Graph Spectral Analysis

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(No Transcript)
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• Graph Spectral Analysis
• Active Site Analysis
• Folding Clusters
• Quaternary Association of Proteins
• Domain Identification Domain Interface Residues
• Recurrence Quantification Analysis
• Structural Identification
• Protein Aggregation/ Folding
• Structural and Topological properties
• Global Network Partitioning
• Protein Structural Organization
• Autonomous Folding Units
• Folding-important residues
• Native vs Decoy discrimination

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• RQA
• Nonlinear technique
• Transform original series into its embedding
  matrix (EM) based on delays
• Rows of embedding matrix correspond to windows of
  length 4
• Based on computation of Euclidean distance
  between the rows of the EM
• Looking for epochs close to each other

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• Recurrence
• Point that repeats itself
• Most basic of relations
• Strictly local and independent of any
  mathematical assumption about the system
• Calculation of recurrence requires no
  transformation of data
• Can be used for both linear and nonlinear systems

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EMBEDDING DIMENSION 4 LAG 1
10 11 21 32 41 35 40 19
10 11 21 32 41 35 40 19
11 21 32 41 35 40 19
21 32 41 35 40 19
32 41 35 40 19
EMBEDDING MATRIX
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• Pairwise distances between each pair of rows of
  the EM are calculated
• If d lt r, then a dot is placed on the recurrence
  plot
• Application of this computation produces a
  Recurrence Plot (RP)
• Symmetrical M x M matrix
• Point placed at (i,j) whenever row Xi - Xj lt
  r
• Graphically represented by a dot

10 11 21 32 41 35 40 19
11 21 32 41 35 40 19
21 32 41 35 40 19
32 41 35 40 19
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Recurrence Quantification Analysis (RQA)
PROTEIN SIGNALS
HISTORY
HISTORY
KOENIGSBERG
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Recurrence Quantification Analysis (RQA)
PROTEIN SIGNALS RECURRENCE PLOT
HISTORY
HISTORY
KOENIGSBERG
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Thermophylic and Mesophylic Proteins
HISTORY
HISTORY
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Recurrence Quantification Analysis (RQA)
THERMAL STABILITY
HISTORY
HISTORY

• Rubredoxins
• Member of the family of redox metalloenzymes
• Relatively short polypeptide chain (53 AA)
• Ferrous/Ferric ion tetrahedrally bound to four
  Cys residues
• Huge difference in stability
• Pyrococcus Furiosus (Rubr Pyrfu)
• Thermophilic
• Clostridium pasteurianum (Rubr Clopa)
• Most similar in terms of 3D structure
• Mesophilic

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Recurrence Quantification Analysis (RQA)
THERMAL STABILITY
HISTORY
HISTORY
Thermophylic Proteins
Mesophylic Proteins
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Recurrence Quantification Analysis (RQA)
THERMAL STABILITY
HISTORY
HISTORY
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Recurrence Quantification Analysis (RQA)
THERMAL STABILITY
HISTORY
HISTORY
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Recurrence Quantification Analysis (RQA)
THERMAL STABILITY
HISTORY
HISTORY

• Kurtosis of recurrence distributions
• Kurtosis Index of relative peaked character of a
  distribution
• Higher kurtosis gt More concentrated recurrence
  patterns

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Protein Aggregation
HISTORY
HISTORY
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Protein Aggregation
THE PROBLEM
HISTORY
HISTORY

• Protein aggregation is an important problem
• Involvement of protein misfolding in Alzheimers,
  Huntingtons and prion diseases
• Possibility of forming aggregates intrinsic to
  proteins
• How to delineate betweeen proteins that form
  aggregates and proteins that dont?
• In principle change of environmental conditions
  could drive any protein to form multimeric
  aggregates

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Protein Aggregation
THE PROBLEM
HISTORY
HISTORY

• Main driving force in protein folding
• Need to be soluble in water
• Proteins fold so as to
• Hide hydrophobic residues in core
• Expose polar residues to solvent
• Protein-Protein interaction can be considered as
  similar to protein folding
• Mainly hydrophobic character of folding processes
  led to the study of hydrophobicity patterning
  along the sequence using RQA

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Protein Aggregation
RQA
HISTORY
HISTORY

• 9 groups studied
• A almost pure natural ?-helix
• B almost pure natural ?-sheet
• C mainly ?-helix polymerizing
• D aggregating systems, eg amyloids
• E natively unfolded proteins
• F Proteins undergoing lot of PPI, eg DNA repair
  systems
• G Synthetic ?-helices
• H Synthetic ?-sheets
• I ?-helices

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Protein Aggregation
RQA
HISTORY
HISTORY

• Self aggregating systems (D) similar to
  DNA-processing proteins (F) and ?-helices (I)
• Connection between D and F maybe linked to the
  capability of forming multimeric aggregates
• Polymerizing proteins (C) have a distinct
  patterning
• G H have clear-cut peaks

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Protein Aggregation
RQA CDA
HISTORY
HISTORY

• Canonical Discriminant Analysis (CDA)
• Variant of linear model
• Y XBe
• Y dependent variables
• X independent variables
• B regression coefficients
• e errors
• Model used to calculate variables which can best
  separate the data into classes

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Protein Aggregation
RQA CDA
HISTORY
HISTORY

• Canonical Variates plot
• CV1 vs CV2
• CV3 vs CV4
• CV1 vs CV2
• CV1 is a measure of shape
• Synthetic ?-helices (G) and natural ?-sheets (B)
  situated at extremes
• Regular arrangement of secondary structures
• H (artificial ?-sheets ) is also unique in that
  it is most periodic
• Natural Amyloid proteins are towards the center

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Protein Aggregation
RQA CDA
HISTORY
HISTORY

• CV3 vs CV4
• CV3 models the oppostion between D and G on one
  hand and all other proteins on the other
• CV4 represents a fine tuning of different spectra
• D (amyloid) is near DNA repair (F) and also of
  artificial ?-sheets (H)

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Protein Aggregation
RQA CLUSTERING
HISTORY
HISTORY

• K-means clustering was carried out on CDA
  variables
• D group goes together with A (natural ?-helices),
  F (DNA repair) and I (?-helices)
• Implications
• ?-sheets not a prerequisite for aggregation
• Oligomerization has a pivotal role to play in
  amyloid forming system
• Formation of high polymers of collagen type (C)
  follow a different mechanism

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Protein Secondary Structures
HISTORY
HISTORY
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Recurrence Quantification Analysis (RQA)
SECONDARY STRUCTURE
HISTORY
HISTORY

• RQA shows that higher dimensional recurrences
  could be captured by single variables
• By creating time/space delayed versions of
  the signal
• Setting a radius
• For multi-dimensional data, no need to do
  embedding
• For PDB structures
• x, y, z co-ordinates available
• No need for embedding matrix
• Recurrent plots are mathematically equivalent to
  simple contact (adjacency ) matrices

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Recurrence Quantification Analysis (RQA)
SECONDARY STRUCTURE
HISTORY
HISTORY

• Distance matrices were built for proteins with
  the C? x,y,z co-ordinates
• Euclidean distance between each pair of residues
  was calculated
• Radius 6 A

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Recurrence Quantification Analysis (RQA)
SECONDARY STRUCTURE
HISTORY
HISTORY
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Recurrence Quantification Analysis (RQA)
SECONDARY STRUCTURE
HISTORY
HISTORY

• Parallel and anti-parallel lines
• However parallel lines
• Both by ?-helices and ?-sheets
• Introduced a new parameter to separate between
  these
• Based on number of recurrences in a 10 C?-atom
  stretch

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Recurrence Quantification Analysis (RQA)
SECONDARY STRUCTURE
HISTORY
HISTORY

• 3D lattice models of proteins
• Construct a 3D model
• 120 residues
• ?-helix (A)
• Antiparallel ?-sheet (B)
• Parallel ?-sheet (C )

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Recurrence Quantification Analysis (RQA)
SECONDARY STRUCTURE
HISTORY
HISTORY
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Recurrence Quantification Analysis (RQA)
SECONDARY STRUCTURE
HISTORY
HISTORY
Trp Repressor
Tumor Necrosis Factor Alpha Protein
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Recurrence Quantification Analysis (RQA)
SECONDARY STRUCTURE
HISTORY
HISTORY
Chloramphenical Acetyltransferase
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Recurrence Quantification Analysis (RQA)
SECONDARY STRUCTURE
HISTORY
HISTORY

• Relationship with DSSP

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Protein Topological Architecture using RQA
HISTORY
HISTORY
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Protein Topological Architecture
THE PROBLEM
HISTORY
HISTORY

• Significant literature exists reporting a general
  scaling of different geometrical features of
  protein 3D structure with chain length
• Contact maps as shown before can be used to study
  this phenomenon
• REC exhibits a scaling law
• This measure allows to detect a general scaling
  of protein structures
• Two different topological modes
• Small protein lt 180 aa residues
• Large Protein gt 320 aa residues
• Between these two, a zone with a rich repertoire
  of topological arrangements

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Protein Topological Architecture
RQA
HISTORY
HISTORY

• Dataset
• 91 monomeric proteins
• Well characterized structures
• Cover almost entire range of secondary structures
• RQA
• Applied to the 91 proteins
• Bend near 100 residues

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Protein Topological Architecture
RQA
HISTORY
HISTORY

• Ratio of Surface Volume (SV) and REC3D plotted
  against protein length
• Divergence beginning near 180 and ending at
  around 320
• Looking at theoretical REC3D, divergence appears
  near 274

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Protein Topological Architecture
RQA IMPLICATIONS
HISTORY
HISTORY

• Explanation of this based on geometric
  considerations of protein folding
• As chain length increases, a critical value of
  surface volume (SV) to compactness (REC3D) is
  achieved and a divergence occurs
• This divergence implies a structural instability
• Maybe due to the natural limit for domain length
• Singularity might
• Suggest a maximal preferred domain size
• Mark the transition between single domain and
  multi-domain proteins

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Protein Topological Architecture
RQA APPLICATIONS
HISTORY
HISTORY

• Used as a check of consistency of predicted
  structures
• Figure shows that almost all correct models fall
  in a tight band around the scaling curve
• If the compactness of the model is far from the
  reference line, then the model is erroneous

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Protein Topological Architecture
HYDROPHOBIC PATCHES
HISTORY
HISTORY

• Understanding of folding dynamics by studying
  hydrophobic patches
• What are minimal sized patches important for
  folding?
• Data
• 1977 single chain proteins
• RQA was applied

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Protein Topological Architecture
HYDROPHOBIC PATCHES
HISTORY
HISTORY

• Histogram of Maximal length recurrence lines
  (MAXL) shows a peak at around 6 residues
• Plot of MAXL vs Entropic contribution also shows
  a similar peak around 6
• Implies that 6-residue words contain maximal
  information for hydrophobicity nucleation in
  globular proteins

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Protein Global Network Partitioning
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Protein Global Network Partitioning
THE PROBLEM
HISTORY
HISTORY

• Protein Folding
• Many models
• However, all in agreement that small regions of
  proteins tend to fold separately
• Stabilized by interactions between these distinct
  units
• Proteins can thus be considered as collections of
  small units
• Autonomous folding units/motifs/domains
• Domain
• Defined as regions containing maximum
  intra-cluster connectivity and minimum
  inter-cluster connectivity

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Protein Global Network Partitioning
ALGORITHM
HISTORY
HISTORY
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Protein Global Network Partitioning
ALGORITHM
HISTORY
HISTORY
Intra-module Connectivity
Participation Coefficient (Inter-module
Connectivity)

• R1 Ultra-peripheral nodes
• R2 Peripheral Nodes
• R3 Non-hub Connector Nodes
• R4 Non-hub Kinless nodes
• R5 Provincial Hubs
• R6 Connector Hubs
• R5 Kinless Hubs

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Protein Global Network Partitioning
ALGORITHM
HISTORY
HISTORY

• Used the Guimera idea for partitioning Protein
  Networks into modules
• Algorithm based on Genetic Algorithm instead of
  on Simulated Annealing
• Algorithm is based on natural partitioning
• Independent of Interaction Strength etc..
• Data
• 1420 single chain proteins
• lt 20 identity with each other
• Resolution lt 2 Å

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Protein Global Network Partitioning
HIERARCHICAL NATURE
HISTORY
HISTORY

• Hierarchical nature of protein organization
• Module frequency greatest for a module size of
  around 12
• Maximal module size scales linearly with protein
  length

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Protein Global Network Partitioning
HIERARCHICAL NATURE
HISTORY
HISTORY

• Scaling of Modularity/Length with protein length
• Similar to scaling of REC3D with protein length
• The curve in this case occurs earlier, around 50
  residues
• Maximal preferred folding unit size?
• Points again to a hierarchical (almost
  fractal-like) behavior

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Protein Global Network Partitioning
HYDROPHOBICITY AND RASA
HISTORY
HISTORY

• P-z space was partitioned into 8 x 10 bins
• The hydrophobicities and Relative Accessible
  Surface Area (RASA) of the residues falling
  within these bins was identified
• Standard deviation of the inter-module
  hydrophobicities and RASA shown
• Variance of both hyd and RASA decreases with
  increasing module size
• Implies that we are moving towards an average
  protein
• At low module sizes pure hyd or pure polar
  modules possible
• Not so at larger module sizes
• Variance flattens out at around 30 residues

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Protein Global Network Partitioning
HYDROPHOBICITY AND RASA
HISTORY
HISTORY

• Maximal connected patches of polar and
  hydrophobic residues identified in each and every
  module
• Frequency plots of intra-module patch size for
  hydrophobic and polar patches
• Surprisingly, they are almost exactly the same
• Maximal patch size peaks at around 12
• Implications
• Hydrophobic and polar residues start aggregating
  together at the beginning of the folding process
  in a like-meets-like behavior
• These mini-clusters must solve the communication
  problem
• Happens around 30 residues
• After this topological constraints come into play
• Our modules thus correspond to foldons, or
  early folding units

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Protein Global Network Partitioning
SECONDARY STRUCTURES
HISTORY
HISTORY

• At smaller module sizes, we observe propensity to
  have a single secondary structural feature
• As the module approaches the 30 residue barrier,
  we observe a breaking of this pattern
• Consistent with the mixing together of the
  all-hydrophobic and all-polar modules

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Protein Global Network Partitioning
INVARIANCE OF P/z
HISTORY
HISTORY

• Plot of protein P-z space
• Invariance of the P-z space
• Typical Dentists-chair sort of behavior
• Notice that high P/z valued residues are
  significant
• Correspond to connector hubs
• We characterized well studied proteins with their
  P/z values.
• High P/z valued residues act as early folding
  residues
• Important for folding

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Protein Global Network Partitioning
INVARIANCE OF P/z
HISTORY
HISTORY

• Ubiquitin
• Show the partitioning of the proteins in modules
  as well as colored by high P/z valued residues
• Very few connector hubs

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Protein Global Network Partitioning
INVARIANCE OF P/z
HISTORY
HISTORY

• Show the top P/z valued hits for ubiquitin
• Most of the top hits are within two residues of a
  residue protected during folding (obtained from
  experimental studies)
• protected residue
• o within one residue of protected residue
• ? within two residues of protected residue

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Protein Global Network Partitioning
INVARIANCE OF P/z
HISTORY
HISTORY

• Distribution of RASA over the 8 x 10 bins of the
  P/z plot for all 1420 proteins
• Contour plot shown here
• Two regions of low RASA (red areas) corresponding
  to buried regions
• Surrounded by polar (blue) residues
• Analogous to the Ramachandran plot?

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Protein Global Network Partitioning
NATIVE vs DECOY DISCRIMINATION
HISTORY
HISTORY

• Native vs Decoy structures distribution
• Clear differences between the native (top) and
  decoy (bottom) distributions
• Can we use this to discriminate between Native
  and Decoy structures?

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Protein Global Network Partitioning
NATIVE vs DECOY DISCRIMINATION
HISTORY
HISTORY

• PCA carried out on the RASA distribution values
• Clear separation visible between native and decoy
  structures

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• Giuliani et. al., Nonlinear Signal Analysis
  Methods in the Elucidation of Protein
  Sequence-Structure Relationships, Chem. Rev.
  2002, 102, 1471-1491
• Giuliani et al., Mapping protein sequence spaces
  by recurrence quantification analysis a case
  study on chimeric structures, Protein Eng., 2000,
  13(10), 671-678
• Webber et al., Elucidating protein secondary
  structures using alpha carbon recurrence
  quantifications, Proteins Stuc. Func. Bioinf.,
  2001, 44292-303
• Giuliani et al., Nonlinear methods in the
  analysis of protein sequences a case study in
  Rubredoxins, Biophysical Journal, 2000, 78,
  136-148
• Zbilut et al., Protein Aggregation/Folding The
  role of deterministic singularities of sequence
  hydrophobicity as determined by nonlinear signal
  analysis of acylphosphatase and AB(1-40),
  Biophysical Journal, 2003, 85, 3544-3557
• Zbilut et al., A topologically related
  singularity suggests a maximum preferred domain
  size for protein domains, Proteins Struct. Func.
  Bioinf., 2006, 65
• Joseph P. Zbilut et al., Entropic criteria for
  protein folding derived from recurrences Six
  residues patch as the basic protein word, FEBS
  Letters, 580 4861-4864, 2006
• Guimera et al., Functional cartography of complex
  metabolic networks, Nature, 2005, 433, 895-899
• Krishnan et al., Network scaling invariants help
  to elucidate basic topological principles of
  proteins., under review, 2007
• Krishnan et al., A topology-energetics nexus
  informs native vs decoy protein distinctions,
  under review, 2007

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• This is the second of a two part assignment.
• For the selected protein for which you carried
  out the graph spectral analysis, employ the 1D
  and 3D RQA and report results
• Can you identify secondary structural features
  using 3D RQA?
• What does the 1D RQA using hydrophobicity tell
  you about the protein?
• For the same protein, partition into modules
  using the GANDivA algorithm and report results
• Look at how the modules are distributed
• Which are the high P/z valued residues
• What does the P/z plot look like?

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• Download RQA.tar.gz from http//www.iab.keio.ac.jp
  /krishnan/downloads/Course_Materials/RQA.tar.gz
• Copy this to some dir say RQA
• Then do
• tar zxf RQA.tar.gz
• make RQA1D
• make RQA3D
• This will create two executables RQA1D and RQA3D
• Run the program to see the USAGE
• Download and install GANDivA as told in the first
  lecture (refer to the slides from the first
  lecture)

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• Protein Modularity Detection
• Download from
• http//www.iab.keio.ac.jp/krishnan/downloads/Cour
  se_Materials/GANDivA.tar.gz
• Copy this to account on cacao.bioinfo.ttck.keio.ac
  .jp
• Installation Instructions
• Tar and unzip the package using
• tar zxf GANDivA.tar.gz
• Change directory to the main GANDivA directory
• cd GANDivA_v1.0
• Run the install script
• perl install_gandiva.pl
• You will be asked a series of questions

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• Installation Instructions
• a) Do you want to install a parallel version of
  the program? Y/N
• Y
• b) Do you have MPI Installed? Y/N
• Y
• c) Where would you like to have PGA installed?
• Give full path of directory in which to install
• Eg /home/krishnan/GANDivA_v1.0/PGA
• d) Please enter the path of the MPI library file
• /usr/local/lib/libmpich.a
• e) Please enter the path of the MPI include
  directory
• /usr/local/include
• This should automatically install the GANDivA
  binary in
• /path/to/GANDivA_v1.0/bin

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• ./GANDivA -a ltAdjmat filegt -n lt of Residuesgt -g
  10000 -G 1000 -o ltOutput filegt
• Output looks like this
• Nodes Modules z(i) P(i) P/z abs(P/z)
  Score 0.524569
• 1 4 0.048422 0.000000
  0.000000 0.000000
• 2 4 0.048422 0.493827
  10.198457 10.198457
• 3 4 1.888448 0.165289
  0.087526 0.087526
• 4 4 0.508428 0.512397
  1.007805 1.007805
• 5 4 0.048422 0.444444
  9.178612 9.178612
• Remember that this is a stochastic algorithm.
  Hence you have to run it a few times (at least 50
  times) and take the result with the highest
  Modularity score. (given on the first line of the
  output file. In the example here, it is 0.524569
• For those of you on cacao, you should submit the
  different runs using a submit script to the job
  scheduler
• For those of you running it on your own machines
  you can use a simple shell script to do it in
  series. Since your proteins are all small, you
  should be able to run this in about 20 minutes
• for I in seq 100 do ./GANDivA -a ltAdjmat filegt
  -n lt of Residuesgt -g 10000 -G 1000 -o RI.out
  done
• This will run the algorithm 100 times and the
  output files will be called R1.out, R2.out, ,
  R100.out