1. 16. Lecture WS 2006/07Bioinformatics III1 V16 Modelling Dynamic Cellular Processes John TysonBela Novak Mathematical description of signalling pathways helps answering questions like: (1) How do the magnitudes of signal output and signal duration depend on the kinetic properties of pathway components? (2) Can high signal amplification be coupled with fast signaling? (3) How are signaling pathways designed to ensure that they are safely off in the absence of stimulation, yet display high signal amplification following receptor activation? (4) How can different agonists stimulate the same pathway in distinct ways to elicit a sustained or a transient response, which can have dramatically different consequences? Heinrich et al. Mol. Cell. 9, 957 (2002)

2. 16. Lecture WS 2006/07Bioinformatics III2 The Cyclin – E2F cell cycle control system as Kohn, Molec. Biol. Cell 1999 10:2703-34

3. 16. Lecture WS 2006/07Bioinformatics III3 Biocarta pathways http://cgap.nci.nih.gov/Pathways/BioCarta/h_mapkPathway

4. 16. Lecture WS 2006/07Bioinformatics III4 Ras signaling pathway http://cgap.nci.nih.gov/Pathways/BioCarta/h_mapkPathway

5. 16. Lecture WS 2006/07Bioinformatics III5 Biocarta pathways Incredible amount of information about signalling pathways! Constantly updated. How can we digest this? Need computational models!

6. 16. Lecture WS 2006/07Bioinformatics III6 Protein synthesis and degradation: linear response S = signal strength (e.g. concentration of mRNA) R = response magnitude (e.g. concentration of protein) Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003) A steady-state solution of a differential equation, dR/dt = f(R) is a constant R ss that satisfies the algebraic equation f(R ss ) = 0. In this case

7. 16. Lecture WS 2006/07Bioinformatics III7 Protein synthesis and degradation Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003) Solid curve: rate of removal of the response component R. Dashed lines: rates of production of R for various values of signal strength. Filled circles: steady-state solutions where rate of production and rate of removal are identical. Wiring diagram rate curve signal-response curve Steady-state response R as a function of signal strength S. Parameters: k 0 = 0.01 k 1 = 1 k 2 = 5

8. 16. Lecture WS 2006/07Bioinformatics III8 phosphorylation/dephosphorylation: hyperbolic response Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003) A steady-state solution: RP: phosphorylated form of the response element R P = [RP] P i : inorganic phosphate R T = R + R P total concentration of R Parameters: k 1 = k 2 = 1 R T = 1

9. 16. Lecture WS 2006/07Bioinformatics III9 phosphorylation/dephosphorylation: sigmoidal response Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003) Modification of (b) where the phosphorylation and dephosphorylation are governed by Michaelis-Menten kinetics: steady-state solution of R P The biophysically acceptable solutions must be in the range 0 < R P < R T : with the „Goldbeter-Koshland“ function G:

10. 16. Lecture WS 2006/07Bioinformatics III10 phosphorylation/dephosphorylation: sigmoidal response This mechanism creates a switch-like signal-response curve which is called zero-order ultrasensitivity. (a), (b), and (c) give „graded“ and reversible behavior of R and S. „graded“: R increases continuously with S reversible: if S is change from S initial to S final, the response at S final is the same whether the signal is being increased (S initial S final ). Element behaves like a buzzer: to activate the response, one must push hard enough on the button. Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003) Parameters: k 1 = k 2 = 1 R T = 1 K m1 = K m2 = 0.05

11. 16. Lecture WS 2006/07Bioinformatics III11 perfect adaptation: sniffer Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003) Here, the simple linear response element is supplemented with a second signaling pathway through species X. The response mechanism exhibits perfect adaptation to the signal: although the signaling pathway exhibits a transient response to changes in signal strength, its steady-state response R ss is independent of S. Such behavior is typical of chemotactic systems, which respond to an abrupt change in attractants or repellents, but then adapt to a constant level of the signal. Our own sense of smell operates this way  we call this element „sniffer“.

12. 16. Lecture WS 2006/07Bioinformatics III12 perfect adaptation Right panel: transient response (R, black) as a function of stepwise increases in signal strength S (red) with concomitant changes in the indirect signaling pathway X (green). The signal influences the response via two parallel pathways that push the response in opposite directions. This is an example of feed-forward control. Alternatively, some component of a response pathway may feed back on the signal (positive, negative, or mixed). Tyson et al., Curr.Opin.Cell.Biol. 15, 221 (2003) Parameters: k 1 = k 2 = 2 k 3 = k 4 = 1

13. 16. Lecture WS 2006/07Bioinformatics III13 Positive feedback: Mutual activation Tyson et al., Curr.Opin.Cell.Biol. 15, 221 (2003) E: a protein involved with R E P : phosphorylated form of E Here, R activates E by phosphorylation, and E P enhances the synthesis of R.

14. 16. Lecture WS 2006/07Bioinformatics III14 mutual activation: one-way switch As S increases, the response is low until S exceeds a critical value S crit at which point the response increases abruptly to a high value. Then, if S decreases, the response stays high.  between 0 and S crit, the control system is „bistable“ – it has two stable steady-state response values (on the upper and lower branches, the solid lines) separated by an unstable steady state (on the intermediate branch, the dashed line). This is called a one-parameter bifurcation. Tyson et al., Curr.Opin.Cell.Biol. 15, 221 (2003)

15. 16. Lecture WS 2006/07Bioinformatics III15 mutual inhibition Tyson et al., Curr.Opin.Cell.Biol. 15, 221 (2003) Here, R inhibits E, and E promotes the degradation of R.

16. 16. Lecture WS 2006/07Bioinformatics III16 mutual inhibition: toggle switch This bifurcation is called toggle switch („Kippschalter“): if S is decreased enough, the switch will go back to the off-state. For intermediate stimulus strengh (S crit1 < S < S crit2 ), the response of the system can be either small or large, depending on how S was changed. This is often called „hysteresis“. Examples: lac operon in bacteria, activation of M-phase promoting factor in frog egg extracts, and the autocatalytic conversion of normal prion protein to its pathogenic form. Tyson et al., Curr.Opin.Cell.Biol. 15, 221 (2003)

17. 16. Lecture WS 2006/07Bioinformatics III17 Negative feedback: homeostasis Tyson et al., Curr.Opin.Cell.Biol. 15, 221 (2003) In negative feedback, the response counteracts the effect of the stimulus. Here, the response element R inhibits the enzyme E catalyzing its synthesis. Therefore, the rate of production of R is a sigmoidal decreasing function of R. Negative feedback in a two-component system X  R  | X can also exhibit damped oscillations to a stable steady state but not sustained oscillations.

18. 16. Lecture WS 2006/07Bioinformatics III18 Negative feedback: oscillatory response Tyson et al., Curr.Opin.Cell.Biol. 15, 221 (2003) There are two ways to close the negative feedback loop: (1) RP inhibits the synthesis of X (2) RP activates the degradation of X. Sustained oscillations require at least 3 components: X  Y  R  |X Left: example for a negative-feedback control loop.

19. 16. Lecture WS 2006/07Bioinformatics III19 Negative feedback: oscillatory response Feedback loop leads to oscillations of X (black), YP (red), and RP (blue). Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003) Within the range S crit1 < S < S crit2, the steady-state response R P,ss is unstable. Within this range, R P (t) oscillates between R Pmin and R Pmax. Again, S crit1 and S crit2 are bifurcation points. The oscillations arise by a generic mechanism called „Hopf bifurcation“. Negative feedback has ben proposed as a basis for oscillations in protein synthesis, MPF activity, MAPK signaling pathways, and circadian rhythms.

20. 16. Lecture WS 2006/07Bioinformatics III20 Positive and negative feedback: Activator-inhibitor oscillations R is created in an autocatalytic process, and then promotes the production of an inhibitor X, which speeds up R removal. Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003) The classic example of such a system is cyclic AMP production in the slime mold. External cAMP binds to a surface receptor, which stimulates adenylate cyclase to produce and excrete more cAMP. At the same time, cAMP-binding pushes the receptor into an inactive form. After cAMP falls off, the inactive form slowly recovers its ability to bind cAMP and stimulate adenylate cyclase again.

21. 16. Lecture WS 2006/07Bioinformatics III21 Substrate-depletion oscillations X is converted into R in an autocatalytic process. Suppose at first, X is abundant and R is scarce. As R builds up, the production of R accelerates until there is an explosive conversion of the entire pool of X into R. Then, the autocatalytic reaction shuts off for lack of substrate, X. R is degraded, and X must build up again. This is the mechanism of MPF oscillations in frog egg extract. Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003)

22. 16. Lecture WS 2006/07Bioinformatics III22 Complex networks All the signal-response elements just described, buzzers, sniffers, toggles and blinkers, usually appear as components of more complex networks. Example: wiring diagram for the Cdk network regulating DNA synthesis and mitosis. The network involving proteins that regulate the activity of Cdk1-cyclin B heterodimers consists of 3 modules that oversee the - G1/S - G2/M, and - M/G1 transitions of the cell cycle. Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003)

23. 16. Lecture WS 2006/07Bioinformatics III23 Cell cycle control system Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003)

24. 16. Lecture WS 2006/07Bioinformatics III24 Cell cycle control system Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003) The G1/S module is a toggle switch, based on mutual inhibition between Cdk1-cyclin B and CKI, a stoichiometric cyclin-dependent kinase inhibitor.

25. 16. Lecture WS 2006/07Bioinformatics III25 Cell cycle control system Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003) The G2/M module is a second toggle switch, based on mutual activation between Cdk1-cyclinB and Cdc25 (a phosphotase that activates the dimer) and mutual inhibition between Cdk1-cyclin B and Wee1 (a kinase that inactivates the dimer).

26. 16. Lecture WS 2006/07Bioinformatics III26 Cell cycle control system Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003) The M/G1 module is an oscillator, based on a negative-feedback loop: Cdk1-cyclin B activates the anaphase-promoting complex (APC), which activates Cdc20, which degrades cyclin B. The „signal“ that drives cell proliferation is cell growth: a newborn cell cannot leave G1 and enter the DNA synthesis/division process (S/G2/M) until it grows to a critical size.

27. 16. Lecture WS 2006/07Bioinformatics III27 Cell cycle control system The signal-response curve is a plot of steady-state activity of Cdk1-cyclin B as a function of cell size. Progress through the cell cycle is viewed as a sequence of bifurcations. A very small newborn cell is attracted to the stable G1 steady state. As it grows, it eventually passes the saddle-point bifurcation SN 3 where the G1 steady state disappears. The cell makes an irreversible transition into S/G2 until it grows so large that the S/G2 steady state disappears, giving way to an infite period oscillation (SN/IP). Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003) Cyclin-B-dependent kinase activity soars, driving the cell into mitosis, and then plummets, as cyclin B is degraded by APC–Cdc20. The drop in Cdk1–cyclin B activity is the signal for the cell to divide, causing cell size to be halved from 1.46 to 0.73, and the control system is returned to its starting point, in the domain of attraction of the G1 steady state.

28. 16. Lecture WS 2006/07Bioinformatics III28 Signal Transduction (II) Study receptor-stimulated kinase/phosphotase signaling cascades as a model case of signal transduction networks. - amplitude of the signal output - rate and duration of signaling

29. 16. Lecture WS 2006/07Bioinformatics III29 Signal Transduction (II) To simplify the analysis, we first consider a simple linear signaling cascade in which stimulation of a receptor leads to the consecutive activation of several downstream protein kinases (Figure 1). The signal output of this pathway is the phosphorylation of the last kinase which, in turn, can elicit a cellular response (e.g., activation of a transcription factor). Signaling is terminated by phosphatases, which dephosphorylate the kinases, by inactivation of the receptor, which can involve receptor dephosphorylation, internalization of the receptor ligand complex, and/or degradation of the receptor or ligand. This general scheme is representative of many signaling pathways. E.g., growth factors such as EGF, PDGF, or NGF stimulate a receptor tyrosine kinase (RTK), which leads to the activation of three or four consecutive downstream kinases (e.g., Raf, MEK, ERK, and RSK). Growth factor signals are terminated by protein-tyrosine phosphatases, RTK endocytosis and degradation, protein serine-threonine phosphatases, and dual-specificity and tyrosine-specific MAP kinase phosphatases. The same type scheme can model pathways that include lipid kinases, such as PI3K, whose reaction products help to activate downstream kinases such as PDK1 and Akt.

30. 16. Lecture WS 2006/07Bioinformatics III30 Linear Signalling Cascades Describe each phosphorylation step as a reaction between the phosphorylated form X i-1 of kinase i – 1 and the non-phosphorylated form X i of a downstream kinase i. Phosphorylation rate: second-order rate constant (2 proteins involved): ~ It is assumed that the concentration of each kinase-substrate complex is small compared with the total concentration of the reaction partners. Assuming that the concentration of active phosphatase is constant, dephosphorylation can be modeled as a first order reaction:

31. 16. Lecture WS 2006/07Bioinformatics III31 Linear Signalling Cascades For all but the first activated kinase in the pathway, the concentration of each activated kinase i as a function of time, X i (t) is: Defineas the total concentration of kinase i and as a pseudo-first order tate constant. The above equation becomes For the first kinase (X 1 ), activation occurs via the stimulated receptor, and inactivation is mediated by phosphatase 1. Therefore, we have instead

32. 16. Lecture WS 2006/07Bioinformatics III32 Linear Signalling Cascades Assume that all pathway components are inactive at the start and the undergo rapid stimulation by setting the concentration of active receptor at t = 0 to R. Model receptor inactivation by 3 key questions are: (1) How fast does the signal arrive at its destination? (2) How long does the signal last? (3) How strong is the signal?

33. 16. Lecture WS 2006/07Bioinformatics III33 Signalling time The signalling time  i is the average time to activate kinase i. I i, the integrated response of X i is the total amount of active kinase i generated during the signaling period. I i corresponds to the area under the curve of X i versus time.

34. 16. Lecture WS 2006/07Bioinformatics III34 Signal duration The signal duration is given by i gives a measure of how extended the signaling response is around the mean time. The signaling amplitude S i is given by In a geometric representation, S i is the height of a rectangle whose length is 2 i and whose area equals the area under the curve X i (t).

35. 16. Lecture WS 2006/07Bioinformatics III35 Weakly activated pathways A pathway is termed „weakly activated“ if all its component kinases are phosphorylated to a low degree (X i << C i ). This may occur when the concentration of activated receptor is low, when the receptor is rapidly inactivated, and/or when the kinases are present at high concentrations. and the key parameters can be calculated explicitly. E.g. S depends on the kinetic properties of all pathway components.

36. 16. Lecture WS 2006/07Bioinformatics III36 Weakly activated pathways High signal amplitudes are obtained with fast kinases and slow phosphatases. However, the kinase rate constants have a stronger effect that the phosphatase rate constants. Thus, in a weakly activated pathway, kinases regulate only signal amplitude, not signaling time or duration. In contrast, phosphatases affect all of these parameters, and in the same direction.

37. 16. Lecture WS 2006/07Bioinformatics III37 Amplification and dampening At any step in a signaling cascade, a signal can be amplified (S i > S i-1 ), dampened (S i < S i-1 ), or remain constant. Amplification at step i will occur if

38. 16. Lecture WS 2006/07Bioinformatics III38 Amplification and dampening

39. 16. Lecture WS 2006/07Bioinformatics III39 Summary Kinetic modelling of signal transduction networks by systems of coupled ordinary differential equations: - challenge consists of deriving the appropriate sets of equations - solving their time-dependent behavior is routine when done numerically, may be challenging if done analytically. Other approaches for „data-driven“ modelling of signal-transduction networks: - clustering - principal component analysis - partial least squares regression See e.g. Janes & Yaffe, Nat. Rev. Mol. Cell. Biol. 7, 820 (2006)