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

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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

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• I. Systems biology

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Systems biology

Molecular biology Systems biology Very active interdisciplinary research field

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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

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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

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• II. Logic modeling of regulatory

networks A) Boolean logic

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Interaction graph

+ _ + _ +

Nodes

(component is active or not)

Arcs

Activation: Inhibition:

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Boolean model

• 1. Boolean variables
• 2. Boolean mapping

Fi(X1,…,Xn) describes how the next state of Xi

depends on the current state of (X1,…,Xn). discrete dynamics

Sugita 61, Kauffman 69

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Two-element negative circuit

Interaction graph

_ +

State transition graph

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Two-element positive circuit

Interaction graph

_ _

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Synchronous vs. asynchronous

Thomas´ 73: Update only one variable at a time. Nondeterminism: Several successor states possible asynchronous synchronous

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Two-element positive circuit

Interaction graph State transition graph

_ _

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• II. Logic modeling of regulatory

networks B) Multi-valued logic

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Thomas/Snoussi 88 If component j acts on nj other components (up to) nj 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 .

Thresholds and activity levels

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Asynchronous update

State State transitions if resp. discrete non-deterministic dynamics

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State transition graph

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Stable states and cycles

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Positive and negative circuits

_ + _

Sign of circuit = Product of signs of arcs

_ + _ _

Positive 2-circuit Negative 2-circuit

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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.]

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Example

• X1 ∈ {0,1}
• X2 ∈ {0,1,2}
• Assume θ12< θ22, i.e., when

activated, X2 acts first on X1, then on itself.

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K12= 1, K21 = 0, K22 = K21+ 22 = 2

2 stable states no cycle 2 separate domains

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K12= 1, K21 = 1, K22 = K21+ 22 = 2

1 stable state 1 cycle 2 separate domains

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Continuous model

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• III. Logical analysis of the dynamics

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State transition graph (Kripke model) exponentially large check properties expressed in some temporal logic.

Model checking

p q r q r p q p q r r r q r

Infinite computation tree

Clarke/Emerson and Sifakis 81

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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

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Linear time operators

p Xp Fp Gp pUq

p p p p p p p p p p p q

Now

p

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Path quantifiers AGp

p p p p p p p

AFp

p p p

EGp

p p p

EFp

p

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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.

CTL model checking for regulatory networks

Bernot/Comet/Richard/Guespin 04

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• IV. Application: Bio-Logic

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Bio-Logic

Source: www.chaosscience.org.uk

Understand the regulatory logic underlying developmental and

• ther biological

processes

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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