2. 1 Modelling Biochemical Pathways in PEPA Muffy Calder Department of Computing Science University of Glasgow Joint work with Jane Hillston and Stephen Gilmore October 2004

3. 2 Are you in the right room? Yes, this is computing science! Question Can we apply computing science theory and tools to biochemical pathways? If so, What analysis do these new models offer? How do these models relate to traditional ones? What are the implications for life scientists? What are the implications for computing science?

4. 3 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

5. 4 A little more complex.. pathways/networks

6. 5

7. 6 RKIP Inhibited ERK Pathway 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 k15 m13 k14 From paper by Cho, Shim, Kim, Wolkenhauer, McFerran, Kolch, 2003.

8. 7 RKIP Inhibited ERK Pathway 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 k15 m13 k14 From paper by Cho, Shim, Kim, Wolkenhauer, McFerran, Kolch, 2003.

9. 8 RKIP protein expression is reduced in breast cancers

10. 9 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 k15 m13 k14

11. 10 RKIP Inhibited ERK Pathway This network seems to be very similar to producer/consumer networks. Why not to try using process algebras for modelling? 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 k15 m13 k14

12. 11 Why process algebras for pathways? Process algebras are high level formalisms that make interactions and constraints explicit. Structure becomes apparent. Reasoning about livelocks and deadlocks. Reasoning with (temporal) logics. Equivalence relations between high level descriptions. Stochastic process algebras allow performance analysis.

13. 12 Process algebra (for dummies) High level descriptions of interaction, communication and synchronisation Event  (simple),  !34 (data offer),  ?x (data receipt) Prefix .S ChoiceS + S Synchronisation P |l| P   l independent concurrent (interleaved) actions   l synchronised action ConstantA = Sassign names to components LawsP1 + P2  P2 + P1 Relations  (bisimulation) a bc a aa a a c b b bc  

14. 13 PEPA Process algebra with performance, invented by Jane Hillston Prefix( ,r).S ChoiceS + Scompetition between components (race) Cooperation/ P |l| Pa  l independent concurrent (interleaved) actions Synchronisation a  l shared action, at rate of slowest ConstantA = Sassign names to components P ::= S | P |l| P S ::= ( ,r).S | S+S | A

15. 14 Rates is a rate, from which a probability is derived

16. 15 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)

17. 16 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

18. 17 Modelling Signalling Dynamics There is an important difference between computing science networks and biochemical networks We have to distinguish between the individual and the population. Previous approaches have modelled at molecular level (individual) –Simulation –State space explosion –Relation to population (what can be inferred?)

19. 18 Signalling Dynamics m1 P1 m2 P2 k1/k2 m5 P5 k6/k7 m6 P6 m4 P5/P6 Reagent view: model whether or not a reagent can participate in a reaction (observable/unobservable). k4 m3 k3 P1/P2

20. 19 Signalling Dynamics m1 P1 m2 P2 k1/k2 m5 P5 k6/k7 m6 P6 m4 P5/P6 Reagent view: 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

21. 20 Signalling Dynamics m1 P1 m2 P2 k1/k2 m5 P5 K6/k7 m6 P6 m4 P5/P6 Reagent view: 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

22. 21 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 k15 m13 k14

23. 22 Signalling Dynamics 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

24. 23 Signalling Dynamics m1 P1 m2 P2 k1/k2 m5 P5 K6/k7 m6 P6 m4 P5/P6 Pathway view: model chains of behaviour flow k4 m3 k3 P1/P2

25. 24 Signalling Dynamics m1 P1 m2 P2 k1/k2 m5 P5 K6/k7 m6 P6 m4 P5/P6 Pathway view: 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

26. 25 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 k15 m13 k14

27. 26 Pathway view: model configuration Pathway10 | k12react,k13react,k14product | Pathway40 | k3react,k4react,k5product,k6react,k7react,k8product | Pathway30 | k1react,k2react,k3react,k4react,k5product | Pathway20 | k9react,k10react,k11product | Pathway10

28. 27 What is the difference? 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 Formal proof shows that those two models are equivalent! This equivalence proof, based on bisimulation, unites two views of the same biochemical pathway.

29. 28 state reagent-view s1 Raf-1*H, RKIPH,Raf-1*/RKIPL,Raf-1*/RKIPERK-PPL, ERKL,RKIP-PL, RKIP-P/RPL, RPH, MEKL,MEK/Raf-1*L,MEK-PPH,MEK-PP/ERKL/ERK-PPH pathway view Pathway50,Pathway40,Pathway20,Pathway10 s2 …... s28 (28 states) State space of reagent and pathway model

30. 29 State space of reagent and pathway model

31. 30 Quantitative Analysis Generate steady-state probability distribution (using linear algebra). 1. Use state finder (in reagent model) to aggregate probabilities. Example increase k1 from 1 to 100 and the probability of being in a state with ERK-PP H drops from.257 to.005 2. Perform throughput analysis (in pathway model)

32. 31 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

33. 32 Quantitative Analysis Effect of increasing the rate of k1 on k14product throughput (rate x probability) i.e. effect of binding of RKIP to Raf-1* on MEK-PP

34. 33 Quantitative Analysis - Conclusion Increasing the rate of binding of RKIP to Raf-1* dampens down the k14product and k8product reactions, In other words, it dampens down the ERK pathway.

35. 34 Signalling Dynamics m1 P1 m2 P2 k1/k2 m5 P5 K6/k7 m6 P6 m4 P5/P6 Activity matrix k1 k2k3 k4k5 k6 k7 P1 -1 +1 0 0 0 0 0 P2 -1 +1 0 +1 0 0 0 P1/P2 +1 -1 0 0 0 0 0 P5 0 0 +1 -1 0 -1 +1 P6 0 0 0 0 0 -1 +1 P5/P6 0 0 0 0 0 +1 -1 Column: corresponds to a single reaction. Row: correspond to a reagent; entries indicate whether the concentration is +/- for that reaction. k4 m3 k3 P1/P2

36. 35 Signalling Dynamics m1 P1 m2 P2 k1/k2 m5 P5 K6/k7 m6 P6 m4 P5/P6 Activity matrix k1 k2k3 k4k5 k6 k7 P1 -1 +1 0 0 0 0 0 P2 -1 +1 0 +1 0 0 0 P1/P2 +1 -1 0 0 0 0 0 P5 0 0 +1 -1 0 -1 +1 P6 0 0 0 0 0 -1 +1 P5/P6 0 0 0 0 0 +1 -1 Differential equations Each row is labelled by a protein concentration. One equation per row. For row r, dr =  column c A[r,c]) *  row x f(A[x,c]) dt where f(A[x,c]) = if (A[x,c]== -) then x else 1 a rate is a product of the rate constant and current concentration of substrates consumed. k4 m3 k3 P1/P2

37. 36 Signalling Dynamics m1 P1 m2 P2 k1/k2 m5 P5 K6/k7 m6 P6 m4 P5/P6 Activity matrix k1 k2k3 k4k5 k6 k7 P1 -1 +1 0 0 0 0 0 P2 -1 +1 0 +1 0 0 0 P1/P2 +1 -1 0 0 0 0 0 P5 0 0 +1 -1 0 -1 +1 P6 0 0 0 0 0 -1 +1 P5/P6 0 0 0 0 0 +1 -1 Differential equations (mass action) dm1 = - k1 + k2 (two terms) dt k4 m3 k3 P1/P2

38. 37 Signalling Dynamics m1 P1 m2 P2 k1/k2 m5 P5 K6/k7 m6 P6 m4 P5/P6 Activity matrix k1 k2k3 k4k5 k6 k7 P1 -1 +1 0 0 0 0 0 P2 -1 +1 0 +1 0 0 0 P1/P2 +1 -1 0 0 0 0 0 P5 0 0 +1 -1 0 -1 +1 P6 0 0 0 0 0 -1 +1 P5/P6 0 0 0 0 0 +1 -1 Differential equations (mass action) dm1 = - k1*m1*m2 + k2 dt k4 m3 k3 P1/P2

39. 38 Signalling Dynamics m1 P1 m2 P2 k1/k2 m5 P5 K6/k7 m6 P6 m4 P5/P6 Activity matrix k1 k2k3 k4k5 k6 k7 P1 -1 +1 0 0 0 0 0 P2 -1 +1 0 +1 0 0 0 P1/P2 +1 -1 0 0 0 0 0 P5 0 0 +1 -1 0 -1 +1 P6 0 0 0 0 0 -1 +1 P5/P6 0 0 0 0 0 +1 -1 Differential equations (mass action) dm1 = - k1*m1*m2 + k2*m3 (nonlinear) dt k4 m3 k3 P1/P2

40. 39 Signalling Dynamics m1 P1 m2 P2 k1/k2 m5 P5 K6/k7 m6 P6 m4 P5/P6 Differential equations (mass action) For RKIP inhibited ERK pathway, change in Raf-1* is: k4 m3 k3 P1/P2 dm1 = - k1*m1*m2 + k2*m3 + k5*m4 – k12*m1*m12 dt +k13*m13 + k14*m13 (catalysis, inhibition, etc. )

41. 40 Discussion & Conclusions Regent-centric view –probabilities of states (H/L) –differential equations –fit with data Pathway-centric view –simpler model –building blocks, modularity approach –no further information is gained from having multiple levels. Life science –(some) see potential of an interaction approach Computing science –individual/population view –continuous, traditional mathematics

42. 41 Further Challenges Derivation of the reagent-centric model from experimental data. Derivation of pathway-centric models from reagent-centric models and vice-versa. Quantification of abstraction over networks –“chop” off bits of network Model spatial dynamics (vesicles).

43. 42 The End Thank you.