Inducing sustained oscillations in mass action kinetic networks of a certain class

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1 The Internatonal Federaton o Automatc Control Inducng sustaned oscllatons n mass acton knetc networks o a certan class Irene Otero-Muras Gábor Szederkény Antono A Alonso Kataln M Hangos ETH Zürch, Department o Bosystems Scence and Engneerng Unverstätstrasse, 6, Zürch, Swtzerland (e-mal: reneotero@bsseethzch) MTA-SZTAKI Process Control Research Group, Hungaran Academy o Scences, Budapest, Hungary (e-mal: szeder@sztakhu) IIM-CSIC BoSystems Engneerng Group, Spansh Councl or Scentc Research, Vgo, Span (e-mal: antono@mcsces) Abstract: Engneerng a synthetc oscllator requres an oscllatory model whch can be mplemented nto practce Some requred condtons or a successul practcal mplementaton nclude the robustness o the oscllaton under realstc parameter values and the accessblty to the varables that need to be manpulated For a partcular class o theoretcal oscllators derved rom mass acton knetc models, oscllatons appear provded that some o the concentratons are kept constant at gven values Ths condton s not trvally accomplshed n practce In ths work we provde two derent realzatons o knetc networks leadng to the desred lmt cycle oscllaton: a mass acton knetc system wth constant nlow and outlow or some o the speces, and a strred tank reactor conguraton wth some entrapped speces where the nonchemcal eect o keepng some concentratons constant s acheved by manpulatng some accessble varables The results, together wth the robustness o the approach and the condtons or urther practcal mplementaton are dscussed and llustrated through the well known Brusselator example Keywords: bochemcal reacton, eedback lnearzaton, nonlnear control systems, oscllators 1 INTRODUCTION One recurrent topc n synthetc bology s the desgn and mplementaton o bologcal networks that perorm desred oscllatons n a predctable manner We ocus on a partcular class o theoretcal oscllators derved rom mass acton knetc models, where sustaned oscllatons appear provded that the concentraton o some o the speces nvolved are kept constant at partcular values A number o reacton networks showng sustaned oscllatory behavour or some ranges o the knetc parameters have been reported n the lterature (Érd and Tóth, 1989; Ncols and Ncols, 1999) where the condton o keepng some concentraton constant s consdered as a merely theoretcal assumpton However, the appearance o sustaned oscllatons n a real system drven by dynamcs o ths knd requres ths condton to be accomplshed expermentally In ths work, we provde two derent realzatons leadng to the desred oscllatory behavour In Secton 3 we descrbe how to obtan a mass acton knetc system (Horn and Jackson, 1972) that n the presence o constant nlow and outlow (or degradaton) o some speces, wll produce the a pror dened oscllaton In Secton 4, the nonchemcal eect o mantanng the concentraton o some speces constant durng a bochemcal reacton process s acheved by nonlnear control We desgn a lnearzatonbased control law able to keep the key concentratons constant by manpulatng some accessble varables The results are llustrated through the well known Brusselator network 2 SYSTEMS UNDER STUDY Let us consder a mass acton knetc network wth m speces and r reactons The evoluton o the speces concentraton vector c n tme s gven by: ċ = N v (1) where N s the m r stochometrc matrx and v s the vector o reacton rates In the systems under study, oscllatons appear when the concentraton o some speces s kept constant In what ollows, we wll denote as key speces the speces to be mantaned constant to make the system to oscllate Let p be the number o key speces, the dynamcs o a system o ths knd s descrbed through the lterature by a reduced order model o m p ordnary derental equatons: ċ = ˆN v (2) where ˆN s the (m p) r matrx contanng the rows o the orgnal stochometrc matrx correspondng to non key speces, and v s the vector contanng the reacton rates or the r reactons The concentratons o the key speces n the expresson or the reacton rates take the values o gven constants /12/$ IFAC / SG

2 Example The well known Brusselator network (Ncols and Prgogne, 1977) comprses our reversble steps: FGGGGGG GGGGGGB X v 1 = k 1 + c 1 k1 c 2 (3) A k+ 1 k 1 FGGGGGG GGGGGGB C v 2 = k 2 + c 2 k2 c 3 (4) X k+ 2 k 2 B + X k+ 3 FGGGGGG GGGGGGB F + Y v 3 = k 3 + c 2c 4 k3 c 5c 6 (5) k 3 k + 4 2X + Y FGGGGGG GGGGGGB 3X v 4 = k 4 + c2 2c 6 k4 c3 2 (6) k 4 Accordng to Ncols and Ncols (1999) oscllatons appear or some knetc parameters when the concentratons o the key speces A,B,C and F are held constant at partcular values denoted by a, b, c and For the values n Table 1 the system shows the lmt cycle oscllaton depcted n Fg 1 To avod conuson the concentratons o X and Y are denoted n what ollows by x and y The correspondng reduced order model reads: ẋ = k + 1 a + k 2 c (k+ 3 b + k 1 + k+ 2 )x + k 3 y + k + 4 x2 y k 4 x3 ẏ = k + 3 bx k 3 y k+ 4 x2 y + k 4 x3 (7) n the long term Thereore, the reduced order model (2) showng oscllatory behavour wll be only vald or a tme perod when the excess o the key speces persst It s mportant to remark here that closed mass acton knetc networks, ulllng the detaled balance condton, evolve towards a unque and stable equlbrum pont (Otero- Muras et al, 2008) Thereore, any mass acton knetc network exhbtng sustaned oscllatons s necessarly open, where some external orces are mantanng the system ar rom the thermodynamc equlbrum by means o matter and/or energy exchange through the boundary Ths exchange may lead to the unulllment o the detaled balance condton and/or o the mass conservaton (wthn the control volume) or any o the atomc speces n the network Startng rom a non mass acton oscllatory network o the class ntroduced n the prevous Secton, we descrbe next how to desgn a mass acton knetc system producng the same sustaned oscllaton To ths purpose, we construct a network dagram by strppng away the key speces rom the orgnal network such that: ) the key speces appearng n a complex 1 together wth other non key speces are strpped away and ther concentraton s embedded n the correspondng knetc constant, ) all speces n a partcular complex have tme nvarant concentratons, the complex s substtuted by the zero complex (where all the speces coecents are zero) representng the envronment In ths way, the oscllatory behavour can be reproduced by a mass acton knetc system by modyng some knetc constants (case R1) and/or addng a constant nput/output low o some non key speces (case R2) Example Next, we construct a mass acton knetc network leadng to the same dynamcs as (7) In rst nstance, by keepng B and F constant at the values b and, respectvely, the reacton step (5) s transormed nto: k + 2 X FGGGGGG GGGGGGB Y ṽ 2 = k 2 + x k 2 y (8) k 2 Fg 1 Brusselator lmt cycle or a, b, c, n Table 1 Table 1 Key speces values (Brusselator) k + 1 k 1 k + 2 k 2 k + 3 k 3 k + 4 k 4 a c b MASS ACTION EQUIVALENT REALIZATION Usually, n order to mantan the concentraton o some speces constant durng a reacton process, one can make them to be n a large excess over the rest The assumpton o constant concentraton or speces n large excess wthn the control volume s largely wdespread n Chemstry and Chemcal Engneerng (Mssen et al, 1998; Othmer, 2003) Oscllatons appearng at constant concentratons o the key speces wll persst n the real system untl the excess o these speces vanshes, as t corresponds to a closed system, where the oscllatons cannot be sustaned wth k 2 + = b k+ 3, k 2 = k3 Keepng A and C constant, the reactons (3) and (4) lead to: k n FGGGGGG GGGGGGB X ṽ 1 = k n k out x (9) k out where represents the envronment and k n, k out are the nlow/producton rate and the outlow/degradaton rate constant o the speces X An equvalent mass acton knetc network or the Brusselator assumng constant concentratons o the speces A, C, B and F reads: k n FGGGGGG GGGGGGB X ṽ 1 = k n k out x k out k + 2 X FGGGGGG GGGGGGB Y k 2 k + 4 2X + Y FGGGGGG GGGGGGB 3X k 4 ṽ 2 = k + 2 x k 2 y ṽ 3 = k + 4 x2 y k 4 x3 1 Complexes are the sets o speces n both sdes o a reacton arrow 476

3 and the dynamcs, descrbed by: ẋ = ṽ 2 + ṽ 3 + k n k out x, ẏ = ṽ 2 ṽ 3, wth k n = k + 1 a + k 2 c, k out = k 2 + k 1 s equvalent to (7) 4 FEEDBACK LINEARIZING CONTROL In ths Secton we desgn a controller that, by keepng the key speces concentratons constant at some specc crtcal values, drves the system to the manold n the state-space where the desred nonlnear dynamcs occurs, n ths case, a sustaned oscllaton The control approach, nspred by Grognard and Canudas de Wt (2004), ncludes an nput-output lnearzaton by eedback Beore statng the man result, let us ntroduce the ollowng Proposton The complete statement and proo can be ound n the Appendx, together wth a descrpton o the needed notaton and geometrc propertes (nternal dynamcs, normal orm and relatve degree) Proposton 1 Consder the Mult Input Mult Output (MIMO) system ẋ = (x) + g(x)u u U (10) y = h(x) y Y (11) wth x R m, U = Y = R p, and vector relatve degree {r 1,, r p }, globally dened and constant The eedback: u = R 1 (x)(α(x) + w) (12) wth R beng the m m matrx: L g1 L r1 1 L gp L r1 1 L g1 L r2 1 h 2 (x) L gp L r2 1 h 2 (x) R(x) = (13) L g1 L rp 1 h p (x) L gp L rp 1 h p (x) and α(x) beng: L (r1) α = L (rp 1) (14) h p 1 (x) L rp h p(x) lnearzes the MIMO system rom the new nput w to the output y Remark 1 The closed loop dynamc system (11) wth eedback (12) can be expressed n (z, y) coordnates such that 2 : ż = (z, y 1,, y (r1 1) 1,, y p,, y (rp 1) p ) 1 = L y (r1) 1 = w 1 p = L h p (x) y p (rp) = w p (15) where ż = (z, y 1,, y (r1 1) 1,, y p,, y p (rp 1) ) represent the nternal dynamcs 2 Let us remnd here the notaton y (k) dy k /d k t Now, we are n the poston to ntroduce the control law drvng the dynamcs to a lmt cycle Proposton 2 Consder a reacton network wth p key speces s S key, such that the speces ormaton uncton : R m R m correspondng to the nner knetcs (c) = N v, (16) exhbts sustaned oscllatons when the concentraton o p key speces s kept constant at some specc crtcal values c R p 0 Then, the system ċ = N v + φ(c n (t) c) c R m 0 y = h(c) (17) where h : R m R p, wth a control law o the orm: c n (t) = c + 1 G(c)u (18) φ and u gven by: [ ] 1 [ ] 1 h u = c G(c) h h c N v + c G(c) w, (19) exhbts a lmt cycle n the nternal dynamcs, provded that: () w R p s a external reerence nput o the orm: w = K(y ỹ), (20) such that (dm y = dm u = p), K R p p s an arbtrary postve dente dagonal matrx and ỹ = c, wth c contanng the crtcal concentraton values () The matrx G(c) R m p s constructed as ollows Let G R m m dened as: { c + 1 or = j, s, s S key G j(c) = c or j, s, s j S key 0 or j = 1,, p s, s j / S key (21) G(c) = G (c)i k, (22) wth I k = {1, 0} m p, where the columns 1,, p o I k are the vectors o the eucldean bass correspondng to key speces 1,, p Proo Substtutng the expresson (18) nto (17), the system reads: ċ = N v + G(c)u c R m 0 y = h(c) (23) where h : R m R p and G(c) s dened by (21) Let us compute now the MIMO lnearzng law ntroduced n Proposton (1) The matrx R(c) n (13) reads: L G1 h 1 (c) L Gp h 1 (c) L G1 h 2 (c) L Gp h 2 (c) R(c) = (24) L G1 h p (c) L Gp h p (c) where by constructon, the relatve degree o the system (23) ullls: r 1 = r 2 = = r p = 1 The expresson or α n (14) becomes: L h 1 (c) α = L h p 1 (c) (25) L h p (c) 477

4 Ater substtutng (24) and (25) n (12), the MIMO lnearzng eedback s transormed nto (19) Accordng to Proposton 1, ths eedback lnearzes the MIMO system (23) rom the nput w to the output y The closed loop system n the (z, y) coordnates can be expressed, accordng to Eq (2), as: y 1 = w 1 y 2 = w 2 y p = w p ż = (z, y 1, y 2,, y p ) Settng the external reerence gven by (20) n the system above, we arrve to the ollowng expresson or the closed loop dynamcs, n compact orm: ż = ζ(z, y) (26) ẏ = K(y ỹ) (27) where (26) s the nternal dynamcs, and the lnear stablzng eedback (20) drves the system exponentally to the manold ỹ The nternal dynamcs (26), as t was assumed n the man statement, shows a lmt cycle oscllaton when the concentratons o the key speces s S key are constant and equal to the crtcal values c Provded that ỹ = c n (20), the controller drves the concentratons o the key speces to the crtcal values, and the oscllaton wll appear or the controlled system n the reduced manold determned by c = c or s S key The system can be mplemented nto practce provded that: ) all concentratons can be measured, ) nlow concentratons o key speces can be manpulated as requred, ) the control volume s constant durng the process, v) the non key speces are entrapped wthn the control volume The eedback lnearzaton technque s known to be senstve to model error However, n the systems under study, oscllatory behavour preval or ranges o knetc parameters and key speces reerence concentratons, hence error wthn certan margns wll be tolerated wthout loosng the desred oscllaton Example The Brusselator network s agan selected as a proo o concept or the proposed control law As reported prevously, a lmt cycle oscllaton appears n the Brusselator when A, B, C and F are held constant at the values shown n Table 1 Usng the results presented a controller s desgned next to drve the dynamcs (3-6) to the sustaned oscllaton shown n Fg 1 The key speces A, B, C and F are chosen to be controlled by ther nlet low rate The selected output vector s: y = (c 1 c 3 c 4 c 5 ) T and the closed loop expresson s o the orm (23) wth: v v N v = v v The expressons or the reacton rates v 1, v 2, v 3 and v 4 are gven by Eqs (3-6), and G(c) s constructed ollowng (21): G(c) = The matrx (13) reads: R(c) = c c 1 c 1 c c 3 c c 3 c 3 c 4 c 4 c c 4 c 5 c 5 c 5 c c c 1 c 1 c 1 c 3 c c 3 c 3 c 4 c 4 c c 4 c 5 c 5 c 5 c and the vector (14) s o the orm: k 1 + c 1 + k1 c 2 k 2 + c 2 k2 c 3 α(c) = k + 3 c 2c 4 + k 3 c 5c 6 k + 3 c 2c 4 k 3 c 5c 6, Accordng to Proposton 1, the eedback law (19) lnearzes the system rom the nput w to the output y The nput w s dened by (20), where the constant reerence or y to be tracked by the controller s ỹ = ( ) T and the lnear eedback gan K selected to stablze the lnear part o the nput-output lnearzed system s the (4 4) dentty matrx The trajectores o the closed loop system evolve to the manold where the lmt cycle appears, as t s shown n Fg 2 (a, b) We choose a Fg 2 Closed loop response (a) n the tme doman (b) n the c 2 c 6 phase space (c),(d),(e),() key speces nlet concentratons resdence tme such that (18) s satsed or postve nlet concentratons In Fg 2 (c-) the nlet concentratons o the key speces drvng the system to the desred oscllaton computed usng (18) are depcted Here t s mportant to 478

5 Fg 3 Vable reerence values or a sustaned oscllaton (b-c), and correspondng locatons o the unstable ocus (a) In lght grey, vable reerence values or a sustaned oscllaton wth ocus wthn a crcle o radus 05 around the nomnal value note that both the requred nlet and outlet concentratons or speces X and Y are zero, thus X and Y must be entrapped n the control volume In order to analyze the capacty o the controller to mantan a gven desred behavour n presence o uncertanty, we explore the space o the reerence values, usng the algorthm by Zamora- Sllero et al (2011) In Fg 3 we depct the values o the reerence ỹ ound to preserve a sustaned oscllaton (blue regons) In lght grey, we depct the values o the reerence ỹ ound to preserve the oscllaton where dstance o the unstable ocus to the reerence value (1, 11) s less or equal than 05 APPENDIX Concepts and results summarzed n ths Appendx are closer dscussed n (Isdor, 1989; Kravars, 1990; Grognard and Canudas de Wt, 2004) Let us dene a nonlnear system ane n the nput and wthout eedthrough term o the orm: ẋ = (x) + g(x)u u U y = h(x) y Y (28) where x R n For a MIMO (Mult Input Mult Output) system wth p outputs and m nputs, U = R m, Y = R p, g(x) s an n m matrx: g(x) = [g 1 (x),, g m (x)] and h(x) a p-vector Basc geometrc propertes Let λ(x) : R n R The dervatve o λ along s denoted: ( ) T λ n λ L λ(x) = (x) = (x) (29) We also dene: =1 L g L h(x) = (L h) g(x) (30) and L k h(x) = (Lk 1 h) (x) (31) Denton 1 The MIMO system (28) wth U = Y = R m s sad to have a (vector) relatve degree {r 1,, r m } at a pont x 0, or all x n a neghborhood o x 0 : () L gj L k h (x) = 0 or all 1, j m and k < (r 1); () the m m matrx: L g1 L r1 1 L gm L r1 1 L g1 L r2 1 h 2 (x) L gm L r2 1 h 2 (x) R(x) = L g1 L rm 1 h m (x) L gm L rm 1 h m (x) (32) s not sngular at x = x 0 Denton 2 Let us consder a dynamc system o the orm (28) n whch relatve degree s dened For the sake o smplcty, let us consder the SISO system (28) wth p = m = 1, and assume that the relatve degree r o the system s globally dened and constant Let us apply a one to one change o coordnates [ξ, z] = Φ(x) (33) beng: ξ = [φ 1 (x),, φ r (x)], z = [φ r+1 (x),, φ n (x)] (34) such that φ (x) = L 1 h(x) or 1 r and L g φ (x) = 0 or r + 1 n, then: ξ 1 = h(x) ξ 2 = L h(x) and thereore: ξ 1 = ξ 2 ξ 2 = ξ 3 ξ r 1 = L r 2 h(x) ξ r = L r 1 h(x) ξ r 1 = ξ r ξ r = Lr 1 h(x) ẋ = L r h(x) + L g L r 1 h(x)u = v (35) Usng the condton L g φ (x) = 0 or r + 1 n, subsystem z takes the orm: ż = F (z, ξ) (36) The expresson o the dynamc system (28) ater the change o coordnates descrbed, consttuted by two subsystems o the orm (35) and (36) s denoted as normal orm n the context o nonlnear eedback theory The subsystem (36) represents the nternal dynamcs o (28) wth respect to the output y Input Output lnearzng eedback Proposton 3 Consder the MIMO system (28) wth U = Y = R m, and vector relatve degree {r 1,, r m }, globally dened and constant The eedback: u = R 1 (x)(α(x) + v) (37) 479

6 wth R dened by (32) and α(x) beng: L (r1) α = L (rm 1) h m 1 (x) h m(x) L rm (38) lnearzes the MIMO system rom the nput v to the output y That s, transorms the system nto a system whose nput-output behavour s dentcal to that o a lnear system havng a transer uncton matrx: s r1 Y (s) V (s) = 1 0 s 0 r s rm (39) proo Frst, let us transorm (28) to normal orm by means o a change o coordnates o the orm: [ξ 1,, ξ m, z] = Φ(x) (40) where, or = 1,, m ξ = [ξ 1,, ξ r ] = [φ 1(x),, φ r (x)] (41) and z = [z 1,, z n r ] = [φ r+1 (x),, φ n (x)] (42) such that, on the one hand: ξ 1 = h (x) ξ 2 = L h (x) ξr 1 = L r 2 h (x) ξr = L r 1 h (x) or = 1,, m and then: ξ 1 = ξ2 ξ 2 = ξ3 ξ r 1 = ξ r ξ r = L r h r (x) + L g j Lr 1 h (x) (43) and, on the other hand, φ r+1 (x),, φ n (x) are such that: ż = (z, ξ 1,, ξ m ) (44) Condtons under whch such a change o coordnates can be ound are descrbed by Isdor (1989) The eedback (37), known as the Standard Nonnteractve Feedback (Isdor, 1989), appled to the system descrbed by (43) and (44) yelds the ollowng system, characterzed, on the one hand, by m sets o equatons o the orm: ξ 1 = ξ 2 ξ 2 = ξ 3 ξ r 1 = ξ r ξ r = v (45) plus an addtonal set o the orm (44) consttuted by n r equatons The nput-output behavour o ths system concdes wth the correspondng to a lnear system havng the transer uncton matrx (39) Remark 2 Takng n account the ollowng equvalences: y 1 = ξ 2 y 2 = ξ 3 = ξr y r = v the system n normal orm descrbed by (45) and (44) can be expressed n an equvalent orm by: y (r 1) ż = (z, y 1,, y (r1 1) 1,, y m,, y (rm 1) m ) 1 = L y (r1) 1 = v 1 m = L h m (x) y m (rm) = v m (46) REFERENCES P Érd and J Tóth Mathematcal Models o chemcal reactons Prnceton Unversty Press, 1984 F Grognard and C Canudas de Wt Desgn o orbtally stable zero dynamcs or a class o nonlnear systems Syst Control Lett, 51:89 103, 2004 F J M Horn and R Jackson General Mass Acton Knetcs Arch Ratonal Mech Anal, 47:81 106, 1972 A Isdor Nonlnear Control Systems Sprnger, 1989 C Kravars Synthess o multvarable nonlnear controllers by nput/output lnearzaton AIChE J, 36(2): , 1990 R W Mssen, C A Mms and B Savlle Introducton to chemcal reacton engneerng and knetcs John Wley & Sons, 1998 G Ncols and C Ncols Thermodynamc dsspaton versus dynamcal complexty J Chem Phys, 18: , 1999 G Ncols and Y Prgogne Sel organzaton n nonequlbrum systems John Wlew & Sons, 1977 I Otero-Muras, G Szederkeny, A A Alonso and K M Hangos Local dsspatve Hamltonan descrpton o reversble reacton networks Syst Control Lett, 57: , 2008 H G Othmer Analyss o complex reacton networks Notes o the lectures gven at the School o Mathematcs Unversty o Mnnesota, 2003 E Zamora-Sllero, M Haner, A Ibg, J Stellng and A Wagner (2011) Ecent characterzaton o hghdmensonal parameter spaces or systems bology BMC Syst Bol, 5,