Reconstructing Chemical Reaction Networks by Solving Boolean Polynomial Systems

Transcription

1 Reconstructing Chemical Reaction Networks by Solving Boolean Polynomial Systems Chenqi Mou Wei Niu LMIB-School of Mathematics École Centrale Pékin and Systems Science Beihang University, Beijing , China chenqi.mou, December 12, 2013 Nanning, China

2 The problem Chemical reaction networks

3 The problem Chemical reaction networks

4 The problem Chemical reaction networks

5 The problem Chemical reaction networks

6 Reconstructing Chemical Reaction Networks Chemical reaction networks

7 Why this problem? S- and R-graphs: easier for detecting Can the same S- and R-graphs lead to different SR-graphs? What do these SR-graphs mean?

8 Why this problem? S- and R-graphs: easier for detecting Can the same S- and R-graphs lead to different SR-graphs? What do these SR-graphs mean? CRR (Compound-Reaction-Reconstruction) problem [Fagerberg et. al. 2013] Existence / NP-hard / SAT, SMT, ILP

9 Why this problem? S- and R-graphs: easier for detecting Can the same S- and R-graphs lead to different SR-graphs? What do these SR-graphs mean? CRR (Compound-Reaction-Reconstruction) problem [Fagerberg et. al. 2013] Existence / NP-hard / SAT, SMT, ILP = CRR + problem: all the potential SR-graphs

10 Why Polynomial System Solving (PoSSo)? CRR problem Existence Hilbert s Nullstellensatz NP-hardness PoSSo is also NP-hard [Garey & Johnson 1979] SAT, SMT, ILP Polynomial system solvers

11 Why Polynomial System Solving (PoSSo)? CRR problem Existence Hilbert s Nullstellensatz NP-hardness PoSSo is also NP-hard [Garey & Johnson 1979] SAT, SMT, ILP Polynomial system solvers All the solutions feasible natural Complexity: Worst: doubly exponential (in #var) [Mayr & Meyer 1982] Dedicated complexity (structured): bidegree (1,1) [Faugère, Safey El Din, Spaenlehauer 2010]

12 Matrix representation R: a reaction = Input species: I(R); Output species: O(R); SR-graph two Boolean matrices

13 Matrix representation R: a reaction = Input species: I(R); Output species: O(R); SR-graph two Boolean matrices E m n { such that 1, Si I(R E i,k := k ) 0, Otherwise P n m { such that 1, Sj O(R P k,j := k ) 0, Otherwise On the Complexity of Reconstructing Chemical Reaction Networks 3 A R1 C R2 E B D F FIGURE 1. A directed hypergraph, consisting of two reactions {R1, R2} and six chemical species {A, B,..., F}. Hyperarc R1 has t(r1) = {A, B} and h(r1) = {C, D}, hyperarc R2 has t(r2) = {C} and h(r2) = {E, F }.

14 Matrix representation S-graphs: Boolean matrix S m m such that { 1, Rk s.t. S S i,j := i I(R k ) and S j O(R k ) 0, Otherwise R-graphs: Boolean matrix R n n such that { 1, Si s.t. S R k,l := i O(R k ) and S i I(R k ) 0, Otherwise

15 Matrix representation S-graphs: Boolean matrix S m m such that { 1, Rk s.t. S S i,j := i I(R k ) and S j O(R k ) 0, Otherwise R-graphs: Boolean matrix R n n such that { 1, Si s.t. S R k,l := i O(R k ) and S i I(R k ) 0, Otherwise Input: S, R = Output: E, P CRR: existence of E and P CRR + : all the possible E and P

16 Relationship S, R, E, and P S i,j = (E i,k P k,j ), R k,l = (P k,i E i,l ). k=1,...,n i=1,...,m Direct translation to PoSSo problem Background Boolean polynomial ring F 2 [E 1,1,..., E m,n, P 1,1,..., P n,m ] x y = x y and x y = x + y + x y Boolean polynomial system

17 Structure S i,j = k=1,...,n (E i,k P k,j ) x y = x y and x y = x + y + x y S i,j = 1 = 1 polynomial equation (degree 2n; variable 2n) = of type s (or r if R i,j = 1) S i,j = 0 = n bivariate quadratic equations = of type 0

18 Structure S i,j = k=1,...,n (E i,k P k,j ) x y = x y and x y = x + y + x y S i,j = 1 = 1 polynomial equation (degree 2n; variable 2n) = of type s (or r if R i,j = 1) S i,j = 0 = n bivariate quadratic equations = of type 0 Structure (p and q: #zeros in S and R) type 0: np + mq type s: m 2 p type r: n 2 q #Solutions #Variables = overdefined

19 PoSSo Methods Gröbner bases [Buchberger 1965, Faugère 1999, 2002] triangular sets [Wang 2001, Moreno Maza 2000, Gao & Huang 2012] XL (overdefined) e.g., [Ars et. al. 2004] Polynomial system = in a better form = solutions Complexity (Gröbner bases): Salvy 2004] O( ( n+d reg n Over F 2 : add the field equations (x 2 k + x k = 0). ) ω)[bardet, Faugère,

20 PoSSo Implementation Gröbner bases: Buchberger algorithm: almost in all Computer Algebra Systems F 4, F 5 : FGb, MAGMA... = MAGMA: optimization for over F 2 (since V2.15) Triangular sets: Epsilon, RegularChains (in Maple)...

21 Randomly generated S and R MAGMA V (F 4 implementation) = V2.20 (released yesterday, F 4 updated) m, n P Density (%) #Var #F Time #Solutions / / / > 1000 unknown / / / > 1000 unknown / / / / /

22 Remarks on the experiments General one: no optimization is made for CRR: (1) Experimentally, not comparable to SMT / SAT in efficiency (with optimization) (2) Problem generation (VS CNF generation) There exist instances with more than 1 solution (not trivial) For real-world examples (Biology): size (m, n 40), sparsity 98%

23 Future work Structure = simplify the problem / dedicated algorithm Complexity analyses: better? CRR: NP-hardness by PoSSo?