Logic-Based Modeling in Systems Biology Alexander Bockmayr - PowerPoint PPT Presentation

Logic-Based Modeling in Systems Biology Alexander Bockmayr LPNMR’09, Potsdam, 16 September 2009 DFG Research Center Matheon Mathematics for key technologies

Outline I. Systems biology II. Logic modeling of regulatory networks A. Boolean logic B. Multi-valued logic III. Logical analysis of network dynamics IV. Application: Bio-Logic A.Bockmayr, FU Berlin/Matheon 2

3 I. Systems biology A.Bockmayr, FU Berlin/Matheon

Systems biology Molecular Systems biology biology Very active interdisciplinary research field A.Bockmayr, FU Berlin/Matheon 4

Modeling in systems biology Various types of biological networks • metabolic • regulatory • signaling, … Various modeling approaches • continuous (ordinary/partial differential equations) • stochastic (chemical master equation) • discrete (logic, Petri nets, process calculi, …) • hybrid (continuous/stochastic, discrete/continuous) Here: Logic-based discrete modeling of regulatory networks A.Bockmayr, FU Berlin/Matheon 5

Logic-based modeling of regulatory networks 1. Logic modeling of the network structure • Boolean logic • Multi-valued logic 2. Logical analysis of the dynamics • Non-determinism • Temporal logic • Model checking A.Bockmayr, FU Berlin/Matheon 6

II. Logic modeling of regulatory networks A) Boolean logic A.Bockmayr, FU Berlin/Matheon 7

Interaction graph + Nodes _ (component is active or not) Arcs _ + Activation: Inhibition: + A.Bockmayr, FU Berlin/Matheon 8

Boolean model Sugita 61, Kauffman 69 1. Boolean variables 2. Boolean mapping F i ( X 1 , … , X n ) describes how the next state of X i depends on the current state of ( X 1 ,…, X n ). discrete dynamics A.Bockmayr, FU Berlin/Matheon 9

Two-element negative circuit Interaction graph State transition graph + _ A.Bockmayr, FU Berlin/Matheon 10

11 Two-element positive circuit _ Interaction graph A.Bockmayr, FU Berlin/Matheon _

Synchronous vs. asynchronous Thomas´ 73: Update only one variable at a time. Nondeterminism: Several successor states possible asynchronous synchronous A.Bockmayr, FU Berlin/Matheon 12

Two-element positive circuit Interaction graph State transition graph _ _ A.Bockmayr, FU Berlin/Matheon 13

II. Logic modeling of regulatory networks B) Multi-valued logic A.Bockmayr, FU Berlin/Matheon 14

Thresholds and activity levels Thomas/Snoussi 88 If component j acts on n j other components (up to) n j thresholds: activity level of component j is above the k -th threshold and below the ( k + 1)-th. discrete update function with discrete parameter vector . A.Bockmayr, FU Berlin/Matheon 15

Asynchronous update State State transitions if resp. discrete non-deterministic dynamics A.Bockmayr, FU Berlin/Matheon 16

State transition graph A.Bockmayr, FU Berlin/Matheon 17

Stable states and cycles A.Bockmayr, FU Berlin/Matheon 18

Positive and negative circuits + _ _ _ Positive 2-circuit _ + _ Negative 2-circuit Sign of circuit = Product of signs of arcs A.Bockmayr, FU Berlin/Matheon 19

Thomas´ rules Thomas´ 81 A positive circuit in the interaction graph is a necessary condition for multistationarity. A negative circuit in the interaction graph is a necessary condition for stable periodic behavior. [Proofs exist in various scenarios.] A.Bockmayr, FU Berlin/Matheon 20

Example • X 1 ∈ {0,1} • X 2 ∈ {0,1,2} • Assume θ 12 < θ 22 , i.e., when activated, X 2 acts first on X 1 , then on itself. A.Bockmayr, FU Berlin/Matheon 21

K 12 = 1, K 21 = 0, K 22 = K 21+ 22 = 2 2 stable states no cycle 2 separate domains A.Bockmayr, FU Berlin/Matheon 22

K 12 = 1, K 21 = 1, K 22 = K 21+ 22 = 2 1 stable state 1 cycle 2 separate domains A.Bockmayr, FU Berlin/Matheon 23

Continuous model A.Bockmayr, FU Berlin/Matheon 24

25 III. Logical analysis of the dynamics A.Bockmayr, FU Berlin/Matheon

Model checking Clarke/Emerson and Sifakis 81 State transition graph Infinite computation tree (Kripke model) p q p q q r r r q r r r p q exponentially large check properties expressed in some temporal logic. A.Bockmayr, FU Berlin/Matheon 26

Computation Tree Logic (CTL) Atomic formulae : p, q, r, …, e.g. Linear time operators : • X p : p holds next time • F p : p holds sometimes in the future • G p : p holds globally in the future • p U q : p holds until q holds Path quantifiers : • A : for every path • E : there exists a path A.Bockmayr, FU Berlin/Matheon 27

28 Linear time operators pUq Gp Xp Fp p p p p p q p p A.Bockmayr, FU Berlin/Matheon p p p Now p p p

Path quantifiers AGp AFp p p p p p p p p p p EGp EFp p p p p A.Bockmayr, FU Berlin/Matheon 29

CTL model checking for regulatory networks Bernot/Comet/Richard/Guespin 04 Input • Interaction graph / state transition graph • Temporal logic formula (CTL) Output Set of states in which the formula is true Example Can also be used for network inference. A.Bockmayr, FU Berlin/Matheon 30

31 IV. Application: Bio-Logic A.Bockmayr, FU Berlin/Matheon

Bio-Logic Understand the regulatory logic underlying developmental and other biological processes Source: www.chaosscience.org.uk A.Bockmayr, FU Berlin/Matheon 32

Conclusion • Molecular systems biology • Logical modeling of regulatory structures � Boolean logic � Multi-valued logic • Logical analysis of the dynamics � Non-determinism � Temporal logic � Model checking • Bio-Logic A.Bockmayr, FU Berlin/Matheon 33