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Short answer
A sigmoidal graph is an S-shaped curve: flat at one end, steep through the middle, and flat again at the other end. Sigmoidal curves are also called sigmoid or logistic curves. The same shape appears in dose-response fits and in the haemoglobin oxygen dissociation curve, and in each case the middle of the curve is described by a different named parameter.
What is a sigmoid shape curve?
A sigmoid shape curve rises slowly, then quickly, then slowly again until it levels off at an upper plateau. MathWorld notes that the sigmoid function is also called the sigmoidal curve or the logistic function. The curve has a single inflection point, where the slope stops increasing and starts decreasing.
In laboratory work the shape matters because many measured responses saturate. A response cannot keep rising forever, so it has a lower plateau, an upper plateau and a steep working range in between.
What is the equation for the sigmoid function?
The standard logistic sigmoid is:
y = 1 / (1 + e−x)
MathWorld gives this form. At x = 0 the value is y = 1/2, and that is also the inflection point. Because e−x is always positive, y always lies between 0 and 1, approaching 0 for large negative x and 1 for large positive x.
Laboratory software rarely fits this bare form. It fits a version with extra parameters that move and stretch the curve to match real data, as described in the next section.
Sigmoidal curves in dose-response fitting
GraphPad Prism documents the variable-slope dose-response model as:
Y = Bottom + (XHillSlope × (Top − Bottom)) / (XHillSlope + EC50HillSlope)
- Top and Bottom are the plateaus, in the units of the Y axis.
- EC50 is the concentration of agonist that gives a response half way between Bottom and Top.
- HillSlope describes the steepness of the family of curves. A value of 1.0 is the standard; values above 1.0 give a steeper curve and values below 1.0 a shallower one.
Prism describes the log(dose) version as the familiar symmetrical sigmoidal shape, and lists the four-parameter fit with X as log(concentration) as “log(agonist) vs. response — Variable slope”. It lists a five-parameter fit as the asymmetric sigmoidal (5PL). For a worked immunoassay application of the four-parameter model, see four-parameter logistic fitting for ELISA standard curves. For the mechanism behind HillSlope, see the Hill coefficient guide.
The variable-slope model fits HillSlope from the data instead of assuming 1.0. Prism’s guidance is that constraining it to 1.0 may be sensible when there are few data points.
Why is the oxygen dissociation curve sigmoidal?
The oxyhaemoglobin dissociation curve is sigmoidal because of positive cooperativity: Deranged Physiology attributes the sigmoid shape to positive cooperativity between the haemoglobin subunits, which enhance each other’s binding of oxygen. Saturation therefore climbs steeply over the middle of the PaO2 range.
The same source notes that in the absence of all allosteric effectors the curve is actually a rather boring hyperbolic one. Effectors such as pH, pCO2 and 2,3-DPG shift and shape the curve in the body.
Two parts of the curve have practical meaning:
- The plateau decreases variability in blood oxygen content even with large changes of PaO2.
- The steep section allows increased release of oxygen from haemoglobin with only a small change in PaO2.
The midpoint here is called P50, the PaO2 at 50% saturation, which corresponds to 24-28 mmHg under normal circumstances. The source also notes that an abnormal ferric or carboxyhaem subunit increases affinity (a left shift), while sulfhaemoglobin shifts the curve right.
One S-shape, three midpoints
| Context | Curve | Name of the midpoint | Source |
|---|---|---|---|
| Logistic sigmoid | y = 1/(1 + e−x) | Inflection point at x = 0, where y = 1/2 | MathWorld |
| Dose-response fit | Bottom to Top plateaus, HillSlope steepness | EC50, the response half way between Bottom and Top | GraphPad Prism guide |
| Haemoglobin oxygen binding | Saturation against PaO2 | P50, 24-28 mmHg under normal circumstances | Deranged Physiology |
What is the opposite of a sigmoidal curve?
There is no single opposite. The usual contrast is the hyperbolic curve: Deranged Physiology describes the effector-free haemoglobin curve as hyperbolic, the shape that results without cooperative binding.
Common misunderstandings
- Steepness and position are different things. HillSlope sets how steep the curve is; EC50 (or P50) sets where along the X axis the midpoint sits.
- Not every sigmoid is symmetrical. Prism describes the four-parameter log(dose) curve as symmetrical and treats the five-parameter 5PL as the asymmetric version.
- “Sigmoid” is not one fixed equation. The bare logistic function has no parameters; lab models add Top, Bottom, EC50 and HillSlope.
FAQ
What is a sigmoid shape curve?
An S-shaped curve with a lower plateau, a steep middle section and an upper plateau, also called a logistic curve.
What is the equation for the sigmoid function?
y = 1/(1 + e−x), which equals 1/2 at x = 0.
What is the reason for the sigmoidal pattern of the oxygen dissociation curve?
Positive cooperativity between the haemoglobin subunits, each of which enhances the others’ binding of oxygen.
What is the opposite of a sigmoidal curve?
No single opposite exists; the usual contrast is the hyperbolic, non-cooperative curve.
Sources
- MathWorld, “Sigmoid Function”: https://mathworld.wolfram.com/SigmoidFunction.html, checked 2026-10-29.
- GraphPad Prism guide, “Dose-response – Stimulation, variable slope”: https://www.graphpad.com/guides/prism/latest/curve-fitting/reg_dr_stim_variable_2.htm, checked 2026-10-29.
- Deranged Physiology, “The oxyhaemoglobin dissociation curve”: https://derangedphysiology.com/main/cicm-primary-exam/required-reading/respiratory-system/Chapter%20112/oxyhaemoglobin-dissociation-curve, checked 2026-10-29.
Related CASRAI guides: logistic regression and the logit model and the Chou-Talalay median-effect method.
