Exploring Variation in Biochemical Pathways with the Continuous pi-Calculus Ian Stark Marek Kwiatkowski Chris Banks Laboratory for Foundations of Computer Science SynthSys: Synthetic & Systems Biology The University of Edinburgh Evolutionary Ecology Institute of Integrative Biology Department of Environmental Sciences EAWAG & ETH Zürich Thursday 16 June 2012 N I V E U R S E I H T T Y O H F G R E U D I B N
Summary The continuous pi-calculus (c-pi) is a process algebra for modelling behaviour and variation in biomolecular systems: e.g. enzyme activation and inhibition; circadian clocks; signalling pathways. Expressions in c-pi represent mixtures of chemical reagents, and can be compiled to conventional ODE models for fast numerical simulation. With a language of potential changes in c-pi processes we systematically explore evolutionary neighbourhoods of a specific signalling pathway, and observe instances of robustness, neutrality and evolvability. A complementary temporal logic for behaviour in context gives a language to classify these variations in behaviour. Marek Kwiatkowski and Ian Stark. On Executable Models of Molecular Evolution. In Proc. 8th International Workshop on Computational Systems Biology WCSB 2011 , pp. 105–108. Stark, Kwiatkowski, Banks Exploring Variation in Biochemical Pathways with c-pi 2012-06-14
The Continuous pi-Calculus Continuous pi is a name-passing process algebra for modelling behaviour and variation in molecular systems. Based on Milner’s pi-calculus, it introduces real-valued variability in: rates of reaction; affinity between interacting names; and quantities of processes. Although sharing an approach common to process algebras for biomodelling, some features are distinctive. For example, by comparison with the stochastic pi-calculus: ODEs are the primary mode of execution, not stochastic simulation Continuous concentrations of chemicals replace discrete individuals End-to-end channels are replaced by multiple competing names Stark, Kwiatkowski, Banks Exploring Variation in Biochemical Pathways with c-pi 2012-06-14
Basics of Continuous pi Continuous pi has two levels of system description: Species Individual molecules (proteins) Transition system semantics Processes Bulk population (concentration) Differential equations Process space arises as a real-valued vector space over species, with each point the state of a system and behaviours as trajectories through that. Stark, Kwiatkowski, Banks Exploring Variation in Biochemical Pathways with c-pi 2012-06-14
Example: Enzyme Catalysis S = s ( x , y ) . ( x . S + y . ( P | P ′ )) E = ν ( u , r , t : M ) . ( e � u , r � . t . E ) P = P ′ = τ @ k degrade .0 E | S k bind s u r νM ( t . E | ( u . S + r . ( P | P ′ ))) k bind k unbind k react e M t k unbind k react E | P | P ′ E | S Stark, Kwiatkowski, Banks Exploring Variation in Biochemical Pathways with c-pi 2012-06-14
Example: Enzyme Catalysis S = s ( x , y ) . ( x . S + y . ( P | P ′ )) E = ν ( u , r , t : M ) . ( e � u , r � . t . E ) P = P ′ = τ @ k degrade .0 Cpi tool Octave enzyme.cpi ODEs . . . x ′ 2 = − k 1 x 4 x 2 + . . . species E() = { . site u, r, t; . . . . . Stark, Kwiatkowski, Banks Exploring Variation in Biochemical Pathways with c-pi 2012-06-14
Process Space: Substrate & Product & Enzyme 10 9.5 9 Enzyme 8.5 8 7.5 0 200 400 0 Substrate 600 100 200 800 300 400 Product 1000 500 Stark, Kwiatkowski, Banks Exploring Variation in Biochemical Pathways with c-pi 2012-06-14
Biomodelling in Continuous pi For individual species, continuous pi uses a modelling idiom based on that of Regev and Shapiro: Reagent-centric rather than rule-based Individual species are represented by processes Complexes are modelled by name restriction νx . ( A | B ) Interaction is modelled by communication between names ...but with competition between multiple alternatives Stark, Kwiatkowski, Banks Exploring Variation in Biochemical Pathways with c-pi 2012-06-14
From Species to Processes Take a language for interaction between individual species and raise it into one for reactions in mixtures: Species A , B :: = Σα . A | A | B | νM . A | . . . Processes P , Q :: = 0 c · A P � Q | | Component c · A of species A at concentration c ∈ R � 0 . Mixture of processes P � Q . We can identify processes with elements of process space P = R S , where S is the set of species. Stark, Kwiatkowski, Banks Exploring Variation in Biochemical Pathways with c-pi 2012-06-14
Process Semantics dP dt : Immediate behaviour ∂P : Interaction potential d Vector field dt over process space P Captures available reactivity Element of R N × S × C Equivalent to an ODE system ∂ ( P � Q ) = ∂P + ∂Q d ( P � Q ) = dP dt + dQ dt + ∂P � ∂Q dt Both dP dt and ∂P are defined by induction on the structure of processes; and beneath that, from the transitions of component species c · A . With this, we are able to compose the phase portraits of our systems. Stark, Kwiatkowski, Banks Exploring Variation in Biochemical Pathways with c-pi 2012-06-14
Example: Synechococcus Elongatus S. Elongatus circadian clock proteins, effective in vitro: KaiA, KaiB and KaiC. (Tomita et al. 2005) Several mechanisms have been proposed: one is the cyclic six-fold phosphorylation of KaiC hexamers in two alternative conformations, stabilised by KaiA and KaiB. (van Zon et al. 2007) Sherman&Sherman, Purdue k p k p k p · · · C 0 C 1 C 6 f ′ f 6 0 C ′ C ′ C ′ · · · 0 1 6 k ′ k ′ k ′ d d d RCSB Protein Data Bank Stark, Kwiatkowski, Banks Exploring Variation in Biochemical Pathways with c-pi 2012-06-14
Execution and Modification Stark, Kwiatkowski, Banks Exploring Variation in Biochemical Pathways with c-pi 2012-06-14
Process Algebras for Molecular Evolution One way to model molecular evolution is by specific modifications of concrete mathematical models. Process algebras, and similar intermediate languages, offer a framework to generalise this model for variation and selection. Process ∼ Genotype Execution ∼ Development Behaviour ∼ Phenotype Relevant features of models like continuous pi include: Reagent-centric models to match genetic variation Free formation of new terms, particularly novel complexes Computable behaviour of created components Stark, Kwiatkowski, Banks Exploring Variation in Biochemical Pathways with c-pi 2012-06-14
Variation Operators Variation operators are transformations of c-pi models which correspond to evolutionary events. Ideally, a suite of such operations should: Maintain the biological idiom Be biologically meaningful Be expressive enough to build new reaction networks from scratch Stark, Kwiatkowski, Banks Exploring Variation in Biochemical Pathways with c-pi 2012-06-14
Variation Operators Variation operators are transformations of c-pi models which correspond to evolutionary events. For example: site reconfiguration b b k 1 a a k 3 d k 3 d k 2 k 4 k 4 c c Stark, Kwiatkowski, Banks Exploring Variation in Biochemical Pathways with c-pi 2012-06-14
Variation Operators Variation operators are transformations of c-pi models which correspond to evolutionary events. For example: site reconfiguration b b k 1 a a k 3 d k 3 d k 2 k 4 k 4 c c We have defined a dozen such operators modelling gene duplications, gene knockouts, changes in activity rates within complexes, and more. Stark, Kwiatkowski, Banks Exploring Variation in Biochemical Pathways with c-pi 2012-06-14
Simplified MAPK Cascade Ras Raf Raf* PP2A1 MEK MEK* MEK** PP2A2 ERK ERK* ERK** MKP3 Stark, Kwiatkowski, Banks Exploring Variation in Biochemical Pathways with c-pi 2012-06-14
MAPK in Continuous pi Ras = ( ν x � x ) ras ( x ; y ) . ( x . Ras + y . Ras ) Raf = ( ν x � x ) raf ( x ; y ) . ( x . Raf + y . Raf ∗ ) . . . ERK ∗∗ = ( ν x � x ) erk ∗∗ b ( x ; y ) . ( x . ERK ∗∗ + y . ERK ∗ ) MKP3 = ( ν x � x ) mkp3 ( x ; y ) . ( x . MKP3 + y . MKP3 ) Π = c 1 · Raf � c 2 · Ras � . . . � c 4 · ERK � c 7 · MKP3 ras raf ∗ mek ∗ mek ∗∗ erk ∗ raf mek erk erk ∗∗ raf ∗ mek ∗ mek ∗∗ erk ∗ b b b b pp2a1 pp2a2 mkp3 Stark, Kwiatkowski, Banks Exploring Variation in Biochemical Pathways with c-pi 2012-06-14
MAPK Behaviour This MAPK model compiles into 23 differential equations, which are then solved with Octave. The signalling cascade correctly transmits initial presence of Ras into a peak of ERK** via Raf* and MEK** . Stark, Kwiatkowski, Banks Exploring Variation in Biochemical Pathways with c-pi 2012-06-14
Evolutionary Analysis of MAPK ras raf ∗ mek ∗ mek ∗∗ erk ∗ raf mek erk erk ∗∗ raf ∗ mek ∗ mek ∗∗ erk ∗ b b b b pp2a1 pp2a2 mkp3 Reconfigure every site in every way possible (16 × 2 16 ≈ 10 6 ). Generate ODEs and hence behaviour traces for every variant. Qualitative analysis Classify phenotypes with LTL model-checking Find evolutionarily fragile and robust sites Quantitative analysis Compute the fitness of every variant using signal integration Find the distribution of mutation effects on fitness Stark, Kwiatkowski, Banks Exploring Variation in Biochemical Pathways with c-pi 2012-06-14
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