constraint-based modeling in systems biology · 2015-07-28 · constraint-based modeling in systems...
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Constraint-based modeling in systems biology
Alexander Bockmayr
WCB-10, 21 July 2010
DFG Research Center MatheonMathematics for key technologies
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Outline
I. Systems biology
II. Constraint-based modeling
III. Regulatory networks: structure
IV. Regulatory networks: dynamics
V. Temporal logic and model checking
VI. Temporal constraints and time delays
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I. Systems biology
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Molecular biology
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Systems biology
Molecularbiology
Systemsbiology
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Molecular networks
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Modeling in systems biology
Various network types• 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)
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Traditional approach
• Kinetic modeling
• Deterministic or stochastic mathematical model
• Numerical simulation (ODE, Gillespie, …)
• Requires detailed knowledge of the network (rate laws, kinetic parameters, …)
• Often not available
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II. Constraint-based modeling
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Constraint-based methods in systems biology
Palsson, Nature Biotech.,2000
“ Because biological information is incomplete, it is necessary to take into account the fact that cells are subject to certain constraints that limit their possible behaviors. By imposing these constraints in a model, one can then determine what is possible and what is not, and determine how a cell is likely to behave, but never predict its behavior precisely.”
COBRA toolbox
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Constraints in computer science
Constraint system = system of inference with pieces of partial information
… in systems biology
Saraswat´89
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Constraint-based modeling in systems biology
• State constraints on a biological networkstructure/topologydynamics
• Make inferences about the set of possible behaviors non-determinism
• Forward + backward reasoning structure dynamicsdynamics structure
• Formal reasoning vs. numerical simulation
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Constraint-based modeling of regulatory networks
Discrete modeling formalism of René Thomas (1973)• Interaction graphs Structure• State transition graphs Dynamics
Constraint-based analysis of the dynamics• Temporal logic• Model checking Formal reasoning
Adding time delays• Temporal constraints• Hybrid discrete-continuous modeling
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III. Regulatory networks: structure
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Regulatory network
http://www.zaik.uni-koeln.de/AFS/
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Interaction graph
+_
_+
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Regulatory networks
Regulatory components: Variables
represents activity level of component j.
Regulatory interactions: Activation/inhibition
Component i is influenced by component j only if the activity level is above a certain threshold .
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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.
Thresholds and activity levels
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: weight given to the action on
Xi by Xj
: weight given to the
common action on Xi by Xj and Xk
finitely many possible parameter values
discrete update function
Logical parameters
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Annotated interaction graph
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IV. Regulatory networks: dynamics
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Asynchronous dynamics
Thomas 73, Thomas/Snoussi 89
State
State transitions
if resp.
(where is the update value for ).
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Synchronous vs. asynchronous
Only one variable updated at a time.
Nondeterminism: Several successor states possible
asynchronous synchronous
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State transition graph
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Set of possible behaviors
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Stable states and cycles
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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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V. Temporal logic and model checking
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State transition graph (Kripke model)
exponentially large
check dynamics properties expressed in suitable temporal logic
Model checking
p q
r
q rp q
p q
r
rr
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
Nowp
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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
Very efficient software available (e.g. NuSMV)
Forward reasoning: Structure dynamics
What are the possible trajectories/dynamics compatible with the given structure ?
CTL model checking
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Specify dynamic properties using CTL formulas
Find compatible logical parameter values (SMBioNet)
Alternative: Infer constraints on logical parameters
Backward reasoning: Dynamics structure
Network inference
Bernot/Comet/Richard/Guespin 04
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VI. Temporal constraints on time delays
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Time delays
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Reducing non-determinism
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Temporal constraints
• Time delays may depend on • the component• the activity level• activation/inhibition
• Temporal constraints on time delays
• Temporal constraints along a pathway
• Hybrid discrete/continuous modeling
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Formal analysis using timed automataSiebert/Bockmayr 08
reduce non-determinism
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Biological applications
• Elimination of pathways violating temporal constraints
• Feasibility, stability of certain dynamic behaviors
• Additional information on gene activitystationaryintermediate
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Conclusion
• Constraint-based modeling of regulatory networks
• Interaction graph Structure
• State transition graph Dynamics
• Temporal logic and model checking
• Reducing non-determinism: time delays, temporal constraints, hybrid automata
• Deterministic/stochastic vs. constraint-based modeling
• Numerical simulation vs. formal reasoning
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