Cita● on:Mlmowsk P,Carter TO,Weber OF(2013)Enzyme Cata ysis and he Outcome of 0 0Chem cal Reaい ons U P,oteonics B:o nform 6i 432-

141 doi:10 417211pb 1000271

to allect outcome by turning periodic subslrate inp!t into fl on-periodicprodr.rct generetion has been uncertain. Enamples were found in the(xidase peroxidase reaction J6] and a hemin hydrogen perodde-suirtite model of.atal)$is [7], bul broad potential for non-linearity inother emyme-calallzed systems has not been established. Enzlrnaticreactions.an be described in general tenns by a set of ordinarydifferential equatioDs for aII reactants (Figure 1). Thek format has someresernblance to the differential equations descrihing the t,orellz andRdssler systems ofr}on-lillear dFami.s [8,9], and therelore implies the

?otential to generate cotupiex, possibly chaodc bchavior.

Here we test the h,lothesis that eru,)lnc calallsis can affcct theoutcome of biochemicxl reactions We investigate the panmeterspace for the rate aonstants in substrate-activated and general enzlmecatalysis, and we $tudy enzynre calalysis in a brarched reaction wirltr,o competing arms.

Materials and Methods

L sirico modeling

'Ihe chemical reactioos under study can bc dcfined as sets ofordioary dillerential equations (ODLS) thal describe t})e rate ofchangeof each reactant per time step- TIre soluiions oftllose equations showconstant, peaiodiq ol non-periodic flow over time- Here, a solverftom tr SciPy [t0] package (scipy.intcgrate.odeint) was used to modelen4ane catal)sis. To solve the equations under study, the time step wassct to 0.01 seconds, resulting in a samp[ng frequency of 100 Hz. 'Ihe

inpui fluctuated at a constani rate of0.01 l{2, i.e. each oscil}atioo took100 seconds or 10000 points. A double precision arilhmetic (64 bit)was applied

Non-lir€ar systcrns can be rcpresented in a hifurcation diagramas a set of densely packed poirts- the use of a Fourier transform

Bives another method of visualization. lhe Fourier hansfbrm is

a sigral represe$ation in a liequency space. Wlrereas a siglal int-he time domain shous how much energy is carried in a given timeperiod tLe Fourier spectrum shows how 1i!_h energy is carried in a

given frequency range. 'Ihe periodicity of a siglal is particularly wellrepresented on a Fourier spectrum-a-s a peak at a spediic frequency(and at ha Donic frequencies). Because in the chaos regime everyperiod is presenl, the Irourier spectnfil of a chaotic signal has pealcfor every frequencp

A bilurcation djagram is a graphical method of presefltilrg a

change itl a dyiamical s]Btcm depending on a change in one of theundcrlying parameters. A biiurcation plot represents a map, not a flow(the q itcm is representcd by a recurrencq not difierential, equation). Ifa system has oriy fixed pofurts in sofle parameter rarge, this should be

rcprose[ted on the diagram a5 a solid [ine. Limit cycles are representedas nrultiple, vertical lines. The number of lines depends on the peiiodof oscillations. Chaos in this case is represented as a block of denselypacked points. lo define the reaction states of the enz)'rne catallsis, wegenerxted bjfurcatioB maps by solving the sel of differential equations(Figure 2) fbr each of its parameters At every ite.alion,local minimaand maxima qlues of a chosen variable v.ere plotted on a vertjcal a.xis.

Upon completion of lhe desired range o[ parameters, the resultinggraph pres€nts the change of the system's variables depending on thedelrcd parameter change.

Agent-based modeling

For agent based modeling, a basic Netlrgo model on enzymckineli.s was cdiied to allow for contjnuous rh)4llmic substrate input

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dt

dES・ kl l【 ]lS〕+К4[E]【 P〕 ―(k2+k3)【 ES〕

Ot

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dP・ k311印 〔Ell― k4:ElIP〕

at

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B ochen■cal∞tat,on DD,ffere威●l elualonS COmmontt k.-O h ihe sP∝ おized“se ofsubsmtemnb面。nL_o The nrsttem m“e●た面 積l equalon tor suDま rate cttnge

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Asubirate

input

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dt

dP‐ k`[61dt

i[.1‐ [ヽ 1【呵 lFSl

Foure 2:A)Poss'わ le rea“ on 3cheme for enzyme ca● 呼●S ttth subslrate

advalon ″Vang et al t341) B,Set Of direren,al equations desc10ing lherea“on schett ofttbstrate ad"節 on(E=嘔

.―E ES)The l嵐 lem hthe

reacton of substale cha"e assumes th/11m e substrate inplt n the brm ol

and the宙sualintlo“ of rate tl・ angts(“ atlVanted enゅ 唖e ttneuぽ .

PrOgram in Nellogo Model Llbrary httP′ ′cd nOrth■ estern edl1/

n.・tlogO/1110dels′ )nc PrOgram models ule rttral en″ nc tnCuc

reac● o“ (■諄 re l)m■ eaぬ rttctltln constant αl hrougllり i"ほastlbttaF volmle,ald the rate orthe lnupg lor∝ ●mplilldc ad

llequen")tO be set by the invc■ gator

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