rizk 2009 v6
TRANSCRIPT
-
8/13/2019 Rizk 2009 v6
1/16
David Fuente Herraiz
Review of
A general computational method
for robustness analysis withapplications to synthetic gene networks
Riez et al., 2009
-
8/13/2019 Rizk 2009 v6
2/16
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
-
8/13/2019 Rizk 2009 v6
3/16
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
-
8/13/2019 Rizk 2009 v6
4/16
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
-
8/13/2019 Rizk 2009 v6
5/16
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?
-
8/13/2019 Rizk 2009 v6
6/16
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()
-
8/13/2019 Rizk 2009 v6
7/16
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)
-
8/13/2019 Rizk 2009 v6
8/16
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):
-
8/13/2019 Rizk 2009 v6
9/16
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
-
8/13/2019 Rizk 2009 v6
10/16
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.
-
8/13/2019 Rizk 2009 v6
11/16
ODE model and perturbation model
ODE model with Hill functions :
Perturbation model: (log-)normally distributed parameters
Exp. Data Coeff. of Variation Simulation Satisf. Degree
-
8/13/2019 Rizk 2009 v6
12/16
Improving robustness
More robust system after non linear optimization (5000 simulations)
-
8/13/2019 Rizk 2009 v6
13/16
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
-
8/13/2019 Rizk 2009 v6
14/16
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
-
8/13/2019 Rizk 2009 v6
15/16
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
-
8/13/2019 Rizk 2009 v6
16/16
Thanks for your attention!
Questions?