Computing similarity between multiscale biological systems under - PowerPoint PPT Presentation

Computing similarity between multiscale biological systems under uncertainty Kris Ghosh Miami University, Ohio The Eighth International Workshop on Static Analysis for Systems Biology

An Example: Model of chemical reactions c e R5 c b R8 R1 R4 R2 c a c c start Reactions: R1,R2,R3,..R8. R3 Concentration of chemicals: R6 c a , c b , . . . , c e . c d R7 Finite State System representing chemical reactions Kris (Miami University) Computing Similarity SASB,2017 2 / 27

Motivation: Scales in models Why multiscale model? Biological systems need an integration of different scales-cellular, molecular , atomic levels... Kris (Miami University) Computing Similarity SASB,2017 3 / 27

Motivation: Scales in models Why multiscale model? Biological systems need an integration of different scales-cellular, molecular , atomic levels... a a a a b b c c g c a c b c c c d c e c f c h start (A) a b c c a c e c g c h start (B) a,b and c are biological processes. Kris (Miami University) Computing Similarity SASB,2017 3 / 27

Multiscales to Discrete Time Markov Chains Discrete Time Markov Chains representing identical partial ordering of pathways represented by edge labels a,b,c and d, respectively. p ij represent the probabilities on the edges where i , j ∈ N . Kris (Miami University) Computing Similarity SASB,2017 4 / 27

Challenges in Modeling in Biology Imprecise information Incomplete information Kris (Miami University) Computing Similarity SASB,2017 5 / 27

Challenges in Modeling in Biology Imprecise information Incomplete information A potential solution could be to use: nondeterminism and using probabilistic models. Kris (Miami University) Computing Similarity SASB,2017 5 / 27

Challenges in Modeling in Biology Imprecise information Incomplete information A potential solution could be to use: nondeterminism and using probabilistic models. Computational challenge: Large state space of the models. Kris (Miami University) Computing Similarity SASB,2017 5 / 27

Outline Related Work: theories and formalisms 1 Formalization 2 Current Work 3 Kris (Miami University) Computing Similarity SASB,2017 6 / 27

Preliminaries Definition (Kripke structure) Given a set of propositions, AP , a Kripke structure , K = � S , S 0 , E , L � consists of 1 S is the set of states. 2 S 0 ⊆ S is the initial set of states. 3 E ⊆ S × S is the transition relation. 4 L : S → 2 AP where L is the labeling function that labels each state with a subset from the set, AP . Kris (Miami University) Computing Similarity SASB,2017 7 / 27

Stuttering Equivalence on Paths α 0 α 1 Two infinite paths in Kripke structure K , µ = s o ֌ s 1 ֌ s 2 . . . and β 0 β 1 ν = r 0 ֌ r 1 ֌ . . . are stuttering equivalent ( ≡ s )ent if there are two infinite ordered sequences of positive integers, i = 0 < i 0 < i 1 < . . . and j = 0 < j 0 < j 1 < . . . such that ∀ k ≥ 0 L ( s i k ) = L ( s i k + 1 ) = . . . = L ( s i k + 1 − 1 ) = L ( r j k ) = L ( r j k + 1 ) = . . . = L ( r j k + 1 − 1 ) . The indices i k and j k are the starting points of µ and ν , respectively. Stuttering Equivalence Two Kripke structures K and K ′ are stuttering equivalent iff 1 The initial states of K and K ′ are the same. 2 For all paths, µ from an initial state, s 0 ∈ S 0 of K , there exists a path ν of K ′ from the same initial state of s 0 such that µ ≡ s ν . 3 For all paths, ν from an initial state of s 0 ∈ S 0 of K ′ , there exists a path µ of K from the same initial state of s such that ν ≡ s µ . Ref: Clarke, E.M., Grumberg and Peled, Model Checking.

Theories Interleaving asynchronous Ref: Clarke et al, State space reduction using partial order techniques, 1999. Bounded Asnchrony Ref: J. Fisher et al Bounded asynchrony: Concurrency for modeling cell-cell interactions, 2008. Computing bisimulation on structures Ref: Paige et al Three partition refinement algorithms, 1987. Kullback Leibler Divergence in Systems Biology Ref: Petrov Formal reductions of stochastic rule-based models of biochemical systems, 2013. Model Reduction in Systems Biology Ref: Feret et al, Lumpability Abstractions of Rule-based Systems, 2012. Kris (Miami University) Computing Similarity SASB,2017 9 / 27

Labeled transition system (LTS) Given a set of propositions, AP being the set of labels for states and EL , a set of labels for edges a labeled state transition system is defined as M = � S 0 , S , E , L e , L � where, 1 � S , S 0 , E , L � forms a Kripke structure. 2 L e : E �→ EL is an edge-labeling function. Kris (Miami University) Computing Similarity SASB,2017 10 / 27

Labeled Probabilistic System (LPS) a LPS is a tuple, W = � S , S 0 , ι init , P , L e , L , E � where: � S , S 0 , ι init , P , L � is DTMC. L e : S × S → E where, E is the set of edge labels. Kris (Miami University) Computing Similarity SASB,2017 11 / 27

Measures on Probability Distributions Kullback Leibler Divergence (KLD) of two distributions: P ( i ) log P ( i ) H ( P || Q ) = � Q ( i ) . i Jensen-Shannon Divergence (JSD) is symmetric verson of KLD: JSD ( P || Q ) = 1 2 H ( P || M ) + 1 2 H ( Q || M ) where M = 1 2 ( P + Q ) . KLD can only be computed on same state space. Kris (Miami University) Computing Similarity SASB,2017 12 / 27

Formalization of System Read For an infinite path, π = e 0 , e 1 , e 2 , e 3 , . . . in a LPS W , ( α 0 , α 1 , α 2 , . . . ) is the sequence of reaction labels in π . The read of a path is the subsequence of reaction labels ˜ π = α 0 , α i 1 , α i 2 where 0 ≤ i 1 ≤ i 2 ≤ . . . , α i j is in ˜ π iff α i j � = α i j − 1 and α 0 � = α i 1 . A finite path segment σ = e 0 ֌ e 1 ֌ e 2 ֌ e 3 . . . → e m ֌ . . . , is identically labeled ( il ) if the reactions are identical. We explicitly allow m = 0; in that case we write e 0 � e 0 Kris (Miami University) Computing Similarity SASB,2017 13 / 27

Compact Probability on Paths P c ( e , e ′ ) between two edges is computed by the following equations dependent on the label of the successive edge. � P ( e , e ′ ) if, e � = e ′ P c ( e , e ′ ) = P ( e ) × P ( e 1 ) × · · · P ( e k ) if e ֌ e 1 ֌ . . . , e k ֌ e ′ The compact probability for an il path fragment is computed by the products of the probabilities. Kris (Miami University) Computing Similarity SASB,2017 14 / 27

Read equivalence on paths Paths π 1 ∈ Π( W 1 ) , π 2 ∈ Π( W 2 ) are read equivalent iff their reads are identical and denoted by π 1 ≡ r π 2 . Read equivalence on reactions Given two LPSs, W 1 and W 2 , the relation read on edges ( ≡ r ) is defined on reaction labels, e 1 ∈ E 1 and e 2 ∈ E 2 . e 1 ≡ r e 2 if and only if the following conditions hold: 1 L ( e 1 ) = L e ( e 2 ) . 2 For all paths, π e 1 ∈ Π( e 1 ) ∃ a path π e 2 ∈ Π( e 2 ) such that π e 1 ≡ r π e 2 . 3 For all paths, π e 2 ∈ Π( e 2 ) ∃ a path π e 1 ∈ Π( e 1 ) such that π e 1 ≡ r π e 2 . Kris (Miami University) Computing Similarity SASB,2017 15 / 27

Problem Statement Given two LTSs, M 1 and M 2 , construct two LPSs, W 1 and W 2 Are W 1 and W 2 read equivalent? If yes, compute the KLD between the two structures. Important: Only takes account of information of the edges. Compares the partial ordering of the two LPSs based on the edge labels. Kris (Miami University) Computing Similarity SASB,2017 16 / 27

Ordered Pairs A relation, R e defined on the edges of W 1 and W 2 is given by ( e 1 , e 2 ) ∈ R e , e 1 ∈ E 1 and e 2 ∈ E 2 where, L e ( e 1 ) = L e ( e 2 ) . Predecessor The subset of ordered pairs, Predecessor Pred r ( Y ) is defined from the set of ordered pairs, ( e 1 , e 2 ) ∈ R e represented by the Y is: Pred r ( Y ) = { ( e 1 , e 2 ) ∈ Y | ∀ e ′ 1 , e 1 ֌ e ′ 1 implies ∃ an il -path fragment e 2 ֌ . . . ֌ e m , 2 ֌ e ′ 2 , ∀ i ≤ m , ( e 1 , e i , 2 ) ∈ Y ∧ ( e ′ 1 , e ′ 2 ) ∈ Y , and ∀ e ′ 2 , e 2 ֌ e ′ 2 implies ∃ an ilpath-fragment e 1 ֌ . . . ֌ e m , 1 ֌ e ′ 1 , ∀ i ≤ m , ( e i , 1 , e 2 ) ∈ Y ∧ ( e ′ 1 , e ′ 2 ) ∈ Y } .

a a a a b b c c a c b c c c d c e c f c g c h start a b c c g c a c e c h start Kris (Miami University) Computing Similarity SASB,2017 18 / 27

a a a a b b c c a c b c c c d c e c f c g c h start a b c c g c a c e c h start a a a a b b c c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 start Kris (Miami University) Computing Similarity SASB,2017 18 / 27

Model Assumptions Comparing LPSs is to be compared on the same state space (edge labels) as required by KLD. There is no self loop on the states. Kris (Miami University) Computing Similarity SASB,2017 19 / 27

Computing:Greatest Fixed Point Input: Set of Ordered Pairs, R e Output: Set of ordered pairs in the greatest fixed point, Y ∞ . 1 Y := R e ; 2 Y ′ := 0 ; 3 H ( W 1 �W 2 ) = 0 ; 4 while ( Y � = Y ′ ) { 5 Y ′ := Y ; 6 Y := Y ∩ Pred r ( Y ); 7 H ( W 1 �W 2 ) = H ( W 1 || 8 W 2 ) + P c ( Fst ( Y ) , Pre ( Fst ( Y )) log ( P c ( Fst ( Y ) , Pre ( Fst ( Y ))) P c ( Snd ( Y ) , Pre ( Snd ( Y ))) } 9 10 Y ∞ = Y ′ Kris (Miami University) Computing Similarity SASB,2017 20 / 27