True/False valuation of temporal logic formulae

(1)

True/False valuation of temporal logic formulae

The True/False valuation of temporal logic formulae is not well adapted to several problems :

parameter search, optimization and control of continuous models quantitative estimation of robustness

sensitivity analyses

1 / 39

(2)

True/False valuation of temporal logic formulae

The True/False valuation of temporal logic formulae is not well adapted to several problems :

parameter search, optimization and control of continuous models quantitative estimation of robustness

sensitivity analyses

→ need for a continuous degree of satisfaction of temporal logic formulae How far is the system from verifying the specification ?

2 / 39

(3)

Model-Checking Generalized to Constraint Solving

QF LT L( R )

Φ*=F([A]≥x F([A]≤y))

Constraint solving

the formula is true for any x≤10 y≥2 Φ=F([A]≥7

F([A]≤0)) Model-checking

the formula is false

LT L( R )

D

^φ^∗

(T )

2 10

[A]

time

T

y

φ

x

D

^φ^∗

(T )

vd=2 sd=1/3

3 / 39

(4)

Model-Checking Generalized to Constraint Solving

QF LT L( R )

Φ*=F([A]≥x F([A]≤y))

LT L( R )

D

^φ^∗

(T )

2 10

[A]

time

T

y

φ

x

D

^φ^∗

(T )

vd=2 sd=1/3

4 / 39

(5)

Model-Checking Generalized to Constraint Solving

QF LT L( R )

Φ*=F([A]≥x F([A]≤y))

LT L( R )

D

^φ^∗

(T )

2 10

[A]

time

T

y

φ

x

D

^φ^∗

(T )

vd=2 sd=1/3

Validity domain D

φ^∗

(T ) for the free variables in φ

^∗ [Fages Rizk CMSB’07]

5 / 39

(6)

Model-Checking Generalized to Constraint Solving

QF LT L( R )

Φ*=F([A]≥x F([A]≤y))

LT L( R )

D

^φ^∗

(T )

2 10

[A]

time

T

y

φ

x

D

^φ^∗

(T )

vd=2 sd=1/3

Validity domain D

φ^∗

(T ) for the free variables in φ

^∗ [Fages Rizk CMSB’07]

Violation degree vd(T , φ) = distance(val(φ), D

φ^∗

(T )) Satisfaction degree sd (T , φ) =

_1+vd(T,φ)¹

∈ [0, 1]

6 / 39

(7)

Violation degree of an LTL formula

Definition of violation degree vd(T , φ) and satisfaction degree sd (T , φ) In the variable space of φ

^∗

, original formula φ is single point var(φ).

vd(T , φ) = min

_v∈D_φ∗(T)

d(v, var(φ)) sd (T , φ) =

_1+vd(T,φ)¹

∈ [0, 1]

[A]

time

(✕) (✓) (✕) vd=0

vd=2 vd=2√2

= F([A]≥6 F([A]≤5))

= F([A]≥6 F([A]≤0))

= F([A]≥12 F([A]≤0))

φ

a

φ

_b

φ

c

(x,y)= F([A]≥x F([A]≤y))

φ

^∗

(6,5)

φ

^∗

(6,0)

φ

^∗

(12,0)

φ

^∗

y

x (10,2)

φ

a

φ

b

φ

c

D

^φ^∗

(T )

10

2

T

7 / 39

(8)

Violation degree of an LTL formula

Definition of violation degree vd(T , φ) and satisfaction degree sd (T , φ) In the variable space of φ

^∗

, original formula φ is single point var(φ).

vd(T , φ) = min

_v∈D_φ∗(T)

d(v, var(φ)) sd (T , φ) =

_1+vd(T,φ)¹

∈ [0, 1]

[A]

time

(✕) (✓) (✕) vd=0

vd=2 vd=2√2

= F([A]≥6 F([A]≤5))

= F([A]≥6 F([A]≤0))

= F([A]≥12 F([A]≤0))

φ

a

φ

_b

φ

c

(x,y)= F([A]≥x F([A]≤y))

φ

^∗

(6,5)

φ

^∗

(6,0)

φ

^∗

(12,0)

φ

^∗

y

x (10,2)

φ

a

φ

b

φ

c

D

^φ^∗

(T )

10

2

T

6 5

8 / 39

(9)

Violation degree of an LTL formula

Definition of violation degree vd(T , φ) and satisfaction degree sd (T , φ) In the variable space of φ

^∗

, original formula φ is single point var(φ).

vd(T , φ) = min

_v∈D_φ∗(T)

d(v, var(φ)) sd (T , φ) =

_1+vd(T,φ)¹

∈ [0, 1]

[A]

time

(✕) (✓) (✕) vd=0

vd=2 vd=2√2

= F([A]≥6 F([A]≤5))

= F([A]≥6 F([A]≤0))

= F([A]≥12 F([A]≤0))

φ

a

φ

_b

φ

c

(x,y)= F([A]≥x F([A]≤y))

φ

^∗

(6,5)

φ

^∗

(6,0)

φ

^∗

(12,0)

φ

^∗

y

x (10,2)

φ

a

φ

b

φ

c

D

^φ^∗

(T )

10

2

T

6

0

9 / 39

(10)

Violation degree of an LTL formula

Definition of violation degree vd(T , φ) and satisfaction degree sd (T , φ) In the variable space of φ

^∗

, original formula φ is single point var(φ).

vd(T , φ) = min

_v∈D_φ∗(T)

d(v, var(φ)) sd (T , φ) =

_1+vd(T,φ)¹

∈ [0, 1]

[A]

time

(✕) (✓) (✕) vd=0

vd=2 vd=2√2

= F([A]≥6 F([A]≤5))

= F([A]≥6 F([A]≤0))

= F([A]≥12 F([A]≤0))

φ

a

φ

_b

φ

c

(x,y)= F([A]≥x F([A]≤y))

φ

^∗

(6,5)

φ

^∗

(6,0)

φ

^∗

(12,0)

φ

^∗

y

x (10,2)

φ

a

φ

b

φ

c

D

^φ^∗

(T )

10

2

T

12

0

10 / 39

(11)

Learning kinetic parameter values from LTL specifications

simple model of the yeast cell cycle from

[Tyson PNAS 91]

models Cdc2 and Cyclin interactions (6 variables, 8 kinetic parameters)

0 0.1 0.2 0.3 0.4 0.5

0 20 40 60 80 100 120 140

Cdc2 Cdc2~{p1}

Cyclin Cdc2-Cyclin~{p1,p2}

Cdc2-Cyclin~{p1}

Cyclin~{p1}

0 0.1 0.2 0.3 0.4 0.5

0 20 40 60 80 100 120 140

Cdc2 Cdc2~{p1}

Cdc2-Cyclin~{p1}

Cyclin~{p1}

0.3

p p^∗ [MPF]

Pb : find values of 8 parameters such that amplitude is ≥ 0.3 φ

^∗

: F( [A]>x ∧ F([A]<y) )

amplitude z=x-y goal : z = 0.3

→ solution found after 30s (100 calls to the fitness function)

11 / 39

(12)

Learning kinetic parameter values from LTL specifications

simple model of the yeast cell cycle from

[Tyson PNAS 91]

models Cdc2 and Cyclin interactions (6 variables, 8 kinetic parameters)

0 0.1 0.2 0.3 0.4 0.5

0 20 40 60 80 100 120 140

Cdc2 Cdc2~{p1}

Cdc2-Cyclin~{p1}

Cyclin~{p1}

0 0.1 0.2 0.3 0.4 0.5

0 20 40 60 80 100 120 140

Cdc2 Cdc2~{p1}

Cdc2-Cyclin~{p1}

Cyclin~{p1}

0.3

p p^∗ [MPF]

Pb : find values of 8 parameters such that amplitude is ≥ 0.3 φ

^∗

: F( [A]>x ∧ F([A]<y) )

amplitude z=x-y goal : z = 0.3

→ solution found after 30s (100 calls to the fitness function)

12 / 39

(13)

Learning kinetic parameter values from LTL specifications

simple model of the yeast cell cycle from

[Tyson PNAS 91]

models Cdc2 and Cyclin interactions (6 variables, 8 kinetic parameters)

0 0.1 0.2 0.3 0.4 0.5

0 20 40 60 80 100 120 140

Cdc2 Cdc2~{p1}

Cdc2-Cyclin~{p1}

Cyclin~{p1}

0 0.1 0.2 0.3 0.4 0.5

0 20 40 60 80 100 120 140

Cdc2 Cdc2~{p1}

Cdc2-Cyclin~{p1}

Cyclin~{p1}

0.3

p p^∗ [MPF]

Pb : find values of 8 parameters such that amplitude is ≥ 0.3 φ

^∗

: F( [A]>x ∧ F([A]<y) )

amplitude z=x-y goal : z = 0.3

→ solution found after 30s (100 calls to the fitness function)

13 / 39

(14)

LTL Continuous Satisfaction Diagram

Example with :

yeast cell cycle model

[Tyson PNAS 91]

oscillation of at least 0.3

φ

^∗

: F( [A]>x ∧ F([A]<y) ); amplitude x-y≥0.3

k4

k6 .

Violation degree in parameter space .

. .

.

pApB

pC

Proc.Natl. Acad. Sci. USA 88(1991) 1000r

E 1001

10I

0.1 1.0

k6^min1

10

FIG. 2. Qualitativebehavior of thecdc2-cyclin modelofcell- cycleregulation. The control parametersarek4, therate constant describingthe maximumrateofMPFactivation,andk6,therate constantdescribing dissociation oftheactiveMPFcomplex.Regions A andCcorrespondtostable steady-state behavior of the model;

regionBcorrespondsto spontaneouslimit cycle oscillations.Inthe stippled areatheregulatorysystem is excitable.Theboundaries between regions A, B,and C were determinedby integrating the differentialequations inTable 1, for the parameter values in Table 2.

Numericalintegrationwascarriedoutby using Gear's algorithm for solving stiffordinary differentialequations (32).The"developmental path"1... 5is described inthe text.

sok6 abruptly increases 2-fold. Continued cell growth causes k6(t) again todecrease,and the cycle repeats itself. The interplaybetween the controlsystem,cellgrowth,and DNA replication generatesperiodic changesink6(t)andperiodic burstsof MPFactivitywith acycletimeidentical to the mass-doubling time ofthegrowingcell.

Figs.2and 3 demonstratethat,dependingonthevaluesof k4 andk6, the cell cycle regulatorysystem exhibits three

b

0.4a 100

0 20 40 60 80 100 0 20 40 60 80 100

t,min t,min

differentmodes ofcontrol.For small values of k6,thesystem displaysa stablesteadystate ofhighMPFactivity,whichI associate with metaphase arrest of unfertilizedeggs. For moderate Values of k6,thesystem executesautonomous oscillationsreminiscent of rapid cell cycling in earlyem- bryos.Forlargevaluesof k6, the system is attractedto an excitablesteadystateof low MPF activity, whichcorre- spondstointerphasearrestof resting somatic cellsor to growth-controlled bursts ofMPFactivity in proliferating somatic cells.

MidblastulaTraiisiton

A possible developmental scenario is illustrated by the path 1 ... 5inFig.2.Uponfertilization, the metaphase-arrested egg (at point 1)isstimulated to rapid cell divisions (2) byan increaseintheactivity of the enzyme catalyzing step 6 (28).

Duringtheearlyembryoniccellcycles (2-+ 3),theregulatory systemisexecutingautonomousoscillations,and thecontrol parameters,k4and k6, increaseasthenucleargenescoding fortheseenzymes areactivated.Atmidblastula (3),auton- omousoscillationscease,andthe regulatorysystem enters theexcitable domain. Cell divisionnowbecomesgrowth controlled.Ascellsgrow,k6decreases(inhibitor diluted) and/or k4 increases (activator accumulates), which drives the regulatorysystem back into domain B(4 -S5).Thesubse- quent burstofMPFactivity triggers mitosis, causes k6to increase(inhibitor synthesis)and/or k4todecrease(activator degradation), andbrings the regulatorysystemback into the excitable domain (5^-*4).

Although there isaclear and abrupt lengthening of inter- division timesatMBT, there isnovisible increase in cell volumeimmediately thereafter (6, 20), so how can we enter- tain the ideathatcelldivision becomes growth controlled afterMBT? Intheparadigm ofgrowth controlofcell division, cell "size" isneverpreciselyspecified, becauseno one knowswhatmolecules, structures,orpropertiesareusedby cells tomonitortheirsize.Thus,eventhough post-MBT cells

C rk6'^min-1

0 100 200 300 400 500

t, min

FIG. 3. Dynamical behavior of thecdc2-cyclinmodel. The curves are totalcyclin([YT]⁼[Y]⁺[YP] +[pM]+[M])andactive MPF[Ml relativetototal cdc2([CT]=[C2]+[CP]+[pM]+[MI).Thedifferential equationsin Table 1,forthe parameter values in Table2,weresolved numerically by usingafourth-order Adams-Moulton integrationroutine(33)withtimestep=0.001min.(Theadequacy ofthenumerical integrationwascheckedby decreasingthe time step and alsoby comparisontosolutionsgenerated byGear's algorithm.)(a) Limit cycle oscillations fork4⁼180min-',k6=1min- (pointx inFig. 2).A"limitcycle"solutionofa setof ordinarydifferential equations is aperiodic solution that isasymptoticallystable withrespecttosmallperturbationsinanyof thedynamical variables. (b) Excitable steadystatefork4= 180min1,k6=2min'(point+inFig. 2). Noticethat theordinateis alogarithmic scale.Thesteadystateof lowMPFactivity ([M]/[CT]

=0.0074,[YT]/[CT]=0.566)is stable withrespecttosmallperturbations ofMPFactivity (at1and2)but asufficiently large perturbationof [Ml(at 3)triggersatransient activation of MPF andsubsequent destructionofcyclin.Theregulatory systemthenrecoversto thestable steady state.(c)Parametervalues as in bexceptthatk6is now afunction of time(oscillatingnearpoint+inFig.2). See textforanexplanationof therulesfork6(Q).Notice that theperiodbetween cell divisions(burstsin MPFactivity)isidentical to the mass-doublingtime(Td⁼116 min in thissimulation).Simulations with different values ofTd(notshown)demonstrate that theperiodbetween MPF bursts istypicallyequalto themass-doublingtime-i.e.,the celldivisioncycleisgrowthcontrolled under thesecircumstances.Growth control canalsobeachieved (simulationsnotshown),holding k6 constant, by assumingthatk4 increaseswithtime between divisions and decreasesabruptlyafter an MPF burst.

7330 CellBiology: Tyson

Bifurcation diagram LTL satisfaction diagram

14 / 39

(15)

Black-box Randomized Non-linear Optimization Method

Use existing non-linear optimization toolbox for kinetic parameter search using satisfaction degree as fitness function

We use the state-of-the-art Covariance Matrix Adaptation Evolution Strategy (CMA-ES) [Hansen Osermeier 01, Hansen 08]

CMA-ES maximizes an objective function in continuous domain in a black box scenario

CMA-ES uses a probabilistic neighborhood and updates information in covariance matrix at each move

15 / 39

(16)

Black-box Randomized Non-linear Optimization Method

Use existing non-linear optimization toolbox for kinetic parameter search using satisfaction degree as fitness function

We use the state-of-the-art Covariance Matrix Adaptation Evolution Strategy (CMA-ES) [Hansen Osermeier 01, Hansen 08]

CMA-ES maximizes an objective function in continuous domain in a black box scenario

CMA-ES uses a probabilistic neighborhood and updates information in covariance matrix at each move

16 / 39

(17)

Black-box Randomized Non-linear Optimization Method

Use existing non-linear optimization toolbox for kinetic parameter search using satisfaction degree as fitness function

We use the state-of-the-art Covariance Matrix Adaptation Evolution Strategy (CMA-ES) [Hansen Osermeier 01, Hansen 08]

CMA-ES maximizes an objective function in continuous domain in a black box scenario

CMA-ES uses a probabilistic neighborhood and updates information in covariance matrix at each move

17 / 39

(18)

Black-box Randomized Non-linear Optimization Method

Use existing non-linear optimization toolbox for kinetic parameter search using satisfaction degree as fitness function

We use the state-of-the-art Covariance Matrix Adaptation Evolution Strategy (CMA-ES) [Hansen Osermeier 01, Hansen 08]

CMA-ES maximizes an objective function in continuous domain in a black box scenario

CMA-ES uses a probabilistic neighborhood and updates information in covariance matrix at each move

18 / 39

(19)

Learning Parameter Values from Period Constraints in LTL

0 0.1 0.2 0.3 0.4 0.5

0 20 40 60 80 100 120 140

Cdc2 Cdc2~{p1}

Cdc2-Cyclin~{p1}

Cyclin~{p1}

0 0.1 0.2 0.3 0.4 0.5

0 20 40 60 80 100 120 140

Cdc2 Cdc2~{p1}

Cdc2-Cyclin~{p1}

Cyclin~{p1}

p p^∗ [MPF]

Pb : find values of 8 parameters such that period is 20

φ

^∗

:F(MPF

localmaximum

∧Time=t1∧ F(MPF

localmaximum

∧Time=t2) )

( with MPFlocalmaximum : d([MPF])/dt>0∧X(d([MPF])/dt<0) )

period z=t2-t1 goal z=20

→ Solution found after 60s (200 calls to the fitness function)

19 / 39

(20)

Learning Parameter Values from Period Constraints in LTL

0 0.1 0.2 0.3 0.4 0.5

0 20 40 60 80 100 120 140

Cdc2 Cdc2~{p1}

Cdc2-Cyclin~{p1}

Cyclin~{p1}

0 0.1 0.2 0.3 0.4 0.5

0 20 40 60 80 100 120 140

Cdc2 Cdc2~{p1}

Cdc2-Cyclin~{p1}

Cyclin~{p1}

p p^∗ [MPF]

Pb : find values of 8 parameters such that period is 20

φ

^∗

:F(MPF

localmaximum

∧Time=t1∧ F(MPF

localmaximum

∧Time=t2) )

period z=t2-t1 goal z=20

→ Solution found after 60s (200 calls to the fitness function)

20 / 39

(21)

Learning Parameter Values from Period Constraints in LTL

0 0.1 0.2 0.3 0.4 0.5

0 20 40 60 80 100 120 140

Cdc2 Cdc2~{p1}

Cdc2-Cyclin~{p1}

Cyclin~{p1}

0 0.1 0.2 0.3 0.4 0.5

0 20 40 60 80 100 120 140

Cdc2 Cdc2~{p1}

Cdc2-Cyclin~{p1}

Cyclin~{p1}

p p^∗ [MPF]

Pb : find values of 8 parameters such that period is 20

φ

^∗

:F(MPF

localmaximum

∧Time=t1∧ F(MPF

localmaximum

∧Time=t2) )

period z=t2-t1 goal z=20

→ Solution found after 60s (200 calls to the fitness function)

21 / 39

(22)

Oscillations in MAPK signal transduction cascade

MAPK signaling model

[Huang Ferrel PNAS 96]

search for oscillations in 37 dimensions (30 parameters and 7 initial conditions)

→ solution found after 3 min (200 calls to the fitness function) Oscillations already observed by simulation

[Qiao et al. 07]

No negative feedback in the reaction graph, but negative circuits in the influence graph

[Fages Soliman FMSB’08, CMSB’06]

22 / 39

(23)

Oscillations in MAPK signal transduction cascade

MAPK signaling model

search for oscillations in 37 dimensions (30 parameters and 7 initial conditions)

→ solution found after 3 min (200 calls to the fitness function) Oscillations already observed by simulation

[Qiao et al. 07]

No negative feedback in the reaction graph, but negative circuits in the influence graph

23 / 39

(24)

Oscillations in MAPK signal transduction cascade

MAPK signaling model

search for oscillations in 37 dimensions (30 parameters and 7 initial conditions)

→ solution found after 3 min (200 calls to the fitness function) Oscillations already observed by simulation

[Qiao et al. 07]

No negative feedback in the reaction graph, but negative circuits in the influence graph

24 / 39

(25)

Coupled Models of Cell Cycle, Circadian Clock, DNA repair

Context of colorectal cancer chronotherapies

EU FP6 TEMPO, EraSysBio C5Sys, coord. F. L´ evi INSERM Villejuif Coupled model of the cell cycle

[Tyson Novak 04][Gerard Goldbeter 09]

and the circadian clock

[Leloup Goldbeter 99]

with condition of entrainment in period

[Calzone Soliman 06]

Coupled model with DNA repair system p53/Mdm2

[Cilberto et al.04]

, metabolism of irinotecan, and drug administration optimization

[De Maria Soliman Fages 09 CMSB]

25 / 39

(26)

Basis of Operators of LTL(R)

Atomic propositions: arithmetic expressions with <, ≤, =, ≥, > over the state variables (closed by negation)

Duality: ¬Xφ = X¬φ, ¬Fφ = G¬φ, ¬Gφ = F¬φ,

¬(φ U ψ) = (¬ψ W ¬φ), ¬(φ W ψ) = (¬ψ U ¬φ),

Properties: Fφ = true U φ, Gφ = φ W false, φWψ = Gφ ∨ (φU(φ ∧ ψ))

Negation free formulae: expressed with ∧, ∨, F, G, U, X with negations eliminated down to atomic propositions.

26 / 39

(27)

LTL(R) model-checking

Given a finite trace T and an LTL(R) formula φ

1

label each state with the atomic sub-formulae of φ that are true at this state;

2

add sub-formulae of the form φ

₁

U φ

₂

to the states labeled by φ

₂

and to the predecessors of states labeled with φ

₂

as long as they are labeled by φ

1

;

3

add sub-formulae of the form φ

1

W φ

2

to the last state if it is labeled by φ

₁

, and to the states labeled by φ

₁

and φ

₂

, and to their predecessors as long as they are labeled by φ

₁

;

4

add sub-formulae of the form Xφ to the last state if it is labeled by φ and to the immediate predecessors of states labeled by φ;

5

return the vertices labeled by φ.

27 / 39

(28)

QFLTL(R) Formulae with Variables

Quantifier free LTL formulae, noted φ(y) with free variables y

The satisfaction domain of φ(y) in a trace T is the set of y values for which φ(y) holds:

D

_T,φ(y)

= {y ∈ R

^q

| T | = φ(y)} (1)

For linear constraints over R, satisfaction domains can be computed with polyhedral libraries.

Biocham uses the Parma Polyhedral Library PPL

28 / 39

(29)

QFLTL(R) constraint solving

The satisfaction domains of QFLTL formulae satisfy the equations:

D

_T,φ(y)

= D

_s₀_,φ(y)

,

D

s_i,π(y)

= {y ∈ R

^m

| s

i

| =

R

π(y)}, D

_s_i_{,φ(y)∧ψ(y)}

= D

_s_i_,φ(y)

∩ D

_s_i_,ψ(y)

, D

s_i,φ(y)∨ψ(y)

= D

s_i,φ(y)

∪ D

s_i,ψ(y)

, D

_s_i_,Fφ(y)

= ∪

_j∈[i,n]

D

_s_j_,φ(y)

, D

s_i,Gφ(y)

= ∩

_j∈[i,n]

D

s_j,φ(y)

,

D

_s_i_,φ(y)Uψ(y)

= ∪

_j∈[i,n]

(D

_s_j_,ψ(y)

∩ ∩

_{k∈[i,j−1]}

D

_s_k_,φ(y)

), D

_s_i_,Xφ(y)

=

D

_s_i+1_,φ(y)

, if i < n, D

_s_i_,φ(y)

, if i = n,

29 / 39

(30)

Complexity with bound constraints x > b, x < b

Bound constraints define boxes R

i

∈ R

^v

.

Let the size of a union of boxes be the least integer k such that D = S

k

i=1

R

i

.

Proposition (complexity of the solution domain)

The validity domain of a QFLTL formula of size f containing v variables on a trace of length n is a union of boxes of size less than (nf )

^2v

.

The maximum number of bounds for a variable x is n × f E.g; F([A] = u ∨ [A] + 1 = u ∨ · · · ∨ [A] + f = u) F([A

1

] = X

1

∨ [A

1

] + 1 = X

1

∨ ... ∨ [A

1

] + f = X

1

) ∧ ...

∧ F([A

v

] = X

v

∨ [A

v

] + 1 = X

v

∨ ... ∨ [A

v

] + f = X

v

) has a solution domain of size (nf )

^v

on a trace of n values with [A

_i

] + k all different for 1 ≤ i ≤ v , 0 ≤ k ≤ f .

30 / 39

(31)

Robustness Measure Definition

Robustness defined with respect to : a biological system

a functionality property D

_a

a set P of perturbations

General notion of robustness proposed in [Kitano MSB 07]:

R

a,P

= Z

p∈P

D

a

(p) prob(p) dp

Computational measure of robustness w.r.t. LTL( R ) spec: R

φ,P

=

Z

p∈P

sd (T (p), φ) prob(p) dp

where T (p) is the trace obtained by numerical integration of the ODE for perturbation p

31 / 39

(32)

Robustness Measure Definition

Robustness defined with respect to : a biological system

a functionality property D

_a

a set P of perturbations

General notion of robustness proposed in [Kitano MSB 07]:

R

a,P

= Z

p∈P

D

a

(p) prob(p) dp

Computational measure of robustness w.r.t. LTL( R ) spec:

R

φ,P

= Z

p∈P

sd (T (p), φ) prob(p) dp

where T (p) is the trace obtained by numerical integration of the ODE for perturbation p

32 / 39

(33)

Robustness analysis w.r.t parameter perturbations

Example with :

cell cycle model

[Tyson PNAS 91]

oscillation of amplitude at least 0.2

φ

^∗

: F( [A]>x ∧ F([A]<y) ); amplitude x-y≥0.2 parameters normally distributed, µ = p

ref

, CV=0.2

k4

k6 .

Violation degree in parameter space .

. .

.

pApB

pC

Proc.Natl. Acad. Sci. USA 88(1991) 1000r

E 1001

10I

0.1 1.0

k6min1 10

FIG. 2. Qualitativebehavior of thecdc2-cyclin modelofcell- cycleregulation. The control parametersarek4, therate constant describingthe maximumrateofMPFactivation,andk6,therate constantdescribing dissociation oftheactiveMPFcomplex.Regions A andCcorrespondtostable steady-state behavior of the model;

regionBcorrespondsto spontaneouslimit cycle oscillations.Inthe stippled areatheregulatorysystem is excitable.Theboundaries between regions A, B,and C were determinedby integrating the differentialequations inTable 1, for the parameter values in Table 2.

Numericalintegrationwascarriedoutby using Gear's algorithm for solving stiffordinary differentialequations (32).The"developmental path"1... 5is described inthe text.

sok6 abruptly increases 2-fold. Continued cell growth causes k6(t) again todecrease,and the cycle repeats itself. The interplaybetween the controlsystem,cellgrowth,and DNA replication generatesperiodic changesink6(t)andperiodic burstsof MPFactivitywith acycletimeidentical to the mass-doubling time ofthegrowingcell.

Figs.2and 3 demonstratethat,dependingonthevaluesof k4 andk6, the cell cycle regulatorysystem exhibits three

b

0.4a 100

0 20 40 60 80 100 0 20 40 60 80 100

t,min t,min

differentmodes ofcontrol.For small values of k6,thesystem displaysa stablesteadystate ofhighMPFactivity,whichI associate with metaphase arrest of unfertilizedeggs. For moderate Values of k6,thesystem executesautonomous oscillationsreminiscent of rapid cell cycling in earlyem- bryos.Forlargevaluesof k6, the system is attractedto an excitablesteadystateof low MPF activity, whichcorre- spondstointerphasearrestof resting somatic cellsor to growth-controlled bursts ofMPFactivity in proliferating somatic cells.

MidblastulaTraiisiton

A possible developmental scenario is illustrated by the path 1 ... 5inFig.2.Uponfertilization, the metaphase-arrested egg (at point 1)isstimulated to rapid cell divisions (2) byan increaseintheactivity of the enzyme catalyzing step 6 (28).

Duringtheearlyembryonic^cellcycles (2-+ 3),theregulatory systemisexecutingautonomousoscillations,and thecontrol parameters,k4and k6, increaseasthenucleargenescoding fortheseenzymes areactivated.Atmidblastula (3),auton- omousoscillationscease,andthe regulatorysystem enters theexcitable domain. Cell divisionnowbecomesgrowth controlled.Ascellsgrow,k6decreases(inhibitor diluted) and/or k4 increases (activator accumulates), which drives the regulatorysystem back into domain B(4 -S5).Thesubse- quent burstofMPFactivity triggers mitosis, causes k6to increase(inhibitor synthesis)and/or k4todecrease(activator degradation), andbrings the regulatorysystemback into the excitable domain (5^-*4).

Although there isaclear and abrupt lengthening of inter- division timesatMBT, there isnovisible increase in cell volumeimmediately thereafter (6, 20), so how can we enter- tain the ideathatcelldivision becomes growth controlled afterMBT? Intheparadigm ofgrowth controlofcell division, cell "size" isneverpreciselyspecified, becauseno one knowswhatmolecules, structures,orpropertiesareusedby cells tomonitortheirsize.Thus,eventhough post-MBT cells

C rk6'^min-1

0 100 200 300 400 500

t, min

FIG. 3. Dynamical behavior of thecdc2-cyclinmodel. The curves are totalcyclin([YT]⁼[Y]⁺[YP] +[pM]+[M])andactive MPF[Ml relativetototal cdc2([CT]⁼[C2]+[CP]+[pM]+[MI).Thedifferential equationsin Table 1,forthe parameter values in Table2,weresolved numerically by usingafourth-order Adams-Moulton integrationroutine(33)withtimestep⁼0.001min.(Theadequacy ofthenumerical integrationwascheckedby decreasingthe time step and alsoby comparisontosolutionsgenerated byGear's algorithm.)(a) Limit cycle oscillations fork4⁼180min-',k6⁼1min- (pointx inFig. 2).A"limitcycle"solutionofa setof ordinarydifferential equations is aperiodic solution that isasymptoticallystable withrespecttosmallperturbationsinanyof thedynamical variables. (b) Excitable steadystatefork4= 180min1,k6⁼2min'(point+inFig. 2). Noticethat theordinateis alogarithmic scale.Thesteadystateof lowMPFactivity ([M]/[CT]

=0.0074,[YT]/[CT]⁼0.566)is stable withrespecttosmallperturbations ofMPFactivity (at1and2)but asufficiently large perturbationof [Ml(at 3)triggersatransient activation of MPF andsubsequent destructionofcyclin.Theregulatory systemthenrecoversto thestable steady state.(c)Parametervalues as in bexceptthatk6is now afunction of time(oscillatingnearpoint+inFig.2). See textforanexplanationof therulesfork6(Q).Notice that theperiodbetween cell divisions(burstsin MPFactivity)isidentical to the mass-doublingtime(Td⁼116 min in thissimulation).Simulations with different values ofTd(notshown)demonstrate that theperiodbetween MPF bursts istypicallyequalto themass-doublingtime-i.e.,the celldivisioncycleisgrowthcontrolled under thesecircumstances.Growth control canalsobeachieved (simulationsnotshown),holding k6 constant, by assumingthatk4 increaseswithtime between divisions and decreasesabruptlyafter an MPF burst.

7330 CellBiology: Tyson

R

_φ,p_A

= 0.83, R

_φ,p_B

= 0.43, R

_φ,p_C

= 0.49

33 / 39

(34)

Application to Synthetic Biology in E. Coli

Cascade of transcriptional inhibitions added to E.coli

[Weiss et al PNAS 05]

input small molecule aTc output protein EYFP

Specification: EYFP has to remain below 10

³

for at least 150 min., then exceeds 10

⁵

after at most 450 min.,

and switches from low to high levels in less than 150 min.

34 / 39

(35)

Specifying the expected behavior in QFLTL( R )

The timing specifications can be formalized in temporal logic as follows:

φ(t

1

, t

2

) = G(time < t

1

→ [EYFP] < 10

³

)

∧ G(time > t

2

→ [EYFP] > 10

⁵

)

∧ t

1

> 150 ∧ t

2

< 450 ∧ t

2

− t

1

< 150 which is abstracted into

φ(t

1

, t

2

, b

1

, b

2

, b

3

) = G(time < t

1

→ [EYFP] < 10

³

)

∧ G(time > t

2

→ [EYFP] > 10

⁵

)

∧ t

1

> b1 ∧ t

2

< b

2

∧ t

2

− t

1

< b

3

for computing validity domains for b

1

, b

2

, b

3

with the objective b

1

= 150, b

2

= 450, b

3

= 150 for computing the satisfaction degree in a given trace.

35 / 39

(36)

Improving robustness

Variance-based global sensitivity indices

S

_i

=

^Var(E(R|P_Var(R)ⁱ⁾⁾

∈ [0, 1]

S_γ 20.2 % S_κ

eyfp,γ 8.7 % S_κ

eyfp 7.4 % S_κ

cI,γ 6.2 %

S_κ_cI 6.1 % S_κ₀

cI,γ 5.0 % S_κ0

lacI

3.3 % S_κ0

cI,κeyfp 2.8 % S_κ0

cI

2.0 % SκcI,κeyfp 1.8 % S_κ_lacI 1.5 % S_κ₀

eyfp,γ 1.5 % S_κ0

eyfp

0.9 % S_κ0

cI,κcI 1.1 %

S_uaTc 0.4 % S_κ0

cI,κlacI 0.5 % total first order 40.7 % total second order 31.2 %

degradation factor γ has the strongest impact on the cascade.

aTc variations have a very low impact

different importance of the basal κ

⁰_eyfp

and regulated κ

_eyfp

EYFP production rates

the basal production of EYFP is due to an incomplete repression of the promoter by CI (high effect of κ

cI

) rather than a constitutive leakage of the promoter (low effect of κ

⁰_eyfp

).

36 / 39

(37)

Improving robustness

Variance-based global sensitivity indices

S

_i

=

∈ [0, 1]

S_γ 20.2 % S_κ

eyfp,γ 8.7 % S_κ

eyfp 7.4 % S_κ

cI,γ 6.2 %

S_κ_cI 6.1 % S_κ₀

cI,γ 5.0 % S_κ0

lacI

3.3 % S_κ0

cI

eyfp,γ 1.5 % S_κ0

eyfp

0.9 % S_κ0

cI,κcI 1.1 %

S_uaTc 0.4 % S_κ0

degradation factor γ has the strongest impact on the cascade.

aTc variations have a very low impact

different importance of the basal κ

⁰_eyfp

and regulated κ

_eyfp

EYFP production rates

the basal production of EYFP is due to an incomplete repression of the promoter by CI (high effect of κ

cI

) rather than a constitutive leakage of the promoter (low effect of κ

⁰_eyfp

).

37 / 39

(38)

Improving robustness

Variance-based global sensitivity indices

S

_i

=

∈ [0, 1]

S_γ 20.2 % S_κ

eyfp,γ 8.7 % S_κ

eyfp 7.4 % S_κ

cI,γ 6.2 %

S_κ_cI 6.1 % S_κ₀

cI,γ 5.0 % S_κ0

lacI

3.3 % S_κ0

cI

eyfp,γ 1.5 % S_κ0

eyfp

0.9 % S_κ0

cI,κcI 1.1 %

S_uaTc 0.4 % S_κ0

degradation factor γ has the strongest impact on the cascade.

aTc variations have a very low impact

different importance of the basal κ

⁰_eyfp

and regulated κ

eyfp

EYFP production rates

the basal production of EYFP is due to an incomplete repression of the promoter by CI (high effect of κ

cI

) rather than a constitutive leakage of the promoter (low effect of κ

⁰_eyfp

).

38 / 39

(39)

Improving robustness

Variance-based global sensitivity indices

S

_i

=

∈ [0, 1]

S_γ 20.2 % S_κ

eyfp,γ 8.7 % S_κ

eyfp 7.4 % S_κ

cI,γ 6.2 %

S_κ_cI 6.1 % S_κ₀

cI,γ 5.0 % S_κ0

lacI

3.3 % S_κ0

cI

eyfp,γ 1.5 % S_κ0

eyfp

0.9 % S_κ0

cI,κcI 1.1 %

S_uaTc 0.4 % S_κ0

degradation factor γ has the strongest impact on the cascade.

aTc variations have a very low impact

different importance of the basal κ

⁰_eyfp

and regulated κ

eyfp

EYFP production rates

the basal production of EYFP is due to an incomplete repression of the promoter by CI (high effect of κ

cI

) rather than a constitutive leakage of the promoter (low effect of κ

⁰_eyfp

).

39 / 39