Rizk 2009 v6

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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?