Towards a general computational method for robustness analysis Grgory Batt with Aurlien Rizk, Franoi

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Towards a general computational method for
robustness analysis Grégory Batt with
Aurélien Rizk, François Fages, Sylvain
SolimanContraintes project team, INRIA
Paris-Rocquencourt
April 14, 2009
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Robust behaviour of transcriptional cascade
synthetic transcriptional cascade
Hooshangi et al, PNAS, 05

• Design problem obtain a rapid and high-amplitude
  response to aTc

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Robust behaviour of transcriptional cascade
synthetic transcriptional cascade
Hooshangi et al, PNAS, 05

• Design problem obtain a rapid and high-amplitude
  response to aTc

Could be used as a timer enforce correct
sequencing of events

• But
• cells subject to environmental fluctuations
• robustness capacity of a system to maintain a
  function in the face of perturbations
• ? explore large ranges of parameter values and
  determine robustness of cell response

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Describing the functioning of biological systems

• Robustness is threefold definition system,
  function, perturbation
• Large variety of function of interest ad hoc
  definitions
• growth signatures
• steady state values
• expression patterns
• presence, frequence, amplitude and phase of
  oscillations
• Definition of robustness as mean functionality
• where is probability of perturbation p and
  is "evaluation" function
• Robustness computation/estimation difficult to
  automate
• General problem with no general formalism

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Temporal logic as specification language

• Temporal logics are general-purpose languages for
  specifying dynamical properties of discrete
  transition systems
• Paves the way for developing general methods for
  robustness estimate
• Linear time logic (LTL) syntax
• finite set of atomic propositions
• usual logical operators
• temporal operators
• Semantics of LTL formulas defined over traces
• Pnueli, FOCS 77

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Biological properties formalized in LTL

• Formalizing properties of (finite) numerical
  traces in LTL
• But true/false valuation of TL not adapted to
  robustness analysis

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Biological properties formalized in LTL

• Formalizing properties of (finite) numerical
  traces in LTL
• But true/false valuation of TL not adapted to
  robustness analysis

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Biological properties formalized in LTL

• Formalizing properties of (finite) numerical
  traces in LTL
• But true/false valuation of TL not adapted to
  robustness analysis
• Need to quantify how far is the system from
  verifying the specification

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Validity domain of QFLTL formulae

• Validity domainD?(T) set of values of
  variables in formula ? making it true on trace
  T Fages and Rizk, CMSB'07

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Continuous valuation of LTL formulae

• Violation degree vd(T,?) distance between
  specification and validity domain

Rizk et al, CMSB'08
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Robustness definition using violation degree

• System's evaluation function defined using
  satisfaction degree
• Instantiation of Kitano's definition of
  robustness
• Temporal-logic based definition generic approach
  and computational method
• Computational estimation of robustness

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Various notions of robustness

• Robustness average functionality

same mean behaviour
but significant difference between mean and
nominal behaviour
Ex von Dassow et al.
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Various notions of robustness

• Robustness average functionality
• Relative robustness average and nominal
  functionality ratio

same mean behaviour
but significant difference between mean and
nominal behaviour
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Various notions of robustness

• Robustness average functionality
• Relative robustness average and nominal
  functionality ratio
• same relative robustness different
  robustness

same mean behaviour
but significant difference between mean and
nominal behaviour
Ex Gonze et al.
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Various notions of robustness

• Robustness average functionality
• Relative robustness average and nominal
  functionality ratio
• same relative robustness different robustness
• Formal definitions discriminate between different
  robustness notions

same mean behaviour
but significant difference between mean and
nominal behaviour
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Computation of validity domain
?
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Computation of validity domain

• Computation by induction on the trace and on the
  formula

?
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Computation of validity domain

• Computation by induction on the trace and on the
  formula

?
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Computation of validity domain

• Computation by induction on the trace and on the
  formula
• D?(T) computation unions and intersections of
  polytopes/orthotopes

?
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Implementation in BIOCHAM

• BIOCHAM is a modeling environment for analysis of
  biochemical systems
• New features using satisfaction degree of
  temporal logic formula
• robustness computation wrt versatile
  specifications? promotes genericity
• parameter search wrt high level specifications
  (currently up to 50 parameters)? ? avoids over
  specification

http//contraintes.inria.fr/BIOCHAMRizk et al,
CMSB'08
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Application design of transcriptional cascade
synthetic transcriptional cascade
Hooshangi et al, PNAS, 05

• Find parameters for which a fast and
  high-amplitude response to aTc is robustly
  obtained

• Approach
• specify expected behavior as TL formula
• develop ODE model
• find a perturbation model
• ? explore large ranges of parameter values
  and compute robustness of cell response

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Describing timed behavior of cascade
1) Specification fast and high-amplitude
response to aTc
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Describing timed behavior of cascade
1) Specification fast and high-amplitude
response to aTc
"well timed" if t1gt150, t2lt450 and ?tlt150
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Describing timed behavior of cascade
1) Specification fast and high-amplitude
response to aTc
"well timed" if t1gt150, t2lt450 and ?tlt150
2) ODE model with Hill functions
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Selection of perturbation model

• 3) Perturbation model
• Many kinds of perturbations are considered SDEs
  with additive/multiplicative noise, random ODEs
  with (log-)normally distributed parameters...

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Selection of perturbation model

• 3) Perturbation model
• Many kinds of perturbations are considered SDEs
  with additive/multiplicative noise, random ODEs
  with (log-)normally distributed parameters...
• Which perturbation model is most faithful to
  reality?
• Compare predicted and observed coefficients of
  variations for various fluorescence intensities

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Selection of perturbation model

• 3) Perturbation model
• Many kinds of perturbations are considered SDEs
  with additive/multiplicative noise, random ODEs
  with (log-)normally distributed parameters...
• Which perturbation model is most faithful to
  reality?
• Compare predicted and observed coefficients of
  variations for various fluorescence intensities
• LogN parameter distribution provides
  experimentally-consistent perturbation model

experimental measurement
coefficient of variations as function of mean
fluorescence
Hooshangi et al, PNAS, 05
computed using ODEs with LogN parameters
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Satisfaction degree and robustness landscapes

• 2D landscapes of satisfaction degree and
  robustness (16D)
• Large variations (2 orders of magnitude) of 2
  parameters?
• Parameter variations may alter correct
  functioning of system
• Robustness and relative robustness provide
  important design information
• identification of regions of robust behaviour
• identification of regions of discrepancy between
  reference and mean behaviours
• 8D robustness analysis
• Most robust parameters close to experimental best
  fit parameters

robustness
satisfaction degree
relative robustness
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Robustness analysis by variance decomposition

• Determination of how each parameter globally
  affects robust functioning using variance
  decomposition methods

(first order) global sensitivity index
Saltelli et al, Wiley, 04
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Robustness analysis by variance decomposition

• Determination of how each parameter globally
  affects robustfunctioning using variance
  decomposition methods
• Parameter contributions to robustness of temporal
  behaviours

(first order) global sensitivity index
Saltelli et al, Wiley, 04
first and second order global sensitivity indices
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Robustness analysis by variance decomposition

• Determination of how each parameter globally
  affects robustfunctioning using variance
  decomposition methods
• Parameter contributions to robustness of temporal
  behaviours
• parameters closely-related to output value
  generally have higher sensitivities

(first order) global sensitivity index
Saltelli et al, Wiley, 04
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Robustness analysis by variance decomposition

• Determination of how each parameter globally
  affects robustfunctioning using variance
  decomposition methods
• Parameter contributions to robustness of temporal
  behaviours
• parameters closely-related to output value
  generally have higher sensitivities
• sensitivity of expression rate of eyfp repressor
  cI higher than that of basal expression rate of
  eyfp

(first order) global sensitivity index
Saltelli et al, Wiley, 04
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Robustness analysis by variance decomposition

• Determination of how each parameter globally
  affects robustfunctioning using variance
  decomposition methods
• Parameter contributions to robustness of temporal
  behaviours
• parameters closely-related to output value
  generally have higher sensitivities
• sensitivity of expression rate of eyfp repressor
  cI higher than that of basal expression rate of
  eyfp
• relatively low influence of degradation parameter
  ? and input concentration uaTC

(first order) global sensitivity index
Saltelli et al, Wiley, 04
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Conclusion

• Instantiation of Kitano's definition of
  robustness to temporal logic setting
• ? general framework for robustness analysis
• ? implemented in publicly-available tool BIOCHAM

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Conclusion

• Instantiation of Kitano's definition of
  robustness to temporal logicsetting
• ? general framework for robustness analysis
• ? implemented in publicly-available tool BIOCHAM
• Temporal logic provides rich conceptual
  environment formethodological developments

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Conclusion

• Instantiation of Kitano's definition of
  robustness to temporal logicsetting
• ? general framework for robustness analysis
• ? implemented in publicly-available tool BIOCHAM
• Temporal logic provides rich conceptual
  environment formethodological developments
• Application to design of synthetic
  transcriptional cascade
• biologically consistent results validates method
• unexpected results suggests usefulness for
  system design

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Discussion

• Scalability is the issue for global robustness
  analysis
• curse of dimensionality exponential increase of
  computational time and space
• representation problem huge matrices do not
  provide much intuition on phenomenon of interest

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Discussion

• Scalability is the issue for global robustness
  analysis
• curse of dimensionality exponential increase of
  computational time and space
• representation problem huge matrices do not
  provide much intuition on phenomenon of interest
• Global sensitivity methods provide solutions on
  two counts
• efficient sampling techniques (e.g. Fourier
  amplitude sensitivity test)?
• information aggregation via use of sensitivity
  indices

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Discussion

• Scalability is the issue for global robustness
  analysis
• curse of dimensionality exponential increase of
  computational time and space
• representation problem huge matrices do not
  provide much intuition on phenomenon of interest
• Global sensitivity methods provide solutions on
  two counts
• efficient sampling techniques (e.g. Fourier
  amplitude sensitivity test)?
• information aggregation via use of sensitivity
  indices
• For parameter search problem possibility to
  integrate robustness infitness function

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Thank you for your attention!