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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Alexander Bockmayr, FU Berlin/Matheon 2
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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Alexander Bockmayr, FU Berlin/Matheon 3
I. Systems biology
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Alexander Bockmayr, FU Berlin/Matheon 4
Molecular biology
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Alexander Bockmayr, FU Berlin/Matheon 5
Systems biology
Molecularbiology
Systemsbiology
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Alexander Bockmayr, FU Berlin/Matheon 6
Molecular networks
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Alexander Bockmayr, FU Berlin/Matheon 7
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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Alexander Bockmayr, FU Berlin/Matheon 8
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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Alexander Bockmayr, FU Berlin/Matheon 9
II. Constraint-based modeling
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Alexander Bockmayr, FU Berlin/Matheon 10
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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Alexander Bockmayr, FU Berlin/Matheon 11
Constraints in computer science
Constraint system = system of inference with pieces of partial information
… in systems biology
Saraswat´89
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Alexander Bockmayr, FU Berlin/Matheon 12
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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Alexander Bockmayr, FU Berlin/Matheon 13
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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Alexander Bockmayr, FU Berlin/Matheon 14
III. Regulatory networks: structure
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Alexander Bockmayr, FU Berlin/Matheon 15
Regulatory network
http://www.zaik.uni-koeln.de/AFS/
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Alexander Bockmayr, FU Berlin/Matheon 16
Interaction graph
+_
_+
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Alexander Bockmayr, FU Berlin/Matheon 17
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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Alexander Bockmayr, FU Berlin/Matheon 18
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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Alexander Bockmayr, FU Berlin/Matheon 19
: 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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Alexander Bockmayr, FU Berlin/Matheon 20
Annotated interaction graph
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Alexander Bockmayr, FU Berlin/Matheon 21
IV. Regulatory networks: dynamics
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Alexander Bockmayr, FU Berlin/Matheon 22
Asynchronous dynamics
Thomas 73, Thomas/Snoussi 89
State
State transitions
if resp.
(where is the update value for ).
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Alexander Bockmayr, FU Berlin/Matheon 23
Synchronous vs. asynchronous
Only one variable updated at a time.
Nondeterminism: Several successor states possible
asynchronous synchronous
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Alexander Bockmayr, FU Berlin/Matheon 24
State transition graph
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Alexander Bockmayr, FU Berlin/Matheon 25
Set of possible behaviors
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Alexander Bockmayr, FU Berlin/Matheon 26
Stable states and cycles
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Alexander Bockmayr, FU Berlin/Matheon 27
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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Alexander Bockmayr, FU Berlin/Matheon 28
K12=1, K21 = 0, K22 = K21+22 = 2
2 stable states
no cycle
2 separate domains
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Alexander Bockmayr, FU Berlin/Matheon 29
K12=1, K21 = 1, K22 = K21+22 = 2
1 stable state
1 cycle
2 separate domains
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Alexander Bockmayr, FU Berlin/Matheon 30
Continuous model
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Alexander Bockmayr, FU Berlin/Matheon 31
V. Temporal logic and model checking
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Alexander Bockmayr, FU Berlin/Matheon 32
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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Alexander Bockmayr, FU Berlin/Matheon 33
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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Alexander Bockmayr, FU Berlin/Matheon 34
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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Alexander Bockmayr, FU Berlin/Matheon 35
Path quantifiers
AGp p
p
p
p
p p p
AFp
p
p p
EGp p
p
p
EFp
p
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Alexander Bockmayr, FU Berlin/Matheon 36
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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Alexander Bockmayr, FU Berlin/Matheon 37
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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Alexander Bockmayr, FU Berlin/Matheon 38
VI. Temporal constraints on time delays
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Alexander Bockmayr, FU Berlin/Matheon 39
Time delays
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Alexander Bockmayr, FU Berlin/Matheon 40
Reducing non-determinism
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Alexander Bockmayr, FU Berlin/Matheon 41
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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Alexander Bockmayr, FU Berlin/Matheon 42
Formal analysis using timed automataSiebert/Bockmayr 08
reduce non-determinism
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Alexander Bockmayr, FU Berlin/Matheon 43
Biological applications
• Elimination of pathways violating temporal constraints
• Feasibility, stability of certain dynamic behaviors
• Additional information on gene activitystationaryintermediate
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Alexander Bockmayr, FU Berlin/Matheon 44
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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Alexander Bockmayr, FU Berlin/Matheon 45