EKM 31

Analysis of the thermodynamic equilibrium relationships for a ligand binding to two different receptors

U__U model

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Goals

Here I will analyze numeric solutions I derived in U__U_derivation.mn for equilibrium between two different receptors competing for the same ligand.

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Clean up

reset()

Path to previous results

ProjectName:="U__U";

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

Assume some values for testing operation

Total_P:=1e-3:
Total_L:=10e-3:
Eq_L:=9e-3:
Ka1:=1e3:
Ka2:=2e4:
Total_R:=0.000006138735421:

test operation of all procedures

pnLeq_U_U(Total_R, Total_P, Total_L, Ka1, Ka2);
pnReq_U_U(Total_R, Total_P, Total_L, Ka1, Ka2);
pnPeq_U_U(Total_R, Total_P, Total_L, Ka1, Ka2);
pnRLeq_U_U(Total_R, Total_P, Total_L, Ka1, Ka2);
pnPLeq_U_U(Total_R, Total_P, Total_L, Ka1, Ka2);

=> OK

Make wrapper functions for plotting

fnLeq:=Total_L -> pnLeq_U_U(Total_R, Total_P, Total_L, Ka1, Ka2):
fnReq:=Total_L -> pnReq_U_U(Total_R, Total_P, Total_L, Ka1, Ka2):
fnPeq:=Total_L -> pnPeq_U_U(Total_R, Total_P, Total_L, Ka1, Ka2):
fnRLeq:=Total_L -> pnRLeq_U_U(Total_R, Total_P, Total_L, Ka1, Ka2):
fnPLeq:=Total_L -> pnPLeq_U_U(Total_R, Total_P, Total_L, Ka1, Ka2):

Test plotting

Total_R:=1e-3:
Total_P:=1e-3:
Total_L_max:=2e-3:
Ka1:=1e3:
Ka2:=1e3:

LineW:=1.5: //line width

// create plots

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

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

plot(pLeq, pRLeq, LegendVisible=TRUE)

=> OK

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Assume some constants and evaluate titrations

Total_R:=1e-3:
Total_P:=1e-3:
Ka1:=1e4:
Ka2:=1e4:

Total_L_max:=2e-3: // plotting range

LineW:=1.5: // plot line width

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

pReq:=  plot::Function2d(
Function=(fnReq),
LegendText="[R]",
Color = RGB::Black,
XMin=(0),
XMax=(Total_L_max),
XName=(L),
TitlePositionX=(0),
LineWidth=LineW):

pPeq:=  plot::Function2d(
Function=(fnPeq),
LegendText="[P]",
Color = RGB::Magenta,
XMin=(0),
XMax=(Total_L_max),
XName=(L),
TitlePositionX=(0),
LineWidth=LineW):

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

pPLeq:=  plot::Function2d(
Function=(fnPLeq),
LegendText="[PL]",
Color = RGB::Green,
XMin=(0),
XMax=(Total_L_max),
XName=(L),
TitlePositionX=(0),
LineWidth=LineW):

// Text report
print(Unquoted,"Model: ".ProjectName);
print(Unquoted,"Total_R=".Total_R);
print(Unquoted,"Total_P=".Total_P);
Kd1:=1/Ka1:
print(Unquoted,"Ka1=".Ka1." 1/M,   Kd1=".Kd1." M");
Kd2:=1/Ka2:
print(Unquoted,"Ka2=".Ka2." 1/M,   Kd2=".Kd2." M");

// plot all together
plot(//pLeq,
pReq, pPeq, pRLeq, pPLeq,
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]);

Model: U__U
Total_R=0.001
Total_P=0.001
Ka1=10000.0 1/M,   Kd1=0.0001 M
Ka2=10000.0 1/M,   Kd2=0.0001 M

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Test of the model: titration of a mixture of P and R with L

 R only (P not present) P only (R not present) Both P and R present at identical concentrations Amount of R is reduced two-fold. Model: U_UTotal_R=0.001Total_P=0.0Ka1=20000.0 1/M,   Kd1=0.00005 MKa2=10000.0 1/M,   Kd2=0.0001 M Model: U_UTotal_R=0.0Total_P=0.001Ka1=20000.0 1/M,   Kd1=0.00005 MKa2=10000.0 1/M,   Kd2=0.0001 M Model: U_UTotal_R=0.001Total_P=0.001Ka1=20000.0 1/M,   Kd1=0.00005 MKa2=10000.0 1/M,   Kd2=0.0001 M Model: U_UTotal_R=0.0005Total_P=0.001Ka1=20000.0 1/M,   Kd1=0.00005 MKa2=10000.0 1/M,   Kd2=0.0001 M Conclusion:   1. Normal saturation curve of RL Conclusion:   1. Normal saturation curve of PL.   2. Addinity of P to L is weaker so we have lower saturation. Conclusion:   1. Overall degree of saturation of RL and PL is smaller than when a competitor is not present.   2. R is binding to L twice as tight. Therefore, degree of saturation of RL is higher than of PL.   3. Ratio between RL and PL is not constant---a consequence of binding reactions being bimolecular and both dependent on concentration of L. Conclusions:   1. Total concentration of binding sites for L is reduced---reflected by higher [L].   2. Total concentration of PL is increased due to reduced competition---fractional saturation of PL is also increased.   3. Fractional saturation of RL is increased at lower concentration of R because of higher ratio of free L to free R For C(L)=0.0012 ---from 65% at 1 mM R to 80% at 0.5 mM R).

Saturation of R with L at different concentrations of R

print(Unquoted,"Plot a comparative series for fractional saturation of RL at different total concentrations of R (concentration of P is held constant).");

Total_R1:=1.5e-3:
print(Unquoted,"Total_R1=".Total_R1." M");
Total_R2:=1.0e-3:
print(Unquoted,"Total_R2=".Total_R2." M");
Total_R3:=0.5e-3:
print(Unquoted,"Total_R3=".Total_R3." M");

// Other parameters
Total_P:=1e-3:
Ka1:=1e4:
Ka2:=1e5:
print(Unquoted,"Total_P=".Total_P." M");
Kd1:=1/Ka1:
print(Unquoted,"Ka1=".Ka1." 1/M,   Kd1=".Kd1." M");
Kd2:=1/Ka2:
print(Unquoted,"Ka2=".Ka2." 1/M,   Kd2=".Kd2." M");

// Plot parameters
Total_L_max:=2e-3: // plotting range
LineW:=1.5: // plot line width

print(Unquoted,"Model: ".ProjectName);

// Define normalized functions [RL]/Rtot, [PL]/Ptot and [L]/Ltot
fnRLeqNorm1:=Total_L -> pnRLeq_U_U(Total_R1, Total_P, Total_L, Ka1, Ka2)/Total_R1:
fnRLeqNorm2:=Total_L -> pnRLeq_U_U(Total_R2, Total_P, Total_L, Ka1, Ka2)/Total_R2:
fnRLeqNorm3:=Total_L -> pnRLeq_U_U(Total_R3, Total_P, Total_L, Ka1, Ka2)/Total_R3:

// Generate three plots

pRLeq1:=  plot::Function2d(
Function=(fnRLeqNorm1),
LegendText="[RL1]",
Color = RGB::Red,
XMin=(0),
XMax=(Total_L_max),
XName=(L),
TitlePositionX=(0),
LineWidth=LineW):

pRLeq2:=  plot::Function2d(
Function=(fnRLeqNorm2),
LegendText="[RL2]",
Color = RGB::Green,
XMin=(0),
XMax=(Total_L_max),
XName=(L),
TitlePositionX=(0),
LineWidth=LineW):

pRLeq3:=  plot::Function2d(
Function=(fnRLeqNorm3),
LegendText="[RL3]",
Color = RGB::Blue,
XMin=(0),
XMax=(Total_L_max),
XName=(L),
TitlePositionX=(0),
LineWidth=LineW):

// plot all together
plot(pRLeq1, pRLeq2, pRLeq3,
YAxisTitle="[X]", Header=("Model: ".ProjectName.", Normalized Y, variable R"),
Height=160, Width=160,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=1);

Plot a comparative series for fractional saturation of RL at different total co\
ncentrations of R (concentration of P is held constant).
Total_R1=0.0015 M
Total_R2=0.001 M
Total_R3=0.0005 M
Total_P=0.001 M
Ka1=10000.0 1/M,   Kd1=0.0001 M
Ka2=100000.0 1/M,   Kd2=0.00001 M
Model: U__U

Conclusion:  Upon reducing total concentration of R we have higher degree of saturaton of R (and P) with L due to reduction in total amount of binding sites in the solution.

Plot a comparative series for fractional saturation of RL at different total co\
ncentrations of R (concentration of P is held constant).
Total_R1=0.0015 M
Total_R2=0.001 M
Total_R3=0.0005 M
Total_P=0.001 M
Ka1=20000.0 1/M,   Kd1=0.00005 M
Ka2=10000.0 1/M,   Kd2=0.0001 M
Model: U_U

Switched affinities

Plot a comparative series for fractional saturation of RL at different total co\
ncentrations of R (concentration of P is held constant).
Total_R1=0.0015 M
Total_R2=0.001 M
Total_R3=0.0005 M
Total_P=0.001 M
Ka1=10000.0 1/M,   Kd1=0.0001 M
Ka2=100000.0 1/M,   Kd2=0.00001 M
Model: U__U

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