Charts

Dose Response

The standard way to characterize the potency or efficacy of a compound.

What is a dose–response curve?

The dose-response relationship represents the association between the concentration of a substance (i.e. drug or toxin) and the magnitude of the response it elicits in a biological system. The dose is plotted on the x-axis and the response on the y-axis, allowing for the visualization of the effect of increasing exposure.

While dose-response curves are commonly thought of in the context of pharmacology and toxicology for determining effective doses, toxic levels, and mechanisms of action, their usefulness extends to all kinds of experiments. The “dose” in dose response is used loosely, and the term is also used to describe experiments where you might not even be manipulating concentration, but rather some other variable, like pressure, temperature, light intensity, or voltage. The common factor being these experiments display a sigmoidal shape, having a range where the response increases proportionally with the "dose" before reaching a plateau.

Default dose-response relationship
An S-curve characteristic of many biological phenomena
Two sigmoidal dose-response curves. Both show increasing response with dose before plateauing, but the gray curve is more potent than the red curve.

Four Parameter Logistic Regression - (4PL)

The four-parameter logistic (4PL) curve is a widely used regression model, particularly prevalent in binding-assays and dose-response experiments. It defines a sigmoidal (S-shaped) curve, which is characteristic of many biological phenomena. The "4 parameters" of the model represent: the lower asymptote (minimum response), the upper asymptote (maximum response), the inflection point (the dose at which the response is halfway between the asymptotes, often called EC50 or IC50), and the Hill slope (the curve's steepness).

This model is often used to interpolate unknown concentrations, and characterize the potency and efficacy of compounds. The equation for the model is:

Y=bot+topbot1+⁢(EC50X)HillSlope
WARNING: Despite sharing the word 'logistic,' 4PL regression should not be confused with simple or multiple logistic regression, which are distinct statistical models used to predict the probability of a binary outcome (e.g., yes/no, true/false).
Dose-response curve annotated with its key parameters: the upper and lower asymptotes (maximum and minimum responses), the EC50 (concentration for half-maximal response), and the Hill slope (steepness of the curve).

X-axis values

Graphmatik will plots "doses" on the x-axis using a logarithmic scale. To effectively characterize a compound's dose-response relationship, it's best practice to test at least 5-6 doses that are relatively equally spaced when plotted on a logarithmic scale. This ensures broad coverage and optimal curve fitting.

CAUTION: Do NOT pre-transform concentrations to logarithmic doses. This will lead to improper fitting. Rest assured, Graphmatik will handle the logarithmic scaling for you, so no manual transformation is required.

Y-axis values

The biological "response" is typically plotted as raw values on the y-axis, though normalization to 0-100% can be applied for comparability across successive experiments.

If you normalize your data, you can constrain the bottom and top asymptotes to 0% and 100%, respectively, provided you have robust control data to justify this.

Initializing parameters used during fitting

Rather than directly calculating parameters like linear models, non-linear regression employs an iterative process that starts with initial guesses and uses computational algorithms to progressively refine parameter values until the 'sum of squared residuals' is minimized."

Graphmatik streamlines this process by auto-initializing parameters with estimated good values. Nevertheless, this process isn't infallible, especially for datasets that don't fully depict the dose-response curve, as poor initial guesses can sometimes cause the algorithm to settle for a local minimum instead of the desired global minimum.

Should this happen during your analysis, manually adjusting the initial values can help Graphmatik find the best fit for your data.

Constraining the model

When working with incomplete dose-response curves, you often won't have enough data to precisely determine the upper and lower asymptotes of the 4-parameter logistic regression. If you have robust control data and a strong reason to believe your compound of interest should elicit a full (not partial) response, you can constrain the top and bottom parameters to 0% and 100% controls. This should allow you to still derive a meaningful fit from your limited data.

Dose-response curve with limited data preventing accurate observation of the curve's maximum response.Dose-response curve with insufficient data to capture the full range of the response, but constrained by 0% and 100% controls.

Interpolate from the curve

A dose-response curve is most often used to measure a compound's potency by interpolating either the EC50, half-maximal effective concentration, for curves that go uphill or the IC50, half-maximal inhibitory concentration, for downhill curves. In other words, the EC50/IC50 is a relative metric and reflects the concentration at which a substance elicits 50% of its maximal effect, halfway between the upper and lower asymptotes.

After running a dose-response in Graphmatik, the EC50/IC50 will be reported as a parameter estimate inside the Analyze tab of the stats workspace alongside the estimates for the top, bottom, and Hill slope (β1).

But Graphmatik also makes it really easy to interpolate other unknowns (i.e EC10, EC90, IC80, etc.) from the regression curve. After running an analysis, switch to the stats workspace and select the interpolate tab. A table will appear where you can enter values into X or Y columns.

Provide a value for X, and Graphmatik will interpolate the response value for Y. Enter a value for Y, and Graphmatik will estimate the corresponding "dose" of X.

How to calculate the absolute IC50

In some fields, you might need to report IC50 values relative to a standard inhibitor dose that represents 0% response. When working with a complete (or full) inhibitor—one that totally blocks enzyme activity (meaning the top of the curve aligns with the blank control and the bottom plateau matches the standard inhibitor)—the relative IC50 calculated by Graphmatik should be equivalent to the "absolute" IC50.

However, if you are working with a partial or incomplete inhibitor that does not reach the standard control's minimum (i.e., it bottoms out before the standard control), the relative IC50 calculated by Graphmatik will not equal the "absolute" IC50.

In such scenarios, you must first determine if the inhibitor's maximal effect is above 50% inhibition. If the inhibition does not reach at least 50%, an absolute IC50 cannot be determined and it is considered undefined. However, if your partial inhibitor reduces the response below 50%, then the absolute IC50 can be calculated by interpolating from the dose-response curve.

To calculate the absolute IC50, you must first determine the Y-value representing 50% inhibition. This is achieved by subtracting the standard control (0% activity) from the signal of the blank control (100% activity) and dividing by 2. Next, within the interpolate tab of Graphmatik's stats workspace, enter this calculated 50% response value into the Y column. Graphmatik will then automatically interpolate the corresponding concentration (X-value) required to achieve this level of inhibition, which is the absolute IC50.

Dose-response curve showing full (or complete) inhibition such that the relative IC50 and absolute IC50 are identical.Dose-response curve showing partial (or incomplete) inhibition such that the relative IC50 is less than the absolute IC50.

Chart properties

PropDefaultDescription
central tendencymean
mean
The sum of a set of values divided by the number of values in the set.
median
The middle most value of a sorted set of numbers.
errorSEM
standard error of the mean (SEM) mean
How much the sample means vary from the population mean.
standard deviation (SD) mean
A measure of the variation of a set of values around their mean.
95% confidence interval (95% CI) mean or median
95% probability that the population parameter lies within this range.
range mean or median
The difference between the highest and lowest values within a set.
Interquartile range (IQR) median
The middle 50% of a set of values (i.e. 3rd quartile - 1st quartile).
equation4pl
4-parameter logistic (4pl)
Fit a four-parameter logistic regression curve to the data.
auto-initializetrue
true
Auto-fit initial parameter estimates based on the data.
false
Manually configure initial parameter estimates for the bottom asymptote (min), top asymptote (max), hill slope (β), and EC50/IC50.
constrainfalse
false
The model is NOT constrained.
true
Constrain the upper and lower asymptotes of the model.