SAT Approximations of PSPACE Problems for Cellular Reprogramming - - PowerPoint PPT Presentation

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SAT Approximations of PSPACE Problems for Cellular Reprogramming - - PowerPoint PPT Presentation

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SAT Approximations of PSPACE Problems for Cellular Reprogramming Loc Paulev CNRS/LRI, Univ. Paris-Sud, Univ. Paris-Saclay BioInfo team loic.pauleve@lri.fr http://loicpauleve.name Journes BIOSS-IA - 23 Juin 2017 SAT Approximations of

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SLIDE 1

SAT Approximations of PSPACE Problems for Cellular Reprogramming

Loïc Paulevé

CNRS/LRI, Univ. Paris-Sud, Univ. Paris-Saclay – BioInfo team loic.pauleve@lri.fr http://loicpauleve.name Journées BIOSS-IA - 23 Juin 2017

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SLIDE 2

SAT Approximations of PSPACE Problems for Cellular Reprogramming

Cellular Dynamics

Cell state at t=0 Cell state

  • f interest

Initial state(s)/Goal state(s)

  • Trajectory existence (reachability)
  • Reasoning on all trajectories: e.g., common features
  • Control: perturbations to avoid/enforce goal reachability

Loïc Paulevé 2/9

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SLIDE 3

SAT Approximations of PSPACE Problems for Cellular Reprogramming

Cellular Dynamics

Cell state at t=0 Cell state

  • f interest

Initial state(s)/Goal state(s)

  • Trajectory existence (reachability)
  • Reasoning on all trajectories: e.g., common features
  • Control: perturbations to avoid/enforce goal reachability

Loïc Paulevé 2/9

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SLIDE 4

SAT Approximations of PSPACE Problems for Cellular Reprogramming

Cellular Dynamics

Cell state at t=0 Cell state

  • f interest

Initial state(s)/Goal state(s)

  • Trajectory existence (reachability)
  • Reasoning on all trajectories: e.g., common features
  • Control: perturbations to avoid/enforce goal reachability

Loïc Paulevé 2/9

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SLIDE 5

SAT Approximations of PSPACE Problems for Cellular Reprogramming

Cellular Dynamics

Cell state at t=0 Cell state

  • f interest

Initial state(s)/Goal state(s)

  • Trajectory existence (reachability)
  • Reasoning on all trajectories: e.g., common features
  • Control: perturbations to avoid/enforce goal reachability

Loïc Paulevé 2/9

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SLIDE 6

SAT Approximations of PSPACE Problems for Cellular Reprogramming

Qualitative Models for Cellular Reprogramming

(source: Crespo et al. Stem cells 2013; 31:2127-2135) a b c

fa(x) = xc ∨ (¬xa ∧ ¬xb) fb(x) = ¬xa ∨ xb fc(x) = xc ∨ (xa ∧ ¬xb)

010 110 000 100 011 111 001 101

Loïc Paulevé 3/9

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SLIDE 7

SAT Approximations of PSPACE Problems for Cellular Reprogramming

PSPACE problems for cellular reprogramming

Examples

initial state(s) goal state(s) initial state(s) goal state(s) initial state(s) goal state(s)

reachability cut sets mutations bifurcations

initial state(s) goal state(s) {a=0,b=1} c 0 -> 1 when b=1 c 0 -> 1 when b=1 KO b (lock b=0)

Loïc Paulevé 4/9

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SLIDE 8

SAT Approximations of PSPACE Problems for Cellular Reprogramming

State Transition Graph

initial state state reaching goal goal state (e.g., c=2)

⇒ avoid building it! (even symbolically): abstractions

Loïc Paulevé 5/9

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SLIDE 9

SAT Approximations of PSPACE Problems for Cellular Reprogramming

Local Causality Graph (LCG)

  • Initial state s = {a → 0; b → 0; c → 0; d → 0}.

c2 c0 c2 a1 b0 a1 a1 a0 a1 b0 b0 d1 d0 d1 e1 e0 e1 Node state State change Local cause - prior steps

Loïc Paulevé 6/9

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SLIDE 10

SAT Approximations of PSPACE Problems for Cellular Reprogramming

Local Causality Graph (LCG)

  • Initial state s = {a → 0; b → 0; c → 0; d → 0}.

c2 c0 c2 a1 b0 a1 a1 a0 a1 b0 b0 d1 d0 d1 e1 e0 e1 Node state State change Local cause - prior steps Necessary condition for reachability OA(s →∗ c2) ≡ there is an acyclic traversal from c2 s.t.

  • node state change → follow at least one child;
  • other nodes → follow all children;
  • terminates on empty “local cause” (leafs).

(can be verified linearly in the size of the LCG).

Loïc Paulevé 6/9

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SLIDE 11

SAT Approximations of PSPACE Problems for Cellular Reprogramming

Local Causality Graph (LCG)

  • Initial state s = {a → 0; b → 0; c → 0; d → 0}.

c2 c0 c2 a1 b0 a1 a1 a0 a1 b0 b0 d1 d0 d1 e1 e0 e1 Node state State change Local cause - prior steps

Loïc Paulevé 6/9

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SLIDE 12

SAT Approximations of PSPACE Problems for Cellular Reprogramming

Local Causality Graph (LCG)

  • Initial state s = {a → 0; b → 0; c → 0; d → 0}.

c2 c0 c2 a1 b0 a1 a1 a0 a1 b0 b0 d1 d0 d1 e1 e0 e1 Node state State change Local cause - prior steps Sufficient condition for reachability UA(s →∗ c2) ≡ ∃ particular acyclic sub-LCG NP formulation (find the right combination of local paths).

Loïc Paulevé 6/9

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SLIDE 13

SAT Approximations of PSPACE Problems for Cellular Reprogramming

Local Causality Graph (LCG)

  • Initial state s = {a → 0; b → 0; c → 0; d → 0}.

c2 c0 c2 a1 b0 a1 a1 a0 a1 b0 b0 d1 d0 d1 e1 e0 e1 Node state State change Local cause - prior steps Formal approximations of reachability UA(s →∗ c2) ⇒ s →∗ c2 ⇒ OA(s →∗ c2)

Loïc Paulevé 6/9

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SAT Approximations of PSPACE Problems for Cellular Reprogramming

Summary of the approach

Abstract interpretation of automata networks (1-bounded Petri nets)

  • LCG is P w/ nb automata; EXP w/ nb discrete levels (generally 2-5).

note: if from Boolean networks, translation is EXP w/ in-degree

  • Verifying OA is P; UA is NP (with LCG size)
  • SAT implementation of LCG generation and UA/OA.
  • Very low number of variables compared to nb reachable states

⇒ highly tractable for large networks of small automata Compared to Bounded Model-Checking (BMC):

  • BMC is an under-approximation only, no necessary condition
  • Can lead to a huge, when not intractable, number of variables (states reachable

in less than n transitions)

  • Incomplete capture of trajectories (important for control)

Loïc Paulevé 7/9

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SLIDE 15

SAT Approximations of PSPACE Problems for Cellular Reprogramming

Summary of the approach

Abstract interpretation of automata networks (1-bounded Petri nets)

  • LCG is P w/ nb automata; EXP w/ nb discrete levels (generally 2-5).

note: if from Boolean networks, translation is EXP w/ in-degree

  • Verifying OA is P; UA is NP (with LCG size)
  • SAT implementation of LCG generation and UA/OA.
  • Very low number of variables compared to nb reachable states

⇒ highly tractable for large networks of small automata Compared to Bounded Model-Checking (BMC):

  • BMC is an under-approximation only, no necessary condition
  • Can lead to a huge, when not intractable, number of variables (states reachable

in less than n transitions)

  • Incomplete capture of trajectories (important for control)

Loïc Paulevé 7/9

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SLIDE 16

SAT Approximations of PSPACE Problems for Cellular Reprogramming

Examples of Formal Approximations

Reachability [LP, M Magnin, O Roux in MSCS 2012; M Folschette, LP, M Magnin, O Roux in TCS 2015]

  • Over-approximation (necessary condition): OA(s →∗ g)
  • Under-approximation (sufficient condition): UA(s →∗ g)

Cut-sets [LP, G Andrieux, H Koeppl at CAV 2013]

  • UA: ai, bj, · · · : disable(ai, bj, · · · ) ∧ ¬ OA(s →∗ g)

Mutations for blocking g

  • UA: ai, bj, · · · : lock(ai, bj, · · · ) ∧ ¬ OA(s →∗ g)

Bifurcations [L F Fitime, C Guziolowski, O Roux, LP in BMC Algorithms for Mol Bio, 2017]

  • UA: sb, tb : UA(s →∗ sb) ∧ UA(sb →∗ g) ∧ ¬ OA(sb · tb →∗ g)

Implemented in Pint - http://loicpauleve.name/pint

  • Input: Boolean/discrete networks; automata networks; 1-bounded Petri nets
  • ASP implementation for solution enumeration (clingo)
  • Scalable to networks between 100 to 10,000 variables

Loïc Paulevé 8/9

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SLIDE 17

SAT Approximations of PSPACE Problems for Cellular Reprogramming

Examples of Formal Approximations

Reachability [LP, M Magnin, O Roux in MSCS 2012; M Folschette, LP, M Magnin, O Roux in TCS 2015]

  • Over-approximation (necessary condition): OA(s →∗ g)
  • Under-approximation (sufficient condition): UA(s →∗ g)

Cut-sets [LP, G Andrieux, H Koeppl at CAV 2013]

  • UA: ai, bj, · · · : disable(ai, bj, · · · ) ∧ ¬ OA(s →∗ g)

Mutations for blocking g

  • UA: ai, bj, · · · : lock(ai, bj, · · · ) ∧ ¬ OA(s →∗ g)

Bifurcations [L F Fitime, C Guziolowski, O Roux, LP in BMC Algorithms for Mol Bio, 2017]

  • UA: sb, tb : UA(s →∗ sb) ∧ UA(sb →∗ g) ∧ ¬ OA(sb · tb →∗ g)

Implemented in Pint - http://loicpauleve.name/pint

  • Input: Boolean/discrete networks; automata networks; 1-bounded Petri nets
  • ASP implementation for solution enumeration (clingo)
  • Scalable to networks between 100 to 10,000 variables

Loïc Paulevé 8/9

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SAT Approximations of PSPACE Problems for Cellular Reprogramming

Current projects and perspectives

ANR-FNR 2017-2020 “AlgoReCell” (porteur) Computational Models and Algorithms for the Prediction of Cell Reprogramming Determinants

  • Partners: LRI, LSV, Curie, Univ Luxembourg
  • Application to trans-differentiation from adipocytes to osteoblasts
  • Experimental validations

Towards predictions for Temporal Reprogramming of Boolean networks

[H Mandon, S Haar, LP at CMSB 2017]

  • Theoretical framework for Boolean network control
  • Explore use of incremental SAT and Just-in-time compilation of knowledge bases

Other research directions:

  • Parametric models (uncertainty on model specification),
  • .. combine with model reduction which preserves reachability properties

[LP at CMSB 2016; T Chatain, LP at CONCUR 2017]

  • Quantify number of trajectories from abstraction, . . .

Loïc Paulevé 9/9