Models

Logistic models

These models are calculated by the fit_50 function. You can use these models for any dose-response curves without knowing any concentrations in the system. Parameters ymin, ymax and slope could be fixed. If you fix slope to -1 (for inhibition) or 1 (for activation), the model will be reduced from a model with variable slope to a model with fixed slope.

Model name	Description	Complexity	Input parameters	Fitted parameters
IC50	Fits midpoint from c	logistic	-	ymin, ymax, slope, IC50
logIC50	Fits midpoint from log(c)	logistic	-	ymin, ymax, slope, logIC50

Direct binding

These models are calculated by the fit_Kd_direct function. It is assumed that LsT is fixed and RT is titrated. Parameters ymin and ymax could be fixed.

Model name	Description	Complexity	Input parameters	Fitted parameters
dir_simple	Simplified direct binding	\([R] = [R]_T\)	-	ymin, ymax, Kds
dir_specific	Specific direct binding	quadratic	LsT	ymin, ymax, Kds
dir_total	Total direct binding	quadratic	LsT, Ns	ymin, ymax, Kds

Competitive binding

These models are calculated using the fit_Kd_competition function. It is assumed that RT and LsT are fixed and LT is titrated. Parameters ymin and ymax could be fixed. When fitting comp_4st_specific or comp_4st_total, it is recommended to fix the ymin parameter (minimal asymptote of the model).

Model name	Description	Complexity	Input parameters	Fitted parameters
comp_3st_specific	Three-state, specific	cubic	RT, LsT, Kds	ymin, ymax, Kd
comp_3st_total	Three-state, total	cubic	RT, LsT, Kds, N	ymin, ymax, Kd
comp_4st_specific	Four-state, specific binding	quintic	RT, LsT, Kds	ymin, ymax, Kd, Kd3
comp_4st_total	Four-state, total binding	quintic	RT, LsT, Kds, N	ymin, ymax, Kd, Kd3

Warning

The quintic models comp_4st_specific and comp_4st_total are solved numerically. Choosing the physical root is not always straightforward. The current implementation might give unphysical results in some situations, for example in strongly cooperative regimes.

IC50 -> Kd conversion models

These models are calculated using the convert function. No fitting is performed, only a single-point conversion of IC50 values from provided dataframe to Kd.

Model name	Description	Complexity	Input parameters	Output
cheng_prusoff	Cheng-Prusoff	-	LsT, Kds, IC50	Kd
cheng_prusoff_corr	Corrected Cheng-Prusoff	-	LsT, Kds, y0, IC50	Kd
coleska	Nikolovska-Coleska	quadratic	RT, LsT, Kds, IC50	Kd

Parameter fixing

BindCurve allows fixing certain parameters during fitting to user-defined value. In all models, you can fix both the ymin and/or ymax parameters which will constrain the lower and/or upper asymptote of the model. For example, if your data are normalized between 0 and 1, you can simply fix ymin to 0 and ymax to 1 and the fit will be constrained to these values. You can also fix these parameters to any other arbitrary value depending on your situation. The fit_50 function also allows fixing the slope parameter for the logistic models, which could be used to simplify the model. Typically, you might want to fix the slope to either -1 for inhibition, or 1 for activation.

Fixing of ymin and ymax parameters is controled by fix_ymin and fix_ymax arguments when calling any fitting function in BindCurve. The slope parameter can be fixed by fix_slope argument (only available for fit_50 function). All of these parameters are set to False by default, which means the parameter is fitted freely. When you change the fix_ymin, fix_ymax or fix_slope to any numerical value, the parameter will be fixed to the given value.

# This will fix minimum to 0, maximum to 1 and slope to -1
IC50_results = bc.fit_50(input_data, model="IC50", fix_ymin=0, fix_ymax=1, fix_slope=-1)

Fixing can also be useful when fitting Kd using the exact polynomial models. For example, if your competitive inhibition curve is not titrated all the way to the minimum asymptote, you can just fix the ymin parameter to the value of the lower asymptote, which you have determined previously from direct binding.

# This will fix minimum to 20, maximum is fitted freely
Kd_results = bc.fit_Kd_competition(input_data, model="comp_3st_specific", fix_ymin=20)

Note

The ymin and ymax parameters have different meaning for the logistic models and for the exact Kd models. In the logistic models, ymin and ymax will always correspont to the actual minimum and maximum of the curve. In the exact Kd models, ymin and ymax define the asymptotes of the model which is often not be the same as the minimum and maximum of the curve.

Uncertainties

BindCurve offers two kinds of uncertainty estimates for the fitted parameters, both obtained from the underlying LMFIT library. For every fit, standard error (SE) is outputted. Additionally, the calculation of 95% confidence intervals is enabled by default, but could be disabled by setting ci=False. The 95% confidence interval is outputted as loCL and upCL, which correspond to lower and upper confidence limits. Please note that the 95% confidence interval can only be calculated if more than one parameter is freely fitted. In case only one parameter is freely fitted, BindCurve will automatically set ci=False and only SE will be calculated.

Goodness of fit

To quantify the goodness of fit, \(\chi^2\) and \(R^2\) metrics are outputted for every fit. Good fit is characterized by low value of \(\chi^2\) and high value (close to 1) of \(R^2\).