U-5R-RL
Derivation of differential equations describing evolution of spin concentrations
NOTE: This document is based on IDAP/Mathematical_models/NMR_line_shape_models/2D/U_5R_RL/U_5R_RL.mn
NOTE 2: The model with one isomerization, U-1R-RL, is not identical to U-R-RL because the latter contains a transition connecting R* and RL* directly, which is absent from U-nR-RL family models!
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**, R***, R****, R*****, RL* to simplify expansion of the model. It is different from U-R model in the existing IDAP code!
Accurate extraction of K matrix:
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. NOTE: It is more convenient to do arrange terms with increasing coefficients in MATLAB, place commas after groups with the same coefficient (and zeros for the absent coefficients) and only then paste into the MuPad.
- execute cell after every equation to see how matrix is filled
- delete coefficients (keep negative signs!!!)
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-RL mechanism
K matrix for U-R-RL model with new species order
Derivation for K matrix of U-2R-RL mechanism
- Final expression for U-2R-RL kinetic matrix
Derivation of K matrix for U-3R-RL mechanism
- Final expression for U-3R-RL kinetic matrix
Derivation of K matrix for U-4R-RL mechanism
- Final expression for U-4R-RL kinetic matrix
Derivation of K matrix for U-5R-RL mechanism
- Final expression for U-5R-RL kinetic matrix
clean up workspace
reset()
Definition of transitions and strategy
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 B1
R <=> R*
Constants: k_1_B_1_s_1 (forward), k_2_B_1_s_1 (reverse).
R <=> R**
Constants: k_1_B_1_s_2 (forward), k_2_B_1_s_2 (reverse).
R <=> R***
Constants: k_1_B_1_s_3 (forward), k_2_B_1_s_3 (reverse).
R <=> R****
Constants: k_1_B_1_s_4 (forward), k_2_B_1_s_4 (reverse).
R <=> R*****
Constants: k_1_B_1_s_5 (forward), k_2_B_1_s_5 (reverse).
Interconversion of R-starred isomers: transitions C
-- R* --
R* <=> R**
Constants: k_1_C_s_1_2 (forward), k_2_C_s_1_2 (reverse).
R* <=> R***
Constants: k_1_C_s_1_3 (forward), k_2_C_s_1_3 (reverse).
R* <=> R****
Constants: k_1_C_s_1_4 (forward), k_2_C_s_1_4 (reverse).
R* <=> R*****
Constants: k_1_C_s_1_5 (forward), k_2_C_s_1_5 (reverse).
-- R** --
R** <=> R***
Constants: k_1_C_s_2_3 (forward), k_2_C_s_2_3 (reverse).
R** <=> R****
Constants: k_1_C_s_2_4 (forward), k_2_C_s_2_4 (reverse).
R** <=> R*****
Constants: k_1_C_s_2_5 (forward), k_2_C_s_2_5 (reverse).
-- R*** --
R*** <=> R****
Constants: k_1_C_s_3_4 (forward), k_2_C_s_3_4 (reverse).
R*** <=> R*****
Constants: k_1_C_s_3_5 (forward), k_2_C_s_3_5 (reverse).
-- R**** --
R****<=> R*****
Constants: k_1_C_s_4_5 (forward), k_2_C_s_4_5 (reverse).
Isomerization of RL to RL-starred species: transition B2
RL <=> R*L
Constants: k_1_B_2 (forward), k_2_B_2 (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 the 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 conversion 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.
- Summary of equations for R**
- Summary of equations for R***
- Summary of equations for R****
- Summary of equations for R*****
- Summary of equations for RL*
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_1_s_1 (forward), k_2_B_1_s_1 (reverse).
Equations subgroup: Bs1
a forward reaction rate for the transition
eq1_Bs1_1a:= Rate_1_B_1_s_1 = k_1_B_1_s_1*R
a partial conversion rate of R in this transition
molecularity:=-1:
eq1_Bs1_1b:= dcRdt_1_B_1_s_1 = molecularity*Rate_1_B_1_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_1_s_1 = k_2_B_1_s_1*R_s_1
a partial conversion rate of R in this transition
molecularity:=1:
eq1_Bs1_2b:= dcRdt_2_B_1_s_1 = molecularity*Rate_2_B_1_s_1
The final form
eq1_Bs1_2c:= eq1_Bs1_2b | eq1_Bs1_2a
R <=> R**
Constants: k_1_B_1_s2 (forward), k_2_B_1_s2 (reverse).
Equations subgroup: Bs2
a forward reaction rate for the transition
eq1_Bs2_1a:= Rate_1_B_1_s_2 = k_1_B_1_s_2*R
a partial conversion rate of R in this transition
molecularity:=-1:
eq1_Bs2_1b:= dcRdt_1_B_1_s_2 = molecularity*Rate_1_B_1_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_1_s_2 = k_2_B_1_s_2*R_s_2
a partial conversion rate of R in this transition
molecularity:=1:
eq1_Bs2_2b:= dcRdt_2_B_1_s_2 = molecularity*Rate_2_B_1_s_2
The final form
eq1_Bs2_2c:= eq1_Bs2_2b | eq1_Bs2_2a
R <=> R***
Constants: k_1_B_1_s3 (forward), k_2_B_1_s3 (reverse).
Equations subgroup: Bs3
a forward reaction rate for the transition
eq1_Bs3_1a:= Rate_1_B_1_s_3 = k_1_B_1_s_3*R
a partial conversion rate of R in this transition
molecularity:=-1:
eq1_Bs3_1b:= dcRdt_1_B_1_s_3 = molecularity*Rate_1_B_1_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_1_s_3 = k_2_B_1_s_3*R_s_3
a partial conversion rate of R in this transition
molecularity:=1:
eq1_Bs3_2b:= dcRdt_2_B_1_s_3 = molecularity*Rate_2_B_1_s_3
The final form
eq1_Bs3_2c:= eq1_Bs3_2b | eq1_Bs3_2a
R <=> R****
Constants: k_1_B_1_s4 (forward), k_2_B_1_s4 (reverse).
Equations subgroup: Bs4
a forward reaction rate for the transition
eq1_Bs4_1a:= Rate_1_B_1_s_4 = k_1_B_1_s_4*R
a partial conversion rate of R in this transition
molecularity:=-1:
eq1_Bs4_1b:= dcRdt_1_B_1_s_4 = molecularity*Rate_1_B_1_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_1_s_4 = k_2_B_1_s_4*R_s_4
a partial conversion rate of R in this transition
molecularity:=1:
eq1_Bs4_2b:= dcRdt_2_B_1_s_4 = molecularity*Rate_2_B_1_s_4
The final form
eq1_Bs4_2c:= eq1_Bs4_2b | eq1_Bs4_2a
R <=> R*****
Constants: k_1_B_1_s5 (forward), k_2_B_1_s5 (reverse).
Equations subgroup: Bs5
a forward reaction rate for the transition
eq1_Bs5_1a:= Rate_1_B_1_s_5 = k_1_B_1_s_5*R
a partial conversion rate of R in this transition
molecularity:=-1:
eq1_Bs5_1b:= dcRdt_1_B_1_s_5 = molecularity*Rate_1_B_1_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_1_s_5 = k_2_B_1_s_5*R_s_5
a partial conversion rate of R in this transition
molecularity:=1:
eq1_Bs5_2b:= dcRdt_2_B_1_s_5 = molecularity*Rate_2_B_1_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_1_s_1 + dcRdt_2_B_1_s_1
Substitute
eq1_R_N__U_R_RL:= % | 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_1_s_1 + dcRdt_2_B_1_s_1 + dcRdt_1_B_1_s_2 + dcRdt_2_B_1_s_2
Substitute
eq1_R_N__U_2R_RL:= % | 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_1_s_1 + dcRdt_2_B_1_s_1 + dcRdt_1_B_1_s_2 + dcRdt_2_B_1_s_2 \
+ dcRdt_1_B_1_s_3 + dcRdt_2_B_1_s_3
Substitute
eq1_R_N__U_3R_RL:= % | 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_1_s_1 + dcRdt_2_B_1_s_1 + dcRdt_1_B_1_s_2 + dcRdt_2_B_1_s_2 \
+ dcRdt_1_B_1_s_3 + dcRdt_2_B_1_s_3 + dcRdt_1_B_1_s_4 + dcRdt_2_B_1_s_4
Substitute
eq1_R_N__U_4R_RL:= % | 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_1_s_1 + dcRdt_2_B_1_s_1 + dcRdt_1_B_1_s_2 + dcRdt_2_B_1_s_2 \
+ dcRdt_1_B_1_s_3 + dcRdt_2_B_1_s_3 + dcRdt_1_B_1_s_4 + dcRdt_2_B_1_s_4 + dcRdt_1_B_1_s_5 + dcRdt_2_B_1_s_5
Substitute
eq1_R_N__U_5R_RL:= % | 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;
eq1_R_N__U_R_RL
eq1_R_N__U_2R_RL
eq1_R_N__U_3R_RL
eq1_R_N__U_4R_RL
eq1_R_N__U_5R_RL
Back to Equations for each species
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
RL <=> RL*
Constants: k_1_B_2 (forward), k_2_B_2 (reverse).
Equations subgroup: B2
a forward reaction rate for the transition
eq2_B2_1a:= Rate_1_B_2 = k_1_B_2*RL
a partial conversion rate of R in this transition
molecularity:=-1:
eq2_B2_1b:= dcRLdt_1_B_2 = molecularity*Rate_1_B_2
The final form
eq2_B2_1c:= eq2_B2_1b | eq2_B2_1a
a reverse reaction rate for the transition
eq2_B2_2a:= Rate_2_B_2 = k_2_B_2*RL_s
a partial conversion rate of R in this transition
molecularity:=1:
eq2_B2_2b:= dcRLdt_2_B_2 = molecularity*Rate_2_B_2
The final form
eq2_B2_2c:= eq2_B2_2b | eq2_B2_2a
Net conversion rate for the species
Same in all alternative mechanisms
dcRLdt_N = dcRLdt_1_A + dcRLdt_2_A + dcRLdt_1_B_2 + dcRLdt_2_B_2
Substitute
eq2_RL_N:= % | eq2_A_1c | eq2_A_2c | eq2_B2_1c | eq2_B2_2c
All versions of mechanism will have it the same
eq2_RL_N
Back to Equations for each species
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: B1
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_1_s_1 = molecularity*Rate_1_B_1_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_1_s_1 = molecularity*Rate_2_B_1_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_1_s_1 + dcRs1dt_2_B_1_s_1
Substitute
eq3_Rs1_N__U_R_RL:= % | 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_1_s_1 + dcRs1dt_2_B_1_s_1 + dcRs1dt_1_C_s_1_2 + dcRs1dt_2_C_s_1_2
Substitute
eq3_Rs1_N__U_2R_RL:= % | 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_1_s_1 + dcRs1dt_2_B_1_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_RL:= % | 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_1_s_1 + dcRs1dt_2_B_1_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_RL:= % | 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_1_s_1 + dcRs1dt_2_B_1_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_RL:= % | 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;
eq3_Rs1_N__U_R_RL;
eq3_Rs1_N__U_2R_RL;
eq3_Rs1_N__U_3R_RL;
eq3_Rs1_N__U_4R_RL;
eq3_Rs1_N__U_5R_RL;
Back to Equations for each species
Equations group: 4
Equations subgroup: B1
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_1_s_2 = molecularity*Rate_1_B_1_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_1_s_2 = molecularity*Rate_2_B_1_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_RL:= 0
U-2R-RL
dcRs2dt_N = dcRs2dt_1_B_1_s_2 + dcRs2dt_2_B_1_s_2 + dcRs2dt_1_C_s_1_2 + dcRs2dt_2_C_s_1_2
Substitute
eq4_Rs2_N__U_2R_RL:= % | 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_1_s_2 + dcRs2dt_2_B_1_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_RL:= % | 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_1_s_2 + dcRs2dt_2_B_1_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_RL:= % | 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_1_s_2 + dcRs2dt_2_B_1_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_RL:= % | 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;
eq4_Rs2_N__U_R_RL;
eq4_Rs2_N__U_2R_RL;
eq4_Rs2_N__U_3R_RL;
eq4_Rs2_N__U_4R_RL;
eq4_Rs2_N__U_5R_RL;
Back to Equations for each species
Equations group: 5
Equations subgroup: B1
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_1_s_3 = molecularity*Rate_1_B_1_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_1_s_3 = molecularity*Rate_2_B_1_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_RL:= 0
U-2R-RL (species not present)
eq5_Rs3_N__U_2R_RL:= 0
U-3R-RL
dcRs3dt_N = dcRs3dt_1_B_1_s_3 + dcRs3dt_2_B_1_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_RL:= % | 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_1_s_3 + dcRs3dt_2_B_1_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_RL:= % | 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_1_s_3 + dcRs3dt_2_B_1_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_RL:= % | 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;
eq5_Rs3_N__U_R_RL;
eq5_Rs3_N__U_2R_RL;
eq5_Rs3_N__U_3R_RL;
eq5_Rs3_N__U_4R_RL;
eq5_Rs3_N__U_5R_RL;
Back to Equations for each species
Equations group: 6
Equations subgroup: B1
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_1_s_4 = molecularity*Rate_1_B_1_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_1_s_4 = molecularity*Rate_2_B_1_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_RL:= 0
U-2R-RL (species not present)
eq6_Rs4_N__U_2R_RL:= 0
U-3R-RL (species not present)
eq6_Rs4_N__U_3R_RL:= 0
U-4R-RL
dcRs4dt_N = dcRs4dt_1_B_1_s_4 + dcRs4dt_2_B_1_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_RL:= % | 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_1_s_4 + dcRs4dt_2_B_1_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_RL:= % | 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_RL;
eq6_Rs4_N__U_2R_RL;
eq6_Rs4_N__U_3R_RL;
eq6_Rs4_N__U_4R_RL;
eq6_Rs4_N__U_5R_RL;
Back to Equations for each species
Equations group: 7
Equations subgroup: B1
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_1_s_5 = molecularity*Rate_1_B_1_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_1_s_5 = molecularity*Rate_2_B_1_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_RL:= 0
U-2R-RL (species not present)
eq7_Rs5_N__U_2R_RL:= 0
U-3R-RL (species not present)
eq7_Rs5_N__U_3R_RL:= 0
U-4R-RL
eq7_Rs5_N__U_4R_RL:= 0
U-5R-RL
dcRs5dt_N = dcRs5dt_1_B_1_s_5 + dcRs5dt_2_B_1_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_RL:= % | 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_RL;
eq7_Rs5_N__U_2R_RL;
eq7_Rs5_N__U_3R_RL;
eq7_Rs5_N__U_4R_RL;
eq7_Rs5_N__U_5R_RL;
Back to Equations for each species
Equations group: 8
Equations subgroup: B2
RL <=> RL*
(towards species)
a forward reaction rate for the transition
eq8_B2_1a:= eq2_B2_1a;
a partial conversion rate of R**** in this transition
molecularity:=1:
eq8_B2_1b:= dcRLsdt_1_B_2 = molecularity*Rate_1_B_2
The final form
eq8_B2_1c:= eq8_B2_1b | eq8_B2_1a
a reverse reaction rate for the transition
eq8_B2_2a:= eq2_B2_2a;
a partial conversion rate of R**** in this transition
molecularity:=-1:
eq8_B2_2b:= dcRLsdt_2_B_2 = molecularity*Rate_2_B_2
The final form
eq8_B2_2c:= eq8_B2_2b | eq8_B2_2a
Net conversion rate for the species (SAME FOR ALL MODELS)
I will create equations for all five versions of the mechanism.
Net rate
dcRLsdt_N = dcRLsdt_1_B_2 + dcRLsdt_2_B_2
Substitute
eq8_RLs_N:= % | eq8_B2_1c | eq8_B2_2c
All versions of mechanism will have it the same
eq8_RLs_N
Back to Equations for each species
not needed here because we do not have oligomerization reactions.
Expession of K matrix for U-1R-RL mechanism
This is derivation for comparison with existing U-R-RL mechanism matrix: order of species as R, R*, RL, RL*
Derivation with more convenient new order of species is below
Summary list of the net rate equations for the mechanism
eq1_R_N__U_R_RL;
eq2_RL_N;
eq3_Rs1_N__U_R_RL;
eq8_RLs_N
Assign sequential names to species
feq_1a:= R = C1;
feq_1b:= R_s_1= C2;
feq_1c:= RL = C3;
feq_1d:= RL_s = C4;
Assign the same order to net rate equations
feq_2a:= dcRdt_N = dC1dt;
feq_2b:= dcRs1dt_N = dC2dt;
feq_2c:= dcRLdt_N = dC3dt;
feq_2d:= dcRLsdt_N = dC4dt;
Restate the equations in terms of new sequential species names (rename free ligand concentration too)
R
eq1_R_N__U_R_RL;
feq_3a:= % | feq_1a | feq_1b | feq_1c | feq_2a | feq_2b | feq_2c | feq_1d | feq_2d | L = Leq
R*
eq3_Rs1_N__U_R_RL;
feq_3b:= % | feq_1a | feq_1b | feq_1c | feq_2a | feq_2b | feq_2c | feq_1d | feq_2d | L = Leq
RL
eq2_RL_N;
feq_3c:= % | feq_1a | feq_1b | feq_1c | feq_2a | feq_2b | feq_2c | feq_1d | feq_2d | L = Leq
RL*
eq8_RLs_N;
feq_3d:= % | feq_1a | feq_1b | feq_1c | feq_2a | feq_2b | 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)
See Workflow for accurate extraction of the K matrix
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_1_s_1-Leq*k_1_A, k_2_B_1_s_1, k_2_A , 0 ],
[ k_1_B_1_s_1 , -k_2_B_1_s_1, 0, 0 ],
[ Leq*k_1_A, 0, -k_1_B_2-k_2_A, k_2_B_2 ],
[ 0, 0, k_1_B_2, -k_2_B_2 ]
])
![matrix([[- k_1_B_1_s_1 - Leq*k_1_A, k_2_B_1_s_1, k_2_A, 0], [k_1_B_1_s_1, -k_2_B_1_s_1, 0, 0], [Leq*k_1_A, 0, - k_2_A - k_1_B_2, k_2_B_2], [0, 0, k_1_B_2, -k_2_B_2]])](U_5R_RL_images/math333.png)
Test the K matrix entry
Create a column vector of species concentrations
P:=matrix(4,1,[C1, C2, C3, C4])
![matrix([[C1], [C2], [C3], [C4]])](U_5R_RL_images/math334.png)
Multiply K and P:
dCdt_manual_input:= K*P
![matrix([[C3*k_2_A + C2*k_2_B_1_s_1 - C1*(k_1_B_1_s_1 + Leq*k_1_A)], [C1*k_1_B_1_s_1 - C2*k_2_B_1_s_1], [C4*k_2_B_2 - C3*(k_2_A + k_1_B_2) + C1*Leq*k_1_A], [C3*k_1_B_2 - C4*k_2_B_2]])](U_5R_RL_images/math335.png)
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)])
![matrix([[C3*k_2_A - C1*k_1_B_1_s_1 + C2*k_2_B_1_s_1 - C1*Leq*k_1_A], [C1*k_1_B_1_s_1 - C2*k_2_B_1_s_1], [C4*k_2_B_2 - C3*k_1_B_2 - C3*k_2_A + C1*Leq*k_1_A], [C3*k_1_B_2 - C4*k_2_B_2]])](U_5R_RL_images/math336.png)
Compare derivation result to manual input
dCdt_mupad=dCdt_manual_input:
normal(%);
bool(%)
![matrix([[C3*k_2_A - C1*k_1_B_1_s_1 + C2*k_2_B_1_s_1 - C1*Leq*k_1_A], [C1*k_1_B_1_s_1 - C2*k_2_B_1_s_1], [C4*k_2_B_2 - C3*k_1_B_2 - C3*k_2_A + C1*Leq*k_1_A], [C3*k_1_B_2 - C4*k_2_B_2]]) = matrix([[C3*k_2_A - C1*k_1_B_1_s_1 + C2*k_2_B_1_s_1 - C1*Leq*k_1_A], [C1*k_1_B_1_s_1 - C2*k_2_B_1_s_1], [C4*k_2_B_2 - C3*k_1_B_2 - C3*k_2_A + C1*Leq*k_1_A], [C3*k_1_B_2 - C4*k_2_B_2]])](U_5R_RL_images/math337.png)
=> Typed K-matrix is correct and corresponds to U-R-RL matrix derived earlier for IDAP if A2 transition is removed.
The following derivation is uses new order of species: R, RL, R*, RL* (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;
feq_1d:= RL_s = 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:= dcRLsdt_N = dC4dt;
Restate the equations in terms of new sequential species names (rename free ligand concentration too)
R
eq1_R_N__U_R_RL:
feq_3a:= % | feq_1a | feq_1b | feq_1c | feq_2a | feq_2b | feq_2c | feq_1d | feq_2d | L = Leq
RL
eq2_RL_N:
feq_3b:= % | feq_1a | feq_1b | feq_1c | feq_2a | feq_2b | feq_2c | feq_1d | feq_2d | L = Leq
R*
eq3_Rs1_N__U_R_RL:
feq_3c:= % | feq_1a | feq_1b | feq_1c | feq_2a | feq_2b | feq_2c | feq_1d | feq_2d | L = Leq
RL*
eq8_RLs_N:
feq_3d:= % | feq_1a | feq_1b | feq_1c | feq_2a | feq_2b | 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)
See Workflow for accurate extraction of the K matrix
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_1_s_1-Leq*k_1_A, k_2_A, k_2_B_1_s_1, 0 ],
[ Leq*k_1_A, -k_1_B_2-k_2_A , 0 , k_2_B_2 ],
[ k_1_B_1_s_1 , 0, -k_2_B_1_s_1, 0 ],
[ 0, k_1_B_2 , 0, -k_2_B_2 ]
])
![matrix([[- k_1_B_1_s_1 - Leq*k_1_A, k_2_A, k_2_B_1_s_1, 0], [Leq*k_1_A, - k_2_A - k_1_B_2, 0, k_2_B_2], [k_1_B_1_s_1, 0, -k_2_B_1_s_1, 0], [0, k_1_B_2, 0, -k_2_B_2]])](U_5R_RL_images/math351.png)
Test the K matrix entry
Create a column vector of species concentrations
P:=matrix(4,1,[C1, C2, C3, C4])
![matrix([[C1], [C2], [C3], [C4]])](U_5R_RL_images/math352.png)
Multiply K and P:
dCdt_manual_input:= K*P
![matrix([[C2*k_2_A + C3*k_2_B_1_s_1 - C1*(k_1_B_1_s_1 + Leq*k_1_A)], [C4*k_2_B_2 - C2*(k_2_A + k_1_B_2) + C1*Leq*k_1_A], [C1*k_1_B_1_s_1 - C3*k_2_B_1_s_1], [C2*k_1_B_2 - C4*k_2_B_2]])](U_5R_RL_images/math353.png)
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)])
![matrix([[C2*k_2_A - C1*k_1_B_1_s_1 + C3*k_2_B_1_s_1 - C1*Leq*k_1_A], [C4*k_2_B_2 - C2*k_1_B_2 - C2*k_2_A + C1*Leq*k_1_A], [C1*k_1_B_1_s_1 - C3*k_2_B_1_s_1], [C2*k_1_B_2 - C4*k_2_B_2]])](U_5R_RL_images/math354.png)
Compare derivation result to manual input
dCdt_mupad=dCdt_manual_input:
normal(%);
bool(%)
![matrix([[C2*k_2_A - C1*k_1_B_1_s_1 + C3*k_2_B_1_s_1 - C1*Leq*k_1_A], [C4*k_2_B_2 - C2*k_1_B_2 - C2*k_2_A + C1*Leq*k_1_A], [C1*k_1_B_1_s_1 - C3*k_2_B_1_s_1], [C2*k_1_B_2 - C4*k_2_B_2]]) = matrix([[C2*k_2_A - C1*k_1_B_1_s_1 + C3*k_2_B_1_s_1 - C1*Leq*k_1_A], [C4*k_2_B_2 - C2*k_1_B_2 - C2*k_2_A + C1*Leq*k_1_A], [C1*k_1_B_1_s_1 - C3*k_2_B_1_s_1], [C2*k_1_B_2 - C4*k_2_B_2]])](U_5R_RL_images/math355.png)
=> Typed K-matrix is correct.
K matrix for the U-R model with the new species order
K;
![matrix([[- k_1_B_1_s_1 - Leq*k_1_A, k_2_A, k_2_B_1_s_1, 0], [Leq*k_1_A, - k_2_A - k_1_B_2, 0, k_2_B_2], [k_1_B_1_s_1, 0, -k_2_B_1_s_1, 0], [0, k_1_B_2, 0, -k_2_B_2]])](U_5R_RL_images/math357.png)
Expression for K matrix of U-2R-RL mechanism
Summary list of the net rate equations for the mechanism
eq1_R_N__U_2R_RL;
eq2_RL_N;
eq3_Rs1_N__U_2R_RL;
eq4_Rs2_N__U_2R_RL;
eq8_RLs_N
Assign sequential names to species R, RL, R*, R**, R***, R****, R*****, RL*
feq_1a:= R = C1;
feq_1b:= RL = C2;
feq_1c:= R_s_1= C3;
feq_1d:= R_s_2= C4;
feq_1e:= RL_s= 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:= dcRLsdt_N = dC5dt;
Restate the equations in terms of new sequential species names (rename free ligand concentration too)
R
eq1_R_N__U_2R_RL;
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_2R_RL;
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_2R_RL;
feq_3d:= % | feq_1a | feq_2a | feq_1b | feq_2b | feq_1c | feq_2c | feq_1d | feq_2d | feq_1e | feq_2e | L = Leq
RL*
eq8_RLs_N:
feq_3e:= % | feq_1a | feq_1b | feq_1c | feq_2a | feq_2b | 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)
See Workflow for accurate extraction of the K matrix
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_1_s_1-k_1_B_1_s_2-Leq*k_1_A, k_2_A , k_2_B_1_s_1, k_2_B_1_s_2, 0 ],
[ Leq*k_1_A, -k_1_B_2-k_2_A, 0, 0, k_2_B_2 ],
[ k_1_B_1_s_1 , 0, -k_1_C_s_1_2-k_2_B_1_s_1, k_2_C_s_1_2, 0 ],
[ k_1_B_1_s_2 , 0, k_1_C_s_1_2, -k_2_B_1_s_2-k_2_C_s_1_2, 0 ],
[ 0, k_1_B_2, 0, 0, -k_2_B_2 ]
])
![matrix([[- k_1_B_1_s_1 - k_1_B_1_s_2 - Leq*k_1_A, k_2_A, k_2_B_1_s_1, k_2_B_1_s_2, 0], [Leq*k_1_A, - k_2_A - k_1_B_2, 0, 0, k_2_B_2], [k_1_B_1_s_1, 0, - k_1_C_s_1_2 - k_2_B_1_s_1, k_2_C_s_1_2, 0], [k_1_B_1_s_2, 0, k_1_C_s_1_2, - k_2_B_1_s_2 - k_2_C_s_1_2, 0], [0, k_1_B_2, 0, 0, -k_2_B_2]])](U_5R_RL_images/math382.png)
Test the K matrix entry
Create a column vector of species concentrations
P:=matrix(5,1,[C1, C2, C3, C4, C5])
![matrix([[C1], [C2], [C3], [C4], [C5]])](U_5R_RL_images/math383.png)
Multiply K and P:
dCdt_manual_input:= K*P
![matrix([[C2*k_2_A + C3*k_2_B_1_s_1 + C4*k_2_B_1_s_2 - C1*(k_1_B_1_s_1 + k_1_B_1_s_2 + Leq*k_1_A)], [C5*k_2_B_2 - C2*(k_2_A + k_1_B_2) + C1*Leq*k_1_A], [C1*k_1_B_1_s_1 + C4*k_2_C_s_1_2 - C3*(k_1_C_s_1_2 + k_2_B_1_s_1)], [C1*k_1_B_1_s_2 + C3*k_1_C_s_1_2 - C4*(k_2_B_1_s_2 + k_2_C_s_1_2)], [C2*k_1_B_2 - C5*k_2_B_2]])](U_5R_RL_images/math384.png)
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)])
![matrix([[C2*k_2_A - C1*k_1_B_1_s_1 - C1*k_1_B_1_s_2 + C3*k_2_B_1_s_1 + C4*k_2_B_1_s_2 - C1*Leq*k_1_A], [C5*k_2_B_2 - C2*k_1_B_2 - C2*k_2_A + C1*Leq*k_1_A], [C1*k_1_B_1_s_1 - C3*k_1_C_s_1_2 - C3*k_2_B_1_s_1 + C4*k_2_C_s_1_2], [C1*k_1_B_1_s_2 + C3*k_1_C_s_1_2 - C4*k_2_B_1_s_2 - C4*k_2_C_s_1_2], [C2*k_1_B_2 - C5*k_2_B_2]])](U_5R_RL_images/math385.png)
Compare derivation result to manual input
dCdt_mupad=dCdt_manual_input:
normal(%);
bool(%)
![matrix([[C2*k_2_A - C1*k_1_B_1_s_1 - C1*k_1_B_1_s_2 + C3*k_2_B_1_s_1 + C4*k_2_B_1_s_2 - C1*Leq*k_1_A], [C5*k_2_B_2 - C2*k_1_B_2 - C2*k_2_A + C1*Leq*k_1_A], [C1*k_1_B_1_s_1 - C3*k_1_C_s_1_2 - C3*k_2_B_1_s_1 + C4*k_2_C_s_1_2], [C1*k_1_B_1_s_2 + C3*k_1_C_s_1_2 - C4*k_2_B_1_s_2 - C4*k_2_C_s_1_2], [C2*k_1_B_2 - C5*k_2_B_2]]) = matrix([[C2*k_2_A - C1*k_1_B_1_s_1 - C1*k_1_B_1_s_2 + C3*k_2_B_1_s_1 + C4*k_2_B_1_s_2 - C1*Leq*k_1_A], [C5*k_2_B_2 - C2*k_1_B_2 - C2*k_2_A + C1*Leq*k_1_A], [C1*k_1_B_1_s_1 - C3*k_1_C_s_1_2 - C3*k_2_B_1_s_1 + C4*k_2_C_s_1_2], [C1*k_1_B_1_s_2 + C3*k_1_C_s_1_2 - C4*k_2_B_1_s_2 - C4*k_2_C_s_1_2], [C2*k_1_B_2 - C5*k_2_B_2]])](U_5R_RL_images/math386.png)
=> Typed K-matrix is correct.
Final expression for U-2R-RL kinetic matrix
K
![matrix([[- k_1_B_1_s_1 - k_1_B_1_s_2 - Leq*k_1_A, k_2_A, k_2_B_1_s_1, k_2_B_1_s_2, 0], [Leq*k_1_A, - k_2_A - k_1_B_2, 0, 0, k_2_B_2], [k_1_B_1_s_1, 0, - k_1_C_s_1_2 - k_2_B_1_s_1, k_2_C_s_1_2, 0], [k_1_B_1_s_2, 0, k_1_C_s_1_2, - k_2_B_1_s_2 - k_2_C_s_1_2, 0], [0, k_1_B_2, 0, 0, -k_2_B_2]])](U_5R_RL_images/math388.png)
Derivation of K matrix for U-3R-RL mechanism
Summary list of the net rate equations for the mechanism
eq1_R_N__U_3R_RL;
eq2_RL_N;
eq3_Rs1_N__U_3R_RL;
eq4_Rs2_N__U_3R_RL;
eq5_Rs3_N__U_3R_RL;
eq8_RLs_N
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:= RL_s = 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:= dcRLsdt_N = dC6dt;
Restate the equations in terms of new sequential species names (rename free ligand concentration too)
R
eq1_R_N__U_3R_RL;
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_3R_RL;
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_3R_RL;
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_3R_RL;
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
RL*
eq8_RLs_N:
feq_3f:= % | feq_1a | feq_1b | feq_1c | feq_2a | feq_2b | 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)
See Workflow for accurate extraction of the K matrix
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_1_s_1-k_1_B_1_s_2-k_1_B_1_s_3-Leq*k_1_A, k_2_A , k_2_B_1_s_1, k_2_B_1_s_2, k_2_B_1_s_3, 0 ],
[ Leq*k_1_A, -k_1_B_2-k_2_A, 0, 0, 0, k_2_B_2 ],
[ k_1_B_1_s_1, 0, -k_1_C_s_1_2-k_1_C_s_1_3-k_2_B_1_s_1, k_2_C_s_1_2, k_2_C_s_1_3,0 ],
[ k_1_B_1_s_2, 0, k_1_C_s_1_2, -k_1_C_s_2_3-k_2_B_1_s_2-k_2_C_s_1_2, k_2_C_s_2_3, 0 ],
[ k_1_B_1_s_3, 0, k_1_C_s_1_3, k_1_C_s_2_3, -k_2_B_1_s_3-k_2_C_s_1_3-k_2_C_s_2_3, 0 ],
[ 0, k_1_B_2, 0, 0, 0, -k_2_B_2 ]
])
![matrix([[- k_1_B_1_s_1 - k_1_B_1_s_2 - k_1_B_1_s_3 - Leq*k_1_A, k_2_A, k_2_B_1_s_1, k_2_B_1_s_2, k_2_B_1_s_3, 0], [Leq*k_1_A, - k_2_A - k_1_B_2, 0, 0, 0, k_2_B_2], [k_1_B_1_s_1, 0, - k_1_C_s_1_2 - k_1_C_s_1_3 - k_2_B_1_s_1, k_2_C_s_1_2, k_2_C_s_1_3, 0], [k_1_B_1_s_2, 0, k_1_C_s_1_2, - k_1_C_s_2_3 - k_2_B_1_s_2 - k_2_C_s_1_2, k_2_C_s_2_3, 0], [k_1_B_1_s_3, 0, k_1_C_s_1_3, k_1_C_s_2_3, - k_2_B_1_s_3 - k_2_C_s_1_3 - k_2_C_s_2_3, 0], [0, k_1_B_2, 0, 0, 0, -k_2_B_2]])](U_5R_RL_images/math418.png)
Test the K matrix entry
Create a column vector of species concentrations
P:=matrix(6,1,[C1, C2, C3, C4, C5, C6])
![matrix([[C1], [C2], [C3], [C4], [C5], [C6]])](U_5R_RL_images/math419.png)
Multiply K and P:
dCdt_manual_input:= K*P
![matrix([[C2*k_2_A + C3*k_2_B_1_s_1 + C4*k_2_B_1_s_2 + C5*k_2_B_1_s_3 - C1*(k_1_B_1_s_1 + k_1_B_1_s_2 + k_1_B_1_s_3 + Leq*k_1_A)], [C6*k_2_B_2 - C2*(k_2_A + k_1_B_2) + C1*Leq*k_1_A], [C1*k_1_B_1_s_1 + C4*k_2_C_s_1_2 + C5*k_2_C_s_1_3 - C3*(k_1_C_s_1_2 + k_1_C_s_1_3 + k_2_B_1_s_1)], [C1*k_1_B_1_s_2 + C3*k_1_C_s_1_2 + C5*k_2_C_s_2_3 - C4*(k_1_C_s_2_3 + k_2_B_1_s_2 + k_2_C_s_1_2)], [C1*k_1_B_1_s_3 + C3*k_1_C_s_1_3 + C4*k_1_C_s_2_3 - C5*(k_2_B_1_s_3 + k_2_C_s_1_3 + k_2_C_s_2_3)], [C2*k_1_B_2 - C6*k_2_B_2]])](U_5R_RL_images/math420.png)
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)])
![matrix([[C2*k_2_A - C1*k_1_B_1_s_1 - C1*k_1_B_1_s_2 - C1*k_1_B_1_s_3 + C3*k_2_B_1_s_1 + C4*k_2_B_1_s_2 + C5*k_2_B_1_s_3 - C1*Leq*k_1_A], [C6*k_2_B_2 - C2*k_1_B_2 - C2*k_2_A + C1*Leq*k_1_A], [C1*k_1_B_1_s_1 - C3*k_1_C_s_1_2 - C3*k_1_C_s_1_3 - C3*k_2_B_1_s_1 + C4*k_2_C_s_1_2 + C5*k_2_C_s_1_3], [C1*k_1_B_1_s_2 + C3*k_1_C_s_1_2 - C4*k_1_C_s_2_3 - C4*k_2_B_1_s_2 - C4*k_2_C_s_1_2 + C5*k_2_C_s_2_3], [C1*k_1_B_1_s_3 + C3*k_1_C_s_1_3 + C4*k_1_C_s_2_3 - C5*k_2_B_1_s_3 - C5*k_2_C_s_1_3 - C5*k_2_C_s_2_3], [C2*k_1_B_2 - C6*k_2_B_2]])](U_5R_RL_images/math421.png)
Compare derivation result to manual input
dCdt_mupad=dCdt_manual_input:
normal(%);
bool(%)
![matrix([[C2*k_2_A - C1*k_1_B_1_s_1 - C1*k_1_B_1_s_2 - C1*k_1_B_1_s_3 + C3*k_2_B_1_s_1 + C4*k_2_B_1_s_2 + C5*k_2_B_1_s_3 - C1*Leq*k_1_A], [C6*k_2_B_2 - C2*k_1_B_2 - C2*k_2_A + C1*Leq*k_1_A], [C1*k_1_B_1_s_1 - C3*k_1_C_s_1_2 - C3*k_1_C_s_1_3 - C3*k_2_B_1_s_1 + C4*k_2_C_s_1_2 + C5*k_2_C_s_1_3], [C1*k_1_B_1_s_2 + C3*k_1_C_s_1_2 - C4*k_1_C_s_2_3 - C4*k_2_B_1_s_2 - C4*k_2_C_s_1_2 + C5*k_2_C_s_2_3], [C1*k_1_B_1_s_3 + C3*k_1_C_s_1_3 + C4*k_1_C_s_2_3 - C5*k_2_B_1_s_3 - C5*k_2_C_s_1_3 - C5*k_2_C_s_2_3], [C2*k_1_B_2 - C6*k_2_B_2]]) = matrix([[C2*k_2_A - C1*k_1_B_1_s_1 - C1*k_1_B_1_s_2 - C1*k_1_B_1_s_3 + C3*k_2_B_1_s_1 + C4*k_2_B_1_s_2 + C5*k_2_B_1_s_3 - C1*Leq*k_1_A], [C6*k_2_B_2 - C2*k_1_B_2 - C2*k_2_A + C1*Leq*k_1_A], [C1*k_1_B_1_s_1 - C3*k_1_C_s_1_2 - C3*k_1_C_s_1_3 - C3*k_2_B_1_s_1 + C4*k_2_C_s_1_2 + C5*k_2_C_s_1_3], [C1*k_1_B_1_s_2 + C3*k_1_C_s_1_2 - C4*k_1_C_s_2_3 - C4*k_2_B_1_s_2 - C4*k_2_C_s_1_2 + C5*k_2_C_s_2_3], [C1*k_1_B_1_s_3 + C3*k_1_C_s_1_3 + C4*k_1_C_s_2_3 - C5*k_2_B_1_s_3 - C5*k_2_C_s_1_3 - C5*k_2_C_s_2_3], [C2*k_1_B_2 - C6*k_2_B_2]])](U_5R_RL_images/math422.png)
=> Typed K-matrix is correct.
Final expression for U-3R-RL kinetic matrix
K
![matrix([[- k_1_B_1_s_1 - k_1_B_1_s_2 - k_1_B_1_s_3 - Leq*k_1_A, k_2_A, k_2_B_1_s_1, k_2_B_1_s_2, k_2_B_1_s_3, 0], [Leq*k_1_A, - k_2_A - k_1_B_2, 0, 0, 0, k_2_B_2], [k_1_B_1_s_1, 0, - k_1_C_s_1_2 - k_1_C_s_1_3 - k_2_B_1_s_1, k_2_C_s_1_2, k_2_C_s_1_3, 0], [k_1_B_1_s_2, 0, k_1_C_s_1_2, - k_1_C_s_2_3 - k_2_B_1_s_2 - k_2_C_s_1_2, k_2_C_s_2_3, 0], [k_1_B_1_s_3, 0, k_1_C_s_1_3, k_1_C_s_2_3, - k_2_B_1_s_3 - k_2_C_s_1_3 - k_2_C_s_2_3, 0], [0, k_1_B_2, 0, 0, 0, -k_2_B_2]])](U_5R_RL_images/math424.png)
Derivation of K matrix for U-4R-RL mechanism
Summary list of the net rate equations for the mechanism
eq1_R_N__U_4R_RL;
eq2_RL_N;
eq3_Rs1_N__U_4R_RL;
eq4_Rs2_N__U_4R_RL;
eq5_Rs3_N__U_4R_RL;
eq6_Rs4_N__U_4R_RL;
eq8_RLs_N
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:= RL_s = 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:= dcRLsdt_N = dC7dt;
Restate the equations in terms of new sequential species names (rename free ligand concentration too)
R
eq1_R_N__U_4R_RL;
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_4R_RL;
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_4R_RL;
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_4R_RL;
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_4R_RL;
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
RL*
eq8_RLs_N:
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)
See Workflow for accurate extraction of the K matrix
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_1_s_1-k_1_B_1_s_2-k_1_B_1_s_3-k_1_B_1_s_4-Leq*k_1_A, k_2_A, k_2_B_1_s_1, k_2_B_1_s_2, k_2_B_1_s_3, k_2_B_1_s_4, 0 ],
[ Leq*k_1_A, -k_1_B_2-k_2_A, 0,0,0,0, k_2_B_2 ],
[ k_1_B_1_s_1, 0, -k_1_C_s_1_2-k_1_C_s_1_3-k_1_C_s_1_4-k_2_B_1_s_1, k_2_C_s_1_2, k_2_C_s_1_3, k_2_C_s_1_4, 0 ],
[ k_1_B_1_s_2, 0, k_1_C_s_1_2, -k_1_C_s_2_3-k_1_C_s_2_4-k_2_B_1_s_2-k_2_C_s_1_2, k_2_C_s_2_3, k_2_C_s_2_4 , 0 ],
[ k_1_B_1_s_3, 0, k_1_C_s_1_3, k_1_C_s_2_3, -k_1_C_s_3_4-k_2_B_1_s_3-k_2_C_s_1_3-k_2_C_s_2_3, k_2_C_s_3_4, 0],
[ k_1_B_1_s_4, 0, k_1_C_s_1_4, k_1_C_s_2_4, k_1_C_s_3_4, -k_2_B_1_s_4-k_2_C_s_1_4-k_2_C_s_2_4-k_2_C_s_3_4, 0 ],
[ 0, k_1_B_2, 0, 0, 0, 0, -k_2_B_2 ]
])
![matrix([[- k_1_B_1_s_1 - k_1_B_1_s_2 - k_1_B_1_s_3 - k_1_B_1_s_4 - Leq*k_1_A, k_2_A, k_2_B_1_s_1, k_2_B_1_s_2, k_2_B_1_s_3, k_2_B_1_s_4, 0], [Leq*k_1_A, - k_2_A - k_1_B_2, 0, 0, 0, 0, k_2_B_2], [k_1_B_1_s_1, 0, - k_1_C_s_1_2 - k_1_C_s_1_3 - k_1_C_s_1_4 - k_2_B_1_s_1, k_2_C_s_1_2, k_2_C_s_1_3, k_2_C_s_1_4, 0], [k_1_B_1_s_2, 0, k_1_C_s_1_2, - k_1_C_s_2_3 - k_1_C_s_2_4 - k_2_B_1_s_2 - k_2_C_s_1_2, k_2_C_s_2_3, k_2_C_s_2_4, 0], [k_1_B_1_s_3, 0, k_1_C_s_1_3, k_1_C_s_2_3, - k_1_C_s_3_4 - k_2_B_1_s_3 - k_2_C_s_1_3 - k_2_C_s_2_3, k_2_C_s_3_4, 0], [k_1_B_1_s_4, 0, k_1_C_s_1_4, k_1_C_s_2_4, k_1_C_s_3_4, - k_2_B_1_s_4 - k_2_C_s_1_4 - k_2_C_s_2_4 - k_2_C_s_3_4, 0], [0, k_1_B_2, 0, 0, 0, 0, -k_2_B_2]])](U_5R_RL_images/math459.png)
Test the K matrix entry
Create a column vector of species concentrations
P:=matrix(7,1,[C1, C2, C3, C4, C5, C6, C7])
![matrix([[C1], [C2], [C3], [C4], [C5], [C6], [C7]])](U_5R_RL_images/math460.png)
Multiply K and P:
dCdt_manual_input:= K*P
![matrix([[C2*k_2_A + C3*k_2_B_1_s_1 + C4*k_2_B_1_s_2 + C5*k_2_B_1_s_3 + C6*k_2_B_1_s_4 - C1*(k_1_B_1_s_1 + k_1_B_1_s_2 + k_1_B_1_s_3 + k_1_B_1_s_4 + Leq*k_1_A)], [C7*k_2_B_2 - C2*(k_2_A + k_1_B_2) + C1*Leq*k_1_A], [C1*k_1_B_1_s_1 + C4*k_2_C_s_1_2 + C5*k_2_C_s_1_3 + C6*k_2_C_s_1_4 - C3*(k_1_C_s_1_2 + k_1_C_s_1_3 + k_1_C_s_1_4 + k_2_B_1_s_1)], [C1*k_1_B_1_s_2 + C3*k_1_C_s_1_2 + C5*k_2_C_s_2_3 + C6*k_2_C_s_2_4 - C4*(k_1_C_s_2_3 + k_1_C_s_2_4 + k_2_B_1_s_2 + k_2_C_s_1_2)], [C1*k_1_B_1_s_3 + C3*k_1_C_s_1_3 + C4*k_1_C_s_2_3 + C6*k_2_C_s_3_4 - C5*(k_1_C_s_3_4 + k_2_B_1_s_3 + k_2_C_s_1_3 + k_2_C_s_2_3)], [C1*k_1_B_1_s_4 + C3*k_1_C_s_1_4 + C4*k_1_C_s_2_4 + C5*k_1_C_s_3_4 - C6*(k_2_B_1_s_4 + k_2_C_s_1_4 + k_2_C_s_2_4 + k_2_C_s_3_4)], [C2*k_1_B_2 - C7*k_2_B_2]])](U_5R_RL_images/math461.png)
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)])
![matrix([[C2*k_2_A - C1*k_1_B_1_s_1 - C1*k_1_B_1_s_2 - C1*k_1_B_1_s_3 - C1*k_1_B_1_s_4 + C3*k_2_B_1_s_1 + C4*k_2_B_1_s_2 + C5*k_2_B_1_s_3 + C6*k_2_B_1_s_4 - C1*Leq*k_1_A], [C7*k_2_B_2 - C2*k_1_B_2 - C2*k_2_A + C1*Leq*k_1_A], [C1*k_1_B_1_s_1 - C3*k_1_C_s_1_2 - C3*k_1_C_s_1_3 - C3*k_1_C_s_1_4 - C3*k_2_B_1_s_1 + C4*k_2_C_s_1_2 + C5*k_2_C_s_1_3 + C6*k_2_C_s_1_4], [C1*k_1_B_1_s_2 + C3*k_1_C_s_1_2 - C4*k_1_C_s_2_3 - C4*k_1_C_s_2_4 - C4*k_2_B_1_s_2 - C4*k_2_C_s_1_2 + C5*k_2_C_s_2_3 + C6*k_2_C_s_2_4], [C1*k_1_B_1_s_3 + C3*k_1_C_s_1_3 + C4*k_1_C_s_2_3 - C5*k_1_C_s_3_4 - C5*k_2_B_1_s_3 - C5*k_2_C_s_1_3 - C5*k_2_C_s_2_3 + C6*k_2_C_s_3_4], [C1*k_1_B_1_s_4 + C3*k_1_C_s_1_4 + C4*k_1_C_s_2_4 + C5*k_1_C_s_3_4 - C6*k_2_B_1_s_4 - C6*k_2_C_s_1_4 - C6*k_2_C_s_2_4 - C6*k_2_C_s_3_4], [C2*k_1_B_2 - C7*k_2_B_2]])](U_5R_RL_images/math462.png)
Compare derivation result to manual input
dCdt_mupad=dCdt_manual_input:
normal(%);
bool(%)
![matrix([[C2*k_2_A - C1*k_1_B_1_s_1 - C1*k_1_B_1_s_2 - C1*k_1_B_1_s_3 - C1*k_1_B_1_s_4 + C3*k_2_B_1_s_1 + C4*k_2_B_1_s_2 + C5*k_2_B_1_s_3 + C6*k_2_B_1_s_4 - C1*Leq*k_1_A], [C7*k_2_B_2 - C2*k_1_B_2 - C2*k_2_A + C1*Leq*k_1_A], [C1*k_1_B_1_s_1 - C3*k_1_C_s_1_2 - C3*k_1_C_s_1_3 - C3*k_1_C_s_1_4 - C3*k_2_B_1_s_1 + C4*k_2_C_s_1_2 + C5*k_2_C_s_1_3 + C6*k_2_C_s_1_4], [C1*k_1_B_1_s_2 + C3*k_1_C_s_1_2 - C4*k_1_C_s_2_3 - C4*k_1_C_s_2_4 - C4*k_2_B_1_s_2 - C4*k_2_C_s_1_2 + C5*k_2_C_s_2_3 + C6*k_2_C_s_2_4], [C1*k_1_B_1_s_3 + C3*k_1_C_s_1_3 + C4*k_1_C_s_2_3 - C5*k_1_C_s_3_4 - C5*k_2_B_1_s_3 - C5*k_2_C_s_1_3 - C5*k_2_C_s_2_3 + C6*k_2_C_s_3_4], [C1*k_1_B_1_s_4 + C3*k_1_C_s_1_4 + C4*k_1_C_s_2_4 + C5*k_1_C_s_3_4 - C6*k_2_B_1_s_4 - C6*k_2_C_s_1_4 - C6*k_2_C_s_2_4 - C6*k_2_C_s_3_4], [C2*k_1_B_2 - C7*k_2_B_2]]) = matrix([[C2*k_2_A - C1*k_1_B_1_s_1 - C1*k_1_B_1_s_2 - C1*k_1_B_1_s_3 - C1*k_1_B_1_s_4 + C3*k_2_B_1_s_1 + C4*k_2_B_1_s_2 + C5*k_2_B_1_s_3 + C6*k_2_B_1_s_4 - C1*Leq*k_1_A], [C7*k_2_B_2 - C2*k_1_B_2 - C2*k_2_A + C1*Leq*k_1_A], [C1*k_1_B_1_s_1 - C3*k_1_C_s_1_2 - C3*k_1_C_s_1_3 - C3*k_1_C_s_1_4 - C3*k_2_B_1_s_1 + C4*k_2_C_s_1_2 + C5*k_2_C_s_1_3 + C6*k_2_C_s_1_4], [C1*k_1_B_1_s_2 + C3*k_1_C_s_1_2 - C4*k_1_C_s_2_3 - C4*k_1_C_s_2_4 - C4*k_2_B_1_s_2 - C4*k_2_C_s_1_2 + C5*k_2_C_s_2_3 + C6*k_2_C_s_2_4], [C1*k_1_B_1_s_3 + C3*k_1_C_s_1_3 + C4*k_1_C_s_2_3 - C5*k_1_C_s_3_4 - C5*k_2_B_1_s_3 - C5*k_2_C_s_1_3 - C5*k_2_C_s_2_3 + C6*k_2_C_s_3_4], [C1*k_1_B_1_s_4 + C3*k_1_C_s_1_4 + C4*k_1_C_s_2_4 + C5*k_1_C_s_3_4 - C6*k_2_B_1_s_4 - C6*k_2_C_s_1_4 - C6*k_2_C_s_2_4 - C6*k_2_C_s_3_4], [C2*k_1_B_2 - C7*k_2_B_2]])](U_5R_RL_images/math463.png)
=> Typed K-matrix is correct.
Final expression for U-4R -RLkinetic matrix
K
![matrix([[- k_1_B_1_s_1 - k_1_B_1_s_2 - k_1_B_1_s_3 - k_1_B_1_s_4 - Leq*k_1_A, k_2_A, k_2_B_1_s_1, k_2_B_1_s_2, k_2_B_1_s_3, k_2_B_1_s_4, 0], [Leq*k_1_A, - k_2_A - k_1_B_2, 0, 0, 0, 0, k_2_B_2], [k_1_B_1_s_1, 0, - k_1_C_s_1_2 - k_1_C_s_1_3 - k_1_C_s_1_4 - k_2_B_1_s_1, k_2_C_s_1_2, k_2_C_s_1_3, k_2_C_s_1_4, 0], [k_1_B_1_s_2, 0, k_1_C_s_1_2, - k_1_C_s_2_3 - k_1_C_s_2_4 - k_2_B_1_s_2 - k_2_C_s_1_2, k_2_C_s_2_3, k_2_C_s_2_4, 0], [k_1_B_1_s_3, 0, k_1_C_s_1_3, k_1_C_s_2_3, - k_1_C_s_3_4 - k_2_B_1_s_3 - k_2_C_s_1_3 - k_2_C_s_2_3, k_2_C_s_3_4, 0], [k_1_B_1_s_4, 0, k_1_C_s_1_4, k_1_C_s_2_4, k_1_C_s_3_4, - k_2_B_1_s_4 - k_2_C_s_1_4 - k_2_C_s_2_4 - k_2_C_s_3_4, 0], [0, k_1_B_2, 0, 0, 0, 0, -k_2_B_2]])](U_5R_RL_images/math465.png)
Derivation of K matrix for U-5R-RL mechanism
Summary list of the net rate equations for the mechanism
eq1_R_N__U_5R_RL;
eq2_RL_N;
eq3_Rs1_N__U_5R_RL;
eq4_Rs2_N__U_5R_RL;
eq5_Rs3_N__U_5R_RL;
eq6_Rs4_N__U_5R_RL;
eq7_Rs5_N__U_5R_RL;
eq8_RLs_N
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;
feq_1h:= RL_s = C8;
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;
feq_2h:= dcRLsdt_N = dC8dt;
Restate the equations in terms of new sequential species names (rename free ligand concentration too)
R
eq1_R_N__U_5R_RL;
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 | feq_1h | feq_2h | 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 | feq_1h | feq_2h | L = Leq
R*
eq3_Rs1_N__U_5R_RL;
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 | feq_1h | feq_2h | L = Leq
R**
eq4_Rs2_N__U_5R_RL;
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 | feq_1h | feq_2h | L = Leq
R***
eq5_Rs3_N__U_5R_RL;
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 | feq_1h | feq_2h | L = Leq
R****
eq6_Rs4_N__U_5R_RL;
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 | feq_1h | feq_2h | L = Leq
R*****
eq7_Rs5_N__U_5R_RL;
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 | feq_1h | feq_2h | L = Leq
RL*
eq8_RLs_N:
feq_3h:= % | 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 | feq_1h | feq_2h | 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)
See Workflow for accurate extraction of the K matrix
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(8,8,[
[ -Leq*k_1_A-k_1_B_1_s_1-k_1_B_1_s_2-k_1_B_1_s_3-k_1_B_1_s_4-k_1_B_1_s_5, k_2_A, k_2_B_1_s_1, k_2_B_1_s_2, k_2_B_1_s_3, k_2_B_1_s_4, k_2_B_1_s_5 ,0 ],
[ Leq*k_1_A, -k_1_B_2-k_2_A, 0, 0, 0, 0, 0, k_2_B_2 ],
[ k_1_B_1_s_1,0, -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_B_1_s_1, k_2_C_s_1_2, k_2_C_s_1_3, k_2_C_s_1_4, k_2_C_s_1_5, 0 ],
[ k_1_B_1_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_B_1_s_2 -k_2_C_s_1_2, k_2_C_s_2_3, k_2_C_s_2_4, k_2_C_s_2_5, 0 ],
[ k_1_B_1_s_3,0, k_1_C_s_1_3, k_1_C_s_2_3, -k_1_C_s_3_4 -k_1_C_s_3_5 -k_2_B_1_s_3 -k_2_C_s_1_3 -k_2_C_s_2_3, k_2_C_s_3_4, k_2_C_s_3_5,0 ],
[ k_1_B_1_s_4,0, k_1_C_s_1_4, k_1_C_s_2_4, k_1_C_s_3_4, -k_1_C_s_4_5 -k_2_B_1_s_4 -k_2_C_s_1_4 -k_2_C_s_2_4 -k_2_C_s_3_4, k_2_C_s_4_5,0 ],
[ k_1_B_1_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_1_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_5,0 ],
[ 0, k_1_B_2,0,0,0,0,0, -k_2_B_2 ]
])
![matrix([[- k_1_B_1_s_1 - k_1_B_1_s_2 - k_1_B_1_s_3 - k_1_B_1_s_4 - k_1_B_1_s_5 - Leq*k_1_A, k_2_A, k_2_B_1_s_1, k_2_B_1_s_2, k_2_B_1_s_3, k_2_B_1_s_4, k_2_B_1_s_5, 0], [Leq*k_1_A, - k_2_A - k_1_B_2, 0, 0, 0, 0, 0, k_2_B_2], [k_1_B_1_s_1, 0, - 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_B_1_s_1, k_2_C_s_1_2, k_2_C_s_1_3, k_2_C_s_1_4, k_2_C_s_1_5, 0], [k_1_B_1_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_B_1_s_2 - k_2_C_s_1_2, k_2_C_s_2_3, k_2_C_s_2_4, k_2_C_s_2_5, 0], [k_1_B_1_s_3, 0, k_1_C_s_1_3, k_1_C_s_2_3, - k_1_C_s_3_4 - k_1_C_s_3_5 - k_2_B_1_s_3 - k_2_C_s_1_3 - k_2_C_s_2_3, k_2_C_s_3_4, k_2_C_s_3_5, 0], [k_1_B_1_s_4, 0, k_1_C_s_1_4, k_1_C_s_2_4, k_1_C_s_3_4, - k_1_C_s_4_5 - k_2_B_1_s_4 - k_2_C_s_1_4 - k_2_C_s_2_4 - k_2_C_s_3_4, k_2_C_s_4_5, 0], [k_1_B_1_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_1_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, 0], [0, k_1_B_2, 0, 0, 0, 0, 0, -k_2_B_2]])](U_5R_RL_images/math505.png)
Test the K matrix entry
Create a column vector of species concentrations
P:=matrix(8,1,[C1, C2, C3, C4, C5, C6, C7, C8])
![matrix([[C1], [C2], [C3], [C4], [C5], [C6], [C7], [C8]])](U_5R_RL_images/math506.png)
Multiply K and P:
dCdt_manual_input:= K*P
![matrix([[C2*k_2_A + C3*k_2_B_1_s_1 + C4*k_2_B_1_s_2 + C5*k_2_B_1_s_3 + C6*k_2_B_1_s_4 + C7*k_2_B_1_s_5 - C1*(k_1_B_1_s_1 + k_1_B_1_s_2 + k_1_B_1_s_3 + k_1_B_1_s_4 + k_1_B_1_s_5 + Leq*k_1_A)], [C8*k_2_B_2 - C2*(k_2_A + k_1_B_2) + C1*Leq*k_1_A], [C1*k_1_B_1_s_1 + C4*k_2_C_s_1_2 + C5*k_2_C_s_1_3 + C6*k_2_C_s_1_4 + C7*k_2_C_s_1_5 - C3*(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_B_1_s_1)], [C1*k_1_B_1_s_2 + C3*k_1_C_s_1_2 + C5*k_2_C_s_2_3 + C6*k_2_C_s_2_4 + C7*k_2_C_s_2_5 - C4*(k_1_C_s_2_3 + k_1_C_s_2_4 + k_1_C_s_2_5 + k_2_B_1_s_2 + k_2_C_s_1_2)], [C1*k_1_B_1_s_3 + C3*k_1_C_s_1_3 + C4*k_1_C_s_2_3 + C6*k_2_C_s_3_4 + C7*k_2_C_s_3_5 - C5*(k_1_C_s_3_4 + k_1_C_s_3_5 + k_2_B_1_s_3 + k_2_C_s_1_3 + k_2_C_s_2_3)], [C1*k_1_B_1_s_4 + C3*k_1_C_s_1_4 + C4*k_1_C_s_2_4 + C5*k_1_C_s_3_4 + C7*k_2_C_s_4_5 - C6*(k_1_C_s_4_5 + k_2_B_1_s_4 + k_2_C_s_1_4 + k_2_C_s_2_4 + k_2_C_s_3_4)], [C1*k_1_B_1_s_5 + C3*k_1_C_s_1_5 + C4*k_1_C_s_2_5 + C5*k_1_C_s_3_5 + C6*k_1_C_s_4_5 - C7*(k_2_B_1_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)], [C2*k_1_B_2 - C8*k_2_B_2]])](U_5R_RL_images/math507.png)
Collect right-hand-side parts of net rate equations expressed in sequential species names
dCdt_mupad:=matrix(8,1,[ rhs(feq_3a), rhs(feq_3b), rhs(feq_3c), rhs(feq_3d), rhs(feq_3e), rhs(feq_3f), rhs(feq_3g), rhs(feq_3h)])
![matrix([[C2*k_2_A - C1*k_1_B_1_s_1 - C1*k_1_B_1_s_2 - C1*k_1_B_1_s_3 - C1*k_1_B_1_s_4 - C1*k_1_B_1_s_5 + C3*k_2_B_1_s_1 + C4*k_2_B_1_s_2 + C5*k_2_B_1_s_3 + C6*k_2_B_1_s_4 + C7*k_2_B_1_s_5 - C1*Leq*k_1_A], [C8*k_2_B_2 - C2*k_1_B_2 - C2*k_2_A + C1*Leq*k_1_A], [C1*k_1_B_1_s_1 - C3*k_1_C_s_1_2 - C3*k_1_C_s_1_3 - C3*k_1_C_s_1_4 - C3*k_1_C_s_1_5 - C3*k_2_B_1_s_1 + C4*k_2_C_s_1_2 + C5*k_2_C_s_1_3 + C6*k_2_C_s_1_4 + C7*k_2_C_s_1_5], [C1*k_1_B_1_s_2 + C3*k_1_C_s_1_2 - C4*k_1_C_s_2_3 - C4*k_1_C_s_2_4 - C4*k_1_C_s_2_5 - C4*k_2_B_1_s_2 - C4*k_2_C_s_1_2 + C5*k_2_C_s_2_3 + C6*k_2_C_s_2_4 + C7*k_2_C_s_2_5], [C1*k_1_B_1_s_3 + C3*k_1_C_s_1_3 + C4*k_1_C_s_2_3 - C5*k_1_C_s_3_4 - C5*k_1_C_s_3_5 - C5*k_2_B_1_s_3 - C5*k_2_C_s_1_3 - C5*k_2_C_s_2_3 + C6*k_2_C_s_3_4 + C7*k_2_C_s_3_5], [C1*k_1_B_1_s_4 + C3*k_1_C_s_1_4 + C4*k_1_C_s_2_4 + C5*k_1_C_s_3_4 - C6*k_1_C_s_4_5 - C6*k_2_B_1_s_4 - C6*k_2_C_s_1_4 - C6*k_2_C_s_2_4 - C6*k_2_C_s_3_4 + C7*k_2_C_s_4_5], [C1*k_1_B_1_s_5 + C3*k_1_C_s_1_5 + C4*k_1_C_s_2_5 + C5*k_1_C_s_3_5 + C6*k_1_C_s_4_5 - C7*k_2_B_1_s_5 - C7*k_2_C_s_1_5 - C7*k_2_C_s_2_5 - C7*k_2_C_s_3_5 - C7*k_2_C_s_4_5], [C2*k_1_B_2 - C8*k_2_B_2]])](U_5R_RL_images/math508.png)
Compare derivation result to manual input
dCdt_mupad=dCdt_manual_input:
normal(%);
bool(%)
![matrix([[C2*k_2_A - C1*k_1_B_1_s_1 - C1*k_1_B_1_s_2 - C1*k_1_B_1_s_3 - C1*k_1_B_1_s_4 - C1*k_1_B_1_s_5 + C3*k_2_B_1_s_1 + C4*k_2_B_1_s_2 + C5*k_2_B_1_s_3 + C6*k_2_B_1_s_4 + C7*k_2_B_1_s_5 - C1*Leq*k_1_A], [C8*k_2_B_2 - C2*k_1_B_2 - C2*k_2_A + C1*Leq*k_1_A], [C1*k_1_B_1_s_1 - C3*k_1_C_s_1_2 - C3*k_1_C_s_1_3 - C3*k_1_C_s_1_4 - C3*k_1_C_s_1_5 - C3*k_2_B_1_s_1 + C4*k_2_C_s_1_2 + C5*k_2_C_s_1_3 + C6*k_2_C_s_1_4 + C7*k_2_C_s_1_5], [C1*k_1_B_1_s_2 + C3*k_1_C_s_1_2 - C4*k_1_C_s_2_3 - C4*k_1_C_s_2_4 - C4*k_1_C_s_2_5 - C4*k_2_B_1_s_2 - C4*k_2_C_s_1_2 + C5*k_2_C_s_2_3 + C6*k_2_C_s_2_4 + C7*k_2_C_s_2_5], [C1*k_1_B_1_s_3 + C3*k_1_C_s_1_3 + C4*k_1_C_s_2_3 - C5*k_1_C_s_3_4 - C5*k_1_C_s_3_5 - C5*k_2_B_1_s_3 - C5*k_2_C_s_1_3 - C5*k_2_C_s_2_3 + C6*k_2_C_s_3_4 + C7*k_2_C_s_3_5], [C1*k_1_B_1_s_4 + C3*k_1_C_s_1_4 + C4*k_1_C_s_2_4 + C5*k_1_C_s_3_4 - C6*k_1_C_s_4_5 - C6*k_2_B_1_s_4 - C6*k_2_C_s_1_4 - C6*k_2_C_s_2_4 - C6*k_2_C_s_3_4 + C7*k_2_C_s_4_5], [C1*k_1_B_1_s_5 + C3*k_1_C_s_1_5 + C4*k_1_C_s_2_5 + C5*k_1_C_s_3_5 + C6*k_1_C_s_4_5 - C7*k_2_B_1_s_5 - C7*k_2_C_s_1_5 - C7*k_2_C_s_2_5 - C7*k_2_C_s_3_5 - C7*k_2_C_s_4_5], [C2*k_1_B_2 - C8*k_2_B_2]]) = matrix([[C2*k_2_A - C1*k_1_B_1_s_1 - C1*k_1_B_1_s_2 - C1*k_1_B_1_s_3 - C1*k_1_B_1_s_4 - C1*k_1_B_1_s_5 + C3*k_2_B_1_s_1 + C4*k_2_B_1_s_2 + C5*k_2_B_1_s_3 + C6*k_2_B_1_s_4 + C7*k_2_B_1_s_5 - C1*Leq*k_1_A], [C8*k_2_B_2 - C2*k_1_B_2 - C2*k_2_A + C1*Leq*k_1_A], [C1*k_1_B_1_s_1 - C3*k_1_C_s_1_2 - C3*k_1_C_s_1_3 - C3*k_1_C_s_1_4 - C3*k_1_C_s_1_5 - C3*k_2_B_1_s_1 + C4*k_2_C_s_1_2 + C5*k_2_C_s_1_3 + C6*k_2_C_s_1_4 + C7*k_2_C_s_1_5], [C1*k_1_B_1_s_2 + C3*k_1_C_s_1_2 - C4*k_1_C_s_2_3 - C4*k_1_C_s_2_4 - C4*k_1_C_s_2_5 - C4*k_2_B_1_s_2 - C4*k_2_C_s_1_2 + C5*k_2_C_s_2_3 + C6*k_2_C_s_2_4 + C7*k_2_C_s_2_5], [C1*k_1_B_1_s_3 + C3*k_1_C_s_1_3 + C4*k_1_C_s_2_3 - C5*k_1_C_s_3_4 - C5*k_1_C_s_3_5 - C5*k_2_B_1_s_3 - C5*k_2_C_s_1_3 - C5*k_2_C_s_2_3 + C6*k_2_C_s_3_4 + C7*k_2_C_s_3_5], [C1*k_1_B_1_s_4 + C3*k_1_C_s_1_4 + C4*k_1_C_s_2_4 + C5*k_1_C_s_3_4 - C6*k_1_C_s_4_5 - C6*k_2_B_1_s_4 - C6*k_2_C_s_1_4 - C6*k_2_C_s_2_4 - C6*k_2_C_s_3_4 + C7*k_2_C_s_4_5], [C1*k_1_B_1_s_5 + C3*k_1_C_s_1_5 + C4*k_1_C_s_2_5 + C5*k_1_C_s_3_5 + C6*k_1_C_s_4_5 - C7*k_2_B_1_s_5 - C7*k_2_C_s_1_5 - C7*k_2_C_s_2_5 - C7*k_2_C_s_3_5 - C7*k_2_C_s_4_5], [C2*k_1_B_2 - C8*k_2_B_2]])](U_5R_RL_images/math509.png)
=> Typed K-matrix is correct.
Final expression for U-5R-RL kinetic matrix
K;
![matrix([[- k_1_B_1_s_1 - k_1_B_1_s_2 - k_1_B_1_s_3 - k_1_B_1_s_4 - k_1_B_1_s_5 - Leq*k_1_A, k_2_A, k_2_B_1_s_1, k_2_B_1_s_2, k_2_B_1_s_3, k_2_B_1_s_4, k_2_B_1_s_5, 0], [Leq*k_1_A, - k_2_A - k_1_B_2, 0, 0, 0, 0, 0, k_2_B_2], [k_1_B_1_s_1, 0, - 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_B_1_s_1, k_2_C_s_1_2, k_2_C_s_1_3, k_2_C_s_1_4, k_2_C_s_1_5, 0], [k_1_B_1_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_B_1_s_2 - k_2_C_s_1_2, k_2_C_s_2_3, k_2_C_s_2_4, k_2_C_s_2_5, 0], [k_1_B_1_s_3, 0, k_1_C_s_1_3, k_1_C_s_2_3, - k_1_C_s_3_4 - k_1_C_s_3_5 - k_2_B_1_s_3 - k_2_C_s_1_3 - k_2_C_s_2_3, k_2_C_s_3_4, k_2_C_s_3_5, 0], [k_1_B_1_s_4, 0, k_1_C_s_1_4, k_1_C_s_2_4, k_1_C_s_3_4, - k_1_C_s_4_5 - k_2_B_1_s_4 - k_2_C_s_1_4 - k_2_C_s_2_4 - k_2_C_s_3_4, k_2_C_s_4_5, 0], [k_1_B_1_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_1_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, 0], [0, k_1_B_2, 0, 0, 0, 0, 0, -k_2_B_2]])](U_5R_RL_images/math511.png)
K matrices for U-nR-RL models were successfully developed.
NOTE: U-1R-RL is not the same as IDAP's U-R-RL!