[ON]’ = kH[Hbif]m[OFF] - kF[FimE]n[ON]

[OFF]’ = kF[FimE]n[ON] - kH[Hbif]m[OFF]

[FimE]’ = 0

[Hbif]’ = 0

Reduction:

[ON], [OFF], [FimE], [Hbif], kH, kF ≥ 0

[ON] + [OFF] = 1

Reduced Model:

[ON]’ = kH[Hbif]m(1 - [ON]) - kF[FimE]n[ON] = kH[Hbif]m - kH[Hbif]m[ON] - kF[FimE]n[ON]

= kH[Hbif]m - (kH[Hbif]m + kF[FimE]n)[ON]

[Hbif]’ = 0

[FimE]’ = 0

Solution:

[FimE] = [FimE]0 = FimE

[Hbif] = [Hbif]0 = Hbif

[ON]

= e-kH•Hbif^m•t - kF•FimE^n•t([ON]0 - (kH•Hbifm)/(kH•Hbifm + kF•FimEn)) + (kH•Hbifm)/(kH•Hbifm + kF•FimEn)

[OFF] = 1 - [ON]

= e-kH•Hbif^m•t - kF•FimE^n•t((kH•Hbifm)/(kH•Hbifm + kF•FimEn) - [ON]0) + 1 - (kH•Hbifm)/(kH•Hbifm + kF•FimEn)

Steady State:

0 = kH•Hbifm - (kH•Hbifm + kF•FimEn)[ON]

kH•Hbifm = (kH•Hbifm + kF•FimEn)[ON]

• if kH•Hbifm + kF•FimEn ≠ 0:

[ON]∞ = kH•Hbifm/(kH•Hbifm + kF•FimEn) = 1 - kF•FimEn/(kH•Hbifm + kF•FimEn)

[OFF]∞ = 1 - kH•Hbifm/(kH•Hbifm + kF•FimEn) = kF•FimEn/(kH•Hbifm + kF•FimEn)

• if kH•Hbifm = kF•FimEn = 0:

[ON]∞ = [ON]0

[OFF]∞ = 1 - [ON]∞ = 1 - [ON]0 = [OFF]0

• otherwise, there’s no steady state due to unrealistic values

Stability:

d[ON]’/d[ON] = - kH•Hbifm - kF•FimEn < 0

d[OFF]’/d[OFF] = d(1 - [ON])’/d(1 - [ON]) = d[ON]’/d[ON] = - kH•Hbifm - kF•FimEn < 0

Calibration:

0 = kH•Hbifm - (kH•Hbifm + kF•FimEn)[ON]∞

(kH•Hbifm + kF•FimEn)[ON]∞ = kH•Hbifm

kF•FimEn[ON]∞ = kH•Hbifm - kH•Hbifm[ON]∞ = kH•Hbifm(1 - [ON]∞)

kF =  kH•Hbifm(1 - [ON]∞)/(FimEn[ON]∞) = kH•Hbifm([OFF]∞)/(FimEn(1 - [OFF]∞))

• if [ON] ≠ [ON]∞:

        [ON] = e-kH•Hbif^m•t - kF•FimE^n•t([ON]0 - [ON]∞) + [ON]∞

        kH•Hbifm + kF•FimEn = -ln(([ON] - [ON]∞)/([ON]0 - [ON]∞))/t

        = kH•Hbifm + kH•Hbifm(1 - [ON]∞)/([ON]∞)

= kH•Hbifm(1 + (1 - [ON]∞)/([ON]∞)) = kH•Hbifm/[ON]∞

• otherwise not calibratable

Optimization:

• if kH•Hbifm + kF•FimEn ≠ 0:

d[ON]∞/dkH = Hbifm•kF•FimEn/(kH•Hbifm + kF•FimEn)2 = 0

d[OFF]∞/dkH = -Hbifm•kF•FimEn/(kH•Hbifm + kF•FimEn)2  = 0

kF•FimEn•Hbifm = 0

d[ON]∞/dkF = -FimEn•kH•Hbifm/(kH•Hbifm + kF•FimEn)2 = 0

d[OFF]∞/dkF = FimEn•kH•Hbifm/(kH•Hbifm + kF•FimEn)2 = 0

kH•Hbifm•FimEn = 0

d[ON]∞/dm = ln(Hbif)kH•Hbifm•kF•FimEn/(kH•Hbifm + kF•FimEn)2 = 0

d[OFF]∞/dm = -ln(Hbif)kH•Hbifm•kF•FimEn/(kH•Hbifm + kF•FimEn)2 = 0

ln(Hbif)kH•kF•Hbifm•FimEn = 0

d[ON]∞/dn = -ln(FimE)kF•FimEn•kH•Hbifm/(kH•Hbifm + kF•FimEn)2 = 0

d[OFF]∞/dn = ln(FimE)kF•FimEn•kH•Hbifm/(kH•Hbifm + kF•FimEn)2 = 0

ln(FimE)kF•kH•FimEn•Hbifm = 0

kF•FimEn = 0 or kH•Hbifm = 0

• otherwise not optimizable