Order and chaos in biochemistry

TIBS 1 2 - February 1987

45

cal School,USA Chennstry, Austraha 17 Sun, T-T Department of Dermatology and 24 Dale, B Department of Penodonttcs UmverPharmacology,New York Unwerstty Medtcal sltyof Wadungton,USA School.USA 2~ Osbom, I~ =d~x Planck Instntole for Bk318 Fuchs, E V DepanmentofMolecularGenephyst~l Chenuslry,Gottmgen,FRG ttcs and Cell Btology.Umversntyof Chicago, USA ROBERTD B FRASER 12 McKeon, F D Department of Physmlogy, 19 laem, R Department of Pharmacologyand Department of Protein Chermstry CSIRO HarvardUmverqty,USA Cell Btology, New York UmversatyMedncal Parkvdle, Vnctona 3052, Austrahd 13 Wdlnams,R D Departmem of MolecularBtSchool,USA ology,VanderbdtUmversnty,USA 20 Roop, D R Laboratory of Cellular P M STEINERT 14 Rsher, D Z Laboratory of Cen Bmlogy, Carcmogenests and Tumor Promotmn, NCi/ Dermatology Branch of the Natmnal Cancer InstiRockefellerUmversnty,USA NIH, USA tute, N IH, Bcthesda, MD 20892 USA 15 Aebs, U Department of Cell Bmlogy and 21 Rogers. G E Department of Bmchenustry Anatomy, Johns Hoplons Medical School UmversttyofAdela,de,Australia A C STEVEN USA 22 Sargent,T Laboratoryof MolecularGenetncs. Laboratory of Phystcal Btology of the NaUonal In16 Goldman, R D Department of Cell Btology NICHD/NIH,USA stitute of Arthnus and Musculo-skeletal and Skin and Anatomy, Northwest~,r UmversmjMedP 23 Gdlespte, J M CSIRO Ehvlsmn of Protein Diseases, N IH, Bethesda MD20892. USA Branch, N I D R / N I H , USA 9 I-hrokawa, N Department ofC.ellB,ologyand Anatomy, Umversay of Tokyo, Japan I0 Stemert, P M Dermatology Branch, NCI/ N I H , USA II Eagles, P A M Department of Btophystcs, I~ang'sCollege. London, UK

Features 3

Order and chaos in biochemistry Benno Hess and Mado Mark'us

Re.fly, oscdlanomhaving no r e c o g n ~ frequency (chao~ oa~tlons) have beenmeasured m enzymu: systems These oscd.l_~ons may be useful m enhancing biological vanabday, but they may also be related to ~ g u ~ ! phenomena Periodic and chaonc oscdla~ns can be mterconverted by changes m the control parameters and by pulses of the oscdlatmg metabohtes Oscdlatmg reactions m a biological context have been widely studied in the last few decades t.2 In blochemtstry, oscdlatmg dynamics play a prominent role nn biological clock functions, m rtter- and mtracellular signal transmtss, on, and nn ceflular differenttatton. In addition to periodic reacttons, certain btochemtcal systems, namely the peg3xldase reaction 3-5 and glycolysts6"8, have recently been shown to display osc Ilattons with ever-changing and unpredictable frequenaes and amphtudes These oscdlattons, winch are called 'chaotic', are macroscoplc phenomena, as they occur on a much larger scale than thermal nonse Here we revtew bnefly ~hat ts currently known about dynamic processes tn biochemistry. We shall refer not only to periodic and chaottc oscdlatioas, but also to related subjects, namely quaslperodiclty, crises and mulUple oscillatory states, Then, we wdi present an outlook for further exploration m ~ s field

The diversity of dynamic precesses In the general treatment of bnochemn-

B HessandM MarkusareatthoMax-Plancklns~nae for Numv.on PhysuJIogy, Rhemlanddamm~!, 4600Dortmund I. FRG

cal reacuon systems, one usually assumes steady-state dynamics, winch are described by conunuous and differentlable mathematical functions Thus, the dependence of some system feature, such as flow rate, o r a n independent variable, such as concentration, can be descnbed by a paraal derivative, for example a control coeffiaent (see Ref 9 and references therein) This formahsm can be apphed to a large class of phenomena However, due to the nonhneannes m bmchenucal reaction sequences, one must cope v~th discontmutties, occumng at the so-called btfurcabon points. At these points, a small change of an independent variable causes the system to jump into a different state The simplest case is the transtuon from one steady state into another A more complex case is the transmon from a steady state into apenodic oscdlatlont, for example, nf the enzyme ns a ~ vated by its product under statable turnover condtUon~l°. Transmons into other types of oscdlattons are also possible. (1) quasnpenodtc oscdlations, which can be decomposed into a sum of two or more periodic osollatmns wdh mcommensurate frequencies; and (2) chaotic oscdlattons, which

cannot be decomposed into a sum of a fimte number of penodic oscdlations Quasnpenodic and chaotic oscdlations occur, for example, tf two product-acti rated enzymes are coupled m senes u, or nf a product-acuvated enzyme has a penodic source of substrate t2-15 Although at first slght chaotnc behavtour displays the features of nonse, nt can be described by deterrmmsUc laws It ts therefore called 'deternumstic chaos' In sptte of flus detenmmsm, long-term predictmns are not posstble m a chaonc system, because the underlying laws are so sensmve to mmal condmons that the unavondable microscopic random fluctuations become rapndly amplnfled to a macroscopnc scale However, tt should be kept nn mind that the amphficanon process proceeds m such a way that the macrescop¢ randomness becomes constrained to well-defined intervals of metaboltte concentrauons Thus, one often describes deternumsuc chaos as 'constrained randomness' (A general rewew of detemumstic chaos ns found m Ref 16 ) For a better understanding of the different types of oscdlations as well as the transdtons between them nt ts useful to define a so-called phase space The coordmates of such a space are timedependent variables that characterize the state of the system, for example metabohte concentrations, and are called phase variables In contrast, the vanables that rem=n constant m time and are set from outude are called control parameters In an oscdlatory state, the penodic change of the concentration of metabohtes isdescnbed m phasespace b y d o s e d tralectones (or orbnts) For quasiperiodic oscdlatmns, the trajectones are

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46

TIBS 1 2 - February 1987

Periodic

"if

Chaotic

0 I----- 50 rn~n - - - I

Fig

!

Measured O~ concentranon

m ~he

perozulase..oxutase~.acnon The condmons of the penodu: oscdlat~n and the chaotic osctllatmn differ m enzyme concentration fO 9 pM perwd~c, 0 55 pu choz~nc) Reproduced, vdth pemusslon, from Ref 3

not closed, but fill up a surface Chaotic oscdlatlons are characterized neither by closed orbits nor by surfaces, but by weird 'fuzzy' dommns (see below, Fig 3c) These domains have a structure that ~s s~mdar at any scale of magnification" a feature called 'self-sundanty' The peno&c orbRs, the quastpenochc surfaces and the chaotic, self-sundar domains are called'attractors', indicating that the reaction system ts 'attracted' to the~ phase space regsons when the process ts mmated somewhere else In parttcurar, the chaouc regions are called 'strange attractors' Attractors are characterized by assigning them a dimension, winch ts called 'fractal dimension' (see Ref 17) and has non-integer values for strange attractors. The fractal dimension of a periodic orbit is one, and that of a quastperiodic surface ~stwo. The fractal dimension of the strange attractor m F~g 3c hes somewhere between two and three, as the attractor ts an object 13ang between a surface and a sohd. A mathemaucal analysts of Fig. 3c y~eldsthe value 2 2 for its fractal dunensmn In a phase space of experimentally accessible coorchnates, attractors can be constructed from measurements If all expenmental attractors have a fractal dimension less than some fixed integer, n, then we know that n variables are enough to describe the system dynanucs In tins way, R could be deduced from experimental data that n=3 for glycolysts ts In other words, three van ables are enough to describe the complex dynarmc states of glycolysls, m spite of the much larger number of metabohtes revolved A speaal feature of nonhnear systehts ts the existence of the so-called multiple

oscillatory states These are defined by attractors that coexist m phase space for the same set of control parametersu-~s Any of these coextsung attractors can be reached by the system, depandmg on the starting point m phase space. Swncinng between coexisting attractors can be accomphshed through jumps m phase space by externally driven abrupt changes of the phase vanables In a chenucal system, such lamps are possible through pulses of reactants (see Fig 4) Conversion processes of tins type have been yet'tried experimentally for the chlonde-bromate-iodtde oscdlatory system t9 They are expected to occur also m btocbemtcal and b,ologlcal systems Not long ago, a new type of phenomenon called 'cnsts' was reported 2° At a crisis, chaotic oscillations break down into different (e g peno&c) oscdlatmns This breakdown occurs autonomously, m contrast to the transtuons discussed before which occur by changing control parameters or by pulses of the phase variables Cnses are possible ~fat certain control parameter values a strange attractor colhdes w~th a coexisting unstable steady-state or periodic orint The chaouc oscdlauons break down as soon as the system reaches the point of colhston wlale orhmng on the strange attractor Chaotic transients that occur before a crisis may be extremely Iong-hved since ~t may take a long t~me untd the point of coll,ston is reached by the system. Long. hved chaotic transients m~ght indeed look hke chaos; however, they do not correspond to a strange attractor, thai ts a defimttve dynamic state of the system Such transients have recently been studied m a glycolyttc modeltS. Order and chaos in the peroxidast~ oxidase reaction The first chemical reaction reported t o

Periodic

be chaotic was the enzymic perox~dase reaction, catalysmg the reducuon of oxygen by l q A D H 3,4 Figure 1 dlastrates expanmental results obtmned wsth two &fferent enzyme concentrauons NADH ts supphed at a constant rate, while oxygen enters the solutmn by dtff,Jston. Penothc oscdlattons occur at the higher enzyme concentration (0 9 IJu) and chaos at the lower erL_-ymeconcentration (0.55 ltM) At a stdllower enzyme concentration (0 4 laM), the oscdlattons are penochc again A mmnnal model of tlas reactmn was constructed yielding oscdlattons closely resemblmg the expenmental observatmos (see Ref. 5 and references thereto).

Order and chaos in gl.veelysis Giycolysas is known to oscdlate m mtact cellst2 and couples to membrane transport functions21 Such couphng can be sunulated experimentally m yeast extracts by a penochc input flux of glucose to oscdlafing glycolysts Early experime~it~ voth these extracts showed that glycolyttc c,.,cfilatJons can be forced by the glucose input to thsplay mnluples of the input period T, namely periods T, 2T and 3T (Refs 1 and 10) Recently, Ingher multiples of the input parted were also found, such as 4T, 5 T, 7T and 9T as well as chaotic and quastpeno&c responses6-s. A penochc and a chaoUc reactson are dlustrated m Fig. 2 A number of techniques (see Refs 6-8 and 18) were apphed to prove that the observed disordered patterns are indeed deterministic chaos, and not quastpenodic oscillauons or noise. In spite of the complexity of the glycolyuc pathway, these expenmental observations can be descnbed by a slmphfied glycolyttc model mcludmg only the key enzymes phosphofructoiunase and pyruvate kmase, on the basis of detaded enzylmc rate laws. Model

Chao~c

!

o

--I

I.-2o~,.

Fig 2 Measured NADH fluorescence (upper) of a glycolysmB yeast extract under smusmdal mpm flux of glucose Olin, lower) "/'he condmons of the pmodu: response and the clmoac response differ m the input frequence (data from Refs 6 and 8)

TIBS 12- February 1987

47

[F6P]

.tt

[PEP]

[ADP]

I

I

I xlttl

,,...~ IATPI

(a)

(b)

(c)

Fig 3 Penothc(b)andchao~(c)attractorsobtamedfromaglycolyacmodel Theauractorsaredtsplayedbyrotatmgthetrapeztumshownm(a)aroundthe[PEp] a ~ This representat~ allows the sumthaneous msplay o f four metabohle concemrmtons The angle of rotation ts equal to the mpm flux phase Reproduced ruth pernmsmn, from Ref 14

calculations predict penodtc, quaslpenodic and chaotic oscdlattons m agreement vnth biochemtcal expen-

mentsr,-s The complex trajectones of glycol) uc processes can be represented using a chsplay techmque where four metabohte concentratmns and the mput flux phase are shown m a single ptcture. This representauo, m based on the constmctton of the trapezmm shown in Fig. 3a and reties on the fact, that only two metabohte concentrations are independent variables, whereas the other two are obtained from conservatton laws 12-14 The concentrattons at a gwen ttme are represented by the point x(t). The four dashed lines starhng at the point x(t) detenmne the current values of [ADP], [PEP], [F6P] and [ATP] on the four edges of the trapezmm (PEP, phosphoenolpyruvate, F6P, fructose 6-phosphate) The phase of the mput flux ts represented as the angle of rotatton of flus *~,~e_~q,lm around the [PEP] axis. Thus, o, ie ro~,,on ,s complet,:d after one input l g n o d (for d e ~ l s of flus representation see Refs 12-14) A periodic omllatmn (closed curve) on Fig 3b and a chaotic oscdlatmn (strange attractor) on Fig 3c dlustrate flus technique for the dtsplay of dynamic events Here tt should Je noted that the glycolyttc model also predicts ,ault~ple oscdlatory statest2-i5 (up to four attractors coextstmg m phase space) Figure 4 shows switching processes between oscdlatmns corresponding to coexisting

attractors The symbols S, and S2 represent [PEP] and [ADP], respectwely In thts example, a chaottc and a penochc attractor coexist m phase space m such a manner that each attractor can be touched starting from the other one, both by addltton or by subtractton of S t Thus, there are four switchmg posslbdmes m flus case (Ftg 4) A remarkable property of these processes ts that switching occurs without any transtents, since the final attractors are reached dtrectly through the pulses of S t

51

4.o

*S,

-S 1

,

Outlook

The observatton, analystsand modelImg of ordered and chaobc oscdlatzons in the peroxJdase-oxtdase reactton and .n glycolysm is of interest for other biochemical systems that have been observed to oscdlate under suitable condmons, such as photosynthests and mttochondnal resptrauon It remmns to future research to show whether chaos ~s a more widespread biochemical phenomenon

At a level of htgber bzologzca] corn-

-$1

I

+S+ T

3.8'

3.6, 0

Fig 4 Trans~onsbetweenpenod,cand chaotu: oscd~ato " v th,,,,jgh metabohte Pulses

48 plexaty such as intact cells and cellular aggregates, chaottc oscdlaUons have been observed m cluck heart 22, algae ~, molluscan neurons24, and the aggregating shme mould D~cty~ "album a~scmdeum (see Ref 25 and ~e£erences thereto) At stdl luther levels o] organtzauon revolving organs and orgamsms. the hst of phenomena that have been assocaated with chaos ~s long tt includes respiratory, hematologtcal and cardmc arrhythnuas 26, as well as neurological d,sorders 27 In spite of ~ts pathological connotation, tt should be kept in rmnd ths~ chaos may be of functtonal tmporlance Recently. ~t has been found that cell cycle vanab~hty (the broad d~stnbut~on of cell cycle umes m a cell populht~on) can be s~mulated by chaos in model calculauons In expenments, th~s dlstnbutton has the properttes of an attractor. rapidly recovenng after a perturbation 2s A n o t h e r functional importance of chaos may be related to 'trial and error'

n~echamsms, for example chemotax~s, where chaos may act as a 'random generator' In general, ~t ~s of interest to estahhsh the regulatory processes controlhng the avoidance as well as the occurrence of chaos The poss~bd~ty of multiple oscallatory states, a newly investigated property of nonhnear systems, m~ght be of biological slgmficance Ttus multlphc~ty allows switching between osc,llatory modes without intermediate transients Also. ~t pray,des a memory capacRy to the systeml4 Finally. we would like to refer to spaual order and disorder, which IS of interest in chemical, b~ochemlcal and b~ologtcal reaction systems As far -s apenodtc processes are concerned, temporal and spatial disorder has been observed m the mutant F R I 7 of D dtscoMeum In fact. thts orgamsm displays both chaotic c A M P oscdlatmns and morphogenetlc defects, such as aberrant stalks and frumng bodies (see R e f 25) A transition to ordered behavtour may be accomphshed by the addition of phosphod~esterase 29 These results are first examples m a large, new field of research on the mechamsm of spaual pattern formaUon The couphng of time patterns to spatial phenomena ~s of fundamental Importance m morphogenesls and morphopathology References I Hess B (1977) Trends Btochem Scr 2 193--195 2 Rapp, P E (1979)J Erp Biol 81 281-306 30lsen L F and Degn, H (1977) Nature267, 177--178 40lsen, L F (198'~)Ph)s Let] 94A 454-457

TIBS 1 2 - February 1987 50lsen. L F and Degn. H 11985) Q Rev Blophys 18. 165-225 6 Markus, M , Kuschmltz, D and Hess. B 11984) FEBS Let] 172.235-238 7 Markus. M. Kuschnmz D and Hess. B 11985) B.ophys (?hem 22. 95-105 8 Markus. M. Muller. S C and Hess. B 0985) Bet Bunsenges Phys ('hem 89.651-654 9 Burns. J A . Cormsh-Bowden. A . Grnen. A K, Helnnch, R , Kacser, H , Par]anus. J W. Rapoport. S M. Rapoport. T A . Stuckl. J W. Tater. J M. Wanders. R J A and Westerhoff. H V (1985) Trends Btochem Set 10.16 10 Bmteux. A . Goldbeler. A and Hess. B 0975) Proc Natl Acad Scl "USA 72. 3829-3833

I l Decroly. O and Goldbeter. A (1982) Proc Nad Acad Scl USA 79.6917--6921 12 Hess. B and Markus. M 11984).n Synergetlcs - f r o m Microscopic to Macroscoplt Order

(Frehland. E . ed ). pp 6-16. Spnnger-Verlag 13 Markus M and Hess. B 0984) Proc Nad Acad Scl USA 81. 4394--4398 14 Hess B and Markus. M 11985) Bet Bunsenges Phys Chem 89 642--651 15 Markus M and Hess, B (1986),mDynamws of Biochemtcal Systems (Damianowch, S, Keleil, T and Tron, L, ads), pp 11-23, Elsewer 16 Schuster. H G (lq~) Dewrnuntstw Chaos

Physlk Verlag 17 Mandelbro¢, B B (1977) Fractab Form, Chance, Dinenslon, N H Freeman 18 Markus, M and Hess, B 0985) m Temporal Order (Rensmg,L and Jaeger.N i , ads),pp 191-19"3,Sp~lnger-Verlag 19 Alamg~r. M and Epstein. ! R (1983)J Am Chem 3oc 105. 2500-2502 20 Grebogl. C. Oft. E and Yorke. J A 11983) PI4yslca7D. 181-200 21 Hess. B. BoReu.x. A and Kuschnutz. D (1983) m B~oiog~caiOxtdanans (8und. H and UIlnch. U .eds).pp 249-266.Sprlngcr-Verlag 22 Guevara. M R , Glass, L and Shner. A (1981) Scence 214.1350-1353 23 Hayaslu. H . Nakao. M and Hirakawa. K (1983) J Phea Sac Japan 52. 344-351 24 Hayaslu. H . Ish~zuka. S and HIrakawa. K 11983) Phy~ Let] 98A. 474-476 25 CoukeU.M B and Chan. F K 0980) FEBS Lea II0 39-42 26 Glass. L and Mackey. M C 0979) Ann NY Acad Sc~ 316. 214--234 27 King. R. Barchas. J D and Huberman. B A (1984) Proc Natl Acad Scl USA 81. 1244--1247 28 Maekcy. M C (1985) m Temporal Order (Rensmg. L and Jaeger. N I. ads). pp 315-320 Spnnger-Vedag 29 Marael, J L and Goldbeter, A 11985)Nature 313, 590-592

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