NETWORK IDENTIFICATION AND TIME SERIES ANALYIS USING S-SYSTEMS

Transcription

1 NETWORK IDENTIFICATION AND TIME SERIES ANALYIS USING S-SYSTEMS SIREN RØST VEFLINGSTAD Norwean Unversty of Lfe Scences Research done n collaboraton wth JONAS ALMEIDA and EBERHARD VOIT durn a vst at the Medcal Unversty of South-Carolna.

2 Concentraton (normalsed) MOTIVATION Tme seres of bochemcal systems NMR mass spectrometry mcroarrays. Queston: can we dentfy the underlyn network? Tme Y Z Z Y Z Z? Y Y??

3 MOTIVATION Tme seres of bochemcal systems NMR mass spectrometry mcroarrays. Queston: can we dentfy the underlyn network? Inverse problem. Often two components needed: Model structure. Method for fttn data to model. 3

4 NETWORK IDENTIFICATION BY DIRECT OBSERVATION Assume a pulse perturbaton from steady state. Some oberservatons (Vance et al (00) PNAS): Tme of an extremum ncreases and ampltude decreases wth dstance from perturbaton. Rate of ntal slope s non-zero f drectly coupled to perturbed speces; zero otherwse. Idented the frst steps of an n vtro lycolytc system (Torralba et al. (003) PNAS). 4

5 CHALLENGES Choce of model structure: General vs specfc. Lnear vs non-lnear relatons. Need for a pror nformaton. Numercal ssues: Multple local mnma falure to convere. Interaton of dfferental equatons. 5 Data: Nose. Number of samples (densty of tme seres). Mssn varables.

6 6 VARIOUS APPROACHES TO NETWORK IDENTIFICATION Models Boolean (Lan Akutsu Ideker Mak ) Dfferental equatons Lnear (D haeseleer Chen ) S-systems Methods of nference Dstance measures (tme-laed) Correlaton metrc (Arkn Samolov ) Mutual nformaton (Butte Lan ) Genetc alorthms (Mak Kkuch ) Smulated annealn (Mjolsness ) Neural networks nterval methods lnear prorammn Bayesan networks (Fredman Hartemnk Pe er )

7 SUGGESTIONS HERE Model structure: Use canoncal models. Numercal ssues: Most lkely soluton: ntensve and parallelzed computatonal effort. Alorthms perform better when close to true soluton. Preprocess data to obtan a prelmnary network structure (ood ntal uesses) possble constrants on the parameter values. 7

8 8 CANONICAL MODEL: S-SYSTEMS. 3 3 f e d c b a Z Y dt dz Y Z dt dy Z dt d β α β α β α Y Z x 0 d b Z Z d b Y Z Parameter estmaton network dentfcaton:. 3 3 f e c a Z Y dt dz Y dt dy dt d β α β α β α

9 GENERAL APPROACH. Assume a non-lnear model:. Lnearse the model. d f dt ( µ ) 3. Transform the tme seres data accordn to the chosen lnearsaton. 4. Estmate the slopes from the tme seres data. 5. Ft the tme seres data to the lnearsed model by lnear reresson Assumpton: The matrx of reresson coeffcents reflects the connectvty of the network.

10 LINEARISATION We consder 4 ways of lnearsn:. About an absolute devaton from steady state.. About a relatve devaton from steady state.. Pecewse lnearsaton. v. Lotka-Volterra lnearsaton. 0

11 LINEARISATION ABOUT STEADY STATE Absolute devaton z s dz dt Az Relatve devaton u z / s du dt A' u f A j s A ' js s A Lmtatons Requres small perturbatons n order for the lnear models to be a vald approxmaton.

12 THE CORRESPONDING MODELS FOR S-SYSTEMS ARE Absolute devaton Relatve devaton when remembern that S-systems are on the form n j j z c F z&. n ns js s s F h c L L α. n ns js s s F h c L L α ) ( n j j u c F u& m n m n h m n h h m n β α &

13 PIECEWISE LINEARISATION Dvde dataset nto approprate subsets and lnearse about a reference state wthn each subset Choce of subsets s mportant for ood results. Here: use the extreme values as lmt ponts. Advantae: The sze of perturbatons are no loner as mportant. Challene: How to choose the subsets. 3

14 LOTKA-VOLTERRA LINEARISATION Lotka-Volterra model: d dt ( β + α n j K n. Advantae: Independent of reference state and sze of perturbatons. j ) 4

15 REGRESSION MODEL Multple lnear reresson: + y a0 a x j Response varable y : rate of chane for each varable. Predctor varable x : concentraton of each varable. Response/predctor must be transformed accordn to lnearzaton opton. Example: y & r x r r 5 Tmes seres data: only the concentratons. Need to estmate the rate of chane.

16 ESTIMATION OF SLOPES Varous approaches possble. More or less nose-free data: Lnear nterpolaton splnes... Nosy data useful to smooth before estmatn slopes: Splnes varous flters artfcal neural networks... Advantaes nclude unlmted number of samples some resstance to nose. Dsadvantaes nclude a second level of error Vot and Almeda (004): artfcal neural networks. Obtaned smooth traces close to real traces even n the presence of nose. 6

17 EAMPLE (artfcal) Gene 4 mrna Gene mrna + + AA 5 reulator AA enzyme - - substrate 3 nducer 7 Also used n: Hlavacek and Savaeau (996) J. Mol. Bol. 55: -39 Kkuch et al. (003) Bonformatcs 9:

18 8 S-SYSTEM MODEL & & & & & Intaton: 0 % perturbaton n 3

19 NETWORK TOPOLOGY REFLECTED IN REGRESSION COEFFICIENTS: RELATIVE DEVIATION FROM STEADY STATE Value of reresson coeffcent Computed Estmated Reresson coeffcent #

20 EFFECT OF PERTURBATION SIZE I: RELATIVE DEVIATION % devaton from real value % 5 5% 0 0% 50% 5 00% 0 00% Reresson coeffcent # % devaton from real value % 5% 0% 50% 00% 00% Reresson coeffcent #

21 WHY IS 3 SO DIFFICULT? We lnearse around a steady state: u& F ( n j c u j ) c F α h s s L js L ns n. However the contrbuton from s the same on both producton and deradaton n other words c3 0. Ths ves u& 3 F3c 33u3 c33 < 0. The behavour seen must be explaned by other varables?

22 NETWORK TOPOLOGY REFLECTED IN REGRESSION COEFFICIENTS: PIECEWISE LINEAR REGRESSION Value of reresson coeffcent Computed Estmated -30 Reresson coeffcent #

23 DIFFERENT MODELS SIMILAR RESULTS? COMPARISON OF THE LINEARISATION PROCEDURES (4) % dentfed Pecewse lnear Reresson coeffcent # 3 Identfed wthn 5 % devaton from real value.

24 4 % devaton from real value EFFECT OF PERTURBATION SIZE II: PIECEWISE LINEAR Reresson coeffcent # 5% 00%

25 EFFECT OF NOISE % devaton from real value No nose 4% 0% 30% Reresson coeffcent # Lnearsaton procedure: relatve devaton. Intaton: 0 % perturbaton

26 CONSTRAINTS ON PARAMETER VALUES Consder n the example Assume snfcant effect from 3 and 5 Constrants from the nteracton coeffcents for the S-system model F c F c F c 3 5 b b b 3 5 c c c h h h 3 5 < > <

27 SUMMARY: MAIN RESULTS The network connectvty s reflected n the values of the reresson coeffcents. The dfferent lnear models ve qualtatvely the same results. Absolute/relatve best for small perturbatons. Pecewse may detect other relatons because ndependent of steady state. Deree of smlarty: a measure of how relable the estmates are? The reater the perturbaton the less accurate s the estmaton of the reresson coeffcents. Pecewse lnear lnearsaton performs at reater perturbatons. Lotka-Volterra dd not perform as well as expected wth hher perturbaton levels. 7

28 DISCUSSION Advantaes Fast and easy. Provdes an ntal network structure. Lmtatons May loose nformaton n the lnearsaton around steady state. Sze of perturbaton. Relablty crtera of estmates. 8

29 x y y & x & s ( s s ) LV Rel Abs y & x s 9