Biofísica Teorica
(a brief introduction to some interesting open problems)

Pierpaolo Bruscolini

Instituto BIFI – Universidad de Zaragoza

pier@unizar.es

Molecular biology central dogma

Gene expression and regulation : a complex dynamical process

repressor

binding site

for I

promoter

Without lactose: I is expressed and blocks production of lactose-related

enzymes (Z,Y,A).

Lactose blocks the activity of I, that cannot inhibit the expression of

Z,Y,A

Theoretical contributions from Physics

Single molecule studies:

- DNA denaturation transition (thermodynamics,

dynamics), statistical properties of loops

- RNA folding

- Protein Folding

System level studies

gene/ protein networks

Bioinformatic studies (statistical properties of coding/non coding DNA,

individuation of binding sites, statistical inference from noisy data)

Proteins:

paradigmatic example of functional molecules, at room T

- properties, macroscopic behavior allowing

- microscopic components

- comparison with complex (disordered) systems

PROTEINS

Linear Polymers, 50 ÷ >4000 amino acids (typically ~200 aa,

~3000 atoms)

Variety of biological functions (enzymatic, structural, transport, electron transport, movement, signal transduction...)

Proteins can function only when they are in a particular 3D conformation

(Native State)

In-vivo synthesis in

jelly environment:

very often requires molecular

helpers (chaperons),

to prevent misfolding

or aggregation

...but: KEY OBSERVATION:

- Take a functional protein in its natural solvent (water)

- Heat it (or add chemicals, or change pH): function loss (“denaturation”)

- Restore initial conditions: it functions again!

The amino acid sequence completely specify the native structure

(and the way to reach it)!!

Protein Folding is, in principle, a well posed problem from a physical

perspective

The Native State

...is not so ugly as it seems,

there are some regular features:

QTDVIAQRKAILKQMGEATKPIAAMLKGEA

KFDQAVVQKSLAAIADDSKKLPALFPADSK

TGGDTAALPKIWEDKAKFDDLFAKLAAAA

TAAQGTIKDEASLKANIGGVLGNCKSCHDDF

RAKKS

It's “unique” in a dynamical sense:

an ensemble of structurally similar

conformations

...For each sequence, a native structure, but:

for each structure, many possible sequences...

~100000 sequences, ~23000 structures, ~ 1000 families

A closer look to the structure: let's see the details...

primary structure: the sequence of covalently bonded a.a.

secondary structure: hydrogen bonds

tertiary structure: 3-D overall arrangement of one chain

quaternary structure: assembly of several chains

The polypeptide chain (primary structure)

peptide bond

side chain

Secondary structure:

determined by (strongly directional)

hydrogen bonds (20-40 kJ/mol)

-helices

-β-sheets (parallel, antiparallel)

-turns

α- helices

β - sheets

Energetics: strong, local interactions: covalent bonds

NB: RT (at room temperature) = 2.5 kJ/mol

Energetics: weak, non-local interactions

Hydrophobic interactions:

nonpolar molecules induce ordering in water to preserve hydrogen bonds:

entropy favors effective attraction between nonpolar (hydrophobic) residues

Protein Folding:

how does a protein fold? and why? what are the driving forces?

where does the stability come from? How unfolded is the denatured

state? Is there a unique folding pathway? (physics,biology,medicine)

Given a sequence, can we predict the native structure?(biology)

Given a structure, can we design a sequence that folds into it? (biol, pharm.)

Experimentally: few populated states in Thermodynamics an Kinetics

Two-state thermodynamics:

kinetics is related to barriers as in simple chemical reactions!

But why is it so?

A generic polypeptide is expected to have a complex (glassy-like)

behavior: frustration, rugged energy landscape, exponential

number of minima, glass transition, slow dynamics,...

This would be incompatible with function, and hence, life.

Evolution has optimized sequences/structures for producing quick

and stable folders.

Digression: Complex (disordered) systems

- Example: spin glasses

- frustration: ground-state highly

degenerate, many different structures

- properties depend on details of the

hamiltonian

- new statistical mechanics

(Jxy > 0)

(Jxy < 0)

Example of a complex, frustrated system:

spin glass. In ?, both σ = 1 and σ = -1 are

bad choices. Ground state is highly

degenerate.

Biofísica Teorica
(a brief introduction to some interesting open problems)

Pierpaolo Bruscolini

Instituto BIFI – Universidad de Zaragoza

pier@unizar.es

Protein Folding:

how does a protein fold? and why? what are the driving forces?

where does the stability come from? How unfolded is the denatured

state? Is there a unique folding pathway? (physics,biology,medicine)

Given a sequence, can we predict the native structure?(biology)

Given a structure, can we design a sequence that folds into it? (biol, pharm.)

Born to Function:

Native state must be:

- global thermodynamical minimum (or a long lived metastable:

a few cases exist)

- quickly accessible by dynamics in a reasonable time scale (ms ÷s)

NB: random search is ruled out: assuming 10 conformations/residue

a 100-residues protein would have 10100 conformations: at least

1077 years to explore the configuration space!)

- stable against denaturation, but not too much (function implies

switching between states, degradation must be possible)

- “resistant” to mutations: highly designable!

(A polymer of 100 a.a can have 10130 sequences: if structures weren't

highly designable, life wouldn't have evolved)

Digression: some thermodynamic concepts

Free energy:

F= U -T S =

-(1/β) log Σstates exp(-β E(state))=

-(1/β) log ΣE g(E) exp(-β E)

S~ log (# states at the system energy)

Stable thermodyn. state = minimum of F

a rugged, “frustrated” lanscape

(p.e. random sequence)

Proteins: simple behavior. Random seq. heteropolymers: complex behavior

Current understanding (from experiments and theory):

Hydrophobicity main driving “force”, and important for stability.

Stability is a result of huge cancellations of terms

A principle of minimal frustration is at work:

- native state= free energy minimum.

- no competing states at similar energy (energy gap ΔE

to the other relevant minima)

- small energy barriers (little frustration: small δEmisfold/ΔE)

- huge funnel reduces the search in configurational space

(NS must be reached by “all” initial conditions)

- multiple pathways, transition state ensemble, entropic barriers

The folding “funnel”

Theoretical models: from lattice to all-atoms description.

Usually: two-body potentials.

Very often: effective interactions between residues replace solvent.

Theoretical models: several techniques

Monte Carlo (random moves proposed from a predetermined sets,

accepted or refused according to a probabilistic rule)

Molecular Dynamics (implement Newton eqs. with a given potential)

Analytical Techniques (mean field, master equation) in case of simple

models

All-atoms empirical potentials (CHARMM, AMBER,...)

Results: realistic, but short

- MD simulations with “realistic” force fields can describe a ns life

of a small protein (45 residue), at physiological T with explicit solvent,

10÷100 ns with implicit solvent.

(NB: bond formation: ~ 0.1 ps, so that time step ~10-15 s is necessary.

Folding times > 1ms approx)

- just a few runs: little statistics. Equilibrium properties?

- simulations are reliable, if potential is reliable: though realistic,

these potential are empirical. Quantum physical approach: impossible.

On the other hand, maybe proteins not too sensitive to details!

- Very useful to optimize structures, and to simulate high T unfolding

- structure prediction relies on ad-hoc potentials, with extra statistical

terms

Alternative approach: Just study the process, without predicting

(only when native state is known)

Gō-like models: take into account only native contacts, disregard

the others

this approach eliminates energetic frustration, and study if the

structures are geometrically frustrated (= difficult to reach)

It would be realistic if:

- real native contacts are better than non natives and/or

- during folding, dynamics only pass through good contacts

Is it realistic? Compare with experiments...

Go-like models (all atoms, reduced represent., Cα)

Cα case:

if i,j native pair: ε(i,j) > 0, ε2(i,j) = 0,

otherwise ε(i,j) = 0, ε2(i,j) > 0

σij=contact distance if i,j native pair,

otherwise σij=4 Å

Results:

- with good tuning of parameters, these models can recover NS

- substantially correct identification of experimental folding

nucleus and behavior (which structure forms first, etc.) in a

number of proteins (CI2, SH3, Barnase, circular permutations...)

- with all atoms, improved cooperativity

Small proteins appear indeed to have little energetic frustration!

Ising like, Go-like models: discrete monomer states

Eaton model

To (native) interaction ij, energy

. To native unit, entropic cost

Finkelstein model.

qR: entropy gain for

unfolding a residue

entropy of a disordered

stretch (0's from i+1 to j-1)

Simple models allow an analytical treatment:

Eaton model's thermodynamics can be known exactly

(exact evaluation of the partition function)

Finkelstein model requires mean field / simulations.

Analytical approach to dynamics still an open issue

Results:

Reasonable estimate of the barrier height for simple proteins.

Claim of good agreement with experimental results for identification

of the folding nucleus (that is, the top of the barrier, that determines

the speed of folding)

More in general:

simple models are valuable to understand the mechanisms, and

hopefully to give a quick hint to experimentalists on what aspects

should be investigated (eg, what residues to mutate)

Conclusions for protein folding

Proteins have evolved to cope with a number of requirements:

fast folding, stable but not too much, robust against mutations.

Simple macroscopic behavior from an amazing microscopic complexity.

Structure is such that energetic frustration is minimized, and the folding

process amounts to a search of a few good pathways.

Structure prediction is still a black box, even when successful.

Great advancements, from theory and exp., in understanding/predicting.

Proteins: a tiny subset of “simple” systems, within a huge sea of “complex”

random polymers. Is it possible to define “proteinicity”? To define an

equivalence relation between sequences folding to the same structure?

To find similar simple behavior in subset of other glassy systems?

A comprehensive theory for their existence is still waiting to be found...

Physics in genomics and proteomics

Revolutionary high throughput techniques:

- DNA chips → gene expression levels

Allow to detect the amount of each mRNA molecule in a cell

= take a picture of which genes are co-expressed or silenced

= take some snapshot of the chemical dynamics of a cell

Result in a huge amount of (noisy) data witnessing the patterns of

gene activity

- Yeast two-hybrid → protein interactions

Allow to probe if two proteins interact

= to understand directly who is involved in what

Result in a huge interaction network

- MS (mass spectrometry) and MS/MS (tandem MS) techniques:

Allow to identify protein and protein complexes in a cell

= take a picture of which proteins are around at a certain time

Problems (some examples):

- Interpretation of signals

- Characterization of protein/gene functions

- Characterization of regulatory networks

- Further on: characterization of dynamics of chemical processes (mitosis,

embryo development, ...)

Example: interpretation of MS/MS signals

The challenge: given the spectra of its fragment (+ noise), find the

original peptide

Protein networks...

Example of inference of protein function from interaction network:

given the network and the functions of a subset of proteins, find

the functions of the other proteins (possibly adding/removing some links)

Xi=1,...F possible functions

i=1...N nodes of unknown function

(white)

Jij = 1 if there is a link, 0 otherwise

In all these cases, it is possible to map the inference problem on a

statistical mechanics one.

Tipically, one ends up with a graph where the nodes carry some possible

state, and the links specify constraints/interactions/causality. The known

data act as external fields, and the task is to find out the configuration

minimizing an appropriate function.

Statistical mechanics is valuable because:

- it provides tools to work at T≠ 0 (optimization is equivalent to find

the most likely state at T=0, which is hard)

- it provides controlled recipes to factorize probabilities

P(X1,...XN)= P(X1) P(X2,X3) P (...)

Conclusions

New high throughput biological techniques are providing a huge number

of data that need to be assessed and rationalized into models of

biochemical behavior.

The understanding of cells' “states” would have enormous practical

consequences (i.e.: detection/understanding of cancer, drug design beyond

the single molecule level,...).

Statistical mechanics on graphs may have a significant role in the process.

Advancements are needed, especially in finding better mean field

approximations and reliable algorithms (the configuration space is tipically

crowded with local minima, hindering optimization)

We're just at the beginning (and we need smart PhD students...)

Physics at BIFI (Instituto de Biocomputación y Física de Sistemas

complejos – Zaragoza)

- spin glasses

- complex networks

- non-linear dynamics

- protein folding (simplified and realistic)

- computational chemistry

Collaborations with the biologists of the Institute:

protein folding, ligand binding, drug design.

Facilities:

- cluster of ~200 computers

- in the next few years: SSUE (dedicated computer): presently under

development.

- biochemical lab.

http://bifi.unizar.es (still under development)

2004-06-13T17:47:43