Bell-Shaped and Biphasic Dose Response Curve Analysis
Examine a synthetic response that rises and then falls as concentration increases. A monotone four-parameter logistic curve cannot describe both arms of that shape. The existing exploratory bell-shaped model provides a way to inspect two transitions while keeping its reporting limits and the ordinary 4PL result visible.
Two observations at each of fourteen concentrations, generated independently for the seven-parameter oracle. This is a mathematical shape example, not a PROTAC degradation experiment. It carries no target, exposure time, ternary-complex measurement or degradation endpoint.
Find evidence for both arms
The reference includes low concentrations, a rising transition, an intermediate maximum, a falling transition and high concentrations. These regions give the exploratory model information that a short series around the peak would lack. Inspect replicate scatter within each region and ask whether the reversal is larger than that scatter. A single high-dose point should not carry the entire claim of a second phase.
An ordinary 4PL is monotone. Its residuals therefore provide a useful starting diagnostic when it is applied to a true bell. Systematic residual structure can show that the model misses the observed shape, but it does not identify the cause. The richer model should be chosen because the scientific question and observations justify exploring it, not automatically because adding parameters reduces residual error.
Separate model adequacy from mechanism
The two-phase solver searches a deterministic set of starting points and retains the best admissible fit. Ordered midpoints remove one labelling ambiguity, but they do not guarantee that the data determine every parameter. A broad plateau, missing high-concentration arm or minor phase can leave several parameter combinations with nearly equal fit. The reporting policy deliberately withholds numbers in these cases.
For a real degrader assay, record whether the readout is protein abundance, viability, binding or a reporter signal, and specify exposure time. Re-measure the high-concentration region with appropriate vehicle and interference controls. Orthogonal target-abundance measurements and time courses can test explanations that a curve shape alone cannot. This page provides numerical inspection of a reference shape; it does not supply those experimental observations.
Equations and variables
Bell-shaped sum of two transitions
y = D + (P1 − D)[1 − L(h1(x − m1))] + (P2 − D)L(h2(x − m2)); L(z) = 1/(1 + 10⁻ᶻ)
- x
- Log10 concentration in mol/L
- P1, P2
- Low- and high-concentration asymptotic response levels
- D
- Intermediate level parameter; may describe a peak under the fitted ordering
- m1, m2
- Ordered log-concentration transition midpoints
- h1, h2
- Positive transition steepness parameters
Four-parameter logistic response
y = B + (T − B) / [1 + 10^((m − x)h)]
- y
- Expected response (response units)
- B, T
- Lower and upper response asymptotes (response units)
- x
- Base-10 logarithm of concentration in mol/L
- m
- Base-10 logarithm of the relative midpoint in mol/L
- h
- Signed Hill slope; positive for activation, negative for inhibition
Scope and limitations
- Bell-shaped and biphasic are distinct model forms: a bell reverses direction, whereas the implemented biphasic response combines two transitions in the same direction.
- The exploratory fit uses unweighted least squares with all seven parameters free. It does not inherit fixed plateaus or weighting from the 4PL.
- No confidence intervals are provided for the seven-parameter model. The F and AICc comparisons are informational; boundary and identifiability issues prevent reading a small p value as proof of a second mechanism.
Sources and provenance
- Published numerical verification inputs and independent referencesOriginal synthetic inputs; see individual fixture notices
The generated inputs are demonstrations, not experimental measurements. Existing versioned fixtures are retained unchanged.
- Independent two-phase fixture and oracle resultsOriginal synthetic reference data
SciPy multistart fits and a separate R comparison are documented in the existing verification record.
Technical questions
Does a bell-shaped curve prove a PROTAC hook effect?
No. A reversal could also reflect assay interference, toxicity, aggregation, reagent depletion or another process. This reference contains no degradation biology. Establishing a hook mechanism requires orthogonal measurements and controls appropriate to the proposed ternary-complex explanation.
Why might the two-phase fit be withheld?
Seven parameters require substantial information. The existing checks examine concentration count, residual degrees of freedom, transition separation, bracketing and numerical identifiability. A bend without both well-sampled arms may not support reportable phase values, even when a flexible line passes close to the observations.
Can I quote the two midpoints as IC50 values?
Each belongs to a transition in the chosen empirical model. They are not automatically inhibitory potencies or binding constants. Name the model, response and phase, retain the refusal diagnostics, and do not replace the ordinary reported result with an unsupported mechanistic label.
Connected workflows
- 4PL IC50 curve fitting
See the monotone model whose limitations the reference exposes.
- Whole-molecule PROTAC property inspection
Inspect a degrader structure without inferring its dose-response mechanism.
- EC50 and Hill slope fitting
Compare a monotone activation series with the reversing shape.