Analysis of U-R2 model

 

 

A:    R + L <=> RL

 

B:    R <=> R2

 

 

Contents

 

Goals

 

1. Definitions

 

2. Derivation of working equation

 

3. Express concentrations of equilibrium species in terms of a found solution

 

4. Define functions for equilibrium concentrations

 

5. Test if solution is meaningful

 

6. Check whether the solution satisfies all initial equation and conditions

 

7. Save results on disk for future use

 

8. 2D plotting

 

9. ITC curve simulation

 

 

Conclusions

 

 

 

 

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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/";

math

math

 

 

 

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_):

math

 

K_a_B - Dimer formation formation constant

K_a_B;

assumeAlso(K_a_B>=0):

assumeAlso(K_a_B,R_):

math

 

 

 

Total concentrations

 

 

Rtot - total concentration of the receptor

Rtot;

assumeAlso(Rtot>=0):

assumeAlso(Rtot,R_):

math

Ltot - total concentration of a ligand

Ltot;

assumeAlso(Ltot>=0):

assumeAlso(Ltot,R_):

math

 

 

 

 

 

Common equilibrium concentrations

 

 

 

 

Req - equilibrium concentration of a receptor monomer

Req;

assumeAlso(Req>=0):

assumeAlso(Req<=Rtot):

assumeAlso(Req,R_):

math

 

Leq - equilibrium concentration of a receptor monomer

Leq;

assumeAlso(Leq>=0):

assumeAlso(Leq<=Ltot):

assumeAlso(Leq,R_):

math

 

RLeq - equilibrium concentration of a receptor monomer

RLeq;

assumeAlso(RLeq>=0):

assumeAlso(RLeq<=Rtot):

assumeAlso(RLeq,R_):

math

 

Other species will be defined in the sections of specific models.

 

 

 

 

 

anames(All,User);

anames(Properties,User);

 

math

math

 

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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_):

math

 

 

Total concentrations of protein monomers and a ligand

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

eq2_2:= Ltot = Leq + RLeq;

math

math

 

Transition A: Equilibrium constant of ligand binding

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

math

Transition B: Equilibrium constant of dimerization

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

math

 

Let's get rid of [RR]

solve(eq2_4,RReq);

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

math

math

Let's get rid of [R]

solve(eq2_3,Req);

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

math

math

Let's get rid of [RL]

eq2_2;

solve(%,RLeq);

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

math

math

math

 

Substitute

eq2_1 | eq2_5;

% | eq2_6;

% | eq2_7;

eq2_8:= %;

math

math

math

math

 

 

Final equation for [L] in terms of all constants

eq2_8

math

 

 

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

 

 

Solve it for [L]

solutions2:=solve(eq2_8, Leq)

math

 

Analytically insoluble: go to Matlab

 

 

Summary of equations for equilibrium concentrations:

 

eq2_8

math

 

 

eq2_7;

math

 

 

eq2_6;

math

 

 

eq2_5;

math

 

 

 

 

 

 

 

 

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Conclusions

 

1. I derived a formula for numeric solution

 

 

 

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