Analysis of U-R2L2 model

A:    R + L <=> RL

B:    RL <=> (RL)2

U-R2L2 is a model with a receptor dimerizes via ligand interactions (receptor lacks ability to dimerize in the absence of a ligand) and binding affinity to ligand in this quaternary complex becomes very high.

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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_R2L2";

CurrentPath:="/Users/kovrigin/Documents/Workspace/Data/Data.XV/EKM16.Analysis_of_multistep_kinetic_mechanisms/Equilibria/";  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

[(RL)2] - equilibrium concentration of a receptor-ligand complex dimer

R2L2eq;

assumeAlso(R2L2eq>=0):

assumeAlso(R2L2eq<=Rtot):

assumeAlso(R2L2eq,R_): Total concentrations of protein and a ligand

eq2_1:= Rtot = Req + 2*R2L2eq + RLeq;

eq2_2:= Ltot = Leq + RLeq  + 2*R2L2eq;  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 =  R2L2eq/RLeq^2; Let's get rid of [R2L2]

solve(eq2_4,R2L2eq);

eq2_5:=  R2L2eq = %  Let's get rid of [R]

solve(eq2_3,Req);

eq2_6:= Req = %  Let's get rid of [RL]

(choose one solution)

eq2_2;

% | eq2_5;

solve(%,RLeq) assuming K_a_B>0;

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 For the sake of my speed - use numerical solutions leaving analytical for future analysis.

Solve it for [L]

solutions2:=solve(eq2_8, Leq) Analytically insoluble:  go to Matlab

Summary of equations for equilibrium concentrations:

eq2_8 eq2_7; eq2_6; eq2_5; Conclusions

1. I successfully derived equation for numeric analysis

2. System is analytically insoluble

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