Equations for determining the TM and enthalpy of folding from
changes in ellipticity as a function of temperature.  These
equations are in a format suitable for SigmaPlot, but can
easily be converted to work with other plotting programs such as
PSIPLOT.  Note that all calculations are in calories/mole.

I. Calcuate Tm and enthalpy of folding which best fit the data of
a two state transition of a monomer from a folded to unfolded
state.  This treatment assumes that the heat capacity of the
folded and unfolded states are equal.

[ Parameters]

h=-20000 ;starting enthalpy in cal/mol
tm=50    ;starting tm (observed) in deg. C.
u=-40000; ellipticity of 100% folded peptide
l=-5000; ellipticity of unfolded peptide

[Variables]

v=col(1)   ;temperature in degrees C
w=col(2)   ; mean residue ellipticity, absorbance,etc

[Equations]

m=tm+273.15 ; convert Centigrade to Kelvin
t=v+273.15; convert Centigrade to Kelvin
k=exp((h/(1.987*t))*((t/m)-1)); calculate folding constant at any given temperature
y=k/(1+k); calculate fraction folded at any given temperature
f=((u-l)*y)+ l ; calculate ellipticity at any given temperature

fit f to w; fit the calculated ellipticity to the observed ellipticity

II.  Two state transition of a monomer between a folded and
unfolded state assumining that there is a change in
heat capacity of the folded and unfolded forms.  
These equations calculate the Tm of folding at K=1, the enthalpy 
of folding and the change in heat capacity (Delta Cp) which best 
fit the data.

jsv4R
[Parameters]
h=-25000 ; starting enthalpy
tm=30  ;starting tm deg. centigrade
u=-30000; mean residue ellipticity folded form
l=-5000;mean residue ellipticity unfolded form
p=-100; delta Cp in cal/(K.mol)

[Variables]
v=col(1)   ;temperature
w=col(2)   ;absorbance CD etc

[Equations]
m=tm+273.15; convert centigrade to kelvin
t=v+273.15; convert centigrade to kelvin
g=h*(1-t/m)-p*((m-t)+t*ln(t/m))
k=exp(-g/(1.987*t))
y=k/(1+k)
f=((u-l)*y)+ l 
fit f to w
[Constraints]

[Options]


III.  Two state transition between a folded dimer and an unfolded
monomer.  Assume Delta Cp=0.  Calculate Tm apparent at a given
protein concentration and enthalpy of folding.

[Parameters]

h=-20000; starting enthalpy of folding
tm=25;'starting Tm (observed) in degrees centigrade
u=-32000; ellipticity when peptide is fully folded
l=-5000; ellipticity when peptide is fully unfolded

[Variables]

v=col(1)   ;temperature
w=col(2)   ;absorbance CD etc

[Equations]

t=v+273.15; convert temperature to deg. K
m=tm+273.15; convert starting Tm observed to deg. K.
c=200e-6   ; molar concentration of folded two-chain species
k=exp(((h/(1.987*t))*((t/(m))-1))-ln(c)); calculated folding constant
q=(4*c*k+1)
e=(8*c*k+1)^(1/2)
d=4*c*k
y=(q-e)/d; calculated fraction folded
f=(u-l)*y+l; calculated ellipticity

fit f to w; fit calculated ellipticity to observed CD data.

IV.  Two state transition between folded dimer and unfolded
monomer assuming that there is a change in heat capacity , Delta
Cp upon folding.  Fit data to best Tm at K=1, enthalpy of folding
and Delta Cp.



[Parameters]

h=-20000 ;starting enthalpy
tm=72  ;starting Tm at K=1 in deg C
u=-32000; mean residue ellipticity of folded form
l=-5000; mean residue ellipticity of unfolded form
cp=-100; heat capacity change upon folding

[Variables]

v=col(1)   ;temperature deg C.
z=col(2)   ;absorbance CD etc

[Equations]

p=200e-6; protein concentration (put your own protein conc. here)
t=v+273.15  ;absolute temperature deg Kelvin
m=tm+273.15 ; tm in deg Kelvin
g=h+(cp*(t-m))-t*((h/m)+(cp*ln(t/m))); free energy of folding
k=exp(-g/(1.987*t)); folding constant
a=4*k*p^2
b=-8*k*p^2-p
c=4*k*p^2
y=((-b)-sqrt((b^2)-(4*a*c)))/(2*a); fraction folded
f=y*(u-l) +l; calculated ellipticity change at each temperature

fit f to z; fit the calculated ellipticity to the raw data



V.  Calculation of Tm(apparent) and Enthalpy of folding assuming
a folded trimer converts to an unfolded monomer in a two-state
transition.  


[Parameters]

h=-86000        ;starting enthalpy
tm=25           ;starting Tm apparent in deg C.
u=45            ;ellipticity value of 100% folded trimer
l=-1            ;ellipticity value of 100% unfolded monomer

[Variables]

v=col(1)   ;temperature deg C.
w=col(2)   ;ellipticity, absorbance change, etc.

[Equations]

m=tm+273.5; convert Tm in centrigrade to kelvin
t=v+273.15; convert centrigrade to kelvin
c=0.1e-3       ; concentration in molar
k=exp(((h/(1.987*t))*((t/m)-1))-(ln(.75)*c^2)); calculated folding constant
z=3*c^2
q=((3*k*z)+1)/(k*z)
e=(1/3)*((3*q)-(-3)^2)
d=(1/27)*((2*(-3)^3)-(9*(-3)*q)+(27*(-1)))
A=(((-1)*(d/2))+((d^2/4)+(e^3/27))^(1/2))^(1/3)
B=(-1)*((d/2)+((d^2/4)+(e^3/27))^(1/2))^(1/3)
X=A+B
y=X+1; calculated fraction folded at any temperature
f=((u-l)*y)+ l; calculated ellipticity

fit f to w; fit calculated ellipticity to to raw data

VI.  Multiple transitions.
Fit the unfolding of a dimeric protein such as tropomyosin by
three unfolding transitions, assuming that dissociation occurs at
the transition with the highest Tm.  Find the apparent Tm values
of folding of each transition, the enthalpy of folding of each
transition, and the proportion of the ellipticity change
associated with each transition. Assume Delta Cp=0.

[Parameters]

'normalize data between 1 =folded and 0=unfolded
 fit data by three transitions,
'assume that third transition associated
'with dissociation of dimer to monomer
'fix lower limit at 0
'starting enthalpies
h1=-30000
h2=-70000
h3=-100000
'starting TM values in degrees Centigrade
m1=25
m2=40
m3=46 
'starting extinction coefficients
u1=.2
u2=.4
u3=.4

[Variables]

v=col(1)   ;temperature in Deg. C.
w=col(2)   ;absorbance CD etc

[Equations]

t=v+273.15 ; convert Centigrade to Kelvin
k1=exp((h1/(1.987*t))*((t/(m1+273.15))-1))
k2=exp((h2/(1.987*t))*((t/(m2+273.15))-1))
c=10e-6     ; molar concentration of folded dimer
k3=exp(((h3/(1.987*t))*((t/(m3+273.15))-1))-ln(c))
q=(4*c*k3+1)
e=(8*c*k3+1)^(1/2)
d=4*c*k3
y3=(q-e)/d
y1=k1/(1+k1)
y2=k2/(1+k2)
f=(u1*y1)+(u3*y3)+(u2*y2)
fit f to w

[Constraints]

u1>0
u2>0
u3>0
m2>m1
m3>m2


VII.  Calculate the Zimm-Bragg parameters of folding of a
monomeric peptide assuming one folding transition.  

[Parameters]

'B.H. Zimm and J.K. Bragg
'J. Chem Phys 31:526-535 (1959)
'equation 3b
a=0.03 ;initial sigma value
h=-1200  ;initial enthalpy of transition in cal/residue
tm=40; initial Tm of transition
uu=-40000 ;ellipticity of fully folded helical peptide.
ll=-5000 ;ellipticity of fully unfolded peptide.
[Variables]

x=col(1); temperature in deg. C.
y=col(2); fraction folded at any temperature C. 

[Equations]

n=29 ; number of residues in peptide (put your number here)
m=tm+273.15; convert Centigrade to Kelvin.
t=x+273.15; convert Centigrade to Kelvin
s=exp((h/1.987)*((t-m)/(t*m))); calculated helical propensity at any given temperature
u=(n-3)*(s-1)-2+((n-3)*(s-1)+2*s)*(s^(-n+2))
j=(((s-1)^2)*(s^(-n+1)))/a
i=(1+j-((n-3)*(s-1)+s)*(s^(-n+2)))
k=u/((n-3)*(s-1)*i); calculated fraction folded
k2=((uu-ll)*k)+ll

fit k2 to y ;fit calculated CD curve to observed data.


VIII.  Alternate treatment of Zimm-Bragg equation.  
Zipper Model of Cantor and Shimmel assuming one 
helical region in a short peptide unfolds as a function of 
temperature. 

[Parameters]

;Cantor and Shimmel- Biophysical Chemistry Vol III pg 1059
;equation 20-37
a=.0035; starting sigma value
h=-1100; starting enthalpy in calories/peptide H-bond
tm=53.8; Tm in deg. C.
uu=-44000; Ellipticity of fully helical peptide
ll=-5000; Ellipticity of fully unfolded peptide

[Variables]
x=col(1)
y=col(2)

[Equations]
n=26; number of H-bonds in backbone of folded peptide-put your value here
; NOTE H-bonds =amide groups in peptides -3.
m=tm+273.15 ; convert centigrade to Kelvin
t=x+273.15  ; convert centigrade to Kelvin
s=exp((h/1.987)*((t-m)/(t*m))); variation of helical propensity with temp.
d=(a*s)/((s-1)^3)
e=(n*s^(n+2)) -(n+2)*(s^(n+1)) +s*(n+2) -n
f=n*(1+ (a*s/((s-1)^2))* ((s^(n+1)) +n -((n+1)*s)))
k=d*e/f ; fraction folded
w=((uu-ll)*k)+ll; calculated mean residue ellipticity
fit w to y; fit calculated to observed ellipticity

[Constraints]
a>0
a<1
tm>0
h<0