Rule-Based Modeling of Bio-Chemical Networks Workshop on Modelling in Biology and Medicine – MBM2019 Sandro Stucki Computer Science Engineering (CSE) Gothenburg University | Chalmers Gothenburg, 9 May 2019 1 / 25
Why use programming or modeling languages? 2 / 25
Why? Why use programming or modeling languages? Syntax • formal, standardized knowledge representation, • shareable, • executable. 3 / 25
Why? Why use programming or modeling languages? Syntax • formal, standardized knowledge representation, • shareable, • executable. Formal semantics • precise mathematical meaning of programs/models, • enables formal reasoning. 3 / 25
Why? Why use programming or modeling languages? Syntax • formal, standardized knowledge representation, • shareable, • executable. Formal semantics • precise mathematical meaning of programs/models, • enables formal reasoning. Tooling • execution, simulation, • translation, transformation, reduction, • analysis, verification. 3 / 25
CrnABK-data.csv 100 A Number of Molecules 80 B K 60 40 20 0 0 0.5 1 1.5 2 Time How? Formal modeling languages – my wish list: 4 / 25
CrnABK-data.csv 100 A Number of Molecules 80 B K 60 40 20 0 0 0.5 1 1.5 2 Time How? Formal modeling languages – my wish list: • Simple yet expressive syntax/formalism. 4 / 25
CrnABK-data.csv 100 A Number of Molecules 80 B K 60 40 20 0 0 0.5 1 1.5 2 Time How? Formal modeling languages – my wish list: • Simple yet expressive syntax/formalism. • Formal semantics. 4 / 25
CrnABK-data.csv 100 A Number of Molecules 80 B K 60 40 20 0 0 0.5 1 1.5 2 Time How? Formal modeling languages – my wish list: • Simple yet expressive syntax/formalism. • Formal semantics. • Automation and tooling for manipulating models. 4 / 25
CrnABK-data.csv 100 A Number of Molecules 80 B K 60 40 20 0 0 0.5 1 1.5 2 Time How? Formal modeling languages – my wish list: • Simple yet expressive syntax/formalism. • Formal semantics. • Automation and tooling for manipulating models. Example: Chemical Reaction Networks (CRNs) 2 A + K B + K k on , k off ⇋ B ∅ k deg ⇀ 4 / 25
How? Formal modeling languages – my wish list: • Simple yet expressive syntax/formalism. • Formal semantics. • Automation and tooling for manipulating models. Example: Chemical Reaction Networks (CRNs) 2 A + K B + K k on , k off ⇋ B ∅ k deg ⇀ CrnABK-data.csv 100 A Number of Molecules 80 B K 60 40 20 0 0 0.5 1 1.5 2 Time 4 / 25
Chemical Reaction Networks (CRNs) 2 A + K B + K k on , k off ⇋ B k deg ∅ ⇀ Syntax • many formats, graphical textual, etc. • for example, SBML http://sbml.org/ . 5 / 25
Chemical Reaction Networks (CRNs) 2 A + K B + K k on , k off ⇋ B k deg ∅ ⇀ Syntax • many formats, graphical textual, etc. • for example, SBML http://sbml.org/ . Formal semantics • stochastic: Markov processes (CTMC), • differential: rate equations (ODEs), • others (e.g. Boolean). 5 / 25
Chemical Reaction Networks (CRNs) 2 A + K B + K k on , k off ⇋ B k deg ∅ ⇀ Syntax • many formats, graphical textual, etc. • for example, SBML http://sbml.org/ . Formal semantics • stochastic: Markov processes (CTMC), • differential: rate equations (ODEs), • others (e.g. Boolean). Lots of tooling! (See e.g. http://sbml.org/SBML_Software_Guide ) • stochastic simulation (Monte Carlo/Gillespie), • numerical integration, • analysis, verification, . . . 5 / 25
CRNs as a stochastic process C A B A A C A 6 / 25
CRNs as a stochastic process 1. Pick a reaction α at random (weighted by k α × # matches). k α A A B A B C A B A A C A 6 / 25
CRNs as a stochastic process 1. Pick a reaction α at random (weighted by k α × # matches). 2. Pink a match in the current state M at random. k α A A B A B C A B A A C A 6 / 25
CRNs as a stochastic process 1. Pick a reaction α at random (weighted by k α × # matches). 2. Pink a match in the current state M at random. 3. Update M according to α to obtain a future state M ′ . k α A A B A B C C A A B A B A A A A C C A A 6 / 25
CRNs as a stochastic process 1. Pick a reaction α at random (weighted by k α × # matches). 2. Pink a match in the current state M at random. 3. Update M according to α to obtain a future state M ′ . 4. Advance time (Poisson process). k α A A B A B C C A A B A B A A A A C C A A 6 / 25
Rate equations (continuous semantics) • A system of ordinary differential equations (ODEs). • State space is a vector of concentrations/densities. • Derivatives are proportional to production/consumption of molecules by rules (mass action). 2 A + K B + K k on , k off ⇋ B k deg ∅ ⇀ 7 / 25
Rate equations (continuous semantics) • A system of ordinary differential equations (ODEs). • State space is a vector of concentrations/densities. • Derivatives are proportional to production/consumption of molecules by rules (mass action). 2 A + K B + K k on , k off ⇋ B k deg ∅ ⇀ − 1 d 2 k on [ A ] 2 [ K ] dt [ A ] = 2 k off [ B ][ K ] dt [ B ] = 1 d 2 k on [ A ] 2 [ K ] − 2 k off [ B ][ K ] − k deg [ B ] 7 / 25
Rate equations (continuous semantics) • A system of ordinary differential equations (ODEs). • State space is a vector of concentrations/densities. • Derivatives are proportional to production/consumption of molecules by rules (mass action). − 1 d 2 k on [ A ] 2 [ K ] dt [ A ] = 2 k off [ B ][ K ] dt [ B ] = 1 d 2 k on [ A ] 2 [ K ] − 2 k off [ B ][ K ] − k deg [ B ] • The REs approximate the behavior of the CTMC semantics in the limit of large molecule counts (abstraction). • Can solved by numerical integration (more efficient than stochastic simulation for large systems). 7 / 25
The challenge Combinatorial Explosion MAPK pathway , diagram by Kosigrim (Wikipedia), 2007. 8 / 25
Biochemical reaction networks EGF EGF GRB2 GRB2 r r b b a a l l d d Y68 r Y68 r SOS EGFR SOS EGFR Y7 Y7 Y48 Y48 c p SHC SHC Maps for the early EGF model , Figure 5, [Danos et al., 2010]. Biochemical reaction networks can suffer from high combinatorial complexity: • Molecules interact through domains. • A single chemical species may display multiple domains. • The number of species exhibits a combinatorial explosion w.r.t. the number of domain interactions. 9 / 25
Biochemical reaction networks k A , k ′ A + B AB ⇋ A k A , k ′ A + BC ABC ⇋ A k C , k ′ B + C BC ⇋ C k C , k ′ AB + C ABC ⇋ C Biochemical reaction networks can suffer from high combinatorial complexity: • Molecules interact through domains. • A single chemical species may display multiple domains. • The number of species exhibits a combinatorial explosion w.r.t. the number of domain interactions. 9 / 25
Polymers Worst case: polymerization reactions involve an infinite number of species/reactions. k A , k ′ A + A AA ⇋ A AA + A k A , k ′ AAA ⇋ A k A , k ′ AAA + A AAAA ⇋ A . . . 10 / 25
Rule-Based Models (RBMs) Rule-based modeling languages such as Kappa and BNGL 1 have been introduced to deal with this combinatorial complexity. • Rules describe interaction on the domain level. • A single rule captures a (possibly infinite) set of reactions. k A , k ′ p y p y A x B A x B ⇋ A k A , k ′ p y q p y q A x B C A x B C ⇋ A k C , k ′ y q y q x B C x B C ⇋ C k C , k ′ p y q p y q A x B C A x B C ⇋ C 1 See e.g. [Danos et al., 2007] and [Blinov et al., 2004] 11 / 25
Rule-Based Models (RBMs) Rule-based modeling languages such as Kappa and BNGL 1 have been introduced to deal with this combinatorial complexity. • Rules describe interaction on the domain level. • A single rule captures a (possibly infinite) set of reactions. k A , k ′ p y p y A x B A x B ⇋ A k A , k ′ p y q p y q A x B C A x B C ⇋ A k C , k ′ y q y q x B C x B C ⇋ C k C , k ′ p y q p y q A x B C A x B C ⇋ C 1 See e.g. [Danos et al., 2007] and [Blinov et al., 2004] 11 / 25
Rule-Based Models (RBMs) Rule-based modeling languages such as Kappa and BNGL 1 have been introduced to deal with this combinatorial complexity. • Rules describe interaction on the domain level. • A single rule captures a (possibly infinite) set of reactions. k A , k ′ p p A x B A x B ⇋ A k C , k ′ y q y q B C B C ⇋ C 1 See e.g. [Danos et al., 2007] and [Blinov et al., 2004] 11 / 25
Rule-Based Models (RBMs) Rule-based modeling languages such as Kappa and BNGL 1 have been introduced to deal with this combinatorial complexity. • Rules describe interaction on the domain level. • A single rule captures a (possibly infinite) set of reactions. k A , k ′ p p A x B A x B ⇋ A k C , k ′ y q y q B C B C ⇋ C A ( p 1 ) , B ( x 1 ) A ( p ) , B ( x ) k A , k ′ ⇋ A B ( y 2 ) , C ( q 2 ) B ( y ) , C ( q ) k C , k ′ ⇋ C 1 See e.g. [Danos et al., 2007] and [Blinov et al., 2004] 11 / 25
Polymers (cont.) Polymerization reactions can be expressed compactly. k A , k ′ y y A x A A x A ⇋ A 12 / 25
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