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1 Modelling Cell Signalling Pathways in PEPA Muffy Calder Department of Computing Science University of Glasgow Jane Hillston and Stephen Gilmore Laboratory for Foundations of Computer Science University of Edinburgh February 2005

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2 Cell Signalling or Signal Transduction * fundamental cell processes (growth, division, differentiation, apoptosis) determined by signalling most signalling via membrane receptors signalling molecule receptor gene effects * movement of signal from outside cell to inside

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3 A little more complex.. pathways/networks

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5 RKIP Inhibited ERK Pathway proteins/complexes forward /backward reactions (associations/disassociations) products (disassociations) m1, m2.. concentrations of proteins k1,k2..: rate (performance) coefficients m1 Raf-1* m2 k1 m3 Raf-1*/RKIP m12 MEK k12/k13 m7 MEK-PP k6/k7 m5 ERK m8 MEK-PP/ERK-P k8 m9 ERK-PP k3 m4 k5 m6 RKIP-P m10 RP k9/k10 m11 RKIP-P/RP k11 m2 k1 m3 k3 Raf-1*/RKIP/ERK-PP m2 RKIP k1/k2 m3 K3/k4 k15 m13 k14 From paper by Cho, Shim, Kim, Wolkenhauer, McFerran, Kolch, 2003.

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6 RKIP Inhibited ERK Pathway Pathways have computational content! Producers and consumers. Feedback. m1 Raf-1* m2 k1 m3 Raf-1*/RKIP m12 MEK k12/k13 m7 MEK-PP k6/k7 m5 ERK m8 MEK-PP/ERK-P k8 m9 ERK-PP k3 m4 k5 m6 RKIP-P m10 RP k9/k10 m11 RKIP-P/RP k11 m2 k1 m3 k3 Raf-1*/RKIP/ERK-PP m2 RKIP k1/k2 m3 k3/k4 k15 m13 k14

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7 RKIP Inhibited ERK Pathway Why not use process algebras for modelling? High level formalisms that make interactions and event rates explicit. m1 Raf-1* m2 k1 m3 Raf-1*/RKIP m12 MEK k12/k13 m7 MEK-PP k6/k7 m5 ERK m8 MEK-PP/ERK-P k8 m9 ERK-PP k3 m4 k5 m6 RKIP-P m10 RP k9/k10 m11 RKIP-P/RP k11 m2 k1 m3 k3 Raf-1*/RKIP/ERK-PP m2 RKIP k1/k2 m3 k3/k4 k15 m13 k14

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8 Process algebra (for dummies) High level descriptions of interaction, communication and synchronisation Event Prefix .P ChoiceP1 + P2 Synchronisation P1 |l| P2¬( l) independent concurrent (interleaved) actions l synchronised action ConstantA = Passign names to components Relations (bisimulation ) LawsP1 + P2 P2 + P1 a bc a aa a a c b b bc

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9 PEPA Markovian process algebra invented by Jane Hillston with workbench by Stephen Gilmore. PEPA descriptions denote continuous Markov chains. Prefix( , ).P ChoiceP1 + P2competition between components (race) Cooperation/ P1 |l| P1¬(a l) independent concurrent (interleaved) actions Synchronisation a l shared action, at rate of slowest Constant A = Passign names to components is a rate, from which a probability is derived - exponential distribution.

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10 Modelling the ERK Pathway in PEPA Each reaction is modelled by an event, which has a performance coefficient. Each protein is modelled by a process which synchronises others involved in a reaction. (reagent-centric view) Each sub-pathway is modelled by a process which synchronises with other sub-pathways. (pathway-centric view)

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11 Signalling Dynamics m1 P1 m2 P2 k1/k2 m5 P5 K6/k7 m6 P6 m4 P5/P6 Reaction Producer(s) Consumer(s) k1react {P2,P1} {P1/P2} k2react {P1/P2} {P2,P1} k3product {P1/P2} {P5} … k1react will be a 3-way synchronisation, k2react will be a 3-way synchronisation, k3product will be a 2-way synchronisation. k4 m3 k3 P1/P2

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12 Reagent View m1 P1 m2 P2 k1/k2 m5 P5 k6/k7 m6 P6 m4 P5/P6 Model whether or not a reagent can participate in a reaction (observable/unobservable): each reagent gives rise to a pair of definitions. P1 H = (k1react,k1). P1 L P1 L = (k2react,k1). P2 H P2 H = (k1react,k1). P2 L P2 L = (k2react,k2). P2 H + (k4react). P2 H P1/P2 H = (k2react,k2). P1/P2 L + (k3react, k3). P1/P2 L P1/P2 L = (k1react,k1). P1/P2 H P5 H = (k6react,k6). P5 L + (k4react,k4). P5 L P5 L = (k3react,k3). P5 H +(k7react,k7). P5 H P6 H = (k6react,k6). P6 L P6 L = (k7react,k7). P6 H P5/P6 H = (k7react,k7). P5/P6 L P5/P6 L = (k6react,k6). P5/P6 H k4 m3 k3 P1/P2

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13 Reagent View m1 P1 m2 P2 k1/k2 m5 P5 K6/k7 m6 P6 m4 P5/P6 Model configuration P1 H | k1react,k2react | P2 H | k1react,k2react,k4react | P1/P2 L | k1react,k2react,k3react | P5 L | k3react,k6react,k4react | P6 H | k6react,k7react | P5/P6 L Assuming initial concentrations of m1,m2,m6. k4 m3 k3 P1/P2

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14 Reagent view: Raf-1* H = (k1react,k1). Raf-1* L + (k12react,k12). Raf-1* L Raf-1* L = (k5product,k5). Raf-1* H +(k2react,k2). Raf-1* H + (k13react,k13). Raf-1* H + (k14product,k14). Raf-1* H … (26 equations) m1 Raf-1* m2 k1 m3 Raf-1*/RKIP m12 MEK k12/k13 m7 MEK-PP k6/k7 m5 ERK m8 MEK-PP/ERK-P k8 m9 ERK-PP k3 m4 k5 m6 RKIP-P m10 RP k9/k10 m11 RKIP-P/RP k11 m2 k1 m3 k3 Raf-1*/RKIP/ERK-PP m2 RKIP k1/k2 m3 k3/k4 k15 m13 k14

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15 Reagent View model configuration Raf-1* H | k1react,k12react,k13react,k5product,k14product | RKIP H | k1react,k2react,k11product | Raf-1* H /RKIP L | k3react,k4react | Raf-1*/RKIP/ERK-PP L | k3react,k4react,k5product | ERK-P L | k5product,k6react,k7react | RKIP-P L | k9react,k10react | RKIP-P/RP L | k9react,k10react,k11product | RP H || MEK L | k12react,k13react,k15product | MEK/Raf-1* L | k14product | MEK-PP H | k8product,k6react,k7react | MEK-PP/ERK L | k8product | MEK-PP H | k8product | ERK-PP H

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16 Pathway View m1 P1 m2 P2 k1/k2 m5 P5 K6/k7 m6 P6 m4 P5/P6 Model chains of behaviour flow. Two pathways, corresponding to initial concentrations: Path10 = (k1react,k1). Path11 Path11 = (k2react).Path10 + (k3product,k3).Path12 Path12 = (k4product,k4).Path10 + (k6react,k6).Path13 Path13 = (k7react,k7).Path12 Path20 = (k6react,k6). Path21 Path21 = (k7react,k6).Path20 Pathway view: model configuration Path10 | k6react,k7react | Path20 (much simpler!) k4 m3 k3 P1/P2

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17 Pathway view: Pathway10 = (k9react,k9). Pathway11 Pathway11 = (k11product,k11). Pathway10 + (k10react,k10). Pathway10 … (5 pathways) m1 Raf-1* m2 k1 m3 Raf-1*/RKIP m12 MEK k12/k13 m7 MEK-PP k6/k7 m5 ERK m8 MEK-PP/ERK-P k8 m9 ERK-PP k3 m4 k5 m6 RKIP-P m10 RP k9/k10 m11 RKIP-P/RP k11 m2 k1 m3 k3 Raf-1*/RKIP/ERK-PP m2 RKIP k1/k2 m3 k3/k4 k15 m13 k14

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18 model configuration Pathway10 | k12react,k13react,k14product | Pathway40 | k3react,k4react,k5product,k6react,k7react,k8product | Pathway30 | k1react,k2react,k3react,k4react,k5product | Pathway20 | k9react,k10react,k11product | Pathway10 Pathway View

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19 What is the difference between the two views/models? reagent-centric view is a fine grained view pathway-centric view is a coarse grained view –reagent-centric is easier to derive from data –pathway-centric allows one to build up networks from already known components The two models are equivalent! The equivalence proof, based on bisimulation between steady state solutions, unites two views of the same biochemical pathway.

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Reagent view Pathway view

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21 State space of reagent and pathway model

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22 What do you do with these two models? -investigate properties of the underlying Markov model. Generate steady-state probability distribution (using linear algebra) and then perform; -Transient analysis e.g. analysis to determine whether a state will be reached. OR -Steady state analysis (more appropriate here) e.g. analysis of the steady state solution. Note: there isn’t one steady state, but a very large “cycle”!

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23 Quantitative Analysis Effect of increasing the rate of k1 on k8product throughput (rate x probability)i.e. effect of binding of RKIP to Raf-1* on ERK-PP. Increasing the rate of binding of RKIP to Raf-1* dampens down the k8product reactions, i.e. it dampens down the ERK pathway.

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24 Quantitative Analysis – by logic Steady state analysis Formula S=? [ERK_PP_H_STATE = 0] PRISM result (after translation):

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25 Quantitative Analysis – by logic Now reduce backward rates (.5 3 ) Formula S=? [ERK_PP_H_STATE = 0]

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26 Reagent view and ODEs m1 P1 m2 P2 k1/k2 m5 P5 K6/k7 m6 P6 m4 P5/P6 Activity matrix k1 k2k3 k4k5 k6 k7 P P P1/P P P P5/P Column: corresponds to a single reaction. Row: correspond to a reagent; entries indicate whether the concentration is +/- for that reaction. mass action dynamics: dm1 = - k1*m1*m2 + k2*m3 (nonlinear) dt Reagent views tells us producer or consumer. k4 m3 k3 P1/P2

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27 Big Picture Benefits Interactions Relative change Abstraction Behaviour patterns Quantitative analysis stochastic process algebra pathway view reagent view mass action differential equations Continuous time Markov chains abstraction pathway composition throughput analysis denote derive

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28 Bigger Picture Benefits Interactions Relative change Abstraction Behaviour patterns Quantitative analysis stochastic process algebra pathway view Matlab reagent view mass action differential equations multilevel reagent view PRISM Continuous time Markov chains experimental data simulate logic denote derive abstraction pathway composition throughput analysis

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29 Further Challenges Derivation of the reagent-centric model from experimental data Quantification of abstraction over networks –zoom in or out Other dynamics (inhibition) Functional rates Very large scale pathways Model spatial dynamics (vesicles).

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