Wun-Sin Jhang, Shih-Chiang Lo, Chen-Chao Yeh,Che-Chi Shu*

Supplementary Material

Inhibitors Alter the Stochasticity of Regulatory Proteins to Force Cells to Switch to the Other State in the Bistable System Wun-Sin Jhang&, Shih-Chiang Lo&, Chen-Chao Yeh, Che-Chi Shu*

Table S1: The reactions of the system

DA1揪kRA1 井 R 1+ DA 1 (1)

DI1揪kRI1 � R 1 DI 1 (2)

R1揪kP1 � P 1 R 1 (3)

k 2P 1揪P12 井 P 1 k 揪 2 (4) - P12

k P1+ P 1揪P13 井 P 1 2k 揪 3 (5) - P13

揪kDI 2 井 P13 + DA 2 揪 DI 2 (6) k- DI 2

DNAy揪kRy � RNAy DNAy (7)

RNAy揪kPy � RNAy Py (8)

揪 kY P13 + Py Y (9) k-Y

DA2揪kRA2 井 R 2+ DA 2 (10)

DI2揪kRI 2 井 R 2+ DI 2 (11)

R2揪kP 2 � P 2 R 2 (12)

k 2P 2揪P 22 井 P 2 k 揪 2 (13) - P22

k P2+ P 2揪P23 井 P 2 2k 揪 3 (14) - P 23

1 揪kDI 1 井 P23 + DA 1 揪 DI 1 (15) k- DI1

DNAx揪kRx � RNAx DNAx (16)

RNAx揪kPx � RNAx Px (17)

揪 kX P23 + Px X (18) k- X

For Table S1, the reactions 1st and 2nd describe the transcription of the active and inactive conformation of gene 1, respectively. The reaction 3rd indicates the translation of P1. The reactions 4th and 5th depict the process of protein monomer to dimer, dimer to trimer, respectively. The subscript 2 and 3 indicates the dimer and the trimer. The reaction 6th describes the configuration change of gene 2 from active to inactive due to the binding of P1 trimer. The reactions 7th and 8th are the production of inhibitor Py, the transcription and translation processes, respectively. The interaction of inhibitor Py to the trimer of P1 is described by reaction 9 th where

Y is the complex formed by Py and P1 timer. The reactions 10th to 18th are for gene 2. They include the expression of gene 2, the repression of gene 1 by P2 trimer and the complex X formed by Px and P2 trimer.

Table S2: The mass-action equations of the system d[ DA 1] =k[ DI 1] - k [ P 2 ][ DA 1] (1) dt -DI1 DI 1 3 d[ DI 1] =k[ P 2 ][ DA 1] - k [ DI 1] (2) dt DI1 3- DI 1 d[ R 1] =k[ DA 1] + k [ DI 1] - ( k + m )[ R 1] (3) dt RA1 RI 1 dR 1 d[ P 1] 2 =k[ R 1] - 2 k [ P 1] + 2 k [ P 1 ] - k [ P 1 ][ P 1] + k [ P 13 ] - ( k + m )[ P 1] dt P1 P 12- P 1 2 2 P 1 3 2 - P 1 3 dP 1 (4)

2 d[ P 1 ] 2 =k[ P 1]2 - k [ P 1 ] - k [ P 1 ][ P 1] + k [ P 1 ] - ( k + m )[ P 1 ] (5) dt P12- P 1 2 2 P 1 3 2 - P 1 3 3 dP 1 2 2 d[ P 1 ] 3 =k[ P 1 ][ P 1] - k [ P 1 ] - k [ P 1 ][ DA 2] + k [ DI 2] - k [ P 1 ][ Py ] + k [ Y ] - ( k + m )[ P 1 ] dt P13 2- P 1 3 3 DI 2 3 - DI 2 Y 3 - Y dP 1 3 3 (6) d[ Ry ] =k[ DNAy ] - ( k + m )[ Ry ] dt Ry dRy (7)d[ Py ] =k[ Ry ] - k [ P 1 ][ Py ] + k [ Y ] - ( k + m )[ Py ] dt Py Y3 - Y dPy (8)d[ Y ] =k[ P 1 ][ Py ] - k [ Y ] - ( k + m )[ Y ] (9) dt Y3 - Y dY d[ DA 2] =k[ DI 2] - k [ P 1 ][ DA 2] dt -DI2 DI 2 3 (10)d[ DI 2] =k[ P 1 ][ DA 2] - k [ DI 2] (11) dt DI2 3- DI 2 d[ R 2] =k[ DA 2] + k [ DI 2] - ( k + m )[ R 2] dt RA2 RI 2 dR 2 (12)d[ P 2] =k[ R 2] - 2 k [ P 2]2 + 2 k [ P 2 ] - k [ P 2 ][ P 2] + k [ P 2 ] - ( k +m + k )[ P 2] dt P2 P 22- P 2 2 2 P 2 3 2 - P 2 3 3 dP 2 r (13)d[ P 2 ] 2 =k[ P 2]2 - k [ P 2 ] - k [ P 2 ][ P 2] + k [ P 2 ] - ( k + m )[ P 2 ] dt P22- P 2 2 2 P 2 3 2 - P 2 3 3 dP 2 2 2 (14)d[ P 2 ] 3 =k[ P 2 ][ P 2] - k [ P 2 ] - k [ P 2 ][D A 1] + k [ DI 1] - k [ P 2 ][ Px ] + k [ X ] - ( k + m )[ P 2 ] dt P23 2- P 2 3 3 DI 1 3 - DI 1 X 3 - X dP 2 3 3 (15) d[ Rx ] =k[ DNAx ] - ( k + m )[ Rx ] (16) dt Rx dRx d[ Px ] =k[ Rx ] - k [ P 2 ][ Px ] + k [ X ] - ( k + m )[ Px ] (17) dt Px X3 - X dPx d[ X ] =k[ P 2 ][ Px ] - k [ X ] - ( k + m )[ X ] (18) dt X3 - X dX

For Table S2, the 1st equation describes the DNA in an active conformation and the 2nd equation indicates an inactive conformation. The equation 3rd depicts the RNA of gene 1. The equations

4th to 6th are the changes of P1 monomer, dimer, and trimer. Note that the influence of Py on the trimer of P1 is included in equation 6th as fifth and sixth terms. The equations 7th and 8th describe the formation of RNA of gene Y as well as the inhibitor Py, respectively. The equation 9 th

3 describes the behavior of complex Y composed of P1 trimer and inhibitor Py. The equations 10th to 18th are similar to equations 1st to 9th but written for gene 2 and inhibitor Px.

Table S3: Nomenclature of the variables Annotation Description

DA1 DNA of gene 1 in active formation

DI1 DNA of gene 1 in inactive formation

R1 RNA from gene 1 P1 Protein encoded in gene 1

P12 P1 dimer

P13 P1 trimer DNAy Gene y Ry RNA from gene y Py Peptide Y which is the inhibitor to P1 trimer Y Complex Y

DA2 DNA of gene 2 in active formation

DI2 DNA of gene 2 in inactive formation

R2 RNA from gene 2 P2 Protein encoded in gene 2

P22 P2 dimer

P23 P2 trimer DNAx Gene x Rx RNA from gene x Px Peptide X which is the inhibitor to P2 trimer X Complex X

4 Table S4: The values of parameters

5 Parameter Description Value Units Ref. 6 -1 -1 31 3 kDI1 Binding rate constant of P2 trimer to promoter 1 10 M S * Dissociation rate constant of P trimer and k 2 10-3 S-1 31*3 -DI1 promoter 1 -1 -1 32 kRA1 Transcription rate constant of active gene 1 4.2x10 S

-2 -1 1 kRI1 Transcription rate constant of inactive gene 1 2x10 S *

-3 -1 30 kP1 Translation rate constant of P1 2.32x10 S k 2 M-1S-1 P12 Rate constant of forming P1 dimer 7.4x10 k -3 S-1 *2 -P12 Dissociation rate constant of P1 dimer 7.4x10 k 2 -1 -1 P13 Rate constant of forming P1 trimer 7.4x10 M S k -3 -1 2 -P13 Dissociation rate constant of P1 trimer 7.4x10 S *

6 -1 -1 35 3 kY Binding rate constant of P1 trimer to Py 10 M S * -3 -1 35 3 k-Y Unbinding rate constant of P1 trimer from Py 10 S * -3 -1 22 3 kRy Transcription rate constant of Ry from DNAy 10 S * k -2 -1 36 Py Translation rate constant of Py 10 S

6 -1 -1 31 3 kDI 2 Binding rate constant of P1 trimer to promoter 2 10 M S * Dissociation rate constant of P trimer and k 1 10-3 S-1 31*3 -DI 2 promoter 2 -1 -1 32 kRA2 Transcription rate constant of active gene 2 4.2x 10 S

-2 -1 1 kRI2 Transcription rate constant of inactive gene 2 2 x 10 S *

-3 -1 30 kP2 Translation rate constant of P2 2.32x10 S k 2 M-1S-1 P22 Rate constant of forming P2 dimer 7.4x10 k -3 S-1 *2 -P22 Dissociation rate constant of P2 dimer 7.4x10 k 2 M-1S-1 P23 Rate constant of forming P2 trimer 7.4x10 k -3 S-1 *2 -P 23 Dissociation rate constant of P2 trimer 7.4x10 6 -1 -1 35 3 kX Binding rate constant of P2 trimer to Px 10 M S * -3 -1 35 3 k- X Unbinding rate constant of P2 trimer from Px 10 S * -3 -1 22 3 kRx Transcription rate constant of Rx from DNAx 10 S * k -2 S-1 36 Px Translation rate constant of Px 10 -4 -1 kdR1 Degradation rate constant of R1 6.15x10 S

-6 -1 3 kdP1 Degradation rate constant of P1 1.63x10 S * k -4 S-1 37*3 dP12 Degradation rate constant of P1 dimer 10 k -4 S-1 37*3 dP13 Degradation rate constant of P1 trimer 10 -4 -1 30 kdRy Degradation rate constant of Ry 6.15x10 S k -6 -1 30 dPy Degradation rate constant of Py 1.63x10 S -4 -1 37 3 kdY Degradation rate constant of Y complex 10 S * -4 -1 kdR2 Degradation rate constant of R2 6.15x10 S

-6 -1 3 kdP2 Degradation rate constant of P2 1.63x10 S * k -4 S-1 37*3 dP22 Degradation rate constant of P2 dimer 10 k -4 S-1 37*3 dP23 Degradation rate constant of P2 trimer 10 6 -4 -1 30 kd Rx Degradation rate constant of Rx 6.15x10 S k -6 S-1 30 dPx Degradation rate constant of Px 1.63x10 -4 -1 37 3 kdX Degradation rate constant of X complex 10 S * μ Specific growth rate constant 3.85x10-4 S-1 30 *1 We took the transcription rate constant of the active gene from 32. In order to have the same ratio of active to inactive rate constant in 30, the rate constant of the inactive gene is decided. *2 We took the rate constant of the forward reaction from 33. In order to keep the equilibrium rate constant the same as that in 30, the rate constant of the reverse reaction is decided. *3 Instead of taking the whole value, we took only the order of the parameter value.

7 Text S1

Complex X is involved in two equations, Eqs (15) and (18), in Table S2. With the assumption that the only consumption of complex X is the reaction of dissociation, the equations were rewritten as follows.

d[ P 23 ] =kP2[ P 2 2 ][ P 2] - k- P 2 [ P 2 3 ] - k DI 1 [ P 2 3 ][D A 1] + k - DI 1 [ DI 1] - k X [ P 2 3 ][ Px ] + k - X [ X ] - ( k dP 2 + m )[ P 2 3 ] (S1) dt 3 3 3

d[ X ] =k[ P 2 ][ Px ] - k [ X ] (S2) dt X3 - X

d[ P 2 ] d[ X ] If the system reaches steady state, 3 and are zero. Then the mass balance becomes: dt dt

0=k [ P 2 ][ P 2] - k [ P 2 ] - k [ P 2 ][D A 1] + k [ DI 1] - k [ P 2 ][ Px ] + k [ X ] - ( k + m )[ P 2 ] P23 2- P 2 3 3 DI 1 3 - DI 1 X 3 - X dP 2 3 3 (S3)

0=kX [ P 23 ][ Px ] - k- X [ X ] (S4)

Then we substituted Eq (S4) into Eq (S3) and got Eq (S5).

0=k [ P 2 ][ P 2] - k [ P 2 ] - k [ P 2 ][D A 1] + k [ DI 1] - ( k + m )[ P 2 ] P23 2- P 2 3 3 DI 1 3 - DI 1 dP 2 3 3

Eq (S5) suggests that the concentration of complex X shows no effect on P2 trimer.