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Models for Chemical and Temperature Denaturation of Proteins

Mike Blaber2019

Isothermal Equilibrium Denaturation of a Protein by Denaturant

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Guanidine Hydrochloride (M)

Fibroblast Growth Factor-1 Denaturation by GuHCl

(298K)

Example: • Fibroblast growth factor-1 (FGF-1), a monomeric globular protein• Denaturant: Guanidine hydrochloride (GuHCl)• Temperature: 298 K• Spectroscopic signal: Fluorescence (λex = 295nm; λem=300-500 nm integrated)

The overall spectroscopic signal of the system as a function of denaturant (D) is:

S(D) = (SF * XF) + (SU * XU)

Where

• SF is the spectroscopic signal of the folded protein

• SU is the spectroscopic signal of the unfolded protein

Spectroscopic probes of protein unfolding are typically fluorescence or circular dichrosim

• The key point is that the spectroscopic signal must discriminate between the F and U states

• If the F and U states have the same spectroscopic signal, then that signal cannot be used to monitor the unfolding process

Denaturant-induced protein unfolding monitored by spectroscopic signal

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Protein is unfolded(denatured state)

Protein is folded(native state)

Protein is undergoinga transition from

native to denatured state

Question:

What proportion (i.e. fraction) of the total protein is in the

folded/native (F or N) state, and in the unfolded/denatured (U or D)

state, as a function of denaturant concentration [D]?

Key to answering this question is to define the fluorescent signal N

and D state baselines

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Folded state baseline:Spectroscopic signal of the

folded stateSF = (mf*[D])+bf

Unfolded state baseline:Spectroscopic signal of

the unfolded stateSU = (mu*[D])+bu

Baselines for the F and U states are modeled as

linear functions (independent slopes)

Fraction folded (XF) as a function of denaturant can now be determined:

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SF = (mf*[D])+bf

SU = (mu*[D])+bu

Y(u-f) = (mu*[D])+bu – ((mf*[D])+bf)= (mu-mf)*[D] + bu - bf

(mu*[D])+bu-Yexp

Yexp-((mf*[D])+bf)

XF =(mu-mf)*[D]+bu-bf

XU =(mu-mf)*[D]+bu-bf

Yexp-((mf*[D])+bf)

Fraction folded (XF) as a function of denaturant:

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Fraction folded (XF) as a function of denaturant:

XF =(mu-mf)*[D]+bu-bf

XU =(mu-mf)*[D]+bu-bf

Yexp – ((mf*[D])+bf)

1 – XF = (mu-mf)*[D]+bu-bf

–(mu-mf)*[D]+bu-bf

(mu-mf)*[D]+bu-bf


(mu*[D]) – (mf*[D]) + bu – bf – (mu*[D]) – bu + Yexp


(mu*[D]) – (mf*[D]) + bu – bf – (mu*[D]) – bu + Yexp


– (mf*[D]) – bf + Yexp=

(mu-mf)*[D]+bu-bf

Yexp – ((mf*[D]) + bf)

1 – XF = XU 1 – XU = XF

Simple two-state unfolding:

F ↔ U

Ku = [U]/[F]

[U] = ΧU * [Co]

Where ΧU is the mole fraction of U state, and Co is the total protein concentration

[F] = ΧF * [Co]

Where ΧF is the mole fraction of F state, and Co is the total protein concentration

Ku = (ΧU * [Co]) / (ΧF * [Co]) = ΧU / ΧF

Since ΧF = (1 - ΧU)

Ku = ΧU /(1 - ΧU)

XF and XU define the equilibrium unfolding constant Ku

ΔGu is related to Ku:

ΔGu = -R*T*ln(Ku)

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Region of greatest error

Region of greatest error

Region of greatest accuracy

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For most proteins the ΔGu vs. [D] relationship is adequately modeled by a linear function:

ΔGu = (m-value*[D]) + ΔGu0

• m-value is the slope of the ΔGu(D) function (often reported omitting the negative sign)

• ΔGu0 is the value of ΔGu extrapolated to 0M denaturant (the derived value of ΔGu “in water”)

Pace, C. N. (1986) Methods in Enzymology 131: 266-280

Recall that the overall spectroscopic signal of the system is:

S(D) = (SF * XF) + (SU * XU)And

XF = (1 – XU)

Thus, if we can solve the general expression for XU as a function of [D] (and if we know the linear baseline functions) we can obtain expression describing the observed spectroscopic signal.

Ku = exp(ΔGu /(-R*T))

Since: ΔGu = ((m-value*[D]) + ΔGu0) (this is the linear extrapolation model of ΔGu(D))

Ku = exp(((m-value*[D]) + ΔGu0) /(-R*T))

Since: XU = KU /(KU+1)

XU = exp(((m-value*[D]) + ΔGu0)/(-R*T)) / (exp(((m-value*[D]) + ΔGu0)/(-R*T)) + 1)

And:

XF = 1/ (exp(((m-value*[D]) + ΔGu0)/(-R*T)) + 1)

Putting the complete model together for simple F ↔ U two-state protein unfolding:

Recall that in spectroscopic signal of the system:S(D) = (SF * XF) + (SU * XU)

andSU = ((mu*[D])+bu)SF = ((mf*[D])+bf)

The spectroscopic signal (i.e. experimental data) is thus modeled as:

S(D) = (((mf*[D])+bf)*(1/(exp(((m-value*[D])+Gu0)/(-R*T)) + 1))) +(((mu*[D])+bu)*(exp(((m-value*[D])+Gu0)/(-R*T))/(exp(((m-value*[D])+Gu0)/(-R*T))+1)))

This equation becomes easier to manage if we define sub-functions:• F1 = (m-value*[D])+Gu0 (note: this is the linear Gu(D) function)• F2 = exp(F1/(-R*T)) (note: this is Ku)• F3 = F2/(F2 + 1) (note: this is XU)• F4 = 1/(F2 + 1) (note: this is XF)• F5 = (mf*[D])+bf (note: this is SF)• F6 = (mu*[D])+bu (note: this is SU)

S(D) = (F5 * F4) + (F6 * F3)

The completed S(D) function has the following 6 parameters:

1. m-value (mv) the slope of the Gu(D) function (a measure of folding cooperativity)2. Gu0 (dg) the value of Gu at 0M denaturant (a measure of protein stability)3. mf the slope of the folded state spectroscopic signal4. bf the y-int of the folded state spectroscopic signal5. mu the slope of the unfolded state spectroscopic signal6. bf the y-int of the unfolded state spectroscopic signal

Plus:R gas constant (8.314 J mol-1 K-1)T temperature in K (typically 298)

The above equation is put into a non-linear least squares program and the raw data is fit to this model to generate the above parameters.

S(D) = (((mf*[D])+bf)*(1/(exp(((m-value*[D])+Gu0)/(-R*T)) + 1))) +(((mu*[D])+bu)*(exp(((m-value*[D])+Gu0)/(-R*T))/(exp(((m-value*[D])+Gu0)/(-R*T))+1)))

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dg 21,840 J mol-1

mv -19,371 J mol-1 M

-1

mf 418,208

bf 1,424,426

mu 831,325

bu 30,808,352

R 8.314 J mol-1 K

-1

T 298 K

Exp data

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Data fitting and analysis

• The concentration of denaturant [D] where the protein unfolding is half-completed is an important parameter:

Ku = ΧU /(1 - ΧU) = 0.5/0.5 = 1.0

• At this Ku the value of ΔGu is:

ΔGu = -R*T*ln(Ku) = 0

• In the linear definition of the ΔGu function:

ΔGu = (m-value*[D]) + ΔGu0

• This = 0 at the [D] where XU = 0.5

0 = (m-value*[D]) + ΔGu0

[D]XU=0.5 = -(ΔGu0/m-value) = Cm (midpoint of denaturation)

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Cm = (21,840/19,371)

= 1.13 M

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(298K)

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The important parameters to report in the analysis are the following:• ΔGu0

• the value of ΔGu the absence of denaturant (i.e. “in water”)• Units are J mol-1 (or kJ mol-1)

• m-value• Indicates how ΔGu changes with denaturant; sometimes referred to as the “folding

cooperativity”• Units are J mol-1 M-1

• Cm• Midpoint of the unfolding transition• Calculated by Cm = -(ΔGu0/m-value), units are M• The higher the Cm, the more stable the protein

In a variant of the fitting model, the ΔGu linear model ΔGu = (m-value*[D])+ ΔGu0is defined differently:

ΔGu = m-value*([D]-Cm)• This refines Cm parameter, but not ΔGu0• ΔGu0 is calculated by (-Cm*m-value)• This alternative definition for the ΔGu function (while mathematically equivalent) is

considered more accurate in fitting since the Cm parameter is located at the most accurately defined region of the ΔGu function (i.e. at the Cm)

ΔGu = m-value*([D]-Cm)Param Value ErrorCm 1.127 1.822E-03R 8.314 0T 298 0bf 1,424,426 60,816bu 30,808,352 234,331mf 418,208 167,263mu 831,325 111,242mv -19,371 215ΔGu0 (calc)21,831

ΔGu = (m-value*[D])+ΔGu0Param Value ErrorR 8.314 0T 298 0bf 1,424,426 60,816bu 30,808,352 234,331dg 21,840 250mf 418,208 167,263mu 831,325 111,242mv -19,371 215Cm (calc) 1.127

Comparison of fitting models:

Modifications for oligomeric folded states

If folded state is a dimer:F ↔ 2U

Ku = [U]2/[F]

[U] = ΧU * Co


[F] = ΧF * Co/2


Ku = (ΧU * Co)2 / ΧF * Co/2

Ku = 2*ΧU2*Co / ΧF


Define the protein concentration as Co (in terms of total monomer concentration)

Ku = 2*ΧU2*Co / (1 - ΧU)

Notice now that Ku includes a term for the total concentration Co

Ku = 2*ΧU2*Co / (1 - ΧU)

Solving for XU, there are two possible solutions:

XU = -(sqrt(KU2+8*Co*KU)+KU)/(4*Co)

Or

XU = (sqrt(KU2+8*Co*KU)-KU)/(4*Co) (Correct solution)

Former relationships derived for the simple monomer two-state model still hold:

XF = (1 – XU)

Ku = exp(((m-value*[D])+Gu0)/(-R*T))

S(D) = (SF * XF) + (SU * XU)

However, an important difference now exists in the dimer model compared to simple monomer:

• The midpoint of the transition, Cm, is where XF = XU = 0.5

• In other words, the concentration of denaturant where half the protein had unfolded

• In the simple monomer model this was also the denaturant concentration where Gu = 0, because

Ku = ΧU / ΧF = 0.5/0.5 = 1

• However, in the dimer model, the denaturant concentration where ΔGu = 0 is not the same denaturant concentration where XU, XF = 0.5

At the denaturant concentration where XU, XF = 0.5 in the dimer model:

Ku = 2*ΧU2*Co / (1 - ΧU)

Ku = 2*(0.5)2*Co / 0.5

Ku = Co

At the denaturant concentration where XU, XF = 0.5 in the dimer model:

Ku = Co

And the relationship between Ku and ΔGu:

ΔGu = -R*T*ln(Ku)

(m-value*[D])+Gu0 = -R*T*ln(Co)

m-value*[D] = -R*T*ln(Co) - Gu0

[D]Xu=0.5 = (-R*T*ln(Co) - Gu0)/m-value

The denaturant concentration where ΔGu = 0 in the dimer model:

ΔGu = (m-value * [D]) + ΔGu0

0 = (m-value * [D]) + ΔGu0

[D]ΔGU=0 = -ΔGu0 / (m-value)

Ku = 2*ΧU2*Co / (1 - ΧU)

Interpreting the equations

• Ku is the unfolding equilibrium constant, and for the dimer model it can be considered as the dissociation constant for the dimer (as dissociation corresponds to unfolding)

• The protein concentration can therefore influence the dissociation • In fact, Ku is directly proportional to Co:

Stable dimer = lower Co at which it dissociates = Smaller Ku (less dissociation/unfolding)• This indicates that the dimer can dissociate even in the absence of denaturant (we just

need to go to a really low protein concentration)

F ↔ 2U

Ku = [U]2/[F]

[D]Xu=0.5 = (-R*T*ln(Co) - Gu0)/m-value

• Since the unfolding equilibrium is affected by concentration, the [D] at which XU=0.5 is dependent upon concentration

[D]ΔGu=0 = -ΔGu0 / (m-value)

• ΔGu = 0 is unaffected by protein concentration – it is a Gibbs energy term intrinsic to the protein stability• [D]Xu=0.5 = [D]ΔGu=0 when Co = 1.0 Molar

Protein concentration affects the unfolding equilibrium of a multimer, but does not affect the intrinsic protein stability (Gibbs energy)


If folded state is a trimer:F ↔ 3U

Ku = [U]3/[F]

[U] = ΧU * Co


[F] = ΧF * Co/3


Ku = (ΧU * Co)3 / ΧF * Co/3

Ku = 3*ΧU3*Co2 / ΧF



Ku = 3*ΧU3*Co2 / (1 - ΧU)

Ku = 3*ΧU3*Co2 / (1 - ΧU)

Solving for XU there are three possible solutions; however, two of these have imaginary numbers, so the solution is:

XU = ((Ku*sqrt(4*Ku+81*Co^2))/(54*Co^3)+Ku/(6*Co^2))^(1/3)-Ku/(9*Co^2*((Ku*sqrt(4*Ku+81*Co^2))/(54*Co^3)+Ku/(6*Co^2))^(1/3))


XF = (1 – XU)


S(D) = (SF * XF) + (SU * XU)


ΔGu = -R*T*ln(Ku)

(m-value*[D])+Gu0 = -R*T*ln(3*Co2/4)

m-value*[D] = -R*T*ln(3*Co2/4) - Gu0

[D]Xu=0.5 = (-R*T*ln(3*Co2/4) - Gu0)/m-value

At XU = 0.5Ku = 3*Co2/4

Co at which [D]Xu=0.5 = [D]ΔGu=0

3*Co2/4 = 1Co = SQRT(4/3)Co = 1.1547 M



If folded state is a tetramer:F ↔ 4U

Ku = [U]4/[F]

[U] = ΧU * Co


[F] = ΧF * Co/4


Ku = (ΧU * Co)4 / ΧF * Co/4




Ku = 4*ΧU4*Co3 / (1 - ΧU)

Ku = 4*ΧU4*Co3 / (1 - ΧU)

Solving for XU

• There are four solutions, the correct one is:

XU=sqrt((sqrt(3)*Ku)/(2*Co^2*sqrt((3*Co^3*((Ku*sqrt(Ku*(27*Ku+1024*Co^3)))/(32*3^(3/2)*Co^6)+Ku^2/(32*Co^6))^(2/3)-Ku)/(Co*((Ku*sqrt(Ku*(27*Ku+1024*Co^3)))/(32*3^(3/2)*Co^6)+Ku^2/(32*Co^6))^(1/3))))-((Ku*sqrt(Ku*(27*Ku+1024*Co^3)))/(32*3^(3/2)*Co^6)+Ku^2/(32*Co^6))^(1/3)+Ku/(3*Co^3*((K

u*sqrt(Ku*(27*Ku+1024*Co^3)))/(32*3^(3/2)*Co^6)+Ku^2/(32*Co^6))^(1/3)))/2-sqrt((3*Co^3*((Ku*sqrt(Ku*(27*Ku+1024*Co^3)))/(32*3^(3/2)*Co^6)+Ku^2/(32*Co^6))^(2/3)-Ku)/(Co*((Ku*sqrt(Ku*(27*Ku+1024*Co^3)))/(32*3^(3/2)*Co^6)+Ku^2/(32*Co^6))^(1/3)))/(2*sqrt(3)*Co)


XF = (1 – XU)


S(D) = (SF * XF) + (SU * XU)

At XU = 0.5

Ku = 4*(0.5)4*Co3 / 0.5

Ku = Co3/2


ΔGu = -R*T*ln(Ku)

(m-value*[D])+Gu0 = -R*T*ln(Co3/2)

m-value*[D] = -R*T*ln(Co3/2) - Gu0

[D]Xu=0.5 = (-R*T*ln(Co3/2) - Gu0)/m-value

Co at which [D]Xu=0.5 = [D]ΔGu=0

Co3/2 = 1Co3 = 2

Co = 1.26 M


Unfolding with an intermediate state (three state unfolding)

The unfolding pathway of some monomeric proteins proceeds through an observable intermediate (stable and significantly populated)

F I UKFI KIU

KFI =[I]

[F]=

XI * Co

XF * Co=

XI

XF

KIU =[U]

[I]=

XU * Co

XI * Co=

XU

XI

XF + XI + XU = 1

KFI =XI

XF

=1 – XF – XU

XF

KIU =XU

XI

=1 – XF – XU

XU

KFI in terms of XF and XU

KIU in terms of XF and XU

XI = 1 – XF – XU

KFI = exp(((mfi*[D])+bfi)/(-R*T))

KIU = exp(((miu*[D])+biu)/(-R*T))

GFI = (mfi*[D])+bfi

GIU = (miu*[D])+biu

GFI = -R*T*ln(KFI)and

KFI = exp(GFI/-R*T)

Likewise

GIU = -R*T*ln(KIU)and

KIU = exp(GIU/-R*T)

Adopting a linear extrapolation model for GFI(D) and GIU(D):

Therefore

Parameters defined so far:• mfi slope of GFI(D) function• bfi GU0 of N state• miu slope of GIU(D) function• biu GU0 of I state

Known relationships:

XF + XI + XU = 1 KFI =XI

XF

KIU =XU

XI

thus KIU =XU

KFI*XF

XU

XF

KFI*KIU= XU = XF*KFI*KIU

Solve for XF in terms of equilibrium constants only:

XF = 1 – XI – XU

Divide both sides by XF

1 = (1/XF) – (XI/XF) – (XU/XF)

1 = (1/XF) – (KFI) – (KFI*KIU)

1 + KFI + (KFI*KIU) = 1/XF

XF = 1/(1 + KFI + (KFI*KIU))or

XF = 1/(1 + KFI*(1 + KIU))



XF

KIU =XU

XI

thus KIU =XU

KFI*XF

XU

XF


Solve for XU in terms of equilibrium constants only:

XU = 1 – XI – XF

Divide both sides by XU

1 = (1/XU) – (XI/XU) – (XF/XU)

1 = (1/XU) – (1/KIU) – (1/KFI*KIU)

1 + (1/KIU) + (1/KFI*KIU) = 1/XU

(KFI*KIU/KFI*KIU) + (KFI/KFI*KIU) + (1/KFI*KIU) = 1/XU

(1 + KFI + KFI*KIU)/(KFI*KIU) = 1/XU

XU = (KFI*KIU)/(1 + KFI + KFI*KIU) or XU = (KFI*KIU)/(1 + KFI*(1 + KIU))

XF = 1/(1 + KFI*(1 + KIU)) XU = (KFI*KIU)/(1 + KFI*(1 + KIU)) XI + = 1 – XF – XU

S = (SF*XF)+(SI*XI)+(SU*XU)

SF = (mf*[D])+bf

SI = (mi*[D])+bi

SU = (mu*[D])+bu

The spectroscopic signal of the F, I and U states are modeled as linear functions:

Parameters:• mf slope of the N(D) spectroscopic function• bf Y-int of the N(D) spectroscopic linear function• mi slope of the I(D) spectroscopic function• bi Y-int of the I(D) spectroscopic linear function• mu slope of the U(D) spectroscopic function• bu slope of the U(D) spectroscopic linear function

KFI = exp(GFI /(-R*T)) = exp(F1/(-R*T)) = F3

KIU = exp(GIU /(-R*T)) = exp(F2/(-R*T)) = F4

GFI = (mfi*[D])+bfi = F1

GIU = (miu*[D])+biu = F2


XF = 1/(1 + KFI*(1 + KIU)) = 1/(1 + F3*(1 + F4)) = F6

XU = (KFI*KIU)/(1 + KFI*(1 + KIU)) = (F3*F4)/(1+F3*(1+F4)) = F5

XI = 1 – XF – XU = 1 – F6 – F5


SF = (mf*[D])+bf SI = (mi*[D])+bi SU = (mu*[D])+bu

Final parameters:• mfi m-value for GFI function• bfi G0 for F state• miu m-value for GIU function• biu G0 for I state• mf slope of F spectroscopic function• bf y-int of F spectroscopic function• mi slope of I spectroscopic function• bi y-int of I spectroscopic function• mu slope of U spectroscopic function• bu y-int of U spectroscopic function• R Gas constant• T Temp (K)

Denaturation of a Protein by Temperature and Monitored by Spectroscopy

Example: • Fibroblast growth factor-1 (FGF-1), a monomeric globular protein• Spectroscopic signal: Fluorescence (λex = 295nm; λem=300-500 nm integrated)• Scanning temperature: 280 - 370

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Protein Denaturation by Temperature

(Monitored by Fluorescence)

Gu function

Gu = -R*T*ln(Ku) = Hu – T*Su

There are two energetic terms contributing to the overall Gibbs energy of protein unfolding:

1. ΔHu (the “enthalpy of unfolding”)

2. -T*ΔSu (the “entropy of unfolding” scaled by the (negative) temperature)

ΔHu is the “Enthalpy of unfolding”

• Its value is large and positive

• It is an unfavorable contributor to ΔGu (i.e. the change in enthalpy opposes unfolding)

• Unfolding requires energy (i.e. heat) input in order to overcome the numerous energetically favorable non-covalent interactions (e.g. H-bonds, salt bridges, van der Waals) that maintain the folded structure

ΔSu is the “Entropy of unfolding”

• Entropy (i.e. disorder of the system) increases as a protein unfolds

• This is principally due to an increase in conformational disorder of the polypeptide chain

• ΔSu is positive for protein unfolding

• Entropy change is scaled by the temperature at which such change in disorder occurs in order to quantify the associated Gibbs energy change

• A given increase in disorder is more energetically significant when it occurs at a higher temperature

• Consider a system near absolute 0 K

• No disorder

• A large change in disorder is associated with a small energy change

• Consider the surface of the sun (10,000K)

• Incredibly high disorder

• A small change in disorder would be associated with a large energy change

• For a given change in disorder (ΔS) the associated Gibbs energy change is proportional to the T at which such change in disorder occurs

• ΔS is scaled by negative T, since an increase in disorder (i.e. positive ΔS) is a favorable contributor (i.e. negative) to the overall Gibbs energy change (ΔG)

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Hu(T)

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Su(T)

Heat energy needed to unfold (J mol-1) Increase in disorder upon unfolding (J mol-1 K-1)

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-450,000

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-350,000

-300,000

-250,000

-200,000

-150,000

-100,000

-50,000

0

273 293 313 333 353 373

-T*Su(T)

-80,000

-60,000

-40,000

-20,000

0

20,000

40,000

60,000

80,000

273 293 313 333 353 373

ΔGu(T)

Gibbs energy associated with -T*S (J mol-1) Resultant overall G (J mol-1)

Disorder is increasingly favored at higher T

Gu = Hu – T*Su

• ΔHu and -T*ΔSu are large, and opposite energetic terms

• The prior example was typical:

• ΔHu = 400,000 J mol-1 (400 kJ mol-1)

• -T*ΔSu = ~-375,000 J mol-1 (-375 kJ mol-1) at around room temperature

• The net balance (i.e. ΔGu ) is a small magnitude (~+25 kJ mol-1) keeping the protein in the folded state

• This energetic balance shows why some point mutations are sufficiently destabilizing to prevent protein folding

• One really strong salt bridge can approach 25 kJ mol-1 of stability

Another effect upon the temperature denaturation of proteins:Cp

Hu is the heat energy absorbed by proteins during unfolding

• There is another process that occurs during denaturation that also absorbs heat energy

• The folded (native) state and the unfolded (denatured) state have unique heat capacities and these are not identical

• The denatured state exposes a large number of hydrophobic groups that are buried (i.e. solvent inaccessible) in the native state

• Exposure of hydrophobic groups causes water to organize around such groups (sometimes termed an “ice-like clathrate” water structure)

• The organized water molecules have a higher ability to absorb heat energy than regular water (since the organized clathrate can increase in disorder in response to the absorption of such heat energy)

• The denatured state has a higher heat capacity than the native state

• The difference in native and denatured state heat capacity is Cp (“the difference in constant pressure heat capacity”)

• A more accurate description of Hu and Su takes into account the effect of Cp in the thermal unfolding process

Accounting for Cp in thermal unfolding

Hu(T) = Hu(Tm) + Cp*(T – Tm)

C

p (

J m

ol-1

K-1

)

T (K)

Tm

CpU (unfolded)

CpF (folded)

Cp

• Above the melting temperature (Tm) more heat energy than Hu is associated with unfolding the protein due to the effects of Cp• Below the melting temperature (Tm) less heat energy than Hu is associated with unfolding the protein due to the effects of Cp

0

50,000

100,000

150,000

200,000

250,000

300,000

350,000

400,000

450,000

500,000

273 293 313 333 353 373

Hu(T)


Red: Cp = 0Blue: Cp = 1000 J mol-1 K-1

Tm = 333

Hu(Tm) 400,000 J mol-1


• Su is the increase in disorder associated with protein unfolding (mostly the increase in configurational entropy of the polypeptide chain)

• The increase in heat capacity above the Tm (associated with formation of water clathratearound hydrophobic groups) is indicative of an increased ability of the system to “soak up heat energy” via increasing disorder (of organized clathrates)

• The value of Su(T) is greater than Su when T is above the Tm, and less when it is below the Tm

Su(T) = Su(Tm) + Cp*(ln(T) – ln(Tm))

0

200

400

600

800

1,000

1,200

1,400

273 293 313 333 353 373

Su(T)

Tm

Cp = 0

Cp = 1000

-600,000

-500,000

-400,000

-300,000

-200,000

-100,000

0

273 293 313 333 353 373

-T*Su(T)


• -T*Su(T) is the Gibbs energy contribution of the entropy change

• The increase in heat capacity above the Tm (associated with formation of water clathratearound hydrophobic groups) is indicative of an increased ability of the system to “soak up heat energy” via increasing disorder (of organized clathrates)

• The value of Su(T) is greater than Su when T is above the Tm, and less when it is below the Tm

Su(T) = Su(Tm) + Cp*(ln(T) – ln(Tm))

Tm

Cp = 0

Cp = 1000


Su(T) = Su(Tm) + Cp*(ln(T) – ln(Tm)) = Su(Tm) + Cp*ln(T/Tm)


At T = Tm, Gu(Tm) = 0 = Hu(Tm) – Tm*Su(Tm)So

Su(Tm) = Hu(Tm)/Tm

ThusSu(T) = Hu(Tm)/Tm + Cp*ln(T/Tm)

Gu(T)= Hu(Tm) + Cp(T – Tm) – T*(Hu(Tm)/Tm + Cp*ln(T/Tm))

Gu(T)= Hu(Tm) + Cp(T – Tm) – T*(Hu(Tm)/Tm) – T*(Cp*ln(T/Tm))

Gu(T)= Hu(Tm)*(1 – T/Tm) + Cp*(T – Tm– T*ln(T/Tm))


Gu(T)= Hu(Tm)*(1 – T/Tm) + Cp*(T – Tm– T*ln(T/Tm))

The Cp term introduces curvature into the Gu(T) function: (higher Cp, greater curvature)

-60,000

-40,000

-20,000

0

20,000

40,000

60,000

80,000

273 283 293 303 313 323 333 343 353 363 373

ΔGu(T)

Tm

Cp = 0

Cp = 2000 J mol-1 K-1

Simple two-state monomer heat unfolding monitored by spectroscopic signal: the model

F1 = HTm + Cp*(T – Tm) Hu(T)

Parameters:Tm: Tm (the melting temperature where XU = XF = 0.5; unfolding is half completed)DHtm: HTm (the value of Hu(T) at the Tm)DCp: Cp (the value of Cp, modeled as a constant) R: Gas constant 8.314 J mol-1 K-1

T: Temperature (X values of dataset)

F2 = -T*(Hu(Tm)/Tm + Cp*ln(T/Tm)) -T*Su(T)

F3 = F1 + F2 Gu(T)

F4 = exp(F3/-R*T) Ku

F5 = F4(F4+1) XU

F6 = 1-F5 XF

Parameters:Tm: Tm (the melting temperature where XU = XF = 0.5; unfolding is half completed)DHtm: HTm (the value of Hu(T) at the Tm)DCp: Cp (the value of Cp, modeled as a constant) R: Gas constant 8.314 J mol-1 K-1

mf: slope of folded state spectroscopic baselinebf: Y-int of folded state spectroscopic signalmu: slope of folded state spectroscopic baselinebu: Y-int of folded state spectroscopic signalT: Temperature (X values of dataset)

F5 = F4(F4+1) XU

F6 = 1-F5 XF

F7 = (mf*T)+bf SF (folded spectroscopic baseline)

F8 = (mu*T)+bu SU (unfolded spectroscopic baseline)

F9 = (F7*F6)+(F8*F5) S(T) exp spectroscopic signal

The folded (native) and unfolded (denatured) spectroscopic baselines are modeled as linear functions:

Simple two-state monomer heat unfolding monitored by spectroscopic signal: the model

Modification to heat unfolding spectroscopic model: dimer folded state


Ku = [U]2/[F]

[U] = ΧU * Co


[F] = ΧF * Co/2


Ku = (ΧU * Co)2 / ΧF * Co/2




Ku = 2*ΧU2*Co / (1 - ΧU)

Hu(T) = HTm + Cp*(T – Tm) = F1

Parameters:Co: Protein concentration (in terms of monomer)Tm: Tm (the melting temperature where XU = XF = 0.5; unfolding is half completed)DHtm: HTm (the value of Hu(T) at the Tm)DCp: Cp (the value of Cp, modeled as a constant) R: Gas constant 8.314 J mol-1 K-1


-T*Su(T) = -T*(Hu(Tm)/Tm + Cp*ln(T/Tm)) = F2

Gu(T) = F1 + F2 = F3

Ku = exp(F3/-R*T) = F4

XF = (1-F5) = F6

Ku = 2*ΧU2*Co / (1 - ΧU)

XU = (sqrt(KU2+8*Co*KU)-KU)/(4*Co)

XU = (sqrt(F42+8*Co*F4)-F4)/(4*Co) = F5


Ku = [U]3/[F]

[U] = ΧU * Co


[F] = ΧF * Co/3


Ku = (ΧU * Co)3 / ΧF * Co/3




Ku = 3*ΧU3*Co2 / (1 - ΧU)

Modification to heat unfolding spectroscopic model: trimer folded state

Hu(T) = HTm + Cp*(T – Tm) = F1




Gu(T) = F1 + F2 = F3


XF = (1-F5) = F6

Ku = 3*ΧU3*Co2 / (1 - ΧU)


XU = ((F4*sqrt(4*F4+81*Co^2))/(54*Co^3)+F4/(6*Co^2))^(1/3)-F4/(9*Co^2*((F4*sqrt(4*F4+81*Co^2))/(54*Co^3)+F4/(6*Co^2))^(1/3)) = F5


Ku = [U]4/[F]

[U] = ΧU * Co


[F] = ΧF * Co/4


Ku = (ΧU * Co)4 / ΧF * Co/4




Ku = 4*ΧU4*Co3 / (1 - ΧU)

Modification to heat unfolding spectroscopic model: tetramer folded state

Hu(T) = HTm + Cp(T – Tm) = F1


Gu(T) = F1 + F2 = F3


Ku = 4*ΧU4*Co3 / (1 - ΧU)



XF = (1-F5) = F6

XU=sqrt((sqrt(3)*F4)/(2*Co^2*sqrt((3*Co^3*((F4*sqrt(F4*(27*F4+1024*Co^3)))/(32*3^(3/2)*Co^6)+F4^2/(32*Co^6))^(2/3)-F4)/(Co*((F4*sqrt(F4*(27*F4+1024*Co^3)))/(32*3^(3/2)*Co^6)+F4^2/(32*Co^6))^(1/3))))-((F4*sqrt(F4*(27*F4+1024*Co^3)))/(32*3^(3/2)*Co^6)+F4^2/(32*Co^6))^(1/3)+F4/(3*Co^3*((F4*sqrt(F4*(27*F4+1024*Co^3)))/(32*3^(3/2)*Co^6)+F4^2/(32*Co^6))^(1/3)))/2-sqrt((3*Co^3*((F4*sqrt(F4*(27*F4+1024*Co^3)))/(32*3^(3/2)*Co^6)+F4^2/(32*Co^6))^(2/3)-F4)/(Co*((F4*sqrt(F4*(27*F4+1024*Co^3)))/(32*3^(3/2)*Co^6)+F4^2/(32*Co^6))^(1/3)))/(2*sqrt(3)*Co) = F5

Thermal unfolding monitored by spectroscopy with an intermediate state (three state unfolding)

The unfolding pathway of some monomeric proteins proceeds through an observable intermediate (stable and significantly populated)

F I UKFI KIU

KFI =[I]

[F]=

XI * Co

XF * Co=

XI

XF

KIU =[U]

[I]=

XU * Co

XI * Co=

XU

XI

XF + XI + XU = 1

KFI =XI

XF

=1 – XF – XU

XF

KIU =XU

XI

=1 – XF – XU

XU



XI = 1 – XF – XU

Parameters defined so far:• HFI TmF Enthalpy of unfolding for F to I states at TmF

• HIU TmU Enthalpy of unfolding for I to U states at TmU

• TmF Temp where XF = 0.5• TmU Temp where XU = 0.5• CpFI Cp between F and I states• CpIU Cp between I and U states• R Gas constant• T Temp (x values of experimental data)

HFI(T) = HFI TmF + CpFI*(T – TmF)

SFI(T) = (HFI TmF/TmF) + CpFI*ln(T/TmF)

and

HIU(T) = HIU TmU + CpIU*(T – TmU)

SIU(T) = (HIU TmU/TmU) + CpIU*ln(T/TmU)

GFI(T) = HFI(T) – (T * SFI(T))

and

GIU(T) = HIU(T) – (T * SIU(T))

andKFI = exp(GFI(T)/(-R*T))

KIU = exp(GIU(T)/(-R*T))



XF

KIU =XU

XI

thus KIU =XU

KFI*XF

XU

XF



XF = 1 – XI – XU


1 = (1/XF) – (XI/XF) – (XU/XF)

1 = (1/XF) – (KFI) – (KFI*KIU)



XF = 1/(1 + KFI*(1 + KIU))



XF

KIU =XU

XI

thus KIU =XU

KFI*XF

XU

XF



XU = 1 – XI – XF


1 = (1/XU) – (XI/XU) – (XF/XU)

1 = (1/XU) – (1/KIU) – (1/KFI*KIU)

1 + (1/KIU) + (1/KFI*KIU) = 1/XU






SF = (mf*[D])+bf

SI = (mi*[D])+bi

SU = (mu*[D])+bu

The spectroscopic signal of the F, I and U states are modeled as linear functions:

Parameters:• mf slope of the N(D) spectroscopic function• bf Y-int of the N(D) spectroscopic linear function• mi slope of the I(D) spectroscopic function• bi Y-int of the I(D) spectroscopic linear function• mu slope of the U(D) spectroscopic function• bu slope of the U(D) spectroscopic linear function

TmF 300 T where XF = 0.5

TmU 340 T where XU = 0.5

CpFI 1000 Cp for F to I

CpIU 1000 Cp for I to U

HFI 300000 H for F to I at TmF

HIU 200000 H for I to U at TmU

mf 0 slope F baseline

bf 1000 y-int F baseline

mi 0 slope I baseline

bi 2500 y-int I baseline

mu 0 slope U baseline

bu 4000 y-int U baseline

R 8.314 Gas const.

T 298 Temp (K)

Denaturation of a Protein by Temperature and Monitored by Heat Capacity

A hypothetical differential scanning calorimetry “endotherm”

280 300 320 340 360

Cp

(J m

ol-1

K-1

)

Temp (K)

Differential Scanning Calorimetry "Endotherm":

Heat Capacity as a Function of Temperature

Native state heat capacity function: CpN(T)

280 300 320 340 360

Cp

(J

mo

l-1 K

-1)

Temp (K)

Native state heat capacity

CpN(T)

Denatured state heat capacity function: CpD(T)

280 300 320 340 360

Cp

(J

mo

l-1 K

-1)

Temp (K)

Denatured state heat capacity

CpD(T)

CpN(T) to CpD(T) baseline transition during protein unfolding

280 300 320 340 360

Cp

(J

mo

l-1 K

-1)

Temp (K)

Baseline transition during protein unfolding

Excess enthalpy: “heat of fusion” (protein “melting”)

0 Tm

CpN(T) = (mf*T) + bf

(0,bf)

(Tm,(mf*Tm)+bf)

(0,(mf*Tm)+bf)

Define CpN(T) in terms of mf, Tm and CpN(Tm):

CpN(T) = (mf * T) + bf

CpN(Tm) = (mf * Tm) + bf

bf = CpN(Tm) – (mf * Tm)

CpN(T) = (mf * T) + CpN(Tm) – (mf * Tm)

CpN(T) = mf*(T – Tm) + CpN(Tm)

The native state (i.e. folded) heat capacity function CpN(T)

• CpN(T) is modeled as a linear function• But 0 K is not a convenient reference for Y-int (long extrapolation)• Tm is the temperature (T) where ΔGu=0 (the “melting temperature”)

• Tm is a more convenient reference T for the Y-int of the CpN(T) linear function

0 Tm

• CpD(T) is modeled as a linear function• Define CpD(T) in terms of CpN(Tm), mu, Tm and ΔCp(Tm):

CpD(T) = (mu*T) + bu

CpD(Tm) = (mu*Tm) + bu

CpN(Tm) + ΔCp(Tm) = (mu*Tm) + bu

bu = CpN(Tm) + ΔCp(Tm) - (mu*Tm)

CpD(T) = (mu * T) + CpN(Tm) + ΔCp(Tm) - (mu*Tm)

CpD(T) = mu*(T - Tm) + CpN(Tm) + ΔCp(Tm)


(0,bu) ΔCp(Tm)

(Tm, CpN(Tm))

(Tm, CpN(Tm)+ΔCp(Tm))

The denatured state (i.e. unfolded) heat capacity function CpD(T)

0 T

ΔCp(T) = mu*(T – Tm) + CpN(Tm) + ΔCp(Tm) - mf*(T – Tm) - CpN(Tm)

ΔCp(T) = CpD(T) – CpN(T)

ΔCp(T) = (mu – mf)*(T – Tm) + ΔCp(Tm)

CpD(T) = mu*(T - Tm) + CpN(Tm) + ΔCp(Tm)

CpN(T) = mf*(T – Tm) + CpN(Tm)

• ΔCp(T) = CpD(T) – CpN(T)

• If CpN(T) and CpD(T) are both linear functions, then ΔCp(T) is also a linear function

The general form of the ΔCp(T) function:

ΔCp(T) = CpD(T) – CpN(T)

ΔGu(T) = ΔHu(T) – T*ΔSu(T)

ΔHu (T) = ΔHu (0K) – ΔCp(T)*(0-T)

Thermodynamic Functions

Gibbs energy of unfolding has both enthalpy and entropy terms:

Enthalpy of unfolding is affected by ΔCp(T)

• Basically, with a positive value for ΔCp(T), there is an increase in heat capacity upon unfolding, and enthalpy increases

• Here we are referencing enthalpy at Tm, but consider the behavior if we chose to reference at T~0K

ΔHu (T) = ΔHu (0K) + ΔCp(T)*T

ΔHu(T) = ΔHu (Tm) – ΔCp(T)*(Tm-T)

ΔSu (T) = (ΔHu(Tm)/Tm) + ΔCp(T)*ln(T/Tm)

ΔGu(T) = (ΔH(Tm) – ΔCp(T)*(Tm-T)) – T*((ΔH(Tm)/Tm)+ΔCp(T)*ln(T/Tm))

Thermodynamic Functions

Entropy of unfolding is also affected by ΔCp(T):

• Basically, the entropy change (i.e. change in system disorder) is proportional to the temperature at which an enthalpy change (heat input) occurs, and ΔCp(T) affects the enthalpy change

The overall Gibbs energy equation taking into account the effects of ΔCp(T):

ΔSu (T) = (ΔHu(Tm)/Tm) + ΔCp(T)*ln(T/Tm)

ΔGu(T) = (ΔH(Tm) – ΔCp(T)*(Tm-T)) – T*((ΔH(Tm)/Tm)+ΔCp(T)*ln(T/Tm))

Summary of Thermodynamic Functions

ΔHu(T) = ΔHu (Tm) – ΔCp(T)*(Tm-T)

Tm The temperature at which XU=XF=0.5 (protein is half unfolded)ΔHu(Tm) The value of ΔHu (referenced at the melting temperature Tm)ΔCp(T) Difference in heat capacity between native and denatured states

Monomer two-state unfolding

Simple two-state unfolding:F ↔ U

Ku = [U]/[F]

[U] = ΧU * [Co]


[F] = ΧF * [Co]


Ku = (ΧU * [Co]) / (ΧF * [Co]) = ΧU / ΧF


Ku = ΧU /(1 - ΧU)

XF and XU define the equilibrium unfolding constant Ku

Solving for XU in terms of Ku:

XU = Ku /(Ku+ 1)

280 300 320 340 360

Cp

(J

mo

l-1 K

-1)

Temp (K)

Differential Scanning Calorimetry "Endotherm":

Heat Capacity as a Function of Temperature

Note: although in the monomer case the concentration terms drops out in the equilibrium constant definition, the Cp(T) function references the molar concentration of protein

• Thus, for the Cp(T) function to be correct, the raw heat capacity data must be normalized to the protein concentration (which must be accurately determined)

XU = Ku /(Ku+ 1)

ΔGu = -R*T*ln(Ku)

ΔGu

-R*TKu = exp

ΔGu(T) = (ΔHu (Tm) – ΔCp(T)*(Tm-T)) – T*((ΔHu (Tm)/Tm)+ΔCp(T)*ln(T/Tm))

• Substitute the expression for ΔGu into the expression for Ku

• Substitute the expression for Ku into the expression for XU

• XU(T) is now defined in terms of:

• ΔHu(Tm) (enthalpy of unfolding referenced at Tm)

• ΔCp(T), a linear function defined in terms of:

• mu (slope of denatured state heat capacity)

• mf (slope of native state heat capacity)

• Tm (melting temperature)

• ΔCp(Tm) (CpD-CpN referenced at Tm)

• R

• T

Cp(T) Function

The experimentally-observed Cp(T) heat capacity function (i.e. DSC Endotherm) is the sum of:

• The CpN(T) and CpD(T) baseline transition function

• The excess enthalpy function associated with the “melting” transition

The CpN(T) and CpD(T) baseline transition function:

(XF * CpN(T)) + (XU * CpD(T))

Or, the mathematically equivalent

CpN(T) + (XU * ΔCp(T))

The excess enthalpy function:

ΔHu(T)2

R * T2* XU * (1 – XU)

Cp(T) Function

Monomer (2-state):

Cp(T) = CpN(T) + ΔCp*XU + ΔH(T)2

R * T2

CpN(T) to CpD(T) baseline transition

Excess enthalpy(F↔U phase transition)

n Oligomer (2-state):

Cp(T) = CpN(T) + ΔCp*XU) + ΔH(T)2

R * T2

CpN(T) to CpD(T) baseline transition

Excess enthalpy(F↔U phase transition)

* XU * (1 – XU)

*XU * (1 – XU)

n – XU*(n-1)

DSC Endotherm: Modification for Dimer oligomer (2-state unfolding)


Ku = [U]2/[F]

[U] = ΧU * Co


[F] = ΧF * Co/2


Ku = (ΧU * Co)2 / ΧF * Co/2




Ku = 2*ΧU2*Co / (1 - ΧU)


Dimer (2-state) modification to excess enthalpy:


R * T2

Excess enthalpyF↔2U phase transition

*XU * (1 – XU)

2 – XU

Dimer (2-state) modification to XU:


Note that the “Tm” term refers to the temperature where ΔGu = 0 (and NOT the temperature where XU=XF=0.5)

DSC Endotherm: Modification for Trimer oligomer (2-state unfolding)


Ku = [U]3/[F]

[U] = ΧU * Co


[F] = ΧF * Co/3


Ku = (ΧU * Co)3 / ΧF * Co/3




Ku = 3*ΧU3*Co2 / (1 - ΧU)


Ku = 3*ΧU3*Co2 / (1 - ΧU)

Trimer (2-state) modification to excess enthalpy:


R * T2


*XU * (1 – XU)

3 – 2*XU

DSC Endotherm: Modification for Tetramer (2-state unfolding)


Ku = [U]4/[F]

[U] = ΧU * Co


[F] = ΧF * Co/4


Ku = (ΧU * Co)4 / ΧF * Co/4




Ku = 4*ΧU4*Co3 / (1 - ΧU)



Ku = 4*ΧU4*Co3 / (1 - ΧU)

XU * (1 – XU)

4 – 3*XU

Tetramer (2-state) modification to excess enthalpy:


R * T2


*

Thermal unfolding monitored by heat capacity with an intermediate state (three state unfolding)

F I UKFI KIU

KFI =[I]

[F]=

XI * Co

XF * Co=

XI

XF

KIU =[U]

[I]=

XU * Co

XI * Co=

XU

XI

XF + XI + XU = 1

KFI =XI

XF

=1 – XF – XU

XF

KIU =XU

XI

=1 – XF – XU

XU



XI = 1 – XF – XU

Note: KFI will equal 1.0 (and ΔGFI = 0) when [XI] = [XF], this is not necessarily where [XI] and [XF] = 0.5 since another state exists ([XU])

0 Tmn

• CpN(T), CpI(T) and CpD(T) are all modeled as linear functions

CpD(T) = (md*T) + bd

(0,bu)

The denatured state (i.e. unfolded) heat capacity function CpD(T)

CpN(T) = (mn*T) + bn

CpI(T) = (mi*T) + bi

Tmi

CpFI(T)

CpIU(T)

CpFI(T) = (mi*T) + bi – (mn*T) – bn = T*(mi – mn) + bi – bn

CpIU(T) = (md*T) + bd – (mi*T) – bi = T*(md – mi) + bd – bi

Parameters for N to I transition:• mn Slope of the CpN(T) function• bn Y-int (at 0K) of the CpN(T) function• mi Slope of the CpI(T) function• bi Y-int (at 0K) of the CpI(T) function• HFI TmF Enthalpy of unfolding for F to I states at TmF

• TmF Temp where N state XF = 0.5• R Gas constant• T Temp (x values of experimental data)




KFI = exp(GFI(T)/(-R*T))

CpFI(T) = T*(mi – mn) + bi – bn





CpIU(T) = T*(md – mi) + bd – bi

Parameters for I to U transition:• mi Slope of the CpI(T) function• bi Y-int (at 0K) of the CpI(T) function• md Slope of the CpU(T) function• bd Y-int (at 0K) of the CpU(T) function•HIU TmI Enthalpy of unfolding for I to U states at TmI

• TmI Temp where I state XF = 0.5• R Gas constant• T Temp (x values of experimental data)

All parameters for N to I transition and I to U transition:• mn Slope of the CpN(T) function• bn Y-int (at 0K) of the CpN(T) function• mi Slope of the CpI(T) function• bi Y-int (at 0K) of the CpI(T) function• mu Slope of the CpU(T) function• bu Y-int (at 0K) of the CpU(T) function• HFI TmF Enthalpy of unfolding for F to I states at TmF

• TmF Temp where N state XF = 0.5•HIU TmI Enthalpy of unfolding for I to U states at TmI




XF

KIU =XU

XI

thus KIU =XU

KFI*XF

XU

XF



XF = 1 – XI – XU


1 = (1/XF) – (XI/XF) – (XU/XF)

1 = (1/XF) – (KFI) – (KFI*KIU)



XF = 1/(1 + KFI*(1 + KIU))



XF

KIU =XU

XI

thus KIU =XU

KFI*XF

XU

XF



XU = 1 – XI – XF


1 = (1/XU) – (XI/XU) – (XF/XU)

1 = (1/XU) – (1/KIU) – (1/KFI*KIU)

1 + (1/KIU) + (1/KFI*KIU) = 1/XU








KFI = exp(GFI(T)/(-R*T))

CpFI(T) = T*(mi – mn) + bi – bn





CpIU(T) = T*(md – mi) + bd – bi

All parameters for N to I transition and I to U transition:• mn Slope of the CpN(T) function• bn Y-int (at 0K) of the CpN(T) function• mi Slope of the CpI(T) function• bi Y-int (at 0K) of the CpI(T) function• md Slope of the CpU(T) function• bd Y-int (at 0K) of the CpU(T) function• HFI TmF Enthalpy of unfolding for F to I states at TmF

• TmF Temp where N state XF = 0.5•HIU TmI Enthalpy of unfolding for I to U states at TmI


The CpN(T) and CpI(T) baseline transition function:

(XF * CpN(T)) + (XI * CpI(T))


CpN(T) + (XI * ΔCpFI)


ΔHFI(T)2

R * T2* XF * XI

The CpI(T) and CpU(T) baseline transition function:

(XI * CpI(T)) + (XU * CpU(T))


CpI(T) + (XU * ΔCpIU)


ΔHIU(T)2

R * T2* XI * XU

0 Tmf


(0,bn)

(Tmf,(mn*Tmf)+bn)

(0,(mn*Tmf)+bn)

CpN(T) = mn*(T – Tmf) + CpN(Tmf)

Modification to define CpN(T), CpI(T), and CpD(T) functions in terms of Cp

New variables to define CpN(T):• mn slope of CpN(T)• Tmf reference Temp (reference Tm of F state)• CpN(Tmf) value of CpN(T) at Tmf

0 Tm

• Define CpI(T) in terms of CpN(Tmf), mi, Tmf and ΔCp(Tmf):

CpI(T) = mi*(T - Tmf) + CpN(Tmf) + ΔCpNI(Tmf)


(0,bu) ΔCpNI(Tmf)

(Tm, CpN(Tmf))

(Tmf, CpN(Tmf)+ΔCpNI(Tmf))

The intermediate state heat capacity function CpI(T)

0 T

ΔCpNI(T) = mi*(T - Tmf) + CpN(Tmf) + ΔCpNI(Tmf) - mn*(T – Tmf) - CpN(Tmf)

ΔCpNI(T) = CpI(T) – CpN(T)

ΔCpNI(T) = (mi – mn)*(T – Tmf) + ΔCpNI(Tmf)

The general form of the ΔCpNI(T) function:

ΔCpNI(T) = CpI(T) – CpN(T)



0 Tmi

(0,bi)

(Tmi,(mi*Tmi)+bi)

(0,(mi*Tmi)+bi)

CpI(T) = mi*(T – Tmi) + CpI(Tmi)

Modification to define CpN(T), CpI(T), and CpD(T) functions in terms of Cp

New variables to define CpI(T):• mi slope of CpI(T)• Tmi reference Temp (reference Tm of I state)• CpI(Tmi) value of CpI(T) at Tmi

CpI(T) = mi*(T - Tmf) + CpN(Tmf) + ΔCp(Tmf)

Tmf

(Tmf, (mi*Tmf) +bi)

(mi*(Tmi-Tmf))+bi

0 Tmi

• Define CpD(T) in terms of CpI(Tmf), mu, Tmi and ΔCpID(Tmi):

CpD(T) = mu*(T - Tmi) + CpI(Tmi) + ΔCpID(TmI)


(0,bu) ΔCpID(Tmi)

(Tmi, CpI(Tmi))

(Tmi, CpI(Tmi)+ΔCpID(Tmi))

The denatured state heat capacity function CpD(T)

• Note: Tmf is not Tmi

0 T

ΔCpID(T) = mu*(T - Tmi) + CpI(Tmi) + ΔCpID(Tmi) - mi*(T – Tmi) - CpI(Tmi)

ΔCpID(T) = CpD(T) – CpI(T)

ΔCpNI(T) = (mu – mi)*(T – Tmi) + ΔCpID(Tmi)

The general form of the ΔCpID(T) function:

ΔCpID(T) = Cpd(T) – CpI(T)

CpD(T) = mu*(T - Tmi) + CpI(Tmi) + ΔCpID(Tmi)


Summary of CpN(T), CpI(T), CpD(T), DCpNI(T) and DCpID(T) functions (all linear):



CpD(T) = mu*(T - Tmi) + CpI(Tmi) + ΔCpID(TmI)

ΔCpNI(T) = (mi – mn)*(T – Tmf) + ΔCpNI(Tmf)

ΔCpID(T) = (mu – mi)*(T – Tmi) + ΔCpID(Tmi)

Term Meaning Assigned variable• mn slope of CpN(T) mn• mi slope of CpI(T) mi• mu slope of CpD(T) mu • Tmf Tm of N state tmf• Tmi Tm of I state tmi• CpN(Tmf) Value of CpN(T) at Tmf cntf• CpI(Tmi) Value of CpI(T) at Tmi citi• CpNI(Tmf) value of CpNI at Tmf dctf• CpID(Tmi) value of CpID at Tmi dcti

0 Tmf Tmi



CpD(T) = (md*T) + bd

CpFI

CpIU

bn

bi

bu


CpI(T) = mi*(T – Tmf) + CpI(Tmf)

CpD(T) = md*(T – Tmf) + CpD(Tmf)

CpN(T) = mn*(T – Tmi) + CpN(Tmi)


CpD(T) = md*(T – Tmi) + CpD(Tmi)


CpI(T) = mi*(T – Tmf) + CpI(Tmf)

CpD(T) = md*(T – Tmf) + CpD(Tmf)

CpN(T) = mn*(T – Tmi) + CpN(Tmi)


CpD(T) = md*(T – Tmi) + CpD(Tmi)

CpNI(T) = mi*(T – Tmf) + CpI(Tmf) – (mn*(T – Tmf) + CpN(Tmf))CpNI(T) = (mi – mn)*(T – Tmf) + CpI(Tmf) - CpN(Tmf)CpNI(T) = (mi – mn)*(T – Tmf) + CpNI(Tmf) mi, mn, Tmf, CpNI(Tmf)

CpID(T) = md*(T – Tmi) + CpD(Tmi) – (mi*(T – Tmi) + CpI(Tmi))CpID(T) = (md – mi) * (T – Tmi) + CpD(Tmi) – CpI(Tmi)CpID(T) = (md – mi) * (T – Tmi) + CpID(Tmi) mi, md, Tmi, CpID(Tmi)

CpNI(Tmf) = CpI(Tmf) - CpN(Tmf)CpI(Tmf) = CpNI(Tmf) + CpN(Tmf)CpI(T) = mi*(T – Tmf) + CpI(Tmf)CpI(T) = mi*(T – Tmf) + CpN(Tmf) + CpNI(Tmf) mi, Tmf, CpN(Tmf), CpNI(Tmf)

CpID(Tmi) = CpD(Tmi) - CpI(Tmi)CpI(Tmi) = CpD(Tmi) - CpID(Tmi)CpI(T) = mi*(T – Tmi) + CpI(Tmi)CpI(T) = mi*(T – Tmi) + CpD(Tmi) - CpID(Tmi) mi, Tmi, CpD(Tmi), CpID(TmI)

mi*Tmi - CpI(Tmi) = mi*Tmf - CpI(Tmf)

CpI(T) = mi*(T – Tmf) + CpN(Tmf) + CpNI(Tmf)

CpI(T) = mi*(T – Tmi) + CpD(Tmi) - CpID(Tmi)

mi*(T – Tmf) + CpN(Tmf) + CpNI(Tmf) = mi*(T – Tmi) + CpD(Tmi) - CpID(Tmi)

mi*T – mi*Tmf + CpN(Tmf) + CpNI(Tmf) = mi*T – mi*Tmi + CpD(Tmi) - CpID(Tmi)

mi*T – mi*Tmf + CpN(Tmf) + CpNI(Tmf) = mi*T – mi*Tmi + CpD(Tmi) - CpID(Tmi)

mi*Tmi – mi*Tmf + CpN(Tmf) + CpNI(Tmf) = CpD(Tmi) - CpID(Tmi)

mi*Tmi – mi*Tmf + CpN(Tmf) + CpI(Tmf) - CpN(Tmf) = CpD(Tmi) - CpID(Tmi)

mi*Tmi – mi*Tmf + CpN(Tmf) + CpI(Tmf) - CpN(Tmf) = CpD(Tmi) - CpD(Tmi) + CpI(Tmi)

mi*Tmi – mi*Tmf + CpI(Tmf) = CpI(Tmi)

CpI(Tmf) – mi*Tmf = CpI(Tmi) – mi*Tmi

mi*(Tmi –Tmf) = CpI(Tmi) - CpI(Tmf)

mi*Tmi - CpI(Tmi) = mi*Tmf - CpI(Tmf)

Baseline equations summary:

CpN(T) = mn*(T – Tmf) + CpN(Tmf) mn, Tmf, CpN(Tmf)

CpI(T) = mi*(T – Tmf) + CpI(Tmf) mi, Tmf, CpI(Tmf)

CpD(T) = md*(T – Tmf) + CpD(Tmf) md, Tmf, CpD(Tmf)

CpN(T) = mn*(T – Tmi) + CpN(Tmi) mn, Tmi, CpN(Tmi)

CpI(T) = mi*(T – Tmi) + CpI(Tmi) mi, Tmi, CpI(Tmi)

CpD(T) = md*(T – Tmi) + CpD(Tmi) md, Tmi, CpD(Tmi)

CpI(T) = mi*(T – Tmf) + CpN(Tmf) + CpNI(Tmf) mi, Tmf, CpN(Tmf), CpNI(Tmf)

CpI(T) = mi*(T – Tmi) + CpD(Tmi) - CpID(Tmi) mi, Tmi, CpD(Tmi), CpID(TmI)

CpD(T) - md*(T – Tmi) = CpD(Tmi)

CpI(T) - mi*(T – Tmi) + CpID(Tmi) = CpD(Tmi)

CpD(T) - md*(T – Tmi) = CpI(T) - mi*(T – Tmi) + CpID(Tmi)

CpD(T) = CpI(T) - mi*(T – Tmi) + CpID(Tmi) + md*(T – Tmi)

CpD(T) = CpI(T) + (md – mi)*(T - Tmi)+ CpID(Tmi) md, mi, Tmi, CpID(Tmi)

CpN(T) = mn*(T – Tmf) + CpN(Tmf) mn, Tmf, CpN(Tmf)

CpI(T) = mi*(T – Tmf) + CpN(Tmf) + CpNI(Tmf) mi, Tmf, CpN(Tmf), CpNI(Tmf)

CpD(T) = CpI(T) + (md – mi)*(T - Tmi)+ CpID(Tmi) md, mi, Tmi, CpID(Tmi)

Variables that define the CpN(T), CpI(T), CpD(T), CpNI(T), and CpID(T) functions

Variable Meaning Abbreviation

mn CpN(T) slope mn

mi CpI(T) slope mi

md CpD(T) slope md

Tmf Melting temperature of folded (native) state (XF=0.5) tmf

Tmi Melting temperature of intermediate state (XI=0.5) ? tmi

CpN(Tmf) Value of CpN(T) at Tmf cntf

CpNI(Tmf) Delta Cp at Tmf for FI transition dcni

CpID(Tmi) Delta Cp at Tmi for IU transition dciu

CpNI(T) = (mi – mn)*(T – Tmf) + CpNI(Tmf) mi, mn, Tmf, CpNI(Tmf)

CpID(T) = (md – mi) * (T – Tmi) + CpID(Tmi) mi, md, Tmi, CpID(Tmi)

models for chemical and temperature denaturation of proteins …mikeblaber.org/methods/models for...

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