cold denaturation in proteins

Cold Denaturation in Proteins.

Olivier Collet

LPM, Nancy-Universite

CompPhys07Leipzig, Nov. 2007

Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 1 / 19

Plan

1 Introduction.

2 The Problem.

3 Model.

4 Results 1. : Two-states phase diagram.

5 Results 2 : Four-states phase diagram.

6 Conclusion.


Introduction.

Biochemistry

Amino-acids and Bond between amino acids.


Introduction.

Biochemistry

Proteins are large and linear chains made of amino acids

CH2

OH

−NH−C−CO=

H

CH3

−NH−C−CO=

H


Introduction.

Biochemistry

Put into water proteins fold in a unique compact structure

CH2

OH

−NH−C−CO=

H

CH3

−NH−C−CO=

H

Water

Hydrophobic

Hydrophilic

Main folding force is the Temperature Dependent Hydrophobic Effecta.

aKauzmann 1959, Balwin 1987, Pratt-Pohorille 2002


The Problem.

Warm and Cold Denaturations 1.

1Privalov, 1989Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 6 / 19

Model.

Statistical Physics approach.

Hmic = E (m)intr + E (mm′)

solv

m : protein conformationm′ : water configuration

⇒ Z (T ) =ΩX

m=1

Ω′(m)X

m′=1

exp

−E (m)

intr + E (mm′)solv

T

!

Ω′(m)X

m′=1

exp

−E (mm′)

solv

T

!

= z(m)solv(T ) = exp

−F (m)

solv(T )

T

!

⇒ Z (T ) =ΩX

m=1

exp

−H

(m)eff (T )

T

!

with H(m)eff (T ) = E (m)

intr + F (m)solv(T )


Model.

Effective hamiltonian.

first shell solvent cell

neat solvent cell

solvent nod

monomer

∆(m)ij = 1 if i and j are first neighbors.

Bij : coupling between i and jn(m)

i : number of cells arround ifi(T ): free energy of this solvent cell.n(m)

s : total number of solvent cellsfs(Bs, T ) : free energy of neat solvent

H(m)eff =

X

i>j+1

Bij ∆(m)ij

| z

previous works

+X

i

n(m)i fi(T ) + 2n(m)

s fs(Bs, T )

| z

Collet 2001;2005

More complicated form than the usual : H(m)eff =

X

i>j+1

Bij ∆(m)ij !

Constant of the model.

total lattice links :X

i

X

j

12

∆(m)ij +

X

i

n(m)i + n(m)

s = K1

links of monomeri :X

j

∆(m)ij + n(m)

i = 4Olivier Collet ( LPM, Nancy-Universit e) Cold Denaturation in Proteins. CompPhys07 Leipzig, Nov. 2007 8 / 19

Model.

Effective hamiltonian.

(

n(m)i = 4 −

P

j ∆(m)ij

n(m)s = 1

2

P

i

P

j ∆(m)ij + K1 − 4N

H(m)eff =

X

i>j+1

Bij ∆(m)ij +

X

i

n(m)i fi(T ) + 2n(m)

s fs(Bs, T )

Effective Couplings.

H(m)eff (Bs, T ) =

X

i

X

j>i

Beffij (Bs, T ) ∆

(m)ij

with Beffij (Bs, T ) = Bij − fi(T ) − fj(T ) + 2fs(Bs, T )

- takes a simple form- which forms for fi(T ) and fs(Bs, T ) ?


Model.

Solvation Model

fs(Bs, T ) = Bs − αT ln Ns

Small Bs ⇒ bad solventLarge Bs ⇒ good solvent

n(Bi) =2Ns exp

−B2

i2σ

2

!

σ

√2π erfc

Bmini

σ

√2

!

zi(T ) =

Z ∞

Bmini

n(Bi) exp„

−Bi

T

«

dBi

fi(T ) = −σ

2

2T− T ln

0

B@Ns

erfc“

Bmini

σ

√2− σ

√2

2T

”

erfc“

Bmini

σ

√2

”

1

CA


Model.

The Chain and the Statistical averages.

The chain.

16-mersin 2D lattice

802 075 conformations116 579 extended conformations69 more maximally compact conf

Nat

Statistical average 〈X (T )〉 =PΩ

m=1 X (m)(T )P(m)eq (T ) with

P(m)eq (T ) = 1

Z(T )exp

„

−H(m)

effT

«

.

Compactness: 〈Nc(Bs, T )〉 where N(m)c = 1

9

PNi>j ∆

(m)ij

Order parameter: 〈Q(Bs, T )〉 where Q(m) = 19

PNi>j ∆

(m)ij ∆Nat

ij

Chain entropy, Sch: −〈ln P(m)eq (Bs, T )〉


Results 1. : Two-states phase diagram.

Results 1. : Two-states phase diagram


Results 2 : Four-states phase diagram.


α = 0.5 → 0.9

-15

-10

-5

1 2 34 5

0.0

0.5

1.0

B

T

Q

s

< >

-15

-10

-5

1 2 34 5

0.0

0.5

1.0

B

T

N

s

c< >

-15

-10

-5

1 2 34 5

0

5

10

15

B

T

S

s

ch




0 1 2 3 4 5T

0.00

0.01<

Nc>

Bs=0Bs=2Bs=4Bs=6

0 1 2 3 4 5T

11.70

12.00

S ch

Bs=0Bs=2Bs=4Bs=6

0 1 2 3 4 5T

0.0

0.1

0.2

0.3

0.4

0.5

TdS

ch/d

T

Bs=0Bs=2Bs=4Bs=6



Result 2 : Four-states phase diagram.


Conclusion.

Conclusions.

Calculation of H(m)eff (T ) also simple than E (m)

intr

Cold Denaturation due to hydrophobic effect

Realistic couplings must

take into account of the temperature dependenceof the hydrophobic effect.


Conclusion.


Conclusion.

Effective Couplings as function of T .

0,2 0,4 0,6 0,8T

-25

-20

-15

free

ene

rgy

fi+f

j

Bij+2f

s ; B

s=-4.0

Bij+2f

s ; B

s=-5.5

Bij+2f

s ; B

s=-6.0

Bij+2f

s ; B

s=-7.0

Tij

Tij

-

+

(Bs=-6.0)

(Bs=-6.0)

Figure: Curves of the different contributions to the effective coupling between the monomers 1and 4 as function of the temperature for several values of the solvent quality. The twotemperatures for which the coupling vanishes are shown for Bs = −6.0.

Beffij (Bs, T ) = Bij − fi(T ) − fj(T ) + 2fs(Bs, T ) may be :

positive at low Tnegative at medium Tpositive at high T


cold denaturation in proteins

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