Efficient Reduction of Kappa Models by Static Inspection of the Rule-Set
Andreea Beica, Calin C. Guet, and Tatjana Petrov
Total Spoilage!
\text{(if } k_2 << k_3 \text{ or concentration of E is low)}
Stochastic Chemical
Reaction Networks
Rule Based Model (Kappa)
Enzimatic Reduction (Reactions)
Enzimatic Reduction (Rules)
Stochastic Chemical Reaction Networks
\text{Well mixed system with molecular species:}
S = \{S_1, ... , S_n \}
\text{State is a multiset of species:}
x = (x_1, ... , x_n) \in \mathbb{N}^n
\text{Reaction defined by }rate\text{ and }production\text{ and }consumption\text{ vector:}
r_j \equiv (a_j, \nu_j, c_j) \in \mathbb{N}^n \times \mathbb{N}^n \times \mathbb{R}_{\geq 0}
\text{Reaction semantics:}
\text{Reaction defined by }rate\text{ and }production\text{ and }consumption\text{ vector:}
r_j \equiv (a_j, \nu_j, c_j) \in \mathbb{N}^n \times \mathbb{N}^n \times \mathbb{R}_{\geq 0}
??!
\mathbb{Z}^n
Stochastic Semantics (Markov Chain)
Deterministic Semantics (ODE)
N \to \infty
Reduction Algorithm
Modifier Elimination
A + B + C \to A:C + B
A + C \to A:C
~
Similar Reaction Composition
A + 2B \to A:B + C
3A + 2B \to 2A:B + 4C
~
3A + 4B \to 3A:B + 5C
Fast Dimerisation Reduction
M + M \to M_2
(assuming the reaction rate is fast)
~
\text{New specie } M_T \text{ replaces both } M \text{ and } M_2
Generalised Enzymatic Reduction
\lambda-phage \text{ Model}
\text{92 species, 13 rules, 61 species } \rightarrow \text{ 5 proteins and 11 rules}
Conclusions
- Technique designed for stochastic reaction networks applied to Kappa models
- Tailored for GRNs - applications to other types of models are an open question (so far, it is not looking great)
- Implementation in OCaml
Kappa reductions
By Samuel Pastva
