# Curve Fitting

Sphinx offer multiple kinds of curve fitting. This page provides a brief overview of the different types of curve fitting available in Sphinx and explain how results are calculated.

The result of each curve fit will result in a new data table in the analysis that provides the fit parameters.

# Log(inhibitor) vs. response

This curve describes how the response to an inhibitor varies as the logarithm of the inhibitor concentration increases. It is commonly used to assess the potency of inhibitors, with the midpoint of the curve ($value_{50}$) indicating the concentration needed to achieve half-maximal inhibition. This curve is particularly useful when analyzing the efficacy of competitive inhibitors.

$e_{min} + \frac{(e_{max} - e_{min})}{(1 + 10^{(x - \log_{10}(value_{50}))})}$# Log(agonist) vs. response

This curve shows the response to an agonist as a function of the logarithm of the agonist concentration. It is frequently used to determine the potency of agonists, where the $value_{50}$ indicates the concentration required for half-maximal activation. This type of curve is essential in pharmacology for assessing receptor activation by various agonists.

$e_{min} + \frac{(e_{max} - e_{min})}{(1 + 10^{(\log{10}(value_{50}) - x)})}$# Log(agonist) or log(inhibitor) vs. response — Variable slope

This curve type accounts for variations in slope, allowing for more flexibility in modeling different biological systems. The variable slope parameter can describe situations where the steepness of the response varies, which is important for analyzing complex interactions in dose-response studies. This model should be used when the standard Hill slope (e.g. $slope = 1$ or $slope = -1$) does not adequately describe the data.

$e_{min} + \frac{(e_{max} - e_{min})}{(1 + 10^{((\log{10}(value_{50}) - x) \cdot slope)})}$# [Inhibitor] vs. response

This curve illustrates the direct relationship between inhibitor concentration and the response, without using a logarithmic scale. It is particularly useful when the inhibitor’s effect is linear or when working within a narrow concentration range. This model is preferred when dealing with systems where the logarithmic transformation does not enhance interpretability.

$e_{min} + \frac{(e_{max} - e_{min})}{(1 + \frac{x}{value_{50}})}$# [Agonist] vs. response

Similar to the previous curve, this one depicts the direct relationship between agonist concentration and the response. It is typically used when the response to the agonist is proportional to its concentration, and the effect saturates at higher concentrations. This curve is applied in cases where the response mechanism is straightforward and does not require logarithmic transformation.

$e_{min} + \frac{x \cdot (e_{max} - e_{min})}{(value_{50} + x)}$# [Inhibitor] vs. response — Variable slope

This curve introduces a variable slope to better fit data where the inhibitory response does not follow a simple sigmoidal pattern. It is particularly useful for analyzing non-linear inhibitory effects, providing a more nuanced understanding of how inhibitors function across different concentration ranges. This model is essential when the inhibition mechanism is complex or involves multiple binding sites.

$e_{min} + \frac{(e_{max} - e_{min})}{(1 + (\frac{value_{50}}{x})^{slope})}$# [Agonist] vs. response — Variable slope

This model applies a variable slope to the agonist response curve, allowing for more accurate modeling when the response does not follow a standard sigmoidal curve. It is particularly useful in cases where the agonist’s effect changes more steeply or shallowly than expected. This curve type is critical in analyzing systems with variable receptor activation profiles.

$e_{min} + \frac{(x^{slope}) \cdot (e_{max} - e_{min})}{(value_{50}^{slope} + x^{slope})}$