_Bs5_1c |   eq1_Bs5_2c;

U-3R-RL

dcRdt_N = dcRdt_1_A + dcRdt_2_A + dcRdt_1_B_s_1 + dcRdt_2_B_s_1  + dcRdt_1_B_s_2 + dcRdt_2_B_s_2 \
+ dcRdt_1_B_s_3 + dcRdt_2_B_s_3

Substitute

eq1_R_N__U_3R:= % | eq1_A_1c | eq1_A_2c | \
eq1_Bs1_1c |   eq1_Bs1_2c | \
eq1_Bs2_1c |   eq1_Bs2_2c | \
eq1_Bs3_1c |   eq1_Bs3_2c | \
eq1_Bs4_1c |   eq1_Bs4_2c | \
eq1_Bs5_1c |   eq1_Bs5_2c;

U-4R-RL

dcRdt_N = dcRdt_1_A + dcRdt_2_A + dcRdt_1_B_s_1 + dcRdt_2_B_s_1  + dcRdt_1_B_s_2 + dcRdt_2_B_s_2 \
+ dcRdt_1_B_s_3 + dcRdt_2_B_s_3  + dcRdt_1_B_s_4 + dcRdt_2_B_s_4

Substitute

eq1_R_N__U_4R:= % | eq1_A_1c | eq1_A_2c | \
eq1_Bs1_1c |   eq1_Bs1_2c | \
eq1_Bs2_1c |   eq1_Bs2_2c | \
eq1_Bs3_1c |   eq1_Bs3_2c | \
eq1_Bs4_1c |   eq1_Bs4_2c | \
eq1_Bs5_1c |   eq1_Bs5_2c;

U-5R-RL

dcRdt_N = dcRdt_1_A + dcRdt_2_A + dcRdt_1_B_s_1 + dcRdt_2_B_s_1  + dcRdt_1_B_s_2 + dcRdt_2_B_s_2 \
+ dcRdt_1_B_s_3 + dcRdt_2_B_s_3  + dcRdt_1_B_s_4 + dcRdt_2_B_s_4  + dcRdt_1_B_s_5 + dcRdt_2_B_s_5

Substitute

eq1_R_N__U_5R:= % | eq1_A_1c | eq1_A_2c | \
eq1_Bs1_1c |   eq1_Bs1_2c | \
eq1_Bs2_1c |   eq1_Bs2_2c | \
eq1_Bs3_1c |   eq1_Bs3_2c | \
eq1_Bs4_1c |   eq1_Bs4_2c | \
eq1_Bs5_1c |   eq1_Bs5_2c;

Summary equations for R

eq1_R_N__U_R

eq1_R_N__U_2R

eq1_R_N__U_3R

eq1_R_N__U_4R

eq1_R_N__U_5R

Back to  Equations for each species

Species: RL

Equations group: 2

R+L<=>RL

Constants: k_1_A (forward), k_2_A (reverse).

Equations subgroup: A

a forward reaction rate

eq1_A_1a

a partial conversion rate in this transition

molecularity:=1:
eq2_A_1b:= dcRLdt_1_A = molecularity*Rate_1_A

The final form

eq2_A_1c:= eq2_A_1b | eq1_A_1a

a reverse reaction rate for the transition

eq1_A_2a

a partial conversion rate in this transition

molecularity:=-1:
eq2_A_2b:= dcRLdt_2_A = molecularity*Rate_2_A

The final form

eq2_A_2c:= eq2_A_2b | eq1_A_2a

Net conversion rate for the species

Same in all alternative mechanisms

dcRLdt_N = dcRLdt_1_A + dcRLdt_2_A

Substitute

eq2_RL_N:= % |  eq2_A_1c |  eq2_A_2c

Summary equation for RL

All versions of mechanism will have it the same

eq2_RL_N

Back to  Equations for each species

Species: R*

Equations group: 3

Consider all processes contributing or removing this species

R <=> R*

R* <=> R**

R* <=> R***

R* <=> R****

R* <=> R*****

-------------------------

R <=> R*

Equations subgroup: B

a forward reaction rate for the transition

eq1_Bs1_1a

a partial conversion rate of R* in this transition

molecularity:=1:
eq3_Bs1_1b:= dcRs1dt_1_B_s_1 = molecularity*Rate_1_B_s_1

The final form

eq3_Bs1_1c:= eq3_Bs1_1b | eq1_Bs1_1a

a reverse reaction rate for the transition

eq1_Bs1_2a

a partial conversion rate of R* in this transition

molecularity:=-1:
eq3_Bs1_2b:= dcRs1dt_2_B_s_1 = molecularity*Rate_2_B_s_1

The final form

eq3_Bs1_2c:= eq3_Bs1_2b | eq1_Bs1_2a

Equations subgroup: C

R* <=> R**

a forward reaction rate for the transition

eq3_Cs12_1a:= Rate_1_C_s_1_2 = R_s_1 * k_1_C_s_1_2

a partial conversion rate of R* in this transition

molecularity:=-1:
eq3_Cs12_1b:= dcRs1dt_1_C_s_1_2 = molecularity*Rate_1_C_s_1_2

The final form

eq3_Cs12_1c:= eq3_Cs12_1b | eq3_Cs12_1a

a reverse reaction rate for the transition

eq3_Cs12_2a:= Rate_2_C_s_1_2 = R_s_2 * k_2_C_s_1_2

a partial conversion rate of R* in this transition

molecularity:=1:
eq3_Cs12_2b:= dcRs1dt_2_C_s_1_2 = molecularity*Rate_2_C_s_1_2

The final form

eq3_Cs12_2c:= eq3_Cs12_2b | eq3_Cs12_2a

R* <=> R***

a forward reaction rate for the transition

eq3_Cs13_1a:= Rate_1_C_s_1_3 = R_s_1 * k_1_C_s_1_3

a partial conversion rate of R* in this transition

molecularity:=-1:
eq3_Cs13_1b:= dcRs1dt_1_C_s_1_3 = molecularity*Rate_1_C_s_1_3

The final form

eq3_Cs13_1c:= eq3_Cs13_1b | eq3_Cs13_1a

a reverse reaction rate for the transition

eq3_Cs13_2a:= Rate_2_C_s_1_3 = R_s_3 * k_2_C_s_1_3

a partial conversion rate of R* in this transition

molecularity:=1:
eq3_Cs13_2b:= dcRs1dt_2_C_s_1_3 = molecularity*Rate_2_C_s_1_3

The final form

eq3_Cs13_2c:= eq3_Cs13_2b | eq3_Cs13_2a

R* <=> R****

a forward reaction rate for the transition

eq3_Cs14_1a:= Rate_1_C_s_1_4 = R_s_1 * k_1_C_s_1_4

a partial conversion rate of R* in this transition

molecularity:=-1:
eq3_Cs14_1b:= dcRs1dt_1_C_s_1_4 = molecularity*Rate_1_C_s_1_4

The final form

eq3_Cs14_1c:= eq3_Cs14_1b | eq3_Cs14_1a

a reverse reaction rate for the transition

eq3_Cs14_2a:= Rate_2_C_s_1_4 = R_s_4 * k_2_C_s_1_4

a partial conversion rate of R* in this transition

molecularity:=1:
eq3_Cs14_2b:= dcRs1dt_2_C_s_1_4 = molecularity*Rate_2_C_s_1_4

The final form

eq3_Cs14_2c:= eq3_Cs14_2b | eq3_Cs14_2a

R* <=> R*****

a forward reaction rate for the transition

eq3_Cs15_1a:= Rate_1_C_s_1_5 = R_s_1 * k_1_C_s_1_5

a partial conversion rate of R* in this transition

molecularity:=-1:
eq3_Cs15_1b:= dcRs1dt_1_C_s_1_5 = molecularity*Rate_1_C_s_1_5

The final form

eq3_Cs15_1c:= eq3_Cs15_1b | eq3_Cs15_1a

a reverse reaction rate for the transition

eq3_Cs15_2a:= Rate_2_C_s_1_5 = R_s_5 * k_2_C_s_1_5

a partial conversion rate of R* in this transition

molecularity:=1:
eq3_Cs15_2b:= dcRs1dt_2_C_s_1_5 = molecularity*Rate_2_C_s_1_5

The final form

eq3_Cs15_2c:= eq3_Cs15_2b | eq3_Cs15_2a

Summary of partial conversion rates for the species

eq3_Bs1_1c; eq3_Bs1_2c;
eq3_Cs12_1c; eq3_Cs12_2c;  eq3_Cs13_1c;  eq3_Cs13_2c;  eq3_Cs14_1c;  eq3_Cs14_2c;  eq3_Cs15_1c;  eq3_Cs15_2c;

Net conversion rate for the species

I will create equations for all five versions of the mechanism.

U-R-RL

dcRs1dt_N = dcRs1dt_1_B_s_1 + dcRs1dt_2_B_s_1

Substitute

eq3_Rs1_N__U_R:= % | eq3_Bs1_1c |  eq3_Bs1_2c | \
eq3_Cs12_1c |  eq3_Cs12_2c |  \
eq3_Cs13_1c |   eq3_Cs13_2c |  \
eq3_Cs14_1c |   eq3_Cs14_2c |  \
eq3_Cs15_1c |   eq3_Cs15_2c;

U-2R-RL

dcRs1dt_N = dcRs1dt_1_B_s_1 + dcRs1dt_2_B_s_1 + dcRs1dt_1_C_s_1_2 + dcRs1dt_2_C_s_1_2

Substitute

eq3_Rs1_N__U_2R:= % | eq3_Bs1_1c |  eq3_Bs1_2c | \
eq3_Cs12_1c |  eq3_Cs12_2c |  \
eq3_Cs13_1c |   eq3_Cs13_2c |  \
eq3_Cs14_1c |   eq3_Cs14_2c |  \
eq3_Cs15_1c |   eq3_Cs15_2c;

U-3R-RL

dcRs1dt_N = dcRs1dt_1_B_s_1 + dcRs1dt_2_B_s_1 \
+ dcRs1dt_1_C_s_1_2 + dcRs1dt_2_C_s_1_2 \
+ dcRs1dt_1_C_s_1_3 + dcRs1dt_2_C_s_1_3

Substitute

eq3_Rs1_N__U_3R:= % | eq3_Bs1_1c |  eq3_Bs1_2c | \
eq3_Cs12_1c |  eq3_Cs12_2c |  \
eq3_Cs13_1c |   eq3_Cs13_2c |  \
eq3_Cs14_1c |   eq3_Cs14_2c |  \
eq3_Cs15_1c |   eq3_Cs15_2c;

U-4R-RL

dcRs1dt_N = dcRs1dt_1_B_s_1 + dcRs1dt_2_B_s_1 \
+ dcRs1dt_1_C_s_1_2 + dcRs1dt_2_C_s_1_2 \
+ dcRs1dt_1_C_s_1_3 + dcRs1dt_2_C_s_1_3  \
+ dcRs1dt_1_C_s_1_4 + dcRs1dt_2_C_s_1_4

Substitute

eq3_Rs1_N__U_4R:= % | eq3_Bs1_1c |  eq3_Bs1_2c | \
eq3_Cs12_1c |  eq3_Cs12_2c |  \
eq3_Cs13_1c |   eq3_Cs13_2c |  \
eq3_Cs14_1c |   eq3_Cs14_2c |  \
eq3_Cs15_1c |   eq3_Cs15_2c;

U-5R-RL

dcRs1dt_N = dcRs1dt_1_B_s_1 + dcRs1dt_2_B_s_1 \
+ dcRs1dt_1_C_s_1_2 + dcRs1dt_2_C_s_1_2 \
+ dcRs1dt_1_C_s_1_3 + dcRs1dt_2_C_s_1_3  \
+ dcRs1dt_1_C_s_1_4 + dcRs1dt_2_C_s_1_4 \
+ dcRs1dt_1_C_s_1_5 + dcRs1dt_2_C_s_1_5

Substitute

eq3_Rs1_N__U_5R:= % | eq3_Bs1_1c |  eq3_Bs1_2c | \
eq3_Cs12_1c |  eq3_Cs12_2c |  \
eq3_Cs13_1c |   eq3_Cs13_2c |  \
eq3_Cs14_1c |   eq3_Cs14_2c |  \
eq3_Cs15_1c |   eq3_Cs15_2c;

Summary of equations for R*

eq3_Rs1_N__U_R;
eq3_Rs1_N__U_2R;
eq3_Rs1_N__U_3R;
eq3_Rs1_N__U_4R;
eq3_Rs1_N__U_5R;

Back to  Equations for each species

Species: R**

Equations group: 4

Equations subgroup: B

R <=> R**

(towards species)

a forward reaction rate for the transition

eq4_Bs2_1a:= eq1_Bs2_1a;

a partial conversion rate of R** in this transition

molecularity:=1:
eq4_Bs2_1b:= dcRs2dt_1_B_s_2 = molecularity*Rate_1_B_s_2

The final form

eq4_Bs2_1c:= eq4_Bs2_1b | eq4_Bs2_1a

a reverse reaction rate for the transition

eq4_Bs2_2a:= eq1_Bs2_2a;

a partial conversion rate of R* in this transition

molecularity:=-1:
eq4_Bs2_2b:= dcRs2dt_2_B_s_2 = molecularity*Rate_2_B_s_2

The final form

eq4_Bs2_2c:= eq4_Bs2_2b | eq4_Bs2_2a

Equations subgroup: C

R* <=> R**

(towards species)

eq3_Cs12_1a

a forward reaction rate for the transition

eq4_Cs12_1a:= eq3_Cs12_1a

a partial conversion rate of R** in this transition

molecularity:=1:
eq4_Cs12_1b:= dcRs2dt_1_C_s_1_2 = molecularity*Rate_1_C_s_1_2

The final form

eq4_Cs12_1c:= eq4_Cs12_1b | eq4_Cs12_1a

a reverse reaction rate for the transition

eq4_Cs12_2a:= eq3_Cs12_2a

a partial conversion rate of R** in this transition

molecularity:=-1:
eq4_Cs12_2b:= dcRs2dt_2_C_s_1_2 = molecularity*Rate_2_C_s_1_2

The final form

eq4_Cs12_2c:= eq4_Cs12_2b | eq4_Cs12_2a

R** <=> R***

(away from species)

a forward reaction rate for the transition

eq4_Cs23_1a:= Rate_1_C_s_2_3 = R_s_2 * k_1_C_s_2_3

a partial conversion rate of R** in this transition

molecularity:=-1:
eq4_Cs23_1b:= dcRs2dt_1_C_s_2_3 = molecularity*Rate_1_C_s_2_3

The final form

eq4_Cs23_1c:= eq4_Cs23_1b | eq4_Cs23_1a

a reverse reaction rate for the transition

eq4_Cs23_2a:= Rate_2_C_s_2_3 = R_s_3 * k_2_C_s_2_3

a partial conversion rate of R** in this transition

molecularity:=1:
eq4_Cs23_2b:= dcRs2dt_2_C_s_2_3 = molecularity*Rate_2_C_s_2_3

The final form

eq4_Cs23_2c:= eq4_Cs23_2b | eq4_Cs23_2a

R** <=> R****

(away from species)

a forward reaction rate for the transition

eq4_Cs24_1a:= Rate_1_C_s_2_4 = R_s_2 * k_1_C_s_2_4

a partial conversion rate of R** in this transition

molecularity:=-1:
eq4_Cs24_1b:= dcRs2dt_1_C_s_2_4 = molecularity*Rate_1_C_s_2_4

The final form

eq4_Cs24_1c:= eq4_Cs24_1b | eq4_Cs24_1a

a reverse reaction rate for the transition

eq4_Cs24_2a:= Rate_2_C_s_2_4 = R_s_4 * k_2_C_s_2_4

a partial conversion rate of R** in this transition

molecularity:=1:
eq4_Cs24_2b:= dcRs2dt_2_C_s_2_4 = molecularity*Rate_2_C_s_2_4

The final form

eq4_Cs24_2c:= eq4_Cs24_2b | eq4_Cs24_2a

R** <=> R*****

(away from species)

a forward reaction rate for the transition

eq4_Cs25_1a:= Rate_1_C_s_2_5 = R_s_2 * k_1_C_s_2_5

a partial conversion rate of R** in this transition

molecularity:=-1:
eq4_Cs25_1b:= dcRs2dt_1_C_s_2_5 = molecularity*Rate_1_C_s_2_5

The final form

eq4_Cs25_1c:= eq4_Cs25_1b | eq4_Cs25_1a

a reverse reaction rate for the transition

eq4_Cs25_2a:= Rate_2_C_s_2_5 = R_s_5 * k_2_C_s_2_5

a partial conversion rate of R** in this transition

molecularity:=1:
eq4_Cs25_2b:= dcRs2dt_2_C_s_2_5 = molecularity*Rate_2_C_s_2_5

The final form

eq4_Cs25_2c:= eq4_Cs25_2b | eq4_Cs25_2a

Net conversion rate for the species

I will create equations for all five versions of the mechanism.

U-R-RL (species not present)

eq4_Rs2_N__U_R:= 0

U-2R-RL

dcRs2dt_N = dcRs2dt_1_B_s_2 + dcRs2dt_2_B_s_2 + dcRs2dt_1_C_s_1_2 + dcRs2dt_2_C_s_1_2

Substitute

eq4_Rs2_N__U_2R:= % | eq4_Bs2_1c |  eq4_Bs2_2c | \
eq4_Cs12_1c |  eq4_Cs12_2c |  \
eq4_Cs23_1c |   eq4_Cs23_2c |  \
eq4_Cs24_1c |   eq4_Cs24_2c |  \
eq4_Cs25_1c |   eq4_Cs25_2c;

U-3R-RL

dcRs2dt_N = dcRs2dt_1_B_s_2 + dcRs2dt_2_B_s_2 + \
dcRs2dt_1_C_s_1_2 + dcRs2dt_2_C_s_1_2 + \
dcRs2dt_1_C_s_2_3 + dcRs2dt_2_C_s_2_3

Substitute

eq4_Rs2_N__U_3R:= % | eq4_Bs2_1c |  eq4_Bs2_2c | \
eq4_Cs12_1c |  eq4_Cs12_2c |  \
eq4_Cs23_1c |   eq4_Cs23_2c |  \
eq4_Cs24_1c |   eq4_Cs24_2c |  \
eq4_Cs25_1c |   eq4_Cs25_2c;

U-4R-RL

dcRs2dt_N = dcRs2dt_1_B_s_2 + dcRs2dt_2_B_s_2 + \
dcRs2dt_1_C_s_1_2 + dcRs2dt_2_C_s_1_2 + \
dcRs2dt_1_C_s_2_3 + dcRs2dt_2_C_s_2_3 + \
dcRs2dt_1_C_s_2_4 + dcRs2dt_2_C_s_2_4

Substitute

eq4_Rs2_N__U_4R:= % | eq4_Bs2_1c |  eq4_Bs2_2c | \
eq4_Cs12_1c |  eq4_Cs12_2c |  \
eq4_Cs23_1c |   eq4_Cs23_2c |  \
eq4_Cs24_1c |   eq4_Cs24_2c |  \
eq4_Cs25_1c |   eq4_Cs25_2c;

U-5R-RL

dcRs2dt_N = dcRs2dt_1_B_s_2 + dcRs2dt_2_B_s_2 + \
dcRs2dt_1_C_s_1_2 + dcRs2dt_2_C_s_1_2 + \
dcRs2dt_1_C_s_2_3 + dcRs2dt_2_C_s_2_3 + \
dcRs2dt_1_C_s_2_4 + dcRs2dt_2_C_s_2_4 + \
dcRs2dt_1_C_s_2_5 + dcRs2dt_2_C_s_2_5

Substitute

eq4_Rs2_N__U_5R:= % | eq4_Bs2_1c |  eq4_Bs2_2c | \
eq4_Cs12_1c |  eq4_Cs12_2c |  \
eq4_Cs23_1c |   eq4_Cs23_2c |  \
eq4_Cs24_1c |   eq4_Cs24_2c |  \
eq4_Cs25_1c |   eq4_Cs25_2c;

Summary of equations for R**

eq4_Rs2_N__U_R;
eq4_Rs2_N__U_2R;
eq4_Rs2_N__U_3R;
eq4_Rs2_N__U_4R;
eq4_Rs2_N__U_5R;

Back to  Equations for each species

Species: R***

Equations group: 5

Equations subgroup: B

R <=> R***

(towards species)

a forward reaction rate for the transition

eq5_Bs3_1a:= eq1_Bs3_1a;

a partial conversion rate of R*** in this transition

molecularity:=1:
eq5_Bs3_1b:= dcRs3dt_1_B_s_3 = molecularity*Rate_1_B_s_3

The final form

eq5_Bs3_1c:= eq5_Bs3_1b | eq5_Bs3_1a

a reverse reaction rate for the transition

eq5_Bs3_2a:= eq1_Bs3_2a;

a partial conversion rate of R* in this transition

molecularity:=-1:
eq5_Bs3_2b:= dcRs3dt_2_B_s_3 = molecularity*Rate_2_B_s_3

The final form

eq5_Bs3_2c:= eq5_Bs3_2b | eq5_Bs3_2a

Equations subgroup: C

R* <=> R***

(towards species)

a forward reaction rate for the transition

eq5_Cs13_1a:= eq3_Cs13_1a

a partial conversion rate of R** in this transition

molecularity:=1:
eq5_Cs13_1b:= dcRs3dt_1_C_s_1_3 = molecularity*Rate_1_C_s_1_3

The final form

eq5_Cs13_1c:= eq5_Cs13_1b | eq5_Cs13_1a

a reverse reaction rate for the transition

eq5_Cs13_2a:= eq3_Cs13_2a

a partial conversion rate of R** in this transition

molecularity:=-1:
eq5_Cs13_2b:= dcRs3dt_2_C_s_1_3 = molecularity*Rate_2_C_s_1_3

The final form

eq5_Cs13_2c:= eq5_Cs13_2b | eq5_Cs13_2a

R** <=> R***

(towards species)

a forward reaction rate for the transition

eq5_Cs23_1a:= eq4_Cs23_1a

a partial conversion rate of R** in this transition

molecularity:=1:
eq5_Cs23_1b:= dcRs3dt_1_C_s_2_3 = molecularity*Rate_1_C_s_2_3

The final form

eq5_Cs23_1c:= eq5_Cs23_1b | eq5_Cs23_1a

a reverse reaction rate for the transition

eq5_Cs23_2a:= eq4_Cs23_2a

a partial conversion rate of R** in this transition

molecularity:=-1:
eq5_Cs23_2b:= dcRs3dt_2_C_s_2_3 = molecularity*Rate_2_C_s_2_3

The final form

eq5_Cs23_2c:= eq5_Cs23_2b | eq5_Cs23_2a

R*** <=> R****

(away from species)

a forward reaction rate for the transition

eq5_Cs34_1a:= Rate_1_C_s_3_4 = R_s_3 * k_1_C_s_3_4

a partial conversion rate of R** in this transition

molecularity:=-1:
eq5_Cs34_1b:= dcRs3dt_1_C_s_3_4 = molecularity*Rate_1_C_s_3_4

The final form

eq5_Cs34_1c:= eq5_Cs34_1b | eq5_Cs34_1a

a reverse reaction rate for the transition

eq5_Cs34_2a:= Rate_2_C_s_3_4 = R_s_4 * k_2_C_s_3_4

a partial conversion rate of R** in this transition

molecularity:=1:
eq5_Cs34_2b:= dcRs3dt_2_C_s_3_4 = molecularity*Rate_2_C_s_3_4

The final form

eq5_Cs34_2c:= eq5_Cs34_2b | eq5_Cs34_2a

R*** <=> R*****

(away from species)

a forward reaction rate for the transition

eq5_Cs35_1a:= Rate_1_C_s_3_5 = R_s_3 * k_1_C_s_3_5

a partial conversion rate of R** in this transition

molecularity:=-1:
eq5_Cs35_1b:= dcRs3dt_1_C_s_3_5 = molecularity*Rate_1_C_s_3_5

The final form

eq5_Cs35_1c:= eq5_Cs35_1b | eq5_Cs35_1a

a reverse reaction rate for the transition

eq5_Cs35_2a:= Rate_2_C_s_3_5 = R_s_5 * k_2_C_s_3_5

a partial conversion rate of R** in this transition

molecularity:=1:
eq5_Cs35_2b:= dcRs3dt_2_C_s_3_5 = molecularity*Rate_2_C_s_3_5

The final form

eq5_Cs35_2c:= eq5_Cs35_2b | eq5_Cs35_2a

Net conversion rate for the species

I will create equations for all five versions of the mechanism.

U-R-RL (species not present)

eq5_Rs3_N__U_R:= 0

U-2R-RL (species not present)

eq5_Rs3_N__U_2R:= 0

U-3R-RL

dcRs3dt_N = dcRs3dt_1_B_s_3 + dcRs3dt_2_B_s_3 + \
dcRs3dt_1_C_s_1_3 + dcRs3dt_2_C_s_1_3 + \
dcRs3dt_1_C_s_2_3 + dcRs3dt_2_C_s_2_3

Substitute

eq5_Rs3_N__U_3R:= % | eq5_Bs3_1c |  eq5_Bs3_2c | \
eq5_Cs13_1c |  eq5_Cs13_2c |  \
eq5_Cs23_1c |   eq5_Cs23_2c |  \
eq5_Cs34_1c |   eq5_Cs34_2c |  \
eq5_Cs35_1c |   eq5_Cs35_2c;

U-4R-RL

dcRs3dt_N = dcRs3dt_1_B_s_3 + dcRs3dt_2_B_s_3 + \
dcRs3dt_1_C_s_1_3 + dcRs3dt_2_C_s_1_3 + \
dcRs3dt_1_C_s_2_3 + dcRs3dt_2_C_s_2_3  + \
dcRs3dt_1_C_s_3_4 + dcRs3dt_2_C_s_3_4

Substitute

eq5_Rs3_N__U_4R:= % | eq5_Bs3_1c |  eq5_Bs3_2c | \
eq5_Cs13_1c |  eq5_Cs13_2c |  \
eq5_Cs23_1c |   eq5_Cs23_2c |  \
eq5_Cs34_1c |   eq5_Cs34_2c |  \
eq5_Cs35_1c |   eq5_Cs35_2c;

U-5R-RL

dcRs3dt_N = dcRs3dt_1_B_s_3 + dcRs3dt_2_B_s_3 + \
dcRs3dt_1_C_s_1_3 + dcRs3dt_2_C_s_1_3 + \
dcRs3dt_1_C_s_2_3 + dcRs3dt_2_C_s_2_3  + \
dcRs3dt_1_C_s_3_4 + dcRs3dt_2_C_s_3_4  + \
dcRs3dt_1_C_s_3_5 + dcRs3dt_2_C_s_3_5

Substitute

eq5_Rs3_N__U_5R:= % | eq5_Bs3_1c |  eq5_Bs3_2c | \
eq5_Cs13_1c |  eq5_Cs13_2c |  \
eq5_Cs23_1c |   eq5_Cs23_2c |  \
eq5_Cs34_1c |   eq5_Cs34_2c |  \
eq5_Cs35_1c |   eq5_Cs35_2c;

Summary of equations for R***

eq5_Rs3_N__U_R;
eq5_Rs3_N__U_2R;
eq5_Rs3_N__U_3R;
eq5_Rs3_N__U_4R;
eq5_Rs3_N__U_5R;

Back to  Equations for each species

Species: R****

Equations group: 6

Equations subgroup: B

R <=> R****

(towards species)

a forward reaction rate for the transition

eq6_Bs4_1a:= eq1_Bs4_1a;

a partial conversion rate of R**** in this transition

molecularity:=1:
eq6_Bs4_1b:= dcRs4dt_1_B_s_4 = molecularity*Rate_1_B_s_4

The final form

eq6_Bs4_1c:= eq6_Bs4_1b | eq6_Bs4_1a

a reverse reaction rate for the transition

eq6_Bs4_2a:= eq1_Bs4_2a;

a partial conversion rate of R**** in this transition

molecularity:=-1:
eq6_Bs4_2b:= dcRs4dt_2_B_s_4 = molecularity*Rate_2_B_s_4

The final form

eq6_Bs4_2c:= eq6_Bs4_2b | eq6_Bs4_2a

Equations subgroup: C

R* <=> R****

(towards species)

a forward reaction rate for the transition

eq6_Cs14_1a:= eq3_Cs14_1a

a partial conversion rate of R**** in this transition

molecularity:=1:
eq6_Cs14_1b:= dcRs4dt_1_C_s_1_4 = molecularity*Rate_1_C_s_1_4

The final form

eq6_Cs14_1c:= eq6_Cs14_1b | eq6_Cs14_1a

a reverse reaction rate for the transition

eq6_Cs14_2a:= eq3_Cs14_2a

a partial conversion rate of R**** in this transition

molecularity:=-1:
eq6_Cs14_2b:= dcRs4dt_2_C_s_1_4 = molecularity*Rate_2_C_s_1_4

The final form

eq6_Cs14_2c:= eq6_Cs14_2b | eq6_Cs14_2a

R** <=> R****

(towards species)

a forward reaction rate for the transition

eq6_Cs24_1a:= eq4_Cs24_1a

a partial conversion rate of R**** in this transition

molecularity:=1:
eq6_Cs24_1b:= dcRs4dt_1_C_s_2_4 = molecularity*Rate_1_C_s_2_4

The final form

eq6_Cs24_1c:= eq6_Cs24_1b | eq6_Cs24_1a

a reverse reaction rate for the transition

eq6_Cs24_2a:= eq4_Cs24_2a

a partial conversion rate of R**** in this transition

molecularity:=-1:
eq6_Cs24_2b:= dcRs4dt_2_C_s_2_4 = molecularity*Rate_2_C_s_2_4

The final form

eq6_Cs24_2c:= eq6_Cs24_2b | eq6_Cs24_2a

R*** <=> R****

(towards species)

a forward reaction rate for the transition

eq6_Cs34_1a:= eq5_Cs34_1a

a partial conversion rate of R**** in this transition

molecularity:=1:
eq6_Cs34_1b:= dcRs4dt_1_C_s_3_4 = molecularity*Rate_1_C_s_3_4

The final form

eq6_Cs34_1c:= eq6_Cs34_1b | eq6_Cs34_1a

a reverse reaction rate for the transition

eq6_Cs34_2a:= eq5_Cs34_2a

a partial conversion rate of R**** in this transition

molecularity:=-1:
eq6_Cs34_2b:= dcRs4dt_2_C_s_3_4 = molecularity*Rate_2_C_s_3_4

The final form

eq6_Cs34_2c:= eq6_Cs34_2b | eq6_Cs34_2a

R**** <=> R*****

(away from species)

a forward reaction rate for the transition

eq6_Cs45_1a:= Rate_1_C_s_4_5 = R_s_4 * k_1_C_s_4_5

a partial conversion rate of R** in this transition

molecularity:=-1:
eq6_Cs45_1b:= dcRs4dt_1_C_s_4_5 = molecularity*Rate_1_C_s_4_5

The final form

eq6_Cs45_1c:= eq6_Cs45_1b | eq6_Cs45_1a

a reverse reaction rate for the transition

eq6_Cs45_2a:= Rate_2_C_s_4_5 = R_s_5 * k_2_C_s_4_5

a partial conversion rate of R** in this transition

molecularity:=1:
eq6_Cs45_2b:= dcRs4dt_2_C_s_4_5 = molecularity*Rate_2_C_s_4_5

The final form

eq6_Cs45_2c:= eq6_Cs45_2b | eq6_Cs45_2a

Net conversion rate for the species

I will create equations for all five versions of the mechanism.

U-R-RL (species not present)

eq6_Rs4_N__U_R:= 0

U-2R-RL (species not present)

eq6_Rs4_N__U_2R:= 0

U-3R-RL (species not present)

eq6_Rs4_N__U_3R:= 0

U-4R-RL

dcRs4dt_N = dcRs4dt_1_B_s_4 + dcRs4dt_2_B_s_4 + \
dcRs4dt_1_C_s_1_4 + dcRs4dt_2_C_s_1_4 + \
dcRs4dt_1_C_s_2_4 + dcRs4dt_2_C_s_2_4 + \
dcRs4dt_1_C_s_3_4 + dcRs4dt_2_C_s_3_4

Substitute

eq6_Rs4_N__U_4R:= % | eq6_Bs4_1c |  eq6_Bs4_2c | \
eq6_Cs14_1c |  eq6_Cs14_2c |  \
eq6_Cs24_1c |   eq6_Cs24_2c |  \
eq6_Cs34_1c |   eq6_Cs34_2c ;

U-5R-RL

dcRs4dt_N = dcRs4dt_1_B_s_4 + dcRs4dt_2_B_s_4 + \
dcRs4dt_1_C_s_1_4 + dcRs4dt_2_C_s_1_4 + \
dcRs4dt_1_C_s_2_4 + dcRs4dt_2_C_s_2_4 + \
dcRs4dt_1_C_s_3_4 + dcRs4dt_2_C_s_3_4 + \
dcRs4dt_1_C_s_4_5 + dcRs4dt_2_C_s_4_5

Substitute

eq6_Rs4_N__U_5R:= % | eq6_Bs4_1c |  eq6_Bs4_2c | \
eq6_Cs14_1c |  eq6_Cs14_2c |  \
eq6_Cs24_1c |   eq6_Cs24_2c |  \
eq6_Cs34_1c |   eq6_Cs34_2c |  \
eq6_Cs45_1c |   eq6_Cs45_2c ;

Summary of equations for R****

eq6_Rs4_N__U_R;
eq6_Rs4_N__U_2R;
eq6_Rs4_N__U_3R;
eq6_Rs4_N__U_4R;
eq6_Rs4_N__U_5R;

Back to  Equations for each species

Species: R*****

Equations group: 7

Equations subgroup: B

R <=> R*****

(towards species)

a forward reaction rate for the transition

eq7_Bs5_1a:= eq1_Bs5_1a;

a partial conversion rate of R**** in this transition

molecularity:=1:
eq7_Bs5_1b:= dcRs5dt_1_B_s_5 = molecularity*Rate_1_B_s_5

The final form

eq7_Bs5_1c:= eq7_Bs5_1b | eq7_Bs5_1a

a reverse reaction rate for the transition

eq7_Bs5_2a:= eq1_Bs5_2a;

a partial conversion rate of R**** in this transition

molecularity:=-1:
eq7_Bs5_2b:= dcRs5dt_2_B_s_5 = molecularity*Rate_2_B_s_5

The final form

eq7_Bs5_2c:= eq7_Bs5_2b | eq7_Bs5_2a

Equations subgroup: C

R* <=> R*****

(towards species)

a forward reaction rate for the transition

eq7_Cs15_1a:= eq3_Cs15_1a

a partial conversion rate of R***** in this transition

molecularity:=1:
eq7_Cs15_1b:= dcRs5dt_1_C_s_1_5 = molecularity*Rate_1_C_s_1_5

The final form

eq7_Cs15_1c:= eq7_Cs15_1b | eq7_Cs15_1a

a reverse reaction rate for the transition

eq7_Cs15_2a:= eq3_Cs15_2a

a partial conversion rate of R***** in this transition

molecularity:=-1:
eq7_Cs15_2b:= dcRs5dt_2_C_s_1_5 = molecularity*Rate_2_C_s_1_5

The final form

eq7_Cs15_2c:= eq7_Cs15_2b | eq7_Cs15_2a

R** <=> R*****

(towards species)

a forward reaction rate for the transition

eq7_Cs25_1a:= eq4_Cs25_1a

a partial conversion rate of R***** in this transition

molecularity:=1:
eq7_Cs25_1b:= dcRs5dt_1_C_s_2_5 = molecularity*Rate_1_C_s_2_5

The final form

eq7_Cs25_1c:= eq7_Cs25_1b | eq7_Cs25_1a

a reverse reaction rate for the transition

eq7_Cs25_2a:= eq4_Cs25_2a

a partial conversion rate of R***** in this transition

molecularity:=-1:
eq7_Cs25_2b:= dcRs5dt_2_C_s_2_5 = molecularity*Rate_2_C_s_2_5

The final form

eq7_Cs25_2c:= eq7_Cs25_2b | eq7_Cs25_2a

R*** <=> R*****

(towards species)

a forward reaction rate for the transition

eq7_Cs35_1a:= eq5_Cs35_1a

a partial conversion rate of R***** in this transition

molecularity:=1:
eq7_Cs35_1b:= dcRs5dt_1_C_s_3_5 = molecularity*Rate_1_C_s_3_5

The final form

eq7_Cs35_1c:= eq7_Cs35_1b | eq7_Cs35_1a

a reverse reaction rate for the transition

eq7_Cs35_2a:= eq5_Cs35_2a

a partial conversion rate of R***** in this transition

molecularity:=-1:
eq7_Cs35_2b:= dcRs5dt_2_C_s_3_5 = molecularity*Rate_2_C_s_3_5

The final form

eq7_Cs35_2c:= eq7_Cs35_2b | eq7_Cs35_2a

R**** <=> R*****

(towards species)

a forward reaction rate for the transition

eq7_Cs45_1a:= eq6_Cs45_1a

a partial conversion rate of R***** in this transition

molecularity:=1:
eq7_Cs45_1b:= dcRs5dt_1_C_s_4_5 = molecularity*Rate_1_C_s_4_5

The final form

eq7_Cs45_1c:= eq7_Cs45_1b | eq7_Cs45_1a

a reverse reaction rate for the transition

eq7_Cs45_2a:= eq6_Cs45_2a

a partial conversion rate of R***** in this transition

molecularity:=-1:
eq7_Cs45_2b:= dcRs5dt_2_C_s_4_5 = molecularity*Rate_2_C_s_4_5

The final form

eq7_Cs45_2c:= eq7_Cs45_2b | eq7_Cs45_2a

Net conversion rate for the species

I will create equations for all five versions of the mechanism.

U-R-RL (species not present)

eq7_Rs5_N__U_R:= 0

U-2R-RL (species not present)

eq7_Rs5_N__U_2R:= 0

U-3R-RL (species not present)

eq7_Rs5_N__U_3R:= 0

U-4R-RL

eq7_Rs5_N__U_4R:= 0

U-5R-RL

dcRs5dt_N = dcRs5dt_1_B_s_5 + dcRs5dt_2_B_s_5 + \
dcRs5dt_1_C_s_1_5 + dcRs5dt_2_C_s_1_5 + \
dcRs5dt_1_C_s_2_5 + dcRs5dt_2_C_s_2_5 + \
dcRs5dt_1_C_s_3_5 + dcRs5dt_2_C_s_3_5 + \
dcRs5dt_1_C_s_4_5 + dcRs5dt_2_C_s_4_5

Substitute

eq7_Rs5_N__U_5R:= % | eq7_Bs5_1c |  eq7_Bs5_2c | \
eq7_Cs15_1c |  eq7_Cs15_2c |  \
eq7_Cs25_1c |   eq7_Cs25_2c |  \
eq7_Cs35_1c |   eq7_Cs35_2c |  \
eq7_Cs45_1c |   eq7_Cs45_2c ;

Summary of equations for R*****

eq7_Rs5_N__U_R;
eq7_Rs5_N__U_2R;
eq7_Rs5_N__U_3R;
eq7_Rs5_N__U_4R;
eq7_Rs5_N__U_5R;

Back to  Equations for each species

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not needed here because we do not have oligomerization reactions.

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Expession of K matrix for U-R mechanism

This is derivation for comparison with existing U-R mechanism matrix: order of species R, R*, RL.

Derivation with more conventient new order of species is below

Summary list of the net rate equations for the mechanism

eq1_R_N__U_R;
eq2_RL_N;
eq3_Rs1_N__U_R;

Assign sequential names to species

feq_1a:= R    = C1;
feq_1b:= R_s_1= C2;
feq_1c:= RL   = C3;

Assign the same order to net rate equations

feq_2a:= dcRdt_N      = dC1dt;
feq_2b:= dcRs1dt_N    = dC2dt;
feq_2c:= dcRLdt_N     = dC3dt;

Restate the equations in terms of new sequential species names (rename free ligand concentration too)

R

eq1_R_N__U_R;
feq_3a:= % | feq_1a | feq_1b | feq_1c | feq_2a | feq_2b | feq_2c |  L = Leq

R*

eq3_Rs1_N__U_R;
feq_3b:= % | feq_1a | feq_1b | feq_1c | feq_2a | feq_2b | feq_2c |  L = Leq

RL

eq2_RL_N;
feq_3c:= % | feq_1a | feq_1b | feq_1c | feq_2a | feq_2b | feq_2c |  L = Leq

Prepare results for transfer to MATLAB

(To avoid typing errors when transfering the derived K matrix to MATLAB, I will type it in here and then directly test against the  derivation result. After that the K matrix may be transfered to MATLAB by cut-and-paste  operation of the MuPad output of the following cell)

Simple rules that allow catching mistakes in K matrix derivation:

(1) a sum of each column should be zero (so each constant must appear with both positive and negative sign), and

(2) each row has to have complete pairs of constants (i.e., if k12

appears there must be k21 in the same row with an opposite sign and so on).

K:=matrix(3,3,[
[    -k_1_B_s_1 - Leq*k_1_A,             k_2_B_s_1,        k_2_A        ],
[     k_1_B_s_1,                        -k_2_B_s_1,          0          ],
[                 Leq*k_1_A,                 0,           -k_2_A        ]
])

Test the K matrix entry

Create a column vector of species concentrations

P:=matrix(3,1,[C1, C2, C3])

Multiply K and P:

dCdt_manual_input:= K*P

Collect right-hand-side parts of net rate equations expressed in sequential species names

dCdt_mupad:=matrix(3,1,[ rhs(feq_3a), rhs(feq_3b), rhs(feq_3c)])

Compare derivation result to manual input

normal(%);
bool(%)

=> Typed K-matrix is correct and identical to U-R matrix derived earlier for IDAP.

This derivation is uses order of species: R, RL, R* to be able to expand mechanism without renumbering equations.

Summary list of the net rate equations for the mechanism

Assign sequential names to species

feq_1a:= R    = C1;
feq_1b:= RL   = C2;
feq_1c:= R_s_1= C3;

Assign the same order to net rate equations

feq_2a:= dcRdt_N      = dC1dt;
feq_2b:= dcRLdt_N     = dC2dt;
feq_2c:= dcRs1dt_N    = dC3dt;

Restate the equations in terms of new sequential species names (rename free ligand concentration too)

R

eq1_R_N__U_R:
feq_3a:= % | feq_1a | feq_1b | feq_1c | feq_2a | feq_2b | feq_2c |  L = Leq

RL

eq2_RL_N:
feq_3b:= % | feq_1a | feq_1b | feq_1c | feq_2a | feq_2b | feq_2c |  L = Leq

R*

eq3_Rs1_N__U_R:
feq_3c:= % | feq_1a | feq_1b | feq_1c | feq_2a | feq_2b | feq_2c |  L = Leq

Prepare results for transfer to MATLAB

(To avoid typing errors when transfering the derived K matrix to MATLAB, I will type it in here and then directly test against the  derivation result. After that the K matrix may be transfered to MATLAB by cut-and-paste  operation of the MuPad output of the following cell)

Simple rules that allow catching mistakes in K matrix derivation:

(1) a sum of each column should be zero (so each constant must appear with both positive and negative sign), and

(2) each row has to have complete pairs of constants (i.e., if k12

appears there must be k21 in the same row with an opposite sign and so on).

K:=matrix(3,3,[
[    -k_1_B_s_1 - Leq*k_1_A,            k_2_A ,             k_2_B_s_1         ],
[                 Leq*k_1_A,           -k_2_A  ,                 0            ],
[     k_1_B_s_1,                            0 ,            -k_2_B_s_1         ]
])

Test the K matrix entry

Create a column vector of species concentrations

P:=matrix(3,1,[C1, C2, C3])

Multiply K and P:

dCdt_manual_input:= K*P

Collect right-hand-side parts of net rate equations expressed in sequential species names

dCdt_mupad:=matrix(3,1,[ rhs(feq_3a), rhs(feq_3b), rhs(feq_3c)])

Compare derivation result to manual input

normal(%);
bool(%)

=> Typed K-matrix is correct.

K matrix for the U-R model with the new species order

K;

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Expression for K matrix of U-2R mechanism

Summary list of the net rate equations for the mechanism

eq1_R_N__U_2R;
eq2_RL_N;
eq3_Rs1_N__U_2R;
eq4_Rs2_N__U_2R;

Assign sequential names to species

feq_1a:= R    = C1;
feq_1b:= RL   = C2;
feq_1c:= R_s_1= C3;
feq_1d:= R_s_2= C4;

Assign the same order to net rate equations

feq_2a:= dcRdt_N      = dC1dt;
feq_2b:= dcRLdt_N     = dC2dt;
feq_2c:= dcRs1dt_N    = dC3dt;
feq_2d:= dcRs2dt_N    = dC4dt;

Restate the equations in terms of new sequential species names (rename free ligand concentration too)

R

eq1_R_N__U_2R;
feq_3a:= % | feq_1a | feq_2a | feq_1b | feq_2b | feq_1c | feq_2c | feq_1d | feq_2d |  L = Leq

RL

eq2_RL_N;
feq_3b:= % | feq_1a | feq_2a | feq_1b | feq_2b | feq_1c | feq_2c | feq_1d | feq_2d |  L = Leq

R*

eq3_Rs1_N__U_2R;
feq_3c:
= % | feq_1a | feq_2a | feq_1b | feq_2b | feq_1c | feq_2c | feq_1d | feq_2d |  L = Leq

R**

eq4_Rs2_N__U_2R;
feq_3d:
= % | feq_1a | feq_2a | feq_1b | feq_2b | feq_1c | feq_2c | feq_1d | feq_2d |  L = Leq

Prepare results for transfer to MATLAB

(To avoid typing errors when transfering the derived K matrix to MATLAB, I will type it in here and then directly test against the  derivation result. After that the K matrix may be transfered to MATLAB by cut-and-paste  operation of the MuPad output of the following cell)

Simple rules that allow catching mistakes in K matrix derivation:

(1) a sum of each column should be zero (so each constant must appear with both positive and negative sign), and

(2) each row has to have complete pairs of constants (i.e., if k12

appears there must be k21 in the same row with an opposite sign and so on).

K:=matrix(4,4,[
[ -k_1_B_s_1-k_1_B_s_2-Leq*k_1_A,      k_2_A,     k_2_B_s_1,                          k_2_B_s_2      ],
[                      Leq*k_1_A,     -k_2_A,         0 ,                             0              ],
[  k_1_B_s_1,                            0,       -k_2_B_s_1-k_1_C_s_1_2,   k_2_C_s_1_2              ],
[            k_1_B_s_2,                  0,                  k_1_C_s_1_2,  -k_2_B_s_2-k_2_C_s_1_2    ]
])

Test the K matrix entry

Create a column vector of species concentrations

P:=matrix(4,1,[C1, C2, C3, C4])

Multiply K and P:

dCdt_manual_input:= K*P

Collect right-hand-side parts of net rate equations expressed in sequential species names

dCdt_mupad:=matrix(4,1,[ rhs(feq_3a), rhs(feq_3b), rhs(feq_3c), rhs(feq_3d)])

Compare derivation result to manual input

normal(%);
bool(%)

=> Typed K-matrix is correct.

Final expression for U-2R kinetic matrix

K

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Derivation of K matrix for U-3R mechanism

Summary list of the net rate equations for the mechanism

eq1_R_N__U_3R;
eq2_RL_N;
eq3_Rs1_N__U_3R;
eq4_Rs2_N__U_3R;
eq5_Rs3_N__U_3R;

Assign sequential names to species

feq_1a:= R    = C1;
feq_1b:= RL   = C2;
feq_1c:= R_s_1= C3;
feq_1d:= R_s_2= C4;
feq_1e:= R_s_3= C5;

Assign the same order to net rate equations

feq_2a:= dcRdt_N      = dC1dt;
feq_2b:= dcRLdt_N     = dC2dt;
feq_2c:= dcRs1dt_N    = dC3dt;
feq_2d:= dcRs2dt_N    = dC4dt;
feq_2e:= dcRs3dt_N    = dC5dt;

Restate the equations in terms of new sequential species names (rename free ligand concentration too)

R

eq1_R_N__U_3R;
feq_3a:= % | feq_1a | feq_2a | feq_1b | feq_2b | feq_1c | feq_2c | feq_1d | feq_2d | feq_1e | feq_2e |  L = Leq

RL

eq2_RL_N;
feq_3b:= % | feq_1a | feq_2a | feq_1b | feq_2b | feq_1c | feq_2c | feq_1d | feq_2d | feq_1e | feq_2e |  L = Leq

R*

eq3_Rs1_N__U_3R;
feq_3c:
= % |  feq_1a | feq_2a | feq_1b | feq_2b | feq_1c | feq_2c | feq_1d | feq_2d | feq_1e | feq_2e |  L = Leq

R**

eq4_Rs2_N__U_3R;
feq_3d:
= % |  feq_1a | feq_2a | feq_1b | feq_2b | feq_1c | feq_2c | feq_1d | feq_2d | feq_1e | feq_2e |  L = Leq

R***

eq5_Rs3_N__U_3R;
feq_3e:= % |  feq_1a | feq_2a | feq_1b | feq_2b | feq_1c | feq_2c | feq_1d | feq_2d | feq_1e | feq_2e |  L = Leq

Prepare results for transfer to MATLAB

(To avoid typing errors when transfering the derived K matrix to MATLAB, I will type it in here and then directly test against the  derivation result. After that the K matrix may be transfered to MATLAB by cut-and-paste  operation of the MuPad output of the following cell)

Simple rules that allow catching mistakes in K matrix derivation:

(1) a sum of each column should be zero (so each constant must appear with both positive and negative sign), and

(2) each row has to have complete pairs of constants (i.e., if k12

appears there must be k21 in the same row with an opposite sign and so on).

K:=matrix(5,5,[
[ -k_1_B_s_1-k_1_B_s_2-k_1_B_s_3-Leq*k_1_A,      k_2_A,      k_2_B_s_1,                          k_2_B_s_2 ,                           k_2_B_s_3                            ],
[                                Leq*k_1_A,     -k_2_A,         0 ,                                 0,                                     0                                ],
[  k_1_B_s_1,                                      0,       -k_2_B_s_1-k_1_C_s_1_2-k_1_C_s_1_3,             k_2_C_s_1_2,                         k_2_C_s_1_3                ],
[            k_1_B_s_2,                            0,                  k_1_C_s_1_2,              -k_2_B_s_2-k_2_C_s_1_2-k_1_C_s_2_3,                         k_2_C_s_2_3    ],
[                      k_1_B_s_3,                  0,                              k_1_C_s_1_3,                         k_1_C_s_2_3,  -k_2_B_s_3-k_2_C_s_1_3-k_2_C_s_2_3    ]
])

Test the K matrix entry

Create a column vector of species concentrations

P:=matrix(5,1,[C1, C2, C3, C4, C5])

Multiply K and P:

dCdt_manual_input:= K*P

Collect right-hand-side parts of net rate equations expressed in sequential species names

dCdt_mupad:=matrix(5,1,[ rhs(feq_3a), rhs(feq_3b), rhs(feq_3c), rhs(feq_3d), rhs(feq_3e)])

Compare derivation result to manual input

normal(%);
bool(%)

=> Typed K-matrix is correct.

Final expression for U-3R kinetic matrix

K

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Derivation of K matrix for U-4R mechanism

Summary list of the net rate equations for the mechanism

eq1_R_N__U_4R;
eq2_RL_N;
eq3_Rs1_N__U_4R;
eq4_Rs2_N__U_4R;
eq5_Rs3_N__U_4R;
eq6_Rs4_N__U_4R;

Assign sequential names to species

feq_1a:= R    = C1;
feq_1b:= RL   = C2;
feq_1c:= R_s_1= C3;
feq_1d:= R_s_2= C4;
feq_1e:= R_s_3= C5;
feq_1f:= R_s_4= C6;

Assign the same order to net rate equations

feq_2a:= dcRdt_N      = dC1dt;
feq_2b:= dcRLdt_N     = dC2dt;
feq_2c:= dcRs1dt_N    = dC3dt;
feq_2d:= dcRs2dt_N    = dC4dt;
feq_2e:= dcRs3dt_N    = dC5dt;
feq_2f:= dcRs4dt_N    = dC6dt;

Restate the equations in terms of new sequential species names (rename free ligand concentration too)

R

eq1_R_N__U_4R;
feq_3a:= % | feq_1a | feq_2a | feq_1b | feq_2b | feq_1c | feq_2c | feq_1d | feq_2d | feq_1e | feq_2e | feq_1f | feq_2f |  L = Leq

RL

eq2_RL_N;
feq_3b:= % | feq_1a | feq_2a | feq_1b | feq_2b | feq_1c | feq_2c | feq_1d | feq_2d | feq_1e | feq_2e | feq_1f | feq_2f  |  L = Leq

R*

eq3_Rs1_N__U_4R;
feq_3c:
= % |  feq_1a | feq_2a | feq_1b | feq_2b | feq_1c | feq_2c | feq_1d | feq_2d | feq_1e | feq_2e | feq_1f | feq_2f  |  L = Leq

R**

eq4_Rs2_N__U_4R;
feq_3d:
= % |  feq_1a | feq_2a | feq_1b | feq_2b | feq_1c | feq_2c | feq_1d | feq_2d | feq_1e | feq_2e | feq_1f | feq_2f |  L = Leq

R***

eq5_Rs3_N__U_4R;
feq_3e:= % |  feq_1a | feq_2a | feq_1b | feq_2b | feq_1c | feq_2c | feq_1d | feq_2d | feq_1e | feq_2e | feq_1f | feq_2f  |  L = Leq

R****

eq6_Rs4_N__U_4R;
feq_3f:= % |  feq_1a | feq_2a | feq_1b | feq_2b | feq_1c | feq_2c | feq_1d | feq_2d | feq_1e | feq_2e | feq_1f | feq_2f  |  L = Leq

Prepare results for transfer to MATLAB

(To avoid typing errors when transfering the derived K matrix to MATLAB, I will type it in here and then directly test against the  derivation result. After that the K matrix may be transfered to MATLAB by cut-and-paste  operation of the MuPad output of the following cell)

Simple rules that allow catching mistakes in K matrix derivation:

(1) a sum of each column should be zero (so each constant must appear with both positive and negative sign), and

(2) each row has to have complete pairs of constants (i.e., if k12

appears there must be k21 in the same row with an opposite sign and so on).

K:=matrix(6,6,[
[ -k_1_B_s_1-k_1_B_s_2-k_1_B_s_3-k_1_B_s_4-Leq*k_1_A,   k_2_A,         k_2_B_s_1,                                      k_2_B_s_2 ,             k_2_B_s_3   ,                k_2_B_s_4                                    ],
[                                          Leq*k_1_A,  -k_2_A,         0 ,                                                   0,                                          0  ,                              0                       ],
[  k_1_B_s_1,                                             0,          -k_2_B_s_1-k_1_C_s_1_2-k_1_C_s_1_3-k_1_C_s_1_4,            k_2_C_s_1_2,                                    k_2_C_s_1_3,                k_2_C_s_1_4                         ],
[            k_1_B_s_2,                                   0,                     k_1_C_s_1_2,                         -k_2_B_s_2-k_2_C_s_1_2-k_1_C_s_2_3-k_1_C_s_2_4,                         k_2_C_s_2_3 ,                     k_2_C_s_2_4             ],
[                      k_1_B_s_3,                         0,                                 k_1_C_s_1_3,                                    k_1_C_s_2_3,              -k_2_B_s_3-k_2_C_s_1_3-k_2_C_s_2_3-k_1_C_s_3_4 ,          k_2_C_s_3_4 ],
[                                k_1_B_s_4,               0,                                             k_1_C_s_1_4,                                     k_1_C_s_2_4,                   k_1_C_s_3_4,   -k_2_B_s_4-k_2_C_s_1_4-k_2_C_s_2_4-k_2_C_s_3_4  ]
])

Test the K matrix entry

Create a column vector of species concentrations

P:=matrix(6,1,[C1, C2, C3, C4, C5, C6])

Multiply K and P:

dCdt_manual_input:= K*P

Collect right-hand-side parts of net rate equations expressed in sequential species names

dCdt_mupad:=matrix(6,1,[ rhs(feq_3a), rhs(feq_3b), rhs(feq_3c), rhs(feq_3d), rhs(feq_3e), rhs(feq_3f)])

Compare derivation result to manual input

normal(%);
bool(%)

=> Typed K-matrix is correct.

Final expression for U-4R kinetic matrix

K

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Derivation of K matrix for U-5R mechanism

Summary list of the net rate equations for the mechanism

eq1_R_N__U_5R;
eq2_RL_N;
eq3_Rs1_N__U_5R;
eq4_Rs2_N__U_5R;
eq5_Rs3_N__U_5R;
eq6_Rs4_N__U_5R;
eq7_Rs5_N__U_5R;

Assign sequential names to species

feq_1a:= R    = C1;
feq_1b:= RL   = C2;
feq_1c:= R_s_1= C3;
feq_1d:= R_s_2= C4;
feq_1e:= R_s_3= C5;
feq_1f:= R_s_4= C6;
feq_1g:= R_s_5= C7;

Assign the same order to net rate equations

feq_2a:= dcRdt_N      = dC1dt;
feq_2b:= dcRLdt_N     = dC2dt;
feq_2c:= dcRs1dt_N    = dC3dt;
feq_2d:= dcRs2dt_N    = dC4dt;
feq_2e:= dcRs3dt_N    = dC5dt;
feq_2f:= dcRs4dt_N    = dC6dt;
feq_2g:= dcRs5dt_N    = dC7dt;

Restate the equations in terms of new sequential species names (rename free ligand concentration too)

R

eq1_R_N__U_5R;
feq_3a:= % | feq_1a | feq_2a | feq_1b | feq_2b | feq_1c | feq_2c | feq_1d | feq_2d | feq_1e | feq_2e | feq_1f | feq_2f  | feq_1g | feq_2g |  L = Leq

RL

eq2_RL_N;
feq_3b:=   % | feq_1a | feq_2a | feq_1b | feq_2b | feq_1c | feq_2c | feq_1d | feq_2d | feq_1e | feq_2e | feq_1f | feq_2f  | feq_1g | feq_2g |  L = Leq

R*

eq3_Rs1_N__U_5R;
feq_3c:
=  % | feq_1a | feq_2a | feq_1b | feq_2b | feq_1c | feq_2c | feq_1d | feq_2d | feq_1e | feq_2e | feq_1f | feq_2f  | feq_1g | feq_2g |  L = Leq

R**

eq4_Rs2_N__U_5R;
feq_3d:
=   % | feq_1a | feq_2a | feq_1b | feq_2b | feq_1c | feq_2c | feq_1d | feq_2d | feq_1e | feq_2e | feq_1f | feq_2f  | feq_1g | feq_2g |  L = Leq

R***

eq5_Rs3_N__U_5R;
feq_3e:=  % | feq_1a | feq_2a | feq_1b | feq_2b | feq_1c | feq_2c | feq_1d | feq_2d | feq_1e | feq_2e | feq_1f | feq_2f  | feq_1g | feq_2g |  L = Leq

R****

eq6_Rs4_N__U_5R;
feq_3f:=   % | feq_1a | feq_2a | feq_1b | feq_2b | feq_1c | feq_2c | feq_1d | feq_2d | feq_1e | feq_2e | feq_1f | feq_2f  | feq_1g | feq_2g |  L = Leq

R*****

eq7_Rs5_N__U_5R;
feq_3g:=  % | feq_1a | feq_2a | feq_1b | feq_2b | feq_1c | feq_2c | feq_1d | feq_2d | feq_1e | feq_2e | feq_1f | feq_2f  | feq_1g | feq_2g |  L = Leq

Prepare results for transfer to MATLAB

(To avoid typing errors when transfering the derived K matrix to MATLAB, I will type it in here and then directly test against the  derivation result. After that the K matrix may be transfered to MATLAB by cut-and-paste  operation of the MuPad output of the following cell)

Workflow for accurate extraction of the K matrix

- prepare empty matrix

- copy-paste one equation at a time to MATLAB script

- cut-and-paste terms (together with signs and coefficients) into MuPad matrix

- execute cell after every equation to see how matrix is filled

- delete coefficients

Simple rules that allow catching mistakes in K matrix derivation:

(1) a sum of each column should be zero (so each constant must appear with both positive and negative sign), and

(2) each row has to have complete pairs of constants (i.e., if k12

appears there must be k21 in the same row with an opposite sign and so on).

K:=matrix(7,7,[
[    -k_1_B_s_1 -k_1_B_s_2 -k_1_B_s_3 -k_1_B_s_4 -k_1_B_s_5 -Leq*k_1_A,    k_2_A,  k_2_B_s_1, k_2_B_s_2, k_2_B_s_3, k_2_B_s_4, k_2_B_s_5                           ],
[     Leq*k_1_A,  - k_2_A, 0,0,0,0,0                           ],
[    k_1_B_s_1 , 0, -k_2_B_s_1 -k_1_C_s_1_2 -k_1_C_s_1_3 -k_1_C_s_1_4 -k_1_C_s_1_5 ,  k_2_C_s_1_2,  k_2_C_s_1_3, k_2_C_s_1_4,   k_2_C_s_1_5      ],
[    k_1_B_s_2, 0, k_1_C_s_1_2,  -k_1_C_s_2_3 -k_1_C_s_2_4 -k_1_C_s_2_5 -k_2_C_s_1_2  -k_2_B_s_2, k_2_C_s_2_3, k_2_C_s_2_4, k_2_C_s_2_5                            ],
[    k_1_B_s_3, 0, k_1_C_s_1_3,   k_1_C_s_2_3, -k_2_B_s_3  -k_1_C_s_3_4 -k_1_C_s_3_5 -k_2_C_s_1_3 -k_2_C_s_2_3,  k_2_C_s_3_4,  k_2_C_s_3_5                        ],
[    k_1_B_s_4, 0, k_1_C_s_1_4,  k_1_C_s_2_4 , k_1_C_s_3_4 , -k_2_B_s_4    -k_1_C_s_4_5 -k_2_C_s_1_4 -k_2_C_s_2_4 -k_2_C_s_3_4, k_2_C_s_4_5                           ],
[   k_1_B_s_5, 0, k_1_C_s_1_5,   k_1_C_s_2_5, k_1_C_s_3_5, k_1_C_s_4_5,  -k_2_B_s_5     -k_2_C_s_1_5 -k_2_C_s_2_5 -k_2_C_s_3_5 -k_2_C_s_4_5                             ]
])

Test the K matrix entry

Create a column vector of species concentrations

P:=matrix(7,1,[C1, C2, C3, C4, C5, C6, C7])

Multiply K and P:

dCdt_manual_input:= K*P

Collect right-hand-side parts of net rate equations expressed in sequential species names

dCdt_mupad:=matrix(7,1,[ rhs(feq_3a), rhs(feq_3b), rhs(feq_3c), rhs(feq_3d), rhs(feq_3e), rhs(feq_3f), rhs(feq_3g)])

Compare derivation result to manual input

normal(%);
bool(%)

=> Typed K-matrix is correct.

Final expression for U-5R kinetic matrix

K;

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K matrices for U-R to U-5R models were successfully developed.

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