CSL model checking of biochemical networks with Interval Decision Diagrams

Transcription

1 CSL model checking of biochemical networks with Interval Decision Diagrams Brandenburg University of Technology Cottbus Computer Science Department September 01, 2009

2 Contents 10 mr 10 transl_r deg_c transc_dr deg_mr transc_dr_a dr_a dr r c 10 rel_r bind_r deg_r deactive a 10 rel_a bind_a deg_a transl_a da_a da deg_ma transc_da_a transc_da ma Background IDD-based matrix-free analysis Interval Decision Diagrams Matrix-free approach Implementation - IDD-CSL Static variable order Further techniques Benchmarks Conclusions

3 Stochastic Petri net (SPN) 3way-oscillator model [Ballarini2009] A R 1 : A + B r B A B B + B R 2 : B + C r C B C C + C R 3 : C + A r A C A A + A R3 C R2 R1 B

4 Continuous Time Markov Chain (CTMC) 0,6,0 0,5,1 0,4,2 0,3,3 0,2,4 0,1,5 0,0,6 1,5,0 1,4,1 1,3,2 1,2,3 1,1,4 1,0,5 2,4,0 2,3,1 2,2,2 2,1,3 2,0,4 3,3,0 3,2,1 3,1,2 3,0,3 4,2,0 4,1,1 4,0,2 5,1,0 5,0,1 6,0,0 marking-dependent state transition rates: e.g. (2, 2, 2) r A 2 2 (3, 2, 1)

5 CTMC analysis Transient analysis computation of π(α, τ) uniformisation method Steady state analysis computation of π(α) iterative methods (Jacobi, Gauss-Seidel) Model checking Continuous Stochastic Logic (CSL) can be reduced to transient and steady state analysis

6 CTMC analysis Transient analysis computation of π(α, τ) uniformisation method Steady state analysis computation of π(α) iterative methods (Jacobi, Gauss-Seidel) Model checking Continuous Stochastic Logic (CSL) can be reduced to transient and steady state analysis In any case, basic operation: multiplication of a matrix and a vector over real values.

7 CTMC analysis Problem possibly huge matrix and vector explicit (sparse) matrix representation techniques fail Solutions Kroneker algebra Multi Terminal Binary Decision Diagrams (MTBDD) matrix-free sophisticated swapping (disc-based)

8 CTMC analysis Kronecker based techniques (SMART) require regular models (partitioning of the place set) MTBDD based techniques (PRISM) require prior knowledge of the boundedness degree become inefficient when high amount of variables must be doubled to encode a matrix boundedness degree increases (binary encoding) #(BDD variables) increases #(different matrix entries) increases #(terminal nodes) increases, destroys BDD compression effect

9 CTMC analysis Biochemical network models often non-regular structure accurate analysis requires an high amount of tokens (molecules, concentration levels) many different real-valued matrix entries because of the marking-dependent rate functions

10 CTMC analysis Biochemical network models often non-regular structure accurate analysis requires an high amount of tokens (molecules, concentration levels) many different real-valued matrix entries because of the marking-dependent rate functions Our contribution: a matrix-free technique based on Interval Decision Diagrams (IDD)

11 Interval Decision Diagrams Interval Decision Diagrams Characteristics #(variables) = #(places) = IDD height always two terminal nodes arcs are label with intervals over N intervals of the outgoing arcs of a nonterminal node are a partitioning of N 0 [0, 6) [6, 7) x 1 [8, ) [7, 8) x 2 x 2 [0, 1) [0, 1) [1, ) [1, ) 1

12 Matrix-free approach Matrix-free IDD-based approach Basic ideas no storage of the rate matrix the SPN and its state space (symbolically encoded by an IDD) represent the CTMC CTMC = state space + net structure + rate functions for a state s all state transitions (s,s ) and their rates are computed when needed defined by the firing of the transitions of the net forward (backward) firing means: extracting the rows (columns) of the CTMC

13 Matrix-free approach Matrix-free IDD-based approach n7(p1) [0,1) # 0 [1,2) # 2 [2,3) #4 [3,4) #5 [4,oo) # 6 [2,3) # 0 n3(p3) [1,2) # 0 [3,4) # 1 n4(p3) [2,3) # 1 [3,oo) # 2 [1,2) # 0 n5(p3) [0,1) # 0 [0,1) # 0 n6(p3) 3 t1 p1 2 t2 p2 t3 2 [4,oo) # 2 n1(p2) [2,oo) # 1 [1,2) # 0 [0,1) # 0 [0,1) # 0 n2(p2) [1,oo) # 1 [0,1) #0 [2,oo) # 1 [1,oo) # 1 t4 3 p3 2 t5 [0,2) 1 # 0 0

14 IDD-CSL Basics prototype implemented in C++ command line tool Features implemented so far subset of CSL (time bounded operators) Jacobi-based steady state analysis ordinary Place/Transition nets

15 Static variable order Variable order variable order affects the size of decision diagrams finding an optimal order is NP complete solution: net structure based heuristics; dependent places should be close together W (p) := Σ t p t Q t + Σ t p t Q t p p Q is the set of already used places in each step select p P \ Q such that p P \ Q,p p : W(p) W(p )

16 Further techniques Further techniques Caching (like PRISM) no traversal of the whole IDD introducing a cache layer stopping traversal at the cache layer and processing the cached data (index pairs and rates) problem: memory consumption for the cached data finding the best layer Parallelisation parallelised multiplication partitioning of the state space applying the traversal to the subsets

17 Benchmarks comparison with the probabilistic model checker PRISM 4 2.8GHz Intel XEON, 4 GB RAM, 64bit Linux only runtime maximal runtime 24 h three biochemical networks different levels by multipilying the initial marking with a constant

18 PRISM vs. IDD-CSL IDD-CSL is inspired by PRISM s hybrid engine, there are several Similarities based on Decision Diagrams multiplication by DD traversal, state indices are computed using offsets during the traversal explicit storage of probability vectors and the diagonal entries for IDD-CSL the limiting factor similar caching strategies

19 PRISM vs. IDD-CSL Differences PRISM IDD-CSL

20 PRISM vs. IDD-CSL Differences PRISM IDD-CSL state space BDD IDD

21 PRISM vs. IDD-CSL Differences PRISM IDD-CSL state space BDD IDD rate matrix MTBDD IDD + SPN

22 PRISM vs. IDD-CSL Differences PRISM IDD-CSL state space BDD IDD rate matrix MTBDD IDD + SPN prior knowledge of the boundedness degree yes no

23 PRISM vs. IDD-CSL Differences PRISM IDD-CSL state space BDD IDD rate matrix MTBDD IDD + SPN prior knowledge of the boundedness degree yes no #(variables) depends on the boundedness degree #(places)

24 PRISM vs. IDD-CSL Differences PRISM IDD-CSL state space BDD IDD rate matrix MTBDD IDD + SPN prior knowledge of the boundedness degree yes no #(variables) depends on the boundedness degree #(places) variable order plain heuristics

25 PRISM vs. IDD-CSL Differences PRISM IDD-CSL state space BDD IDD rate matrix MTBDD IDD + SPN prior knowledge of the boundedness degree yes no #(variables) depends on the boundedness degree #(places) variable order plain heuristics rates on-the-fly terminal nodes computation

26 Mitogen Activated Protein Kinase [Levchenko2000] RasGTP Raf_RasGTP k1/k2 k3 Raf 4 RafP k6 k4/k5 Comment RafP_Phase1 MEK_RafP MEKP_RafP Phase1 k7/k8 k9 k12 k10/k11 MEK k18 MEKP k16/k17 k15 MEKPP k13/k14 MEKP_Phase2 MEKPP_Phase2 ERK_MEKPP ERKP_MEKPP Phase2 ERK k21 k19/k20 ERKP k28/k29 k30 k24 k22/k23 ERKPP k25/k26 k27 ERKP_Phase3 ERKPP_Phase3 Phase3

27 CTMC size for different levels levels states edges a) 4 24, , ,110,643 78,948, ,647,600 4,958,809, ,920,337, ,381,517, ,125,763,956 1,689,018,298, ,371,342,640 15,635,976,824, ,084,605,436, ,065,356,604, ,124,071,792, ,236,499,605,178 a) i.e., #(non-zero entries) in rate matrix; For level 520 the MAPK model exhibits 3.40e+30 states (IDD-CTL).

28 Influence of variable order MTBDD size (PRISM) for different levels levels terminal original order good order nodes a) time nodes time nodes a) i.e.,#(different entries) in the matrix;

29 Influence of variable order MTBDD size (PRISM) for different levels levels terminal original order good order nodes a) time nodes time nodes , ,881, a) i.e.,#(different entries) in the matrix; exceeds physical memory;

30 Influence of variable order MTBDD size (PRISM) for different levels levels terminal original order good order nodes a) time nodes time nodes , , ,881, , , , , ,029, ,771, ,015,521 a) i.e.,#(different entries) in the matrix; exceeds physical memory;

31 P >0.0 [F [0,1] RafP = 2], for 8 levels PRISM a) cl b) total c) iter d) IDD-CSL cl 1 thread 2 threads total iter total iter a) using a good variable order, determined by the network structure; b) cache layers; c) includes time for state space construction, initialisation, computation and determining the satisfying states; d) effective probability computation time;

32 P >0.0 [F [1,1] RafP = 2], for 8 levels PRISM a) cl b) total c) iter d) IDD-CSL cl 1 thread 2 threads total iter total iter a) using a good variable order, determined by the network structure; b) cache layers; c) includes time for state space construction, initialisation, computation and determining the satisfying states; d) effective probability computation time;

33 RKIP inhibited ERK pathway [Cho2003] Raf1Star RKIP r1 r2 s9 ERKPP s3 Raf1Star_RKIP r11 r8 r3 r4 s8 s4 s11 RKIPP_RP MEKPP_ERK Raf1Star_RKIP_ERKPP r6 r7 r5 r9 r10 s6 MEKPP ERK RKIPP RP

34 P >0.0 [F [1,1] Raf 1Star = 1] level CTMC size PRISM a) IDD-CSL states edges hybrid sparse 1 thread 2 threads 5 1,974 12, , , ,220 3,213, ,696,618 15,609, ,723,991 54,438, ,721, ,964,146 a) using a good variable order, determined by the network structure;

35 P >0.0 [F [1,1] Raf 1Star = 1] level CTMC size PRISM a) IDD-CSL states edges hybrid sparse 1 thread 2 threads 5 1,974 12, , , ,220 3,213, ,696,618 15,609, ,723,991 54,438, ,721, ,964,146 a) using a good variable order, determined by the network structure; time for initialisation exceeds 24 hours;

36 P >0.0 [F [1,1] Raf 1Star = 1] level CTMC size PRISM a) IDD-CSL states edges hybrid sparse 1 thread 2 threads 5 1,974 12, , , ,220 3,213, ,696,618 15,609, ,723,991 54,438, ,721, ,964,146 3, , a) using a good variable order, determined by the network structure; time for initialisation exceeds 24 hours;

37 Circadian clock [Barkai2000] 10 mr 10 transc_dr deg_mr transc_dr_a transl_r deg_c dr_a rel_r 10 bind_r dr r deg_r deactive 10 c 10 a 10 rel_a bind_a deg_a transl_a da_a da deg_ma transc_da_a transc_da ma For the experiments an ordinary SPN was used, the capacities being modelled by complementary places.

38 P >0.0 [F [1,1] a = 1] level CTMC size PRISM a) IDD-CSL states edges hybrid sparse 1 thread 2 threads 5 31, , ,204 6,766, ,194,304 45,972, ,336, ,032, ,525, ,650, ,516,604 1,312,110,960 a) original model from the PRISM case study suite

39 P >0.0 [F [1,1] a = 1] level CTMC size PRISM a) IDD-CSL states edges hybrid sparse 1 thread 2 threads 5 31, , ,204 6,766, ,194,304 45,972,480 1, ,336, ,032,640 5, , ,525, ,650, ,516,604 1,312,110,960 - a) original model from the PRISM case study suite time for initialisation exceeds 24 hours; exceeds the physical memory;

40 P >0.0 [F [1,1] a = 1] level CTMC size PRISM a) IDD-CSL states edges hybrid sparse 1 thread 2 threads 5 31, , ,204 6,766, ,194,304 45,972,480 1, ,336, ,032,640 5, , , , ,525, ,650,800-8, , ,516,604 1,312,110,960-26, , a) original model from the PRISM case study suite time for initialisation exceeds 24 hours; exceeds the physical memory;

41 Summary new symbolic matrix-free multiplication technique for the CTMC of an SPN and a probability vector very efficient prototype implementation of a CSL model checker efficiency is achieved by IDD-based encoding of the state space considering the Petri net structure variable order computation transition firing to extract matrix entries parallelisation our variable order helps tuning PRISM

42 Outlook non-ordinary stochastic Petri nets with extended arcs steady states analysis based on Gauss-Seidel increasing performance caching parallelisation controlled swapping porting to a cluster environment with distributed memory

43 Thanks for your attention. Questions? Demonstration?