2. S E n  1 1 E 1 E T T 2 E 2 E 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: Local Response: the sensitivity (r) of the level i to the preceding level i-1. Active protein forms (E i ) are “communicating” intermediates: Kholodenko et al (1997 ) FEBS Lett. 414: 430

3. 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

4. Cascade response R = d ln[ERK-PP]/d ln[Ras] Cascade with feedback R f = R/(1 – f  R) Feedback strength f =  ln v/  ln[ERK-PP] Cascade with no feedback R = r 1  r 2  r 3 Control and Dynamic Properties of the MAPK Cascades Raf-PRaf MEK-PPMEK-PMEK ERK-PPERK-P ERK Ras f Kholodenko, B.N. (2000) Europ. J. Biochem. 267: 1583.

5. Negative Feedback And Ultrasensitivity Can Cause Oscillations ERK-PP - red line ERK - green line Effects of Feedback Loops on the MAPK Dynamics Kholodenko, B.N. (2000) Europ. J. Biochem. 267: 1583 Multi-stability analysis: Angeli & Sontag, Systems & Control Letters, 2003 (in press) Positive Feedback Can Cause Bistability and Hysteresis

6. Interaction Map of a Cellular Regulatory Network is Quantified by the Local Response Matrix A dynamic system: F = (  f/  x). The Jacobian matrix: Signed incidence matrix If  f i /  x j = 0, there is no connection from variable x j to x i on the network graph. Local response matrix r (Network Map) r = - (dgF) -1  F Relative strength of connections to each x i is given by the ratios, r ij =  x i /  x j = - (  f i /  x j )/(  f i /  x i ) Kholodenko et al (2002) PNAS 99: 12841.

7. System Responses are Determined by ‘Local’ Intermodular Interactions R P =  x/  p system response matrix; p – perturbation parameters (signals); r - local response matrix (interaction map); F =  f/  x the Jacobian matrix r P =  f/  p matrix of intramodular (immediate) responses to signals Bruggeman et al, J. theor Biol. (2002) R p = - r –1  r p = F –1  (dgF)  r p 1 2 3    4   S

8. Untangling the Wires: Tracing Functional Interactions in Signaling, Metabolic, and Gene Networks The goal is to determine F i (t) = (F i1, …, F in ) 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 F ij by diagonal elements, we obtain the network interaction matrix, r = - (dgF) -1  F Problem: N etwork interaction map r cannot be captured in intact cells. Only system responses (R) to perturbations can be measured in intact cells.

9. 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 )) -1  R -1 Steady-State Analysis: Stark et al (2003) Trends Biotechnol. 21:290

10. 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-P ERK Ras - +

11. 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: b) Generation of the global response matrix 10% change in parameters50% change in parameters 123

12. Step 2: Calculating the Ras/MAPK Cascade Interaction Map from the System Responses r = - (dg(R -1 )) -1  R -1 Known Interaction Map Two interaction maps (local response matrices) retrieved from two different system response matrices Raf-PRaf MEK-PPMEK-PMEK ERK-PPERK-P ERK 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

13. Unraveling the Wiring Using Time Series Data Orthogonality theorem: (F i (t), G j (t)) = 0 Sontag E. et al. (submitted) A vector A i (t) is orthogonal to the linear subspace spanned by responses to perturbations affecting either one or multiple nodes different from i Vector G j (t) contains experimentally measured network responses  x i (t) to parameter p j perturbation, G j (t) = (  x i /  t,  x i, …,  x n ) dx/dt = f(x,p) F(t) = (  x/  p) Perturbation in p j affects any nodes except node i. The goal is to determine F i (t) = (1, F i1, …, F in ) the Jacobian elements that quantify connections to x i

14. Inferring dynamic connections in MAPK pathway successfully MAPK pathway kinetic diagram Deduced 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)

15. Oscillatory dynamics of the feedback connection strengths is successfully deduced Oscillations in MAPK pathway Raf-PRaf MEK-PPMEK-PMEK ERK-PPERK-P ERK Ras f The Jacobian element F 15 quantifies the negative feedback strength  (red) - 1%,  (black) - 10% perturbation

16. Unraveling the Wiring of a Gene Network Kholodenko et al. (2002) PNAS 99: 12841. Known Interaction Map Calculated Interaction Map System Response Matrix

17. Unraveling The Wiring When Some Genes Are Unknown: The Case of Hidden Variables Kholodenko et al. (2002) PNAS 99: 12841. System Response Matrix Calculated Interaction Map Existing Network

18. Reverse engineering of dynamic gene interactions F ij =  f i /  x j

19. r 12 1 2 3 r 13 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

20. Special Thanks to: Jan B. Hoek Anatoly Kiyatkin (Thomas Jefferson University, Philadelphia) Hans Westerhoff Frank Bruggeman (Free University, Amsterdam) Eduardo Sontag Michael Andrec Ronald Levy (Rutgers University, NJ, USA) Supported by the NIH/ NIGMS grant GM59570

21. Numbers on the top (with superscript a) are the correct “theoretical” values of the Jacobian elements F ij 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