Computational Methods for Systems and Synthetic Biology

Computational Methods for Systems and Synthetic Biology

François Fages

The French National Institute for Research in Computer Science and Control

Inria Saclay – Ile de France

Lifeware Team

http://lifeware.inria.fr/

09/02/2017 François Fages - MPRI C2-19 2

Overview of the Lecture

Quantitative modeling of biological processes with formal specifications 1. Complex dynamics of GPCR signaling

2. Coupled model of cell cycle and circadian clock

3. Trace simplification for FO-LTL(Rlin) model-checking

The Question of 7TMR Activation

The Question of GPCR Signalling Dynamics

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Data under Perturbed Conditions

Hypothesized 7TMR Pathways

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Formal Reaction Rules in Biocham

Vm9*GRK23*[R-H]/(Km10+[R-H]) for R-H => R~{P1}-H.

Vm18*GRK56*[R-H]/(Km19+[R-H]) for R-H => R~{P2}-H.

MA(k1) for G =[R~{P1}-H]=> G_a.

MA(k2) for G =[R-H]=> G_a.

MA(k0) for G_a => G.

MA(k26) for G => G_a.

MA(k3) for DAG =[G_a]=> DAG_a.

MA(k4) for DAG_a => DAG.

MA(k5) for PKC =[DAG_a]=> PKC_a.

MA(k6) for PKC_a => PKC.

MA(k7) for ERK =[PKC_a]=> GpERK.

MA(k8) for GpERK => ERK.

(MA(k12),MA(k13)) for R~{P1}-H + barr1 <=> R~{P1}-H-barr1. %+fort

MA(k14) for R~{P1}-H-barr1 => R-H + barr1. % recyclage

(MA(k15),MA(k16)) for R~{P1}-H + barr2 <=> R~{P1}-H-barr2. %+faible

MA(k17) for R~{P1}-H-barr2 => R-H + barr2. % recyclage

(MA(k11),MA(k20)) for barr2 + R-H <=> barr2-R-H. %tres faible.

(MA(k21),MA(k22)) for barr2 + R~{P2}-H <=> barr2-H-R~{P2}. %++fort

MA(k23) for ERK =[barr2-R-H]=> bpERK.

MA(k24) for ERK =[barr2-H-R~{P2}]=> bpERK.

MA(k27) for bpERK => ERK.

MA(k25) for R~{P2}-H => R-H.

Translation in FO-LTL(R) Constraints for Automated Parameter Search

Parameters in [1E-8,200.0]

& 50*k21/k11 > cc1 & 50*k12/k15 > cc2

…

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Translation in FO-LTL(R) Constraints for Automated Parameter Search

Parameters in [1E-8,200.0]

& 50*k21/k11 > cc1 & 50*k12/k15 > cc2

…

& G( (5=<Time & Time=<35) -> (db1=<nDAG_a*100 & 100*nDAG_a=<db2))

& G( (20=<Time & Time=<80) -> (pb7=<nPKC_a*100 & 100*nPKC_a=<pb8

& curve_fit_err(…)

…

Translation in FO-LTL(R) Constraints for Automated Parameter Search

Parameters in [1E-8,200.0]

& 50*k21/k11 > cc1 & 50*k12/k15 > cc2

…

& G( (5=<Time & Time=<35) -> (db1=<nDAG_a*100 & 100*nDAG_a=<db2))

& G( (20=<Time & Time=<80) -> (pb7=<nPKC_a*100 & 100*nPKC_a=<pb8

& curve_fit_err(…)

…

& add_search_condition( curve_fit_err([pERKtotal,pERKtotal,pERKtotal,pERKtotal,pERKtotal],

[0.26,73.12,24.27,15.65,5.58], [0,2,10,30,60], [x1,x2,x3,x4,x5], 0.1), [x1,x2,x3,x4,x5], [0.16,7.42,6.02,4.91,2.84], [(barr2_0_si,barr2_0)])

& add_search_condition( curve_fit_err([pERKtotal,pERKtotal,pERKtotal,pERKtotal,pERKtotal],

[1.18,25.36,58.99,52.41,51.91], [0,2,10,30,60], [x1,x2,x3,x4,x5], 0.1), [x1,x2,x3,x4,x5], [0.72,5.77,7.96,7.8,12.86], [(P0_Ro,P0),(P0_a_Ro,P0_a)] ).

…

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Translation in FO-LTL(R) Constraints for Automated Parameter Search

Parameters in [1E-8,200.0]

& 50*k21/k11 > cc1 & 50*k12/k15 > cc2

…

& G( (5=<Time & Time=<35) -> (db1=<nDAG_a*100 & 100*nDAG_a=<db2))

& G( (20=<Time & Time=<80) -> (pb7=<nPKC_a*100 & 100*nPKC_a=<pb8

& curve_fit_err(…)

…

& add_search_condition( curve_fit_err([pERKtotal,pERKtotal,pERKtotal,pERKtotal,pERKtotal],

[0.26,73.12,24.27,15.65,5.58], [0,2,10,30,60], [x1,x2,x3,x4,x5], 0.1), [x1,x2,x3,x4,x5], [0.16,7.42,6.02,4.91,2.84], [(barr2_0_si,barr2_0)])

& add_search_condition( curve_fit_err([pERKtotal,pERKtotal,pERKtotal,pERKtotal,pERKtotal],

[1.18,25.36,58.99,52.41,51.91], [0,2,10,30,60], [x1,x2,x3,x4,x5], 0.1), [x1,x2,x3,x4,x5], [0.72,5.77,7.96,7.8,12.86], [(P0_Ro,P0),(P0_a_Ro,P0_a)] ).

…

& F(G(pERKtotal> 3.5-v & pERKtotal< 3.5 + v))

& F(G(2*([DAG_a]/0.009) < v2 & 2*(0.009/[DAG_a])< v2))

& F(G(2*([PKC_a]/0.002)< v3 & 2*(0.002/[PKC_a])< v3))

GPCR Signaling Revisited

1. Reduced model containing the 4 observables, 4 mutations, known interactions

2. Failure of finding parameters satisfying the FO-LTL(Rlin) specification in BIOCHAM 3. Model revision, revised interactions (in blue) verified a posteriori

[Heitzler, …, FF , Lefkowitz, Reiter 2012 Molecular Systems Biology 8(590)]

09/02/2017 François Fages - MPRI C2-19 François Fages

1. coexistence of two distinct phosphorylated forms of the receptor, HRP1 and HRP2;

2. reversibility of b-arrestin-dependent ERK phosphorylation;

3. b-arrestin-dependent ERK phosphorylation undergoes enzymatic amplification;

4. non-phosphorylated ligand-bound receptor still has the ability to recruit b-arrestin 2 and to induce ERK phosphorylation;

5. differential phosphorylation of the ligand-bound receptor by GRK2/3 versus GRK5/6 leading to HRP1 and HRP2.

Model Hypotheses Verified

Experimental observations on periods and phases suggest bidirectional

influence between cell divisions and the autonomous cellular circadian clock

Modeling the coupling between

the cell cycle and the circadian clock

Bidirectional coupled model of the cell cycle and the circadian clock

Model building assisted with formal methods

(model calibration)

Predictions: mechanisms and perturbations, optimization

Context: Optimizing cancer treatment with chronotherapy

What are the mechanisms behind these observations?

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• Dynamical behaviors for oscillatory systems:

-

period, amplitude, phase - oscillations regularity

• Formalised with temporal logic:

Ex: period

• Applications:

• Data analysis: extracting meaningful information from a trace

• Model checking: verifying that a model satisfies some constraints

• Model analysis: comparing how the properties of a model evolve when some parameters vary

• Parameter inference: continuous satisfaction degree of a temporal logic formula, powerful optimization algorithm CMA-ES

Temporal logic specification

t1 t3 t2

diff

t M

t M M

t

A, B

t

Result:

-Possible values: p = 23 | p=24.5 - Satisfaction degree (with

objective p=24): 0.1

Generic FO-LTL(Rlin) Constraint Solver

D

= { s < max[A] }

φ = F( [A] > s )

‒ Operator F (finally) → union:

‒ Operator G (globally) → intersection:

‒ Operator X (next) → next domain if valid:

‒ Operator U (until) → union of intersections:

Computational cost: up to O(n

) (v = number of variables)

How to find a simplified trace that will keep the same validity domain?

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• Decomposition of φ in sub-formulas

• For each constraint and each time point, computing of a domain of possible variables

• Combination of the subdomains with the logical operators :

• Domain for φ = combinated domain for the first point of the trace

• Decomposition of φ in sub-formulas

• For each constraint and each time point, computing of a domain of possible variables

• Combination of the subdomains with the logical operators :

• Domain for φ = combinated domain for the first point of the trace

Generic FO-LTL(Rlin) Constraint Solver

φ = F( [A] > s )

‒ Operator F (finally) → union:

‒ Operator G (globally) → intersection:

‒ Operator X (next) → next domain if valid:

‒ Operator U (until) → union of intersections:

Computational cost: up to O(n

) (v = number of variables)

How to find a simplified trace that will keep

the same validity domain?

• Decomposition of φ in sub-formulas

• For each constraint and each time point, computing of a domain of possible variables

• Combination of the subdomains with the logical operators :

• Domain for φ = combinated domain for the first point of the trace

Generic FO-LTL(Rlin) Constraint Solver

φ = F( [A] > s )

‒ Operator F (finally) → union:

‒ Operator G (globally) → intersection:

‒ Operator X (next) → next domain if valid:

‒ Operator U (until) → union of intersections:

D

= { s < max[A] }

Computational cost: up to O(n

) (v = number of variables)

How to find a simplified trace that will keep the same validity domain?

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• Decomposition of φ in sub-formulas

• For each constraint and each time point, computing of a domain of possible variables

• Combination of the subdomains with the logical operators :

• Domain for φ = combinated domain for the first point of the trace

Generic FO-LTL(Rlin) Constraint Solver

φ = F( [A] > s )

‒ Operator F (finally) → union:

‒ Operator G (globally) → intersection:

‒ Operator X (next) → next domain if valid:

‒ Operator U (until) → union of intersections:

Computational cost: up to O(n

) (v = number of variables)

How to find a simplified trace that will keep

the same validity domain?

• Decomposition of φ in sub-formulas

• For each constraint and each time point, computing of a domain of possible variables

• Combination of the subdomains with the logical operators :

• Domain for φ = combinated domain for the first point of the trace

Generic FO-LTL(Rlin) Constraint Solver

φ = F( [A] > s )

‒ Operator F (finally) → union:

‒ Operator G (globally) → intersection:

‒ Operator X (next) → next domain if valid:

‒ Operator U (until) → union of intersections:

D

= { s < max[A] }

Computational cost: up to O(n

) (v = number of variables)

How to find a simplified trace that will keep the same validity domain?

09/02/2017 20

• Decomposition of φ in sub-formulas

• For each constraint and each time point, computing of a domain of possible variables

• Combination of the subdomains with the logical operators :

• Domain for φ = combinated domain for the first point of the trace

Generic FO-LTL(Rlin) Constraint Solver

φ = F( [A] > s )

‒ Operator F (finally) → union:

‒ Operator G (globally) → intersection:

‒ Operator X (next) → next domain if valid:

‒ Operator U (until) → union of intersections:

Computational cost: up to O(n

) (v = number of variables)

How to find a simplified trace that will keep

the same validity domain?

• Decomposition of φ in sub-formulas

• For each constraint and each time point, computing of a domain of possible variables

• Combination of the subdomains with the logical operators :

• Domain for φ = combinated domain for the first point of the trace

Generic FO-LTL(Rlin) Constraint Solver

φ = F( [A] > s )

‒ Operator F (finally) → union:

‒ Operator G (globally) → intersection:

‒ Operator X (next) → next domain if valid:

‒ Operator U (until) → union of intersections:

Computational cost: up to O(n

) (v = number of variables)

How to find a simplified trace that will keep the same validity domain?

D

= { s < max[A] }

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• Specific function for a dynamical behavior

• Direct computing of the validity domain on the trace

Dedicated Solver for FO-LTL(Rlin) Patterns

Computational cost: O(n²) O(n)

distanceSuccPeaks(A,B,dist)

A

t

Result: p = 23 | p=24.5 p = 23 | p=24.5

Specification:

Trace simplification

Under which condition on the constraints is it safe to use this simplification ?

Trace simplification: local extrema

For the case when there is no dedicated solver, how to make the generic algorithm more efficient?

t A

T

A,B

t

T

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Proof of validity for peak formula

T

Trace simplification:

An optimal trace simplification is T

with

T

is a simplification of T for φ.

Period formula

Same trace simplification

T

Trace simplification:

An optimal trace simplification is T

with

T

is a simplification of T for φ.

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Proof of validity for amplitude

t

Trace simplification:

An optimal trace simplification is T

where J = {minA, maxA}.

T

is a simplification of T for φ.

Trace simplification:

The optimal trace simplification is T

with T

is a simplification of T for φ.

Phase formula

A,B

t

T

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Fitting to Cell Cycle and Circadian Clock Fibroblast Data

• Unicellular data on NIH3T3 fibroblasts

• The cell cycle and the circadian clock: two coupled oscillators involving:

‒ qualitative properties: oscillations, stability

‒ quantitative properties: period of each oscillator, phase

• Constraints on one molecule:

‒ Minimum ampitude

‒ Distance between successive peaks

‒ Regularity of the distances between peaks

‒ Regularity of the peak amplitudes

• Constraints on two molecules: Phase

à Prediction of Reverb-alpha up-regulation during mitosis Cell cycle: MPF , Wee1

Circadian clock: Bmal1, PerCry , Rev-erbα

Evaluation

Minimum ampitude of PerCry Reachability of PerCry

Local maxima of PerCry Distance betw. PerCry peaks

Distance betw. succ. PerCry peaks Regularity of PerCry peaks

Phase betw. PerCry and MPF

Trace simplification:

- Extrema subtrace implemented in BIOCHAM - Computing times:

• Rosenbrock’s variable step-size simulation: 8-16 ms

• 4th order Runge-Kutta fixed step-size simulation: 160-250 ms

• Validity domain computing time (in ms):