Two examples

1
Liquid Association (LA)

• Two examples
• A challenge

2
Liquid Association (LA)

• LA is a generalized notion of association for
  describing certain kind of ternary relationship
  between variables in a system. (Li 2002 PNAS)

• Green points represent four conditions for
  cellular state 1.
• Red points represent four conditions for cellular
  state 2.
• Blue points represent the transit state between
  cellular states 1 and 2.
• (X,Y) forms a LA.

Profiles of genes X and Y are displayed in the
above scatter plot.
Important! Correlation between X and Y is 0
3
Mathematical Statistics on LA

• EX0, EY0, SD(X)SD(Y)1
• LA is defined by following equation. g(Z) is the
  conditional expectation of the correlation
  between X and Y. LA(X,YZ) is the expected
  changes of the correlation between X and Y.

4
Stein Lemma

• To compute E(g(Z)) is not easy. With help from
  mathematical statistics theory, the LA(X,YZ) can
  be simplified as E(XYZ) when Z follows normal
  distribution.

Stein lemma
5
Human Genome Program, U.S. Department of Energy,
Genomics and Its Impact on Medicine and Society
A 2001 Primer, 2001
6
gene-expression data
cond1 cond2 .. condp
x11 x12 .. x1p x21 x22 ..
x2p
gene1gene2 gene n

7
Correlation Coefficient has been used by Gauss,
Bravais, Edgeworth Sweeping impact in data
analysis is due to Galton(1822-1911) Typical
laws of heredity in man Karl Pearson modifies
and popularizes its use. A building block in
multivariate analysis, of which clustering,
classification, dimension reduction are recurrent
themes

8
(No Transcript)
9
Two classes problem
An application
ALL (acute lymphoblastic leukemia) AML(acute
myeloid leukemia)
10
Why clustering make sense biologically?
The rationale is
Genes with high degree of expression similarity
are likely to be functionally related. may form
structural complex, may participate in common
pathways. may be co-regulated by common
upstream regulatory elements.
Simply put,
Profile similarity implies functional association
11
However, the converse is not true

• The expression profiles of majority of
  functionally associated genes are indeed
  uncorrelated

• Microarray is too noisy
• Biology is complex

12
Why no correlation?

• Protein rarely works alone
• Protein has multiple functions
• Different biological processes or pathways have
  to be synchronized
• Competing use of finite resources metabolites,
  hormones,
• Protein modification Phosphorylation,
  proteolysis, shuttle,
• Transcription factors serving both as
  activators and repressors

13
Transcription factors proteins that bind to DNA
Activator repressors
14
Going subtleProtein modification Histone
inhibits transcription To activate transcription,
the lysine side chain must be acetylated.
Weaver(2001)
15
Corepressor histone deacetylase
Thyroid hormone
Coactivator Histone acetyltransferase
16
Math. Modeling a nightmare
Current
Next
mRNA
F I T N E S S
mRNA
mRNA
Observed
protein kinase
hidden
ATP, GTP, cAMP, etc
Cytoplasm Nucleus Mitochondria Vacuolar
localization
F U N C T I O N
Statistical methods become useful
DNA methylation, chromatin structure
Nutrients- carbon, nitrogen sources Temperature Wa
ter
17
What is LA? PLA?

• Concept of mediator

18
Schematic illustration of LA
19
Example 1. Positive-to-negative

• XARP4,YLAS17, ZMCM1
• Corr 0 in each plot
• For low Z (marked points in A), X and Y are
  coexpressed
• (B). For high Z (marked points in B), X and Y are
  contra-expressed

Arp4 Protein that interacts with core histones,
member of the NuA4 histone acetyltransferase
complex actin related protein Las17 Component
of the cortical actin cytoskeleton
20
(No Transcript)
21
Example 2 -Negative to Positive

• XQCR9, Y ROX1, ZMCM1
• Corr0 in each plot
• For low Z (marked points in A), X and Y are
  contra-expressed
• (B). For high Z (marked points in B), X and Y are
  co-expressed

Rox1 Heme-dependent transcriptional repressor of
hypoxic genes including CYC7(iso-2-cytochrome c
) and ANB1(translation initiation, ribosome) Qcr9
Ubiquinol cytochrome c reductase subunit 9
22
(No Transcript)
23
A Challenge

• What genes behave like that ?
• Can we identify all of them ?
• N5878 ORFs
• N choose 3 33.8 billion triplets to inspect

24
Statistical theory for LA

• X, Y, Z random variables with mean 0 and variance
  1
• Corr(X,Y)E(XY)E(E(XYZ))Eg(Z)
• g(z) an ideal summary of association pattern
  between X and Y when Z z
• g(z)derivative of g(z)
• Definition. The LA of X and Y with respect to Z
  is LA(X,YZ) Eg(Z)

25
Statistical theory-LA

• Theorem. If Z is standard normal, then
  LA(X,YZ)E(XYZ)
• Proof. By Steins Lemma Eg(Z)Eg(Z)Z
• E(E(XYZ)Z)E(XYZ)
• Additional math. properties
• bounded by third moment
• 0, if jointly normal
• transformation

26
Normality ?

• Convert each gene expression profile by taking
  normal score transformation
• LA(X,YZ) average of triplet product of three
  gene profiles
• (x1y1z1 x2y2z2 . ) / n

27
How does LA work in yeast?

• Urea cycle/arginine biosynthesis

28
Yeast Cell Cycle(adapted from Molecular Cell
Biology, Darnell et al)
Most visible event
29
ARG1
Glutamate
ARG2
30
ARG1
Glutamate
ARG2
31
ARG1
8th place negative
Y
Head
X
Compute LA(X,YZ) for all Z
Backdoor
Rank and find leading genes
Adapted from KEGG
32
Why negative LA?high CPA2 signal for
arginine demand. up-regulation of ARG2
concomitant with down-regulation of CAR2
prevents ornithine from leaving the urea
cycle.When the demand is relieved, CPA2 is
lowered, CAR2 is up-regulated, opening up the
channel for orinthine to leave the urea cycle.

33
Other examples (see Li 2002)

• XGLN3(transcription factor), YCAR1, ZARG4 (8th
  place negative end)
• Electron transport XCYT1(cytochome c1), gives
  ATP1 (11 times), ATP5 (subunits of ATPase)
• Calmodulin CMD1, NUF1 (binding target of CMD1),
  CMK1(calmodulin-regulated kinase), YGL149W
• Glycolysis genes PFK1, PFK2 (6-phospho-fructokinas
  e)
• CYR1(adenylate cyclase) , GSY1 (glycogen
  synthase), GLC2( glucan branching),
  SCH9(serine/threonine protein kinase longevity)

34
SCH9

• Protein kinase that regulates signal transduction
  activity and G1 progression, controls cAPK
  activity, required for nitrogen activation of the
  FGM pathway, involved in life span regulation,
  homologous to mammalian Akt/PKB (SGD summary)
• Science. 2001 Apr 13292(5515)288-90.  
  Regulation of longevity and stress resistance by
  Sch9 in yeast.Fabrizio P, Pozza F, Pletcher SD,
  Gendron CM, Longo VD.
• The protein kinase Akt/protein kinase B (PKB) is
  implicated in insulin signaling in mammals and
  functions in a pathway that regulates longevity
  and stress resistance in Caenorhabditis elegans.
  We screened for long-lived mutants in nondividing
  yeast Saccharomyces cerevisiae and identified
  mutations in adenylate cyclase and SCH9, which is
  homologous to Akt/PKB, that increase resistance
  to oxidants and extend life-span by up to
  threefold. Stress-resistance transcription
  factors Msn2/Msn4 and protein kinase Rim15 were
  required for this life-span extension. These
  results indicate that longevity is associated
  with increased investment in maintenance and show
  that highly conserved genes play similar roles in
  life-span regulation in S. cerevisiae and higher
  eukaryotes.

35

• Blue low SCH9
• Red high SCH9