Analysis of U-RL model

A:    R + L <=> RL

B:    RL <=> R*L

U-RL is a model with a receptor-ligand complex isomerization such that only one of the isomers appreciably dissociates

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Goals

In this notebook I will write out equations for equilibrium concentrations and either solve them or generate expressions for numeric solutions for a number of models derived in   /Users/kovrigin/Documents/Workspace/Data/Data.XV/EKM16.Analysis_of_multistep_kinetic_mechanisms/LRIM/Specific_models/Models.pdf

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1. Definitions

clean up workspace

reset()

Set path to save results into:

ProjectName:="LRIM_U_RL";
CurrentPath:="/Users/kovrigin/Documents/Workspace/Global Analysis/code_development/EKM14.BiophysicsLab/BiophysicsLab_v1.5/Mathematical_models/Equilibrium_thermodynamic_models/U-RL/";  Binding constants:

All binding constants I am using are formation constants so I denote them all as Ka and add a label for the transition.

K_a_A

K_a_A;
assume(K_a_A > 0):
assumeAlso(K_a_A, R_): K_a_B

K_a_B;
assumeAlso(K_a_B > 0):
assumeAlso(K_a_B,R_): Total concentrations

Rtot - total concentration of the receptor

Rtot;
assumeAlso(Rtot > 0):
assumeAlso(Rtot,R_): Ltot - total concentration of a ligand

Ltot;
assumeAlso(Ltot > 0):
assumeAlso(Ltot,R_): Common equilibrium concentrations

Req - equilibrium concentration of a receptor monomer

Req;
assumeAlso(Req>=0):
assumeAlso(Req<=Rtot):
assumeAlso(Req,R_): Leq - equilibrium concentration of a receptor monomer

Leq;
assumeAlso(Leq>=0):
assumeAlso(Leq<=Ltot):
assumeAlso(Leq,R_): RLeq - equilibrium concentration of a receptor monomer

RLeq;
assumeAlso(RLeq>=0):
assumeAlso(RLeq<=Rtot):
assumeAlso(RLeq,R_): Other species will be defined in the sections of specific models.

anames(All,User);
anames(Properties,User);  Back to Contents

2. Derivation of working equation

Working equation: I will try to express analytical [L] from equation for a total concentration of a receptor or use it for numeric solution if analytical is not possible

[R*L] - equilibrium concentration of a receptor non-binding isomer

RLstareq;
assumeAlso(RLstareq>=0):
assumeAlso(RLstareq<=Rtot):
assumeAlso(RLstareq,R_): Total concentrations of protein and a ligand

eq2_1:= Rtot = Req + RLstareq + RLeq;
eq2_2:= Ltot = Leq + RLeq  + RLstareq;  Transition A: Equilibrium constant of ligand binding

eq2_3:= K_a_A = RLeq / (Req*Leq); Transition B: Equilibrium constant of isomerization

eq2_4:= K_a_B =  RLstareq/RLeq; Let's get rid of [ R*L]

solve(eq2_4, + RLstareq);
eq2_5:=  + RLstareq = %  Let's get rid of [R]

solve(eq2_3,Req);
eq2_6:= Req = %  Let's get rid of [RL]

eq2_2;
% | eq2_5;
solve(%,RLeq);
eq2_7:= RLeq = %    Substitute

eq2_1;
% | eq2_5;
% | eq2_6;
% | eq2_7;
eq2_8:= %;     Final equation for [L] in terms of all constants

eq2_8 Solve it for Leq

solutions2:=solve(eq2_8, Leq) Extract solutions

eq2_9:=  solutions2[i,1] \$ i=1..nops(solutions2);
nops(%)  Find unique solutions:

Is 1st solution a combination of 2nd and 3rd?

solution1:=eq2_9;   // a sequence of roots
solution2:=eq2_9;  // extract equation out of a sequence
solution3:=eq2_9;   // extract equation out of a sequence

if solution2 in solution1
then print(Unquoted,"First set of roots contains the second root.");
else print(Unquoted,"First  set of roots  DOES NOT contain the second root!");
end_if;

if solution3 in solution1
then print(Unquoted,"First set of roots contains the third root.");
else print(Unquoted,"First  set of roots  DOES NOT contain the third root!");
end_if;   First set of roots contains the second root.
First set of roots contains the third root.

Check correctness of the solutions by substitution into original equation solved:

Check first root

test1:= eq2_8 | Leq=solution2;
normal(%);  -> OK

Check second root

test2:= eq2_8 | Leq=solution3;
normal(%);  -> OK

Both solutions are correct.

Test which solution is meaningful

solution2 | K_a_A=1 | K_a_B=1 |  Rtot=1 | Ltot=1;
float(%)  -> meaningful

solution3 | K_a_A=1 | K_a_B=1 |  Rtot=1 | Ltot=1;
float(%)  -> meaningless

Choose as a final solution

eq2_10:= Leq = solution2 Summary of equations for all species

eq2_10 eq2_7; eq2_6; eq2_5; Back to Contents

3. Define functions for equilibrium concentrations

Define functions for plotting and analysis

Conclusions

1. I successfully derived equation for numeric analysis

2. System is analytically soluble

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