ORMAT Supporting Information
Model details and parameters
Results in Fig. 2 of the main context show the behaviors of the motif in Fig. 1a (upper right panel). The enzyme has two basic conformations with different activities, E and E’. Each conformation can bind to substrate S and product P. Below are the detailed ordinary differential equations (ODEs) describing the motif,
1)
where the parameters except the ones prescribed in Fig. 2 are set as , , kES_S = 10, kS_ES = 10, kE’S_S = 5, k1 = 4, kE’P_P = 20, kEP_E’P = 10, kE’P_EP = 10, kES_EP = 10, kEP_ES = 10, kout = 0.5, Etotal = 10.
For the stochastic simulation (Fig. 2a-b), [S] can be written as
2)
where z(t) refers to white noise drawn from a normal Gaussian distribution, and <z(t)z(t’)> = d(t-t’). For monochromatic sinuous [S] fluctuations, we have
3)
The value of A and can be read directly from Fig. 2c-e.
For the network showed in figure 3, the equations can be written as
4)
5)
6)
7)
8)
9)
Where the fluctuation is defined similarly as the last term in Eq. (S2) or (S3) depending on stochastic system or not. In Figure 2, the fluctuation is on substrate S while in this simulation, the fluctuation is on pH. Et = 6; k1 = 1; KM1 = 20; k2 = 5; KM2 = 20; k3 = 1; KM3 = 20; Vin = 0.5; Vout = 5; KMout = 10; a12 = 0.1; a21 = 0.1; a23 = 0.1; a32 = 0.1; pH012 = 8.2; pH023 = 7.8; a = 5; pH = 8; a1p = 1; ap1 = 1; a2p = 1; ap2 = 1; a3p = 1; ap3 = 1;
For the network shown in Fig 4a, the corresponding equations are
10)
11)
12)
13)
14)
Where n = 3; k21 = 1; Km1 = 16.4; Ki1 = 0.91; k22 = 1; Km2 = 34.4; Ki2 = 0.17; Vm = 2; Kmatp = 10; kin = 0.1; Vmout = 1; Kmout = 10; a12 = Amp(AmpIndex); a21 = 0.5; Tat = 2; pH0 = 7; a = 5.
For the metabolic networks mentioned in Fig. 5, they are abstracted from Threonine (F)/Lysin (F’) synthesis pathway. The equations to describe the network2 (Lower Panel) in Fig. 5 are
d[A]/dt = - vEAA1I - vEAA1III+vIn1
d[A1]/dt = vEAA1I + vEAA1III - vEA1B
d[B]/dt = vEA1B - vEBCI - vEBCII - vEBF’
d[C]/dt = vEBCI + vEBCII - vECD
d[D]/dt = vECD - vEDF
d[F’]/dt = vEBF’ - vOut2
d[EAA1I]/dt = -ka*[EAA1I] +kb*[EAA1III] +kfb*[EAA1I_F]-kf*[EAA1I]*[F]^4
d[EAA1I_[F]]/dt = -kfb*[EAA1I_F] + kf*[EAA1I]*[F]^4
d[EAA1III]/dt = ka*[EAA1I]-kb*[EAA1III]+kfb3*[EAA1III_F]-kf3*[EAA1III]*[F]^4
v[EAA1I] = k[EAA1I] * [EAA1I]t * ( [A] / (K[A]1 + [A]) )
v[EAA1III] = k[EAA1III] * [EAA1III]t * [A] / (K[A]3 + [A])
v[EA1B] = Vm[EA1B] * [A1] / (K[A1] + [A1])
v[EBCI] = k[EBCI] * [EBCI]t * [B] / ( K[B]1 + [B] )
v[EBCII] = k[EBCII] * [EBCII]t * [B] / ( K[B]2 + [B] )
v[ECD] = Vm[ECD] * [C] / ( K[C] + [C] )
v[EDF] = Vm[EDF] * [D] / (K[D] +[D])
vEBF’ = VmE[F’] * [B] / (K[F’] +[B])
vOut2 = (kOut2) * [F’]/(KMout2+[F’])
vIn1 = kIn1
[EBCII]t = [EBC]t - [EBCI] – [EBCI_F]
[EBCI]t = [EBCI]+[EBCI_F]
[EBCII_Thr] = [EBC]t – [EBCI] – [EBCI_F]-[EBCII]
[EAA1III]t = [EAA1]t- [EAA1I]- [EAA1I_F]
[ EAA1I]t = [EAA1I]+ [EAA1I_F]
[EAA1III_F] = [EAA1]t- [EAA1I]- [EAA1I_F]- [EAA1III]
where kf = 100; kf3 = 100; keb = 1; ke = 1; ke3 = 0.1; kb = ka*keb; kfb = kf*ke; kfb3 = kf3*ke3; k[EAA1]I = 0.6; k[EAA1]III = 0.06; k[EBC]I = 0.4; k[EBC]II = 0.04; Vm[EA1B] = 0.25; Vm[ECD] = 0.4; Vm[EDF] = 0.2; VmE[F’] = 0.4; K[A]1 = 0.3; K[A]3 = 0.3; K[A1] = 0.2; K[B]1 = 0.25; K[B]2 = 0.25; K[C] = 0.1; K[D] = 0.3; K[F’] = 0.5; kOut1 = 0.4; KMout1 = 4; kOut2 = 0.6; KMout2 = 4; kIn1 = 0.1; ka (Enzyme conformation change rate) is changeable in the model. [F] is fluctuating as indicated in the Fig. 5.
The equations describing the Upper panel of Fig. 5 are
dAsp/dt = - vAKI - vAKIII+vIn1*x
daspp/dt = vAKI + vAKIII - vASD
dASA/dt = vASD - vHDHI - vHDHII - vELys
dhs/dt = vHDHI + vHDHII - vHK
dhsp/dt = vHK - vTS
dLys/dt = vELys - vOut2
dHDHI/dt = -kaH*HDHI+kbH*HDHII+kfbH*HDHIThr-kfH*HDHI*Thr^4
dHDHIThr/dt = -kfbH*HDHIThr + kfH*HDHI*Thr^4
dHDHII/dt = kaH*HDHI-kbH*HDHII+kfb3H*HDHIIThr-kf3H*HDHII*Thr^4
tHDHII(i) = HDHt - HDHI(i) - HDHIThr(i);
tHDHI(i) = HDHI(i)+HDHIThr(i);
HDHIIThr(i) = HDHt - HDHI(i) - HDHIThr(i)-HDHII(i);
vAKI = kAKI * tAKI * ( Asp / (KAsp1 + Asp) )
vAKIII = kAKIII * tAKIII * Asp / (KAsp3 + Asp)
vASD = VmASD * aspp / (Kaspp + aspp)
vHDHI = kHDHI *tHDHI * ASA / ( KASA1 + ASA )
vHDHII = kHDHII *tHDHII * ASA / ( KASA2 + ASA )
vHK = VmHK * hs / ( Khs + hs )
vTS = VmTS * hsp / (Khsp +hsp)
vELys = VmELys * ASA / (KLys +ASA)
vOut1 = (kOut1-Zeta) * Thr/(KMout1+Thr)
vOut2 = (kOut2) * Lys/(KMout2+Lys)
vIn1=kIn1
where x = 1; tAKI = 1; tAKIII = 0; AKt = 1; HDHt = 1; kaH = 1; kfH = 100; kf3H = 100; kebH = 1; keH = 1; ke3H = 0.1; kbH = kaH*kebH; kfbH = kfH*keH; kfb3H = kf3H*ke3H; kAKI = 0.6; kAKIII = 0.06; kHDHI = 0.4; kHDHII = 0.04; VmASD = 0.25; VmHK = 0.4; VmTS = 0.2; VmELys = 0.4; KAsp1 = 0.3; KAsp3 = 0.3; Kaspp = 0.2; KASA1 = 0.25; KASA2 = 0.25; Khs = 0.1; Khsp = 0.3; KLys = 0.5; kOut1 = 0.4; KMout1 = 4; kOut2 = 0.6; KMout2 = 4; kIn1 = 0.1;
The related equations for lower panal of Fig. 6a and Fig. 6d are
dE/dt = ( kES_S*ES-kS_ES*S*E + b*Ep -a*E );
dEp/dt = ( - b*Ep +a*E +kEpS_S*EpS - kS_EpS*S*Ep);
dP/dt = ( kEpP_P * EpP - kout*P );
dES/dt = ( - kES_S*ES +kS_ES*S*E + be*EpS -ae*ES + kEP_ES *EP - kES_EP*ES );
dEP/dt = ( -kEP_ES *EP + kES_EP*ES - kEP_EpP*EP + kEpP_EP*EpP );
dEpP/dt = ( kEP_EpP*EP - kEpP_EP*EpP - kEpP_P * EpP );
EpS = ET - E - Ep - ES - EP -EpP;
dP2/dt = ( VmaxP2*(1/(1 + (1/x0 - 1)*exp(-r*(P-t0))))*S2/( KMP2 + S2 ) - koutP2*P2 );
where S and S2 have the similar forms as in Fig. 2. Be=ae=0.5; kES_S=10; kS_ES=10;
kEpS_S = 5; kS_EpS = kEpS_S*0.8; kEpP_P = 20; kEP_EpP = 10; kEpP_EP = 10; kES_EP = 10; kEP_ES = 10; kout = 2; VmaxP2 = 30; KMP2 = 10; KI = 10; koutP2 = 2; r = 200; x0 = 0.00001; t0=1.86;
The corresponding equations for upper panel of Fig. 6a and Fig. b-c are just doubled from the single “standard” module as shown in Fig. 1.
All simulations are performed with Matlab. The codes are available upon request.
Supporting Figure Legends
Figure S1 Detailed signal dependence of the two metabolic networks in Fig 5. (a) The dependence of F’ variation on F fluctuation frequency shows high frequency filtering with network 1, and resonance with network 2. (b) D[F’] versus F fluctuation duration at different enzyme conformational change rates g. With network 1, D[F’] with different g values converge to the same value upon increasing the F fluctuation duration. With network 2, D[F’] is larger with larger g values . (c) D[F’] versus g at different F fluctuation duration. With network 1, the D[F’] value is less sensitive to g with more sustained F fluctuations. With network 2, the D[F’] value is sensitive to g with both short and more sustained F fluctuations. (d) D[F’] versus the increasing rate of F fluctuations. Network 2, but not network 1, response to the increasing rate of F changes.
Figure S1
1