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Aracne
Jorge Viveros
Summer 2006 Workshop
June 29th, 2006
Contents
1.
Overview (the problem, the alternatives, ARACNE’s arlgorithm central
idea)
2.
Demo (reconstruction of gene regulatory networks for affymatrix gene
expression data)
3.
Algorithm details (approximating the mutual information, comparative
study results, ARACNE vs Bayesian and Relevance Networks)
4.
Conclusions
5.
Bibliography
1. Overview:
ARACNE
Algorithm for the Reconstruction of Accurate Cellular Networks
“Reverse engineering” or “deconvolution” problem:
Samples
ga
gb
Information-theory
+
max entropy methods
gc
gd
ga
gb
ge
gc
gd
ge
Gene regulatory network
(overview, cont’d)
Authors
A.A. Margolin [1,2], I. Nemenman [2], K. Basso [3], C. Wiggings [2,4], G.
Stolovitzky [5], R. Dalla-Favera [3], A. Califano [1,2]
[1]
Dept. Biomedical informatics, [2] Joint Centers for Sys Biology, [3] Institute for
Cancer Genetics, [4] Dept. of Appl. Physics and Appl. Math.
Columbia University
[5]
IBM T.J. Watson Research Center.
Main reference:
http://www.arxiv.org/abs/q-bio/0410037
BMC Bioinformatics 2006, 7(Suppl 1):S7
(overview, cont’d)
Goal
Understand mammalian normal cell physiology and complex pathologic
phenotypes through elucidating gene transcriptional regulatory networks.
Thesis
Statistical associations between mRNA abundance levels helps to
uncover gene regulatory mechanisms.
(overview: alternatives)
ARACNE vs Clustering
ARACNE recovers specific transcriptional interactions but does not attempt to
recover all of them (too complex a problem).
Genome-wide clustering of gene expression profiles: cannot discern direct
(irreducible) from “cascade” transcriptional gene interactions.
a
ga
gb
clustering
b
c
ARACNE
gc
d
ga,gb
gc,gd
ge
gd
ge
e
(central idea)
Gene network inference
edge = (direct) statistical dependency
= direct regulatory interaction
nodes = genes
gi
gj
Temporal gene expression data for higher eukaryotes, difficult to obtain.
Only steady-state statistical dependencies are studied.
Accounting for dependence: definition and measurement
Gene expression values
samples from a joint probability distribution
Consider the multi-information = average log-deviation of the joint probability
distribution (JPD) from the product of its marginals (also “Kullback-Leibler
divergence” (KL-div)).
Use maximum entropy methods to approximate JPD by an element of its “mway” marginal Frechet class (m-way maximum-entropy estimate m-MEE)
Use m-MEE to define mth-order connected information (m-cinfo) to account for
m-way statistical dependencies (only!).
Multi-info = sum of all m-cinfo’s.
The multi-information
Multi-information (KL-div)
JPD
{ X i : i  1,..., M } “nodes, “expressions” or “genes”
Integral if conts case; sum if discrete case
Entropy of P(x)
JPD not known, approximate it!
m-way max entropy estimate of JPD
m-MEE , P (m ) , has the same m-marginals as P (x )
m-MEE has the following form:
Lagrange multipliers
Have no analytical solution BUT
can be obtained via an iterative
Proportional fitting proc (IPFP)
Connected and Multi informations
mth-order connected information
Multi-information
Compensate for the lack of knowledge of JPD by using the (truncated!) multi-info
to establish and quantify statistical dependencies
Detecting a particular m-way interaction
M-way interaction   {ii , ... , im } contributes to multi-info, iff minimum of
interaction multi-information (inter multi-info) over  -specific Frechet class is
positive.
Inter multi-info =
Q () and Q* are m-MEE sharing same m-way marginals except for, perhaps,
Q ( )   P
Positivity of minimal inter multi-info   is an irreducible (direct) interaction
Thus draw edges coming from  nodes and meeting at m-edge vertex.
Examples
Regulatory cascade (Markov chain)
X1  X 2  X 3
P( x1 , x2 , x3 )  P( x1 ) P( x2 | x1 ) P( x3 | x2 )
  {1,2}
I *{1, 2}  I X 1 , X 3   I X 2 , X 3   I {1, 2}  I X 1 , X 2   I X 2 , X 3 
Information processing inequalty
*{1, 2}  0  X 1 , X 2
generically dependent
(similarly, X 2 , X 3 )
*{1,3}  0  X 1 , X 3 generically independent
I *{1, 2,3}  I X 1 , X 2   I X 2 , X 3   I {1, 2,3}
No triplet interactions (coregulation)
(examples, cont’d)
Other dependencies
2 regulates 1 and 3 OR 1 and 3 regulate 2 jointly
P123 does not factor
but pairwise marginals do
2. Demo
Platforms
1.
caWorkBench2.0 (downloadable through web site) (JAVA)
Most developed features: microarray data analysis, pathway analysis
and reverse engineering, sequence analysis, transcription factor binding
site analysis, pattern discovery.
http://amdec-bioinfo.cu-genome.org/html/caWorkBench.htm
2.
Cygwin (for windows). Windows and Linux versions available in web site
Sample input data file
(Demo)
Input_file_name.exp
N = 3 # genes
M = 2 # microarrays
Input file has N+1=4 lines
each lines has M+2 (2M+2) fields
Microarray chip names
annotation name
AffyID HG_U95Av2
SudHL6.CHP ST486.CHP
G1 G1
16.477367
0.69939363 20.150969
0.5297595
G2 G2
7.6989274
0.55935365 26.04019
0.5445875
G3 G3
8.8098955
0.5445875
0.31372303
(value,p-value)-chip1
21.554955
header line
(Demo, cont’d)
Syntax (Cygwin)
ARACNE: algorithm for gene regulatory network computation given
microarray data.
Usage:
aracne
aracne GeneExpressionFile [-a | -k | -s | -t | -e | -f]
aracne -adj GeneExpressioFile AdjacencyFile [-t | -e]
-a
-k
-s
-t
-e
-f
accurate | fast [default: accurate]
gaussian kernel width [accurate method only; default: 0.15]
Averaging Window step size [fast method only; default: 6]
Mutual Info. threshold [default: 0]
DPI tolerance (btw 0 and 1) [default: 1]
mean stdev [default: no filtering]
Sample output data file
(Demo, cont’d)
input_data_file_name[non-default_param_vals].adj
# lines = N = # genes
5
AffyID
ID#
Associated gene ID#
MI value
4
G1:0
8
0.064729
G2:1
2
0.0298643
G3:2
1
0.0298643
G4:3
8
0.0427217
G5:4
5
0.403516
G6:5
4
0.403516
6
0.582265
G7:6
5
0.582265
9
0.38039
G8:7
1
0.0521425
8
0.743262
G9:8
0
0.064729
3
0.0427217
G10:9
6
0.38039
8
0.333104
7
1
6
9
7
0.0521425
8
7
0.743262
10
2
3
9
0.333104
3. Algorithm details
Incorporate information-theoretic ideas (Markov networks) to model statistical
dependencies (cf. [2])
= joint prob dist function of stationary expressions of all genes (i=1,…,N)
N = # genes, Z = partition fun (normalization factor),
,
,
model iff
= Hamiltonian,
, … = interaction potentials (e.g., genes i,j,k do not interact in the
= 0.
Aim: identify nonzero potentials.
(Algorithm details)
Aracne’s model
First-order approximation: genes are independent
1st order potentials obtained from marginal probabilities
experimentally).
(estimated
ARACNE’s approximation: truncate joint prob dist fun to pairwise potentials
In this model
non-interacting genes (includes statistically
independent genes
i.e.,
and genes that do not interact directly,
but
).
Reduce number of potential pairwise interactions via realistic biological
assumptions.
(algorithm details, cont’d)
MI estimation
Assume two-way interaction: pairwise potentials determine all statistical
dependencies.
Mutual information (MI) = measure of relatedness
= 0 iff
MI approximation:
G = bivariate standard Gaussian density
h = kernel width
(algorithm details, cont’d)
Some details and technicalities:
Transform x, y so
0  x, y  1 and their marginal distributions seem uniform
There is not a universal way of choosing h, however the ranking of the MI’s
depends only weakly on them.
I ( x' , y ' ) 
p( xi ) 
1
2 M
1
M
 p( xi , yi ) 
log
i  p( x ) p( y ) 
i
i 

 ( xi  x j ) 2 
j exp 2d 2  i  1,2
1


1
p( xi , y j ) 
2
2 Md 2
 ( xi  x j ) 2  ( yi  y j ) 2 

j exp 
2

2d 2


(algorithm details, cont’d)
Establishing the network
Define threshold IO to discard MI’s (lower-bound interaction)
Shuffle genes across microarray profiles & evaluate MIs for seemingly
independent genes, choose IO based on what fraction of MIs falls below the
threshold.
Data processing inequality: if genes g1 and g2 interact thorugh g3 then
ARACNE starts with network so
for every edge
look at gene triplets and remove edge with smallest MI
(algorithm details, cont’d)
Establishing the network
ARACNE’s algorithm complexity:
N = number of genes, M = number of samples
DPI analysis
MI estimation (order
of pairwise interactions N 2 )
Perfect network reconstruction theorems
Thm 1: If MI’s are estimated with no errors and true underlying interaction
network is a tree with only pairwise interactions then ARACNE will
reconstruct it.
Thm 2: If Chow-Liu maximum MI info tree is subnetwork of ARACNE’s network
then this is the true network.
Thm 3: “ARACNE will reconstruct tree-network topologies exactly.”
Comparative study results
Reconstruction of class of synthetic transcriptional networks by Mendes et al
(cf. [1]) and human B lymphocyte genetic network from gene expressions
profile data.
Performance of ARACNE compared against Bayesian Networks (use LibB
package) and Relevance networks (similar to ARACNE but has less accurate
MI estimation procedure and less-developed of assigning statistical
significance).
(results)
Synthetic networks
100 genes, 200 interactions organized in two types of networks
1. Erdos-Renyi: each vertex interaction is equally likely
2. Scale-free topology: distribution of vertex connections obeys a power law
Performance metrics
(results)
Pairwise gene interaction is
“(True) positive” if their statistical regulatory interaction is directly linked.
“(True) negative” if their interaction is not direct.
Precision
NTP
NTP  N FP
Recall
N TP
N TP  N FN
fraction of true interactions correctly inferred
(expected success rate in experimental validation of
predicted interactions)
fraction of true interactions among all inferred ones
Performance to be assessed via Precision-Recall curves (PRCs)
(results cont’d)
PRCs for synthetic data
1
ARACNE’s performance above 40% for both models
2
(result con’td)
Quantitative results on synthetic data
ARACNE recovers far more true connections and predicts far less false ones
(results cont’d)
Results on Human B cells
Assembled expression profile data set of ~340 B lymphocytes from normal,
tumor-related and experimentally manipulated populations.
Data set was deconvoluted by ARACNE to generate B-cell specific regulatory
network of ~129,000 interactions.
Validation of the network’s quality was done by comparing inferred interactions
with those identified through biochemical methods.
See cf [3].
Conclusions and Discussions
1.
Algorithm is robust enough for its application in other network
reconstruction problems in biology and the social and engineering fields.
2.
Pairwise interaction model  higher-order potential interactions will not
be accounted for (ARACNE’s algorithm will open 3-gene loops).
3.
A two-gene interaction will be detected iff there are no alternate paths.
4.
To keep three-gene loops, modify tolerance for edge-removal by
introducing tolerance parameter,
.
5.
ARACNE’s performance deteriorates as local (true) network topology
deviates from a tree (tight loops may be a problem).
6.
ARACNE achieved high precision and substantial recall even for few data
points when compared to BN and RN (synthetic data).
7.
ARACNE cannot predict the orientation of the edges of the networks.
8.
The algorithm is suited for more complex (mammalian) networks.
Bibliography
1.
P. Mendes, W. Sha, K. Ye. Artificial gene networks for objective
comparison of analysis algorithms. Bioinformatics 2003, 19 Suppl 2:
II122-II129.
2.
I. Nemenman. Information theory, multivariate dependence and genetic
network inference. Technical report: arXiv:q-bio/0406015; 2004.
3.
K. Basso, A.A. Margolin, G. Stolovitzky, U. Klein, R. Dalla-Favera, A.
Califano. Reverse engineering of regulatory networks in human B cells.
Nature Genetics, 2005, 37(4):382-390.
Main web site
•
Important documentation and relevant publications, application download
and support.
AMDeC Bionformatics Core Facility at the Columbia Genome Center
AMDeC (Academic Medicine Development Company)
http://amdec-bioinfo.cu-genome.org/html/ARACNE.htm