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Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

Precise Parameter Synthesis for Stochastic Biochemical Systems

Milan ˇ Ceˇ ska1,2, Frits Dannenberg2, Marta Kwiatkowska2, Nicola Paoletti2

Faculty of Informatics, Masaryk University, Brno, Czech Republic1 Department of Computer Science, University of Oxford, UK2 CMSB 2014, Manchester

17.11.2014

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 1 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

Introduction

Biochemical reaction networks

• formalism for modelling biological systems
• signalling pathways, gene regulation, epidemic models
• DNA logic gates, DNA walker circuits
• low molecular counts – stochastic dynamics
• semantics given by Continuous-Time Markov Chains (CTMCs)

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 2 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

Introduction

Biochemical reaction networks

• formalism for modelling biological systems
• signalling pathways, gene regulation, epidemic models
• DNA logic gates, DNA walker circuits
• low molecular counts – stochastic dynamics
• semantics given by Continuous-Time Markov Chains (CTMCs)

Uncertain kinetic parameters

• limited knowledge of rate parameters
• controllable parameters

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 2 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

Introduction

Biochemical reaction networks

• formalism for modelling biological systems
• signalling pathways, gene regulation, epidemic models
• DNA logic gates, DNA walker circuits
• low molecular counts – stochastic dynamics
• semantics given by Continuous-Time Markov Chains (CTMCs)

Uncertain kinetic parameters

• limited knowledge of rate parameters
• controllable parameters

Precise parameter synthesis

• synthesising parameters so that a given property is guaranteed

to hold or the probability of satisfying is maximised/minimized

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 2 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

Running Example

Parameters: P = k1 ∈ [0.1, 0.3], k2 = 0.02, initial state X = 15

CTMCs for biochemical reaction networks

• state - vector of populations/positions
• bounds on molecular counts – finite-state models
• transition rates given by rate parameters using rate functions
• low degree polynomial functions (mass action kinetics, etc.)

Parameter space P

• Cartesian product of intervals of rate parameters
• continuous parameter spaces

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 3 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

Running Example

Parameters: P = k1 ∈ [0.1, 0.3], k2 = 0.02, initial state X = 15

Property specification

• time-bounded fragment of Continuous Stochastic Logic (CSL)
• also applicable to CSL with reward operators
• path formula φ = F [1000,1000]15 ≤ X ≤ 20

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 3 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

Running Example

Parameters: P = k1 ∈ [0.1, 0.3], k2 = 0.02, initial state X = 15

Property specification

• time-bounded fragment of Continuous Stochastic Logic (CSL)
• also applicable to CSL with reward operators
• path formula φ = F [1000,1000]15 ≤ X ≤ 20

Synthesize values of k1 such that the probability of

1 φ being satisfied is above 40% (threshold synthesis) 2 φ being satisfied is maximized (max synthesis)

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 3 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

Problem Formulation

Parametric CTMC

• transition rates depend on a set of variables K
• parametric rate matrix RK – polynomials with variables k ∈ K
• describes set {Cp | p ∈ P} where Cp is the CTMC obtained by

instantiating p in RK

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 4 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

Problem Formulation

Parametric CTMC

• transition rates depend on a set of variables K
• parametric rate matrix RK – polynomials with variables k ∈ K
• describes set {Cp | p ∈ P} where Cp is the CTMC obtained by

instantiating p in RK Satisfaction function Λ

• let φ be a CSL path formula
• Λ : P → [0, 1] such that Λ(p) is the probability of φ being

satisfied over Cp

• analytical computation of Λ is intractable
• Λ can be discontinuous due to nested probabilistic operators

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 4 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

Problem Formulation

Satisfaction function Λ for the running example P = k1 ∈ [0.1, 0.3], k2 = 0.02, initial state X = 15 φ = F [1000,1000]15 ≤ X ≤ 20

k1

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 4 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

Problem Formulation – Threshold Synthesis

For a given P, φ, probability threshold r and volume tolerance ε, the problem is finding a partition {T, U, F} of P such that

1 ∀p ∈ T. Λ(p) ≥ r; and 2 ∀p ∈ F. Λ(p) < r; and 3 vol(U)/vol(P) ≤ ε (vol(A) is the volume of A).

T F U U F

k

1

r = 0.4

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 5 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

Problem Formulation – Max Synthesis

For a given P, φ and probability tolerance ǫ the problem is finding a partition {T, F} of P and probability bounds Λ⊥, Λ⊤ such that:

1 Λ⊤ − Λ⊥ ≤ ǫ; 2 ∀p ∈ T. Λ⊥ ≤ Λ(p) ≤ Λ⊤; and 3 ∃p ∈ T. ∀p′ ∈ F. Λ(p) > Λ(p′).

T F

k

1 F

probability bounds

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

Computing Lower and Upper Probability Bounds

Safe approximation of the lower and upper bounds of Λ

• generalization of a procedure from ˇ

Ceˇ ska et al. CAV’13

• Λmin ≤ minp∈P Λ(p) and Λmax ≥ maxp∈P Λ(p)
• orange box - lower and upper bounds
• purple box - approximation of lower and upper bounds

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 7 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

Computing Lower and Upper Probability Bounds

Parameter space decomposition

• independent computation for each subspace
• same asymptotic time complexity as standard uniformization
• improves the accuracy of approximation
• provides the basis of our synthesis algorithms

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 7 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

Refinement-based Threshold Synthesis

1: T ← ∅, F ← ∅, U ← P 2: repeat 3: R ← decompose(U), U ← ∅ 4: for all R ∈ R do 5: (ΛR

min, ΛR max) ← computeBounds(R, φ)

6: if ΛR

min ≥ r then

7: T ← T ∪ R 8: else if ΛR

max < r then

9: F ← F ∪ R 10: else 11: U ← U ∪ R 12: until vol(U)/vol(P) ≤ ε

F U T

• for our setting we shown Λ is a piecewise polynomial function

with finite number of subdomains → termination is guaranteed

• several heuristics for the parameter space decomposition

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 8 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

Refinement-based Max Synthesis

1: F ← ∅, T ← P 2: repeat 3: R ← decompose(T), T ← ∅ 4: for all R ∈ R do 5: (ΛR

min, ΛR max) ← computeBounds(R, φ)

6: Λ⊤

min ← getMaximalLowerBound(R)

7: for all R ∈ R do 8: if ΛR

max < Λ⊤ min then

9: F ← F ∪ R 10: else 11: T ← T ∪ R 12: Λ⊥ ← min{ΛR

min | R ∈ T}

13: Λ⊤ ← max{ΛR

max | R ∈ T}

14: until Λ⊤ − Λ⊥ ≤ ǫ

F T

getMaximalLowerBound(R) – under-approximation of the maximum

• naive approach – Λ⊤

min = max{ΛR min |R ∈ R}

• sampling-based approach improves Λ⊤

min by

Λ

⊤ min = max {Λ(pi) | pi ∈ {p1, p2, . . .}} – excludes more boxes

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 9 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

Refinement-based Max Synthesis

1: F ← ∅, T ← P 2: repeat 3: R ← decompose(T), T ← ∅ 4: for all R ∈ R do 5: (ΛR

min, ΛR max) ← computeBounds(R, φ)

6: Λ⊤

min ← getMaximalLowerBound(R)

7: for all R ∈ R do 8: if ΛR

max < Λ⊤ min then

9: F ← F ∪ R 10: else 11: T ← T ∪ R 12: Λ⊥ ← min{ΛR

min | R ∈ T}

13: Λ⊤ ← max{ΛR

max | R ∈ T}

14: until Λ⊤ − Λ⊥ ≤ ǫ

F T

• jump discontinuity of Λ may prevent termination
• additional checks are performed to detect and discard regions

containing such jumps (volume tolerance can be used)

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 9 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

SIR Epidemic Model

Epidemic dynamics in population of

• susceptible (S), infected (I) and recovered (R) individuals
• biochemical reaction model with mass action kinetics

S + I

ki

− → I + I (Infection) I

kr

− → R (Recovery)

• ki and kr are uncertain parameters

Inspected property

• infection lasts for at least 100 time units, and dies out before

120 time units – φ = (I > 0)U[100,120](I = 0)

• property and model parameters from Bortolussi et al. ArXiv’14

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 10 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

SIR Epidemic Model - Threshold Synthesis

One-dimensional parameter space

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Ki P

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.05 0.1 0.15 0.2 0.25 0.3 0.35

kr P

Problem ki kr Runtime Subspaces 1. [0.005, 0.3] 0.05 42.2 s 23 2. 0.12 [0.005, 0.2] 26.7 s 15

• probability threshold r = 0.1 and volume tolerance ε = 10%

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 11 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

SIR Epidemic Model - Threshold Synthesis

Two-dimensional parameter space

P

ki kr

kr

ki k

0.05 0.1 0.15 0.2 0.25 0.3 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

r

Problem ki kr Runtime Subspaces 1. [0.005, 0.3] [0.005, 0.2] 29.3 min 1320

• probability threshold r = 0.1 and volume tolerance ε = 10%

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 11 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

SIR Epidemic Model - Max and Min Synthesis

One-dimensional parameter space

0.1 0.2 0.3 0.05 0.1 0.15 0.2 0.25 0.3 0.35

ki P Max Synthesis

0.1 0.2 0.3 0.05 0.1 0.15 0.2 0.25 0.3 0.35

ki P Min Synthesis

0.05 0.1 0.15 0.2 0.05 0.1 0.15 0.2

P kr Max Synthesis

0.05 0.1 0.15 0.2 0.1 0.2 0.3 0.4 0.5

kr P Min Synthesis

Problem ki kr Runtime Subspaces Λ∗[%] T

• 1. Max [0.005, 0.3]

0.05 16.5 s 9 33.94 [0.267, 0.3]

• 2. Min

[0.005, 0.3] 0.05 49.5 s 21 2.91 [0.005, 0.0054]

• 3. Max

0.12 [0.005, 0.2] 99.7 s 57 19.94 [0.071, 0.076]

• 4. Min

0.12 [0.005, 0.2] 10.4 s 5 0.005 [0.005, 0.026]

• sampling-based refinement
• probability tolerance ǫ = 1% (1,3) and ǫ = 0.1% (2,4)
• Λ∗ denotes Λ⊥ (1,3) and Λ⊤ (2,4)

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 12 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

SIR Epidemic Model - Max and Min Synthesis

Max Synthesis for two-dimensional parameter space

0.05 0.1 0.15 0.2 0.25 0.3 0.05 0.1 0.15 0.2 0.1 0.2 0.3 0.4

kr ki P

0.05 0.1 0.15 0.2 0.25 0.3 0.05 0.1 0.15 0.2 0.1 0.2 0.3 0.4

kr ki

P

Problem Runtime Subspaces Λ⊥[%] T

• 1. Max

6.2 h 10249 34.77 [0.209, 0.29]×[0.051, 0.061]

• 2. Max

3.6 h 5817 35.01 [0.217, 0.272]×[0.053, 0.059]

• ki = [0.005, 0.3] and kr = [0.005, 0.2]
• naive refinement (1) and sampling-based refinement (2)
• probability tolerance ǫ = 1%

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 12 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

DNA walkers – reliability analysis

Man-made molecular motor/robot

• traverses a track of DNA strands (anchorages)
• stepping and blocking mechanism
• junctions – circuits evaluating Boolean functions
• stepping rates – given by the distance between anchorages
• unknown parameter ks ∈ [0.005, 0.02]

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 13 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

DNA walkers – reliability analysis

Runtime Subspaces Time bound Min. correct Max. incorrect ∅ Sampling ∅ Sampling T = 15 1.68% 5.94% 0.55 s 0.51 s 22 11 T = 30 14.86% 10.15% 1.43 s 1.35 s 35 15 T = 45 33.10% 12.25% 3.53 s 2.14 s 61 21 T = 200 79.21% 16.47% 213.57s 88.97 s 909 329

• min synthesis for the property P=?[F [T,T] finish-correct]
• max synthesis for the property P=?[F [T,T] finish-incorrect]
• probability tolerance ǫ = 1%

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 13 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

DNA walkers – reliability analysis

Additional stepping parameter

• c ∈ [0.25, 4] – stepping onto non-adjacent anchorages
• volume tolerance ε = 10%

d)

5 20 18 16 14 12 10 8 6 ks c 1 0.25 3 0.5 4 2

×10−3

F U T

c)

5 20 18 16 14 12 10 8 6 ks c 1 0.25 3 0.5 4 2

×10−3

b)

1 0.25 2 4 3 5 20 16 12 8

×10−3

c ks

a)

1 0.25 2 4 3 5 20 16 12 8

×10−3

c ks

T = 30 min T = 200 min

a) Φ1 = P≥0.4[F [30,30] finish-correct], 443.5 s, 2692 subspaces b) Φ2 = P≤0.08[F [30,30] finish-incorrect], 132.3 s, 807 subspaces c) Φ1 ∧ Φ2, 575.7 s, 3499 subspaces d) P≥0.8[F [200,200] finish-correct] ∧ P≤0.16[F [200,200] finish-incorrect], 12.3 h, 47229 subspaces

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 13 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

Conclusion

Practical complexity

• given by the number of required subspaces
• exponential in the number parameters
• linear in the volume of the parameter space
• shape of the satisfaction function, type of the synthesis

Effective methods for precise parameter synthesis

• stochastic biochemical networks and time-bounded CSL
• specifications are guaranteed to hold in synthesized regions
• regions are precise to within an arbitrarily small tolerance value
• based on computation of probability bounds, region

refinement and sampling of parameters points

• application to SIR and DNA walker case studies

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 14 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

Related work

Parameter synthesis

• methods for discrete-time Markov models and time-unbounded

properties – Hahn et al. STTT’11 and TASE’13

• approximate methods for CTMCs and time-bounded

reachability specifications – Han et al. RTSS’08

• reduction to analysis of high degree polynomial functions
• statistical methods for CTMCs and time-bounded linear

properties – Bortolussi et al. ArXiv’14 Parameter inference from time-series

• Wolf et al. CAV’11 and Bortolussi et al. QEST’13

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 15 / 16

Introduction Problem Formulation Parameter Synthesis Case Studies Conclusion

Current and Future Work

• efficient parallelization
• advanced uniformization techniques
• more complicated rate functions (e.g. Hill kinetics)
• extension of the PRISM model checker

Thank you for your attention.

Milan ˇ Ceˇ ska et al. Precise Parameter Synthesis for Stochastic Biochemical Systems 17.11.2014 16 / 16