U-5R-RL

Derivation of equilibrium thermodynamic equations for U-5R-RL system: isomerization in the binding-incompetent state of the receptor (many states) and bound state (induced fit).

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Goals

Here I will analyze numeric solutions from "derivation.mn" document.

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Clean up

reset()

Path to previous results

ProjectName:="U-5R-RL";
CurrentPath:="/Users/kovrigin_laptop/Documents/Workspace/Global_Analysis/IDAP/Mathematical_models/Equilibrium_thermodynamic_models/U-multi-path-models/nR/U-5R-RL";

filename:=CurrentPath."/".ProjectName.".mb";
anames(User)

Assume some values for testing operation

Total_R:=1e-3:
Total_L:=10e-3:
Ka:=1e3:
Kb1_s1:=2:
Kb1_s2:=3:
Kb1_s3:=4:
Kb1_s4:=5:
Kb1_s5:=6:
Kb2:=2:

test operation of all functions

fLeq_U_5R_RL(Total_R,  Total_L, Ka, Kb1_s1, Kb1_s2,  Kb1_s3,  Kb1_s4,  Kb1_s5, Kb2);
fReq_U_5R_RL(Total_R,  Total_L, Ka, Kb1_s1, Kb1_s2,  Kb1_s3,  Kb1_s4,  Kb1_s5, Kb2);
fR_s_1eq_U_5R_RL(Total_R,  Total_L, Ka, Kb1_s1, Kb1_s2,  Kb1_s3,  Kb1_s4,  Kb1_s5, Kb2);
fR_s_2eq_U_5R_RL(Total_R,  Total_L, Ka, Kb1_s1, Kb1_s2,  Kb1_s3,  Kb1_s4,  Kb1_s5, Kb2);
fR_s_3eq_U_5R_RL(Total_R,  Total_L, Ka, Kb1_s1, Kb1_s2,  Kb1_s3,  Kb1_s4,  Kb1_s5, Kb2);
fR_s_4eq_U_5R_RL(Total_R,  Total_L, Ka, Kb1_s1, Kb1_s2,  Kb1_s3,  Kb1_s4,  Kb1_s5, Kb2);
fR_s_5eq_U_5R_RL(Total_R,  Total_L, Ka, Kb1_s1, Kb1_s2,  Kb1_s3,  Kb1_s4,  Kb1_s5, Kb2);
fRLeq_U_5R_RL(Total_R,  Total_L, Ka, Kb1_s1, Kb1_s2,  Kb1_s3,  Kb1_s4,  Kb1_s5, Kb2);
fR_sLeq_U_5R_RL(Total_R,  Total_L, Ka, Kb1_s1, Kb1_s2,  Kb1_s3,  Kb1_s4,  Kb1_s5, Kb2);

=> operative

Make wrapper functions for plotting using L/R as X axis

fLeq:=LRratio ->      fLeq_U_5R_RL     (Total_R,  LRratio*Total_R,  Ka, Kb1_s1, Kb1_s2, Kb1_s3,  Kb1_s4,  Kb1_s5,  Kb2):
fReq:=LRratio ->      fReq_U_5R_RL     (Total_R,  LRratio*Total_R,  Ka, Kb1_s1, Kb1_s2, Kb1_s3,  Kb1_s4,  Kb1_s5,  Kb2):
fR_s_1eq:=LRratio ->  fR_s_1eq_U_5R_RL     (Total_R,  LRratio*Total_R,  Ka, Kb1_s1, Kb1_s2, Kb1_s3,  Kb1_s4,  Kb1_s5,  Kb2):
fR_s_2eq:=LRratio ->  fR_s_2eq_U_5R_RL     (Total_R,  LRratio*Total_R,  Ka, Kb1_s1, Kb1_s2, Kb1_s3,  Kb1_s4,  Kb1_s5,  Kb2):
fR_s_3eq:=LRratio ->  fR_s_3eq_U_5R_RL     (Total_R,  LRratio*Total_R,  Ka, Kb1_s1, Kb1_s2, Kb1_s3,  Kb1_s4,  Kb1_s5,  Kb2):
fR_s_4eq:=LRratio ->  fR_s_4eq_U_5R_RL     (Total_R,  LRratio*Total_R,  Ka, Kb1_s1, Kb1_s2, Kb1_s3,  Kb1_s4,  Kb1_s5,  Kb2):
fR_s_5eq:=LRratio ->  fR_s_5eq_U_5R_RL     (Total_R,  LRratio*Total_R,  Ka, Kb1_s1, Kb1_s2, Kb1_s3,  Kb1_s4,  Kb1_s5,  Kb2):
fRLeq:=LRratio ->     fRLeq_U_5R_RL     (Total_R,  LRratio*Total_R,  Ka, Kb1_s1, Kb1_s2, Kb1_s3,  Kb1_s4,  Kb1_s5,  Kb2):
fR_sLeq:=LRratio ->     fR_sLeq_U_5R_RL     (Total_R,  LRratio*Total_R,  Ka, Kb1_s1, Kb1_s2, Kb1_s3,  Kb1_s4,  Kb1_s5,  Kb2):

Test plotting

Total_R:=1e-3:
LRratio_max:=2:
Ka:=1e6:
Kb1_s1:=2:
Kb1_s2:=3:
Kb1_s3:=4:
Kb1_s4:=5:
Kb1_s5:=6:
Kb2:=2:

LineW:=1.5: //line width

// create plots

pLeq:=  plot::Function2d(
Function=(fLeq),
LegendText="[L]",
Color = RGB::Blue,
XMin=(0),
XMax=(LRratio_max),
XName=(LRratio),
TitlePositionX=(0),
LineWidth=LineW):

pRLeq:=  plot::Function2d(
Function=(fRLeq),
LegendText="[RL]",
Color = RGB::Red,
XMin=(0),
XMax=(LRratio_max),
XName=(LRratio),
TitlePositionX=(0),
LineWidth=LineW):

plot(pLeq, pRLeq, LegendVisible=TRUE)

=> works

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Assume some constants and evaluate titrations.

NOTE: Adjust dependent constant calculation if necessary.

Simulation_name:= "Full model":
Total_R:=1e-3:
LRratio_max:=2:
Ka:=1e5:
Kb1_s1:=2:
Kb1_s2:=3:
Kb1_s3:=4:
Kb1_s4:=5:
Kb1_s5:=6:
Kb2:=2:

LRratio_max:=1.5: // plotting range

LineW:=1.5: // plot line width

pLeq:=  plot::Function2d(
Function=(fLeq),
LegendText="[L]",
Color = RGB::Blue,
XMin=(LRratio_max*1e-6),
XMax=(LRratio_max),
XName=(LRratio),
TitlePositionX=(0),
LineWidth=LineW):

pReq:=  plot::Function2d(
Function=(fReq),
LegendText="[R]",
Color = RGB::Black,
XMin=(LRratio_max*1e-6),
XMax=(LRratio_max),
XName=(LRratio),
TitlePositionX=(0),
LineWidth=LineW):

pR_s_1eq:=  plot::Function2d(
Function=(fR_s_1eq),
LegendText="[R*]",
Color = RGB::Green,
XMin=(LRratio_max*1e-6),
XMax=(LRratio_max),
XName=(LRratio),
TitlePositionX=(0),
LineWidth=LineW):

pR_s_2eq:=  plot::Function2d(
Function=(fR_s_2eq),
LegendText="[R**]",
Color = RGB::Grey,
XMin=(LRratio_max*1e-6),
XMax=(LRratio_max),
XName=(LRratio),
TitlePositionX=(0),
LineWidth=LineW):

pR_s_3eq:=  plot::Function2d(
Function=(fR_s_3eq),
LegendText="[R***]",
Color = RGB::Magenta,
XMin=(LRratio_max*1e-6),
XMax=(LRratio_max),
XName=(LRratio),
TitlePositionX=(0),
LineWidth=LineW):

pR_s_4eq:=  plot::Function2d(
Function=(fR_s_4eq),
LegendText="[R****]",
Color = RGB::Yellow,
XMin=(LRratio_max*1e-6),
XMax=(LRratio_max),
XName=(LRratio),
TitlePositionX=(0),
LineWidth=LineW):

pR_s_5eq:=  plot::Function2d(
Function=(fR_s_5eq),
LegendText="[R*****]",
Color = RGB::Cyan,
XMin=(LRratio_max*1e-6),
XMax=(LRratio_max),
XName=(LRratio),
TitlePositionX=(0),
LineWidth=LineW):

pRLeq:=  plot::Function2d(
Function=(fRLeq),
LegendText="[RL]",
Color = RGB::Red,
XMin=(LRratio_max*1e-6),
XMax=(LRratio_max),
XName=(LRratio),
TitlePositionX=(0),
LineWidth=LineW):

pR_sLeq:=  plot::Function2d(
Function=(fR_sLeq),
LegendText="[R*L]",
Color = RGB::Orange,
XMin=(LRratio_max*1e-6),
XMax=(LRratio_max),
XName=(LRratio),
TitlePositionX=(0),
LineWidth=LineW):

// Text report
print(Unquoted, Simulation_name);
print(Unquoted, "-------------");
print(Unquoted,"Model: ".ProjectName);
print(Unquoted,"Total_R=".Total_R);
Kda:=1/Ka:
print(Unquoted,"Ka=".Ka.",   Kd=".Kda);
print(Unquoted,"Kb*=".Kb1_s1);
print(Unquoted,"Kb**=".Kb1_s2);
print(Unquoted,"Kb***=".Kb1_s3);
print(Unquoted,"Kb****=".Kb1_s4);
print(Unquoted,"Kb*****=".Kb1_s5);
Kc21:=Kb1_s2/Kb1_s1:
Kc31:=Kb1_s3/Kb1_s1:
Kc41:=Kb1_s4/Kb1_s1:
Kc51:=Kb1_s5/Kb1_s1:
print(Unquoted,"Kc*-**=".Kc21);
print(Unquoted,"Kc*-***=".Kc31);
print(Unquoted,"Kc*-****=".Kc41);
print(Unquoted,"Kc*-*****=".Kc51);
print(Unquoted,"Kb2=".Kb2);

// plot all together
plot(pLeq, pReq,  pR_s_1eq, pR_s_2eq,  pR_s_3eq, pR_s_4eq,  pR_s_5eq,  pRLeq, pR_sLeq,
Height=160, Width=100,TicksLabelFont=["Helvetica",12,[0,0,0],Left],
AxesTitleFont=["Helvetica",14,[0,0,0],Left],
XGridVisible=TRUE, YGridVisible=TRUE,
LegendVisible=TRUE, LegendFont=["Helvetica",14,[0,0,0],Left],
ViewingBoxYMax=Total_R);

Full model
-------------
Model: U-5R-RL
Total_R=0.001
Ka=100000.0,   Kd=0.00001
Kb*=2
Kb**=3
Kb***=4
Kb****=5
Kb*****=6
Kc*-**=3/2
Kc*-***=2
Kc*-****=5/2
Kc*-*****=3
Kb2=2

Jump back to the beginning of simulation section

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Jump back to the beginning of simulation section

Test of the model: titration of  R with L

 Reduced to U-------------Model: U-5R-RLTotal_R=0.001Ka=100000.0,   Kd=0.00001Kb*=0.000001Kb**=0.000001Kb***=0.000001Kb****=0.000001Kb*****=0.000001Kc*-**=1.0Kc*-***=1.0Kc*-****=1.0Kc*-*****=1.0Kb2=0 Reduced to U-RL-------------Model: U-5R-RLTotal_R=0.001Ka=100000.0,   Kd=0.00001Kb*=0.000001Kb**=0.000001Kb***=0.000001Kb****=0.000001Kb*****=0.000001Kc*-**=1.0Kc*-***=1.0Kc*-****=1.0Kc*-*****=1.0Kb2=2 Reduced to U-R (use R*)-------------Model: U-5R-RLTotal_R=0.001Ka=100000.0,   Kd=0.00001Kb*=2Kb**=0.000001Kb***=0.000001Kb****=0.000001Kb*****=0.000001Kc*-**=0.0000005Kc*-***=0.0000005Kc*-****=0.0000005Kc*-*****=0.0000005Kb2=0 Reduced to U-R (use R**)-------------Model: U-5R-RLTotal_R=0.001Ka=100000.0,   Kd=0.00001Kb*=0.000001Kb**=2Kb***=0.000001Kb****=0.000001Kb*****=0.000001Kc*-**=2000000.0Kc*-***=1.0Kc*-****=1.0Kc*-*****=1.0Kb2=0 Reduced to U-R (use R***)-------------Model: U-5R-RLTotal_R=0.001Ka=100000.0,   Kd=0.00001Kb*=0.000001Kb**=0.000001Kb***=2Kb****=0.000001Kb*****=0.000001Kc*-**=1.0Kc*-***=2000000.0Kc*-****=1.0Kc*-*****=1.0Kb2=0 Reduced to U-R (use R****)-------------Model: U-5R-RLTotal_R=0.001Ka=100000.0,   Kd=0.00001Kb*=0.000001Kb**=0.000001Kb***=0.000001Kb****=2Kb*****=0.000001Kc*-**=1.0Kc*-***=1.0Kc*-****=2000000.0Kc*-*****=1.0Kb2=0 Reduced to U-R (use R*****)-------------Model: U-5R-RLTotal_R=0.001Ka=100000.0,   Kd=0.00001Kb*=0.000001Kb**=0.000001Kb***=0.000001Kb****=0.000001Kb*****=2Kc*-**=1.0Kc*-***=1.0Kc*-****=1.0Kc*-*****=2000000.0Kb2=0 Full model-------------Model: U-5R-RLTotal_R=0.001Ka=100000.0,   Kd=0.00001Kb*=2Kb**=3Kb***=4Kb****=5Kb*****=6Kc*-**=3/2Kc*-***=2Kc*-****=5/2Kc*-*****=3Kb2=2

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Conclusion:

The model works as expected