U-2R-RL

 

Derivation of equilibrium thermodynamic equations for U-2R-RL system: isomerization in the binding-incompetent state of the receptor (many states) and in the bound state (induced fit)

 

 

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Contents

 

Goals

 

1. Load equations

 

2. Simulation

 

3. Summary of test results

 

 

 

Conclusions

 

 

 

 

 

 

 

 

 

Back to Contents

 

Goals

 

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

 

 

Back to Contents

 

 

1. Load equations

 

Clean up

reset()


Path to previous results

ProjectName:="U-2R-RL";
CurrentPath:="/Users/kovrigin_laptop/Documents/Workspace/Global_Analysis/IDAP/Mathematical_models/Equilibrium_thermodynamic_models/U-multi-path-models/nR/U-2R-RL";

"U-2R-RL"
"/Users/kovrigin_laptop/Documents/Workspace/Global_Analysis/IDAP/Mathematical_models/Equilibrium_thermodynamic_models/U-multi-path-models/nR/U-2R-RL"

 

 

 

 

 

 

Read results of derivations

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

"/Users/kovrigin_laptop/Documents/Workspace/Global_Analysis/IDAP/Mathematical_models/Equilibrium_thermodynamic_models/U-multi-path-models/nR/U-2R-RL/U-2R-RL.mb"
{CurrentPath, Leq_U_2R_RL, ProjectName, RLeq_U_2R_RL, R_sLeq_U_2R_RL, R_s_1eq_U_2R_RL, R_s_2eq_U_2R_RL, Req_U_2R_RL, fLeq_U_2R_RL, fRLeq_U_2R_RL, fR_sLeq_U_2R_RL, fR_s_1eq_U_2R_RL, fR_s_2eq_U_2R_RL, fReq_U_2R_RL, filename}

 

 

 

Assume some values for testing operation

Total_R:=1e-3:
Total_L:=10e-3:
Ka:=1e3:
Kb1_s1:=2:
Kb1_s2:=3:
Kb2:=4:


test operation of all functions

fLeq_U_2R_RL(Total_R,  Total_L, Ka, Kb1_s1, Kb1_s2, Kb2);
fReq_U_2R_RL(Total_R,  Total_L, Ka, Kb1_s1, Kb1_s2, Kb2);
fR_s_1eq_U_2R_RL(Total_R,  Total_L, Ka, Kb1_s1, Kb1_s2, Kb2);
fR_s_2eq_U_2R_RL(Total_R,  Total_L, Ka, Kb1_s1, Kb1_s2, Kb2);
fRLeq_U_2R_RL(Total_R,  Total_L, Ka, Kb1_s1, Kb1_s2, Kb2);
fR_sLeq_U_2R_RL(Total_R,  Total_L, Ka, Kb1_s1, Kb1_s2, Kb2);

0.009116320542
0.00001938675705
0.0000387735141
0.00005816027115
0.0001767358915
0.0007069435662

=> operative

 

 

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

fLeq:=LRratio ->      fLeq_U_2R_RL     (Total_R,  LRratio*Total_R,  Ka, Kb1_s1, Kb1_s2, Kb2):
fReq:=LRratio ->      fReq_U_2R_RL     (Total_R,  LRratio*Total_R,  Ka, Kb1_s1, Kb1_s2, Kb2):
fR_s_1eq:=LRratio ->  fR_s_1eq_U_2R_RL     (Total_R,  LRratio*Total_R,  Ka, Kb1_s1, Kb1_s2, Kb2):
fR_s_2eq:=LRratio ->  fR_s_2eq_U_2R_RL     (Total_R,  LRratio*Total_R,  Ka, Kb1_s1, Kb1_s2, Kb2):
fRLeq:=LRratio ->     fRLeq_U_2R_RL     (Total_R,  LRratio*Total_R,  Ka, Kb1_s1, Kb1_s2, Kb2):
fR_sLeq:=LRratio ->     fR_sLeq_U_2R_RL     (Total_R,  LRratio*Total_R,  Ka, Kb1_s1, Kb1_s2, Kb2):

Test plotting

Total_R:=1e-3:
LRratio_max:=2:
Ka:=1e5:
Kb1_s1:=2:
Kb1_s2:=3:
Kb2:=4:

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)

MuPAD graphics

=> works

 

 

 

Back to Contents

 

 

2. Simulation

 

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:=4:
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):




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::Cyan,
          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,"Kb1*=".Kb1_s1);
print(Unquoted,"Kb1**=".Kb1_s2);
Kc21:=Kb1_s2/Kb1_s1:
print(Unquoted,"Kc*-**=".Kc21);
print(Unquoted,"Kb2=".Kb2);

// plot all together
plot(pLeq, pReq,  pR_s_1eq, pR_s_2eq,  pRLeq, pR_sLeq,
   YAxisTitle="[X]", Header=("Model: ".ProjectName),
   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-2R-RL
Total_R=0.001
Ka=100000.0,   Kd=0.00001
Kb1*=2
Kb1**=4
Kc*-**=2
Kb2=2
MuPAD graphics

 

 

Jump back to the beginning of simulation section

 

 

 

 

 

 

Back to Contents

 

 

3. Summary of test results

 

 

Jump back to the beginning of simulation section

 

Test of the model: titration of  R with L

 

Reduce to U
Model: U-2R-RL
Total_R=0.001
Ka=100000.0,   Kd=0.00001
Kb1*=0
Kb1**=0
Kc*-**=Kb_s2/Kb_s1
Kb2=0
MuPAD graphics

Reduce to U-RL
Model: U-2R-RL
Total_R=0.001
Ka=100000.0,   Kd=0.00001
Kb1*=0
Kb1**=0
Kc*-**=Kb_s2/Kb_s1
Kb2=2
MuPAD graphics

Reduce to U-R (use R*)
Model: U-2R-RL
Total_R=0.001
Ka=100000.0,   Kd=0.00001
Kb1*=2
Kb1**=0
Kc*-**=Kb_s2/Kb_s1
Kb2=0
MuPAD graphics

Reduce to U-R (use R**)
Model: U-2R-RL
Total_R=0.001
Ka=100000.0,   Kd=0.00001
Kb1*=0.000001
Kb1**=2
Kc*-**=2000000.0
Kb2=0
MuPAD graphics

Full model
-------------
Model: U-2R-RL
Total_R=0.001
Ka=100000.0,   Kd=0.00001
Kb1*=2
Kb1**=4
Kc*-**=2
Kb2=2
MuPAD graphics

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Conclusion:

 

The model works as expected