qPCR Efficiency Calculator: ΔΔCt and Pfaffl

Technical Ct replicates are averaged within each assay; their sample SD gives variance of the mean $s^2/n$. Livak and Schmittgen (2001, sections 1.1 and 1.4) motivate the separate-well calculation. Define $d=C_{cal}-C_{sample}$ and $g$ reference genes. Then $\Delta\Delta Ct=-d_t+\mathrm{mean}(d_r)$ and $F_{DD}=2^{-\Delta\Delta Ct}$. Pfaffl (2001, attached s2, p. 3) supplies the efficiency-corrected ratio; here it is extended using the geometric normalization described by Vandesompele (2002, s3, p. 4): $\ln F_P=d_t\ln A_t-g^{-1}\sum_j d_j\ln A_j$. The reference normalization factor is the geometric mean of $A_j^{d_j}$. When all reference assays have the same factor $A_r$, this equals $A_r^{\mathrm{mean}(d_j)}$; choosing base two adds the assumption $A_r=2$.

The standard regression is unweighted least squares on mean Ct against log₁₀ copies/reaction: $Ct=m x+b$. Amplification factor $A=10^{-1/m}$ and efficiency $100(A-1)$ follow MIQE 2009 (s4, p. 8). Slope SE is residual SD divided by $\sqrt{S_{xx}}$, with levels minus two degrees of freedom.

Uncertainty is derived here by first-order propagation of independent variables. For each Ct deviation $V_d=s_{cal}^2/n+s_{sample}^2/n$. Thus $V_{DD}=V_{d,t}+\sum V_{d,j}/g^2$. For a slope SE $s_m$, $V_{\ln A}=(\ln(10)s_m/m^2)^2$. Pfaffl log variance is $(\ln A_t)^2V_{d,t}+d_t^2V_{\ln A_t}+g^{-2}\sum[(\ln A_j)^2V_{d,j}+d_j^2V_{\ln A_j}]$. Fold SE is $F s_{\ln F}$; displayed asymmetric limits are $F\exp(\pm s_{\ln F})$.

For dsDNA, IDT gives an approximate 660 g/mol per base pair. Divide mass concentration by whole-molecule molar mass and multiply by Avogadro's constant; apply template volume and pre-dilution for copies/reaction. A known-copy entry can replace the mass estimate. Absolute sample copies follow $10^{(Ct-b)/m}$.

One biological sample and a distinct calibrator; two or three technical replicates in independent wells for every active assay, with the same replicate count. Reference genes are already validated as stable and equally weighted. No target/reference covariance, inter-assay covariance or Ct/slope covariance is modeled. Fitted target slope uncertainty uses the residual scatter of dilution-level means; it excludes dilution-concentration error. Entered slope SE of zero fixes that efficiency. Standard copy numbers assume pure dsDNA, average base-pair molar mass, one target site per full molecule and accurate concentration, volume and dilution. Those preparation uncertainties are not propagated. Default Ct values are an original illustrative dataset, not experimental measurements. The animation is an original logistic illustration, with relative threshold 0.2 and no fitted fluorescence data.

For detected quantitative Ct values (inputs 1–45), negative slopes from −20 to −1.5, three to five equally spaced serial dilutions, and one to three reference genes. The bounds are numerical operating limits, not assay acceptance limits. ΔΔCt assumes all amplification factors are approximately two; Pfaffl requires stable measured efficiencies across sample and calibrator. A 90–110% efficiency range, R² ≥0.98 and Ct SD ≤0.5 are screening prompts chosen for this tool, not proof of MIQE compliance or universal pass/fail criteria. R² does not establish specificity or lack of inhibition. Technical SE limits are not biological confidence intervals. Correlated multiplex measurements, a sample compared with itself, undetermined wells, group hypothesis tests, extraction/RT efficiency and low-copy Poisson modeling are outside scope. Standard copy interpolation needs samples bracketed by validated standards. Do not use the animation to infer instrument signal or a diagnostic conclusion. Uncertainty propagation is blocked when any fitted or entered calibration slope SE exceeds 20% of its magnitude, or either Pfaffl log-fold limit exceeds ±700. Exponent guards keep invalid-input previews finite; these guarded values are not reportable results.