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David Fuente Herraiz

Review of

A general computational method

for robustness analysis withapplications to synthetic gene networks

Riez et al., 2009

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Introduction to robustness

-An ability of the system to maintain its functions evenunder external and internal perturbations

-All biological systems live in noisy conditions

-Robustness as an evolution consequence

Robustness Variability

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Temporal logics approach

-Temporal logics are general-purpose languages for specifyingdynamical properties of discrete transition systems (Pnueli 1977)

- Automatic verification done by model-checking

- Temporal logic adapted to high level specifications and to

imprecise experimental data obtained in systems biology

- Description of temporal behaviour

Numerical data time series ([A]=0 at t=0, [A]= 45 at t=7, )

Need of formal language to be used by a computer program

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Formalize temporal properties in Linear Time Logic (LTL)

Linear Time Logic add temporal operators to usual logical operators

(,,,):

Fq (finally): q is true at some time point in the future

Gq (globally): q is true at all time points in the future

pUq (until) : p is true until q becomes true

Xq (next) : q is true at the next time point

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Drawback of LTL

True/False valuation of temporal logic formulae not well adapted toseveral problems:

- Parameter search, optimization and control of continuous models

- Quantitative estimation of robustness

- Local and global sensitivity analyses

need for a continuous degree of satisfaction of temporal logic formulae

How far is the system from verifying the specifications?

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Validity domain of free variables in LTL(R) formulae

Evaluation of temporal logic formulae on numerical traces

QFLTL()

f= F([A] 7 F([A] 0)) f* = F([A] x F([A] y))

Constraint solving

the formula is true for any x10 y2the formula is false

Model-checking

Validity domain D (T)

LTL()

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fa= F([A] 6 F([A] 5))

fc= F([A] 12 F([A] 0))

fb= F([A] 6 F([A] 0))

vd = 0 sd= 1

vd = 22 sd= 0,26

vd = 2 sd= 0,33

Violation and satisfaction degree of an LTL(R) formula

f*= F([A] 6 F([A] 5))

vd(T, f) = minvDf*(T) d(v,var(f))1 + vd(T, f)

1sd(T, f) = [0,1]

f*(6,5)

f*(12,0)

f*(6,0)

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Robustness Measure Definition

Robustness defined w. r. t.: a biological system

a functionality property Daa set P of perturbations

a,P =

pPDa(p) prob(p) dp

The proposed computational measure of robustness w.r.t. LTL(R) spec:

pP

sd (T (p), )prob(p) dp,P =

evaluate mean behavior of a system subject to noise, comparerobustness of different designs, use robustness as optimization objective

/ sd (T (p), )*,P = ,P

General notion of robustness (Kitano 2007):

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Application to a Transcriptional cascade in E. Coli

Ultrasensitivity and noise propagation in a synthetic transcriptional

cascade(Weiss et al., 2005)

The output protein EYFP is controlled by the small input molecule aTc

Biological timer for synthetic biology applications

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Specifying the expected behavior in LTL(R)

The timing specifications can be formalized in temporal logic as:

(t1, t2) = G(time < t1 [EYFP] < 103)

G(time > t2 [EYFP] > 105)

t1 > 150 t2 < 450 t2 t1 < 150

which is abstracted into

(t1, t2, b1, b2, b3) = G(time < t1 [EYFP] < 103)

G(time > t2 [EYFP] > 105)

t1 > b1 t2 < b2 t2 t1 < b3

with the objective b1 = 150, b2 = 450, b3 = 150 for computing the

validity domains and the satisfaction degree in a given trace.

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ODE model and perturbation model

ODE model with Hill functions :

Perturbation model: (log-)normally distributed parameters

Exp. Data Coeff. of Variation Simulation Satisf. Degree

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Improving robustness

More robust system after non linear optimization (5000 simulations)

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Improving robustness

- has to remain in a narrow interval,

whereas eyfp simply has to exceed some value

- Blurred: Parameter variations smaller than parameter perturbations

Satisfaction degree Robustness Relative Robustness

2D Parameters space

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Parameter contribution on global robustness

Variance-based global sensitivity indices

Si =Var (E (R |Pi))

Var (R) [0,1]

- Degradation scaling factor has the strongest impact

- aTc inducer has no effect

- Basal production of EYFP is due to an incomplete repression of

the promoter by CI (high effect of cI ) rather than a constitutive

leakage of the promoter (low effect of 0eyfp)

S

Seyfp

ScI

S0LacIS0cI

SLacID

S0eyfp3

SuaTc

20.2 %

7.4 %

6.1 %

3.3 %

2.0 %

1.5 %

0.9 %

0.4 %

8.7 %

6.2 %

5.0 %

2.8 %

1.8 %

1.5 %

1.1 %

0.5 %total first order 40.7 % total second order 31.2 %

Seyfp,

ScI,

S0cI,

S0cI, eyfpScI, eyfpS0eyfp ,

S0cI, cI,sS0cI, LacI

8D Parameters space

Conclusion

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ConclusionsConclusion

Presented a general computation framework for robustness estimation

- Generalization of model-checking to temporal constraint solving

- Defined satisfaction degree for quantitative notion of robustness

Reported an unambiguous definition of robustness

Robustness study for experimental design assessment

Improved robustness of the timed response of transcriptional cascade

- Found parameter modifications for a robust timed behaviour

- Explored the impact of possibly large parameter variations on robustness

Useful paper for learning temporal logics and robustness context

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Thanks for your attention!

Questions?