Model order reduction of biochemical networks by tropical geometry methods Ovidiu Radulescu University of Montpellier
Models of intracellular biochemistry Metabolic networks: synthesis, growth Signalling networks: information transfer Bonn, March 2018, 2 / 27 Gene networks: Decision making
Models of intracellular biochemistry: signalling MAPK signalling, Huang and Ferrell 96 Biochemical reactions: MAPKKK + E 1 → MAPKKK : E 1 → MAPKKK ∗ + E 1 , MAPKKK ∗ + E 2 → MAPKKK ∗ : E 2 → MAPKKK + E 2 ,... Bonn, March 2018, 3 / 27
Chemical reactions networks (CRN) ◮ x ∈ R n + concentrations vector, x i concentration of the species i . Bonn, March 2018, 4 / 27
Chemical reactions networks (CRN) ◮ x ∈ R n + concentrations vector, x i concentration of the species i . ◮ X ⇄ X + ν j reactions ν j ∈ Z n , stoichiometric vector Bonn, March 2018, 4 / 27
Chemical reactions networks (CRN) ◮ x ∈ R n + concentrations vector, x i concentration of the species i . ◮ X ⇄ X + ν j reactions ν j ∈ Z n , stoichiometric vector ◮ reaction rates R + / − ( x ) (reactions / unit volume / unit time) j Bonn, March 2018, 4 / 27
Chemical reactions networks (CRN) ◮ x ∈ R n + concentrations vector, x i concentration of the species i . ◮ X ⇄ X + ν j reactions ν j ∈ Z n , stoichiometric vector ◮ reaction rates R + / − ( x ) (reactions / unit volume / unit time) j ◮ kinetic law, for instance mass action ν + R + j ( x ) = k + ji j ∏ i ∈ reactants ( j ) x , i ν − R − j ( x ) = k − ji j ∏ i ∈ products ( j ) x i Bonn, March 2018, 4 / 27
Chemical reactions networks (CRN) ◮ x ∈ R n + concentrations vector, x i concentration of the species i . ◮ X ⇄ X + ν j reactions ν j ∈ Z n , stoichiometric vector ◮ reaction rates R + / − ( x ) (reactions / unit volume / unit time) j ◮ kinetic law, for instance mass action ν + R + j ( x ) = k + ji j ∏ i ∈ reactants ( j ) x , i ν − R − j ( x ) = k − ji j ∏ i ∈ products ( j ) x i ◮ Deterministic dynamics d x j = 1 ν j ( R + j ( x ) − R − dt = ∑ r j ( x )) . For mass action, these are polynomial ODEs Bonn, March 2018, 4 / 27
Model order reduction produce models with less variables and differential equations; popular method : aggregation of species and reactions Bonn, March 2018, 5 / 27
Multiple separated scales Widely distributed Widely distributed timescales, in concentrations, in log scale log scale Produced with the model in Radulescu et al BMC Systems Biol. 2008 Bonn, March 2018, 6 / 27
Slow/fast systems from Chiavazzo et al Comm.Comp.Phys. 2007 from Hung and Sheperd 12th Ann. Int. Detonation Symp. 2002 In slow/fast systems, fast variables relax quickly, then they are slaved : the system moves on a low dimensional invariant manifold describing the reduced model. The reduced model can change with time. Bonn, March 2018, 7 / 27
Slow/fast systems : Tikhonov theorem dx dt = 1 η f ( x , y ) fast dynamics dy dt = g ( x , y ) slow dynamics η is the fast/slow timescale ratio. Tikhonov : If for any y the fast dynamics has a hyperbolic point attractor, then after a quick transition the system evolves according to: dy dt = g ( x , y ) and f(x,y)=0 fast variables are slaved by slow ones Bonn, March 2018, 8 / 27
Model reduction Several steps: Bonn, March 2018, 9 / 27
Model reduction Several steps: ◮ Determine the slow/fast decomposition (who are the small parameter η , the slow and the fast variables?): Jacobian based numerical methods (CSP , ILDM, COPASI implementation); tropical geometry based methods. Bonn, March 2018, 9 / 27
Model reduction Several steps: ◮ Determine the slow/fast decomposition (who are the small parameter η , the slow and the fast variables?): Jacobian based numerical methods (CSP , ILDM, COPASI implementation); tropical geometry based methods. ◮ Solve f ( x , y ) = 0 for x : hard, few symbolic methods(Grobner bases, sparse polynomial systems (Grigoriev, Weber 2012)); use tropical approximations? Bonn, March 2018, 9 / 27
Model reduction Several steps: ◮ Determine the slow/fast decomposition (who are the small parameter η , the slow and the fast variables?): Jacobian based numerical methods (CSP , ILDM, COPASI implementation); tropical geometry based methods. ◮ Solve f ( x , y ) = 0 for x : hard, few symbolic methods(Grobner bases, sparse polynomial systems (Grigoriev, Weber 2012)); use tropical approximations? ◮ Pool reactions, prune species : use elementary modes (Radulescu et al 2008). Bonn, March 2018, 9 / 27
Model reduction Several steps: ◮ Determine the slow/fast decomposition (who are the small parameter η , the slow and the fast variables?): Jacobian based numerical methods (CSP , ILDM, COPASI implementation); tropical geometry based methods. ◮ Solve f ( x , y ) = 0 for x : hard, few symbolic methods(Grobner bases, sparse polynomial systems (Grigoriev, Weber 2012)); use tropical approximations? ◮ Pool reactions, prune species : use elementary modes (Radulescu et al 2008). ◮ Pool species, prune reactions : use conservation laws (Gorban, Radulescu, Zinovyev 2010). Bonn, March 2018, 9 / 27
Scaling and tropical geometry Polynomial kinetics (mass action law) M i dx i α 1 α n k ij x α ij , x α ij = x ∑ ij ij dt = P i ( x ) = 1 ... x n , 1 ≤ i ≤ n . j = 1 α ij ∈ N n are multi-indices, k ij > 0 are kinetic parameters. x i are positive species concentrations. D = { x i ≥ 0 , 1 ≤ i ≤ n } is a positive invariant set of the ODEs. Bonn, March 2018, 10 / 27
Scaling and tropical geometry Polynomial kinetics (mass action law) M i dx i α 1 α n k ij x α ij , x α ij = x ∑ ij ij dt = P i ( x ) = 1 ... x n , 1 ≤ i ≤ n . j = 1 α ij ∈ N n are multi-indices, k ij > 0 are kinetic parameters. x i are positive species concentrations. D = { x i ≥ 0 , 1 ≤ i ≤ n } is a positive invariant set of the ODEs. Consider ε > 0 a small scaling parameter. Bonn, March 2018, 10 / 27
Scaling and tropical geometry Polynomial kinetics (mass action law) M i dx i α 1 α n k ij x α ij , x α ij = x ∑ ij ij dt = P i ( x ) = 1 ... x n , 1 ≤ i ≤ n . j = 1 α ij ∈ N n are multi-indices, k ij > 0 are kinetic parameters. x i are positive species concentrations. D = { x i ≥ 0 , 1 ≤ i ≤ n } is a positive invariant set of the ODEs. Consider ε > 0 a small scaling parameter. Rescale k ij = ¯ k ij ε γ ij with ¯ k ij ∼ 1, for instance γ ij = round ( d log ( k ij ) / log ( ε )) / d , d ∈ N ∗ . Bonn, March 2018, 10 / 27
Scaling and tropical geometry Polynomial kinetics (mass action law) M i dx i α 1 α n k ij x α ij , x α ij = x ∑ ij ij dt = P i ( x ) = 1 ... x n , 1 ≤ i ≤ n . j = 1 α ij ∈ N n are multi-indices, k ij > 0 are kinetic parameters. x i are positive species concentrations. D = { x i ≥ 0 , 1 ≤ i ≤ n } is a positive invariant set of the ODEs. Consider ε > 0 a small scaling parameter. Rescale k ij = ¯ k ij ε γ ij with ¯ k ij ∼ 1, for instance γ ij = round ( d log ( k ij ) / log ( ε )) / d , d ∈ N ∗ . Suppose the same is possible for concentrations: x i = ε a i ¯ x i . Not possible if x i are zero or get exponentially close to zero. Some persistence condition is needed: ∃ a i , C i > 0 , x i ( t ) > C i ε a i , ∀ t ≥ 0 . Bonn, March 2018, 10 / 27
Scaling and tropical geometry Rescaled equations M i ε a i d ¯ x i ε µ ij ( a ) ¯ ∑ dt = k ij ¯ x α ij , 1 ≤ i ≤ n . j = 1 where n α ij ∑ µ ij ( a ) = γ ij + l a l l = 1 Bonn, March 2018, 11 / 27
Crazy-quilt: the big picture of nonlinear, multiscale dynamics. Abstract representation of metastability as itinerant trajectory in a patchy phase space landscape. Bonn, March 2018, 12 / 27
Tropical equilibration problem ◮ Find a such that there is ( j , j ′ ) , j � = j ′ such that i) µ ij ( a ) = µ ij ′ ( a ) , ii) µ ij ( a ) ≤ µ il ( a ) for all l � = j , j ′ , iii) k ij k ij ′ < 0. at least one negative and one positive dominant monomials have the same orders Bonn, March 2018, 13 / 27
Tropical equilibration problem ◮ Find a such that there is ( j , j ′ ) , j � = j ′ such that i) µ ij ( a ) = µ ij ′ ( a ) , ii) µ ij ( a ) ≤ µ il ( a ) for all l � = j , j ′ , iii) k ij k ij ′ < 0. at least one negative and one positive dominant monomials have the same orders ◮ Tropically truncated system (describes the dominant, fast dynamics) d ¯ x α ij −| ¯ x i x α ij ′ dt = ε ν i ( a ) ( | ¯ k ij | ¯ k ij ′ | ¯ + terms of same order ) ν i ( a ) = µ i ( a ) − a i gives the timescale of x i : small ν i means fast x i . Bonn, March 2018, 13 / 27
Tropical equilibration problem Equivalent formulation k ij > 0 ( µ ij ( a )) = min k ij < 0 ( µ ij ( a )) , 1 ≤ i ≤ n min n α ij ∑ µ ij ( a ) = γ ij + l a l l = 1 system of polynomial equations in min-plus algebra a ⊕ b = min ( a , b ) a ⊗ b = a + b Bonn, March 2018, 14 / 27
Recommend
More recommend
