Mathematical Modeling of Bistable Toggle Switch

Clear["Global`*"]
k1 = 100;
d1 = 0.007;
k2 = 100;
d2 = 0.007;
K1 = k1/d1;
K2 = k2/d2;
Nns = 5*10^6;
delEpd = -1.0421*10^-19;
kb = 1.3806*10^-23;
T = 298;
P = 3000;
\[Alpha] = Nns*E^(delEpd/(kb*T))/P;
n1 = 5;
n2 = 0;
(*Ksrd=;*)
Knsrd = 10000;
Stable1 = 0;
Saddle1 = 0;
Unstable1 = 0;
matUp = {};
matDown = {};
RangeUp = {};
RangeDown = {};
For[Ksud = 0.1*10^-9, Ksud <= 7000*10^-9, Ksud = Ksud*1.1,
Print[Ksud]
 For[Ksvd = 0.1*10^-9, Ksvd <= 7000*10^-9, Ksvd = Ksvd*1.1,
  Ksudweak = Ksud*10;
  Ksvdweak = Ksvd*10;
  Stable2 = 0;
  Saddle2 = 0;
  Unstable2 = 0;
  delEud = kb*T*Log[10, Ksud/Knsrd];
  delEvd = kb*T*Log[10, Ksvd/Knsrd];
  delEudweak = kb*T*Log[10, Ksudweak/Knsrd];
  delEvdweak = kb*T*Log[10, Ksvdweak/Knsrd];
  xdot[x_, y_] := 
   K1/(1 + \[Alpha] ((1 + (y/Nns)*E^(-delEvd/(kb*T)))^
          n1)*((1 + (y/Nns)*E^(-delEvdweak/(kb*T)))^n2)) - x; 
  ydot[x_, y_] := 
   K2/(1 + \[Alpha] ((1 + (x/Nns)*E^(-delEud/(kb*T)))^
          n1)*((1 + (x/Nns)*E^(-delEudweak/(kb*T)))^n2)) - y; 
  Jac[x_, 
    y_] := {{Derivative[1, 0][xdot][x, y], 
     Derivative[0, 1][xdot][x, y]}, {Derivative[1, 0][ydot][x, y], 
     Derivative[0, 1][ydot][x, y]}};
  fps = NSolve[xdot[x, y] == 0 && ydot[x, y] == 0, {x, y}, Reals];
  (*Print[N[fps,3]]*)
  (*Print[Length[fps]-1]*)
  For[i = 1, i < Length[fps] + 1, i++,
    xtemp = x /. fpsi;
    ytemp = y /. fpsi;
    (*Print[N[xtemp,3]]*)
    (*Print[N[ytemp,3]]*)
    If[Im[xtemp] == 0 && Im[ytemp] == 0,
     If[xtemp > 0 && ytemp > 0,
      (*Print["              ",i-1,"  R1=",xtemp,"  R2=",ytemp, 
      "  Ksud=",Ksud,"   Ksvd=", Ksvd]*)
      If[
       Re[Eigenvalues[Jac[xtemp, ytemp]]1] > 0 && 
        Re[Eigenvalues[Jac[xtemp, ytemp]]2] > 0,
       Unstable2 += 1;
       ];
      If[
       Re[Eigenvalues[Jac[xtemp, ytemp]]1] < 0 && 
        Re[Eigenvalues[Jac[xtemp, ytemp]]2] < 0,
       Stable2 += 1;
       ];
      If[
       Re[Eigenvalues[Jac[xtemp, ytemp]]1] < 0.0001 && 
        Re[Eigenvalues[Jac[xtemp, ytemp]]2] > 0.0001,
       Saddle2 += 1;
       ];
      If[
       Re[Eigenvalues[Jac[xtemp, ytemp]]1] > 0.0001 && 
        Re[Eigenvalues[Jac[xtemp, ytemp]]2] < 0.0001,
       Saddle2 += 1;
       ];
      ]
     ]
    ]
   (*Print[Stable2, "  ", Stable1]*)
   If[ (Stable2 == 2 && Stable1 == 1),
    Print["Switch  (UP)"];
    (*Print[{\[Alpha],\[Beta]}];*)
    matUp = Insert[matUp, {Log[10, Ksud], Log[10, Ksvd]}, 1];
    RangeUp = Insert[RangeUp, {Ksud, Ksvd}, 1];
    ];
  If[ ((Stable2 == 1 && Stable1 == 2)),
   Print["Switch (DOWN)"];
   (*Print[{\[Alpha],\[Beta]}];*)
   matDown = Insert[matDown, {Log[10, Ksud], Log[10, Ksvd]}, 1];
   RangeDown = Insert[RangeDown, {Ksud, Ksvd}, 1];
   ];
  If[(Stable2 == 2),
   (*Print["BISTABLE ", \[Alpha], "   ",\[Beta], ",  Saddle: ",
   Saddle2 ];*)
   ];
  If[ (Stable2 == 1),
   (*Print["MONOSTABLE ",  \[Alpha],"   ",\[Beta],",  Saddle: ",
   Saddle2 ];*)
   ];
  Saddle1 = Saddle2;
  Unstable1 = Unstable2;
  Stable1 = Stable2;
  ]
 ]






k1 = 100;
d1 = 0.007;
k2 = 100;
d2 = 0.007;
 K1 = k1/d1;
K2 = k2/d2;
Nns = 5*10^6;
delEpd = -1.0421*10^-19;
kb = 1.3806*10^-23;
T = 298;
P = 3000;
\[Alpha] = Nns*E^(delEpd/(kb*T))/P;
n1 = 5;
n2 = 0;
(*Ksrd=;*)
Knsrd = 10000;
Stable1 = 0;
Saddle1 = 0;
Unstable1 = 0;
matUp1 = {};
matDown1 = {};
RangeUp1 = {};
RangeDown1 = {};
For[Ksud = 0.1*10^-9, Ksud <= 7000*10^-9, Ksud = Ksud*1.1,
 Print[Ksud]
 For[Ksvd = 0.1*10^-9, Ksvd <= 7000*10^-9, Ksvd = Ksvd*1.1,
  Ksudweak = Ksud*10;
  Ksvdweak = Ksvd*10;
  Stable2 = 0;
  Saddle2 = 0;
  Unstable2 = 0;
  delEud = kb*T*Log[10, Ksud/Knsrd];
  delEvd = kb*T*Log[10, Ksvd/Knsrd];
  delEudweak = kb*T*Log[10, Ksudweak/Knsrd];
  delEvdweak = kb*T*Log[10, Ksvdweak/Knsrd];
  xdot[x_, y_] := 
   K1/(1 + \[Alpha] ((1 + (y/Nns)*E^(-delEvd/(kb*T)))^
          n1)*((1 + (y/Nns)*E^(-delEvdweak/(kb*T)))^n2)) - x + 300; 
  ydot[x_, y_] := 
   K2/(1 + \[Alpha] ((1 + (x/Nns)*E^(-delEud/(kb*T)))^
          n1)*((1 + (x/Nns)*E^(-delEudweak/(kb*T)))^n2)) - y + 300; 
  Jac[x_, 
    y_] := {{Derivative[1, 0][xdot][x, y], 
     Derivative[0, 1][xdot][x, y]}, {Derivative[1, 0][ydot][x, y], 
     Derivative[0, 1][ydot][x, y]}};
  fps = NSolve[xdot[x, y] == 0 && ydot[x, y] == 0, {x, y}, Reals];
  (*Print[N[fps,3]]*)
  (*Print[Length[fps]-1]*)
  For[i = 1, i < Length[fps] + 1, i++,
    xtemp = x /. fpsi;
    ytemp = y /. fpsi;
    (*Print[N[xtemp,3]]*)
    (*Print[N[ytemp,3]]*)
    If[Im[xtemp] == 0 && Im[ytemp] == 0,
     If[xtemp > 0 && ytemp > 0,
      (*Print["              ",i-1,"  R1=",xtemp,"  R2=",ytemp, 
      "  Ksud=",Ksud,"   Ksvd=", Ksvd]*)
      If[
       Re[Eigenvalues[Jac[xtemp, ytemp]]1] > 0 && 
        Re[Eigenvalues[Jac[xtemp, ytemp]]2] > 0,
       Unstable2 += 1;
       ];
      If[
       Re[Eigenvalues[Jac[xtemp, ytemp]]1] < 0 && 
        Re[Eigenvalues[Jac[xtemp, ytemp]]2] < 0,
       Stable2 += 1;
       ];
      If[
       Re[Eigenvalues[Jac[xtemp, ytemp]]1] < 0.0001 && 
        Re[Eigenvalues[Jac[xtemp, ytemp]]2] > 0.0001,
       Saddle2 += 1;
       ];
      If[
       Re[Eigenvalues[Jac[xtemp, ytemp]]1] > 0.0001 && 
        Re[Eigenvalues[Jac[xtemp, ytemp]]2] < 0.0001,
       Saddle2 += 1;
       ];
      ]
     ]
    ]
   (*Print[Stable2, "  ", Stable1]*)
   If[ (Stable2 == 2 && Stable1 == 1),
    Print["Switch  (UP)"];
    (*Print[{\[Alpha],\[Beta]}];*)
    matUp1 = Insert[matUp1, {Log[10, Ksud], Log[10, Ksvd]}, 1];
    RangeUp1 = Insert[RangeUp1, {Ksud, Ksvd}, 1];
    ];
  If[ ((Stable2 == 1 && Stable1 == 2)),
   Print["Switch (DOWN)"];
   (*Print[{\[Alpha],\[Beta]}];*)
   matDown1 = Insert[matDown1, {Log[10, Ksud], Log[10, Ksvd]}, 1];
   RangeDown1 = Insert[RangeDown1, {Ksud, Ksvd}, 1];
   ];
  If[(Stable2 == 2),
   (*Print["BISTABLE ", \[Alpha], "   ",\[Beta], ",  Saddle: ",
   Saddle2 ];*)
   ];
  If[ (Stable2 == 1),
   (*Print["MONOSTABLE ",  \[Alpha],"   ",\[Beta],",  Saddle: ",
   Saddle2 ];*)
   ];
  Saddle1 = Saddle2;
  Unstable1 = Unstable2;
  Stable1 = Stable2;
  ]
 ]







 k1 = 100;
d1 = 0.007;
k2 = 100;
d2 = 0.007;
K1 = k1/d1;
K2 = k2/d2;
Nns = 5*10^6;
delEpd = -1.0421*10^-19;
kb = 1.3806*10^-23;
T = 298;
P = 3000;
\[Alpha] = Nns*E^(delEpd/(kb*T))/P;
n1 = 5;
n2 = 0;
(*Ksrd=;*)
Knsrd = 10000;
Stable1 = 0;
Saddle1 = 0;
Unstable1 = 0;
matUp2 = {};
matDown2 = {};
RangeUp2 = {};
RangeDown2 = {};
For[Ksud = 0.1*10^-9, Ksud <= 7000*10^-9, Ksud = Ksud*1.1,
Print[Ksud]
 For[Ksvd = 0.1*10^-9, Ksvd <= 7000*10^-9, Ksvd = Ksvd*1.1,
  Ksudweak = Ksud*10;
  Ksvdweak = Ksvd*10;
  Stable2 = 0;
  Saddle2 = 0;
  Unstable2 = 0;
  delEud = kb*T*Log[10, Ksud/Knsrd];
  delEvd = kb*T*Log[10, Ksvd/Knsrd];
  delEudweak = kb*T*Log[10, Ksudweak/Knsrd];
  delEvdweak = kb*T*Log[10, Ksvdweak/Knsrd];
  xdot[x_, y_] := 
   K1/(1 + \[Alpha] ((1 + (y/Nns)*E^(-delEvd/(kb*T)))^
          n1)*((1 + (y/Nns)*E^(-delEvdweak/(kb*T)))^n2)) - x + 600; 
  ydot[x_, y_] := 
   K2/(1 + \[Alpha] ((1 + (x/Nns)*E^(-delEud/(kb*T)))^
          n1)*((1 + (x/Nns)*E^(-delEudweak/(kb*T)))^n2)) - y + 600; 
  Jac[x_, 
    y_] := {{Derivative[1, 0][xdot][x, y], 
     Derivative[0, 1][xdot][x, y]}, {Derivative[1, 0][ydot][x, y], 
     Derivative[0, 1][ydot][x, y]}};
  fps = NSolve[xdot[x, y] == 0 && ydot[x, y] == 0, {x, y}, Reals];
  (*Print[N[fps,3]]*)
  (*Print[Length[fps]-1]*)
  For[i = 1, i < Length[fps] + 1, i++,
    xtemp = x /. fpsi;
    ytemp = y /. fpsi;
    (*Print[N[xtemp,3]]*)
    (*Print[N[ytemp,3]]*)
    If[Im[xtemp] == 0 && Im[ytemp] == 0,
     If[xtemp > 0 && ytemp > 0,
      (*Print["              ",i-1,"  R1=",xtemp,"  R2=",ytemp, 
      "  Ksud=",Ksud,"   Ksvd=", Ksvd]*)
      If[
       Re[Eigenvalues[Jac[xtemp, ytemp]]1] > 0 && 
        Re[Eigenvalues[Jac[xtemp, ytemp]]2] > 0,
       Unstable2 += 1;
       ];
      If[
       Re[Eigenvalues[Jac[xtemp, ytemp]]1] < 0 && 
        Re[Eigenvalues[Jac[xtemp, ytemp]]2] < 0,
       Stable2 += 1;
       ];
      If[
       Re[Eigenvalues[Jac[xtemp, ytemp]]1] < 0.0001 && 
        Re[Eigenvalues[Jac[xtemp, ytemp]]2] > 0.0001,
       Saddle2 += 1;
       ];
      If[
       Re[Eigenvalues[Jac[xtemp, ytemp]]1] > 0.0001 && 
        Re[Eigenvalues[Jac[xtemp, ytemp]]2] < 0.0001,
       Saddle2 += 1;
       ];
      ]
     ]
    ]
   (*Print[Stable2, "  ", Stable1]*)
   If[ (Stable2 == 2 && Stable1 == 1),
    Print["Switch  (UP)"];
    (*Print[{\[Alpha],\[Beta]}];*)
    matUp2 = Insert[matUp2, {Log[10, Ksud], Log[10, Ksvd]}, 1];
    RangeUp2 = Insert[RangeUp2, {Ksud, Ksvd}, 1];
    ];
  If[ ((Stable2 == 1 && Stable1 == 2)),
   Print["Switch (DOWN)"];
   (*Print[{\[Alpha],\[Beta]}];*)
   matDown2 = Insert[matDown2, {Log[10, Ksud], Log[10, Ksvd]}, 1];
   RangeDown2 = Insert[RangeDown2, {Ksud, Ksvd}, 1];
   ];
  If[(Stable2 == 2),
   (*Print["BISTABLE ", \[Alpha], "   ",\[Beta], ",  Saddle: ",
   Saddle2 ];*)
   ];
  If[ (Stable2 == 1),
   (*Print["MONOSTABLE ",  \[Alpha],"   ",\[Beta],",  Saddle: ",
   Saddle2 ];*)
   ];
  Saddle1 = Saddle2;
  Unstable1 = Unstable2;
  Stable1 = Stable2;
  ]
 ]

RangeUp;
RangeDown;
ListPlot[{RangeUp, RangeDown}];
matUp;
matDown;
ListLinePlot[{matUp, matDown, matUp1, matDown1, matUp2, matDown2}, 
PlotStyle -> {Blue, Blue, Red, Red, Green, Green}, 
PlotRange -> {{-10, -5}, {-10, -5}}, InterpolationOrder -> 1, 
Mesh -> Full, MeshStyle -> Directive[Gray], 
AxesLabel -> {"Log(\!\(\*SubscriptBox[\(K\), \(d, Repressor\\\ \
1\)]\))", "Log(\!\(\*SubscriptBox[\(K\), \(d, Repressor\\\ 2\)]\))"}, 
PlotLabel -> 
 Style["Bistable Regions for Leaky Promoters", 
  Directive[Large, Bold]], LabelStyle -> Directive[Medium, Bold]]