Structure-to-Function Theory for Boolean Networks Henning S. - PowerPoint PPT Presentation

Background Equivalence of Sequential Graph Dynamical Systems Enumeration for κ -equivalence Structure-to-Function Theory for Boolean Networks Henning S. Mortveit Department of Engineering Systems and Environment & NSSAC, Biocomplexity Institute and Initiative University of Virginia IWBN, Concepci´ on, January 2020

Background Equivalence of Sequential Graph Dynamical Systems Enumeration for κ -equivalence What is Structure-to-Function Theory for BNs? ◮ The structure of a Boolean network includes: the vertex functions ( f i ) n i =1 the update mechanism (e.g., parallel, sequential) the variable dependency graph G (defined by the vertex functions) ◮ Structure-to-function theory for BNs relates the properties of the above components to properties of the associated phase spaces: 3 n [4]=(3,4,5,8) 1 f 4 ( x 3 , x 4 , x 5 , x 8 ) 4 − → 5 8 2 7 6 ◮ Most of the theory and results shown in this presentation hold for generalizations of BNs (referred to as for example graph dynamical systems/automata networks/polynomial dynamical systems/finite dynamical systems/sequential dynamical systems).

Background Equivalence of Sequential Graph Dynamical Systems Enumeration for κ -equivalence Terminology and Notation: Sequential Graph Dynamical Systems (I) ◮ Structure: i =1 with f i : K n − A (vertex) function sequence ( f i ) n → K with K a finite set (for example K = { 0 , 1 } .) i =1 : K n − → K n defined by A corresponding function sequence ( F i ) n � x = ( x 1 , x 2 , . . . , x n ) � = ( x 1 , . . . , x i − 1 , f i ( x ) , x i +1 , . . . , x n ) . F i � � A permutation π = π 1 , . . . , π n ∈ S n . Definition The sequential graph dynamical system map F π : K n − → K n given by f = ( f i ) i and π is F π = F π n ◦ F π n − 1 ◦ · · · ◦ F π 2 ◦ F π 1 .

Background Equivalence of Sequential Graph Dynamical Systems Enumeration for κ -equivalence Terminology and Notation: Sequential Graph Dynamical Systems II Definition The variable dependency graph G of ( f i ) i is the simple graph with vertex set V ( G ) = { 1 , 2 , . . . , n } and edge set E ( G ) all undirected edges { i , j } for which f i depends non-trivially on x j or f j depends non-trivially on x i . The symmetric group on V ( G ) is denoted by S G (the set of all permutation update sequences). Definition The phase space of F : K n − → K n is the directed graph Γ with vertex set K n and edge set { ( x , F ( x )) | x ∈ K n } .

Background Equivalence of Sequential Graph Dynamical Systems Enumeration for κ -equivalence Example I: a Structure-to-Function Result for ABNs ◮ Boolean vertex functions f = ( f i ) 4 i =1 defined by (indices modulo 4): f k ( x 1 , x 2 , x 3 , x 4 ) = nor 3 ( x k − 1 , x k , x k +1 ) = (1 + x k − 1 )(1 + x k )(1 + x k +1 ) mod 2 ◮ Dependency graph G is a square. ◮ Example phase spaces Γ( F π ) with π ∈ S G : 1000 1100 0101 0010 1001 0100 0001 1100 0011 0110 0111 0111 0101 0100 1011 0000 1001 1010 0000 1010 1101 1011 1000 1111 1110 1010 0001 0010 1111 1101 0110 1110 ¼ = (1423) ¼ = (1234) Theorem For any π ∈ S G and ABN map F π where each vertex function is a nor -function, Per ( F π ) is in a 1-1 correspondence with the set of independent sets of G. (For I ∈ I define x I = ( x v ) v by x v = 1 if and only if v ∈ I .)

Background Equivalence of Sequential Graph Dynamical Systems Enumeration for κ -equivalence Example II: a Structure-to-Function Result for BNs Definition (Threshold vertex function) Let K = { 0 , 1 } , let A = ( a ij ) n i , j =1 be a real symmetric matrix, let θ = ( θ 1 , . . . , θ n ) ∈ R n , and let F = ( f 1 , . . . , f n ): K n − → K n be the function defined coordinate-wise by  n � 0 , if a ij x j < θ i   f i ( x 1 , . . . , x n ) = j =1  1 , otherwise .  Theorem (Goles & Olivos [1]) If F is a BN map over a graph G where each vertex function is a generalized threshold function as above, then all x ∈ { 0 , 1 } n , are forward asymptotic to a fixed point or a 2 -cycle.

Background Equivalence of Sequential Graph Dynamical Systems Enumeration for κ -equivalence Main Presentation Outline Setup: will consider a fixed list of vertex function ( f v ) v (and therefore a fixed graph G ), and will vary the update sequence π ∈ S G . Goals: Demonstrate how one may compare maps F π and F π ′ using various types of comparisons using properties of G Give algorithms for deriving complete sets of update sequence representatives for exploring the diversity of dynamics under the various comparisons (i.e., equivalence notions) Comparisons: Functional equivalence – identify of maps Dynamical equivalence – topological conjugacy of maps Cycle equivalence – topological conjugacy of maps restricted to their periodic points Associated structures and combinatorics: The set of acyclic orientations of G , denoted by Acyc ( G ) Toric equivalence ∼ κ on Acyc ( G ) and its set of equivalence classes Acyc ( G ) / ∼ κ The automorphism group of G , denoted by Aut ( G ) (if time permits)

Background Equivalence of Sequential Graph Dynamical Systems Enumeration for κ -equivalence Acyclic Orientations and Functional Equivalence I — Acyc ( G ) G = Circle 4 ◮ Question: for π, π ′ ∈ S G , when is F π = F π ′ ? 1 2 ◮ Key insight : F 4 ◦ F 1 ◦ F 3 ◦ F 2 = F 4 ◦ F 3 ◦ F 1 ◦ F 2 4 3 U ( Circle 4 ) Definition ( ∼ α on S G ) (1234) (2341) (3412) (4123) Two permutations π, π ′ ∈ S G are α -related if they differ (4321) (3214) (2143) (1432) by exactly one transposition of two consecutive (1243) (1423) (3241) (3421) (2134) (2314) (4132) (4312) elements π i and π i +1 where { π i , π i +1 } �∈ E ( G ). (1324) (3124) (2413) (4213) The equivalence relation ∼ α on S G is the transitive and (1342) (3142) (2431) (4231) reflexive closure of the α -relation. 1 2 1 2 ◮ The map f ′ G : S G − → Acyc ( G ) is defined by mapping π ∈ S G to the acyclic orientation O ( π ) where each edge is ori- ented according to π (as a linear order.) 4 3 4 3 ¼ = (1,2,3,4) O ( ¼ )

Background Equivalence of Sequential Graph Dynamical Systems Enumeration for κ -equivalence Acyclic Orientations and Functional Equivalence II — Acyc ( G ) ◮ Let f = ( f i ) i . We set α f ( G ) = |{ F π | π ∈ S G }| . Proposition Let f = ( f i ) i be a function sequence with dependency graph G. We have π ∼ α π ′ implies F π = F π ′ . ( i ) f G ( ii ) The map f ′ G extends to a well-defined bijection f G : S G / ∼ α − → Acyc ( X ) by [ π ] �→ O ( π ) . ( iii ) We have α nor ( G ) = α ( G ) . ◮ Implications and results: Nor π = Nor π ′ if and only if π ∼ α π ′ . Have a computationally efficient, graph-based, sufficient condition to guarantee equality of maps F π and F π ′ : if O ( π ) = O ( π ′ ) then F π = F π ′ Can enumerate α ( G ) through the deletion/contraction recursion relation: α ( X ) = α ( X / e ) + α ( X \ e ) Note that α ( G ) = T G (2 , 0). Here T G is the Tutte polynomial of G . (Remark: the point (2 , 0) is in the computationaly intractable domain (D. Welsh).)

Background Equivalence of Sequential Graph Dynamical Systems Enumeration for κ -equivalence Acyclic Orientations and Functional Equivalence III — Acyc ( G ) ◮ Summary: Have linked Acyc ( G ) to functional equivalence of ABN maps F π Have an efficient, sufficent condition to determine if F π = F π ′ using O ( π ) and O ( π ′ ) The condition is valid for any fixed list of vertex funtions ( f v ) v for any state set K (even infinite) Have an upper bound for the number of distinct maps F π that can be constructed by varying π : α ( G ) = | Acyc ( G ) | These results are also valid for directed graphs G

Background Cycle Equivalence I Equivalence of Sequential Graph Dynamical Systems Enumeration for κ -equivalence Cycle Equivalence Definition ( Cycle Equivalence ) Two maps φ and ψ over finite state spaces are cycle equivalent if there is a bijection h such that ψ ◦ h = h ◦ φ holds when restricted to the periodic points of φ . (Or: multi-sets of cycle sizes are equal.) ◮ Example: Nor π for selected permutation update sequences over G = Circle 4 : 1000 1100 0101 0010 1001 0100 0001 1100 0011 0110 0111 0111 0101 0100 1001 1011 0000 1010 0000 1010 1101 1011 1000 1111 1110 1010 0010 0001 1111 0110 1110 1101 ¼ = (1423) ¼ = (1234) 0101 0011 0110 0111 1100 0001 0100 1001 0010 1000 1101 0000 1110 1011 1111 1010 ¼ = (1324) ◮ Note: there are 2 distinct cycle structures in the phase spaces above: { 7(1) } and { 2(2) , 3(1) }

Background Cycle Equivalence I Equivalence of Sequential Graph Dynamical Systems Enumeration for κ -equivalence Theorem (Macauley & Mortveit, Nonlinearity 2009) Let f = ( f i ) i be a sequence of vertex functions and assume that the state space satisfies | K | < ∞ . For any permutation π ∈ S G , the maps F π and F shift( π ) are cycle equivalent. Proof idea: F 1 ◦ ( F n ◦ · · · ◦ F 2 ◦ F 1 ) = ( F 1 ◦ F n ◦ · · · ◦ F 2 ) ◦ F 1 .