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Product form stationary distributions of stochastic reaction networks, and application to constrained averaging of multiscale systems Simon Cotter University of Manchester 19th July 2016 Simon Cotter Product form stationary dist. 0 / 35

  1. QSSA: Simple Example k 2 k 4 / ( k 2 + k 3 ) s k 1 ∅ − − − − → S − − − − − − − − − → ∅ Quantify error by computing error in invariant distribution of S = X 1 + X 2 � k 1 ( k 2 + k 3 ) � S ( ∞ ) ∼ P (QSSA approximation) k 2 k 4 Invariant distribution of full system given by � � k 4 , λ 2 = k 1 ( k 3 + k 4 ) λ 1 = k 1 P k 2 k 4 � � k 1 ( k 2 + k 3 + k 4 ) S ( ∞ ) ∼ P (True invariant distribution) k 2 k 4 QSSA error in Poisson intensity k 1 k 2 Simon Cotter Product form stationary dist. 9 / 35

  2. QSSA: Simple Example k 2 k 4 / ( k 2 + k 3 ) s k 1 ∅ − − − − → S − − − − − − − − − → ∅ Quantify error by computing error in invariant distribution of S = X 1 + X 2 � k 1 ( k 2 + k 3 ) � S ( ∞ ) ∼ P (QSSA approximation) k 2 k 4 Invariant distribution of full system given by � � k 4 , λ 2 = k 1 ( k 3 + k 4 ) λ 1 = k 1 P k 2 k 4 � � k 1 ( k 2 + k 3 + k 4 ) S ( ∞ ) ∼ P (True invariant distribution) k 2 k 4 QSSA error in Poisson intensity k 1 k 2 Simon Cotter Product form stationary dist. 9 / 35

  3. QSSA: Simple Example k 2 k 4 / ( k 2 + k 3 ) s k 1 ∅ − − − − → S − − − − − − − − − → ∅ Quantify error by computing error in invariant distribution of S = X 1 + X 2 � k 1 ( k 2 + k 3 ) � S ( ∞ ) ∼ P (QSSA approximation) k 2 k 4 Invariant distribution of full system given by � � k 4 , λ 2 = k 1 ( k 3 + k 4 ) λ 1 = k 1 P k 2 k 4 � � k 1 ( k 2 + k 3 + k 4 ) S ( ∞ ) ∼ P (True invariant distribution) k 2 k 4 QSSA error in Poisson intensity k 1 k 2 Simon Cotter Product form stationary dist. 9 / 35

  4. Outline Introduction 1 The Quasi Steady-State Assumption 2 The Constrained Approach 3 Product Form Stationary Distributions for Non-Mass Action 4 Kinetics 5 Numerical Example Conclusions 6 Simon Cotter Product form stationary dist. 9 / 35

  5. The Constrained Approach The Constrained philosophy The assumption is that the fast variables equilibrate on a timescale which is negligible with respect to the rate of change of slow species, but that the invariant distribution of the fast variables can be affected by the slow reactions . Slow stoichiometries are not in general perpendicular to the fast stoichiometries! In practice We cannot approximate P ( F | S ) by considering just the fast subsystem, as we need to take into account the effect of the slow reactions on the invariant distribution of the fast variables . This can be approximated by considering the dynamics of the full system, constrained to a particular value of the slow variables . NB: Assymptotically (in the limit of large timescale gaps), this approach is in agreement with the QSSA Simon Cotter Product form stationary dist. 10 / 35

  6. The Constrained Approach The Constrained philosophy The assumption is that the fast variables equilibrate on a timescale which is negligible with respect to the rate of change of slow species, but that the invariant distribution of the fast variables can be affected by the slow reactions . Slow stoichiometries are not in general perpendicular to the fast stoichiometries! In practice We cannot approximate P ( F | S ) by considering just the fast subsystem, as we need to take into account the effect of the slow reactions on the invariant distribution of the fast variables . This can be approximated by considering the dynamics of the full system, constrained to a particular value of the slow variables . NB: Assymptotically (in the limit of large timescale gaps), this approach is in agreement with the QSSA Simon Cotter Product form stationary dist. 10 / 35

  7. The Constrained Approach The Constrained philosophy The assumption is that the fast variables equilibrate on a timescale which is negligible with respect to the rate of change of slow species, but that the invariant distribution of the fast variables can be affected by the slow reactions . Slow stoichiometries are not in general perpendicular to the fast stoichiometries! In practice We cannot approximate P ( F | S ) by considering just the fast subsystem, as we need to take into account the effect of the slow reactions on the invariant distribution of the fast variables . This can be approximated by considering the dynamics of the full system, constrained to a particular value of the slow variables . NB: Assymptotically (in the limit of large timescale gaps), this approach is in agreement with the QSSA Simon Cotter Product form stationary dist. 10 / 35

  8. The Constrained Approach ν ′ R 1 : ν 1 → 1 , ν ′ R 2 : ν 2 → 2 , . . . ν ′ R M : ν M → M , Pick slow variable S = � i a ij X i which is invariant with respect to fast reactions, i.e. perpendicular to fast stoichiometries Form basis with the slow variables, along with chosen fast variables F Define constrained stoichiometric projector P : R d → R d P ([ S , F ]) = [ 0 , F ] Simon Cotter Product form stationary dist. 11 / 35

  9. The Constrained Approach ν ′ R 1 : ν 1 → 1 , ν ′ R 2 : ν 2 → 2 , . . . ν ′ R M : ν M → M , Pick slow variable S = � i a ij X i which is invariant with respect to fast reactions, i.e. perpendicular to fast stoichiometries Form basis with the slow variables, along with chosen fast variables F Define constrained stoichiometric projector P : R d → R d P ([ S , F ]) = [ 0 , F ] Simon Cotter Product form stationary dist. 11 / 35

  10. The Constrained Approach ν ′ R 1 : ν 1 → 1 , ν ′ R 2 : ν 2 → 2 , . . . ν ′ R M : ν M → M , Pick slow variable S = � i a ij X i which is invariant with respect to fast reactions, i.e. perpendicular to fast stoichiometries Form basis with the slow variables, along with chosen fast variables F Define constrained stoichiometric projector P : R d → R d P ([ S , F ]) = [ 0 , F ] Simon Cotter Product form stationary dist. 11 / 35

  11. The Constrained Approach ν ′ R 1 : ν 1 → 1 , ν ′ R 2 : ν 2 → 2 , . . . ν ′ R M : ν M → M , Pick slow variable S = � i a ij X i which is invariant with respect to fast reactions, i.e. perpendicular to fast stoichiometries Form basis with the slow variables, along with chosen fast variables F Define constrained stoichiometric projector P : R d → R d P ([ S , F ]) = [ 0 , F ] Simon Cotter Product form stationary dist. 11 / 35

  12. The Constrained Subsystem ν 1 + P ( ν ′ R 1 : ν 1 → 1 − ν 1 ) = ˆ ν 1 , ν 2 + P ( ν ′ → 2 − ν 2 ) = ˆ R 2 : ν 2 ν 2 , . . . ν M + P ( ν ′ R M : ν M → M − ν M ) = ˆ ν M . Preserves changes to fast variables, removes changes to slow variables Multiply propensities by ✶ X + P ( ν ′ 0 to preserve k − ν k ) ∈ Z d non-negativity Constrains the slow variable to its starting value Invariant distribution a good approximation of P ( F | S ) for full system Invariant distribution of constrained system needs to be found Simon Cotter Product form stationary dist. 12 / 35

  13. The Constrained Subsystem ν 1 + P ( ν ′ R 1 : ν 1 → 1 − ν 1 ) = ˆ ν 1 , ν 2 + P ( ν ′ → 2 − ν 2 ) = ˆ R 2 : ν 2 ν 2 , . . . ν M + P ( ν ′ R M : ν M → M − ν M ) = ˆ ν M . Preserves changes to fast variables, removes changes to slow variables Multiply propensities by ✶ X + P ( ν ′ 0 to preserve k − ν k ) ∈ Z d non-negativity Constrains the slow variable to its starting value Invariant distribution a good approximation of P ( F | S ) for full system Invariant distribution of constrained system needs to be found Simon Cotter Product form stationary dist. 12 / 35

  14. The Constrained Subsystem ν 1 + P ( ν ′ R 1 : ν 1 → 1 − ν 1 ) = ˆ ν 1 , ν 2 + P ( ν ′ → 2 − ν 2 ) = ˆ R 2 : ν 2 ν 2 , . . . ν M + P ( ν ′ R M : ν M → M − ν M ) = ˆ ν M . Preserves changes to fast variables, removes changes to slow variables Multiply propensities by ✶ X + P ( ν ′ 0 to preserve k − ν k ) ∈ Z d non-negativity Constrains the slow variable to its starting value Invariant distribution a good approximation of P ( F | S ) for full system Invariant distribution of constrained system needs to be found Simon Cotter Product form stationary dist. 12 / 35

  15. The Constrained Subsystem ν 1 + P ( ν ′ R 1 : ν 1 → 1 − ν 1 ) = ˆ ν 1 , ν 2 + P ( ν ′ → 2 − ν 2 ) = ˆ R 2 : ν 2 ν 2 , . . . ν M + P ( ν ′ R M : ν M → M − ν M ) = ˆ ν M . Preserves changes to fast variables, removes changes to slow variables Multiply propensities by ✶ X + P ( ν ′ 0 to preserve k − ν k ) ∈ Z d non-negativity Constrains the slow variable to its starting value Invariant distribution a good approximation of P ( F | S ) for full system Invariant distribution of constrained system needs to be found Simon Cotter Product form stationary dist. 12 / 35

  16. The Constrained Subsystem ν 1 + P ( ν ′ R 1 : ν 1 → 1 − ν 1 ) = ˆ ν 1 , ν 2 + P ( ν ′ → 2 − ν 2 ) = ˆ R 2 : ν 2 ν 2 , . . . ν M + P ( ν ′ R M : ν M → M − ν M ) = ˆ ν M . Preserves changes to fast variables, removes changes to slow variables Multiply propensities by ✶ X + P ( ν ′ 0 to preserve k − ν k ) ∈ Z d non-negativity Constrains the slow variable to its starting value Invariant distribution a good approximation of P ( F | S ) for full system Invariant distribution of constrained system needs to be found Simon Cotter Product form stationary dist. 12 / 35

  17. The Constrained Subsystem ν 1 + P ( ν ′ R 1 : ν 1 → 1 − ν 1 ) = ˆ ν 1 , ν 2 + P ( ν ′ → 2 − ν 2 ) = ˆ R 2 : ν 2 ν 2 , . . . ν M + P ( ν ′ R M : ν M → M − ν M ) = ˆ ν M . Preserves changes to fast variables, removes changes to slow variables Multiply propensities by ✶ X + P ( ν ′ 0 to preserve k − ν k ) ∈ Z d non-negativity Constrains the slow variable to its starting value Invariant distribution a good approximation of P ( F | S ) for full system Invariant distribution of constrained system needs to be found Simon Cotter Product form stationary dist. 12 / 35

  18. Constrained Approach: Simple Example Consider the system: k 2 x 1 − − − − → k 1 k 4 x 2 → X 1 X 2 ∅ − − − − ← − − − − − − − − → ∅ k 3 x 2 First construct basis made up of slow/fast variables S = X 1 + X 2 , F = X 1 or F = X 2 We will pick F = X 2 Therefore P ([ X 1 , X 2 ]) = [ − X 2 , X 2 ] Apply to all reactions to arrive at constrained subsystem Simon Cotter Product form stationary dist. 13 / 35

  19. Constrained Approach: Simple Example Consider the system: k 2 x 1 − − − − → k 1 k 4 x 2 → X 1 X 2 ∅ − − − − ← − − − − − − − − → ∅ k 3 x 2 First construct basis made up of slow/fast variables S = X 1 + X 2 , F = X 1 or F = X 2 We will pick F = X 2 Therefore P ([ X 1 , X 2 ]) = [ − X 2 , X 2 ] Apply to all reactions to arrive at constrained subsystem Simon Cotter Product form stationary dist. 13 / 35

  20. Constrained Approach: Simple Example Consider the system: k 2 x 1 − − − − → k 1 k 4 x 2 → X 1 X 2 ∅ − − − − ← − − − − − − − − → ∅ k 3 x 2 First construct basis made up of slow/fast variables S = X 1 + X 2 , F = X 1 or F = X 2 We will pick F = X 2 Therefore P ([ X 1 , X 2 ]) = [ − X 2 , X 2 ] Apply to all reactions to arrive at constrained subsystem Simon Cotter Product form stationary dist. 13 / 35

  21. Constrained Approach: Simple Example Consider the system: k 2 x 1 − − − − → k 1 k 4 x 2 → X 1 X 2 ∅ − − − − ← − − − − − − − − → ∅ k 3 x 2 First construct basis made up of slow/fast variables S = X 1 + X 2 , F = X 1 or F = X 2 We will pick F = X 2 Therefore P ([ X 1 , X 2 ]) = [ − X 2 , X 2 ] Apply to all reactions to arrive at constrained subsystem Simon Cotter Product form stationary dist. 13 / 35

  22. Constrained Approach: Simple Example Consider the system: k 2 x 1 − − − − → k 1 k 4 x 2 → X 1 X 2 ∅ − − − − ← − − − − − − − − → ∅ k 3 x 2 First construct basis made up of slow/fast variables S = X 1 + X 2 , F = X 1 or F = X 2 We will pick F = X 2 Therefore P ([ X 1 , X 2 ]) = [ − X 2 , X 2 ] Apply to all reactions to arrive at constrained subsystem Simon Cotter Product form stationary dist. 13 / 35

  23. Constrained Approach: Simple Example k 1 R 1 : ∅ − − − − → X 1 Reaction 1: P ([ 1 , 0 ]) = [ 0 , 0 ] Constrained version of R 1 has no effect on X k 2 x 1 − − − − → R 2 , R 3 : X 1 X 2 Reactions 2,3: ← − − − − k 3 x 2 No change in S Projection has no effect k 4 x 2 − − − − → ∅ Reaction 4: R 4 : X 2 P ([ 0 , − 1 ]) = [ 1 , − 1 ] k 4 x 2 − − − − − → X 1 Constrained version: R 4 : X 2 Simon Cotter Product form stationary dist. 14 / 35

  24. Constrained Approach: Simple Example k 1 R 1 : ∅ − − − − → X 1 Reaction 1: P ([ 1 , 0 ]) = [ 0 , 0 ] Constrained version of R 1 has no effect on X k 2 x 1 − − − − → R 2 , R 3 : X 1 X 2 Reactions 2,3: ← − − − − k 3 x 2 No change in S Projection has no effect k 4 x 2 − − − − → ∅ Reaction 4: R 4 : X 2 P ([ 0 , − 1 ]) = [ 1 , − 1 ] k 4 x 2 − − − − − → X 1 Constrained version: R 4 : X 2 Simon Cotter Product form stationary dist. 14 / 35

  25. Constrained Approach: Simple Example k 1 R 1 : ∅ − − − − → X 1 Reaction 1: P ([ 1 , 0 ]) = [ 0 , 0 ] Constrained version of R 1 has no effect on X k 2 x 1 − − − − → R 2 , R 3 : X 1 X 2 Reactions 2,3: ← − − − − k 3 x 2 No change in S Projection has no effect k 4 x 2 − − − − → ∅ Reaction 4: R 4 : X 2 P ([ 0 , − 1 ]) = [ 1 , − 1 ] k 4 x 2 − − − − − → X 1 Constrained version: R 4 : X 2 Simon Cotter Product form stationary dist. 14 / 35

  26. Constrained Approach: Simple Example k 1 R 1 : ∅ − − − − → X 1 Reaction 1: P ([ 1 , 0 ]) = [ 0 , 0 ] Constrained version of R 1 has no effect on X k 2 x 1 − − − − → R 2 , R 3 : X 1 X 2 Reactions 2,3: ← − − − − k 3 x 2 No change in S Projection has no effect k 4 x 2 − − − − → ∅ Reaction 4: R 4 : X 2 P ([ 0 , − 1 ]) = [ 1 , − 1 ] k 4 x 2 − − − − − → X 1 Constrained version: R 4 : X 2 Simon Cotter Product form stationary dist. 14 / 35

  27. Constrained Approach: Simple Example k 1 R 1 : ∅ − − − − → X 1 Reaction 1: P ([ 1 , 0 ]) = [ 0 , 0 ] Constrained version of R 1 has no effect on X k 2 x 1 − − − − → R 2 , R 3 : X 1 X 2 Reactions 2,3: ← − − − − k 3 x 2 No change in S Projection has no effect k 4 x 2 − − − − → ∅ Reaction 4: R 4 : X 2 P ([ 0 , − 1 ]) = [ 1 , − 1 ] k 4 x 2 − − − − − → X 1 Constrained version: R 4 : X 2 Simon Cotter Product form stationary dist. 14 / 35

  28. Constrained Approach: Simple Example k 1 R 1 : ∅ − − − − → X 1 Reaction 1: P ([ 1 , 0 ]) = [ 0 , 0 ] Constrained version of R 1 has no effect on X k 2 x 1 − − − − → R 2 , R 3 : X 1 X 2 Reactions 2,3: ← − − − − k 3 x 2 No change in S Projection has no effect k 4 x 2 − − − − → ∅ Reaction 4: R 4 : X 2 P ([ 0 , − 1 ]) = [ 1 , − 1 ] k 4 x 2 − − − − − → X 1 Constrained version: R 4 : X 2 Simon Cotter Product form stationary dist. 14 / 35

  29. Constrained Approach: Simple Example k 1 R 1 : ∅ − − − − → X 1 Reaction 1: P ([ 1 , 0 ]) = [ 0 , 0 ] Constrained version of R 1 has no effect on X k 2 x 1 − − − − → R 2 , R 3 : X 1 X 2 Reactions 2,3: ← − − − − k 3 x 2 No change in S Projection has no effect k 4 x 2 − − − − → ∅ Reaction 4: R 4 : X 2 P ([ 0 , − 1 ]) = [ 1 , − 1 ] k 4 x 2 − − − − − → X 1 Constrained version: R 4 : X 2 Simon Cotter Product form stationary dist. 14 / 35

  30. Constrained Approach: Simple Example k 1 R 1 : ∅ − − − − → X 1 Reaction 1: P ([ 1 , 0 ]) = [ 0 , 0 ] Constrained version of R 1 has no effect on X k 2 x 1 − − − − → R 2 , R 3 : X 1 X 2 Reactions 2,3: ← − − − − k 3 x 2 No change in S Projection has no effect k 4 x 2 − − − − → ∅ Reaction 4: R 4 : X 2 P ([ 0 , − 1 ]) = [ 1 , − 1 ] k 4 x 2 − − − − − → X 1 Constrained version: R 4 : X 2 Simon Cotter Product form stationary dist. 14 / 35

  31. Constrained Approach: Simple Example k 1 R 1 : ∅ − − − − → X 1 Reaction 1: P ([ 1 , 0 ]) = [ 0 , 0 ] Constrained version of R 1 has no effect on X k 2 x 1 − − − − → R 2 , R 3 : X 1 X 2 Reactions 2,3: ← − − − − k 3 x 2 No change in S Projection has no effect k 4 x 2 − − − − → ∅ Reaction 4: R 4 : X 2 P ([ 0 , − 1 ]) = [ 1 , − 1 ] k 4 x 2 − − − − − → X 1 Constrained version: R 4 : X 2 Simon Cotter Product form stationary dist. 14 / 35

  32. Constrained Approach: Simple Example k 2 x 1 − − − − − − − → X 1 X 2 , ← − − − − − − − S = X 1 + X 2 ( k 3 + k 4 ) x 2 Invariant distribution X 2 ∼ B ( S , λ 2 ) = π ( X 2 ) � � ( k 3 + k 4 ) k 2 [ λ 1 , λ 2 ] = k 2 + k 3 + k 4 , steady state solution of mean field k 2 + k 3 + k 4 ODE: k 2 λ 1 = ( k 3 + k 4 ) λ 2 , λ 1 + λ 2 = 1 Compute expectation of the rate of reaction R 4 k 2 k 4 S α 4 = E ( α 4 | S ) = k 4 E ( X 2 | S ) = ˆ k 2 + k 3 + k 4 Simon Cotter Product form stationary dist. 15 / 35

  33. Constrained Approach: Simple Example k 2 x 1 − − − − − − − → X 1 X 2 , ← − − − − − − − S = X 1 + X 2 ( k 3 + k 4 ) x 2 Invariant distribution X 2 ∼ B ( S , λ 2 ) = π ( X 2 ) � � ( k 3 + k 4 ) k 2 [ λ 1 , λ 2 ] = k 2 + k 3 + k 4 , steady state solution of mean field k 2 + k 3 + k 4 ODE: k 2 λ 1 = ( k 3 + k 4 ) λ 2 , λ 1 + λ 2 = 1 Compute expectation of the rate of reaction R 4 k 2 k 4 S α 4 = E ( α 4 | S ) = k 4 E ( X 2 | S ) = ˆ k 2 + k 3 + k 4 Simon Cotter Product form stationary dist. 15 / 35

  34. Constrained Approach: Simple Example k 2 x 1 − − − − − − − → X 1 X 2 , ← − − − − − − − S = X 1 + X 2 ( k 3 + k 4 ) x 2 Invariant distribution X 2 ∼ B ( S , λ 2 ) = π ( X 2 ) � � ( k 3 + k 4 ) k 2 [ λ 1 , λ 2 ] = k 2 + k 3 + k 4 , steady state solution of mean field k 2 + k 3 + k 4 ODE: k 2 λ 1 = ( k 3 + k 4 ) λ 2 , λ 1 + λ 2 = 1 Compute expectation of the rate of reaction R 4 k 2 k 4 S α 4 = E ( α 4 | S ) = k 4 E ( X 2 | S ) = ˆ k 2 + k 3 + k 4 Simon Cotter Product form stationary dist. 15 / 35

  35. Constrained Approach: Simple Example k 2 k 4 / ( k 2 + k 3 + k 4 ) s k 1 ∅ − − − − → S − − − − − − − − − − − → ∅ To compare with QSSA, compute invariant distribution � k 1 ( k 2 + k 3 + k 4 ) � S ( ∞ ) ∼ P (Constrained approximation) k 2 k 4 Invariant distribution identical to true invariant distribution of S = X 1 + X 2 Simon Cotter Product form stationary dist. 16 / 35

  36. Constrained Approach: Simple Example k 2 k 4 / ( k 2 + k 3 + k 4 ) s k 1 ∅ − − − − → S − − − − − − − − − − − → ∅ To compare with QSSA, compute invariant distribution � k 1 ( k 2 + k 3 + k 4 ) � S ( ∞ ) ∼ P (Constrained approximation) k 2 k 4 Invariant distribution identical to true invariant distribution of S = X 1 + X 2 Simon Cotter Product form stationary dist. 16 / 35

  37. Constrained Approach: Simple Example k 2 k 4 / ( k 2 + k 3 + k 4 ) s k 1 ∅ − − − − → S − − − − − − − − − − − → ∅ To compare with QSSA, compute invariant distribution � k 1 ( k 2 + k 3 + k 4 ) � S ( ∞ ) ∼ P (Constrained approximation) k 2 k 4 Invariant distribution identical to true invariant distribution of S = X 1 + X 2 Simon Cotter Product form stationary dist. 16 / 35

  38. Simple Example 0.07 Full generator QSSA generator 0.06 CMA generator 0.05 Probability Mass 0.04 0.03 0.02 0.01 0 0 20 40 60 80 100 S = X 1 + X 2 SLC, “Constrained Approximation of Effective Generators for Multiscale Stochastic Reaction Networks and Application to Conditioned Path Sampling”, in review. Simon Cotter Product form stationary dist. 17 / 35

  39. Outline Introduction 1 The Quasi Steady-State Assumption 2 The Constrained Approach 3 Product Form Stationary Distributions for Non-Mass Action 4 Kinetics 5 Numerical Example Conclusions 6 Simon Cotter Product form stationary dist. 17 / 35

  40. Some Network Properties: Complexes k − → S 3 S 1 + S 2 S 1 + S 2 reactant complex S 3 product complex C number of complexes in network k 1 X 1 ( X 1 − 1 ) − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 C = 4 Simon Cotter Product form stationary dist. 18 / 35

  41. Some Network Properties: Complexes k − → S 3 S 1 + S 2 S 1 + S 2 reactant complex S 3 product complex C number of complexes in network k 1 X 1 ( X 1 − 1 ) − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 C = 4 Simon Cotter Product form stationary dist. 18 / 35

  42. Some Network Properties: Complexes k − → S 3 S 1 + S 2 S 1 + S 2 reactant complex S 3 product complex C number of complexes in network k 1 X 1 ( X 1 − 1 ) − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 C = 4 Simon Cotter Product form stationary dist. 18 / 35

  43. Some Network Properties: Complexes k − → S 3 S 1 + S 2 S 1 + S 2 reactant complex S 3 product complex C number of complexes in network k 1 X 1 ( X 1 − 1 ) − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 C = 4 Simon Cotter Product form stationary dist. 18 / 35

  44. Some Network Properties: Complexes k − → S 3 S 1 + S 2 S 1 + S 2 reactant complex S 3 product complex C number of complexes in network k 1 X 1 ( X 1 − 1 ) − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 C = 4 Simon Cotter Product form stationary dist. 18 / 35

  45. Some Network Properties: Complexes k − → S 3 S 1 + S 2 S 1 + S 2 reactant complex S 3 product complex C number of complexes in network k 1 X 1 ( X 1 − 1 ) − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 C = 4 Simon Cotter Product form stationary dist. 18 / 35

  46. Some Network Properties: Linkage Classes Network can be represented uniquely by graph with C nodes k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 2 S 1 S 2 S 1 ∅ Each connected graph is a linkage class l number of linkage classes (in this example l = 1) Simon Cotter Product form stationary dist. 19 / 35

  47. Some Network Properties: Linkage Classes Network can be represented uniquely by graph with C nodes k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 2 S 1 S 2 S 1 ∅ Each connected graph is a linkage class l number of linkage classes (in this example l = 1) Simon Cotter Product form stationary dist. 19 / 35

  48. Some Network Properties: Linkage Classes Network can be represented uniquely by graph with C nodes k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 2 S 1 S 2 S 1 ∅ Each connected graph is a linkage class l number of linkage classes (in this example l = 1) Simon Cotter Product form stationary dist. 19 / 35

  49. Some Network Properties: Linkage Classes Network can be represented uniquely by graph with C nodes k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 2 S 1 S 2 S 1 ∅ Each connected graph is a linkage class l number of linkage classes (in this example l = 1) Simon Cotter Product form stationary dist. 19 / 35

  50. Some Network Properties: Linkage Classes Network can be represented uniquely by graph with C nodes k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 2 S 1 S 2 S 1 ∅ Each connected graph is a linkage class l number of linkage classes (in this example l = 1) Simon Cotter Product form stationary dist. 19 / 35

  51. Some Network Properties: Stoichiometric Subspace Denote each reaction as: → ν ′ ν m − − − − m Stochiometric subspace given by: S = span m { ν m − ν ′ m } Denote s = dim ( S ) Simon Cotter Product form stationary dist. 20 / 35

  52. Some Network Properties: Stoichiometric Subspace Denote each reaction as: → ν ′ ν m − − − − m Stochiometric subspace given by: S = span m { ν m − ν ′ m } Denote s = dim ( S ) Simon Cotter Product form stationary dist. 20 / 35

  53. Some Network Properties: Stoichiometric Subspace Denote each reaction as: → ν ′ ν m − − − − m Stochiometric subspace given by: S = span m { ν m − ν ′ m } Denote s = dim ( S ) Simon Cotter Product form stationary dist. 20 / 35

  54. Some Network Properties: Deficiency Zero Deficiency δ = C − l − s k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , − − − − → ∅ , S 1 k 2 X 2 δ = 4 − 1 − 2 = 1 Networks with deficiency zero have certain properties and are well studied Simon Cotter Product form stationary dist. 21 / 35

  55. Some Network Properties: Deficiency Zero Deficiency δ = C − l − s k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , − − − − → ∅ , S 1 k 2 X 2 δ = 4 − 1 − 2 = 1 Networks with deficiency zero have certain properties and are well studied Simon Cotter Product form stationary dist. 21 / 35

  56. Some Network Properties: Deficiency Zero Deficiency δ = C − l − s k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , − − − − → ∅ , S 1 k 2 X 2 δ = 4 − 1 − 2 = 1 Networks with deficiency zero have certain properties and are well studied Simon Cotter Product form stationary dist. 21 / 35

  57. Some Network Properties: Weak Reversibility ODE models of zero deficiency weakly reversible networks have a unique complex balance (steady-state of the ODE) System is reversible if all reactions are reversible 1 2 3 C 1 C 2 C 3 5 4 C 5 C 4 Simon Cotter Product form stationary dist. 22 / 35

  58. Some Network Properties: Weak Reversibility ODE models of zero deficiency weakly reversible networks have a unique complex balance (steady-state of the ODE) System is reversible if all reactions are reversible 1 2 3 C 1 C 2 C 3 5 4 C 5 C 4 Simon Cotter Product form stationary dist. 22 / 35

  59. Some Network Properties: Weak Reversibility ODE models of zero deficiency weakly reversible networks have a unique complex balance (steady-state of the ODE) System is reversible if all reactions are reversible 1 2 3 C 1 C 2 C 3 5 4 C 5 C 4 Simon Cotter Product form stationary dist. 22 / 35

  60. Some Network Properties: Weak Reversibility ODE models of zero deficiency weakly reversible networks have a unique complex balance (steady-state of the ODE) System is reversible if all reactions are reversible 1 2 3 C 1 C 2 C 3 5 4 C 5 C 4 Simon Cotter Product form stationary dist. 22 / 35

  61. Some Network Properties: Weak Reversibility ODE models of zero deficiency weakly reversible networks have a unique complex balance (steady-state of the ODE) System is reversible if all reactions are reversible 1 2 3 C 1 C 2 C 3 5 4 C 5 C 4 Simon Cotter Product form stationary dist. 22 / 35

  62. Some Network Properties: Weak Reversibility ODE models of zero deficiency weakly reversible networks have a unique complex balance (steady-state of the ODE) System is reversible if all reactions are reversible 1 2 3 C 1 C 2 C 3 5 4 C 5 C 4 Simon Cotter Product form stationary dist. 22 / 35

  63. Some Network Properties: Weak Reversibility ODE models of zero deficiency weakly reversible networks have a unique complex balance (steady-state of the ODE) System is reversible if all reactions are reversible 1 2 3 C 1 C 2 C 3 5 4 C 5 C 4 Simon Cotter Product form stationary dist. 22 / 35

  64. Preliminaries: form of intensity functions The theorem that follows relies on the following form of the intensity/hazard functions d ν mi − 1 � � α m ( x ) = k m θ i ( x i − j ) , i = 1 j = 0 Mass action kinetics if θ is the identity Simon Cotter Product form stationary dist. 23 / 35

  65. Preliminaries: form of intensity functions The theorem that follows relies on the following form of the intensity/hazard functions d ν mi − 1 � � α m ( x ) = k m θ i ( x i − j ) , i = 1 j = 0 Mass action kinetics if θ is the identity Simon Cotter Product form stationary dist. 23 / 35

  66. Preliminaries: form of intensity functions The theorem that follows relies on the following form of the intensity/hazard functions d ν mi − 1 � � α m ( x ) = k m θ i ( x i − j ) , i = 1 j = 0 Mass action kinetics if θ is the identity Simon Cotter Product form stationary dist. 23 / 35

  67. Product Form Stationary Distributions Theorem (Anderson, Craciun, Kurtz, 2010) Let {S , C , R} be a zero deficiency, weakly reversible reaction network. Modeled deterministically with mass action kinetics and rate constants k m the system has complex balanced equilibrium c ∈ R d > 0 . Then, the stochastically modeled system with the specified intensity functions admits the invariant measure on each stoichiometric class d c x i � i π ( x ) ∝ . � x i − 1 j = 0 θ i ( x i − j ) i = 1 The measure π can be normalized to a stationary distribution so long as it is summable. Simon Cotter Product form stationary dist. 24 / 35

  68. Motivating Example for New Result k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 Multiscale system with reversible dimerisation reactions fast Consider the constrained fast subsystem with slow variable S = X 1 + 2 X 2 k 1 X 1 ( X 1 − 1 )+ k 3 ✶ { X 1 > 1 } − − − − − − − − − − − → 2 S 1 S 2 , ← − − − − − − − − − − − ( k 2 + k 4 ) X 2 Intensity functions do not satisfy the assumptions of the existing results Simon Cotter Product form stationary dist. 25 / 35

  69. Motivating Example for New Result k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 Multiscale system with reversible dimerisation reactions fast Consider the constrained fast subsystem with slow variable S = X 1 + 2 X 2 k 1 X 1 ( X 1 − 1 )+ k 3 ✶ { X 1 > 1 } − − − − − − − − − − − → 2 S 1 S 2 , ← − − − − − − − − − − − ( k 2 + k 4 ) X 2 Intensity functions do not satisfy the assumptions of the existing results Simon Cotter Product form stationary dist. 25 / 35

  70. Motivating Example for New Result k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 Multiscale system with reversible dimerisation reactions fast Consider the constrained fast subsystem with slow variable S = X 1 + 2 X 2 k 1 X 1 ( X 1 − 1 )+ k 3 ✶ { X 1 > 1 } − − − − − − − − − − − → 2 S 1 S 2 , ← − − − − − − − − − − − ( k 2 + k 4 ) X 2 Intensity functions do not satisfy the assumptions of the existing results Simon Cotter Product form stationary dist. 25 / 35

  71. Motivating Example for New Result k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 Multiscale system with reversible dimerisation reactions fast Consider the constrained fast subsystem with slow variable S = X 1 + 2 X 2 k 1 X 1 ( X 1 − 1 )+ k 3 ✶ { X 1 > 1 } − − − − − − − − − − − → 2 S 1 S 2 , ← − − − − − − − − − − − ( k 2 + k 4 ) X 2 Intensity functions do not satisfy the assumptions of the existing results Simon Cotter Product form stationary dist. 25 / 35

  72. Motivating Example for New Result k 1 X 1 ( X 1 − 1 ) − − − − − − − → k 3 k 4 X 1 2 S 1 S 2 , ← − − − − − − − ∅ − − − − → S 2 , S 1 − − − − → ∅ , k 2 X 2 Multiscale system with reversible dimerisation reactions fast Consider the constrained fast subsystem with slow variable S = X 1 + 2 X 2 k 1 X 1 ( X 1 − 1 )+ k 3 ✶ { X 1 > 1 } − − − − − − − − − − − → 2 S 1 S 2 , ← − − − − − − − − − − − ( k 2 + k 4 ) X 2 Intensity functions do not satisfy the assumptions of the existing results Simon Cotter Product form stationary dist. 25 / 35

  73. New Structural Assumption Partition all species S = { S 1 , S 2 , . . . , S N } into S 1 and S 2 . S i ∈ S 2 if for some η i ≥ 2 we have that ν ki is always some nonnegative multiple of η i For all other S i ∈ S 1 , let η i = 1. Assume that all intensity functions satisfy: ν mi η i − 1 d � � α m ( x ) = k m θ i ( X i − j η i ) , i = 1 j = 0 where κ k > 0 and θ i ( x i ) = 0 if and only if x i ≤ α i − 1. Simon Cotter Product form stationary dist. 26 / 35

  74. New Structural Assumption Partition all species S = { S 1 , S 2 , . . . , S N } into S 1 and S 2 . S i ∈ S 2 if for some η i ≥ 2 we have that ν ki is always some nonnegative multiple of η i For all other S i ∈ S 1 , let η i = 1. Assume that all intensity functions satisfy: ν mi η i − 1 d � � α m ( x ) = k m θ i ( X i − j η i ) , i = 1 j = 0 where κ k > 0 and θ i ( x i ) = 0 if and only if x i ≤ α i − 1. Simon Cotter Product form stationary dist. 26 / 35

  75. New Structural Assumption Partition all species S = { S 1 , S 2 , . . . , S N } into S 1 and S 2 . S i ∈ S 2 if for some η i ≥ 2 we have that ν ki is always some nonnegative multiple of η i For all other S i ∈ S 1 , let η i = 1. Assume that all intensity functions satisfy: ν mi η i − 1 d � � α m ( x ) = k m θ i ( X i − j η i ) , i = 1 j = 0 where κ k > 0 and θ i ( x i ) = 0 if and only if x i ≤ α i − 1. Simon Cotter Product form stationary dist. 26 / 35

  76. New Structural Assumption Partition all species S = { S 1 , S 2 , . . . , S N } into S 1 and S 2 . S i ∈ S 2 if for some η i ≥ 2 we have that ν ki is always some nonnegative multiple of η i For all other S i ∈ S 1 , let η i = 1. Assume that all intensity functions satisfy: ν mi η i − 1 d � � α m ( x ) = k m θ i ( X i − j η i ) , i = 1 j = 0 where κ k > 0 and θ i ( x i ) = 0 if and only if x i ≤ α i − 1. Simon Cotter Product form stationary dist. 26 / 35

  77. New Structural Assumption Partition all species S = { S 1 , S 2 , . . . , S N } into S 1 and S 2 . S i ∈ S 2 if for some η i ≥ 2 we have that ν ki is always some nonnegative multiple of η i For all other S i ∈ S 1 , let η i = 1. Assume that all intensity functions satisfy: ν mi η i − 1 d � � α m ( x ) = k m θ i ( X i − j η i ) , i = 1 j = 0 where κ k > 0 and θ i ( x i ) = 0 if and only if x i ≤ α i − 1. Simon Cotter Product form stationary dist. 26 / 35

  78. Theorem: Product Form Stationary Distributions Theorem (Anderson, Cotter, 2016) Let {S 1 ∪ S 2 , C , R} be a zero deficiency, weakly reversible reaction network satisfying the above assumption. Modeled deterministically with mass action kinetics and rate constants k m the system has complex balanced equilibrium c ∈ R d > 0 . Then, the stochastically modeled system with the specified intensity functions admits the invariant measure on each stoichiometric class d c X i � i π ( x ) ∝ . � ⌊ X i /η i ⌋− 1 θ i ( X i − j η i ) i = 1 j = 0 The measure π can be normalized to a stationary distribution so long as it is summable. Simon Cotter Product form stationary dist. 27 / 35

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