Network Inference, With an Application to Yeast Systems Biology

Center for Genomic SciencesCuernavaca, MexicoSeptember 25, 2006

Reinhard Laubenbacher

Virginia Bioinformatics Institute

And

Department of Mathematics

Virginia Tech

http://polymath.vbi.vt.edu

Contributors and Collaborators

Applied Discrete Mathematics Group

(http://polymath.vbi.vt.edu)

• Miguel Colòn-Velez

• Elena Dimitrova (now at Clemson U)

• Luis Garcia (now at Texas A&M)

• Abdul Jarrah

• John McGee (now at Radford U)

• Brandy Stigler (now at MBI)

• Paola Vera-Licona

Collaborators• Diogo Camacho (VBI)• Ana Martins (VBI)• Pedro Mendes (VBI)• Wei Shah (VBI)• Vladimir Shulaev (VBI)• Michael Stillman (Cornell)• Bernd Sturmfels (UC

Berkeley)

Funding: NIH, NSF, Commonwealth of VA

“All processes in organisms,

from the interaction of molecules to the complex

functions of the brain and other whole organs,

strictly obey […] physical laws.

“Where organisms differ from inanimate matter is

in the organization of their systems and especially

in the possession of coded information.”

E. Mayr, 1988

Genome

Molecularnetworks

Organism

Environment

Increasingcomplexity

A multiscale system

Discrete models

“[The] transcriptional control of a gene can be described by a discrete-valued function of several discrete-valued variables.”

“A regulatory network, consisting of many interacting genes and transcription factors, can be described as a collection of interrelated discrete functionsand depicted by a wiring diagram similar to the diagram of a digital logic circuit.”

Karp, 2002

Model Types

Ideker, Lauffenburger, Trends in Biotech 21, 2003

Biochemical Networks

Brazhnik, P., de la Fuente, A. and Mendes, P. Trends in Biotechnology 20, 2002

Gene space

Protein space

Metabolic space

M etabo lite 1 M etabo lite 2

P ro te in 1

P ro te in 2

P ro te in 3

P ro te in 4 C om p lex 3 :4

G ene 1

G ene 2

G ene 3

G ene 4

• Oxidative Stress is a general term used to describe the steady state level of oxidative damage in a cell, tissue, or organ, caused by the species with high oxidative potential.

Introduction to oxidative stress and CHP

+ X + oxidized X

Cumene hydroperoxide (CHP)

Cumyl alcohol (COH)

C CH3CH3

O

O

H

C CH3CH3

O

H

• Cumene hydroperoxide (CHP) is an organic peroxide, thus has high oxidative potential. CHP is very reactive and can easily oxidize molecules such as lipids, proteins and DNA.

• Oxidation by CHP

Courtesy of Wei Sha

Glutathione-glutaredoxin antioxidant defense system

glutathione peroxidase (GPX1, GPX2, GPX3)

GSSG + ROH

(alcohol or water)

Glu + Cys

-GluCys

-glutamylcysteine synthetase (GSH1)

glutathione synthetase (GSH2) + Gly

Feedback inhibition

glutathione S-transferase (GTT1, GTT2)

+

RX

HX + R-SG

glutaredoxin (GRX1, GRX2)

GSHROOH

(peroxides)

+

NADPHNADP+

thioredoxin reductase (TRR1)glutathion oxidoreductase (GLR1)

Courtesy of Wei Sha

Saccharomyces cerevisiae systems biology at VBI Experimentation

Samples for metabolites,

RNA and proteins

Freeze-dry

Metaboliteextraction

GC-MS

Separate cells from the media

DataAnalysis

Quench metabolism in cold buffered methanol

Cell growth in controlled batch (in fermentors)

Experimental treatment(i.e. oxidative stress)

Break cells with high frequency

sound waves

LC-MS CE-MS

RNAextraction

Proteinextraction

AffymetrixGeneChipTM

2D PAGE,MALDI-MS

Sample Prep

Modeling

Courtesy V. Shulaev

CHP treated Samples

Control Samples

Experimental design

Affymetrix Yeast Genome S98 array

0 min 3 min 6 min 12 min 20 min 40 min 70 min 120 min

Cumene hydroperoxide (CHP)

12

3

Wild type yeast cultureWild type yeast culture

Wild type yeast culture

0 min 3 min 6 min 12 min 20 min 40 min 70 min 120 min

Buffer (EtOH)

12

3

Fermentor that contains yeast cell culture

Wild type yeast cultureWild type yeast culture

Wild type yeast culture

Courtesy W. Sha

Comparisons Significantly changed genes (p<0.01)

Up-regulated genes

Down-regulated genes

Cont_3min vs. Cont_0min

1 0 1

Cont_6min vs. Cont_0min

4 2 2

Cont_12min vs. Cont_0min

2 1 1

Cont_20min vs. Cont_0min

18 12 6

Cont_40min vs. Cont_0min

1054 571 483

Cont_70min vs. Cont_0min

2709 1343 1366

Cont_120min vs. Cont_0min

2829 1344 1485

Why is it important to use control samples?

Comparisons Significantly changed genes (p<0.01)

Up-regulated genes

Down-regulated genes

CHP_3min vs. CHP_0min

26 25 1

CHP_6min vs. CHP_0min

235 170 65

CHP_12min vs. CHP_0min

1093 512 581

CHP_20min vs. CHP_0min

1646 867 779

CHP_40min vs. CHP_0min

1643 932 711

CHP_70min vs. CHP_0min

1800 1067 733

CHP_120min vs. CHP_0min

2465 1344 1121

Control samples CHP treated samples

Courtesy W. Sha

Cumene hydroperoxide (CHP) and cumyl alcohol (COH) progress curves

0

50

100

150

200

250

0 20 40 60 80 100 120

Time (min)

Con

cent

ratio

n (

M)

CHP

COH

0

20

40

60

80

100

0 10 20 30 40 50

Time (h)C

on

cen

tra

tion

(m M

)

CHP

COH

In yeast cell culture In medium

Courtesy W. Sha

Pathways induced by oxidative stress were identified

GO term 3min 6min 12min 20min

Response_to_stress >0.1 0.016645 0 0

Carbohydrate_metabolism >0.1 >0.1 0 0

Sporulation >0.1 >0.1 0.02106 0.000644

Protein_catabolism >0.1 >0.1 0.030576 0

Signal_transduction >0.1 >0.1 0.032308 0.000449

KEGG term 3min 6min 12min 20min

Glutathione metabolism >0.1 0.002443 0.013898 5.4E-05

Glycerolipid metabolism >0.1 0.01106 0.00064 0.020554

Starch and sucrose metabolism >0.1 >0.1 0.000485 0.000364

Fructose and mannose metabolism >0.1 >0.1 0.006715 0.028673

Proteasome >0.1 >0.1 >0.1 8.81E-15

Ubiquitin mediated proteolysis >0.1 >0.1 >0.1 0.004418

Courtesy W. Sha

Pathways repressed by oxidative stress were identified

GO term 3min 6min 12min 20min

Nuclear_organization_and_biogenesis >0.1 0.022552 0.000632 0.036472

Ribosome_biogenesis_and_assembly >0.1 0.093905 0 0

Organelle_organization_and_biogenesis >0.1 >0.1 0 0

RNA_metabolism >0.1 >0.1 0 0

Cell_cycle >0.1 >0.1 0.00014 0.036506

Cytokinesis >0.1 >0.1 0 0

Electron_transport >0.1 >0.1 >0.1 0

KEGG term 3min 6min 12min 20min

cell cycle >0.1 0.006487 1.232E-07 0.001035

purine metabolism >0.1 0.009725 5.656E-10 1.133E-09

RNA polymerase >0.1 >0.1 5.396E-13 2.423E-09

pyrimidine metabolism >0.1 >0.1 8.983E-11 6.318E-09

Courtesy W. Sha

k-means clustering analysis result

1 2 3

54

Courtesy W. Sha

Pathway analysis for each cluster

Ribosome

Cell cycle

RNA polymerase

Purine metabolism

Pyrimidine metabolism

Oxidative phosphorylationGalactose metabolism

Starch and sucrose metabolism

-1.5

-1

-0.5

0

0.5

1

1.5

0 10 20

2

-1.5

-1

-0.5

0

0.5

1

1.5

0 10 20

1

ATP synthesis

-1

-0.5

0

0.5

1

0 10 20

3

-1

-0.5

0

0.5

1

1.5

2

0 10 20

4

Proteasome

Ubiquitin mediated proteolysis

MAPK signaling pathway

-1.5

-1

-0.5

0

0.5

1

0 10 20

5

Where are the oxidative

stress defense pathways?

Courtesy W. Sha

YAP1 was successfully knocked out in yap1 mutant yeast

The transformation of CHP to COH

0

50

100

150

200

250

0 20 40 60 80 100 120

Time (min)

Con

cent

ratio

n (

M)

CHP

COH

in wild type

in yap1 mutant

020406080

100120140160

0 4 8 12 16 20

Time (min)

Ex

pre

ss

ion

le

ve

l

Time series of YAP1 gene expression level in

wild type control sample

wild type CHP treated sample

yap1 mutant Control sample

yap1 mutant CHP treated sample

Genotype Phenotype

0

50

100

150

200

250

0 20 40 60 80 100 120

Time (min)

Con

cent

ratio

n (m

M)

CHP

COH

Courtesy W. Sha

Claytor Lake Network

M1 M2

M23

Courtesy P. Mendes

“Bottom-up modeling:” Model individual pathways and aggregate to system-level models

“Top-down modeling:” Develop network inference methods for system-level phenomenological models

Courtesy P. Mendes

Genetic Regulation

Courtesy P. Mendes

I = lac repressor

= protein which regulates transcription of lac mRNA (genes in blue)

Z = beta-galactosidase

= protein which cleaves lactose to produce glucose, galactose, and allolactose

Y = Lactose permease

= protein which transports lactose into the cell

http://web.mit.edu/esgbio/www/pge/lac.html

Discrete Model for lac Operon

fM = A

fB = M

fA = A (L B)

fL = P (L B)

fP = MModel assumptions

• Transcription/translation require 1 time unit

• mRNA/protein degradation require 1 time unit

• Extracellular lactose always available

M = mRNA for lac genes: LacZ, LacY, LacAB = beta-galactosidaseA = allolactose

= isomer of lactose (inducer)L = lactose (intracellular)P = lactose permease

Discrete Model with Dynamics

(M, B, A, L, P)

Variables x1, … , xn with values in a finite set X.(s1, t1), … , (sr, tr) state transition observations with sj, tj

ε Xn.

Goal: Identify a collection of “best” dynamical systems f=(f1, … ,fn): Xn → Xn

such that f(sj)=tj.

(1) Wiring diagram(2) Dynamics

R. Laubenbacher and B. Stigler, A computational algebra approach to the reverse-engineering of gene regulatory networks, J. Theor. Biol. 229 (2004)

A. Jarrah, R. Laubenbacher, B. Stigler, and M. Stillman, Reverse-engineering of polynomial dynamical systems, Adv. in Appl. Math. (2006) in press

Pandapas network• 10 genes, 3 external biochemicals• 17 interactions

Time course data: 9 time points• Generated 8 time series for wildtype, knockouts G1, G2, G5 • 192 data points• G6, G9 constant

Data discretization• 5 states per node• 95 data points

– 49% reduction – < 0.00001% of 513 total states

Method Validation:Simulated gene network

Courtesy B. Stigler

Method Validation:Simulated gene network

Minimal Sets Algorithm

• 77% interactions• Identified targets of P2, P3 (x12, x13)• 11 false positives, 4 false negatives

Pandapas Reverse engineeredCourtesy B. Stigler

Example: Gene Regulatory Networks

)()(01.0

)(1

)(01.0

)(1

)()(01.0

01.0

)())(01.0))((01.0))((01.0(

10

)()(01.0

)(01.0

)(101.0

)()(01.0

)(01.0

)(101.0

53

3

1

15

43

4

3531

63

23

1

1

2

13

1

1

1

tGtG

tG

tG

tG

dt

dG

tGtGdt

dG

tGtGtGtGdt

dG

tGtG

tG

tG

dt

dG

tGtG

tG

tG

dt

dG

Stable steady states: (1.99006, 1.99006, 0.000024814, 0.997525, 1.99994)

(-0.00493694, -0.00493694, -0.0604538, -0.198201, 0.0547545)

Data (discretized to 5 states)

Algorithm input: 7 such time courses, 60 state transitions

1 1 1 1 1

0.203145 0.203145 0.135339 0.169883 3.469657

0.415507 0.415507 0.018334 0.220206 3.478608

1.192199 1.192199 0.002502 0.600941 2.773302

1.760581 1.760581 0.00036 0.883442 2.223943

1.941092 1.941092 7E-05 0.973211 2.047744

1.980977 1.980977 3.09E-05 0.993022 2.008786

1.988499 1.988499 2.56E-05 0.996752 2.001455

1.989805 1.989805 2.49E-05 0.997398 2.000187

1.990021 1.990021 2.48E-05 0.997505 1.999978

1.990056 1.990056 2.48E-05 0.997522 1.999944

3 1 2 2 1

1 0 1 1 2

1 0 0 1 4

3 0 0 1 3

3 1 0 1 2

4 2 0 1 2

4 2 0 2 2

4 2 0 2 2

4 2 0 2 2

4 2 0 2 2

4 2 0 2 2

A model for 1 wildtype time series

f1 = – x4+1f2 = 1f3 = x4+1f4 = 1f5 = – x5

3 – 2x52+2x4 – 2x5 – 2

var1 = {4}

var2 = {}

var3 = {4}

var4 = {}

var5 = {4, 5}

G1

G2 G5G4

G3

G1

G2 G5G4

G3

G1

G2 G5G4

G3

G1

G2 G5G4

G3

Adding another wildtype time series

G1

G2 G5G4

G3

Adding a knockout time series

All time series

Using 10 random variable orders

G1

G2 G5G4

G3

Wiring diagram missing two (20%) edges; includes 5 indirect interactions.

Network has 55 = 3125 possible state transitions. Input: 60 ( = approx. 2%) state transitions.

Dynamics

Stable steady state:

(1.99006, 1.99006, 0.000024814, 0.997525, 1.99994)

Wild type time series 1

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

1 2 3 4 5 6 7 8 9 10 11

Time

Gen

e ex

pre

ssio

n

G1, G2

G3

G4

G5

DiscretizationFixed point: (4, 4, 2, 4,

2)

Dynamics

f1 = 3x3x53+x5

4+4x33+x1

2x5+4x3x52+2x1

2+3x32+4x1x5+x5

2+4x3+4x5+3

f2 = 4x3x54+4x3x5

3+4x54+x3

3+4x12x5+2x3x5

2+4x53+2x1

2+2x1x3+3x32+2x3x5+4x5

2+4x1+4x5+1

f3 = x13x4+4x1x4

3+3x12x4+4x1x4

2+x43+x1

2+x1x4+x42+x1+x4+4

f4 = x3x54+2x3x5

3+3x54+4x3

3+x12x5+3x3

2x5+2x3x52+2x1x3+2x3

2+x52+4x1+x3+4x5+4

f5 = 4x3x54+3x3x5

3+2x53+x1

2+2x1x3+2x32+4x3+4x5+4

Phase space: There are 4 components and 4 fixed point(s) Components Size Cycle Length 1 2200 1 2 890 1 3 10 1 4 25 1

TOTAL: 3125 = 55 nodes

Printing fixed point(s)... [ 0 1 2 1 0 ] lies in a component of size 25. [ 2 2 4 2 3 ] lies in a component of size 10. [ 4 4 2 2 3 ] lies in a component of size 890. [ 4 4 2 4 2 ] lies in a component of size 2200.

Summary

• To use “omics” data set to their full potential network inference methods are useful.

• Cellular processes are dynamical systems, so we need methods for the inference of dynamical systems models.

• Special data requirements.

• Models are useful to generate new hypotheses.

• Validation of modeling technologies is crucial.