Strong Turing Completeness of Continuous Chemical Reaction Networks Amaury Pouly Joint work with Olivier Bournez, François Fages, Guillaume Le Guludec and Daniel Graça 10 october 2018 1 / 20
Chemical Reaction Networks A reaction system is a finite set of ◮ molecular species y 1 , . . . , y n f ◮ reactions of the form � → � i a i y i − i b i y i ( a i , b i ∈ N , f = rate) Example : → 2H + O H 2 O 2 / 20
Chemical Reaction Networks A reaction system is a finite set of ◮ molecular species y 1 , . . . , y n f ◮ reactions of the form � → � i a i y i − i b i y i ( a i , b i ∈ N , f = rate) Example : → 2H + O H 2 O Assumption : law of mass action k y a i � � � − → f ( y ) = k a i y i b i y i � i i i i 2 / 20
Chemical Reaction Networks A reaction system is a finite set of ◮ molecular species y 1 , . . . , y n f ◮ reactions of the form � → � i a i y i − i b i y i ( a i , b i ∈ N , f = rate) Example : → 2H + O H 2 O Assumption : law of mass action k y a i � � � − → f ( y ) = k a i y i b i y i � i i i i Semantics : ◮ discrete ◮ differential ◮ stochastic 2 / 20
Chemical Reaction Networks A reaction system is a finite set of ◮ molecular species y 1 , . . . , y n f ◮ reactions of the form � → � i a i y i − i b i y i ( a i , b i ∈ N , f = rate) Example : → 2H + O H 2 O Assumption : law of mass action k y a i � � � − → f ( y ) = k a i y i b i y i � i i i i y ′ � ( b R i − a R i ) f R ( y ) i = Semantics : reaction R ◮ discrete Example : ◮ differential → [ H 2 O ] ′ = f ( H 2 O ) ◮ stochastic 2 / 20
Chemical Reaction Networks A reaction system is a finite set of ◮ molecular species y 1 , . . . , y n f ◮ reactions of the form � → � i a i y i − i b i y i ( a i , b i ∈ N , f = rate) Example : → 2H + O H 2 O Assumption : law of mass action k y a i � � � − → f ( y ) = k a i y i b i y i � i i i i a j y ′ � ( b R i − a R i ) k R � i = y Semantics : j reaction R j ◮ discrete Example : ◮ differential → [ H 2 O ] ′ = [ O ][ H ] 2 ◮ stochastic � Polynomial ODE! 2 / 20
Chemical Reaction Networks A reaction system is a finite set of ◮ molecular species y 1 , . . . , y n f ◮ reactions of the form � → � i a i y i − i b i y i ( a i , b i ∈ N , f = rate) Example : 2H + O → H 2 O → C + O 2 CO 2 3 / 20
Chemical Reaction Networks A reaction system is a finite set of ◮ molecular species y 1 , . . . , y n f ◮ reactions of the form � → � i a i y i − i b i y i ( a i , b i ∈ N , f = rate) Example : 2H + O → H 2 O → C + O 2 CO 2 Not limited to simple chemical reactions : ◮ DNA strand displacement ◮ RNA ◮ protein reactions 3 / 20
Chemical Reaction Networks A reaction system is a finite set of ◮ molecular species y 1 , . . . , y n f ◮ reactions of the form � → � i a i y i − i b i y i ( a i , b i ∈ N , f = rate) Example : 2H + O → H 2 O → C + O 2 CO 2 Not limited to simple chemical reactions : ◮ DNA strand displacement ◮ RNA ◮ protein reactions Implementing CRNs is a recent and active research field. 3 / 20
Chemical Reaction Networks A reaction system is a finite set of ◮ molecular species y 1 , . . . , y n f ◮ reactions of the form � i a i y i − → � i b i y i ( a i , b i ∈ N , f = rate) Some reactions are unrealistic : y 1 + 26 y 2 + 7 y 3 − → 13 y 4 + y 5 4 / 20
Chemical Reaction Networks A reaction system is a finite set of ◮ molecular species y 1 , . . . , y n f ◮ reactions of the form � i a i y i − → � i b i y i ( a i , b i ∈ N , f = rate) Some reactions are unrealistic : y 1 + 26 y 2 + 7 y 3 − → 13 y 4 + y 5 Only consider elementary reactions : at most two reactants ◮ A + B k − → C ◮ A k − → B + C ◮ A k − → B ◮ A k − → ∅ ◮ ∅ k − → A 4 / 20
Chemical Reaction Networks A reaction system is a finite set of ◮ molecular species y 1 , . . . , y n f ◮ reactions of the form � i a i y i − → � i b i y i ( a i , b i ∈ N , f = rate) Some reactions are unrealistic : y 1 + 26 y 2 + 7 y 3 − → 13 y 4 + y 5 Only consider elementary reactions : at most two reactants ◮ A + B k − → C Example : A + B k ◮ A k → C − − → B + C A ′ = − kAB B ′ = − kAB C ′ = kAB ◮ A k − → B ◮ A k − → ∅ � Quadratic ODE! ◮ ∅ k − → A 4 / 20
Chemical Reaction Networks : what can we compute? Can we use CRNs to compute? 5 / 20
Chemical Reaction Networks : what can we compute? Can we use CRNs to compute? What does it even mean? 5 / 20
Chemical Reaction Networks : what can we compute? Can we use CRNs to compute? What does it even mean? 5 / 20
Chemical Reaction Networks : what can we compute? Can we use CRNs to compute? What does it even mean? It depends a lot on how we define computability, in particular : ◮ rate : dependent/independent ◮ semantics : discrete/stochastic/differential ◮ kinetics : mass action/Michaelis/... ◮ species : finite/unbounded/infinite ◮ encoding : molecule count/concentration/digits ◮ more : robust, stable, ... 5 / 20
Chemical Reaction Networks : what can we compute? Can we use CRNs to compute? What does it even mean? It depends a lot on how we define computability, in particular : ◮ rate : dependent/independent ◮ semantics : discrete/stochastic/differential ◮ kinetics : mass action/Michaelis/... ◮ species : finite/unbounded/infinite ◮ encoding : molecule count/concentration/digits ◮ more : robust, stable, ... Extreme examples : rate-independent, differential, any kinetics, finite species, value is concentration, stable � piecewise linear functions 5 / 20
Chemical Reaction Networks : what can we compute? Can we use CRNs to compute? What does it even mean? It depends a lot on how we define computability, in particular : ◮ rate : dependent/independent ◮ semantics : discrete/stochastic/differential ◮ kinetics : mass action/Michaelis/... ◮ species : finite/unbounded/infinite ◮ encoding : molecule count/concentration/digits ◮ more : robust, stable, ... Extreme examples : rate-dependent, stochastic, rate-independent, differential, any kinetics, finite species, value is Markov, finite species, value is molecule count (must be small) concentration, stable � piecewise linear functions � probabilistic Turing machine 5 / 20
Chemical Reaction Networks : main result A reaction is elementary if it has at most two reactants ⇒ can, in principle, be implemented with DNA, RNA or proteins Theorem (CMSB 2017) Elementary mass-action-law reaction system on finite universes of molecules are Turing-complete under the differential semantics. 6 / 20
Chemical Reaction Networks : main result A reaction is elementary if it has at most two reactants ⇒ can, in principle, be implemented with DNA, RNA or proteins Theorem (CMSB 2017) Elementary mass-action-law reaction system on finite universes of molecules are Turing-complete under the differential semantics. Note : in fact the following elementary reactions suffice : ∅ k k k k → x − x − → x + z x + y − → x + y + z x + y − → ∅ 6 / 20
Chemical Reaction Networks : main result A reaction is elementary if it has at most two reactants ⇒ can, in principle, be implemented with DNA, RNA or proteins Theorem (CMSB 2017) Elementary mass-action-law reaction system on finite universes of molecules are Turing-complete under the differential semantics. Note : in fact the following elementary reactions suffice : ∅ k k k k − → x x − → x + z x + y − → x + y + z x + y − → ∅ We can even say something about the complexity : ◮ polynomial time&space f ∈ FPTIME ⇒ CRN computes f in or equivalently ◮ polynomial length 6 / 20
Chemical Reaction Networks : mathematics mass-action-law reaction system on finite universes of molecules under the differential semantics � 7 / 20
Chemical Reaction Networks : mathematics mass-action-law reaction system on finite universes of molecules under the differential semantics � Polynomial ODE : with constraints : y ′ ◮ nonnegative values (concentration) 1 = p 1 ( y 1 , . . . , y n ) ◮ restricted negative feedback : x ′ = − xyz . . . y ′ n = p n ( y 1 , . . . , y n ) 7 / 20
Chemical Reaction Networks : mathematics Elementary mass-action-law reaction system on finite universes of molecules under the differential semantics � Polynomial ODE : with constraints : y ′ ◮ nonnegative values (concentration) 1 = p 1 ( y 1 , . . . , y n ) ◮ restricted negative feedback : x ′ = − xyz . . . ◮ quadratic : p k ( y ) = � y ′ ij α ij y i y j n = p n ( y 1 , . . . , y n ) 7 / 20
Chemical Reaction Networks : mathematics Elementary mass-action-law reaction system on finite universes of molecules under the differential semantics � Polynomial ODE : with constraints : y ′ ◮ nonnegative values (concentration) 1 = p 1 ( y 1 , . . . , y n ) ◮ restricted negative feedback : x ′ = − xyz . . . ◮ quadratic : p k ( y ) = � y ′ ij α ij y i y j n = p n ( y 1 , . . . , y n ) � clever rewriting � value encoding : y = y + − y − � � Polynomial ODE : y ′ = p ( y ) 7 / 20
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