[Paper Review] Modeling Biphasic, Non-Sigmoidal Dose-Response Relationships: Comparison of Brain-Cousens and Cedergreen Models for a Biochemical Dataset

Venkat D. Abbaraju, Tamaraty L. Robinson+1 more|arXiv (Cornell University)|Aug 16, 2023

Computational Drug Discovery Methods4 citations

TL;DR

This study compares the Brain-Cousens and Cedergreen models for fitting biphasic, non-sigmoidal dose-response relationships in a biochemical dataset, demonstrating both models effectively capture hormetic responses. It provides full datasets, SAS code, and interpretation of key parameters like ED50, ymax, and parameter trends under varying substrate concentrations.

ABSTRACT

Biphasic, non-sigmoidal dose-response relationships are frequently observed in biochemistry and pharmacology, but they are not always analyzed with appropriate statistical methods. Here, we examine curve fitting methods for "hormetic" dose-response relationships where low and high doses of an effector produce opposite responses. We provide the full dataset used for modeling, and we provide the code for analyzing the dataset in SAS using two established mathematical models of hormesis, the Brain-Cousens model and the Cedergreen model. We show how to obtain and interpret curve parameters such as the ED50 that arise from modeling, and we discuss how curve parameters might change in a predictable manner when the conditions of the dose-response assay are altered. In addition to modeling the raw dataset that we provide, we also model the dataset after applying common normalization techniques, and we indicate how this affects the parameters that are associated with the fit curves. The Brain-Cousens and Cedergreen models that we used for curve fitting were similarly effective at capturing quantitative information about the biphasic dose-response relationships.

Motivation & Objective

• To evaluate and compare the performance of the Brain-Cousens and Cedergreen models in fitting biphasic, non-sigmoidal dose-response relationships.
• To provide a comprehensive framework for curve fitting and parameter interpretation in hormetic responses using real biochemical data.
• To examine how normalization techniques affect model parameters and their biological interpretation.
• To investigate the relationships between key hormetic parameters (e.g., d, ymax, ED50) and experimental conditions such as substrate concentration.

Proposed method

• Applied the Brain-Cousens model (equations 1–5) and Cedergreen model (equations 6–10) to a raw biochemical dataset with effector concentrations from 0.01–50 µM and substrate from 0.25–10 µM.
• Used SAS software to perform nonlinear regression fitting and extract parameters including ED50, ymax, d (control response), and e (lower bound of ED50).
• Conducted curve fitting on both raw data and data normalized using common techniques, comparing resulting parameter changes.
• Performed parameter trend analysis across substrate concentrations to assess consistency and biological relevance.
• Graphically and statistically evaluated relationships between parameters (e.g., d vs. ymax, f/a ratio vs. ymax) to assess predictive power.
• Provided full dataset (Table A1) and SAS code for reproducibility and broader application to molecular systems.

Experimental results

Research questions

• RQ1 How do the Brain-Cousens and Cedergreen models compare in fitting biphasic, non-sigmoidal dose-response curves in biochemical systems?
• RQ2 How do normalization techniques affect the estimated parameters of hormetic models?
• RQ3 What are the relationships between key parameters (e.g., d, ymax, ED50, f/a ratio) and substrate concentration?
• RQ4 How do parameter values such as e relate to ED50, and what is their biological interpretability?
• RQ5 To what extent do parameters like a, f, and f/a ratio predict the magnitude of hormetic response (ymax)?

Key findings

• Both the Brain-Cousens and Cedergreen models effectively captured the quantitative features of biphasic dose-response relationships in the biochemical dataset.
• A strong positive correlation was observed between control response (d) and maximum stimulatory response (ymax), with d ≤ ymax consistently satisfied.
• The parameter e was 2.2–5.0-fold lower than ED50 in Cedergreen modeling, but 6–369-fold lower in Brain-Cousens modeling, indicating significant model-dependent variability.
• The f/a ratio showed no reliable association with ymax magnitude in this dataset, contradicting prior reports from plant studies.
• Parameters a, d, ED50, M, LDS, ymax, and ymax% consistently increased with rising substrate concentration, indicating predictable trends.
• Normalization artificially set some parameters to fixed values, altering their trends and reducing interpretability, particularly for the 0.25 µM substrate condition, which was not hormetic.