Constraint-based modeling in systems biology · 2015-07-28 · Constraint-based modeling in systems...

Constraint-based modeling in systems biology

Alexander Bockmayr

WCB-10, 21 July 2010

DFG Research Center MatheonMathematics for key technologies

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

Alexander Bockmayr, FU Berlin/Matheon 3

I. Systems biology

Alexander Bockmayr, FU Berlin/Matheon 4

Molecular biology

Alexander Bockmayr, FU Berlin/Matheon 5

Systems biology

Molecularbiology

Systemsbiology

Alexander Bockmayr, FU Berlin/Matheon 6

Molecular networks

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)

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

Alexander Bockmayr, FU Berlin/Matheon 9

II. Constraint-based modeling

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

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

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

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

Alexander Bockmayr, FU Berlin/Matheon 14

III. Regulatory networks: structure

Alexander Bockmayr, FU Berlin/Matheon 15

Regulatory network

http://www.zaik.uni-koeln.de/AFS/

Alexander Bockmayr, FU Berlin/Matheon 16

Interaction graph

+_

_+

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 .

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

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

Alexander Bockmayr, FU Berlin/Matheon 20

Annotated interaction graph

Alexander Bockmayr, FU Berlin/Matheon 21

IV. Regulatory networks: dynamics

Alexander Bockmayr, FU Berlin/Matheon 22

Asynchronous dynamics

Thomas 73, Thomas/Snoussi 89

State

State transitions

if resp.

(where is the update value for ).

Alexander Bockmayr, FU Berlin/Matheon 23

Synchronous vs. asynchronous

Only one variable updated at a time.

Nondeterminism: Several successor states possible

asynchronous synchronous

Alexander Bockmayr, FU Berlin/Matheon 24

State transition graph

Alexander Bockmayr, FU Berlin/Matheon 25

Set of possible behaviors

Alexander Bockmayr, FU Berlin/Matheon 26

Stable states and cycles

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.

Alexander Bockmayr, FU Berlin/Matheon 28

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

2 stable states

no cycle

2 separate domains

Alexander Bockmayr, FU Berlin/Matheon 29

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

1 stable state

1 cycle

2 separate domains

Alexander Bockmayr, FU Berlin/Matheon 30

Continuous model

Alexander Bockmayr, FU Berlin/Matheon 31

V. Temporal logic and model checking

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

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

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

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

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

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

Alexander Bockmayr, FU Berlin/Matheon 38

VI. Temporal constraints on time delays

Alexander Bockmayr, FU Berlin/Matheon 39

Time delays

Alexander Bockmayr, FU Berlin/Matheon 40

Reducing non-determinism

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

Alexander Bockmayr, FU Berlin/Matheon 42

Formal analysis using timed automataSiebert/Bockmayr 08

reduce non-determinism

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

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

Alexander Bockmayr, FU Berlin/Matheon 45