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5FD Model validation  Available information on structure of network controlling E. coli stress response is incomplete Model is working hypothesis and needs to be tested Model validation is prerequisite for use of model as predictive and explanatory tool Check consistency between model predictions and experimental data Current constraints on model validation: available experimental data essentially qualitative in nature model validation must be automatic and efficienta #5 #V #D #) #q #; 5FD)q  !Objectives and approach of thesis\Objective of thesis: Development of automated and efficient method for testing whether predictions from qualitative models of genetic regulatory networks are consistent with experimental data on dynamics of system Approach based on formal verification of hybrid systems qualitative analysis of piecewise-linear models of genetic networks model checking for testing consistency between predictions and data Expected contributions: scalable method with sound theoretical basis implementation of method in user-friendly computer tool applications to validation of models of networks of biological interest 8  0   Overview@Introduction Method for model validation Piecewise-linear (PL) differential equation models Symbolic analysis using qualitative abstraction Verification of properties by means model-checking techniques Genetic Network Analyzer 6.0 Validation of model of nutritional stress response in E. coli Discussion and conclusionst*vvUvv*S Overview@Introduction Method for model validation Piecewise-linear (PL) differential equation models Symbolic analysis using qualitative abstraction Verification of properties by means model-checking techniques Genetic Network Analyzer 6.0 Validation of model of nutritional stress response in E. coli Discussion and conclusions*vwwwvUvvwwwS PL differential equation modelsGenetic networks modeled by class of differential equations using step functions to describe switch-like regulatory interactions &@ B5 (Qualitative analysis of network dynamicsZAnalysis of the dynamics in phase space: Partition of phase space into mode domains $[M  (Qualitative analysis of network dynamics,Analysis of the dynamics in phase space: $-((Qualitative analysis of network dynamics0Analysis of the dynamics in phase space: @0` ((Qualitative analysis of network dynamics/Analysis of the dynamics in phase space: /@P0(Qualitative analysis of network dynamicsAnalysis of the dynamics in phase space: Extension of PL differential equations to differential inclusions using Filippov approach: ^0\@P($  x (Qualitative analysis of network dynamicsAnalysis of the dynamics in phase space: In every mode domain M, the system either converges monotonically towards focal set, or instantaneously traverses MX(<A\Problem for model validationPartition does not preserve unicity of derivative sign Predictions not adapted to comparison with available experimental data: temporal evolution of direction of change of protein concentrations(77(Qualitative analysis of network dynamicsFiner partition of phase space: flow domains In every domain D, the system either converges monotonically towards focal set, or instantaneously traverses D In every domain D, derivative signs are identical everywhere=   A\a,Continuous transition systemPL system, S = (W,Q,H), associated with continuous PL transition system, S-TS = (W,!,^%), where W continuous phase space _  @@AContinuous transition system&PL system, S = (W,Q,H), associated with continuous PL transition system, S-TS = (W,!,^%), where W continuous phase space ! transition relation 6_1  @@   Continuous transition systemPL system, S = (W,Q,H), associated with continuous PL transition system, S-TS = (W,!,^%), where W continuous phase space ! transition relation ^% satisfaction relation S and S-TS have equivalent reachability properties_H 7 @@      Discrete abstractionQualitative PL transition system, S-QTS = (D, !~,^%~), where D finite set of domains $< Discrete abstractionQualitative PL transition system, S-QTS = (D, !~,^%~), where D finite set of domains !~ quotient transition relation H<;   Discrete abstraction<Qualitative PL transition system, S-QTS = (D, !~,^%~), where D finite set of domains !~ quotient transition relation ^%~ quotient satisfaction relation <]   Discrete abstractionQualitative PL transition system, S-QTS = (D, !~,^%~), where D finite set of domains !~ quotient transition relation ^%~ quotient satisfaction relation Quotient transition system S-QTS is a simulation of S-TS (but not a bisimulation)<bY P   $    Discrete abstractionLImportant properties of S-QTS : S-QTS provides finite and qualitative description of the dynamics of system S in phase space S-QTS is a conservative approximation of S : every solution of S corresponds to a path in S-QTS S-QTS is invariant for all parameters q, k, and g satisfying a set of inequality constraints S-QTS can be computed symbolically using parameter inequality constraints: qualitative simulation Use ofS-QTS to verify dynamical properties of original system S Need for automatic and efficient method to verify properties of S-QTS  @H8G  !F3    X    !Model-checking approach4    Model checking is automated technique for verifying that discrete transition system satisfies certain temporal properties Computation tree logic model-checking framework: set of atomic propositions AP discrete transition system is Kripke structure KS = S, R, L , where S set of states, R transition relation, L labeling function over AP temporal properties expressed in Computation Tree Logic (CTL) p, f1, f1f2, f1f2, f1!f2, EXf1, AXf1, EFf1, AFf1, EGf1, AGf1, Ef1Uf2, Af1Uf2, where pAP and f1, f2 CTL formulas Computer tools are available to perform efficient and reliable model checking (e.g., NuSMV, SPIN, CADP)z1 (aK>wh (l1!  A  A aAIAAIAIAAAIAAAIAAAIAAAIAAAIAAAIAAAIAAIAAAIAAI O   >p"Validation using model checkingAtomic propositions AP = {xa = 0, xa < qa1, ... , xb < maxb, xa < 0, xa= 0, ... , xb > 0} Observed property expressed in CTL G# 2           "                 ,#Validation using model checkingDiscrete transition system computed using qualitative simulation Use of model checkers to check consistency between experimental data and predictions Fairness constraints used to exclude spurious behaviors<80 F$Overview@Introduction Method for model validation Piecewise-linear (PL) differential equation models Symbolic analysis using qualitative abstraction Verification of properties by means model-checking techniques Genetic Network Analyzer 6.0 Validation of model of nutritional stress response in E. coli Discussion and conclusions*vwwwvUvvwww*6% Genetic Network AnalyzerfModel validation approach implemented in version 6.0 of GNA, freely available for academic research &!Genetic Network AnalyzerGNA implemented in Java 1.4 > 17000 lines of code in 6 packages 35% of lines modified with respect to version 5.5 (up to 60% in kernel) 8~~'"Genetic Network AnalyzerRules for symbolic computation of refined partition and corresponding transition relation and domain properties Tailored algorithms and implementation favor upscalability Export functionalities to model checkers (NuSMV, CADP) XpB7 <pB, 2 (#Overview@Introduction Method for model validation Piecewise-linear (PL) differential equation models Symbolic analysis using qualitative abstraction Verification of properties by means model-checking techniques Genetic Network Analyzer 6.0 Validation of model of nutritional stress response in E. coli Discussion and conclusions*vwwwvUvvwww*6)$&Nutritional stress response in E. coli'JEntry into stationary phase is an important developmental decision KDD *%%Model of nutritional stress response &%}Carbon starvation network modeled by PL model 7 PL differential equations, 40 parameters and 54 inequality constraints ^.8IP8 -+v +&#Validation of stress response modelQualitative simulation of carbon starvation: 66 reachable domains (< 1s.) single attractor domain (asymptotically stable equilibrium point) Experimental data on Fis: CTL formulation: Model checking with NuSMV: property true (< 1s.) b- 2` 2 F  3 -`c,-,'#Validation of stress response modelOther properties:  cya transcription is negatively regulated by the complex cAMP-CRP  DNA supercoiling decreases during transition to stationary phase Inconsistency between observation and prediction calls for model revision or model extension Nutritional stress response model extended with global regulator RpoSD C d] -H0 ]HP6 -()Novel prediction of stress response modelQualitative simulation of carbon upshift response: 1143 reachable domains (< 2s) several strongly connected components Are some strongly connected components attractors? Attractor corresponds to damped oscillations towards stable equilibrium point: unexpected prediction Experimental verification of model predictions Time-series measurements of protein concentrations in parallel and at high sampling rate using gene reporter system3 2E0 #30 #0 #e0 </ 2u ! E3O  u J!x    .)Overview@Introduction Method for model validation Piecewise-linear (PL) differential equation models Symbolic analysis using qualitative abstraction Verification of properties by means model-checking techniques Genetic Network Analyzer 6.0 Validation of model of nutritional stress response in E. coli Discussion and conclusions*vwwwvUvvwww*S/*SummaryDevelopment of automated and efficient method for testing whether predictions from qualitative models of genetic regulatory networks are consistent with experimental data on system dynamics Use of discrete abstraction that yields predictions well-adapted to comparison with available experimental data Combination of tailored symbolic analysis and model checking for verification of dynamical properties of hybrid models of large and complex networks Biological relevance demonstrated on validation of models of networks of biological interest  - 0+ DiscussionDiscrete abstractions used for analysis of continuous and hybrid models symbolic reachability analysis of hybrid automata models more precise analysis of system dynamics need for complex decision procedures no treatment of discontinuities in vector field qualitative simulation using qualitative differential equations more general class of model methods are not scalable Model checking used for analysis of discrete models verification of properties of logical models intuitive connection between underlying continuous dynamics and discrete representation no explicit representation of dynamical phenomena at threshold concentrations H:~ZA6Z4 #.=ZZPZH:~@5   4.Q &1, PerspectivesFurther integration of model-checking task into GNA Property specification, verification, interpretation of diagnostics Exploitation of advanced model-checking techniques Partial order reduction, graph minimization, modular model checking, ... 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(  #)  0e0e    B8C!DE4F jJ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| ; Pec!  /8 @    "0e@`ԺԺ ԺԺ ԺԺԺԺ ԺԺ ԺԺԺԺ Ժ@ABC DEFGHIJK5%LMNOPQRSTUWYԺZԺ[ \]Ժ^Ժ_ `aԺbԺN 5%ԺԺ ԺԺ ԺԺN 5%ԺԺ ԺԺ ԺԺN    5%  ԺԺ ԺԺ !Ժ"Ժ?N@ABC DEFGHIJK5%LMNOPQRSTUWYԺZԺ[ \]Ժ^Ժ_ `aԺbԺg  xB $) HD? 0 B %) < D?#" N  z   &)  ,$D 0r ') c 4AFis-Gna"`K T  () H*p?"0@NNN?N'  ? predictions   T pM ))#  k ^ g *) BH;p ?"6@ NNN?Np#M  x = f (x) z 0v0; +) HXAp?"0@NNN?N; 7.   ,) HDp?"0@NNN?NM u 9model H ( 0޽h ? 333WcL+᷷Ҫd\___PPT10<.7P"e+~D ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(0%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*&)%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(+8+0+(p + " 0L(  0x 0 c $Qp@Px  p  0 c $ Xp0 p ())e=H 0 0޽h ? 333WcL+᷷Ҫ___PPT10i.7P"e+D=' = @B +E  8\(  8x 8 c $ kp@XHx  p  8 c $lp0<$ 0 p H 8 0޽h ? 333WcL+᷷Ҫ!___PPT10 .6+FD ' = @B DP ' = @BA?%,( < +O%,( < +Ds' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8S%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8S%(D ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8\%(+8+0+8I + #K0 <P(  <~ < s *Lrp@XHx  p  < s *K'  p  `pH < 0޽h ? 333WcL+᷷Ҫy___PPT10Y+D=' = @B + $K0 DR(  D~ D s * p@XHx  p  D s *p'  p "`@ H D 0޽h ? 333WcL+᷷Ҫy___PPT10Y+D=' = @B +3 %  **0;;L)(  L~ L s *dp@XHx  p x L c $8p p 8z   L S ,$D 0T C ( L#  T  <  L# C  ~ L 6Tp ?/ <  Txa = ka s-(xa , qa2) s-(xb , qb )  ga xa +0 2                          L 0p   O.0   T  <   L# K (  L 6p ?/ <  N:xb = kb s-(xa , qa1)  gb xb 0 2         p           L 0p   O.0   T $   L# x w{  L <$p ?  x : protein concentration:0 &     L <p ? $  ,k , g : rate constants<0   2 L <pp ?W d  8q : threshold concentration<0 &    T 8 Z8l  L#   lB L <D?~B L ND?|<lB L <D?rB L BD?8 L <p ?Z[T  ax00    L L 6$p ?8 Z7T s-(x, ) 0  c&  xB L HDjJ?hh L <`p ?rl  Tq"0    L < ?  a000   B L HDjJ?"0@NNN?NxhxB L HDjJ?"0@NNN?N`x@x L < ? a100    L c $   (,$ 0 CHybrid, piecewise-linear (PL) models of genetic regulatory networksD`7 DC  L H  jJ?"6@ NNN?NF* ,$ 0 (Glass and Kauffman, J. Theor. Biol., 73 6)0  pz |&   L & d ,$D 0lB  L <D?zL L f !L 6(2(2?j  L T (i "L#  2 M f #L 6@@?8hlB $L <D?ixB %L HD?( &L < ? L `  ab00  ,  f2 'L 6(2(2?  r  (L <\ ?r & y  aB00  ,  lB )L <D?L L f *L 6(2(2?" L T (i +LC#  RM f ,L 6@@?8hlB -L <D?ixB .L HD?( /L < ?T @  aa00  ,  f2 0L 6(2(2?Z4  1L <# ?|~)  aA00  ,  rB 2L BD?x  rB 3L BD? lB 4LB <D?pX X X lB 5L <D?pX p lB 6L <D? lB 7L <D? lB 8L <D?  B 9L BDjJ?"0@NNN?N 8 B :L BDjJ?"0@NNN?NQ  B ;L BDjJ?"0@NNN?N * H L 0޽h ? 333WcL+᷷Ҫ  ___PPT10b +!HDF ' = @B D ' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(+9 & 33P88Tf3(  T T c $$30<$ 0   T 64"`G A  tqa1>0      T 6 ~I"`, I&  _0.0     T 6d<"`n ~maxb<0      T 6TA"`T  O  tqa2>0      T 6LG"`M qb>0     x T B o?"`[m i   T 6M"`C y x =  |maxa>0 CI   T 60P @XHx   F    T x O   T 6Q"`H `  tqa1>0       T 6V"`   _0.0     T 6 ["` ~maxb<0      T 6l`"`  tqa2>0      T 6e"`+ |%  qb>0      T 6\k"`  |maxa>0 CI ~B T NDo? ~B T NDo?01 ~B T NDo? \  z  3  T  P ,$D 0 T 6q"`1  tqa1>0      T 6v"` z  _0.0     T 6z"` 3- ~maxb<0      T 6L"`P  tqa2>0      T 6"`~ mMg qb>0      T B o?"` 5  T 6"`  |maxa>0 CI T   T#  3  T 6"`H `  tqa1>0      T 6"`   _0.0      T 6"` ~maxb<0      !T 6"`  tqa2>0      "T 6ĥ"`+ |%  qb>0      #T 6 "`  |maxa>0 CI ~B $T NDo? ~B %T NDo?01 ~B &T NDo? \ 2T Z  'T# "p( (T Z?"6@`NNN?NZ XM1:0 ( )T Z?"6@`NNN?Nz XM2:0 ( *T Z?"6@`NNN?N  XM3:0 ( +T Z$?"6@`NNN?N{   XM4:0 ( ,T Z(Ò?"6@`NNN?Nx   XM5:0 ) -T Zǒ?"6@`NNN?N_   YM10:0 ) .T Z̒?"6@`NNN?Nx   YM15:0 ) /T ZxВ?"6@`NNN?N   YM14:0 ) 0T ZԒ?"6@`NNN?N   YM13:0 ) 1T ZXْ?"6@`NNN?N  YM12:0 ) 2T Zݒ?"6@`NNN?Nz  YM11:0 ( 3T Z8?"6@`NNN?Nd XM6:0 ( 4T Z?"6@`NNN?N~ XM7:0 ( 5T Z?"6@`NNN?N  XM8:0 ( 6T Z?"6@`NNN?N   XM9:0  7T B ?"6@ NNN?N) N,x = h (x), x W \Q 0v0;   8T N ho?"6@ NNN?N  P. 26 0;H T 0޽h ? 333WcL+᷷Ҫg____PPT10?.La+9D' = @B D' = @BA?%,( < +O%,( < +D' =%(Du' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T/Z%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T%(+8+0+T +:B ' 33p55\83(  \ \  6("`0   \ 6\*"`Q K  tqa1>0      \ 6H"`6 I0  _0.0     \ 6"`n ~maxb<0      \ 63"`^  Y  tqa2>0      \ 688"`M qb>0     x \ B o?"`em s   \ 6="`M y x G  |maxa>0 CI   \ 6 @ @XHx   L C (  \#  T  <   \# C  p  \ 6`G ?/ <  Rxa = ka s-(xa , qa2) s-(xb , qb )  ga xa*0 2             ~            \ 0b   O.0   T  <  \# K ( \ 6Xi ?/ <  N:xb = kb s-(xa , qa1)  gb xb 0 2         p          \ 0D}   O.0   F   \ x Y  \ 6"`H `  tqa1>0      \ 6L"`   _0.0     \ 6"` ~maxb<0      \ 6"`  tqa2>0      \ 6"`+ |%  qb>0      \ 6"`  |maxa>0 CI ~B \ NDo? ~B \ NDo?01 ~B \ NDo? \ z   \  ,$D 0 T  <  \#   \ 6Т ?/ <  | xa = ka  ga xa 0 2     V       \ 0   O.0    T  <   \#   !\ 6 ?/ <  | xb = kb  gb xb 0 2     V       "\ 0(ë   O.0   z ]w P  #\ ]w P ,$@ 0~B $\ ND0;o?ml n~B %\ ND0;o?]k C &\ 6Hȫ"` V w P  ka/gax0 0;0;0;0;0;&   Q '\ 6Pϫ"`  kb/gbx0 0;0;0;0;0;4    t2 (\ 6?"`Pl )\ 6?"`dx  G *\# C ,$D 0B +\ TD?"0@NNN?NGB ,\ TD?"0@NNN?Nr  }   -\#   ,$@ 0D .\ 6۫ ? } j  :0 < qa1 < qa2 < ka/ga < maxa 0 2    b       & /\ 6 ?   ,0 < qb < kb/gb < maxb 0 2    |          z z ! 6 0\ y ,$D 0 1\ 6 ?z ! .  N0 = ka  ga xa 0 2    H      2\ 6 ?z ) z 6 N0 = kb  gb xb 0 2    H       3\ NL ho?"6@ NNN?N  P. 26 0; 4\ < ?T_T nM1<0    5\ B ?"6@ NNN?N) N,x = h (x), x W \Q 0v0; H \ 0޽h ? 333WcL+᷷Ҫ:2___PPT10.La+ D ' = @B D ' = @BA?%,( < +O%,( < +D' =%(Df' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<* \%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(DJ' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*\%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#\%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0\%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-\%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<**\%(+?$ ( V#N#''d"(  dx d c $?0   d 6?"`Q K  tqa1>0      d 6D"`6 I0  _0.0     d 6)"`n ~maxb<0      d 6N"`^  Y  tqa2>0      d 6S"`M qb>0     x d B o?"`em s   d 6dY"`M y x G  |maxa>0 CI   d 6[ @XHx   F    d x Y   d 6d_"`H `  tqa1>0       d 6d"`   _0.0     d 6h"` ~maxb<0      d 6$n"`  tqa2>0      d 6\s"`+ |%  qb>0      d 6y"`  |maxa>0 CI ~B d NDo? ~B d NDo?01 ~B d NDo? \  d 6$ ?x 9 Nxa =  ga xa 0 2     <      d 0 , O.0    d 6ؐ ?  | xb = kb  gb xb 0 2     V       d 0 4 O.0   r d <?"`f`q d < ?@I oM11<0   L  }   d#   D d 6x ? } j  :0 < qa1 < qa2 < ka/ga < maxa 0 2    b       & d 6D ?   ,0 < qb < kb/gb < maxb 0 2    |           d NǬ ho?"6@ NNN?N  P. 26 0; d B8ͬ ?"6@ NNN?N) N,x = h (x), x W \Q 0v0; F BS  d ]>B !d TD?"0@NNN?NnB "dB TD?"0@NNN?NBSB #d ZND?"0@NNN?N8<F   $d  ~B %d ND0;o?ml nQ &d 6|"`  kb/gbx0 0;0;0;0;0;4    t2 'd 6?"`VPH d 0޽h ? 333WcL+᷷Ҫ___PPT10i.La+D=' = @B +" ) ''l\(  lx l c $Ԭ0   l 6"`Q K  tqa1>0      l 6"`6 *0  _0.0     l 6L"`O ~maxb<0      l 6"`^ k Y  tqa2>0      l 6l"`. qb>0     x l B o?"`eN s   l 6l "`M Z Y G  |maxa>0 CI   l 6 @XHx   F    l Y Y   l 6"`H `  tqa1>0       l 6"`   _0.0     l 64"` ~maxb<0      l 6!"`  tqa2>0      l 6&"`+ |%  qb>0      l 6+"`  |maxa>0 CI ~B l NDo? ~B l NDo?01 ~B l NDo? \ F f  l f ZB l s *DԔ||f  l <2 ?T nM2<0   l z ]X P  l ]X P ,$D 0t2 l 6?"`W  f N ]X P  l ]X P ~B l ND0;o?]k C l 6H8"`V X P  ka/gax0 0;0;0;0;0;&   Q l 6P?"`M kb/gbx0 0;0;0;0;0;4    B l TD?"0@NNN?NA B l TD?"0@NNN?NlB  l TD?"0@NNN?NtB !l TD?"0@NNN?N(  "l <G ?T nM3<0    #l <L ?T_T nM1<0   ~B $l ND0;o?Pl nt2 %l 6?"`P  &l NQ ho?"6@ NNN?N  P. 26 0; 'l B N ?"6@ NNN?N) N,x = h (x), x W \Q 0v0; H l 0޽h ? 333WcL+᷷Ҫ___PPT10n.La+f6HvDB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*l%(+H * N=F=KKt<(  t t c $$x0<$ 0   t 6y"`Q K  tqa1>0      t 6Y"`6 *0  _0.0     t 6T]"`O ~maxb<0      t 6"`^ k Y  tqa2>0      t 6"`. qb>0     x t B o?"`eN s   t 64"`M Z Y G  |maxa>0 CI   t 6T @XHx   F    t Y Y   t 6"`H `  tqa1>0       t 64"`   _0.0     t 6ؚ"` ~maxb<0      t 6蟰"`  tqa2>0      t 6"`+ |%  qb>0      t 6ર"`  |maxa>0 CI ~B t NDo? ~B t NDo?01 ~B t NDo? \ vB t ND0;o?]k ; t 6p"`V X P  ka/gax0 0;0;0;0;0;&   l2 t 6?"`W  B t TD?"0@NNN?NtB t TD?"0@NNN?N( 5 t 6"`M kb/gbd0 0;0;0;0;4    B t TD?"0@NNN?NA B t TD?"0@NNN?NlB t TD?"0@NNN?NtB t TD?"0@NNN?N(  t < ?T nM3<0     t 6Hư"`H 0HB  tqa1>0      !t 6˰"`- '  _0.0     "t 6ǰ"`  ~maxb<0      #t 6(԰"`U gP  tqa2>0      $t 60ٰ"` d qb>0     x %t B o?"`\ Lj  &t 6ް"`D >  |maxa>0 CI F   't  P  (t 6<"`H `  tqa1>0      )t 6"`   _0.0     *t 6@"` ~maxb<0      +t 6P"`  tqa2>0      ,t 6"`+ |%  qb>0      -t 6"`  |maxa>0 CI ~B .t NDo? ~B /t NDo?01 ~B 0t NDo? \ vB 1t ND0;o?Tb ; 2t 6"`M TG  ka/gax0 0;0;0;0;0;&   l2 3t 6?"`N 8 B 4t TD?"0@NNN?NQfkB 5t TD?"0@NNN?NR B 6t@ TD?"0@NNN?N B 7t@ TD?"0@NNN?N-B 8t TD?"0@NNN?NQfkB 9t TD?"0@NNN?NR  :t < ?wK nM3<0   RB ;t s *DԔ` RB t ND0;o?lP nl2 ?t 6?"`P @t < ?hs nM5<0   l2 At 6?"`P j Bt <H ? P @,$ 0 Gouz and Sari, Dyn. Syst., 02:4 0;( N       3 Ct N8% ho?"6@ NNN?N % " ,$ 0 C.&6 0; Dt <* ?T nM2<0    Et <( ?xL nM4<0   z 2R  Ft 2R ,$D 0B Gt ZN?D?"0@NNN?N2R t2 Ht 6?"`P 0   It N2 ho?"6@ NNN?N  P. 26 0; Jt BG ?"6@ NNN?N) N,x = h (x), x W \Q 0v0;   Kt B< ?"6@ NNN?N` l ; ,$ 0 <"x H (x), x W0v0;0   H t 0޽h ? 333WcL+᷷Ҫ  ___PPT10 .La+C D ' = @B Da ' = @BA?%,( < +O%,( < +D' =%(D@' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t0%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Kt%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Bt%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Ct%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Ft%(++0+t ++0+Bt ++0+Ct ++0+Kt +VK + AAIJ|A(  | | c $H\0<$ 0   | 6p"`  tqa1>0      | 64v"`I  _0.0     | 6z"`2n, ~maxb<0      | 6T"`  tqa2>0      | 6"`lMf qb>0     x | B o?"`m    | 6|"`y x  |maxa>0 CI  F  a   | B B  | TD?"0@NNN?N f * B  | TD?"0@NNN?N B  | TD?"0@NNN?N- 3 B |B TD?"0@NNN?N7 akB |B TD?"0@NNN?NTaRB |B TD?"0@NNN?N  B |B TD?"0@NNN?NiO' B |B TD?"0@NNN?Nv  B |B TD?"0@NNN?Ny  B |B TD?"0@NNN?N|  B | TD?"0@NNN?N! 6( B | TD?"0@NNN?N"h _ B | ZN?D?"0@NNN?N " B | ZND?"0@NNN?NP 1[  | 6 @XHx   F   | 2x  | 6"`H `  tqa1>0      | 60"`   _0.0     | 6`"` ~maxb<0      | 6Ģ"`  tqa2>0      | 64"`+ |%  qb>0       | 6"`  |maxa>0 CI ~B !| NDo? ~B "| NDo?01 ~B #| NDo? \  z  3  $| 3  ,$@ 0 %| 6l"`1  tqa1>0      &| 6t"` z  _0.0     '| 6輲"` 3- ~maxb<0      (| 6²"`P  tqa2>0      )| 6Dz"`~ mMg qb>0      *| B o?"` 5  +| 6̲"`  |maxa>0 CI T   ,|#  3  -| 6pҲ"`H `  tqa1>0      .| 6ײ"`   _0.0     /| 6ܲ"` ~maxb<0      0| 6"`  tqa2>0      1| 6"`+ |%  qb>0      2| 6"`  |maxa>0 CI ~B 3| NDo? ~B 4| NDo?01 ~B 5| NDo? \ 2T Z  6|# "p( 7| Z?"6@`NNN?NZ XM1:0 ( 8| Z?"6@`NNN?Nz XM2:0 ( 9| Zh?"6@`NNN?N  XM3:0 ( :| ZH?"6@`NNN?N{   XM4:0 ( ;| Z ?"6@`NNN?Nx   XM5:0 ) <| Z?"6@`NNN?N_   YM10:0 ) =| Z ?"6@`NNN?Nx   YM15:0 ) >| Z?"6@`NNN?N   YM14:0 ) ?| Z ?"6@`NNN?N   YM13:0 ) @| ZX?"6@`NNN?N  YM12:0 ) A| Z`?"6@`NNN?Nz  YM11:0 ( B| Zt?"6@`NNN?Nd XM6:0 ( C| Z ?"6@`NNN?N~ XM7:0 ( D| Z$?"6@`NNN?N  XM8:0 ( E| Z(?"6@`NNN?N   XM9:0   F| N, ho?"6@ NNN?N  P. 26 0; G| B0 ?"6@ NNN?N* <"x H (x), x W0v0;0    H| Hd: jJ?"6@ NNN?N=PX,$ 0 %de Jong et al., Bull. Math. Biol., 04P&0 j J| <U ?`,$ 0 Gouz and Sari, Dyn. 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Z?"6@`NNN?N   YM14:0 ) @ Z?"6@`NNN?N   YM13:0 ) A ZX?"6@`NNN?N  YM12:0 ) B Zp?"6@`NNN?Nz  YM11:0 ( C Z?"6@`NNN?Nd XM6:0 ( D Z?"6@`NNN?N~ XM7:0 ( E Z?"6@`NNN?N  XM8:0 ( F Z+?"6@`NNN?N   XM9:0 z 5 @  G   q ,$D 0 H Z/jJ?"6@`NNN?N Y @   xa < 0, xb ? "0;0;0;0;$n I T7jJ?"0@NNN?N5X ?  "x M11:~ 0 0;0;0;0; 0;0; J T?jJ?"0@NNN?N j  C.&0  K TCjJ?"0@NNN?N_   C.&0 H  0޽h ? 333WcL+᷷Ҫ___PPT10{.La+=?%D' = @B D' = @BA?%,( < +O%,( < +D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*G%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*>%(+8+0+ +f - KWCW0aaV(    c $@S0<$ 0    6T"`  tqa1>0       6,Z"`I  _0.0      62"`2n, ~maxb<0       68c"`  tqa2>0       6h"`lMf qb>0     x  B o?"`m     6(n"`y x  |maxa>0 CI    6r @XHx   F     2x    6\t"`H `  tqa1>0        6y"`   _0.0      6~"` ~maxb<0       6"`  tqa2>0       6\"`+ |%  qb>0       6"`  |maxa>0 CI ~B  NDo? ~B  NDo?01 ~B  NDo? \ 3z  3   3  ,$D 0  6"`1  tqa1>0       6P"` z  _0.0      6@"` 3- ~maxb<0       6"`P  tqa2>0       6"`~ mMg qb>0       B o?"` 5   6"`  |maxa>0 CI N     3   6P"`H `  tqa1>0       6"`   _0.0       6T"` ~maxb<0      ! 6dõ"`  tqa2>0      " 6$ɵ"`+ |%  qb>0      # 6\ε"`  |maxa>0 CI ~B $ NDo? ~B % NDo?01 ~B & NDo? \ !N   '  ^X~B ( NDo?= ) 6Ե"`  kb/gbd0   4    "N e $  * 6 af  + Ttܵ?"0@NNN?NX ,  ZD12 :0 1 , T?"0@NNN?N  mD22 L0 1 - T4?"0@NNN?NH  mD23 L0 1 . T?"0@NNN?Nz mD24 L0 0 / TP?"0@NNN?N  lD17 L0 0 0 T?"0@NNN?NeF  lD18 L0 0 1 T?"0@NNN?N lD21 L0 0 2 T?"0@NNN?Nm lD20 L0 2 3 T?"0@NNN?N^o C  nD1 N0 0 4 TH ?"0@NNN?Np D  lD3 L0 0 5 T?"0@NNN?Nq E  lD5 L0 0 6 T0?"0@NNN?Nr F  lD7 L0 0 7 T?"0@NNN?N s P G  lD9 L0 1 8 T ?"0@NNN?N# R &  mD15 L0 1 9 T%?"0@NNN?N( I   mD27 L0 2 : Tl+?"0@NNN?NKX   nD26 L0 1 ; TL0?"0@NNN?NL+  mD25 L0  < T5?"0@NNN?NW +  ZD11 :0  = T:?"0@NNN?NQ %  [D13 :0  > T=?"0@NNN?NN % "  [D14 :0 0 ? TlC?"0@NNN?NG   lD2 L0  @ T 0 "0;0;0;0;0;0;0;$n L TmjJ?"0@NNN?N=X ?  "x D17:~ 0 0;0;0;0; 0;0; M TtjJ?"0@NNN?N j  C.&0  N TxjJ?"0@NNN?N_   C.&0  O <?"``,$D 0z ` P `,$D 0B Q ND3fo?"0@NNN?N`B R TZD3fo?"0@NNN?Nmm F  a  S B B T TD?"0@NNN?N f * B U TD?"0@NNN?N B V TD?"0@NNN?N- 3 B WB TD?"0@NNN?N7 akB XB TD?"0@NNN?NTaRB YB TD?"0@NNN?N  B ZB TD?"0@NNN?NiO' B [B TD?"0@NNN?Nv  B \B TD?"0@NNN?Ny  B ]B TD?"0@NNN?N|  B ^ TD?"0@NNN?N! 6( B _ TD?"0@NNN?N"h _ B ` ZN?D?"0@NNN?N " B a ZND?"0@NNN?NP 1[ H  0޽h ? 333WcL+᷷Ҫ ___PPT10.La+nED' = @B DC' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*O%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*3%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*J%(+8+0+ + . P\(  x  c $x@XHx     c $t0<$ 0  H  0޽h ? 333WcL+᷷Ҫ___PPT10. i+[W_DO' = @B D ' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*_x%(+8+0+ +!V / BBp;;AB(  x  c $@XHx   x  c $ԓ0  *z 8  x@,$D 0  NL o?"6@ NNN?NC8 Tand x and x in same or in adjacent domainp+6 0;  N o?"6@ NNN?Np* 0t: transition from x to x iff a solution reaches x from x~;6 0;z p>I @  >pI @,$@ 0  6"`J F I @ |maxa>0 CI    6X"`J F I @ |maxa>0 CI    6"`   tqa1>0        6"`p   _0.0       6ķ"`p>?8  ~maxb<0        6ʷ"` [  tqa2>0       6xϷ"`x r  qb>0       B o?"`>    6Է"`   tqa1>0       6ڷ"`p   _0.0      6׷"`p>?8  ~maxb<0       6H"` [  tqa2>0       6P"`x r  qb>0     =  6"`i =c  kb/gbd0   4    xB  HDo?>  xB  HDo?  xB  HDo?m n xB  HDo?> w   N o?"6@ NNN?NU  ,$ 0 x1 ! x2, 6 0;0;0;0;0;0;0;0;  N8 o?"6@ NNN?NV o ,$ 0 x1 ! x3, 6 0;0;0;0;0;0;0;0;0;  N@  o?"6@ NNN?Nm i ,$  0 x3 ! x46 0;0;0;0;0;0;0;0;  N o?"6@ NNN?Nv  ,$  0 x2 ! x3, 6 0;0;0;0;0;0;0;0;0;F z  a   N  ,$D 0B  TD?"0@NNN?N f * B   TD?"0@NNN?N B ! TD?"0@NNN?N- 3 B "B TD?"0@NNN?N7 akB #B TD?"0@NNN?NTaRB $B TD?"0@NNN?N  B %B TD?"0@NNN?NiO' B &B TD?"0@NNN?Nv  B 'B TD?"0@NNN?Ny  B (B TD?"0@NNN?N|  B ) TD?"0@NNN?N! 6( B * TD?"0@NNN?N"h _ B + ZN?D?"0@NNN?N " B , ZND?"0@NNN?NP 1[ Hz `:  - `: ,$D 0N `:  . `:  / N o?"6@ NNN?N` L2  Px126 0;  0 Nx$ o?"6@ NNN?N :  Px226 0;  1 NT# o?"6@ NNN?N @  Px326 0;  2 N4- o?"6@ NNN?N   Px426 0; 2 3 H?"6@`NNN?N( U%  4 N81 o?"6@ NNN?N$(  Px526 0; 2 N @   5 @   6  0e0e    BC DEF jJ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| `^!$@  "0e@`     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab@  2 7 H?"6@`NNN?NP } 2 8 H?"6@`NNN?NP } 2 9 H?"6@`NNN?N 0 2 : H?"6@`NNN?N   ;  0e0e    B_CDEF jJ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||_dK26_@  "0e@`     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abm  H  0޽h ? 333WcL+᷷Ҫ___PPT10. i+D' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(Df' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(++0+ ++0+ ++0+ ++0+ +H 0 ::00G:(  x  c $H@XHx     c $I0<$ 0      # 1 ,$D 0  NXO o?"6@ NNN?N ~  M 06 0;=  NS o?"6@ NNN?N   y-: describes derivative sign of solutions at x0.6 0;,  6@X"`F J I @ |maxa>0 CI   6\"`F J I @ |maxa>0 CI    64b"`   tqa1>0        6p?8  ~maxb<0        6hq"` [  tqa2>0        6v"`x r  qb>0     x  B o?"` >    6H|"`   tqa1>0       6P"` p  _0.0      6ą"`>p?8  ~maxb<0       6|"` [  tqa2>0       6ܐ"`x r  qb>0     5  6"`i =c  kb/gbd0   4       0e0e    B_CDEF jJ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||_dK26_@  "0e@`     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab m    0e0e    BC DEF jJ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| `^!$@  "0e@`     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab @ pB  HDo? >  pB  HDo?  pB  HDo? mn pB  HDo? >w 2  H?"6@`NNN?N P} 2  H?"6@`NNN?NP } 2  H?"6@`NNN?N 0 2  H?"6@`NNN?N     N o?"6@ NNN?N `L2  Px126 0;     NH o?"6@ NNN?N :  Px226 0;   ! N o?"6@ NNN?N @  Px326 0;   " N o?"6@ NNN?N   Px426 0; z     #   ,$@ 0 $ N o?"6@ NNN?N +  U. 66 0;0; % N o?"6@ NNN?N V   "x1^% xa > 0, 6 0;0;0;0;0;0;0;0;0;0;2 & H?"6@`NNN?N (U%   ' N0 o?"6@ NNN?N$(  Px526 0; z     (  X ,$@ 0 ) Nĸ o?"6@ NNN?N +  U. 66 0;0; * N0ʸ o?"6@ NNN?N V   "x1^% xb > 0, 6 0;0;0;0;0;0;0;0;0;0;z     + Q  ,$@ 0 , NlӸ o?"6@ NNN?N +  U. 66 0;0; - N0ظ o?"6@ NNN?N V   "x4^% xa < 0, 6 0;0;0;0;0;0;0;0;0;0;z     . P X ,$D 0 / N4 o?"6@ NNN?N +  U. 66 0;0; 0 N o?"6@ NNN?N V   "x4^% xb > 0, 6 0;0;0;0;0;0;0;0;0;0;H  0޽h ? 333WcL+᷷Ҫ  ___PPT10 . i+gsDk ' = @B D& ' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(+8+0+ +E 1 8877=8(  x  c $@XHx     c $0<$ 0    N o?"6@ NNN?NT~ <,$ 0 ":D = {D1, & , D27}6 0;    y  # 1 8,$D 0X  T jJ?"0@NNN?N Ny  D1 D ;j0 0; 0;0;0;0;  TjJ?"0@NNN?N "  L&0    T,jJ?"0@NNN?N #  L&0  0z }x     }x  ,$D 0   6T"`y x   |maxa>0 CI    6p"`y x   |maxa>0 CI    6 ("`1 +  tqa1>0        6-"` I  _0.0      6P*"`}nw ~maxb<0       6t6"`> 9  tqa2>0       6;"`M   qb>0       B o?"`mE S   6A"`1 +  tqa1>0       6F"` I  _0.0      6K"`}nw ~maxb<0       6P"`> 9  tqa2>0       6XU"`M   qb>0     =  6Z"`l  kb/gbd0   4    "N e $   #0   Ta?"0@NNN?NX ,  ZD12 :0 1  Tf?"0@NNN?N  mD22 L0 1  Txj?"0@NNN?NH  mD23 L0 1  To?"0@NNN?Nz mD24 L0 0  Tt?"0@NNN?N  lD17 L0 0  Ty?"0@NNN?NeF  lD18 L0 0  T~?"0@NNN?N lD21 L0 0   Tă?"0@NNN?Nm lD20 L0 2 ! TȈ?"0@NNN?N^o C  nD1 N0 0 " T̍?"0@NNN?Np D  lD3 L0 0 # TВ?"0@NNN?Nq E  lD5 L0 0 $ Tԗ?"0@NNN?Nr F  lD7 L0 0 % T؜?"0@NNN?N s P G  lD9 L0 1 & Tܡ?"0@NNN?N# R &  mD15 L0 1 ' T঺?"0@NNN?N( I   mD27 L0 2 ( T䫺?"0@NNN?NKX   nD26 L0 1 ) T谺?"0@NNN?NL+  mD25 L0  * T쵺?"0@NNN?NW +  ZD11 :0  + T\?"0@NNN?NQ %  [D13 :0  , T̾?"0@NNN?NN % "  [D14 :0 0 - T<ú?"0@NNN?NG   lD2 L0  . T@Ⱥ?"0@NNN?NN "  ZD4 :0   / T̺?"0@NNN?NO  #  \D6 :0 0 0 T Ѻ?"0@NNN?N P A $  lD8 L0 1 1 T$ֺ?"0@NNN?NjX ,  mD10 L0 0 2 T(ۺ?"0@NNN?Nt  lD16 L0 3 3 T<?"0@NNN?NH  oD19 N0 xB 4 HDo? xB 5 HDo?01 xB 6 HDo? \ xB 7 HDo?H  0޽h ? 333WcL+᷷Ҫ  ___PPT10e . i+_HD ' = @B D ' = @BA?%,( < +O%,( < +Da' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<U%(D' =%(Du' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+p+0+ ++0+ +?N 2 MDED;;C(  x  c $$@XHx   x  c $0  z A(Z  (AZ,$D 0  Nh o?"6@ NNN?N (R T: transition from D to D iff there exist X+6 0;D  Nx o?"6@ NNN?NA0 Z 6xD, x D such that x ! x 6 0; $  y  # 1 8,$@ 0^  TDjJ?"0@NNN?N y  VD1 D ; D1 !~ D1, D1 !~ D11, D11 !~ D17,(,0 0; 0;0;0;0;0; 0;0;0;0;0; 0;0;0; 0; 0; 0; 0;0; 0;0;0; 0;0;0;0;0; 0;0;0;   T`jJ?"0@NNN?N "  L&0     T4jJ?"0@NNN?N #  L&0  {.z }x     }x  ,$D 0   68"`y x   |maxa>0 CI    6 >"`y x   |maxa>0 CI   60       6H"` I  _0.0      6@M"`}nw ~maxb<0       6PR"`> 9  tqa2>0       6X"`M   qb>0       B o?"`mE S   6]"`1 +  tqa1>0       6b"` I  _0.0      6f"`}nw ~maxb<0       6k"`> 9  tqa2>0       6dp"`M   qb>0     =  6u"`l  kb/gbd0   4      N{ o?"6@ NNN?NF p  Px126 0; N `@ [   `@ [ 0  T?"0@NNN?NFd 8  lD17 L0 2  T?"0@NNN?N, .  nD1 N0   T ?"0@NNN?N`   ZD11 :0 xB  HDo?D[ xB   HDo?@S xB ! HDo?mY [ xB " HDo?mL M `N <x  # <x  $  0e0e    B_CDEF jJ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||_dK26_@  "0e@`     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abX   %  0e0e    BC DEF jJ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| `^!$@  "0e@`     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `aboM X 2 & H?"6@`NNN?N  2 ' H?"6@`NNN?N H 2 ( H?"6@`NNN?NA n 2 ) H?"6@`NNN?N   * N, o?"6@ NNN?NF p  Px126 0;  + N o?"6@ NNN?NN x  Px226 0;  , N  o?"6@ NNN?NT ~  Px326 0;  - N| o?"6@ NNN?N  Px426 0; 2 . H?"6@`NNN?N`6 c  / N좻 o?"6@ NNN?N\<Hf  Px526 0; p z  % .  0 % . ,$D 09 1 Z\?"6@`NNN?N! .  iD1 J0 0;0;0;2 2 B0;0;1?"`n  V 9 3 Z`?"6@`NNN?N!  .  iD1 J0 0;0;) 4 Zd?"6@`NNN?N l ,  YD11:0 0;0;2 5 B0;0;1?"`h  ~B 6B ND0;jJ? 6 ~B 7B ND0;jJ? ) 2 8 H<0;0;1?"`n S  H"00; 2 9 B0;0;1?"`g ) : Z?"6@`NNN?N % $  YD17:0 0;0;2 ; B0;0;1?"`C b H  0޽h ? 333WcL+᷷Ҫ  ___PPT10r . i+EDF ' = @B D ' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(+D 3 D<<<..;(  x  c $pȻ@XHx   x  c $Dɻ0    Nл o?"6@ NNN?N@@ rj,$ 0 Z: D^%~ p iff there exists xD such that x^% p.6 0; , {.z }x    Px R ,$D 0  6߻"`y x   |maxa>0 CI   6"`y x   |maxa>0 CI   6"`1 +  tqa1>0        6"` I  _0.0       6 "`}nw ~maxb<0        6T"`> 9  tqa2>0        6"`M   qb>0        B o?"`mE S   68"`1 +  tqa1>0       6 "` I  _0.0      6"`}nw ~maxb<0       68"`> 9  tqa2>0       6"`M   qb>0     =  6"`l  kb/gbd0   4      N& o?"6@ NNN?NF p  Px126 0; N `@ [   `@ [ 0  T*?"0@NNN?NFd 8  lD17 L0 2  T/?"0@NNN?N, .  nD1 N0   T3?"0@NNN?N`   ZD11 :0 xB  HDo?D[ xB  HDo?@S xB  HDo?mY [ xB  HDo?mL M `N <x   <x    0e0e    B_CDEF jJ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||_dK26_@  "0e@`     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abX     0e0e    BC DEF jJ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| `^!$@  "0e@`     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `aboM X 2   H?"6@`NNN?N  2 ! H?"6@`NNN?N H 2 " H?"6@`NNN?NA n 2 # H?"6@`NNN?N   $ Nx; o?"6@ NNN?NF p  Px126 0;  % N? o?"6@ NNN?NN x  Px226 0;  & ND o?"6@ NNN?NT ~  Px326 0;  ' NH o?"6@ NNN?N  Px426 0; 2 ( H?"6@`NNN?N`6 c  ) N,M o?"6@ NNN?N\<Hf  Px526 0; ] z k2  * 2 k,$D 0 + TTjJ?"0@NNN?Nk  D1 D ; D1 !~ D1, D1 !~ D11, D11 !~ D17, D1^%~ xa>0, D1^%~ xb>0, D4^%~ xa < 0Q0 0; 0;0;0;0;0; 0;0;0;0;0; 0;0;0; 0; 0; 0; 0;0; 0;0;0; 0;0;0;0;0; 0;0;0; 0;0;0;0;0;0;0;0; 0;0;0;0;0;0;0;0;0;0; 0;0;0;0;0;0;>4   , TQjJ?"0@NNN?N 2 7R C.&0 0; - TjJ?"0@NNN?NN5 U C.&0 0; . TXjJ?"0@NNN?N2 R C.&0 0;H  0޽h ? 333WcL+᷷ҪSK___PPT10+. i+9K D' = @B D' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<**%(+8+0+ ++ 4 {(  x  c $@XHx     c $0<$ 0  KL  Z # *G `> 9  Z?"6@`NNN?N > ? iD1 J0 9  Z?"6@`NNN?N=H hI iD3 J0 ;  Z?"6@`NNN?Nf < kD5 L0  (  Z?"6@`NNN?Nf < XD7:0 9   Z0?"6@`NNN?NA >A iD9 J0 )   Z̯?"6@`NNN?Nx M  YD15:0 )   Z<?"6@`NNN?N ?  YD27:0 )   Z?"6@`NNN?N   YD26:0 )   Z?"6@`NNN?N (  YD25:0 )  Z?"6@`NNN?N w AL  YD11:0 )  Zż?"6@`NNN?Ny N  YD12:0 ;  Zlʼ?"6@`NNN?Nfn C  kD13L0 )  Zpϼ?"6@`NNN?Nz ;O  YD14:0 2  B1?"`=  2  B1?"` o- ~B B NDjJ?EW E xB  HDjJ?L ; xB  HDjJ?K H ~B B NDjJ?A, A ~B B NLDjJ?r or ~B  NLDjJ?rr 5r 2  B1?"`7 t 2  B1?"` n 2  B1?"`D B 2  B1?"`a=  ~B  NLDjJ? Z 2  B1?"` 2 ~B   NLDjJ?  2 ! B1?"` M3 ~B " NDjJ?[  2 # B1?"` 8. 2 $ B1?"`\ 4 ~B % NDjJ?Y  ~B & NDjJ?^  2 ' B1?"`J 8 ~B ( NLDjJ?j , 2 ) B1?"`1?  ~B * NDjJ?d` d 2 + B1?"`< - ~B , NDjJ?cV c 2 - B1?"`( ) ~B . NLDjJ?  ~B /B NLDjJ?!q q ~B 0B NLDjJ?q q 2 1 B1?"` u/ 2 2 B1?"` vh 2 3 B1?"`   2 4 B1?"`  0 2 5 B1?"` \ 2 6 B1?"`|  2 7 B1?"` * 2 8 B1?"` b 2 9 B1?"`2 : B1?"`;2 ; B1?"`Z2 < B1?"`2B =B ZDjJ?"0@NNN?NEU : B > ZDjJ?"0@NNN?N  ~B ? NLDjJ?6 [6 ~B @ NLDjJ? [ ~B A NLDjJ?<   B BB ZDjJ?"0@NNN?NI % B CB ZDjJ?"0@NNN?N"  B D ZDjJ?"0@NNN?NY  B EB ZDjJ?"0@NNN?N5 L B F ZDjJ?"0@NNN?N. B B G ZDjJ?"0@NNN?N E B H ZDjJ?"0@NNN?N d ~B IB NLDjJ?"~B JB NLDjJ?~B K NLDjJ?k-~B L NDjJ? 9 M Z8?"6@`NNN?N  iD2 J0 ; N Z<?"6@`NNN?N6 T kD4 L0  ( O Z@?"6@`NNN?N  XD6:0 9 P Z?"6@`NNN?N iD8 J0 2 Q B1?"`  A ~B RB NDjJ? A ) S Z?"6@`NNN?N x M  YD10:0 ) T Z$?"6@`NNN?N c 8  YD16:0 ) U Z?"6@`NNN?N   YD17:0 ) V Z$?"6@`NNN?N t J  YD18:0 ) W Z?"6@`NNN?N  b  YD20:0 ) X Z?"6@`NNN?N 6   YD19:0 ) Y Z?"6@`NNN?N Z/  YD21:0 ) Z Z$ ?"6@`NNN?N :  YD22:0 ) [ Z?"6@`NNN?N> =  YD23:0 ) \ Zd?"6@`NNN?NaJ7  YD24:0 2 ] B1?"` . 2 ^ B1?"`  ( 2 _ B1?"` j.2 ` B1?"`j2 a B1?"`;k^2 b B1?"`< _? 2 c B1?"`  2 d B1?"`  2 e B1?"` 5 2 f B1?"`  2 g B1?"` ; 2 h B1?"`p 2 i B1?"`! D 2 j B1?"` > 2 k B1?"`r  2 l B1?"`l   m 6%"`X y x R  |maxa>0 CI  n 6l"`X y x R  |maxa>0 CI  o 6/"`   tqa1>0      p 6-"` I  _0.0     q 68"`PnJ ~maxb<0      r 6L>"`   tqa2>0      s 6C"` M  qb>0     x t B o?"`m &  u 6I"`   tqa1>0      v 6 N"` I  _0.0     w 6R"`PnJ ~maxb<0      x 68X"`   tqa2>0      y 6]"` M  qb>0     5 z 6b"`{lu  kb/gbd0   4    "F e $  { 0  | Tdj?"0@NNN?NX ,  ZD12 :0 1 } Tn?"0@NNN?N  mD22 L0 1 ~ T$t?"0@NNN?NH  mD23 L0 1  Ty?"0@NNN?Nz mD24 L0 0  T@~?"0@NNN?N  lD17 L0 0  Tz?"0@NNN?NeF  lD18 L0 0  T̀?"0@NNN?N lD21 L0 0  T؄?"0@NNN?Nm lD20 L0 2  T쉾?"0@NNN?N^o C  nD1 N0 0  T?"0@NNN?Np D  lD3 L0 0  T?"0@NNN?Nq E  lD5 L0 0  T(?"0@NNN?Nr F  lD7 L0 0  T<?"0@NNN?N s P G  lD9 L0 1  TP?"0@NNN?N# R &  mD15 L0 1  Td?"0@NNN?N( I   mD27 L0 2  Tx?"0@NNN?NKX   nD26 L0 1  T?"0@NNN?NL+  mD25 L0   T?"0@NNN?NW +  ZD11 :0   Tľ?"0@NNN?NQ %  [D13 :0   T?"0@NNN?NN % "  [D14 :0 0  T¾?"0@NNN?NG   lD2 L0   Tƾ?"0@NNN?NN "  ZD4 :0    Tվ?"0@NNN?NO  #  \D6 :0 0  Ttپ?"0@NNN?N P A $  lD8 L0 1  T޾?"0@NNN?NjX ,  mD10 L0 0  T?"0@NNN?Nt  lD16 L0 3  T?"0@NNN?NH  oD19 N0 xB  HDo? xB  HDo?01 xB  HDo? \ xB  HDo?z o(.x   (o.x ,$D 0   0e0e    B_CDEF jJ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||_dK26_@  "0e@`     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab+     0e0e    BC DEF jJ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| `^!$@  "0e@`     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abo + 9  ZX?"6@`NNN?N! .x  iD1 J0 )  Z\?"6@`NNN?N  YD11:0 2  B1?"`h H ~B B NDjJ? o ~B B NDjJ? " 2  B1?"`n 2  B1?"`g B 2  B1?"`; ~ B  ZDjJ?"0@NNN?N J a )  Z ?"6@`NNN?N $p  YD17:0 )  Z?"6@`NNN?NI (V  YD18:0 2  B1?"`} L jy  ThjJ?"0@NNN?NP`,$ 0 Alur et al., Proc. IEEE, 0080 H  0޽h ? 333WcL+᷷Ҫ___PPT10. i+9.?D$' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(+p+0+ ++0+ + 5 G?0(  x  c $,@XHx     c $\<$ 0  s  TjJ?"0@NNN?N`   ,$ 0 Batt et al., HSCC, 0580 H  0޽h ? 333WcL+᷷Ҫ___PPT10.@n+F½DU' = @B D' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* ~%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*~%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(+p+0+ ++0+ +E  P\(  x  c $+@XHx     c $2.<$ 0  H  0޽h ? 333WcL+᷷Ҫ!___PPT10 .t'+FD ' = @B DP ' = @BA?%,( < +O%,( < +D> ' =%(D ' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*z%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* W%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*W%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* t%(+8+0+ +%T 6 iEaE`11D(  x  c $`^@XHx     c $H_0<$ 0   z >    )>  ,$D 06  <U ?"6@ NNN?N>   ~There Exists a Future state where xa > 0 and xb > 0 and from that state, there Exists a Future state where xa < 0 and xb > 0\0 bgb g bcgogcbgogc b $c$$g$(b(,g,0b04c48888888888888888 0 xb > 0 EF(xa < 0 xb > 0) ),0 gggogGgogg ggggogggog               T8jJ?"0@NNN?Nb   C.&0    TXjJ?"0@NNN?N{   C.&0   TjJ?"0@NNN?N @  C.&0   TjJ?"0@NNN?N o  C.&0 z     ,$D 0  TܶjJ?"0@NNN?N   ?."0   TPjJ?"0@NNN?N |  ?."0   TÿjJ?"0@NNN?Nw ?."0 -z     ,$D 0B  TD1?__ B  TD1?^    B0ǿ ?"`0 T  _0.0      6̿"` xbB0       6Tѿ"` S  `time,0        6ȿ"`1 `time,0      B  TD1?^z^3B  TD1?]36  Bڿ ?"`S} _0.0      6߿"`7  xaB0       s 0e0e    B@CDEF > 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||h 0@@  "0e@`     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab.n   N?"6@`NNN?N ! s 0e0e    BCDEF > 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||`u8@  "0e@`     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab " N?"6@`NNN?N|rT  6 ## 1A G  $ 6"` b6 xa > 0>0     % TX?"0@NNN?N  C.&0 B &B NDjJ?"0@NNN?N` DT ( g  '# ( g lN  6 ( ( >~  ) 6x"` b6 xb > 0>0     * T?"0@NNN?N  C.&0 lN  6 + Q g  , 6"` b6 xb > 0>0     - T?"0@NNN?N  C.&0 rT  6 .# ZB p  / 6`"` b6 xa < 0>0     0 T ?"0@NNN?N  C.&0  1 s 0e0e    BCeDEF > 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||e68FTQ@  "0e@`     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab] H  0޽h ? 333WcL+᷷Ҫd\___PPT10<. i+D ' = @B D ' = @BA?%,( < +O%,( < +D' =%(Du' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*[%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*[~%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(+8+0+ +X 7 (  x  c $@XHx     c $0<$ 0  /  TjJ?"0@NNN?N!  A ,$ 0 ?Yes 0 8  TjJ?"0@NNN?N  ,$D 0 H Consistency? 0  ,F    m B  TD1?__ B  TD1?^     B% ?"`0 T  _0.0       6*"` xbB0        6@/"` S  `time,0         63"`1 `time,0      B   TD1?^z^3B  TD1?]36  B8 ?"`S} _0.0      6="`7  xaB0       s 0e0e    B@CDEF > 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||h 0@@  "0e@`     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab.n  N?"6@`NNN?N  s 0e0e    BCDEF > 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||`u8@  "0e@`     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab  N?"6@`NNN?N|rT  6 # 1A G   6XC"` b6 xa > 0>0      T(I?"0@NNN?N  C.&0 B B NDjJ?"0@NNN?N` DT ( g  # ( g lN  6  ( >~   6M"` b6 xb > 0>0      T0S?"0@NNN?N  C.&0 lN  6  Q g   6PW"` b6 xb > 0>0      TT\?"0@NNN?N  C.&0 rT  6  # ZB p  ! 6`"` b6 xa < 0>0     " Te?"0@NNN?N  C.&0  # s 0e0e    BCeDEF > 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||e68FTQ@  "0e@`     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab] z Q O\  $ 86 ,$D 0 % <@m ?"6@ NNN?NQ b O\  RVEF(xa > 0 xb > 0 EF(xa < 0 xb > 0) ),0 gggogGgogg ggggogggog             & TjJ?"0@NNN?Nb   C.&0  ' TjJ?"0@NNN?N{   C.&0  ( TTjJ?"0@NNN?N @  C.&0  ) T$jJ?"0@NNN?N o  C.&0 mKL  (%  *#  EO 5 + Z@?"6@`NNN?NZ    eD1 F0  5 , Z$?"6@`NNN?N" f"  eD3 F0  % - ZH?"6@`NNN?N ?  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U M   (   ~   s *3@XHx      c $4@<$ 0  ,z ]73    ] ,$D 0   N6jJ?"0@NNN?N d;   Fis concentration decreases and becomes steady in stationary phase E0 EAr   c 4AAliAzam"`]7h 3 ~   N;jJ?"0@NNN?N   "Ali Azam et al., J. Bacteriol., 99N#0   , Gz ? Y     @,$D 0   <E ?"6@ NNN?N?  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.^Oh+'0 `h  4 @ LX`u Validation of Qualitative Models of Genetic Regulatory Networks A Method Based on Formal Verification TechniqueskGregoryGregory17gMicrosoft PowerPointati@q@p8@S~k G:g   ]  -- @ !--'@Arial-. (292 t!Validation of Qualitative Models     !   #  ."Systemdi-@Arial-. (242 of Genetic Regulatory Networks !     !.-@Arial-. (2O2 ]0A Method Based on Formal Verification Techniques           .-@Arial-. 3332 FGrgory    .-@Arial-. 333 2 FBatt .-@Arial-. 3332 rPh.D.r.-@Arial-. 3332 rdefense .-@Arial-. 333 2 --.-@Arial-. 333'2 +under supervision of    .-@Arial-. 3332 Hiddes.-@Arial-. 333 2 \de .-@Arial-. 333 2 ~Jong .-@Arial-. 333 2 ,.-@Arial-. 3332 Helixs .-@Arial-. 3332 research  .-@Arial-. 3332 'groupr.-@Arial-. 3332  INRIA Rhne.-@Arial-. 333 2 -.-@Arial-. 3332 !Alpesr .-@Arial-. 333 2 F--.-@Arial-. 3332 pEcoler  .-@Arial-. 3332 p doctorale   .-@Arial-. 333 2 hMath.-@Arial-. 333 2  .-@Arial-. 3332 matiques  .-@Arial-. 333]2 9, Sciences et technologies de linformation, Informatique       .-@Arial-. 333-2 eUniversit Joseph Fourier    .-՜.+,0    uOn-screen Show  -s!=-' 7ArialTimes New RomanMonotype SortsImpactSymbolArial Unicode MSKunstler Script Wingdingsinria Image Bitmapu Validation of Qualitative Models of Genetic Regulatory Networks A Method Based on Formal Verification Techniques$Stress response in Escherichia coli'Nutritional stress response in E. coli$Network controlling stress responseModeling and simulationModel validationModel validation"Objectives and approach of thesis Overview Overview PL differential equation models)Qualitative analysis of network dynamics)Qualitative analysis of network dynamics)Qualitative analysis of network dynamics)Qualitative analysis of network dynamics)Qualitative analysis of network dynamics)Qualitative analysis of network dynamicsProblem for model validation)Qualitative analysis of network dynamicsContinuous transition systemContinuous transition systemContinuous transition systemDiscrete abstractionDiscrete abstractionDiscrete abstractionDiscrete abstractionDiscrete abstractionModel-checking approach Validation using model checking Validation using model checking OverviewGenetic Network AnalyzerGenetic Network AnalyzerGenetic Network Analyzer Overview'Nutritional stress response in E. coli&Model of nutritional stress response $Validation of stress response model$Validation of stress response model*Novel prediction of stress response model OverviewSummary Discussion Perspectives Slide 45  Fonts UsedDesign TemplateEmbedded OLE Servers Slide Titles-_u/GregoryGregory  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~                           ! 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