A Direct Relationship Between Emax and K Can Be Demonstrated by Considering the Scenario

ABBREVIATED SUMMARY F F  max  F C  CC  b 1  1/ 2  1  e  k 

 C1C   1/2  1  e  k  E  1 2 C  CC   1/ 2  1  e  k 

E F  max 4 max  E

Conformity to the classic Boltzmann sigmoid function (equation 1) posses the question as to how C1/2, k and Fmax relates to ΔE and Emax. Although equation 4 predicts that Fmax is defined by the ratio of Emax to ΔE, it is less clear how k and C1/2 relate. Indeed, a direct relationship between Emax and k can be demonstrated by considering the scenario that if C=C1/2, then EC=Emax/2 (Figure 1A) so that equation 2 becomes:

 C 1C   1/2 1/2  E 1  e  k  E  max  1 C  C C  2  1/2 1/ 2  1  e  k 

 1    E 1  e k  E  max  1 C  0  2   1  e  k 

 1     k  Emax  e 1 or

1 k  6 Ln(Emax  1)

Furthermore, substituting for k using equation 6 and assuming Fb=0, equation 1 simplifies to:

Fmax FC  7 C1/ 2 C 1 Emax 1

When C=0, FC is equal to the target quantity such that: Fmax F0  8 C1/ 2 1 (Emax 1)

where F0 is target quantity expressed in fluorescence units. Rearrangement of equation 7 allows C1/2 to be defined in terms of FC:

 F  C  Log  max 1  C 9 1/ 2 (Emax 1)    FC 

so that substituting for C1/2 and simplifying, equation 8 becomes:

Fmax F0   F   max  C 10 (6) 1  1Emax 1  FC 

This derivative holds great significance for real-time qPCR, in that it allows fluorescence readings to be converted directly into target quantity, once estimates for ΔE and Emax have been obtained. These derivatives can be extended one step further by defining C1/2 in terms of F0 by rearranging equation 8:

 F  C  Log  max 1 11 1/ 2 (Emax 1)    F0 

so that substituting for C1/2 and simplifying, equation 7 becomes:

Fmax FC   F   max  C 12 (7) 1  1Emax 1  F0 

In addition to allowing reaction fluorescence to be predicted for any cycle once an estimate for F0 has been obtained, equation 12 also dictates that the shape of an amplification profile is determined by ΔE and Emax, whereas profile position is determined by target quantity and Emax.

DETAILED SUMMARY Based upon the extensive correlation that can be generated by nonlinear regression of FC datasets to the classic Boltzmann four parametric sigmoid function (equation 1), it was surmised that Fmax, C1/2, and k could be replaced with the two kinetic parameters predicted to govern PCR amplification: ΔE and Emax (equation 3). Indeed equation 4 predicts that Fmax is be derived from the ratio of Emax and ΔE. How then are k and C1/2 related to ΔE and Emax?

F F  max  F C  CC  b 1  1/ 2  1  e  k  Starting with cycle efficiency as described by equation 2:

 C1C   1/ 2  1 e  k  E  1 2 C  CC   1/ 2  1 e  k 

Relating Emax to k

A direct relationship between Emax and k can be demonstrated by considering the scenario that if C=C1/2, then EC=Emax/2 (Figure 1A) so that equation 2 becomes:

 C 1C   1/2 1/2  E 1  e  k  E  max  1 C  C C  2  1/2 1/ 2  1  e  k 

 1   1      E 1 e k  1 e k  max  1  1  0  2   2 1 e k    1        1  1 e k    E  2   1  1 e k   2 max  2       1     k  Emax  e 1  1     k  Emax 1  e  1        k   LnEmax  1  Lne      1 LnE 1  max k 1 k  6 Ln(Emax  1) k substitution Returning to the four-parametric sigmoid function, equation 1: F F  max  F C  CC  b  1/ 2  1 1 e  k 

Substituting for k using equation 6:

F F  max C     CC  1/ 2   1     LnE 1  1 e  max  F F  max C CC LnE 1 1 e 1/ 2 max Fmax FC  LnE 1 C C1/ 2  1 e max  Fmax FC  CC1/ 2  1 Emax 1 Fmax FC  7 C1/ 2 C 1 Emax 1

When C=0, FC is equal to the target quantity such that:

Fmax F0  C1/ 2 0 1 (Emax 1)

Fmax F0  8 C1/ 2 1 (Emax 1)

where F0 is target quantity expressed in fluorescence units.

Relating C1/2 to FC

Rearrangement of equation 7 allows C1/2 to be defined in terms of FC:

Fmax FC  7 C1/ 2 C 1 Emax 1

C1/ 2 C 1 Emax 1  FC  Fmax

C1/ 2 C Fmax 1 Emax 1  FC C1/ 2 C Fmax Emax 1  1 FC

C C   F  1 / 2  max  LogEmax 1  Log 1  FC 

 F   max  C1/ 2  CLogEmax 1  Log 1  FC 

 Fmax  Log 1  FC  C1/ 2  C  LogEmax 1

 F  C  C  Log  max 1 1/ 2 Emax 1   FC 

 F  C  Log  max 1  C 9 1/ 2 (Emax 1)    FC 

Substituting for C1/2 and simplifying equation 8

Fmax F0  8 C1/ 2 1 (Emax 1)

F F  max 0  F  Log  max 1C ( Emax 1)    FC  1 (Emax 1)

Fmax F0   F   max  C 10 (6) 1  1Emax 1  FC 

This derivative holds great significance for real-time qPCR, in that it allows fluorescence readings to be converted directly into target quantity, once estimates for ΔE and Emax have been obtained.

Relating C1/2 to F0

Fmax F  8 0 C1 / 2 1 Emax 1 F C1 / 2 max 1 (Emax 1)  F0 F C1 / 2 max (Emax 1)  1 F0  F  C1 / 2  max  Log(Emax 1)  Log 1  F0   F   max  C1/ 2 Log(Emax 1)  Log 1  F0 

 Fmax  Log 1  F0  C1/ 2  Log(Emax B 1)

 F  C  Log  max 1 11 1/ 2 (Emax 1)    F0 

Substituting for C1/2 and simplifying equation 7

Fmax FC  7 C1/ 2 C 1 Emax 1

F F  max C   F    Log  max 1C   ( Emax 1)  F   1 Emax 1  0  

Fmax FC   F   max  C 12 (7) 1  1Emax 1  F0 

In addition to allowing reaction fluorescence to be predicted for any cycle once an estimate for F0 has been obtained, equation 12 also dictates that the shape of an amplification profile is determined by ΔE and Emax, whereas profile position is determined by target quantity and Emax.