Accurate analysis of real-time PCR (qPCR) amplification curves is essential for estimating initial copy number and amplification efficiency. However, experimental data face fundamental limitations: the early amplification phase is often unobservable due to insufficient fluorescence sensitivity, and the baseline (offset) is inherently uncertain. Consequently, conventional analyses based on log-transformed signals require baseline estimation, which depends on partially observed exponential regions and is intrinsically ill-conditioned.

To address this issue, we focus on the cycle-to-cycle incremental signal, diff(y), which substantially reduces baseline dependence. The logarithm of this incremental signal, log(diff(y)), reveals two linear regimes separated by a transition regime. These regimes correspond to distinct amplification dynamics: the early regime reflects exponential amplification, whereas the late regime reflects approximately exponential decay of the incremental signal as the curve approaches its plateau.

The slopes of these regimes encode different underlying dynamics. The early-regime slope corresponds to amplification efficiency (approximately log(1+p)), while the late-regime slope reflects the approach to the plateau. In real data, these slopes are often unequal, with the late-regime slope typically smaller, indicating asymmetric amplification behavior.

Conventional sigmoid models, described by dy/dx = p·y·(1−y/K), are governed by a single parameter and cannot independently control early- and late-regime slopes. They perform well only when the slopes are similar but fail to capture the asymmetry observed in real data. Although Richards and Gompertz models introduce additional parameters, they remain based on a single functional form, and the two regimes remain coupled, preventing independent control. Thus, they are insufficient for accurately reproducing amplification curves.

Moreover, the number of amplicons follows a stochastic distribution. Simulations suggest that this distribution becomes effectively determined within the first few cycles (approximately six cycles), reflecting a martingale-like property. Early stochastic fluctuations therefore influence subsequent amplification dynamics, highlighting the importance of fitting later-cycle data.

Based on these observations, we propose the AELiSp (Asogawa Exponential Linear-SoftPlus) model. This model represents log(diff(y)) as two linear regimes connected smoothly, allowing independent control of the early- and late-regime slopes and transition width. The SoftPlus function provides a smooth approximation of a broken line, and an additional linear term enables independent slopes.

The model is exponentiated to obtain diff(y) and then integrated to reconstruct y(x). Application to experimental datasets yields consistently lower Akaike Information Criterion (AIC) values than sigmoid, Richards, and Gompertz models, demonstrating a superior balance between goodness-of-fit and model complexity. The proposed framework naturally captures asymmetric amplification dynamics and offers improved flexibility and interpretability.

Author

• Minoru Asogawa (none, Japan)