Inferring Dynamic Architecture of
Cellular Networks by Perturbation
Response Analysis
Boris N. Kholodenko
Department of Pathology, Anatomy and Cell Biology
Thomas Jefferson University
Philadelphia, PA
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
S
E1I
RST  d ln T / d ln S steady state
E1
E2I
E2
En-1
I
T
Local Response: the sensitivity (r) of the
level i to the preceding level i-1. Active
protein forms (Ei) are “communicating”
intermediates:
ri   ln Ei /  ln Ei -1 Level i steady state
System response equals the PRODUCT
of local responses for a linear cascade:
T
RST  r1  r2  ...  rn-1  rn   ( path)
Kholodenko et al (1997) FEBS Lett. 414: 430
Ultrasensitive Response of the MAPK Cascade in
Xenopus Oocytes Extracts Is Explained by Multiplication
of the Local Responses of the Three MAPK Levels
James E. Ferrell, Jr. TIBS 21:460-466 (1996)
Control and Dynamic Properties of the MAPK Cascades
Cascade response
R = d ln[ERK-PP]/d ln[Ras]
Ras
f
Cascade with no feedback
R = r1  r2  r3
Raf
Raf-P
Feedback strength
f =  ln v/ ln[ERK-PP]
MEK
MEK-P
MEK-PP
Cascade with feedback
Rf = R/(1 – fR)
ERK
ERK-P
ERK-PP
Kholodenko, B.N. (2000) Europ. J. Biochem. 267: 1583.
Effects of Feedback Loops on the MAPK Dynamics
•Positive Feedback Can Cause Bistability and Hysteresis
MAPK concentrations (nM)
•Negative Feedback And Ultrasensitivity Can Cause Oscillations
300
250
200
150
100
50
0
0
20
40
60
80
100 120 140
TIME (min)
ERK-PP - red line ERK - green line
Kholodenko, B.N. (2000) Europ. J. Biochem. 267: 1583
Multi-stability analysis: Angeli & Sontag, Systems &
Control Letters, 2003 (in press)
Interaction Map of a Cellular Regulatory Network is
Quantified by the Local Response Matrix
A dynamic system:
dx / dt  f ( x, p). x  x1,..., xn , p  p1,..., pm
The Jacobian matrix:
F = (f/x).
0
0
0
If fi/xj = 0, there is
no connection from
variable xj to xi on
the network graph.
Relative strength of
connections to each xi
is given by the ratios,
rij = xi/xj =
- (fi/xj)/(fi/xi)
Signed incidence
matrix
0
0
0



-
Local response
matrix r
(Network Map)
r = - (dgF)-1F
Kholodenko et al (2002) PNAS 99: 12841.
- 1 r12 0 r14
0 - 1 0 r24
0 r32 - 1 r34
0
0
r43
-1
System Responses are Determined by ‘Local’
Intermodular Interactions
Rp = - r –1 rp = F–1(dgF)rp
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
1

4


2

S


3
Bruggeman et al, J. theor Biol. (2002)
Untangling the Wires: Tracing Functional Interactions in
Signaling, Metabolic, and Gene Networks
Quantitative and predictive biology: the ability to interpret
increasingly complex datasets to reveal underlying interactions
The goal is to determine
Fi(t) = (Fi1 , … , Fin)
the Jacobian elements that quantify
connections to node i
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)-1F
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
Solution: To Measure the Sytem Responses (matrix R)
to Successive Perturbations to All Network Modules
Steady-State Analysis:
r = - (dg(R-1))-1R-1
Kholodenko et al, (2002)
PNAS 99: 12841.
Stark et al (2003) Trends
Biotechnol. 21:290
Testing the Method: Comparing Quantitative
Reconstruction to Known Interaction Maps
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
Ras
-
Raf
Raf-P
MEK
MEK-P
+
ERK
MEK-PP
ERK-P
ERK-PP
Kholodenko et al. (2002), PNAS 99: 12841.
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:  ln X  2( X (1) - X (0) ) /( X (1)  X (0) )
1
2
3
 1 ln  Raf -P  



ln
MEK
PP


 1

  ln  ERK -PP  
 1

  2 ln  Raf -P  



ln
MEK
PP


 2

  ln  ERK -PP  
 2

  3 ln  Raf -P  


  3 ln  MEK -PP  
  ln  ERK -PP  
 3

b) Generation of the global response matrix
- 7.4 6.9
3.7
- 6.2 - 3.1 8.9
- 12.7 - 6.3 - 3.4
10% change in parameters
- 55.5 - 46.3 - 25.0
- 44.8 20.3 - 56.8
- 85.7 39.4
21.8
50% change in parameters
Step 2: Calculating the Ras/MAPK Cascade Interaction
Map from the System Responses
r = - (dg(R-1))-1R-1
Ras
Raf
Two interaction maps (local response
matrices) retrieved from two different
system response matrices
-
Raf-P MEK-PP ERK-PP
Raf-P
- 1 0.0 - 1.1
MEK-PP 1.9 - 1 - 0.6
ERK-PP 0.0 2.0
-1
Raf-P
MEK
+
MEK-P
ERK
MEK-PP
ERK-P
ERK-PP
- 1 0.0 - 1.2
1.8 - 1 - 0.6
0.0 2.0 - 1
Known Interaction Map
Raf-P MEK-PP ERK-PP
Raf-P
MEK-PP
ERK-PP
- 1 0.0 - 1.1
1.9 - 1 - 0.6
0.0 2.0 - 1
Unraveling the Wiring Using Time Series Data
12
Xi(t, X0, pj+pj
PLC (Xi)
10
dx/dt = f(x,p)
F(t) = (x/p)
8
Xi(t)
6
Perturbation in pj affects
any nodes except node i.
4
Xi(t, X0, pj)
2
0
0
30
60
90
120
Time (t) sec
Orthogonality theorem: (Fi(t), Gj(t)) = 0
Sontag E. et al. (submitted)
150
The goal is to determine
Fi(t) = (1, Fi1 , … , Fin)
the Jacobian elements that
quantify connections to xi
Vector Gj(t) contains
experimentally measured
network responses xi(t) to
parameter pj perturbation,
Gj(t) = (xi/t, xi , … , xn)
A vector Ai(t) is orthogonal to the linear subspace spanned by responses
to perturbations affecting either one or multiple nodes different from i
Inferring dynamic connections in MAPK pathway successfu
Transition of MAPK pathway
from resting to stable activity
state
MAPK pathway kinetic diagram
Sontag E. et al. (submitted)
Deduced time-dependent
strength of a negative
feedback for 5, 25, and 50%
perturbations
Oscillatory dynamics of the feedback
connection strengths is successfully
deduced
Oscillations in MAPK
pathway
Ras
f
Raf
Raf-P
MEK
MEK-P MEK-PP
ERK
The Jacobian element F15
quantifies the negative feedback
strength
ERK-P ERK-PP
 (red) - 1%,  (black) - 10% perturbation
Unraveling the Wiring of a Gene Network
System Response Matrix
39.3 - 22.8 2.6 11.4
- 3.3
46.0
5.9 27.3
- 20.7 42.4 43.5 14.0
- 6 .6
10.6 10.9 44.5
Calculated Interaction Map
- 1 - 0 . 6 0. 0 0 . 5
0 .0
- 1 0. 0 0 . 6
- 0.5 0.7 - 1 0.0
0 .0
0.0 0.3 - 1
Kholodenko et al. (2002) PNAS 99: 12841.
Known Interaction Map
- 1 - 0.6 0.0 0.5
0.0
- 1 0.0 0.6
- 0.4 0.6 - 1 0.0
0.0
0.0 0.3 - 1
Unraveling The Wiring When Some Genes Are
Unknown: The Case of Hidden Variables
Existing Network
System Response Matrix
39.2 -22.8 2.6
-3.3 46.0 5.9
-20.7 42.4 43.5
Calculated Interaction Map
-1 -0.5 0.2
0
-1 0.1
-0.4 0.6 -1
Kholodenko et al. (2002) PNAS 99: 12841.
r13  r14  r43
r23  r24  r43
Reverse engineering of
dynamic gene interactions
Fij = fi/xj
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
1
r12
r13
3
2
Special Thanks to:
Jan B. Hoek
Anatoly Kiyatkin
(Thomas Jefferson
University, Philadelphia)
Eduardo Sontag
Michael Andrec
Ronald Levy
(Rutgers University,
NJ, USA)
Hans Westerhoff
Frank Bruggeman
(Free University,
Amsterdam)
Supported by the NIH/ NIGMS grant GM59570
A snapshot of the retrieved dynamics of gene
activation and repression for a four-gene network
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