Reduction of biochemical networks with multiple time scales Ovidiu - PowerPoint PPT Presentation

Reduction of biochemical networks with multiple time scales Ovidiu Radulescu DIMNP UMR 5235 CNRS/UM1/UM2, University of Montpellier 2 jointly with A.N.Gorban,D.Grigoriev,V.Noel,S.Vakulenko,A.Zinovyev NYU, March 2012

Outline ◮ Model reduction for linear networks with separated constants. ◮ Model reduction for non-linear networks with multiple time scales. ◮ Tropical geometry and model reduction. NYU, March 2012

The problem of size Complex, large scale molecular systems. NYU, March 2012 Kitano 2004

Dynamics ◮ State X (numbers of molecules), x = X / V (concentrations), reactions X → X + ν j . ◮ Deterministic dynamics r dx ∑ dt = ν j R j ( x ) j = 1 ◮ Stochastic dynamics X ( t ) is a jump Markov process, of intensity r ∑ λ ( x ) = V R j ( x ) , j = 1 jumps X → X + ν j , jump distribution r ∑ p j ( x ) = R j ( x ) / R j ( x ) j = 1 NYU, March 2012

Model reduction Backward pruning: define Hierarchical model reduction: synthetic parameters that are produce models with less identifiable variables, equations, parameters; use graph rewriting NYU, March 2012

Multiscale networks Our methods apply to molecular networks that have many, well separated, time and concentration 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 Our aim: develop reduction methods for multi-scale models with uncertainty. NYU, March 2012

Linear networks of chemical reactions: digraphs with linear kinetics A i are reagents, c i is concentration of A i . All the reactions are of the type A i → A j (monomolecular). k ji > 0 is the reaction A i → A j rate constant. The reaction rates: w ji = k ji c i . Kinetic equation c i = ∑ ˙ ( k ij c j − k ji c i ) or ˙ c = K c , (1) j , j � = i Relevance for computational biology: ◮ Occur as subsystems of larger, nonlinear networks. ◮ Crude approximations obtained by linearizing networks. NYU, March 2012

Linear networks with separated constants n − 1 c ( t ) = ( l 0 , c ( 0 ))+ r k ( l k , c ( 0 )) exp ( − λ k t ) ∑ k = 1 The eigenvectors of K specify the dynamics. Well separated constants k I 1 ≫ k I 2 ≫ k I 3 ≫ ... Integer labeled digraphs: each reaction arc has an integer label, specifying its position in the sequence of all reactions, ordered by speed; the lowest order is the most rapid. Theorem: the multiscale approximation of an arbitrary linear network with separated constants is an acyclic, deterministic, integer labeled digraph. NYU, March 2012

Auxiliary discrete dynamical systems For each A i , κ i = max j { k ji } , φ ( i ) = argmax j { k ji } ; φ ( i ) = i if there is no outgoing reaction A i → A j . φ determines auxiliary dynamical system on a set A = { A i } . Pruning: keep only the dominating step A 1 A 1 k 1 i κ i Aj Ai κ i Aj Ai k ni An An The auxiliary dynamical system is further decomposed into cycles C j with basins of attraction, Att ( C j ) : A = ∪ j Att ( C j ) . NYU, March 2012

1-st case: acyclic auxiliary dynamic systems All cycles C j are point attractors. κ j r i κ Φ ( j ) − κ i r i Φ ( j ) = j go along the A 6 κ j flow l i κ j − κ i l i j = Φ ( j ) go opposite 1 to the flow. A 1 7 A 2 A 7 5 A 3 3 6 For instance: A 4 l 1 ≈ ( 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ) 2 r 1 ≈ ( 1 , 0 , 0 , 0 , 0 , 0 , 0 , − 1 ) A 5 l 5 ≈ ( 0 , 0 , 0 , 1 , 1 , 1 , 1 , 0 ) r 5 ≈ ( 0 , 0 , 0 , 0 , 1 , 0 , 0 , − 1 ) 4 A 8 NYU, March 2012

Sequence of reduced models A 6 1 A 1 7 A 2 A 7 5 3 A 3 6 A 4 2 A 5 4 A 8 l 6 ≈ ( 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 ) r 6 ≈ ( 0 , 0 , 0 , 0 , 0 , − 1 , 1 , 0 ) NYU, March 2012

Sequence of reduced models A 6 1 A 1 7 A 2 A 7 5 A 3 3 6 A 4 2 A 5 4 A 8 l 7 ≈ ( 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 ) r 7 ≈ ( 0 , 0 , 0 , 0 , 1 , 0 , − 1 , 0 ) NYU, March 2012

Sequence of reduced models A 6 1 A 1 7 A 2 A 7 5 3 A 3 6 A 4 2 A 5 4 A 8 l 5 ≈ ( 0 , 0 , 0 , 1 , 0 , 1 , 1 , 0 ) r 5 ≈ ( 0 , 0 , 0 , 0 , − 1 , 0 , 0 , 1 ) NYU, March 2012

Sequence of reduced models A 6 1 A 1 7 A 2 A 7 5 3 A 3 6 A 4 2 A 5 4 A 8 l 1 ≈ ( 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ) r 1 ≈ ( − 1 , 0 , 0 , 0 , 0 , 0 , 0 , 1 ) NYU, March 2012

2-nd case: C j are sinks in the initial network Delete the limiting steps from cycles C j . The obtained acyclic reaction network A i → A φ ( i ) is the right approximation. A 3 2 4 A 2 6 A 4 1 3 5 A 5 A 1 NYU, March 2012

2-nd case: C j are sinks in the initial network Delete the limiting steps from cycles C j . The obtained acyclic reaction network A i → A φ ( i ) is the right approximation. A 3 A 3 2 4 2 4 A 2 6 A 4 A 2 A 4 1 3 1 3 5 A 5 5 A 5 A 1 A 1 NYU, March 2012

2-nd case: C j are sinks in the initial network Delete the limiting steps from cycles C j . The obtained acyclic reaction network A i → A φ ( i ) is the right approximation. A 3 A 3 A 3 2 4 2 4 2 4 A 2 6 A 4 A 2 A 4 A 2 A 4 1 3 1 3 1 3 5 A 5 5 A 5 A 5 A 1 A 1 A 1 NYU, March 2012

Sequence of reduced models A 3 A 3 A 3 2 4 2 4 2 4 A 2 6 A 4 A 2 A 4 A 2 A 4 1 3 1 3 1 3 A 1 5 A 5 A 1 5 A 5 A 1 A 5 NYU, March 2012

Sequence of reduced models A 3 A 3 A 3 2 4 2 4 2 4 A 2 6 A 4 A 2 A 4 A 2 A 4 1 3 1 3 1 3 A 1 5 A 5 A 1 5 A 5 A 1 A 5 NYU, March 2012

Sequence of reduced models A 3 A 3 A 3 2 4 2 4 2 4 A 2 6 A 4 A 2 A 4 A 2 A 4 1 3 1 3 1 3 A 1 5 A 5 A 1 5 A 5 A 1 A 5 NYU, March 2012

3-rd case: some of C j are not sinks A 4 Pool species in 3 cycles; Compare Slow reactions A 2 2 A 3 renormalized leave the cycle. A 2 + A 3 + A 4 constants to k 5 k 4 / k 3 other constants; A 3 1 2 4 k 5 k 4 / k 3 > k 6 k 5 k 4 / k 3 A 1 A 5 1 A 2 3 A 4 A 4 A 1 6 A 5 3 k 5 k 4 / k 3 < k 6 1 5 Renormalize Prune, restore A 2 2 A 3 constants of cycles A 1 6 A 5 outgoing slow 1 reactions. A 1 6 A 5 NYU, March 2012

Nonlinear networks ◮ Timescales are not inverses of parameters in the model. They involve concentrations and can change with time. ◮ Main ideas: quasi-steady state approximation, quasi-equilibrium. ◮ Given the trajectories c ( t ) of all species solution of d c dt = f ( c ) , the imposed trajectory of the i -th species is a solution c ∗ i ( t ) of the equation f i ( c 1 ( t ) ,..., c i − 1 ( t ) , c ∗ i ( t ) , c i + 1 ( t ) ,..., c n ( t )) = 0. We say that a species i is slaved if the distance between the trajectory c i ( t ) and some imposed trajectory c ∗ i ( t ) is small for some time interval I , sup t ∈ I | log ( c i ( t )) − log ( c ∗ i ( t )) | < δ , for some δ > 0 sufficiently small. NYU, March 2012

Slaved species - NF κ B model NYU, March 2012

Small concentration slaved species - quasi-steady state approximation Example: Michaelis-Menten mechanism. k + k 2 1 S + E → P + E SE ⇋ k − 1 QSS approximation (Briggs-Haldane): quasi-steady state species are low concentration, fast species. Pooling of reactions. R ( S , E tot ) − → S P dP dt = − dS dt = k 2 ES , k 1 S . ( E tot − E ) = ( k − 1 + k 2 ) ES R ( S , E tot ) = k 2 E tot . S / ( k m + S ) NYU, March 2012

Small concentration slaved species - quasi-equilibrium approximation QE approximation (Michaelis-Menten): quasi-equilibrium reactions are fast, reversible reactions. Pooling of species. S tot = S + ES , E tot = E + ES . R ( S tot , E tot ) − → S tot P dP dt = − dS tot = k 2 ES , k 1 ( S tot − ES ) . ( E tot − ES ) = k − 1 ES dt 2 E tot S tot R ( S tot , E tot ) = k 2 � ( E tot + S tot + k − 1 / k 1 )( 1 + 1 − 4 E tot S tot / ( E tot + S tot + k − 1 / k 1 ) 2 E tot S tot R ( S tot , E tot ) ≈ k 2 , if E tot << S tot k − 1 / k 1 + S tot NYU, March 2012

Graph rewriting operations - pooling and pruning of reactions and species Let S f be the stoichiometric matrix of the fast subsystem: fast reactions for QE; reactions producing or consuming fast species for QSS. Pooling For QSS define reaction pools, solutions of S f γ = 0: routes. For QE define species pools, solutions of b T S f = 0: conservation laws. Impose minimality conditions R ( γ ′ ) ⊂ R ( γ ) ⇒ γ ′ = 0 ∨ γ ′ = γ elementary modes. Pruning Quasi-steady state species and quasi-equilibrium reactions can be pruned (singular perturbations). Dominated reactions can be also pruned (regular perturbations). NYU, March 2012

Hierarchy of NF κ B models M ( 39 , 65 , 90 ) , BIOMD0000000227 NYU, March 2012

Hierarchy of NF κ B models M ( 34 , 60 , 82 ) MODEL7743631122 NYU, March 2012

Hierarchy of NF κ B models M ( 24 , 65 , 62 ) MODEL7743608569 NYU, March 2012

Hierarchy of NF κ B models M ( 16 , 34 , 46 ) MODEL7743576806 NYU, March 2012

Hierarchy of NF κ B models M ( 14 , 30 , 41 ) MODEL7743528808 NYU, March 2012

Hierarchy of NF κ B models M ( 14 , 25 , 33 ) MODEL7743444866 NYU, March 2012