Correlated Movements in Proteins:

Insights from the Dynamics of Protein Structure Networks

Saraswathi Vishveshwara

Molecular Biophysics Unit

JNC, Nov 15, 2007

Indian Institute of Science

Bangalore, India

Outline

1.Correlated fluctuations from protein dynamics

a) residue-wise RMSD

b) Pair-wise cross correlations

c) Important modes from essential dynamics

2. Protein structure Networks

3. Dynamics of networks:

Results based on the equilibrium dynamics of

the protein: Methionine tRNA synthetase

Essential modes- effect of ligand binding

Cross-correlation maps

Communication pathways

Depend crucially upon the folded states of the Depend crucially upon the folded states of the

proteins. proteins.

Biological Functions of Proteins

Protein Structure to Function

Biological

multimeric

states

Disease states

mutations

Active sites, enzyme clefts

Antigenic sites

Surface properties

3D STRUCTURE

Protein-Ligand

Interactions

While the rigidity is required to maintain the structure, flexibility is needed to perform the function

Flexibility and correlated motions from MD simulations

Residue-wise RMSD- fluctuations of individual residues

Cross correlation map- correlations between residues

Essential modes- global motions in the protein

Simulation on Eosinophil Cationic Protein (ECP)

Sanjeev & Vishveshwara, JBSD (2004, 2005)

RMSD Trajectories

RMSD as a function of simulation time

continuous line: w.r.t <MD>

broken line: w.r.t crystal structure

RMSD as a function of residue number

( ) ( )( ) ( )222 2

i i j j

ij

i i j j

r r r rC

r r r r

− −=

− −

Pair-wise Cross Correlation

Cij – the cross correlation between the residues i and j

ri, rj the coordinates at a given time point,

<ri>, <rj> are the averages over the trajectory

Collective modes from Essential Dynamics

Molecule of N atoms has 3N degrees of freedom. Essential dynamics is a way of capturing important collective modes

Covariance matrix (M=[mij])

mij=(1/S) Σt(xi(t)-<xi>)(xj(t)-<xj>)

where S = total number of configurations

t = time in picoseconds

i = ith coordinate (i=1,2,...,3N) ('N' being number of atoms)

<xi>= average value of xi over all configurations

Find Eigenvalues and normalized Eigenvectors of Covariance matrix

Reference:

A. Amadei, A.B. Linssen, H.J. Berendsen, Proteins: Struct. Func. Gen., 17, 412 (1993).

Relative positional fluctuation :

RPF(n)= Si=1,nl(i)/ Si =1,3N l(i)

where l(i) is ith eigenvalue.

RPF(n) gives the amount of motion associated in the

subspace spanned by the first n eigenvalues.

Rossmann fold (RED): CP (Green): Anticodon domain (Navy Blue):

Structure of Methionine tRNA Synthetase

Molecular Dynamics (MD) simulations on native and ligand-bound methionine t-RNA synthetase

1. MetRS (550 amino acids) 2. MetRS:Met complex Methionine3. MetRS:Met AMP complex methionyl adenylate

(Amit Ghosh and SV -unpublished)

RMSD Plots

EcMetRS:Met AMP

EcMetRS

EcMetRS:Met

Time(ns)

Time (ns)

Top ten modes account for about 72% of the motion in essential space

RPF Plots

Essential Dynamics

Superposition of the top modesfree (MetRS): Opening mode (red)

ligand-bound MetRS(MetRS:Met AMP) : Closing mode (blue)

Essential Dynamics

captures the global movements in proteins, such as inter-domain motions, hinge bending etc.

Dynamical cross correlation map representing inter-residue cross correlations

MetRS+tRNA+MetAMP

( ) ( )( ) ( )222 2

i i j j

ij

i i j j

r r r rC

r r r r

− −=

− −

Protein structures as Graphs

Protein GraphsProtein Graphs

• Nodes : Amino acid Residues

• Edges : Spatial neighbours

within fixed distance

Main Chain Interaction (back bone level )

S.M.PatraS.M.Patra, , KannanKannan, , VishveshwaraVishveshwara, , BiophysBiophys. . ChemChem (2000); JMB (2000); JMB

(1999)(1999)

High and low contact criteria. A pair of phenylalanine rings interacting with each other are shown. The lines between the phenylalanines indicate the atoms that are within a distance of 4.5Å.

a) High Contact b) Low Contact

Side Chain Interaction

c)No interaction

High (Iij=11%)Iij=0%

Low (Iij=4%)

Iij = (nij÷√÷√÷√÷√(Ni*Nj))××××100

Imin is user defined interaction cutoff.

An (ij) residue pair with Iij > Imin is connected by an edge.

Backbone-based versus the Side-chain-based

Protein Structure Graphs

Backbone-based (coarse grained)

Based on C-alpha-C-alpha distance

The extent of side-chain interaction is not considered

Side-chain-based

The interactions between sidechains are quantified, hence a weighted graph can be constructed or graphs can be constructed on the basis of the strength of interaction

Graph spectral parametersGraph spectral parameters

Provide information on theProvide information on the clusters of interacting residuesclusters of interacting residues

Detect Detect cluster cluster centrescentres, , which play a crucial role in the integrity which play a crucial role in the integrity

of the clusterof the cluster

Centre of a GraphCentre of a Graph

E(v) = max d(v,vi)vi∈G

Clusters (at interaction

strength Imin =6%)

in Ornithine Decarboxylase

Advantages of Graph Spectral Analysis

• Solution to weighted graph

• Identification of Cluster Centre

PSN

Size of largest cluster,Degree distribution, Hubs

Interacting biomolecularresidues

Adjacency Matrix

PSG

Side chain Clusters(DFS)

Construction of Graphs and Networks

Hubs in Protein Structure

Networks

� Hubs - highly connected amino acids in the protein

structure.

� Identified as amino acids having a contact number of >= 4

at a given Imin.

Arg

Pro His

Ser

Asp

Arg

Applications of Graph Spectral Analysis

to Protein structures

A) Clusters of Importance:

a) Active/Binding site

b) Domain identification

c) Determinant of thermal stability-Aromatic clusters

d) Protein-Protein interaction surfaces

B) Cluster centers:

Identification of hot spots, Signature motifs of oligomerization

A case study

Oligomerization in Legume lectins

Brinda, Surolia, Vishveshwara, 2005, 2006

Legume Lectins:

High sequence similarity

Similar 3-D structures

Diverse Quaternary Associations

Interface clusters at different

types of dimeric interfaces

Interface clusters

Different colors represent residues from different chains (note: The residues are not sequential and hence the signature of association type can not be obtained from sequence analysis)

Cluster Center

The residue position in the cluster (interface graph) is obtained from graph spectral analysis. The residues with highest rank are the cluster centers (hot spots). They may be the targets for mutational experiments

Pentraxin is yet another quaternary structure of a lectin

Protein-Protein Interaction Networks

Strong Interface

Centre of interface cluster

Weak Interface

5-aminolevulinic acid dehydratase (1B4K)

Urate Oxidase (1UOX)

Brinda, Vishveshwara, BMC Bioinformatics, 2005

(Analysis of a large dataset)

References on Protein structure networks

Three key residues form a critical contact network in a protein folding transition state.Nature. 409:641-645, Vendruscolo, M., E. Paci, C. M. Dobson, and M. Karplus. 2001.

Small-world view of the amino acids that play a key role in protein folding.Phys. Rev. E. 65:061910 Vendruscolo, M., N. V. Dokholyan, E. Paci, and M. Karplus. 2002.

Topological determinants of protein folding. Proc. Natl. Acad. Sci. USA. 99:8637-8641,Dokholyan, N. V., L. Li, F. Ding, and E. I. Shakhnovich. 2002.

Uncovering network systems within protein structures.J. Mol. Biol. 334:781-791. Greene, L. H., and V. A. Higman. 2003.

Network analysis of protein structures identifies functional residues.J. Mol. Biol. 344:1135-1146 Amitai, G., A. Shemesh, E. Sitbon, M. Shklar, D. Netanely, I. Venger, and S. Pietrokovski. 2004.

Small-world communication of residues and significance for protein dynamics. Biophys. J.86:85-91.Atilgan, A. R., P. Akan, and C. Baysal. 2004.

Network properties of protein structures.Physica A. 346:27-33. Bagler, G., and S. Sinha. 2005.

A Network Represenataion of Protein Structures: Implications for Protein Stability,Biophysical Journal, 6, 296, Brinda K.V. and Vishveshwara.S, 2005

Dynamics of Lysozyme Structure Network: probing the Processes of UnfoldingBiophysical Journal 92, Ghosh Amit, Brinda.K.V., Saraswathi Vishveshwara. , April 1, 2007

Protein Structure Graph and Network Topology

Brinda and S. Vishveshwara, Biophysical Journal, Dec 2005

Investigation of a large number of non-redundant structures

Giant cluster profile

Number of nodes with ‘k’ links

Degree Distribution plots

The Protein network topology exhibits

a complex behaviour

Poisson distribution?

Scale free- power law?

The topology depends on the Interaction strength (Imin)

Poisson Distribution when all the weak interactions are considered

Exponential-like behaviour at Interaction strengths greater than Icritical

Protein Structure Networks: Two proteins performing the same function at different temperatures. Although the overall structures are very much similar, the additional stability of the protein functioning at high temperature is due to increased number of hubs (shown in green). The concept is useful in protein design

Brinda and S.Vishveshwara, Biophysical journal , Dec 2005

Hubs in carboxy peptidase of a thermophilic and a mesophilicorganism

Blue: hubs common to both proteins

Green: Hubs exclusive to the thermophilic

Orange: Hubs exclusive to the mesophelic

Hubs and Thermal Stability of Proteins

Dynamics of structure networks

Analysis of Network parameters from MD trajectories

Insights into biological processes such as

The mechanism of protein folding

Long distance communications (allosteric effect)

Identification of communication pathways in MetRS from network analysis of MD trajectories

Amit Ghosh and S Vishveshwara PNAS September 2007

MD simulations on MetRS complexes

A:MetRS

B: MetRS+MetAMP

C:MetRS+tRNA

D:MetRS+tRNA+MetAMP

Communication between the anticodon region and the active-site region in MetRS is crucial for the faithful translation of the genetic code

RMSD Profiles

MetRS complexes Simulations

The CP domain opens up when MetRS is bound to both the tRNA and the activated methionine

Conformation of tRNA

MetRS+tRNA MetRS+tRNA+MetAMP

Interactions at the active-site

How is the message communicated between the anticodon region and the activation site, which are separated by about 70 A?

Size of the largest cluster in MetRS (averaged over MD snapshots) as a function of interaction strength

Imin (%)

Fluctuations in top three clusters of MetRS during MD simulation

Dynamical cross correlation maps representing the collective atomic fluctuations

MetRS+tRNA+MetAMP MetRS

Cross Correlated residues identified from MD simulations need not be connected in space.

The residues connecting (non-covalent) the correlated ones are identified from the protein structure network, by the analysis of the shortest path between the selected residues

Deduced from dynamic cross correlations and identification of shortest paths connected through non-covalent interactions at Imin=3%

Communication pathways from the anticodon region to the amino-

acylation site

Communication paths between the anti-codon region and the active site in MetRS

Communication paths between the anti-codon region and the active site in MetRS

Summary

Correlated movements are required for the functioning of

proteins. They can be identified at different levels.

Residue-wise RMSD, Cross correlations, Essential dynamics

The global non-covalent connectivity in proteins can be

represented as graphs and networks

Dynamics of such network can be analyzed from MD trajectories .

Cross correlation maps, in combination with network analysis

yield interesting information on important dynamical events such

as: Protein folding, Communication paths

The communication pathways between the anticodon region and the amino acylation site have been deduced from the dynamic cross correlations and the network analysis of the MD trajectories of Methionyl tRNA Synthetase

Acknowledgements

Amit Ghosh: Lysozyme and MetRS

Drs. Brinda, Kannan, swarna M. Patra- structure network

Department of Biotechnology, Government of India

EcMetRS:Met EcMetRS

Closing Mode Opening Mode

Amit Ghosh, Brinda, Vishveshwara,

Biophysical J April 2007

Concept of Structure Network Integrated with Dynamics

Equilibrium and unfolding simulations of T4-Lysozyme

probed from Network perspective

Protein Folding/unfolding

Analysis of the dynamics trajectories

T4 Lysozyme

Simulation details

Simulation Profiles: Root Mean Square Deviation(RMSD)

Time(ns)

RMSD

Interactions across secondary structures (300K)

k

� Non-native contact compared crystal structure

o Native contact retained

a. αααα3 ���� αααα1, ββββ1, ββββ3, αααα2

b. αααα1 ���� αααα5

c. αααα1 ���� αααα10, αααα11

d. αααα5 ���� αααα3, αααα4

e. αααα5 ���� αααα6

f. αααα5 ���� αααα9, αααα10

g. αααα7 ���� αααα8

300K Imin=3.4%

around transition

after transition

Interactions across secondary structures (500K)

* Non-native contact compared with 300K simulation o Native contact retained

500K Imin=2.5%

Unfolding events

Analysis of Network Parameters

Side-Chain Interaction Strength Dependent Analysis

•Degree distribution profiles

•Largest cluster profiles

•Native/Non-native contacts as a function of simulation time

Distribution of nodes with k links (degree distribution)

300K 500K

At 300K the number of nodes with one or two links is higher than the number of nodes with zero link(orphans), for Imin values less than 5%.

The transition from bell shaped curve to a decay-like curve takes place at a lower Imin of 3% in 500K

Size of the Largest Cluster Profiles

The size of the largest cluster in each of the simulations undergoes a transition at Imin = Icritical. (the size of the largest cluster is half of the maximum)

Icritical shifts from 3.4% at 300K to 2.5% at 500K.

Size of the largest cluster at Imin=0% (averaged over the simulation)

Large (135) at 300K Small (104) at 500K

Native/Non native contacts

The average number of contacts

300K 400K

native (0%) 206 160

native (3.5%) 98 72

non-native(0%) 50

non-native(3.5%) 45

native/non-native contacts = 1 at approximate time points 2 ns and 0.7 ns respectively for Imin values 0% and 2.5%. The non-native contacts increase after these time points

Ratio of native to non-native contacts is one

Native/Non native contacts@500K

Size and the composition of the large clusters

• 300K simulation:

At Imin = Icritical, the largest cluster encompasses the residues of both the domains

Average cluster composition (residues present in the largest cluster in >50% snapshots) is fairly constant. The largest cluster includes a considerable amount of hydrophobic residues

At Imin > Icritical , the top 2 largest clusters belong to the larger domain and the 3rd largest cluster belongs to the smaller domain. The participation of hydrophobic residues reduce considerably and aromatic residues dominate

• 500K simulation

Large fluctuations in the size and the composition of the clusters.

Different unfolding states such as the transition state, collapse state, unfolded state can be recognized from the size and the composition of the top large cluster

Lysozyme unfolding monitored through the changes in the size and the composition of large clusters formed by non-covalent interactions of amono acid side-chains

Composition of the top three large clusters (Imin=5%)

1st Largest

cluster

3rd Largest

cluster

2nd Largest cluster

Correlations with Experiments

Mutation studies:

A large number of mutations and their effects on stability have been experimentally carried out.

The destabilizing mutations (Val111,Trp138, Phe153) have been identified as hubs from our study. We predict that the mutation of other residues (Phe104,Arg145,Arg148) will also destabilize the protein, since they are hubs at high Imin and also remain as hubs in 400K simulation

Stages of domain formation during folding:

Our high temperature simulation points to the fact that the domain D2 is formed at an early stage, reinforcing the experimental findings

Outline

1. Introduction to protein structure networks

2. Biologically relevant information from graph/network analysis

3. Dynamics of structure networks

a) Unfolding simulation of Lysozyme

Network changes during folding/unfolding transition

b) Equilibrium simulation of methionyl tRNA synthetase

Network of communication pathways

Size of the largest cluster in PSG is dependent on the

interaction strength and a phase transition is seen at

Imin=Icritical in all the proteins

Property of a small world network

Threshold of the giant cluster appearance

|E| > (n/2) ?

Trajectory of the size of the largest cluster at Imin = 4% (500K simulation)

Rg fluctuates around 13.4Å in the 300K, which is slightly higher at 400K . Large fluctuations, varying from 13 Å to 18 Å are seen in all the three simulations at 500K.

II. Radius of gyration trajectories at different temperatures.

2nd Largest cluster

1st Largest

cluster

3rd Largest

cluster

Composition of the largest cluster(Imin=5%)

Relative Positional Fluctuation(RPF)

Simulations on proteins of Ribonuclease-A family

Essential dynamics is captured by < top 20 modes

Projection of top 3 vectors on to the MD Trajectory

ECP-A (top mode)

backbone

Projection of the Essential Modes

Structures obtained from top 3 modes

along with side chains significantly

contributing to these modes

Displacement along a single eigenvector (S): S= (r-<r>) .e

B.S.Sanjeev and S. Vishveshwara, Eur Phys. J D 20, 601-608 (2002)

II. Essential Dynamics:

WhereS = total number of configurations,t = time in ps.

xi = ith coordinate (i = 1,2,…., 3N) (N being number of atoms).

<xi> = average value of xi over all configurations.

Relative positional fluctuation (RPF) was then calculated as:

Where (i) is ith eigenvalue, RPF(n) gives the amount of motion associated in the subspace spanned by the first n

Essential dynamics captures major part of the dynamics in a few

essential modes.

( )( )(1/ ) ( ) ( )ij i i j j

t

m S x t x x t x= − −∑

1,

1,3

( )

( )( )

i n

i N

i

RPF ni

λ

λ=

=

=∑

∑

Matrix representation: Unweighted and Weighted1

2 3

1

2 3

0.6 0.5

0.7

1 1

1

A= 0 1 1

1 0 1

1 1 0

L= 2 -1 -1

-1 2 -1

-1 -1 2

A= 0.0 0.6 0.5

0.6 0.0 0.7

0.5 0.7 0.0

L= 1.1 -0.6 -0.5

-0.6 1.3 -0.7

-0.5 -0.7 1.2

A=Adjacency; L=Laplacian

Aij = 1, if i≠j and i and j are connected in the graph.

Aij = 0, otherwise.

Protein structure graphs: N×N matrixN=No. of residues

Weights in protein side chain graphs = 1/dij

dij=Cβ-Cβ distance in Å (Cα

in case of Gly).

EigenEigen Spectra of the Laplacian Matrix of a graph Spectra of the Laplacian Matrix of a graph

Eigen-

values

0.0000 0.0800 0.9016 1.5500 1.9500 2.0600 2.7420 4.5164

Node EVC1 EVC EVC EVC EVC EVC EVC EVC

1 0.3536 0.2739 0.1380 -0.0000 0.0000 0.0000 0.5362 0.70242

2 0.3536 0.2739 -0.2560 -0.0000 -0.0000 0.7071 0.2422 -0.4193 3 0.3536 0.2739 -0.4375 -0.0000 -0.0000 -0.0000 -0.7031 0.3380 4 0.3536 0.2739 -0.2560 -0.0000 -0.0000 -0.7071 0.2422 -0.4193 5 0.3536 0.2739 0.8115 0.0000 -0.0000 -0.0000 -0.3175 -0.2018 6 0.3536 -0.4564 -0.0000 -0.4082 -0.7071 0.0000 0.0000 0.0000 7 0.3536 -0.4564 0.0000 -0.4082 0.7071 -0.0000 -0.0000 -0.0000 8 0.3536 -0.4564 -0.0000 0.8165 -0.0000 0.0000 0.0000 0.0000 1 : Eigenvector Components 2 : Largest vector components of top eigenvalues of each cluster are shown in blue