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Conformation Networks: an Application to Protein Folding

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### Conformation Networks: an Application to Protein Folding

Zoltán Toroczkai

Erzsébet Ravasz

Center for Nonlinear Studies

Gnana Gnanakaran (T-10)

Theoretical Biology and Biophysics

Los Alamos National Laboratory

Proteins

- the most complex molecules in nature
- globular or fibrous
- basic functional units of a cell
- chains of amino acids (50 – 103)
- peptide bonds link the backbone

Native state

- unique 3D structure (native physiological conditions)
- biological function
- fold in nanoseconds to minutes
- about 1000 known 3D structures: X-ray crystallography, NMR

153 Residues, Mol. Weight=17181 [D], 1260 Atoms

Main function: primary oxygen storage and carrier in muscle tissue

It contains a heme (iron-containing porphyrin ) group in the center. C34H32N4O4FeHO

Protein conformations

Amino-acid

- defined by dihedral angles
- 2 angles with 2-3 local minima of the torsion energy

- N monomers about 10N different conformations

Levinthal’s paradox

- Anfinsen: thermodynamic hypothesis
- native state is at the global minimum of the free energy

Epstain, Goldberger, & Anfinsen, Cold Harbor Symp. Quant. Biol.28, 439 (1963)

- Levinthal’s paradox, 1968
- finding the native state by random sampling is not possible
- 40 monomer polypeptide 1013 conf/s

3 1019 years to sample all

universe ~ 2 1010 years old

Levinthal, J. Chim. Phys. 65, 44-45 (1968)

Wetlaufer, P.N.A.S. 70, 691 (1973)

- nucleation
- folding pathways

Free energy landscapes

- Bryngelson & Wolynes, 1987
- free energy landscape

Bryngelson & Wolynes, P.N.A.S. 84, 7524 (1987)

- a random hetero-polymer typically does NOT fold

- Experiment:
- random sequences
- GLU, ARG, LEU
- 80-100 amino-acids
- ~ 95% did not fold
- in a stable manner

Davidson & Sauer, P.N.A.S.91, 2146 (1994)

Funnels

Given any amino-acid sequence: can we tell if it is a good folder?

- experiments (X-ray, NMR)
- molecular dynamics simulations
- homology modeling

- Leopold, Mortal & Onuchic, 1992

Leopold, Mortal & Onuchic, P.N.A.S. 89, 8721 (1992)

Energy funnels

Difficult and slow

- many folding pathways

Molecular dynamics

Sanbonmatsu, Joseph & Tung, P.N.A.S. 102 15854 (2005)

- State of the art
- supercomputer (LANL)

- Ribosome in explicit solvent:
- targeted MD
- 2.64x106 atoms (2.5x105 + water)
- Q machine, 768 processors
- 260 days of simulation (event: 2 ns)

~ 1016 times

slower

- distributed computing (Stanford, Folding@home)

- more than 100,000 CPU’s
- simulation of complete folding event
- BBA5, 23-residue, implicit water
- 10,000 CPU days/folding event (~1s)

Shirts & Pande, Science290, 1903 (2000)

Snow, Nguyen, Pande, Gruebele, Nature420,102 (2002)

Configuration networks

Ramachandran map

PDB structures

- Configuration networks

Protein conformations

- dihedral angles have few preferred values

Ramachandran & Sasisekharan, J.Mol.Biol.7, 95 (1963)

- Helix
- Sheet
- other

NODE configuration

LINK change of one degree of freedom (angle)

- refinement of angle values continuous case

Why networks?

- VERY LARGE: 100 monomers 10100 nodes. However:

Generic features of folding are determined

by STATISTICAL properties

of the configuration network

- degree distribution
- average distance
- clustering
- degree correlations

- toolkit from network research
- captures the high dimensionality

Albert & Barabási, Rev. Mod. Phys.74, 67 (2002); Newman, SIAM Rev.45, 167 (2003)

- faster algorithms to simulate folding events
- pre-screening synthetic proteins
- insights into misfolding

A real example

- The Protein Folding Network: F. Rao, A. Caflisch, J.Mol.Biol, 342, 299 (2004)

- beta3s: 20 monomers, antiparallel beta sheets
- MD simulation, implicit water
- 330K, equilibrium folded random coil

NODE -- 8 letters / AA (local secondary struct)

LINK -- 2ps transition

Its native conformation has been studied by NMR experiments:

De Alba et.al. Prot.Sci. 8, 854 (1999).

Beta3s in aqueous solution forms a monomeric triple-stranded antiparallel beta sheet in equilibrium with the denaturated state.

- Simulations @ 330K
- The average folding time from denaturated state ~ 83ns
- The average unfolding time ~83ns
- Simulation time ~12.6s
- Coordinates saved at every 20ps (5105 snapshots in 10s)
- Secondary structures: H,G,I,E,B,T,S,- (-helix, 310 helix, -helix, extended, isolated -bridge, hydrogen-bonded turn, bend and unstructured).
- The native state: -EEEESSEEEEEESSEEEE-
- There are approx. 818 1016 conformations.
- Nodes: conformations, transitions: links.

Scale-free network

Many real-world networks are scale free

- hubs

beta3s

randomized

- co-authorship (=1 - 2.5)
- citations (=3)
- sexual contacts (=3.4)
- movie actors (=2.3)
- Internet (y=2.4)
- World Wide Web (=2.1/2.5)
- Genetic regulation (=1.3)
- Protein-protein interactions ( =2.4)
- Metabolic pathways (=2.2)
- Food webs (=1.1)

Barabási & Albert, Science 286, 509, (1999);

Many reasons

behind SF topology

- Why is the protein network scale free?
- Why does the randomized chain have

similar degree distribution?

- Why is = - 2 ?

Robot arm networks

12

n=0

22

02

11

01

21

0

2

1

20

n=1

00

10

00

22

n=2

01

21

02

20

021

021

10

12

11

020

020

010

010

000

000

100

100

200

200

- Steric constraints?
- missing nodes
- missing links

- n-dimensional hypercube
- binomial degree distribution

Homogeneous

Swiss cheese

A bead-chain model

- Beads on a chain in 3D: robot arm model
- similar to C protein models
- rod-rod angle
- 3 positions around axis

Honeycutt & Thirumalai, Biopolymers32, 695 (1992)

N=6; = 90

N=18; = 120

2212112212111122

- Homogeneous network

Adding monomers not only increases the number of nodes in the network but also its dimensionality!! The combined effect is small-world.

The “dilemma”

SCALE FREE

- from polypeptide MD simulations
- beta3s
- randomized version

HOMOGENEOUS

- from studies of conformation networks
- bead chain
- robot arm

?

Gradients of a scalar (temperature, concentration, potential, etc.) induce flows (heat, particles, currents, etc.).

Naturally, gradients will induce flows on networks as well.

Ex.:

Load balancing in parallel computation and packet routing on the internet

Y. Rabani, A. Sinclair and R. Wanka, Proc. 39th Symp. On Foundations of Computer Science (FOCS), 1998:“Local Divergence of Markov Chains and the Analysis of Iterative Load-balancing Schemes”

References:

Z. T. and K.E. Bassler, “Jamming is Limited in Scale-free Networks”, Nature, 428, 716 (2004)

Z. T., B. Kozma, K.E. Bassler, N.W. Hengartner and G. Korniss “Gradient Networks”, http://www.arxiv.org/cond-mat/0408262

Let G=G(V,E) be an undirected graph, which we call the substrate network.

The vertex set:

The edge set:

A simple representation of E is via the Nx N adjacency (or incidence) matrix

A

(1)

Let us consider a scalar field

Set of nearest neighbor nodes on G of i :

The gradient h(i) of the field {h} in node i is a directed edge:

(2)

Which points from i to that nearest neighbor

for G for which the increase in the

scalar is the largest, i.e.,:

(3)

The weight associated with edge (i,) is given by:

.

.

The self-loop

is a loop through i

with zero weight.

Definition 2

The set F of directed gradient edges on G together with the vertex set V forms the gradient network:

If (3) admits more than one solution, than the gradient in i is degenerate.

In the following we will only consider scalar fields with non-degenerate gradients. This means:

Theorem 1

Non-degenerate gradient networks form forests.

Proof:

0.82

0.67

0.46

0.65

0.6

0.53

0.44

0.5

0.67

0.7

0.2

0.22

0.65

0.1

0.19

0.16

0.87

0.14

0.15

0.32

0.2

0.2

0.18

0.05

0.55

0.44

0.15

0.05

0.24

0.43

0.16

0.8

0.13

0.65

0.65

Theorem 2

The number of trees in this forest = number of local maxima of {h} on G.

For Erdős - Rényi random graph substrates with i.i.d random numbers as scalars, the in-degree distribution is:

A. Clauset, C. Moore, Z.T., E. Lopez, to be published.

K-th Power of a Ring

The energy landscape

- Energy associated with each node (configuration)
- the gradient network
- most favorable transitions
- T=0 backbone of the flow
- MD simulation
- tracks the flow network
- biased walk close to the gradient network
- trees
- basins of local minima

What generates = - 2 ?

The REM generates an exponent of -1.

Model ingredients

k, E increases

constrained (folded)

small kconf

lower energy

loose (random coil)

large kconf

higher energy

- A network model of configuration spaces
- network topology
- homogeneous
- degree correlations

- how to associate energies

Random geometric graph

R=0.113, <k>=20

- Energy proportional to connectivity

E

k

- random geometric graph

Dall & Christensen, Phys.Rev.E 66, 026121 (2002)

- in higher D: similar to hypercube with holes
- degree correlations

Bead-chain model

Lennard-Jones potential

Repulsive

Attractive

- more realistic model: bead-chain
- configuration network
- excluded volume
- energy: Lennard-Jones

The MD traced network

T = 400

More than one simulation

3 different runs: yellow, red and green

The role of temperature

Conclusions

- A network approach was introduced to study sterically constrained conformations of ball-chain like objects.
- This networks approach is based on the “statistical dogma” stating that generic features must be the result of statistical properties of the networks and should not depend on details.
- Protein conformation dynamics happens in high dimensional spaces that are not adequately described by simplistic reaction coordinates.
- The dynamics performs a locally biased sampling of the full conformational network. For low enough temperatures the sampled network is a gradient graph which is typically a scale-free structure.
- The -2 degree exponent appears at and bellow the temperature where the basins of the local energy minima become kinetically disconnected.
- Understanding the protein folding network has the potential of leading to faster simulation algorithms towards closing the gap between nature’s speed and ours.

Coming up: conditions on side chain distributions for the existence of funneled energy landscapes.

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