Analysis of U-R2 model

A:    R + L <=> RL

B:    R <=> R2

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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_R2";

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 - monomer-ligand interaction

K_a_A;

assume(K_a_A >= 0):

assumeAlso(K_a_A, R_):

K_a_B - Dimer formation formation constant

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);

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2. Derivation of working equation

U-R2 is a model with a receptor dimerization such that only a monomer binds ligand.

A:    R + L <=> RL

B:     R <=> R2

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

[RR] - equilibrium concentration of  a non-binding R-dimer

RR;

assumeAlso(RReq>=0):

assumeAlso(RReq<=Rtot):

assumeAlso(RReq,R_):

Total concentrations of protein monomers and a ligand

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

eq2_2:= Ltot = Leq + RLeq;

Transition A: Equilibrium constant of ligand binding

eq2_3:= K_a_A = RLeq / (Req*Leq);

Transition B: Equilibrium constant of dimerization

eq2_4:= K_a_B = RReq/(Req^2);

Let's get rid of [RR]

solve(eq2_4,RReq);

eq2_5:= RReq = %[2][1]

Let's get rid of [R]

solve(eq2_3,Req);

eq2_6:= Req = %[2][1]

Let's get rid of [RL]

eq2_2;

solve(%,RLeq);

eq2_7:= RLeq = %[2][1]

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;

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Conclusions

1. I derived a formula for numeric solution

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