Supervisor:Professor Michel DROZUniversity of GenevaDepartment of Theoretical Physics

The The Biological Physics of Biological Physics of PProtein Folding: rotein Folding:

the Random Energy Model and Beyondthe Random Energy Model and Beyond

PhD Lecture

Péter HANTZ

Geneva, 4 May 2006

Outline of the lecture:

1. Modeling disordered systems• Spin glasses, frustration, Random Energy Model

2. Proteins: Building Elements and Structure • Primary, Secondary and Tertiary Structure, Classification

3. The Problem of Protein Folding • Anfinsen Experiment, Levinthal Paradox

• A Microscopic model: RHP • Phenomenological models: Gō, REM • Sequence Design, Minimal Frustration • Kinetics: Funnel Hypothesis, Nucleation, Reaction Coordinate

What is a spin glass? • interacting system of spins • low-temperature: frozen in random orientations

What is necessary for this? • (at least partially) random interactions • competing interactions

Simple Model Hamiltonians:Sherrington-Kirkpatrick Model

P-spin model

Distribution of coupling constants:2

20...1

1

21

21

2

)/(

2...

...21

..1

...

)(

2)(

})({

})({

J

NJJ

ipi

ipii

ipii

iiIip

ji

jipairsall

ijSK

ipi

p

eJ

NJP

SSSJSH

SSJSH

Spin Glasses

Frustration…

• no configuration is uniqely favoured by all of the interactions

• “fully frustrated” systems: hypercube/hypertetrahedron where the Jij=±1, and

?

J12=1 J13=1

J23= -1

plaquettesall

ijJ 1

And its consequences…

• rugged energy landscape “barrier tree”

of a p-spin model, P=3, N=7 (Fontanari, 2001)

(F=E-TS >> calculating the entropy: restrict to valleys)

0lim N

SN

And its consequences…

• high degree of ground-state degeneracy (Plischke, 1994)

three very different configurations have the same ground state energy

in several models:

And its consequences…

• Great relevance of broken ergodicity (Palmer, 1983)

-pure systems: mean-field theory of ferromagnets

time average≠Gibbs average

<Si>t=±m <Si>G=0

-spin glasses: in the limit of large N, the state space becomes partitioned

into mutually inaccessible “valleys” (Fischer, 1993)

Averages in disordered systems

• quenched average (of the free energy) -“over the realizations of the disorder” -the randomness of a system, Jij, is fixed (time-scale problem)

Note: doing the average of the logarithm is difficult.

• annealed average (of the free energy) -both spins and the randomness Jij are thermodynamic variables

Essential Difference (case of a protein sequence):• q: SUM of the free energies of various sequences • ad: SUM of EXPs of sequences

))},({ln()]([...ln)(

)(

)(

TJZJPdJkTZkTTF

AJDPA

ijij

ijijqq

Jq

}){},({

2

}{ )(

)]([..ln)(ln)( SJijHij

S ijijadad eJPdJkTTZkTTF

N

Averages in disordered systems

• self-averaging quantities

-extensive quantities: macroscopic system and subsystems

• Z is not self-averaging

(eg. one sample with low free energy could dominate the sum)

qsysqsubsr

rsubssys AAJAJA

1

)()(

N

Ff

R

rr

R

r

NfR

r

NfR

rr ZeeZZ rr

1111

qN

J AA

The Random Energy Model (REM)

• the E total energy of a system = sum of independent contributions

• central limit theorem =>

A particular set {E({J})1, E({J})2, … E({J})Ω} represents the energy levels of one particular realization {J}, of the modeled system

• the E({J})i energies of different microstates of a realization are

statistically independent

• number of microstates (eg. in the case of N Ising spins)

N

EN

NE

eN

eN

EP 22

)(

2

2

2

2

2

1

2

1)(

N2

lll

l JJE 2,,})({ })({

Properties of the REM • average density of states (average over the realizations of the disorder)

spectra of two realizations (eg. {J}, polymer chains) (Pande et al., 1997) (1) (2)

• below an average threshold energy EC :

• since

the density n(E) is self-averaging only in the middle region of P(E).

2ln2

2)2ln(

2

12ln,1

2,1)(2

22

2ln2

NE

N

ENNe

N

eEP

TD

C

CN

EN

CN

C

)(2)()})({()()(

)})({()(

2

1})({

2

1})({

EPdEEPJEEEnEn

JEEEn

N

iiiiJ

iiJ

N

N

)(

1

)(

)(,)()(

EnEn

EnEnEn

Properties of the REM • entropy

The entropy cannot be negative. If E< EC, S(E)=0, the system is “frozen”.

• critical temperature

and

for the critical temperature (where S=0, but s=S/N not necessarily 0) we have

N

EkkNekES

EkES

N

EN

TD

2)2ln()2ln()(

)(ln)(2

2

2

E

N

kET

TE

ES 1)(

1)(

2)(

1

2

12ln)(

kTkNTS

2ln2

1,

)(

1

2

12ln

2 kT

kT CC

Properties of the REM • free energy

If T>TC,

However, if T<TC, S=0, and

• partition function

In case if n(E) is self-averaging, Z does not depend on the disorder, and

dEeE kT

E

i

kT

sE

J

i

neJZ )(})({

1

)(

})({

2ln2

)()()( NkTkT

NTTSTETF

2ln2)()( NTETF

)(

22)(2

2

)(22

2

1

TF

eTZ kT

N

NkT

E

N

E

N dEeN

Digression: Order parameters

• distinguishing between HT paramagnetic and LT frozen states (Edwards and Anderson, 1975)

• some other important quantities

Stat. mech. order parameter:

Degree of broken ergodicity:

• “similarity” between states (e.g. phases) of the system

+1: full

a

a

a

S

SH

aS

SH

a

avall

N

iG

aiN

aEAZ

Z

e

ePSPq N

u

u

u

2

}{

})({

)(

. 1

21 ,

qq

SN

q

EA

N

iGi

1

21

N

iG

BiG

AiAB SS

Nq

1

)()(1

Digression: Phase diagram of the SK model (T, J0, H)

• Replica Trick to perform the quenched averaging of F

By simplifying this expression, introducing as new variables qrs,

and performing a saddle-point analysis, we arrive:

Spin glass phase: q≠0, <Si>=0; (Binder, 1986)

(Sherrington and Kirckpatrick, 1978; H-T plane: Almeida and Thouless, 1978)

n

r

N N Nn

r ij

rj

riij SrJijH

S Sr Sn

SSJ

ijijij

qn

ij

ijq

n

n

xn

n

xn

n

qijq

eJDJPeJPdJTJZ

JAJPDJA

n

x

n

ee

dn

dx

TJZkTTF

11 )(

}){},({2

}1{

2

}{

2

}{)(

0

ln

0

ln

0

][....])([...)},({

})({][

1lim

0

1lim)(limln

)},({ln)(

STr

Protein Synthesis

Transcription: DNA A, G, C, T pre-mRNA splicing mRNA A, G, C, U

Translation: ribosomes, tRNA Genetic code (degenerated !) Initiation: usually Met (AUG) Stop: UAA, UAG, UGA

Folding: with or without chaperons Covalent modifications: disulfide bonds proteolytic modifications, glycozylation…

A chaperone

Protein Structure

Primary structure:the amino acid sequence

Ramachanrdan plot L

N C

Protein Structure

Secondary structure: common regular local structures

α-helix β-sheet

RH helixes are more common than LH

Protein Structure and Classification

Tertiary structure: overall three-dimensional structure of a protein molecule motifs=common “blocks”, domains=independently folding regions

Classification:Globular proteins Fibrous proteins

Lysozyme Heat Shock Protein Collagene

Natively Unfolded proteins -substantial regions of disordered structure -usually have a target ligand -disorder-order transition when binding

Protein Structure

Quaternary Structurearrangements of several polymer molecules in a structure

Protein Folding

Interactions stabilizing the proteins• hydrophobic effect -entropic origin• hydrogen bonds - polar molecules• van der Waals interactions - induced dipoles • Coulomb interactions• at some proteins, disulfide bonds

kT = 4 x10-21 J = 0.03 eV

Anfinsen’s experimentDenaturation - Ribonuclease enzymerestoring the original conditions – the enzyme STARTED TO WORK AGAIN

• gentle heating / chemical treatment (urea, mercapto-ethanol)denaturation

• restoring the original conditionsspontaneous refolding (time scale: seconds)

=> Building of the 3D structure is SPONTANEOUS (in many cases)

Levithal’s paradox

• Anfinsen: there is a native state (F=minimum) • small protein, N=100 amino acids• assume 3 rotamers/monomer

Total number of structures:

• one microstate visited in 10-13s

Time necessary for finding the native state:

Thermodynamic + Kinetic problemSolution: Biasing towards the native state is necessary

47100 1033...33

yearsss 27341347 10101010

Microscopic Models

A typically used Hamiltonian

aI monomer species 1...20 (I: index along the chain)

N number of monomers rI position of the monomer IΔ interaction range function

(usual lattice models: 1 for nn., 0 otherwise)

ε(aI, aJ) interactions between amino acids I and J (NxN)

εij amino acid interaction matrix (20x20)

Including hydrophobicity: -the 21th species is the water -in the “empty” sites

)(),( JI

N

JIJI rraaH

)()(),(20

JIji

ijJI ajaiaa

Digression: the Gō model

• assumption: we know the folded, native conformation

• this conformation is energetically very well optimized • energy: function of the native contacts

εIJ= -w if I and J are first neighbors in the native state

εIJ=0 otherwise

η: the number of native contacts

“uses the answer to answer the question” ?

This model does not help the structure prediction, but it is helpful if we are studying how the protein reaches its native state.

)(2

1w E

Energy spectrum of random heteropolymers

The energy spectrum (400 lowest states) looks alike REM (Sali et al., 1994)

Indeed,

O(N) ≈independent terms => Central Limit Theorem => Gaussian distribution

only some sequences would fold repeatedly to the same stateKEY: single low-lying ground energy

neighboursfirst

IJE

)(),( JI

N

JIJI rraaH

Essential: the ground state

Threshold energy of the REM:

Extreme value statistics: Gumbell distribution it can be shown:

width of the energy gap:

Problems with the REM (thermodynamics):

•no flexibility against changing conditions

•no mutation stability matrix elements changed with ±b, energy levels change with (not large enough ΔE for a unique native state /freezing, escape/)

•there must be some correlation between the energy levels…

2ln2NEC

2ln2NE qG

W(Eg)

Eg

Ec

)1(1, OE GG

Nb

A Way Out: Sequence Design

“Pulling down” the energy of a target conformationCanonical design

•Given a 3d conformation C*• Searching for the best sequence of amino acids that minimalizes E for the given C*

Algorythm: the sequence is annealed

Movement in the sequence space: Metropolis MC method

What about Tdes?too high: random walktoo low: can be useless

*)(),(*)( CJI

N

JIJIdes rraaCH

)'()(,

)'()(exp

)()'(,1

)'(SESEif

kT

SESESESEif

SSP

des

Phase Diagram of Designed Proteins

(Pande et al., 2000)“Folded globule”:

•proteins with a stable target conformation

•they are “minimally frustrated”

DigressionInterpretation of a Chaperone Function

avoiding aggregation

e.g. HPhobic-HPhobic residues

(Clark, 2004)

Prion Proteins

•diseases transmitted by proteins

• PrPSC can induce PrPC→PrPSC transition

• PrPC might be an “off-path”

KineticsThe Funnel Hypothesis

How do we solve Levinthal’s Paradox?

Significantly low-energy native state: partially native structures also will have lower energies than others

Bumps: due to competitive interactions

=>FUNNEL

KineticsFree Energy Barriers and Nucleation

Barriers of F : energetic and entropic

Nucleation: • liquid-gas transition: homogeneous shrinking: ΔE and ΔS disadvantages solution: states with non-uniform density • protein folding: folding ~ seems to be a first-order transition nucleus: small, native secondary structure e.g. α-helix subsequent structure formation is speeded up

00

STEF

Digression

Super-Arrhenius behaviour

Most probably energy in the REM:

Assumption: these probable conformations surrounded with ones.

transition-state theory:

the argument is quadratic rather than linear – “Ferry law”

=> roughness (σ) slows down folding

kTEee mp

E

kT

E

dE

d 22 0

2

2

0E

2

2

)(0

kTesc et

Reaction CoordinateSimple (bimolecular) chemical reactions

A+BC→AB+C

PES(rAB, rBC)

reaction coordinate: the minimum energy path via a saddle-point

Protein Folding: the choice is difficult, no general solution

• similarity to the native state, Q

• an alternative choice: Pfold, or “commitment” Pfold: the probability of folding before even touching an unfolded state

total

nativeQ

Digression: Alternative Reaction Coordinate“Development” on the graph• Lattice model

• {C} conformation space ↔ graph

• single “elementary step” difference ↔nodes C1 and C2 connected

• nC – occupation number (eg. # of independent simulations)• mC – degree of the node

• “Potential” on the graph nodes:

• “development”: MMC dynamics

• define:

=>

Ohm’s law!

kT

CE

Cem)(

Ic→c’=(nc/mc) min{1; (mc/mc’)eE(C)-E(C’)} Ic’→c=(nc’/mc’) min{1; (mc’/mc)eE(C’)-E (C)}

Rcc’=max{mCeU(C);mC’eU(C’)} I= Ic→c’- Ic’→c=[Φc- Φc’]/Rcc’

Digression: Alternative Reaction Coordinate“First return” (casino) problem

“particle” (money) at X0

I will end up with 0 money ↔ all the flux is going to 0electric circuit analogy

Pfold: probability to arrive to the folded state FOR THE FIRST TIME

(Grosberg, 2003)

pfold = RCU/(RCU+RCF)punfold = RCF/(RCU+RCF)

moneyx00

( ) ;U

U xCU

C

R e dx( )

( )

UU x

CU Cfold U

U xFU

F

e dxR

pR

e dx

Conclusion

• protein folding: self-assembly

• low-energy ground state

• biased walk – correlations, funnel hypothesis

• “nucleation”

• sequence design

Acknowledgements

I’m indepted to

Michel DROZ, Alexander GROSBERG, Géza GYÖRGYI,Gabriella NETTING,Zoltán RÁCZ, Zoltán SZABÓ,László SZILÁGYI

and many others…