Curve-Fit Kinetic Equations¶
TableCurve 2D's built-in kinetics equations consist of a set of 58 total equations where the reaction begins at time X=0. To make kinetics fitting as flexible as possible, you are first presented with a kinetics configuration dialog which contains custom fit settings specifically for the built-in kinetic equations. You can choose decay or formation equations, one or both of the intercept and zero-intercept versions as well as any one of the four Minimization Criteria available in the non-linear fitting engine. You can choose individual kinetics equations or for decay or formation profiles, you can have the AI Expert feature pick the settings for you based upon an analysis of the data. The non-linear fitting controls override the non-linear fit preferences in the Curve-Fit Preferences option. Your kinetic equations customization will be automatically saved across sessions.
Also fitted with this option are all installed UDFs, and all active TCEX.DLL equations.
Simple Decay and Formation A->B¶
The decay (Concentration of A) or formation (Concentration of B) is covered for orders 1/2, 1, 2, 3, and n. In the order n equations, the order is an adjustable parameter. For an effective determination of the order n, it is recommended that the data have a zero baseline and that the zero intercept version of this function be fitted. For each order there is both an intercept and zero-intercept version as well as both a formation and decay version. There are thus 20 basic built-in kinetics equations, all of which compute true rate constants.
Simple Decay and Formation A->B Hyperbolic Forms¶
The second order decay or formation is often given using a hyperbolic form rather than a true second order form. TableCurve 2D also includes intercept and zero-intercept forms of both decay and saturating hyperbolas. These four equations produce identical goodness-of-fit statistics as the natural second order equations, although the rate constant will not be the actual k in the differential equation defining the reaction.
Decay and Formation A->B Simultaneous 1st,2nd Order¶
This equation models reactions whose mechanisms have a simultaneous first and second order component. Here there is a single term associated with maximum concentration and two rate constants, one for the 1st order reaction pathway and the other for the second order pathway. TableCurve 2D offers four equations, intercept and zero-intercept forms of both the decay and formation profiles.
Formation A->B->C, 1st,1st Order¶
This equation models sequential formation (the concentration of C) when A->B is via a first order mechanism and B->C is also first order. Here there are a single maximum concentration term and the two separate first order rate constants. Note that this kinetic model is identical to the Cascade transition function in the transition equation group except that it assumes the reaction begins at time X=0 (the Cascade transition contains a lag term). There are both intercept and zero-intercept forms.
Decay A+B->C, 1st Order¶
These four equations model the decay of A via a first order mechanism. There are both intercept and zero intercept forms for the case when the molar concentration of A is greater than B as well as when the molar concentration of B is greater than A.
Decay A->B, A->C and Formation A->C, B->C¶
These twelve equations represent the sum of two separate decays or two separate formations. Four equations are based on both being first order, four on both being second order, and four upon one reaction being first order and the other second order. These will probably be the models of choice for data that are well fitted by the sum of two exponentials or the sum of two hyperbolas.
Equilibrium A=B, 1st order forward and reverse¶
TableCurve 2D offers two forms for the standard first order equilibrium concentration of A. One form computes the initial concentration and the forward and reverse rate constants, while the other determines the initial and equilibrium concentrations and the net sum of the rate constants. Both forms will produce identical goodness-of-fit statistics.
Complex Equilibrium A=B+C and A+B=C+D¶
TableCurve 2D offers models for fitting the complex equilibrium reactions A=B+C and A+B=C+D. Both assume first order forward and reverse kinetics and both fit the concentration of A. These equations fit an initial concentration, an equilibrium concentration, and a net rate constant.
Intermediate Equations A->B->C¶
TableCurve 2D offers four equations for fitting the concentration of B. These functions are based upon first order kinetics and produce peak profiles which fully decay. There are intercept and non-intercept forms for the case of the A->B rate constant being greater than the B->C rate constant, and for the case where the A->B rate constant is smaller. Both cases will produce identical goodness-of-fit statistics. Note that this kinetic model is identical to the Intermediate peak function in the peak equation group except that it assumes the reaction begins at time X=0 (the Intermediate peak function contains a lag term).
Equilibrium Intermediate Equations A->B=C¶
TableCurve 2D offers four equations for fitting the concentration of B. These functions are based upon first order kinetics and produce peak profiles which decay to an equilibrium concentration. There are three rate constants, two forward and one reverse. There are intercept and non-intercept forms for the case of the A->B rate constant being greater than the B=C forward rate constant, and for the case where the A->B rate constant is smaller. Both cases produce identical goodness-of-fit statistics. Note that this kinetic model is identical to the Equilibrium peak function in the peak equation group except that it assumes the reaction begins at time X=0 (the Equilibrium peak function contains a lag term).
Michaelis-Menten Equation¶
The Michaelis-Menten model is commonly used to describe the saturable kinetics which occur in many enzyme studies. This model can be readily fit using the IMPLICIT function within a UDF. TableCurve 2D supplies MICHMENT.UDL, which can be used to fit this implicit model. The library also contains an explicit UDF solution. The initial estimates will be automatically determined, provided sufficient data exists for TableCurve 2D to accurately map a transition center and width.
Synthetic Functions¶
TableCurve 2D's built-in kinetics functions have all been constructed to be defined for all values of X, even if the native function has undefined regions or is undefined at certain parameter values. These synthetic functions will either return zero (for zero-intercept versions) or the intercept value (for intercept versions) when an undefined condition is present. Note that this in no way alters the requirement that data begin at time X=0.
AI Expert¶
Select this option only for simple decay and formation profiles. The AI Expert option will reference raw data measurements to determine whether decay or formation versions are appropriate as well as whether zero-intercept, intercept, or both versions should be fitted.
Reading and Saving Kinetics Fitting Configurations¶
The most recent kinetic equation configuration is saved across sessions and presented each time this option is used. If you select the AI Expert and are not satisfied with the suggested fit, press Reset to restore the configuration that was present when first opening this dialog. If you will be fitting several types of kinetic data, you may want to save a configuration specific to each or to a given application or problem. The Save item will save all information shown in the dialog to a binary file with an [KIN] extension. The Read item can then be used to access this configuration whenever desired.
Nomenclature¶
During fitting, a function name with an underscore suffix, such as Decay2_, represents the zero-intercept version of the function. When the underscore is absent, the intercept version of the function is being referenced. The function names discriminate between decay and formation forms. In the Review, non-linear equations can be presented in either a full mathematical representation or in a symbolic notational format. You may want to use the notational format offered in the Preferences option in the main Edit menu or the File menu of the Curve-Fit graph.
Automated Fitting¶
Upon closing the kinetics configuration via the Fit button, TableCurve 2D's Automated Curve-Fit Processing begins and fits the kinetic equations as configured, as well as any installed UDFs and TCEX.DLL functions. The non-linear fitting controls in the kinetics configuration will also apply to such UDFs and external functions.