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Bistability in Biochemical Signaling Models Eric Sobie Pharmacology and Systems Therapeutics Mount Sinai School of Medicine eric.sobie@mssm.edu 1

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Outline: Part 1 Some biological background Biological importance of bistability Qualitative requirements of bistability How to predict if bistability will be present? Rate-balance plots Dynamical systems analyses Examples of bistability An artificial genetic "toggle switch" MAP-kinase pathway in oocyte maturation MAP-kinase pathway in mammalian cells 2

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What is bistability? A situation in which two possible steady-states are both stable. In general, these correspond to a "low activity" state and a "high activity" state. Between stable steady-states, an unstable steady state will be present. Ferrell & Machleder (1998) Science 280:895-898. 3

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Biology: generally monostable and analog Epinephrine increased cardiac output Increased blood glucose insulin production Enzyme conversion of substrate to product stimulus response [epinephrine] [blood glucose] [E] Cardiac output [insulin] d [Product]/ dt Response depends directly on level of stimulus When stimulus is removed, response returns to prior level 4

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When is analog not good enough? For these processes, a graded response is inadequate These phenomena also require persistence Fertilization Cell division Differentiation Action potentials Apoptosis T-cell activation 5

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At the biochemical level, how does bistability arise? Mutual activation Mutual inhibition These types of circuits CAN produce bistability, but they do NOT guarantee bistability This is why we need quantitative analyses Ferrell (2002) Curr. Op. Cell Biol. 14:140–148. 6

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Bistability in terms of dynamical behavior Previously we encountered stable and unstable fixed points and limit cycles K m = 20K m = 13 [G] [ATP] The fixed point is “unstable.” The oscillation is a “stable limit cycle.” The fixed point is “stable.” 7

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Bistability in terms of dynamical behavior A monostable system A bistable system Time [LacY] Time [LacY] Multiple steady states are possible Initial conditions determine which steady state is reached 8

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Quantitative analyses of bistability 1) A simple "Michaelian" system A* = phosphorylated A Total amount of [A] is constant: We want to solve for [A*] in the steady-state or 9

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Rate balance plots Instead of solving equations, find solution graphically At steady-state, FR = BR Forward Rate Backward Rate [A*]/[A] TOTAL Rate Forward Backward analysis based on Ferrell & Xiong (2001) Chaos 11:227-236. 10.50 10

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Rate balance plots Very closely related to phase line plots At steady-state, FR – BR = 0 Intuitively, then, this fixed point is stable [A*]/[A] TOTAL -6 -4 -2 0 2 4 0.5 1 d[A*]/dt 11

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Stability analysis of ODE systems A one-dimensional example Isolated cardiac myocyte Beginning with a myocyte at rest (-85 mV), simulate instantaneous changes in voltage, calculate I ion V (mV) dV/dt (mVms) -100-90-80-70-60 -4 -2 0 2 4 6 8 10 Change to < -85 positive dV/dt Change to ~-70 negative dV/dt Change to >~-58 positive dV/dt Stable fixed point Unstable fixed point 12

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Rate balance plots Now, assume that forward rate is function of stimulus: Plot rate balance for different values of stimulus [S] [A*]/[A] TOTAL Rate Big [S] Backward Small [S] 10.50 13

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Rate balance plots This analysis can be used to plot [S] versus [A*]/[A] TOTAL [A*]/[A] TOTAL Rate [A*]/[A] TOTAL [S] 0102030 0 0.5 1 1 0 14

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Rate balance plots 2) Michaelian system with linear feedback k f determines strength of feedback Strong feedback Weak feedback [A*]/[A] TOTAL Rate [A*]/[A] TOTAL Rate The right plot "looks" bistable. Is it? Forward Backward 1 0 10 15

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How can the "off" state be made stable? Rate Strong feedback [A*]/[A] TOTAL Forward Backward Two ways this can be modified to be truly bistable 1) Nonlinear ("ultrasensitive") feedback 2) Partial saturation of the back reaction 1 0.50 16

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Rate balance plots 3) Michaelian system with ultrasensitive feedback Now we have a bona fide bistable system Rate [A*]/[A] TOTAL 1 0.50 17

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Rate balance plots 3) Michaelian system with ultrasensitive feedback Effects of changes in hill exponent n Rate [A*]/[A] TOTAL 1 0.50 n = 2 n=4 n = 8 18

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Rate balance plots 4) Linear feedback plus saturating back reaction Rate [A*]/[A] TOTAL 1 0.50 19

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Rate balance plots How can the cell change states? Vary the amount of stimulus [S] Last couple of slides assumed [S]=0 00.51 0 1 0 0.3 0.6 0.9 Rate [A*]/[A] TOTAL [S] [A*]/[A] TOTAL Where the system switches between 3 and 1 steady states is a bifurcation 20

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Switching can be reversible or irreversible In either case, bistability implies hysteresis Ferrell (2002) Curr. Op. Cell Biol. 14:140–148. 21

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Analysis of two variable systems 1) Generic example of mutual activation Tyson (2003) Curr. Op. Cell Biol. 15:221-231 Dynamical systems theory and nullclines [S] Steady state [R] low initial [R] high initial [R] 22

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Analysis of two variable systems 2) Generic example of mutual repression Tyson (2003) Curr. Op. Cell Biol. 15:221-231 [S] Steady state [R] low initial [R] high initial [R] 23

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Analysis of two variable systems In 2D phase plane, direction determined by: Plot direction vectors in the Bier model Consider [ATP] big; [G] big: d[G]/dt = 0 d[ATP]/dt = 0 04812 0 4 8 16 [ATP] [Glucose] Each time you cross a nullcline, one of these changes direction! The system will proceed in a clockwise direction (stability is unclear) Nullclines divide the phase space into discrete regions 24

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Analysis of two variable systems What do the nullclines look like? 2) Generic example of mutual repression R nullcline E nullcline 00.51 0 5 10 15 20 25 [E] [R] d[R]/dt = 0 d[E]/dt = 0 [S]=2 [S]=6 [S]=10 Transition from 3 intersections to 1 intersection = bifurcation 25

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Examples of bistable systems An artificial "toggle switch" Gardner, Cantor, & Collins (2000) Nature 339-342 26

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Examples of bistable systems MAPK cascade in oocyte maturation Ferrell & Machleder (1998) Science 280:895-898 27

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Examples of bistable systems MAPK cascade in mammalian cells Bhalla & Iyengar (1999) Science 283:381-387 28

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Part 2 29

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Outline: Part 2 Review of important ideas from Tuesday Concepts underlying rate balance plots How these plots were generated in MATLAB Assessing stability in two-dimensional systems The mathematically rigorous method A qualitative graphical method The E. coli lac operon Basic biology A simple model Homework assignment 30

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At the biochemical level, how does bistability arise? Mutual activation Mutual inhibition These types of circuits CAN produce bistability, but they do NOT guarantee bistability This is why we need quantitative analyses Ferrell (2002) Curr. Op. Cell Biol. 14:140–148. 31

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Rate balance plots Now, assume that forward rate is function of stimulus: Plot rate balance for different values of stimulus [S] [A*]/[A] TOTAL Rate Big [S] Backward Small [S] 10.50 32

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Rate balance plots 3) Michaelian system with ultrasensitive feedback Now we have a bona fide bistable system Rate [A*]/[A] TOTAL 1 0.50 33

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Rate balance plots How can the cell change states? Vary the amount of stimulus [S] Last couple of slides assumed [S]=0 00.51 0 1 0 0.3 0.6 0.9 Rate [A*]/[A] TOTAL [S] [A*]/[A] TOTAL Where the system switches between 3 and 1 steady states is a bifurcation 34

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Switching can be reversible or irreversible In either case, bistability implies hysteresis Why is hysteresis beneficial? Ferrell (2002) Curr. Op. Cell Biol. 14:140–148. 35

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Analysis of two variable systems 2) Generic example of mutual repression Tyson (2003) Curr. Op. Cell Biol. 15:221-231 [S] Steady state [R] low initial [R] high initial [R] 36

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Analysis of two variable systems 2) Generic example of mutual repression [S] Steady state [R] low initial [R] high initial [R] Time course of [R] at different values of [S] 01020304050 0 5 10 15 20 25 30 [R] time [S]=2 [S]=12 [S]=6 Initial conditions determine which steady state is reached 37

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Analysis of two variable systems 00.51 0 5 10 15 20 25 [E] [R] R nullcline E nullcline [S]=6 2) Generic example of mutual repression [R] vs. time at [S] = 6 01020304050 2 4 6 8 10 12 14 [R] Time [R] 0 = 5.2; [E] 0 = 0.5 [R] 0 = 5.4; [E] 0 = 0.3 38

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Analysis of two variable systems What do the nullclines look like? 2) Generic example of mutual repression R nullcline E nullcline 00.51 0 5 10 15 20 25 [E] [R] R nullcline E nullcline [S]=2 [S]=6 [S]=10 How do we tell if the fixed points are stable or unstable? 39

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Analysis of two variable systems On the E nullcline, dE/dt = 0, direction of movement is up/down With these simple rules, we can (often) determine stability R nullcline E nullcline 00.51 0 5 10 15 20 25 [E] [R] A qualitative, graphical analysis of stability On the R nullcline, dR/dt = 0, direction of movement is left/right [dE/dt,dR/dt] defines a direction of "movement" in the [E,R] plane The direction changes when a nullcline is crossed 40

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Graphical analysis of stability On the E nullcline, and above the R nullcline, dR/dt < 0 These considerations suggest that middle steady state is unstable, left and right steady states are stable R nullcline E nullcline 00.51 0 5 10 15 20 25 [E] [R] We plot dR/dt on E nullcline We plot dE/dt on R nullcline Examine equations to determine directions (Remember that R is the ordinate) On the R nullcline, and above the E nullcline, dE/dt < 0 41

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Graphical analysis of stability R nullcline E nullcline [E] [R] Near middle, unstable SS Using the MATLAB "quiver" function Near right, stable SS Epos = linspace(Erange(1),Erange(2),quiverpoints) ; Rpos = linspace(Rrange(1),Rrange(2),quiverpoints) ; [Equiv,Rquiv] = meshgrid(Epos,Rpos) ; dR = k0r*(Etot-Equiv) + k1r*S - k2r*Rquiv ; dE = -k2e*Rquiv.*Equiv./(Km2e + Equiv) + k1e*(Etot - Equiv)./... (Km1e + Etot - Equiv) ; quiver(Equiv,Rquiv,dE,dR,1.2) [E] [R] 42

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Stability analysis of ODE systems How can we understand stable and unstable fixed points mathematically? Evaluate this at the fixed points defined by [E*], [R*] Compute the “Jacobian” matrix: (This is where analytical computations become difficult) 43

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Stability analysis of ODE systems Evaluate Jacobian matrix at the fixed points defined by [E*], [R*] The eigenvalues of the Jacobian (at the fixed points) determine stability The real part of either is positive: the fixed point is unstable Real parts of both are negative: the fixed point is stable 00.51 0 5 10 15 20 25 [E] [R] [S]=6 repression_stability.m computes these eigenvalues = -59.1985 -0.0983 eigenvalues = -1.0282 0.3379 eigenvalues = -21.0618 -0.0915 44

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Examples of bistable systems MAPK cascade in oocyte maturation Ferrell & Machleder (1998) Science 280:895-898 45

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Examples of bistable systems MAPK cascade in mammalian cells Bhalla & Iyengar (1999) Science 283:381-387 46

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Examples of bistable systems The lac operon in E. coli Smits et al. (2006) Nat. Rev. Micro. 4:259-271 With low nutrient levels, LacI will repress transcription of the the LacA, LacY, and LacZ genes. Lactose, allolactose, or IPTG will bind to LacI, relieve repression. LacY encodes a "permease,” which allows lactose into the cell. 47

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Examples of bistable systems A minimal model of the lac operon l = intracellular lactose LacY = expression of LacY/permease β, γ, δ, σ, p, l 0 = constants l ext = external lactose (Note: in most models, dLacY/dt depends on [lactose] 2. We have assumed a dependence on [lactose] 4 to improve the nullcline plots.) 48

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Homework assignment 1) Rate balance plots Linear autocatalytic feedback & saturation of back reaction Rate [A*]/[A] TOTAL K mb [A*]/[A] TOTAL 2) Model of lac operon Nullcline analysis and dynamic simulations [lactose] [LacY] [lactose] Time 49

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www.sciencesignaling.org Slides from a lecture in the course Systems Biology—Biomedical Modeling Citation: E. A. Sobie, Bistability in biochemical signaling models. Sci. Signal. 4, tr10 (2011).

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