﻿ Equation: Three phase decay

# Equation: Three phase decay

## Introduction

An exponential decay equation models many chemical and biological processes. It is used whenever the rate at which something happens is proportional to the amount which is left.

A three-phase model is used when the outcome you measure is the result of the sum of a fast, medium and slow exponential decay. You need lots of data with little scatter to adequately fit a three phase model.

## Entering data

Create an XY data table. Enter time into X, and response (binding, concentration ..) into Y. If you have several experimental conditions, place the first into column A, the second into column B, etc.

After entering data, click Analyze, choose nonlinear regression, choose the panel of exponential equations, and choose Three phase decay.

## Consider constraining Plateau to a constant value of zero

If you have subtracted off any background signal, then you know the curve has to plateau at Y=0. In this case, you should constrain the parameter Plateau to be a constant value equal to zero. To do this, go to the Constrain tab of the nonlinear regression dialog, set the drop down next to Plateau to "Constant equal to" and enter the value 0.0.

## Model

YFast=(Y0-Plateau)*PercentFast*.01*exp(-KFast*X)

YSlow=(Y0-Plateau)*PercentSlow*.01*exp(-KSlow*X)

YMedium=(Y0-Plateau)*(100-PercentFast - PercentSlow)*.01*exp(-Kmedium*X)

Y=Plateau + YFast + YMedium +YSlow

Y0 is the Y value when X (time) is zero. It is expressed in the same units as Y,

Plateau is the Y value at infinite times, expressed in the same units as Y.

Kfast, Kmedium and Kslow are the rate constants, expressed in reciprocal of the X axis time units. If X is in minutes, the rate constants are expressed in inverse minutes.

Half-life (fast, medium and slow) are in the time units of the X axis. they are computed as ln(2) divided by the corresponding rate constant.

PercentFast is the percentage of the span (from Y0 to Plateau) accounted for by the fastest of the three components.

PercentSlow is the percentage of the span (from Y0 to Plateau) accounted for by the slowest of the three components.