This section contains descriptions of the coefficients for each of the nonlinear functions built into the Curve Fitter. You may want to use these as guidelines for estimating starting values for user defined functions.

Power Function

y=a0+a1*x^a2
a0 = y–offset, a1= x–factor, a2 = power

Exponential Function

y=a0+a1*exp(–x/a2)
decay:    
a0 = y–offset,  a1>0 a0+a1 y–intercept
a2>0 decay rate (x @ y = 0.3*a1+a0)
growth:   
a0 = max y (asymptote)
a1<0 a0–a1 y–intercept
a2>0 decay rate (x @ y = 0.3*a1+a0)

Peak Functions

2-Exponential
y=a0exp(–x/a1)+a2exp(–x/a3)

Sum of 2 Gaussians
y=a0exp(–0.5((x–a1)/a2)^2)+a3exp(–0.5((x–a4)/a5)^2)

Gaussian
y=a0+a1exp(–0.5((x–a2)/a3)^2)

Lorentzian
y=a0+a1/(1+((x–a2)/a3)^2)

Logistic Peak
y=a0+a14n/(1+n)^2 n = exp(–(x–a2)/a3)

Erfc Peak
y=a0+a1*erfc(((x–a2)/a3)^2)
a0 = y–offset, a1 = height (y–maximum)
a2= center (x @ peak), a3 = width of peak

Asymmetric

Log-Normal
y=a0+a1*exp(-0.5*(ln(x/a2)/a3)^2)
a0 = y–offset, a1 = height (y–maximum)
a2 = x–value @ peak, a3 = width of peak

S–Shaped Functions

Sigmoidal
y=a0+a1/(1+exp(–(x–a2)/a3))

Cumulative
y=a0+a10.5(1+erf((x–a2)/(sqrt(2)*a3)))
normal transition:
a0 = initial level (y–minimum)
a1 = range of transition (y–maximum – y–minimum)
a2 = middle of transition (x @ inflection point)
a3>0 =  width of transition
reverse transition:
a0 = initial level (y–minimum)
a1 = range of transition (y–maximum – y–minimum)
a2 = middle of transition (x @ inflection point)
a3<0 = width of transition

DoseRspLgstc
y=a0+a1/(1+(x/a2)^a3)
normal transition:
a0 = initial level (y–minimum)
a1 = range of transition (y–maximum – y–minimum)
a2 = middle of transition (x @ inflection point)
a3<0 = width of transition
reverse transition:
a0 = initial level (y–minimum)
a1 = range of transition (y–maximum – y–minimum)
a2 = middle of transition (x @ inflection point)
a3>0 = width of transition

pH Activity
y=(a0+a1*10^(x–a2))/(1+10^(x–a2))
normal S–transition:
a0 y–minimum (y @ low pH), a1 y–maximum (y @ high pH)
a2 –logKa  where Ka is the dissociation constant
reverse S–transition:
a0 y–minimum (y @ high pH), a1 y–maximum (y @ low pH)
a2 –logKa  where Ka is the dissociation constant

1–Site Ligand Binding/Michaelis–Menten Eqn.
y=a0*x/(a1+x)
a0 y–maximum, a1 x @ 1/2 y–maximum

2–Site Ligand Binding plus Nonspecific binding
y=a0*x/(a1+x)+a2*x/(a3+x)+a4*x

Photosynthesis Rate
y=a0*a1*x/(a0+a1*x)
a0 y–maximum, a0/a1 x @ 1/2 y–maximum

Fulcher Equation
y=a0+a1/(x–a2)
a0 y–asymptote, a1 “factor”, a2  x–asymptote

Waveform Functions

SineWave
y=a0+a1*sin(2*pi*x/a3+a2)
a0 y–offset about y=0, a1 amplitude (ymax – ymin)
a3 period, a2 (2*pi/a3)*phase shift

SineWaveSquared
y=a0+a1*(sin(2*pi*x/a3+a2))^2
a0 y–offset about mid of y–range
a1 amplitude (ymax – ymin)
a3 1/2 period, a2 (2*pi/a3)*phase shift