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<div lang="latex" class="equation">
\frac{d[AHL]}{dt}={KAHL}\times{[LuxI]}+2\times KoffAHL \times [AHLLuxR] - 2\times KonAHL \times {[AHL]}^{2}\times {[LuxR]} - KdegAHL\times{[AHL]}
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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, <html><span lang="latex"> {KAHL}\times {[LuxI]} </span></html> represents the transforming rate of LuxI to AHL; <html><span lang="latex"> {2}\times {KoffAHL}\times{ [AHLLuxR]} </span></html> represents the transforming rate of AHL2LuxR complex to AHL and LuxR, the constant 2 means that each complex turns into two AHL; <html><span lang="latex"> -{2}\times {KonAHL}\times {[AHL]^2}\times {[LuxR]} </span></html> represents the transforming rate of 2AHL+LuxR to AHL2LuxR complex; <html><span lang="latex"> {-KdegLuxI}\times {[LuxI]}</span></html> 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.


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