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2. What is the point of a signal transduction pathway? to transmit a signal within a cell, bewteen organelles, or between cells to amplify, process and integrate information from the extracellular environment to the rest of the cell To transduce a signal

4. Why are they so complicated? Multiple levels of regulation to allow for regulation and integration of different signals being recieved Redundancy, crosstalk, regulation of speed and magnitude of response to a stimulus

5. What I want you to get out of this: • 1. Signal processing is just as important as signal transduction • 2. Counterintuitive behavior can arise from simple systems. • 3. Toy Models (and simulations) can aid your intuition. • 4. A little bit of math can tell you a lot about a system

6. A simple system The hydrogen atom of signaling. Let’s start here with a simple system and then use this to learn approaches and principles. Stimulus mRNA

7. Stimulus is added at time point zero. Sketch what you think the mRNA abundance over time will look like (your curve should go through the two data points in red). Steady-state This is the point where The synthesis and degradation Rates are matched Time scale (how long does it take to get “half way” there

8. Steady-state This is the point where The synthesis and degradation Rates are matched Time scale (how long does it take to get “half way” there

9. What controls the level of mRNA? • Synthesis Rate: • Amount of polymerase • Size and length of stimulus • ATP concentration, salt, etc … • Degradation Rate: • mRNA levels • Nuclease, salts ,etc.

10. What do we need to follow if we want to “model/understand” the system? • Only the things that change on the same time scale. • Side Note: • When of the most important parts of modeling is it actually makes us think carefully about what we know, what we don’t know, and what we need to measure better to be able to separate between different types of models

11. Representing our system: simbiology to simulate Pictoral representations synthesis degradation mRNA

14. 3 different synthesis rates Synthesis rate ONLY effect steady-state

15. 3 different degradation rates Degradation effects both steady-state and time scale

16. Even this simple system can have counter intuitive behavior Correct answer

17. How to write a differential equation Pictoral representations synthesis degradation mRNA d(mRNA) dt = Equation synthesis * degradation(mRNA) This just means: how does the mRNA level change at a given moment in time

18. Generic Method k1 k2 A + B C D k-1 One equation for each species (eg each of the letter in your system) One term for each arrow that points towards or away from a letter If the arrow points toward it gets a positive sign; if it points away it gets a negative sign This is multipled by the rate of the arrow (usually written above the arrow) Finally all the species that are at the BACK side of the arrow are multiplied together (if there are none don’t write anything) dC/dt = + k1 * A * B - k-1 * C - k2 * C dD/dt= k2 * C

19. mRNA sythesis and destruction d(mRNA)/dt = a – b*mRNA Steady state d(mRNA)/dt = a – b*mRNA = 0 mRNA = a/b Kinetics mRNA(t) = a/b( 1 – e-b*t) Time scale only depends on b !

20. d(ES)/dt = k1(E)(S) - k-1(ES) – k2(ES)

21. What happens in a chained chemical reaction: think metabolic pathway What will happen to the steady-state rate of production of C if we lower the concentration of E2 two-fold? A. It will increase B. It won't change C. It will decrease D. It depends

22. Kinetic analysis of molecular pathways A. Flux conservation in linear pathways

23. Kinetic analysis of molecular pathways B. Flux diversion

24. Finishing enzyme rates E S -> P d(ES)/dt = k1(E)(S) - k-1(ES) – k2(ES) dP/dt= k2 * (ES) FAST SLOW Separation of time scales – the quick steps will quickly reach equilibrium

25. Separation of time scales • What is the distance between me and my friend? • We both start in San Francisco and go to Boston • My friend take a plane. I walk. • After 1 day you only really need to know where I am to know the distance between us.

26. Finishing enzyme rates E S -> P d(ES)/dt = k1(E)(S) - k-1(ES) – k2(ES) dP/dt= k2 * (ES) FAST SLOW d(ES)/dt = k1(E)(S) - k-1(ES) – k2(ES) = 0 ES * (k-1+k2) = k1(E)(S) ES = k1(E)(S) / (k-1+ k2) Not that useful because E is an unknown (free enzyme concentration)

27. Some more math ES * (k-1+k2) = k1(E)(S) = k1(ET - ES)(S) ET = ES + E E = ET- ES ES * [(k-1+k2)+ k1(S)] = k1ET (S) ES= k1ET (S) / [(k-1+ k2)+ k1(S)] Divide top and bottom by k1 ES = ET(S) / [(k-1+ k2)/k-1+ (S)] Km=(k-1+ k2)/k-1 ES = ET(S) / [Km+ (S)]

28. Two regimes Km>>S ES = ET (S) / Km = a * S ES = ET(S) / [Km+ (S)] Linear range of enzyme S>>Km ES = ET Saturated enzyme

29. In the linear regime At steady state: Equation dA = a – E1*A A = a/E1 dB = E1*A – E2*B B = E1*A/E2 = a/E2 dC = E2*B – E3*C C = E2*B/E3 = a/E3 All the concentrations only depend on a!

30. Wnt signaling is central to stem cell self-renewal but remains poorly understood bCat bCat bCat bCat bCat No Wnt Wnt bCat bCat bCat TCF TCF bCat deg. deg.

31. The core mechanism of b-catenin stabilization by Wnt action is hotly debated U U CK1a GSK3 bTrCP U b-cat b-cat b-cat b-cat • Inhibited (Amit et al., 2002) • Not inhibited (Liu 2002, Li et al. 2012) P P P P P Synthesis Degradation P P P P • Inhibited (Li et al., 2012) Axin/APC • Inhibited (Cselenyi 2008; Piao 2008; Wu et al., 2009; Taelman 2010) • Not inhibited (Li et al., 2012) • Other mechanisms: • Sequestration of Axin1 (Mao 2001) • Axin1 degradation (Mao 2001, Lee 2003) Hernandez*, Klein* and Kirschner, Science 2012

32. The core mechanism of b-catenin stabilization by Wnt action is hotly debated U U CK1a GSK3 bTrCP U b-cat b-cat b-cat b-cat P P P P P Synthesis Degradation P P P P Axin/APC Hernandez*, Klein* and Kirschner, Science 2012

33. Kinetic analysis of molecular pathways A. Flux conservation in linear pathways

34. Response of b-catenin to Wnt stimulation involves a transition between two steady-states transient state initial steady state new steady state Hernandez*, Klein* and Kirschner, Science 2012

35. Kinetic analysis of molecular pathways B. Flux diversion

36. What if regulation is upstream and downstream? Hernandez*, Klein* and Kirschner, Science 2012

37. Kinetic analysis reveals the points of Wnt action b-catenin pT41/S37/S33 b-catenin Hernandez*, Klein* and Kirschner, Science 2012

38. Bacterial chemotaxis • If bacteria sense increasing ligand they swim straight • If they sense decreasing ligand they turn a random direction. • Able to chemotax up a gradient of many orders of magnitude. How?

39. Ligand + receptor isn’t very good k1 L + R LR k-1 L = ligand R = Receptor LR – is the complex and active species Look familiar? This won’t be very responsive

40. Actual System

41. That didn’t help

42. Magic

43. d(RmL + Rm) = Vm * ( (R + RL)/((R + RL) + Km) – VD * ( (Rm+ RmL)/((Rm+ RmL) + KD) Saturated methylase d(RmL + Rm) = Vm * (1) – VD * ( (Rm+ RmL)/((Rm+ RmL) + KD) Demethylase only works on RmL d(RmL + Rm) = Vm * (1) – VD * ( (RmL)/(RmL+ KD) = 0 RmL = Vm * KD/(VD-Vm) Ligand independent!