Inferring Dynamic Architecture of Cellular Networks by Perturbation

Response Analysis

Boris N. KholodenkoDepartment of Pathology, Anatomy and Cell Biology

Thomas Jefferson UniversityPhiladelphia, PA

S

En 1

1E 1E

T T

2E 2E

I

I

I

How to Quantify the Control Exerted by a Signal over a Target?

System Response: the sensitivity (R) of the target T to a change in the signal S

System response equals the PRODUCT of local responses for a linear cascade:

)(... 121 pathrrrrR nnTS

statesteady

TS SdTdR ln/ln

Local Response: the sensitivity (r) of the level i to the preceding level i-1. Active protein forms (Ei) are “communicating” intermediates:

statesteadyiLeveliii EEr 1ln/ln

Kholodenko et al (1997) FEBS Lett. 414: 430

James E. Ferrell, Jr. TIBS 21:460-466 (1996)

Ultrasensitive Response of the MAPK Cascade in Xenopus Oocytes Extracts Is Explained by Multiplication of the Local Responses of the Three MAPK Levels

Cascade response R = d ln[ERK-PP]/d ln[Ras]

Cascade with feedback

Rf = R/(1 – fR)

Feedback strength f = ln v/ ln[ERK-PP]

Cascade with no feedback

R = r1 r2 r3

Control and Dynamic Properties of the MAPK Cascades

Raf-PRaf

MEK-PPMEK-PMEK

ERK-PPERK-PERK

Ras f

Kholodenko, B.N. (2000) Europ. J. Biochem. 267: 1583.

•Negative Feedback And Ultrasensitivity Can Cause Oscillations

TIME (min)

0 20 40 60 80 100 120 140

MA

PK

co

nce

ntr

atio

ns

(nM

)

0

50

100

150

200

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ERK-PP - red line ERK - green line

Effects of Feedback Loops on the MAPK Dynamics

Kholodenko, B.N. (2000) Europ. J. Biochem. 267: 1583Multi-stability analysis: Angeli & Sontag, Systems & Control Letters, 2003 (in press)

•Positive Feedback Can Cause Bistability and Hysteresis

Interaction Map of a Cellular Regulatory Network is Quantified by the Local Response Matrix

A dynamic system: mn pppxxxpxfdtdx ,...,,,...,).,(/ 11

F = (f/x).The Jacobian matrix:

00

0

00

0

Signed incidence matrix

If fi/xj = 0, there is

no connection from variable xj to xi on the network graph.

Local response matrix r

(Network Map)

r = - (dgF)-1F

Relative strength of connections to each xi is given by the ratios, rij = xi/xj =

- (fi/xj)/(fi/xi) 100

10

010

01

43

3432

24

1412

r

rr

r

rr

Kholodenko et al (2002) PNAS 99: 12841.

System Responses are Determined by ‘Local’ Intermodular Interactions

RP = x/p system response matrix; p – perturbation parameters (signals);

r - local response matrix (interaction map); F = f/x the Jacobian matrix

rP = f/p matrix of intramodular (immediate) responses to signals

Bruggeman et al, J. theor Biol. (2002)

Rp = - r –1 rp = F–1(dgF)rp

1

2

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S

Untangling the Wires: Tracing Functional Interactions in Signaling, Metabolic, and Gene Networks

The goal is to determineFi(t) = (Fi1 , … , Fin)

the Jacobian elements that quantify connections to node i

Quantitative and predictive biology: the ability to interpret increasingly complex datasets to reveal underlying interactions

However, at steady-state the F’s can only be determined up to arbitrary scaling factors

Scaling Fij by diagonal elements, we

obtain the network interaction matrix,

r = - (dgF)-1F

Problem: Network interaction map r cannot be captured in intact cells. Only system responses (R) to perturbations can be measured in intact cells.

Untangling the Wires: Tracing Functional Interactions in Signaling and Gene Networks.

Goal: To Determine and Quantify Unknown Network Connections

Kholodenko et al, (2002) PNAS 99: 12841.

Solution: To Measure the Sytem Responses (matrix R) to Successive Perturbations to All Network Modules

r = - (dg(R-1))-1R-1

Steady-State Analysis:

Stark et al (2003) Trends Biotechnol. 21:290

Testing the Method: Comparing Quantitative Reconstruction to Known Interaction Maps

Kholodenko et al. (2002), PNAS 99: 12841.

Problem: To Infer Connections in the Ras/MAPK Pathway

Solution: To Simulate Global Responses to Multiple Perturbations and Calculate the Ras/MAPK Cascade Interaction Map

Raf-PRaf

MEK-PPMEK-PMEK

ERK-PPERK-PERK

Ras -

+

Step 1: Determining Global Responses to Three Independent Perturbations of the Ras/MAPK Cascade.

a). Measurement of the differences in steady-state variables following perturbations:

4.33.67.12

9.81.32.6

7.39.64.7

8.214.397.85

8.563.208.44

0.253.465.55

b) Generation of the global response matrix

10% change in parameters 50% change in parameters

(1) (0) (1) (0)ln 2( ) /( )X X X X X

1

1

1

ln

ln

ln

Raf P

MEK PP

ERK PP

1

2

2

2

ln

ln

ln

Raf P

MEK PP

ERK PP

2

3

3

3

ln

ln

ln

Raf P

MEK PP

ERK PP

3

Step 2: Calculating the Ras/MAPK Cascade Interaction Map from the System Responses

r = - (dg(R-1))-1R-1

10.20.0

6.018.1

2.10.01

10.20.0

6.019.1

1.10.01

10.20.0

6.019.1

1.10.01

Known Interaction Map

Two interaction maps (local response matrices) retrieved from two different system response matrices

Raf-PRaf

MEK-PPMEK-PMEK

ERK-PPERK-PERK

Ras -

+

Raf-P MEK-PP ERK-PP

Raf-P

MEK-PP

ERK-PP

Raf-P MEK-PP ERK-PP

Raf-P

MEK-PP

ERK-PP

Unraveling the Wiring Using Time Series Data

Orthogonality theorem: (Fi(t), Gj(t)) = 0 Sontag E. et al. (submitted)

A vector Ai(t) is orthogonal to the linear subspace spanned by responses

to perturbations affecting either one or multiple nodes different from i

Xi(t)

Time (t) sec

0 30 60 90 120 150

PL

C

( Xi)

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Xi(t, X0, pj+pj

Xi(t, X0, pj)

Vector Gj(t) contains

experimentally measured network responses xi(t) to

parameter pj perturbation,Gj(t) = (xi/t, xi , … , xn)

dx/dt = f(x,p)F(t) = (x/p)

Perturbation in pj affects any nodes except node i.

The goal is to determineFi(t) = (1, Fi1 , … , Fin)

the Jacobian elements that quantify connections to xi

Inferring dynamic connections in MAPK pathway successfully

MAPK pathway kinetic diagramDeduced time-dependent strength of a negative feedback for 5, 25, and 50% perturbations

Transition of MAPK pathway from resting to stable activity state

Sontag E. et al. (submitted)

Oscillatory dynamics of the feedback connection strengths is successfully deduced

Oscillations in MAPK pathway

Raf-PRaf

MEK-PPMEK-PMEK

ERK-PPERK-PERK

Ras f

The Jacobian element F15 quantifies the negative feedback strength

(red) - 1%, (black) - 10% perturbation

Unraveling the Wiring of a Gene Network

Kholodenko et al. (2002) PNAS 99: 12841.

5.449.106.106.6

0.145.434.427.20

3.279.50.463.3

4.116.28.223.39

13.00.00.0

0.017.05.0

6.00.010.0

5.00.06.01

13.00.00.0

0.016.04.0

6.00.010.0

5.00.06.01

Known Interaction Map

Calculated Interaction Map

System Response Matrix

Unraveling The Wiring When Some Genes Are Unknown: The Case of Hidden Variables

Kholodenko et al. (2002) PNAS 99: 12841.

39.2 22.8 2.6

3.3 46.0 5.9

20.7 42.4 43.5

System Response Matrix

Calculated Interaction Map

1 0.5 0.2

0 1 0.1

0.4 0.6 1

13 14 43r r r

23 24 43r r r

Existing Network

Reverse engineering of dynamic gene interactions

Fij = fi/xj

r12

1

2

3

r13

Effect of noise on the ability to infer network

M. Andrec & R. Levy

Probability of misestimating network connections as a function of noise level and connection strengths

Special Thanks to:

Jan B. Hoek Anatoly Kiyatkin(Thomas Jefferson University, Philadelphia)

Hans WesterhoffFrank Bruggeman(Free University, Amsterdam)

Eduardo SontagMichael AndrecRonald Levy (Rutgers University, NJ, USA)

Supported by the NIH/ NIGMS grant GM59570

Numbers on the top (with superscript a) are the correct “theoretical” values of the Jacobian elements Fij

Numbers on the bottom (with superscript b) are the “experimental” estimates deduced using perturbations of the gene synthesis and degradation rates

A snapshot of the retrieved dynamics of gene activation and repression for a four-gene network