LRIM_U

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

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

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/";

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

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

Rtot;
assumeAlso(Rtot > 0):
assumeAlso(Rtot,R_):

Ltot;
assumeAlso(Ltot > 0):
assumeAlso(Ltot,R_):

Req;
assumeAlso(Req>=0):
assumeAlso(Req<=Rtot):
assumeAlso(Req,R_):

Leq;
assumeAlso(Leq>=0):
assumeAlso(Leq<=Ltot):
assumeAlso(Leq,R_):

RLeq;
assumeAlso(RLeq>=0):
assumeAlso(RLeq<=Rtot):
assumeAlso(RLeq,R_):

anames(All,User);
anames(Properties,User);

{CurrentPath, ProjectName}
${K_a_A, K_a_B, Leq, Ltot, RLeq, Req, Rtot}$

RLstareq;
assumeAlso(RLstareq>=0):
assumeAlso(RLstareq<=Rtot):
assumeAlso(RLstareq,R_):

eq2_1:= Rtot = Req + RLstareq + RLeq;
eq2_2:= Ltot = Leq + RLeq + RLstareq;

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

eq2_4:= K_a_B = RLstareq/RLeq;

solve(eq2_4, + RLstareq);
eq2_5:= + RLstareq = %[2][1]

$piecewise([Rtot < K_a_B*RLeq, {}], [K_a_B*RLeq <= Rtot, {K_a_B*RLeq}])$
RLstareq = K_a_B*RLeq

solve(eq2_3,Req);
eq2_6:= Req = %[2][1]

$piecewise([Leq = 0 or RLeq = 0 or K_a_A*Leq*Rtot < RLeq, {}], [Leq <> 0 and RLeq <> 0 and RLeq <= K_a_A*Leq*Rtot, {RLeq/(K_a_A*Leq)}])$
Req = RLeq/(K_a_A*Leq)

eq2_2;
% | eq2_5;
solve(%,RLeq);
eq2_7:= RLeq = %[2][1]

Ltot = Leq + RLeq + RLstareq
Ltot = Leq + RLeq + K_a_B*RLeq
$piecewise([Leq + Rtot + K_a_B*Rtot < Ltot, {}], [Ltot <= Leq + Rtot + K_a_B*Rtot, {-(Leq - Ltot)/(K_a_B + 1)}])$
RLeq = -(Leq - Ltot)/(K_a_B + 1)

eq2_1;
% | eq2_5;
% | eq2_6;
% | eq2_7;
eq2_8:= %;

Final equation for [L] in terms of all constants

eq2_8

Rtot = - (Leq - Ltot)/(K_a_B + 1) - (K_a_B*(Leq - Ltot))/(K_a_B + 1) - (Leq - Ltot)/(K_a_A*Leq*(K_a_B + 1))

solutions2:=solve(eq2_8, Leq)

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

${-((K_a_A^2*K_a_B^2*Ltot^2 - 2*K_a_A^2*K_a_B^2*Ltot*Rtot + K_a_A^2*K_a_B^2*Rtot^2 + 2*K_a_A^2*K_a_B*Ltot^2 - 4*K_a_A^2*K_a_B*Ltot*Rtot + 2*K_a_A^2*K_a_B*Rtot^2 + K_a_A^2*Ltot^2 - 2*K_a_A^2*Ltot*Rtot + K_a_A^2*Rtot^2 + 2*K_a_A*K_a_B*Ltot + 2*K_a_A*K_a_B*Rtot + 2*K_a_A*Ltot + 2*K_a_A*Rtot + 1)^(1/2) - K_a_A*Ltot + K_a_A*Rtot - K_a_A*K_a_B*Ltot + K_a_A*K_a_B*Rtot + 1)/(2*K_a_A + 2*K_a_A*K_a_B), ((K_a_A^2*K_a_B^2*Ltot^2 - 2*K_a_A^2*K_a_B^2*Ltot*Rtot + K_a_A^2*K_a_B^2*Rtot^2 + 2*K_a_A^2*K_a_B*Ltot^2 - 4*K_a_A^2*K_a_B*Ltot*Rtot + 2*K_a_A^2*K_a_B*Rtot^2 + K_a_A^2*Ltot^2 - 2*K_a_A^2*Ltot*Rtot + K_a_A^2*Rtot^2 + 2*K_a_A*K_a_B*Ltot + 2*K_a_A*K_a_B*Rtot + 2*K_a_A*Ltot + 2*K_a_A*Rtot + 1)^(1/2) + K_a_A*Ltot - K_a_A*Rtot + K_a_A*K_a_B*Ltot - K_a_A*K_a_B*Rtot - 1)/(2*K_a_A + 2*K_a_A*K_a_B)}, {((K_a_A^2*K_a_B^2*Ltot^2 - 2*K_a_A^2*K_a_B^2*Ltot*Rtot + K_a_A^2*K_a_B^2*Rtot^2 + 2*K_a_A^2*K_a_B*Ltot^2 - 4*K_a_A^2*K_a_B*Ltot*Rtot + 2*K_a_A^2*K_a_B*Rtot^2 + K_a_A^2*Ltot^2 - 2*K_a_A^2*Ltot*Rtot + K_a_A^2*Rtot^2 + 2*K_a_A*K_a_B*Ltot + 2*K_a_A*K_a_B*Rtot + 2*K_a_A*Ltot + 2*K_a_A*Rtot + 1)^(1/2) + K_a_A*Ltot - K_a_A*Rtot + K_a_A*K_a_B*Ltot - K_a_A*K_a_B*Rtot - 1)/(2*K_a_A + 2*K_a_A*K_a_B)}, {-((K_a_A^2*K_a_B^2*Ltot^2 - 2*K_a_A^2*K_a_B^2*Ltot*Rtot + K_a_A^2*K_a_B^2*Rtot^2 + 2*K_a_A^2*K_a_B*Ltot^2 - 4*K_a_A^2*K_a_B*Ltot*Rtot + 2*K_a_A^2*K_a_B*Rtot^2 + K_a_A^2*Ltot^2 - 2*K_a_A^2*Ltot*Rtot + K_a_A^2*Rtot^2 + 2*K_a_A*K_a_B*Ltot + 2*K_a_A*K_a_B*Rtot + 2*K_a_A*Ltot + 2*K_a_A*Rtot + 1)^(1/2) - K_a_A*Ltot + K_a_A*Rtot - K_a_A*K_a_B*Ltot + K_a_A*K_a_B*Rtot + 1)/(2*K_a_A + 2*K_a_A*K_a_B)}, {}$

solution1:=eq2_9[1]; // a sequence of roots
solution2:=eq2_9[2][1]; // extract equation out of a sequence
solution3:=eq2_9[3][1]; // 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;

${-((K_a_A^2*K_a_B^2*Ltot^2 - 2*K_a_A^2*K_a_B^2*Ltot*Rtot + K_a_A^2*K_a_B^2*Rtot^2 + 2*K_a_A^2*K_a_B*Ltot^2 - 4*K_a_A^2*K_a_B*Ltot*Rtot + 2*K_a_A^2*K_a_B*Rtot^2 + K_a_A^2*Ltot^2 - 2*K_a_A^2*Ltot*Rtot + K_a_A^2*Rtot^2 + 2*K_a_A*K_a_B*Ltot + 2*K_a_A*K_a_B*Rtot + 2*K_a_A*Ltot + 2*K_a_A*Rtot + 1)^(1/2) - K_a_A*Ltot + K_a_A*Rtot - K_a_A*K_a_B*Ltot + K_a_A*K_a_B*Rtot + 1)/(2*K_a_A + 2*K_a_A*K_a_B), ((K_a_A^2*K_a_B^2*Ltot^2 - 2*K_a_A^2*K_a_B^2*Ltot*Rtot + K_a_A^2*K_a_B^2*Rtot^2 + 2*K_a_A^2*K_a_B*Ltot^2 - 4*K_a_A^2*K_a_B*Ltot*Rtot + 2*K_a_A^2*K_a_B*Rtot^2 + K_a_A^2*Ltot^2 - 2*K_a_A^2*Ltot*Rtot + K_a_A^2*Rtot^2 + 2*K_a_A*K_a_B*Ltot + 2*K_a_A*K_a_B*Rtot + 2*K_a_A*Ltot + 2*K_a_A*Rtot + 1)^(1/2) + K_a_A*Ltot - K_a_A*Rtot + K_a_A*K_a_B*Ltot - K_a_A*K_a_B*Rtot - 1)/(2*K_a_A + 2*K_a_A*K_a_B)}$
((K_a_A^2*K_a_B^2*Ltot^2 - 2*K_a_A^2*K_a_B^2*Ltot*Rtot + K_a_A^2*K_a_B^2*Rtot^2 + 2*K_a_A^2*K_a_B*Ltot^2 - 4*K_a_A^2*K_a_B*Ltot*Rtot + 2*K_a_A^2*K_a_B*Rtot^2 + K_a_A^2*Ltot^2 - 2*K_a_A^2*Ltot*Rtot + K_a_A^2*Rtot^2 + 2*K_a_A*K_a_B*Ltot + 2*K_a_A*K_a_B*Rtot + 2*K_a_A*Ltot + 2*K_a_A*Rtot + 1)^(1/2) + K_a_A*Ltot - K_a_A*Rtot + K_a_A*K_a_B*Ltot - K_a_A*K_a_B*Rtot - 1)/(2*K_a_A + 2*K_a_A*K_a_B)
-((K_a_A^2*K_a_B^2*Ltot^2 - 2*K_a_A^2*K_a_B^2*Ltot*Rtot + K_a_A^2*K_a_B^2*Rtot^2 + 2*K_a_A^2*K_a_B*Ltot^2 - 4*K_a_A^2*K_a_B*Ltot*Rtot + 2*K_a_A^2*K_a_B*Rtot^2 + K_a_A^2*Ltot^2 - 2*K_a_A^2*Ltot*Rtot + K_a_A^2*Rtot^2 + 2*K_a_A*K_a_B*Ltot + 2*K_a_A*K_a_B*Rtot + 2*K_a_A*Ltot + 2*K_a_A*Rtot + 1)^(1/2) - K_a_A*Ltot + K_a_A*Rtot - K_a_A*K_a_B*Ltot + K_a_A*K_a_B*Rtot + 1)/(2*K_a_A + 2*K_a_A*K_a_B)
First set of roots contains the second root.
First set of roots contains the third root.

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

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

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

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

eq2_10:= Leq = solution2

eq2_10

Leq = ((K_a_A^2*K_a_B^2*Ltot^2 - 2*K_a_A^2*K_a_B^2*Ltot*Rtot + K_a_A^2*K_a_B^2*Rtot^2 + 2*K_a_A^2*K_a_B*Ltot^2 - 4*K_a_A^2*K_a_B*Ltot*Rtot + 2*K_a_A^2*K_a_B*Rtot^2 + K_a_A^2*Ltot^2 - 2*K_a_A^2*Ltot*Rtot + K_a_A^2*Rtot^2 + 2*K_a_A*K_a_B*Ltot + 2*K_a_A*K_a_B*Rtot + 2*K_a_A*Ltot + 2*K_a_A*Rtot + 1)^(1/2) + K_a_A*Ltot - K_a_A*Rtot + K_a_A*K_a_B*Ltot - K_a_A*K_a_B*Rtot - 1)/(2*K_a_A + 2*K_a_A*K_a_B)

eq2_7;

RLeq = -(Leq - Ltot)/(K_a_B + 1)

eq2_6;

Req = RLeq/(K_a_A*Leq)

eq2_5;

RLstareq = K_a_B*RLeq