Protein Rigidity and Flexibility:Applications to Folding

A.J. Rader

University of PittsburghCenter for Computational Biology & Bioinformatics

How can one characterize the flexibility of proteins?

Molecular dynamics Comparing different

conformations Identifying domains

within proteins

Jacobs, et al. PROT 44 (2001)

Gerstein and Krebs Nucl. Acids. Res. 26 (1998)

http://molmovdb.mbb.yale.edu/molmovdb/

What is Flexibility? Flexibility, as presented here, is determined directly

from the mechanical properties of the structure studied. The number of floppy modes, F, measures the number of degrees of freedom and quantifies the flexibility of the network (protein).

Floppy modes are zero frequency modes corresponding to deformations which cost no energy.

Maxwell counting is a mean field approximation for the fraction of floppy modes as a function of <r>, the mean coordination, of the network.

Inspiration from Network Glasses

Is the universal rigidity phase transition seen in network glasses at <r> = 2.40 also seen in proteins? (suggests mean coordination is a relevant parameter)

Is protein folding/unfolding related to a rigidity phase transition in proteins?

32

2max

2

r

rr

rnNF

2D Maxwell constraint counting

4 atoms, 4 bonds

F=2*4 - 4 - 3 = 1

Floppy

4 atoms, 6 bonds

F=2*4 - 6 - 3 = -1

Rigid & Stressed

4 atoms, 5 bonds

F=2*4 - 5 - 3 = 0

Isostatically Rigid

F = 2N – Nbonds - 3

Laman’s Theorem In 2D: rigidity is uniquely found by building the

network up one bond at a time and checking for redundant bonds in all subgraphs of the network. A redundant bond is found whenever b > 2n-3 in a

subgraph. Laman, J. Engrg. Math. 4:331 (1970).

Led to the pebble game algorithm. Jacobs & Thorpe, Phys. Rev. Lett. 75:4051 (1996).

3D generalization (b > 3n-6) is only valid for the graphs of the type that also contain bond-bending constraints (next-nearest-neighbor constraints). These bond-bending constraints correspond to fixing the

chemical bond angle. Tay & Whiteley, Struct. Topo. 9:31 (1985); Jacobs, J. Phys.

A. 31:6653 (1998).

3D Maxwell constraint counting (on bond-bending networks)

5 Atoms 6 Atoms 7 Atoms

5*3 -5 -5 -6 = -1Rigid

6*3 -6 -6 -6 = 0Isostatic

7*3 -7 -7 -6 = 1Flexible

6322

3max

2

r

rr r

rnNFF = 3N – Nbonds – Nangles -6

Applying Maxwell counting to Proteins FIRST*

*Floppy Inclusions & Rigid Substructure Topography

Jacobs, et al. PROT 44 (2001)http://firstweb.pa.msu.edu/

ModelFix physically inspired constraints and count number of floppy modes and mean coordination.

D

H A

B

Covalent bond

Chemical bond angle

Hydrogen bond

Hydrophobic contact

Dihedral angles left free to rotate degrees of freedom (floppy modes)

Hydrogen Bonds

The energy is assigned according to

CN

H A

d

r

orbitals atomic upon the depends ),,(

Å 8.2 kcal/mol 8

),,(65

00

10

0

12

00

F

RV

Fd

R

d

RVEHB

N

H O

C

Adds 3 constraints per H-bond

R ≤ 0.25ÅR

C C

Adds 2 constraints per tether

Hydrophobic Contact Hydrophobic Tether

Hydrophobic Tethers

The Protein Model

Always present: covalent bonds (with bond-

bending constraints) peptide bonds locked hydrophobic tethers

Variable interactions: inclusion of a hydrogen

bond depends on the temperature

Hespenheide, et al. JMGM (2002)

Rigid Cluster Decomposition

Unfolding with FIRST

Hydrogen bonds are removed one at a time based upon energy to simulate thermal denaturation.

Removal of hydrogen bonds mimics unfolding pathways.(non-covalent interactions in the native state contain

information about folding )

Native State

Hydrophobic Collapse

Formation of hydrogen bonds and salt bridges

Denatured

Folding

Unf

oldi

ng

Breaking of hydrogen bonds and salt bridges

Rigidity Lost

Rader, et al. PNAS 99 (2002)

Hydrogen Bond Dilution

Fraction of floppy modes

Differentiate… once, twice

Protein DatasetCode Protein Name Class Nres Hphob <r>T

1a2p Barnase 108 58 2.411a3k Galectin 137 85 2.401a6m Myoglobin 151 105 2.401ake Adenylate Kinase 214 92 2.401bpi Pancreatic Trypsin Inhib 58 50 2.391bu4 Ribonuclease T1 104 68 2.401hml Ca-binding Protein 123 70 2.401hrc Cytochrome c 105 99 2.421nkr Killer Cell Inhib Receptor 201 160 2.391ruv Ribonuclease A 124 104 2.411rx1 DHFR 159 132 2.411ten Tenascin 90 37 2.401ubi Ubiquitin 76 29 2.392chf Che Y 128 62 2.392ci2 Chymotrypsin Inhib 83 27 2.402liv LIV-binding Protein 344 247 2.40

3lzm Lysozyme 164 75 2.414ilb Interleukin 1- 153 99 2.401bif PFKinase/FBPase 864 633 2.401cku Electron Trans Protein 170 150 2.401hhp HIV-1 Protease 99 120 2.391vls Aspartate Receptor 292 197 2.391ice Interleukin 1- C.E. 518 304 2.411ids Fe-SOD 792 606 2.401szj GAPDH 1332 790 2.402cts Citrate Synthase 874 707 2.40

Folding Results from FIRST Native state hydrogen bonds and hydrophobic

interactions encode information about folding. Results for the rigidity phase transition in glass

networks give a critical mean coordination of <r>c = 2.385.

A similar plot of flexibility versus mean coordination exists for a set of 26 diverse protein structures with the folding transition at <r>T = 2.405 ± 0.015. Higher mean coordination implies rigid and folded, lower implies flexible and unfolded.

The jump in df/d<r> suggests a first-order phase transition within proteins.

Application to Rhodopsin

Question of interest: what is the conformational change induced by retinal isomerization?

Proposed activation mechanism: opening of helical bundle toward cytoplasmic ends of the helices, predominantly involving helices III,VI, & VII.

<r> 2.394 2.390 2.376

Nhb 118 101 31

Rhodopsin Folding Core

Rhodopsin Summary

Greater flexibility observed at the cytoplasmic ends of all transmembrane (TM) helices.

Folding Core is formed by parts of TM helices III,IV,V; 2nd -sheet; and some extracellular loops.

residues 9,10,22-27,102-116,166-171,175-180,185-188,203-207,211*

Flexible portions consistent with proposed activation mechanism: movement of cytoplasmic ends of helices III,VI, & VII.

Acknowledgements

Proteins and Rigidity Claire Vieille Harini Krishnamurthy Ming Lei Maria Zavodszky Mykyta Chubynsky

Funding NSF, DOE, and NIH MSU Center for Biological

Modeling. CCBB, School of Medicine

University of Pittsburgh

MSU and FIRST development Michael F. Thorpe Leslie A. Kuhn Don Jacobs Brandon Hespenheide

CCBB and Rhodopsin Ivet Bahar Judith Klein-Seetharaman Basak Isin