  National Yang Ming University

## Explanation:

\frac{d[mRNACI]}{dt}= \frac{1-[LacI]^nLacI}{ KdLacI^nLacI+ [LacI]^nLacI} \times PoPSconstitutive\times\frac{N}{V}-KdegmRNA\times [mRNACI]

In this equation, PoPSpLac represents the promoter strength of LacI promoter, which is measured by the rate of RNApolymerase binding to the starting site of DNA transcription; \frac{1-[LacI]^nLacI }{ KdLacI^nLacI+[LacI]^nLacI } represents the hill effect of repressor LacI to LacI promoter. Because LacI is a repressor, the numerator is {1-[LacI]^nLacI } . For the section of the equation, \frac{1-[LacI]^nLacI}{ KdLacI^nLacI+ [LacI]^nLacI} \times PoPSconstitutive\times\frac{N}{V}represents the synthesizing rate of mRNACI under the influence of LacI and LacI promoter; -{kdegmRNA}\times {[mRNACI]} represents the degrading rate of mRNA

\frac{d[CI]}{dt}={RBS}\times{[mRNACI]} -KdegCI\times[CI]

In this equation, RBS represents ribosome binding site strength, which is the affinity of ribosome to the starting site of mRNA.

For the section of the equation, RBS\times [mRNAcI] represents the synthesizing rate of CI; -{kdegCI}\times {[CI]} represents the degrading rate of CI.

Since LuxCI hybrid promoter is CI dominant, the presence of CI block the promoter and kill protein is not produced. This situation happens when there is no Nosema in bees.

\frac{d[mRNALacI]}{dt}= \frac{1-[ROSoxyR]^nROSoxyR }{ KdROSoxyR ^nROSoxyR + [ROSoxyR]^nROSoxyR} \times {PoPStrxC}\times\frac{N}{V}-KdegmRNA[mRNALacI]

In this equation, PoPStrxC represents the promoter strength of promotertrxC, which is measured by the rate of RNApolymerase binding to the starting site of DNA transcription; \frac{1-[ROSoxyR]^nROSoxyR }{ KdROSoxyR ^nROSoxyR + [ROSoxyR]^nROSoxyR} represents the hill effect of activator ROSoxyR to trxCpromoter.

For the section of the equation, \frac{1-[ROSoxyR]^nROSoxyR }{ KdROSoxyR ^nROSoxyR + [ROSoxyR]^nROSoxyR} \times {PoPStrxC}\times\frac{N}{V} represents the synthesizing rate of mRNALacI under the influence of activator ROSoxyR complex and trxC promoter; -kdegmRNA\times [mRNAlacI] represents the degrading rate of mRNALacI.

\frac{d[LacI]}{dt}={RBS}\times{[mRNA LacI]} –{Kdeg LacI} \times{[LacI]}

In this equation, RBS represents ribosome binding site strength, which is the affinity of ribosome to the starting site of mRNA. For the section of the equation, RBS\times [mRNALacI] represents the synthesizing rate of LacI; -KdegCI\times [LacI] represents the degrading rate of LacI .

LacI will repress pLac promoter, which leads to no production of CI. Without CI, LuxCI hybrid promoter will no longer be repressed, and kill protein can thus be generated. This situation happens when Nosema exists and the sensor promoter is turned on.

\frac{d[mRNAOxyR]}{dt}=PoPSconstitutive\times\frac{N}{V}-KdegmRNA\times [mRNAOxyR]

In this equation, PoPSconstitutive represents the promoter strength of the constitutive promoter J23102, which is measured by the rate of RNApolymerase binding to the starting site of DNA transcription.

For the section of the equation, PoPSconstitutive\times\frac{N}{V} represents the synthesizing rate of mRNA"oxyR" ; -KdegmRNA\times [mRNAOxyR] represents the degrading rate of mRNA"oxyR" .

\frac{d[mRNALuxI]}{dt}= \frac{1-[ROSoxyR]^n ROSoxyR }{ Kd ROSoxyR ^n ROSoxyR + [ROSoxyR]^n ROSoxyR I} \times PoPStrxC\times\frac{N}{V}+\frac{1-[ LuxRAHL]^nLuxRAHL}{ KdLuxRAHL ^nLuxRAHL + [LuxRAHL]^nLuxRAHL}\times\frac{1-{1-[CI leak]^nCI}}{ KdCI^nCI+[CI]^nCI}\times PoPSLuxCI\times\frac{N}{V}-KdegmRNA[mRNALuxI]

In this equation, PoPStrxC,PoPSLuxcI represents the promoter strength of promoter trxC and LuxcI hybrid promoter, which is measured by the rate of RNApolymerase binding to the starting site of DNA transcription; \frac{1-[ROSoxyR]^n ROSoxyR }{ Kd ROSoxyR ^n ROSoxyR + [ROSoxyR]^n ROSoxyR I} represents the hill effect of activator ROSoxyR complex to trxCpromoter; \frac{1-[ LuxRAHL]^nLuxRAHL}{ KdLuxRAHL ^nLuxRAHL + [LuxRAHL]^nLuxRAHL} represents the hill effect of activator LuxRAHL to LuxcI hybrid promoter; \frac{1-[CI leak]^nCI}{ KdCI^nCI+[CI]^nCI} represents the hill effect of repressor CI, while frac{1-{1-[CI leak]^nCI}}{ KdCI^nCI+[CI]^nCI} takes CI leaking into consideration (promoter pLac will not close thoroughly even though CI exists).

For the section of the equation, \frac{1-[ROSoxyR]^n ROSoxyR }{ Kd ROSoxyR ^n ROSoxyR + [ROSoxyR]^n ROSoxyR I} \times PoPStrxC\times\frac{N}{V} represents the synthesizing rate of mRNALuxI under the influence of activator ROSoxyR complex and trxC promoter; \frac{1-[ LuxRAHL]^nLuxRAHL}{ KdLuxRAHL ^nLuxRAHL + [LuxRAHL]^nLuxRAHL}\times\frac{1-{1-[CI leak]^nCI}}{ KdCI^nCI+[CI]^nCI}\times PoPSLuxCI\times\frac{N}{V} represents the synthesizing rate of mRNALuxI under the influence of LuxRAHL complex, CI repression and leaking, and LuxcI hybrid promoter; -kdegmRNA\times [mRNAluxI]represents the degrading rate of mRNAluxI.

\frac{d[LuxI]}{dt}={RBS}\times{[mRNA LuxI]} -KAHL\times{[LuxI]} -Kdeg {[LuxI]} \times{[LuxI]}

In this equation, RBS represents ribosome binding site strength, which is the affinity of ribosome to the starting site of mRNA; KAHL represents the rate constant of AHL to LuxI. For the section of the equation, {RBS}\times {[mRNALuxI]} represents the synthesizing rate of LuxI; KAHL[LuxI] represents the transforming rate of AHL to LuxI; -KdegLuxI\times [LuxI] represents the degrading rate of LuxI.

LacI will repress pLac promoter, which leads to no production of CI. Without CI, LuxCI hybrid promoter will no longer be repressed, and kill protein can thus be generated. This situation happens when Nosema exist, and when the sensor promoter is turned on.

\frac{d[mRNALuxR]}{dt}= \frac{1-[ROSoxyR]^n ROSoxyR }{ Kd ROSoxyR ^n ROSoxyR + [ROSoxyR]^n ROSoxyR I} \times PoPStrxC\times\frac{N}{V}+\frac{1-[ LuxRAHL]^nLuxRAHL}{ KdLuxRAHL ^nLuxRAHL + [LuxRAHL]^nLuxRAHL}\times\frac{1-{1-[CI leak]^nCI}}{ KdCI^nCI+[CI]^nCI}\times PoPSLuxCI\times\frac{N}{V}-KdegmRNA[mRNALuxR]

In this equation, PoPStrxC,PoPSLuxcI represents the promoter strength of promoter trxC and LuxcI hybrid promoter, which is measured by the rate of RNApolymerase binding to the starting site of DNA transcription; \frac{1-[ROSoxyR]^n ROSoxyR }{ Kd ROSoxyR ^n ROSoxyR + [ROSoxyR]^n ROSoxyR I} represents the hill effect of activator ROSoxyR complex to trxCpromoter; \frac{1-[ LuxRAHL]^nLuxRAHL}{ KdLuxRAHL ^nLuxRAHL + [LuxRAHL]^nLuxRAHL} represents the hill effect of activator LuxRAHL to LuxcI hybrid promoter; \ frac{1-[CI leak]^nCI}{ KdCI^nCI+[CI]^nCI} represents the hill effect of repressor CI, while frac{1-{1-[CI leak]^nCI}}{ KdCI^nCI+[CI]^nCI} takes CI leaking into consideration (promoter pLac will not close thoroughly even though CI exists).

For the section of the equation, \frac{1-[ROSoxyR]^n ROSoxyR }{ Kd ROSoxyR ^n ROSoxyR + [ROSoxyR]^n ROSoxyR I} \times PoPStrxC\times\frac{N}{V} represents the synthesizing rate of mRNALuxR under the influence of activator ROSoxyR complex and trxC promoter; \frac{1-[ LuxRAHL]^nLuxRAHL}{ KdLuxRAHL ^nLuxRAHL + [LuxRAHL]^nLuxRAHL}\times\frac{1-{1-[CI leak]^nCI}}{ KdCI^nCI+[CI]^nCI}\times PoPSLuxCI\times\frac{N}{V} represents the synthesizing rate of mRNALuxR under the influence of LuxRAHL complex, CI repression and leaking, and LuxcI hybrid promoter; -kdegmRNA\times [mRNAluxI]represents the degrading rate of mRNAluxR.

\frac{d[LuxR]}{dt}={RBS}\times{[mRNA LuxR]} – {KmAHL}\times{[AHL]} - {Kdeg LuxR} \times{[LuxR]} \times {KmAHL}\{times[AHL]^{2}} \times{[LuxR]}- KdegAHL\times{[AHL]}

In this equation, RBS represents ribosome binding site strength, which is the affinity of ribosome to the starting site of mRNA; KmAHL represents the rate constant of AHLLuxI; KonAHL represents the rate constant of 2AHL+LuxR to AHL2LuxR complex.

For the section of the equation, { RBS}\times{ [mRNALuxR]} represents the synthesizing rate of LuxR; { -KmAHL}\times{ [AHL]} represents the transforming rate of AHL to LuxI; {-KdegLuxR}\times {[LuxR]} represents the degrading rate of LuxR; { -KonAHL}\times{ [AHL]^2}\times{ [LuxR]} represents the transforming rate of 2AHL+LuxR to AHL2LuxR complex; {-KdegLuxI}\times {[AHL]} represents the degrading rate of AHL.

\frac{d[AHL]}{dt}={KAHL}\times{[LuxI]}+2\times KoffAHL \times [AHLLuxR] - 2\times KonAHL \times {[AHL]}^{2}\times {[LuxR]} - KdegAHL\times{[AHL]}

In this equation, KAHL represents the rate constant of AHL to LuxI; KoffAHL represents the rate constant of AHL2LuxR complex to 2AHL+ LuxR, while KonAHL represents the reverse rate constant.

For the section of the equation, {KAHL}\times {[LuxI]} represents the transforming rate of LuxI to AHL; {2}\times {KoffAHL}\times{ [AHLLuxR]} represents the transforming rate of AHL2LuxR complex to AHL and LuxR, the constant 2 means that each complex turns into two AHL; -{2}\times {KonAHL}\times {[AHL]^2}\times {[LuxR]} represents the transforming rate of 2AHL+LuxR to AHL2LuxR complex; {-KdegLuxI}\times {[LuxI]} represents the degrading rate of LuxI.

Since LuxI will transform to AHL, and 2AHL will combine to LuxR to form AHL2LuxR complex. Because AHL2LuxR will then bind to LuxCI hybrid promoter and activate kill gene, LuxR gene, and LuxI gene. In this way, kill protein will kill Nosema, while LuxR, LuxI will trigger the positive feedback and enhance the production of kill protein.

\frac{d[kill]}{dt}={RBS}\times{[mRNAkill]} –Kdegkill\times{[kill]}

After trxC promoter is turned on by ROSoxyR complex and the downstream genes – LacI, LuxI, and LuxR starts to produce, which blocks the CI promoter and turns on the LuxCI hybrid promoter. After that, the gene encoded with kill protein downstream LuxCI hybrid promoter also begins to transcribe.

This equation shows production rate of mRNAkill. PoPSLuxcI represents the promoter strength of promoter trxC and LuxcI hybrid promoter, which is measured by the rate of RNApolymerase binding to the starting site of DNA transcription; \frac{1-[ LuxRAHL]^nLuxRAHL}{ KdLuxRAHL ^nLuxRAHL + [LuxRAHL]^nLuxRAHL} represents the hill effect of activator LuxRAHL to LuxcI hybrid promoter; \ frac{1-[CI leak]^nCI}{ KdCI^nCI+[CI]^nCI} represents the hill effect of repressor CI, while frac{1-{1-[CI leak]^nCI}}{ KdCI^nCI+[CI]^nCI} takes CI leaking into consideration (promoter pLac will not close thoroughly even though CI exists).

For the section of the equation, \frac{1-[ LuxRAHL]^nLuxRAHL}{ KdLuxRAHL ^nLuxRAHL + [LuxRAHL]^nLuxRAHL}\times \frac {1-[CI leak]^nCI}{ KdCI^nCI+[CI]^nCI}\times {PoPSLuxcI} \times \frac{N}{V}

represents the synthesizing rate of mRNAkill under the influence of LuxRAHLCI repression and leaking, and LuxcI hybrid promoter; -kdegmRNA\times [mRNALuxR] represents the degrading rate of mRNAkill.

As for explanations of equation7, 8, 9, 15, 16, 17, 20, please see the ethanol wiki for more information.