Biophysical Chemistry 18 (1983) 73-87 Elsevier BPC 00790 73 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Review SIMPLIFYING Wlodzimierz PRINCIPLES KLONOWSKI FOR CHEMICAL AND ENZYME REACTION KINETICS * Received 26th January 1983 Accepted 14th April I983 Tihonov’s Theorems for systems of first-order ordinary diffrxential equations containing small Parameters in the derivativea. which for m the mathematical foundation of the stendy-state approximation, are restated. A general procedure for simplifying chemical and enzyme reaction kinetics. based on the difference of characteristic time scales. is presented. Korzuhin’s Theorem. which makes it possible to approximate any kinetic sysrem by il closed chemical system. is also reported. The notions and theorrms are illustrated with examples of Michnehs-Menten enzyme kinetics and of a simple autocatalytic system. Another example illustrates how the differences in the rate constants of different elementary reactions may be exploited 10 simplify reaction kinetics by using Tihonov’s Theorem. AI1 necessary mathematical notions are explained in the appendices. The most simple formulation of Tihonov’s 1st Theorem ‘for beginners’ is also given. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK 1. Introduction In their recent paper, Kijima and Kijima [I] stated in section 1 that “the steady-state assumption or steady-state treatment in chemical reaction kinetics (. _ . ) has been used so far without proof.” And once again in section 3 “_ _ . there has been no general study on the condition when the steadystate approximation holds even on the first-order reaction.” It seems that the authors are not familiar with some important work of such authors as A.N. Tihonov, L.S. Pontryagin. A.B. Vasil’eva, V-F. Bu:uzov. V.M. Volosov, IS. Gradstein and V. Vazov. Especially in Tihonov’s work [2], the mathematical problems which are the very basis of * Gn !eave from: Medical Research Center. Polish Academy of Sciences. Bialobrreska 58. 02-32.5 Warsaw. Poland, and from FacuItti des Sciences. I’Universitti de Kinshasa. B.P. 190. Kinshasa XI, Zaire. N.B. In different sources the name of A.N. Tihonov is often written as Tikhonov, Tichonov. Tichonoff. Tychonoff. etc. the steady-state approximation (i.e.. the theory of systems of ordinary first-order differential equations (SFO) containing small parameters in the derivatives) are considered in a very general manner. The theorems he proved therein, called by other Soviet authors Tihonov’s Thecrems. may be applied to systems with reactions of any order and with any number of components. In appendix A we reproduce the English summary of Tihonov’s paper [2]. by J.L. Massera from Mathematical Reviews [3]. It is interesting to note that the theory of ordinary differential equations with slowly varying coefficients (see. for example, ref. 4) is in some sense equivalent to the theory of differential equations containing small parameters in the derivatives. For example, the equation pdx/dr’=p(r’).r (p” by the transformation (time) I = 1*/p I) of the independent (1) variab!e (2) is transformed dr/dr into the form zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA and its solution, determined by initial conditions: = p(pr)x (3) .K(I”) =.x0. z(rO) = z0 (5) where now the new coefficient.p(~f). varies slowlywhere X=(X, . . . . . x,), x”=(_Y~,___,x,~) and zyxwvutsrqponm f= However. the formulation of Tihonov’s Theorems (f,. . - . .f,,) are vectors in n-dimensional space, has so far been known to the author only in whereas z = ( ~t..._.z,~), z”=(z~,___,z~~), F= Russian f2.5.6] and reported in Polish [7,8]. As far f &, . , F,) are vectors in s-dimensional space. as the literature in EngIish is concerned, even in Putting p = 0 in eq. 4. one obtains the degezzerute the speciahst book by Mtirray [9]_ singtdar perKvsrenr turbation systems are treated heuristically with only a brief mention of Tihonov’s rigorous proof ds/dr=j(x.=,r): _x(J~)=_~~ (63) [2] and a reference to Vasil’eva’s work [lo]. The z=+(_r.r) (6b) latter seems to be the only source in English (apart from the above-mentioned summary [3]) in which where z = +(_K, I) is a root of the system of algeTihonov’s Theorem is form&ted and prove (and braic equations F(x,z,t)= 0 or some similar. more sophisticated mathematical F;(X.-__r)=o (1=1.....5) (7) cases are discussed. everything being treated in zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA a rather complicated manner): even there the more The system of equations general theorem proven by Tihonov in ref. 2 (Tihonov’s 2nd Theorem) is not reported at all. d=/dr=F(s.-_.r): =(r”)=z” (8) As the steady-state approximation is w%dely used in scientific literature comeming. e.g. chemical in which both x and t are taken as parameters. is relaxation (see ref. 11) and enzyme kinetics (see called the adjoi ned sysr enz. Of course. the point r.zf_ 12). we think that it would be interesting to z = Q(S, r) is an isolated singular point (root) of restate here Tihonov’s Theorems (without proofs). the adjoined system. as all terms on the right-hand We will follow Tihonov’s original paper [2]. The sides of eq. 8 are nultified at this point. notions used there are well known to the specialist. We shall assume further that all functions we But_ for the convenience of our readers, we give in use are continuous ones and that the differential appendix B the definitions of al1 the mathematical equations we consider have uniquely determined terms used. solutions. In ref. 2. Tihonov proved two theorems. one for The aim of the work [2]. i.e., the investigation of an SFO. not necessarily autonomous (see appendix zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC the solution of the SF0 (eq. 4) with initial condiC). containing a smail parameter in some derivations (eq. 5) when p + 0 is summarized in tives. and another for SF0 containing sevt ral small parameters in the derivatives_ We shall call them Tihonov’s 1st Theorem and Tihonov’s 2nd Theorem. respectiveIy. The 1st Theorem is in reality a When p + 0 the solution of the original system special c&se of the more comphcated 2nd The(eq. 4). with conditions (eq. 5). tends to the soluorem+ However. because of its greater simplicity tion of the degenerate system (eq. 6) if: and wide applicability \ve give here also the 1st (1) the root z= +(s, t) is the stabIe root of the Theorem. Moreover. in appendix D vve provide adjoined system: possibly most simple formulation of the 1st The(2) the initial values z0 lie in the domain of orem (according to ref. 5) ‘for beginners‘. influence of the root z = Q(X_ I) for initiai values 2. Tihonov’s 1st Theorem Consider an SFO ~l\/J~=/(.~.~.r) pdz/dr= F(x.z.r) with one small parameter IL: (411) (4h) (*X0. rO). This asymptotic equality remains valid for alI times r for which the solution of the adjoined system lies inside the stability domain D of the root z = +(s_ r ). W Klonolvski/SimpIifyi,tg (For the definitions of an isolated, stable root and of the domain of influence. according to Tihonov [2], see appendix B; for the conditions of existence, uniqueness and stability of solutions of an SF0 see appendix C.) This theorem remains valid also if the right-hand sides of eq. 4 depend continuously on the parameter CL,i.e., for the system dx/dt=j(x,z,r,p) pdz/dr (4a') = F( x, -_.I, p) (4b’) 75 zyxwvutsrq reaction kinetics d_r/dr =~(x.=~",...,z~"'.I) #"dz"'/dr= (=a) F"'(x.="'.....='m'.I) (j=l.....(m -I)) (12b) =(m,=*(m,(l.=(I),___.=(m--I)) (12c) with initial conditions .r(rO) =X0; Z(“(Yo)=+” (i=l....,(m-I)) where rcrn’ = #D(“*‘(_Y,z(I). . __ , z(“‘the system of algebraic equations z(““, t) = 0, i.e. (13) -l), t) is a root of i (l ), _ _ _, F(“‘)(r , where f and zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA F are continuous functions of their F:_““(.r,=(” ,.... Z(m). t)=O (/,=I__.._&) arguments 161. (14) As the asymptotic procedure lowers the order The system of equations of the SFO, the initial conditions (eq. 5) generally may not be fulfilled by the solutions of the d;(““/dr = Ftm’(_y. Z(“_..._;(nr’, I); ~(““(0) = +“” (1s) asymptotic (degenerate) system. The solution of in which x, z(I),. . . , z(“‘-“, t are taken as paramethe original system may be approximated by the ters, is called the adjoi ned syst em of the fi r st or der . solution of the degenerate system for times t >> rd. The system is termed doub& degezzerate if it is a where t, = ]plnp] [6]. If the asymptotic system is of singly degenerate system for a degenerate system the second order, one may make a complete disof the first order. The degenerate system of the cussion of its, based e.g., on the phase-plane methfirst order has (m - 1) small parameters @’ (i = ods [S-S]. 1,...,(m1)); by putting $“‘-“=O in this system, one obtains a degenerate system of the second order and similarly the definitions of the other 3. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Tihonov’s 2nd Theorem notions of the second order. Analogously, one Consider now an SF0 with several small defines degenerate systems of the k-th order and parameters p”‘. all other notions of the k-th order. The behavior of the solution of the original d.r/dr=f(x.=(".....=(m'.t) (93) system (eq. 9) with initial conditions. eq. 10, when P(‘“.,jr”‘/& = F’/‘(.~,=“‘,_._,;‘“‘. t); (j= l,...,m)(9b) all p(j) + 0 are under the condition eq. 11, is summarized in and its solution determined by the initial conditions _r(rO)=.G’. $I’(rO)=.+ (_/=l,_._,“I) Tiizorzov’s 2xd Theorenz (10) When p(i) -+ 0 the solution of the whole original and f= where x = (xz ,..., x,), _x”=(x~,___,x~) system (eq. 9), with the initial conditions (eq. IO), ( f, _ ___. f, ) are vectors in n-dimensional zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB space, L(J) = ($‘,_ __, p), =0(J) = (g(i),_ __, tends to the solution of the degenerate (nz-times whereas degenerate) system if: =.:(I)) and F(J)= (I?:‘),. __, ks:-‘)) are vectors in s,(1) the roots z(J) = +(-“, with the aid of which dtmensional spaces. respectively (for i = 1,. . - , zzz)the degenerate system is defined, are stable roots Tihonov investigated the solution of eq. 9 with of adjoined equations of the j-th order for any i the initial conditions, eq. 10, when all /L’*(I)-+0 in such a way that ~~J+“/p’J’+~ Putting (11) in eq. 9 p (“I) = 0 , one obtains deget zer at e syst em of the fi r st or der : the sitzg& (1 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK GjGrn); (2) the initial values z’(” are in the domain of influence of the roots z(i) = +(I’ for initial values (X",io~'~,...,Z~~'~-l~, The asymptotic to). equalities remain valid for all times 1. for which the solution of the totally degenerate system .r( f ). 3” (t). lies inside the stability domains of the roots Z(J) = &I) for ah;. be more than one group of reagents for which an Mc L holds. As far as the reagents for which the system is open are concerned_ suitable flux terms. describing an exchange with the environment, must be in4. Chemicals zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA systems. Korzuhin’s Theorem cluded in the kinetic equations in addition to the terms arising from chemical reactions_ Such terms Tihonov’s 1st and 2nd Theorems have a direct are usually introduced in the form application in the simplification of kinetic equaJ,=k-(R,.,,R) tions_ From the law of mass action (LMA) one (18) obtains generally an SF0 with right-hand sides where .I, is the flux of component R (positive when being polynomials of orders not greater than the R is being supplied to the system. negative when it second. i.e.. the reactions are uni- and bimolecular. is flowing out of the system), R the actual conIn some chemical reaction models trimolecular centration inside the system. and R,,,, the actual reactions are also aswmed (e.g... in the ‘Brusselaconcentration outside the system, i.e.. in the entor’ model [13]). Lvhich leads to terms of the third vironment. Usually. R_,, is assumed to be indeorder. But. for applying Tihonov’s Theorems. pendently controllable (a so-called control varianeither linearity nor positiveness of variables. nor ble). A special case, but one which is very often even autonomy of the SF0 (see appendis B) is utilized. is to keep R,,, constant: zyxwvutsrqponmlkjihgfedcb required. R <,“I = RU First of all. it is necessary to take into account (‘9) the reagents for which the system is clored - for R, is called the reservoir concentration Then the such reagents (or groups of reagents) a mass conflux term (eq. IS) contributes 50 the right-hand servation law (MCL) holds. Biochemical systems side of the kinetic equation for R a constant term are very often closed for some macromolecular equal to enzymes and enzymatic comcomponents_ e-g.. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA -Jr- = X-R,, piexes: (20) and a linear Jr- where actua! E,, is the total enzyme concentration. E the concentration of free enzyme. EC the actual concentration sunl.naiion of the c--th enzyme is taken over complex a11 complexes and the present in For simplicity here after we denote a reagent and its molecular concentration by the s:tmc s_vmhol. ;t capital Latin letter. If one difthe system. ferentiates dE,‘dr’+ From eq. 16. one ~dE<,w=O obtains (17) 17 it is seen that the kinetic equations for for which the MCL exists (e.g.. for an enzyme and its compleses) are linearly dependent. So. if the SF0 has been primarily written down for all reagents present in the system. one equation (for E or one of EC ) must be dropped and replaced by the algebraic equation expressing the MCL (eq. 16). Of course. there may a group eq. = - term kR (21) R ,,“( may also change with time. but the time dependence R,,“,(t) must be known. In such a case the SF0 is no longer an autonomous one. but Tihonov’s Theorems still may be applied. If there are IV reagents in the system under consideration and there are g groups for which MCL are fulfilled (the h-th group composed of IV,, reagents. h = 1.. . _. g). then the system of kinetic equations to be solved is as follows: s .v JR,/dr’= k, i c k,‘R,+ c ,_I= 1-I of reagents kj’R,R, I .v + c k,““‘R,R,R,, ,.l.nr (i=l.....(rV-_R)) = I (nl) \; R, = R:;‘- 2 R_ PI,-1 (i=(A-~+I).___.n-) (h = I......?) (72b) where zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Jz differential equations have been replaced (6) in elementary reactions all stoichiometric by algebraic ones of the type in eq. 16. Hereinafter coefficients are equal to unity or zero. i.e.. even the t’ denotes real time and R, the actual concentrareaction A --+ 2B is treated as a nonelementary one tion of the i-th component; the summation over p,, but as a result of a chain of elementary reactions. denotes summation over all N,, components be+ B: so the kinetic constants in e.g., A+B+C,C longing to the h-th group for which an MCL is eq. 22 do not contain factors related to stoichiofulfilled and Rg” is the total concentration of this metric coefficients. since one assumes that in a group of reagents. The RPh are present also on the CCS only elementary reactions occur. right-hand sides of eq. 22a as their summations are Some authors (e.g.. see ref. 5) define a CCS taken over all N components present in the sysusing only conditions 1-4. It is interesting to tem. report here the theorem proven by Korzuhin (see As usual. each kinetic constant (li,. k{. kj’. refs. 5 and 6): k:““) is taken as h aving a positive or negative sign or being equal to zero if. respectively. R, is proKorzuhin s TJzeorem duced. consumed or is not involved at all in the given elementary step (i.e., in reaction with R,. R,. It is always possible to construct a closed chemetc., or the flux term)_ The terms k, arise from the ical system of kinetic equations. in which the beinflux (eq. 20) or from the decomposition reactions havior of some variables will coincide with any of the zero-th order (e.g.. one may assume that the desired accuracy and for any desired time period reagent is among other things a substrate for an with the behavior of a given system enzyme and that the enzyme is saturated with the dR,/dr’=‘k,(R ,._._. R,v) (i=I _____A’) substrate so that the reaction rate is maximal and where 9, are polynomials of nonnegarive integer Goes not depend on the concentration of the reapowers. gent). The terms k:R, arise from the outflux (eq. 21) or from decomposition reactions of the first For an open system the MCL (eq. 23) is not order_ An SF0 like eq. 22a is called a closed cilemical fulfilled. But any system may always be ‘extended qxrenz (CCS) if the following conditions are to a closed system’ by introducing coupled refulfilled [5]: servoirs of the necessary reagents. Additional variables have to be introduced to fulfill also other (1) all R, are positive (concentrations): conditions imposed on the system (eq. 22) to be a (2) the right-hand sides are of order not greater CCS. Korzuhin’s Theorem is. in some sense, the than two (all ki”” and all higher terms are equal to inverse of Tihonov’s Theorems as it concerns the zero. i.e., at most bimolecular interactions are preproblem of construction of a whole (‘original’) sent): SF0 for the given system treated as a degenerate (3) no autocatalytic terms are present. i.e.. all one. The SF0 eq. 22a. with conditions l-6. repreJ.z:.k:’ and k:’ possess negative signs or are equal to sents a homogeneous CCS which is in general the zero: simplest possible nonlinear SFO. The importance (4) the system is closed. i.e.. there is neither of Korzuhin’s Theorem lies in the fact that it influx (all k, = 0) nor outflux and so the mass demonstrates the possibility of realization of any conservation law given behavior (in particular, of stable auto-oscillations) of a part of the reagents during any given time period just in a homogeneous CCS if only the is valid, where p, denotes the molecular mass of number of variables (reagents) is sufficiently large. In heterogeneous (e.g., compartmental) systems the i-th component; (5) no elementary reactions of decomposition some of the conditions 1-6 are no longer valid. of the zero-th order are present, i.e.. in eq. 22 k: therefore the SF0 representing such systems are and k,J’ are nonnegative for all j # i. I == i: more compiicated. Complex behavior may be ob- served much more frequently in heterogeneous systems than in homogeneous ones as the number of variables may be then considerably smaller. Korzuhin’s algorithm treating any ‘nonchemical’ term in the: SF0 as a complex chemical reaction which unfolds in a chain of elementary reactions - makes it possible to inspect quickly and effectively the different model variants for any given comptex ehemicai or biochemical system. At the same time it shows one to understand better such extremely important phenomena as, for example. autocatalysis. We discuss later the application of Korzuhin’s Theorem to a simple autocataIytic model and its connections with Tihonov’s Theorems. where now a11 kinetic sion [time] - I constants have the dimen- !c,= k,/R)Q’ (254 tcII - zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONML k,,R;“‘/R:” (25b) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJI (2 5 c l IQ 9 X .,,,Rj”R$ “/R!” As a rule, the values for RIO’ it-( eqs. 24 are such taken that the dimensionless variables $- are between 0 and I and of the order of unity at most_ Then the most important term in any equation of the form of eq. 24a is the one for which the constant (K,. nr,,, K ~){, - - _ > ha s the greatest absolute value, yi. Dividing the i-th equation by yi one obtaines from eq. 24a 5. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA General procedure to simplify kinetic equations Small parameters appear in (bio)chemicaI kinetic systems in a natural way if one represents variables in a normalized dimension1es.s form. To do this one puts in eq. 22 R, = r,.:, (31) where 5 denotes the dimensionless normalized concentration of the i-th component and RI’) is a constant having the dimension of concentration. As Rj”’ one may take. for example. the total concentration of a group of reagents for which an MCL is fuffilfed (as EC3in eq. 16 and RF’ in eq. 22b). the Michnelis constant for an enzyme. the inhibition constant for an inhibitor. Constant concentration in the environment {reservoir vaiue. as R,, in eq. 20). initiaL concentration Rf. or concentration z, that the given component would have in the steady state. etc. Smali parameters appear also in a kinetic system if some elcmenta~ reaction steps are rapid as compared to others. Taking into account eq. 23, one obtains eq. 22 in the form s .v dr,/dr’= c Y, + “,,‘r /-’ x + /,I “,,r”,rr Ic,,,“,T)i,~“, c J& fh - t (27) denotes the i-th characteristic constants time and where al1 ~:==h.,/Y‘ (%a) “,I’ = $0, (ZSb) =;,I:,I==z 5/Z/ (zk) 7, are now Less or much Iess than unity and only the one for which x was selected for defining the time q (eq. 27) is exactly equal to unity. If we are interested in the behavior of the system in times of a certain specified order, say of the order of Y-, we introduce the dimensionless time variabIe I by taking T as the unit of time: I = r’/T (2 9 ) Then. changing the differentiation with respect to t’ by differcntintion with respect to dimensionless d c zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA (ret..... (N-g)) (243) ,J.“’ - t .v* $=I- 7; = t/u, time. I - I s + where d z? dr dl’” -i-_dr 1 T d dr (30) denoting fr=(1~“-ni-r) . . . . . IV) (24b) T,/T= “‘* (3’) and eliminating from the right-hand sides of eqs. 26 the variables r,v_,+ ,,, -. , r,\. by using eqs. M b, one obtains finally the system P r?l,-d~/dr=o,t c P a,,/5-+ c a,,,‘/‘,+ whereP=(N-g). In short-hand hereinafter sive classes go the equations iiaving smaller and smaller parameters WI, (i.e., the kinetic equations for variables for which characteristic times T are smaller and smaller than the time unit T). into the j-th class (j = I,. . _, m) go s, equations. The smali parameters (much less than unity) we w ill denote further by p(i) or just by pcI-So ,+~‘=,~J(r notation ,.._.. r,:r,+ ,,__.. r&,:,4/ _*. .._. / fp) (1, = I..... s,:j= J?z=:,= ‘t,(r ,‘.._. rp) where The remaining s equations (S = p - 11) are classified into m classes in such a way that into succes- (33) I.....m) ( 3’b) where denotes zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA dif(37C) ,*-Ito the dimensionless time the dotted variable ferentiation with respect variable I and ‘k, denotes the right-hand side of the i-th ec,uation, eqs. 32. If the time scale T we are interested in is such that some nli are much greater than unity. say these q with indexes i=(pi l).(p-+2).....P. then. by dividing the corresponding equation by III;, one finds that the time derivatives of these variables &==“-p r-1 and t.~“ ’ fulfill eqs. 1 I, i.e., they are small parameters of greater and greater degree. These groups of variables are called quick variables - after a time of order even much smaller than that of T they reach their (quasi)stationary values. In the following the quick. basic and reservoir variables w ill be denoted by z. x and p. respectively. The separation of reservoir variables is often r ,.._ ~_rp) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA (i =(p+I) . .._. P) : =-+( (34) made to some extent automatically, based on the I fact that the components having very great conare close to zero. since I /tlz, -=+zI. They are caIIed zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF centrations as compared to others (that is exactly cery slow or reservoir vuriuhks, i.e., changing only why they are called ‘reservoir variables’) change with characteristic times T, z=- zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA T. For time duramuch slower than other components. If we conl tion of order 7they remain practically unchanged and may be replaced by their initial values sider the simplest bimolecular reaction A I- X 5 B, it is easy to see that in any moment the instantar,(O) = ry= A, (i = (p-f- l )___..P) are dX/ dt a rates (351 transformation neous - exp( - t/tX) and d A/ dr a - exp( - r/ ~.~) where (where ( P -p) = 4) and be treated as parameters. time lh = I/ x-P and r, = I/ kX are instantaneous This is equivalent to treating the system as an constants for X and A, respectively. If one now open one 161. In this way eqs. 33 are reduced to assumes that A 3r) X then r,, B I_~. i.e., in any h~,?,=*,(r ,.-... zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA rp;d,,, . . . . . dP) (i =l ..... p) (36) instant A ci-anges much slower than X. So, if one assumes that some variables are reservoir ones, it The system may now be further reduced using is equivafent to treat them as parameters - they Tihonov’s Theorems. The equations are classified may be replaced in eqs. 22 by their constant in such a way that the first cIass contains all. say (initial) values (cf. eqs. 35). The differential equarr. equations whose relative time scale parameter tions for these variables may be dropped from the #II, is of the order (equal to) unity (i.e., the kinetic very beginning before introducing dimensionless equations for variables for which characteristic concentrations and time variables and separating times q are of the same order as T) - these are so basic and quick variables. In such a case the cahed slog or basic variables constant values of these reservoir variables. treated ?,=+,(r ,..... r,;A,,, ._... d,,) (i ==l . . . . . n) as parameters, are often useful to define the time (37a) unit 1. and/or the small parame:ers $I) or are included directly in kinetic constants. The system eqs. 37. when taken together with the necessary initial conditions, is identica1 with eqs.9 and 10 (or with eqs. 4 and 5 for nr - 1) and Tihonov’s Theorem may be used to simplify it. The conditions of continuity and of uniqueness of solutions (see appendix C) are of course fulfilled for the SFO. eqs. 32). and thus also for eqs. 37. whereas the stability of a solution (see appendices B and C) must be verified in each particular case. However. if the system eqs. 4b or 4b’ is linear in quick variables = (this is the case when reactions between quick reagents. e.g.. between enzyme complexes. may be neglected). then the algebraic system rqs. 7 is a linear one and therefore it has the unique solution: the adjoined system eqs. 8 has then the unique stationary root. This root is stable if there are no bifurcations, i.e.. no reactions of decomposition of the type 2, -+ Z, + Z, of the quick reagents [f;]. In such a case. before applying Tihonov’s 1st Theorem one must verify only a condition concerning the initial values. namcIy. vvhrther they lie in the domain of infIuence of the root (rq. 6b). If we are interested in times of order 7;- we obtain [I41 d.r/dr = -x-+(x--p)=+.\_= fidz/dr = .r - X-z -.x-z with initial conditions s(0) = I: Z(O) = 0 (41c) where r = C/T, are dimensionless variables defined as follows p = T, /l-, = E&S,,: R,,=(k_,+k2)/‘k,: and the constants are 0 = k,,‘k,S,,: k=K,,/S, (42b) t’ denotes real time and K,, the Michaelis constant. If one assumes S,, 3-> E. then T,. > T,. and p < I _ In such a case the enzyme is the f;lst component whereas the substrate is the slow one. The degenerate system is here reduced to one equation obtained from eq. 41b. The root ?=S/(k+_L) (43) (here and in eq_ 44 s is treated as a parameter!) obtained from eq. 41b when p = 0 is the stable root of the adjoined system 6. EsampIes d-_/dr a s - x-z - x-_ (44) AS the first example. let us take the simplest since for any value of s the right-hand side and SO enzymatic reaction - the mechanism of Michaelis the time derivative in eq. 44 is negative for : h ? (: and Menten: zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA wiI1 therefore decrease until the derivative will be equal to 0. i.e._ until = = 5) and positive for = < (in such a case : vvill increase uniil : = _T). When the system starts from the initial value c(O) = 0. = xvith four components: substrate S. enzyme E. will increase until :=?, so the initial conditions enzyme-substrate complrs (ES) and product I’. In (eq. 41~) he in the domain of influence of the root this mechanism no fluxes are t&err into account = == -_. and so there exist two MCL: The assumptions of Tihonov’s 1st Theorem are (39u) &.i ( ES) = E(0) _ E,, fulfilled and the steady-state approximation may ss P;(ES)-=S(O)=S, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA (39b) be used: putting = = ? in eq. 41a. a single equation with one vrariable A- is obtained A,S ;t result, there are only t\vo independent difds/dr = -p-V/( k i- I) ferential kinetic equations. The characteristic time 6453 constants for substrate and snzyme are. respecwhich is easily integrable and gives the transcentivelv. dental algebraic equation for s(r): $ = l/X-,&,: r, = I/x-,s,, (10) .Vi x-. In s = 1- pr (46) 81 zyxwvutsrq Returning to the drmensional variables one obconcentration, i.e., that they are reservoir variables tains from eq. ~5 the we11known Michaelis-Menten and may be treated as parameters (cf. section 5). expression zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA The kinetic equations for 2,. Z, and Z, are dS/dr’ = - k,SE,,/(K,, + S) (47) dZ,/dr’ = -x-,2, dZ,/dr’ = k,Z, + x-,~lZ~ - k ,BZ , i- x-,/I& (5Oa) The steady-state approximation fails. however, for dZ,/dr’= k ,BZ z - k,.-tZ, (SOC) very short times (of the order I < I/p)_ One may see that the initial conditions (eq. 41~) can no Writing eq. 50 in terms of dimensionless variables longer be satisfied in a consistent manner (the - =Z ,/M” (i =1.2.3) -, authors of ref. 1 speak about induction per-iod. (51) T,). If we do not want to use the quasi-stationary one will observe that the characteristic time conapproximation. the solution of the system eq. 41, stants for Z,. Z,, and Z, are equal is very complicated. based on coordinate transfor?-,= l/x-,: T, = T, = I/k zB (52) mation of the rate equations and subsequent solution of an integral equation [ 14:. Introducing dimensionless time It has been demonstrated that the steady-state I = I./T = k,Ar’ (53) approximation for the Michaelis-Menten mechaand assuming that nism may be applied either when E, < .S, (irrespective of the values of rate constants) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA or when X-,‘.t/k2L3 =p; X-,/t/X-, =p (54) the rate constants of decomposition of the en(for simplicity we assume that both quotients have zyme-substrate complex (i.e., k_ , and k,) are much identical values) one finds that all 112,(cf. eqs. 32) greater than the rate constant of its formation (i.e.. are equal to p and so eqs. 50 in dimensionless form k,) even if the concentrations of enzyme and becomes substrate are of the same order. E,, = .S,: the appliof Tihonov’s 1st Theorem to the cations dz,/dr = - z, + z1 (553) Michaelis-Menten mechanism in different cases dz,/dr = z, - z1 C zz (55b) are discussed in refs. 5 and 6. d=,/dr = zz - z3 (55c) As the second example. let us consider the simplest autocatalytic process. describing an exOne may demonstrate that when p -+ 0. the ponential growth solution z3( r) of the system. eqs. 55, coincides with the solution X(I) of eq. 48 (cf. ref. 5). By changing ds/dr = .r (43) the variable - It is not a closed chemical system in the sense of s = pz3 (56) the definition given in section 4. But. following Korzuhin’s Theorem_ it is possibie to find a closed one obtains from eqs. 5.5 chemical system. the behavior of which will coin(573) d.r/dr = - .r + zz cide with any desired accuracy with the behavior pdz,/‘ dr=s-z, (57b) of eq. 48. One may show that this is the case. for example. for the following system: pdz,/dr = .x f -_, - z1 (57c) The &I B+Z,+Z,iC adjoined system is in this case dz,/dT=x-z=, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA (49) dz,/dr = .r f z, - zz *3 ‘Zz A+-B+Ct Z ,+Z 2t Z 3=11fO=con~i r l n~ where x is no longer a function of parameter_ Putting p = 0, one obtains the adjoined system We assume that A and B are present at very high :, =x; z, zz = 2x (50b) (58a) (58b) time but a the root of (59) As eqs. 58 are a linear SFO, it is easy to integrate. The result is: the adjoined system, similar considerations must be made for all roots separately, as the coefficients of the characteristic equation and consequently the =,(z)=i,+(-_p--,)exp(--) (boa) Routh-Hurwitz determinants depend on the value =1(‘)=I,i[(.-:--E,)s~(=4-~z)]exp(--) (60b) of the root (cf. eqs. C7-ClO). From the above considerations. one may see One sees immediately from eqs. 60 that the root. that to sustain an autocatalytic process ‘inexhausrqs. 59, is stable: =I and z2 quickly tend to the tible’ reservoirs of ‘building’ materials (A and B) steady-state values 5, and ZZ, irrespective of their for the reagents 2, and Z2 must exist. In reality initial values z:’ and z:. i.e., all possible initial the system, eqs. 49, modeIs an autocatalytic system values zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA lie in the domain of influence of the root. only for the times for which A and B are practiThe assumptions of Tihonov’s 1st Theorem are caIly constant (of the order I < l/p). After a suffithus fulfilled and. when p - 0, solutions of the cientIy long time any reservoir wiI1 in reaIity be system_ eq. 57. tend to the solution of the degenerexhausted. One ought also to note that the autoate system catalytic process gives a byproduct C. dz/dr = -.r -cZ2 (613) As the third example, let us consider a case when small parameters arise in the system because Z, = s (6’b) of the differences in rate constants (cf.. the remark Z2= 2.r (61c) concerning the applicability of the steady-state and so s (and also z;) behaves as if it were approximation to the Michaelis-Menten mechaproduced in the autocatalytic process (eq. 48). nism in this section)_ This will also demonstrate To demonstrate that the assumptions of another method of separation of quick and basic Tihonov’s Theorem are fulfilled, it is not necessary (slow) variables. Let us assume that the reaction to integrate the adjoined system. In more L R,tR,dR, complicated cases this may be practically impossi(66) ble. Instead. one may use the Routh-Hurwitz critetakes place in the system and that this reaction is a rion for this purpose (cf. appendix C). In the quick one, i.e. above example. the adjoined system (eqs. 58) has the uniqtie root (the uniqueness of a root may be k = l/p where p-z 1 (67) in general checked with the aid of LiFschitz condiThe reagents R,. R, and R, may participate also tions. cf. appendix C). The characteristic equation in slow reactions (Le., with rate constants much (eq. C8) in this case is &I I -1-x , 0 -,-A less than Jc). Assume I =0 (Q) i.e. that in appropriately variables the kinetics 3, = .q,( r. s ) XZ+2X+1=0 />*,==I: h,=Z: and further defined of the system (683) (63) So xve have (eq. C9) Q,=l: dimensionless are given by h2=1 (b-1) i,= -~r,r2~G,(r_x) F2= -_S’,r+G~(r_.x) (68b) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA I D,=Ir D1= zo I i.e.. they are all positive. I, =I! I (65) The Routh-Hunvitz criterion shows that the root is stable and. as it is also unique. any initial condi:ions must lie in its domain of influence. If there is more than one root of wherer=(r,. ‘;. ~-_~)and~=(s,_..._x,,): theterms G, and g, denote contributions of slow reactions_ I denotes slow variables. The SF0 in which the terms ?f the order less than 1.1~ (e.g.. the contributions or slow reactions) W. Kt ono~sk i /Si nt pIl i /yi l l g r eact i on k i net i cs 83 ‘diagrams’ used by Hill [ 181 and other authors are also labelled SFG and therefore the general theory developed for SFG may be applied to these diagrams [20]. Tihonov’s Theorems may be applied to systems in which reactions of the second (and even higher) order take place. If some rate constants are equal to zero, i.e., some reaction steps are irreversible. the fundamental assumption of ref. 1 is not fulfilled. In contrast, Tihonov’s Theorems may be (69x) X,+1 = r, + r, even easier to apply in such cases (because of the x,+2 = r, - ‘2 (@b) smaller number of terms) than in the case when all reactions are reversible and no general assumpBy introducing these integrals into eq. 63 and tions about the number of components in the denoting r, by I, one obtains system are necessary. _?,Tg,(=._r) (i =l .....n) The fast components ‘forget’ their initial values. These values are necessary only to check if the (70a) -% + I =G,(-_._x)+G,(~._x) assumptions of Tihonov’s Theorems are fulfilled Kn+2 = G,(z,x)-G2(z_x) (Le., to check if they are lying in the domain of p?= -z(z-X”LZ ;f~G,(z.X) (‘Ob) influence of the root). However, sums of initial where now X=(X ,,.-.. x,,, x,~, ,, x,,+z). The sysconcentrations of some groups of fast components tem obtained is identical with eqs. 4’ and Tihonov’s (e.g., the totai initial enzyme concentration E,,) 1st Theorem may be applied if the assumptions may determine some characteristic constants of are fulfilled_ The above example illustrates how the system, e.g. small parameters p(j)_ The obvious conclusion from Tihonov’s Theothe differences in rate constants of elementary rems is that the steady-state approximation works reactions may be exploited to simplify reaction kinetics by using Tihonov’s Theorem. well for times (much) greater than the characteristic time constant used as the unit of time (i.e., the time constant of the components which may be 7. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Concluding remarks thought to be slow in the time scale we are interThe method proposed by Kijima and Kijima [I] ested in), however, not so great that the reagents m_ay not be applied directly (without linearlizaassumed to be reservoir variables are exhausted_ tion) even to a Michaelis-Menten mechanism (eq. The fact that the characteristic time constant 41) because of the nonlinear term XI. Their method for a given reagent is small, compared to the time scale we are interested in, constitutes a criterion may be classified between graph-theoretical methfor fast equilibration which is even more imods, however, the word ‘graph’ does not appear in the article. Because of introducing two kinds of portant in practice than the fact that some elemenarrows (edges or branches in graph-theoretical tary reactions are relatively quick (i.e., some reacterminology) their graphs correspond to so-called tion rate constants are much greater than others). colored graphs (two ‘colors’ of edges) and because Only relations between characteristic time scales 7 each edge has given direction and ‘value (the rate of different reagents and their relations to the time constant) they belong to the class of graphs called scale of observation we are interested in (which is taken as the normalizing unit of time T) are of labeiled si gnal flow gr aphs (SFG) or labelled directed graphs [15]. The theory of graphs, deimportance. in section 5 we have subdivided all veloped primarily for the analysis of electrical the reagents into at least three classes, called renetworks, has been extensively used for simplifyservoir (very slow), y. basic (slow), x, and quick, z, ing kinetic (bio)chemical systems (see, for exam&=rL@&(r,_s.=. t,p) (k + I....,q) ple, refs. 16-19). It is interesting to note that the (713) are neglected, is called a t r uncat ed syst em. Usually. the truncated system has certain linear integrals, some of which represent conservation laws. These integrals are slow variables for the original SF0 and taking them as new variables leads to separation of quick and slow variables_ It is easy to see that in the example under consideration there exist two such lineary independent integrals of the truncated system zyxwvutsr _i,=/,(s._~.z.r.p) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA (i =l ..__ .n) r(r) of the adjoined system dz/d?- = F(_Y’, z. to), (71b) ~,=+_.Lz.r _P) (I=l .....s) z(0) = z@, satisfies lim,,, (‘1c) The concentration hierarchy in the system is usualiy in agreement with the time hierarchy 161.i.e. .r; < Y, < z, I( T) = 9(x”, to). Then zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE the solution of (1) through the initial point (72) (x0. z”, r”) tends. as p + 0. to the solution of the degenerate system dx/dr =f(x. Q(x. f). f) as long as the point (x. r) does not leave the region 3. A similar result holds for systems containing several small parameters p,_ ____p,,, tending to 0 in such a way that p, c ,/p, + 0. J. L. Mas~eru (Montevideo)_ we have demonstrated. if we are interested in the behavior of the system in periods of the order of the time scale characteristic for s,. then we put Appendix B _YL= 0 (cf. eqs. 34 and 35) and we eliminate the variables 2, using Tihonov’s Theorem. However_ if B. I. I?fahetnatical dejittitions (from ref- 2) we are interested in the transient phenomena (i.e.. in time interval 0 -Z t CCp), then the variables z, The ttorttz of an s-dimensional vector o= ought to be considered as basic ones and s, as (F,..__. v~) will be denoted by ]o] and defined as reservoir ones. Oppositely. if we are interested in the evolution of the system (I 3-> 1). then the varifL‘I= ,+ (Et) ablesr;, ought t<>be treated as basic ones and X, as /-I quick ones. ReaI systems geurrally have more than three characteristic time scales (i.e.. some groups (Euclidian norm)_ of quick variables. z:” (i = 1.. ___m) and then The root z = +(x, I) of the adjoined system of Tihonov’s 2nd Theorem may be a useful zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB matheequations F(x, z, r) = 0 will be called the i sohr ed As matical tool. it may be said that the mathematical foundations of the steady-state approximation have been well established four more than 30 years. In conclusion. Appendix A if there exists an E for which this system may not be fuIfiIled by any other vector z’ = (z;_ ___ . z-z) with the property roof I-_‘-+(_\‘_r)l’< (=‘-+I) The isolated singular point (root) .Z= (Z,_ __ . , f,) of the acljoitted system d:/ds=F(s.z.r); Tihonov. A.N. Systems of differential equations containing small parameters in the derivatives. Mat. Sbornik N.S. 31 (73). 575-586 ( 1952). (Russian) Consider a system (1) d.ridr =/(_I-. z. I). pd z/d1 = F(x. z. I). where s. f are pr-vectors. z. F i)r-vectors. and f. F satisfy suitable regularity assumptions. Assume -_ = +(s. I ) is an isolated solution of F(_(s. z. I) = 0 vvhose points are (asymptotically) stable equilibrium points of the adjoined system d z/d7 = F(s. z. t) (s. t being considered here as parameters. the point (s. P) belonging to a hounded open region D). Assume that the inittal conditions ( _I-“. z”. t”) are such that the solution (W =(r”)=z” (B3) (where x and r are taken as parameters) is called a srohilir_s point if for any E there exists a 6(e) such that trajectories starting in a point M belonging to the 6(c) neighborhood of the singular point tend to this singular point without leaving the l neighborhood when r -+ so. i.e. (1) the trajectory of any point ?=(z,__.__z,) beronging to the 6(e) neighborhood of the point 5 tends to 5 when r -* co: Iim 7-x Z(T)=: if I=(O)-?lc6(~) (B3) (2) trajectories of points from the 8(e) neighborhood of the point Z do not leave the e neighborhood of this point: f:(r)--_f<r foran~r.ifI=(O)-_l<S(r) (=I The isolated root z = Q(x, t) of the system tinuous and if the Li pscJzi t z condi t i ons F(x, Z, t) = 0 is called zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA st abl e in some bounded region D of the space (x, t), if for all points IF(c=, , . . . . =,)-F(T ; t ,..._, l J,)I -= nr . 2 I=, - o,I (C3) ,--I belonging to 5 the points z = Q(_K, t) are singular stability points of the system (eq. B3). are fulfilled for some A4 independent of z and o. The domai n of influence of the stable root z = Then z(r) is a continuous function of the given +(x, f) in the subspace x = constant, I = constant value z0 = ~(7~). Each solution extends to the is defined as a set of all points z” from which boundary of D. trajectories of the adjoined system (eq. B3) tend to The Lipschitz conditions are satisfied, in partic= = +(x, I) when 7 --j cc. ular. whenever F(z, T ) has bounded and continuThe root z(“‘)= #“‘)(x. z(‘),....P(“~-‘), 2) of the ous derivatives, aF/ar {, (I = 1, ___, s) in D. adjoined system of the first order F(“‘)(_Y_ z(l ). _ _ _. The root of an SF0 is a szabl e root if all z”“‘, I) = 0 (where x, z(‘) ..__,_ -(“‘-‘). t are treated perturbations 6~ around this root diminish to zero as parameters) is called an i sol at ed r oof of the first when 7 * cc or der . if there exists such E that this system may not be fulfilled by any other vector z’(“‘) for which s=0 (C4) 5-x ,-_“““_.*o’“Z’, <c (=?Wi,,+!nl,) (B6) The root of the first order L(“‘) = #“‘)(x. -..(I) ._._,_ -(m-‘). t) is called a st abl e r oar in some closed bounded region 5, if for all (s. I(‘)._ __, :(“I-I), f) of this region z(“‘)= r$(“‘) is a stable singular point of the adjointed system. To investigate if a root 5 of the autonomous SF0 (eq. C2) is stable, one linearizes the system around Z- ; by putting =,==,+s-_, (I=1 . . . . . s) (C5) one obtains Appendis Cl. C Or her d(&,)/dT= wat henzat i cul c a,,:, (I=l.....s) 1-I defi ni t i ons where the matrix A = l l a,,l j is given An ordinary differential equation of the first order or a system of first-order ordinary differential equation (SFO) dz/dr=F(z.i) (~=~,.___.~,) (Cl) is called ar c;ononzous if the right-hand sides do not depend explicitly on the independent variable r, i.e. ” ,, =aF,/a:,I,_: (l.j= (CI) where F and z are vectors of s-dimensional (s= 1,2,...). and uni queness of sol ul i ot zs space I . . . . . .r) by (C7) The root z = 5 is stable if all eigenvalues h,, (I = 1,..., s). of the matrix A have negative real parts. 1, are the zeros of the characteristic equation della,, - h&J= 0 when S,!= 0 for r-i, d-_/di=F(=) C_7_ Exi sr ence (C6) (cs) 6,‘= 1 for i=i (Kronecker’s delta). The characteristic equation, after reso!ving the determinant in eq. C8, is a polynomial of the s-th order: of SF0 b,~~+b,h’-‘+...+b_,_,h+h,=O The system (eqs. Cl ) has a unique solution z = ~(7) through every point (Z = z,,, 7 = ~“0)in the (S + I)-dimensional domain D if for (z, T) E D functions F( =, T ) are single-valued, bounded. con- cc91 zyxwvutsrq Given b, > 0, all roots X, of the characteristic zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB polynomial (all eigenvalues of the system eqs. C2 have negative real parts if and only if the determi- are no others roots); (b) the solution t,,. _. , Zs is a stable isolated 4, = ho zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA singular point of the adjoined system (i.e., any D,=h, perturbation throwing the system out of this point, h, &I diminishes with time to zero) for all values of Dz = I 4 h2 I (x ,, _ . . , x,) which are treated here just as parameters and not as functions of time; bt kl 0 (c) the initial values (zp,. . _, z,“) are in the doD, = 6, b, cc101 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG 6, main of influence of the stable singular point of 6, b., b, the adjoined system (i.e., the system will evolve in _ _ _ _ _ . _ such a way that (z,, . . _, z,), when started from _ _ . (ZPI--.,iI O), will tend to (F,, . . . , z;); 0’ ... 0 (d) the solutions of the whole system (eqs. Dl) ho b, and of the adjoined system (eq. Dl b) are unique hz b, _‘. (where b,,, = 0 if nz > s) D,= h! ________________ :_‘.. ____.____..._____....... and the right-hand sides (f;- and F,) are continuous 6,,_, 6,,_, bZ<--3 ... b, functions. nanis I I all zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA positive (Routh-Hunvitz criterion). This is true only if all 6, and either all even-numbered zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED D, Note added in proof (Received 20th June 1983) or all odd-numbered D, are positive (LiCnardChipart test; [21]. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Professor Benno IHess has recently called the author’s attention to the publication of Reich and Sel’kov [22], in which the authors outline the simAppendis D plest case of using Tihonov’s 1st Theorem (however, without formulation of the theorem itself) Di. Tihonov’s Theorem for Begitmers with one rapid and one slow variable. are Consider a system of p first-order ordinary differential equations. Let us assume that s of the p equations have a small parameter p. multiplying the time derivatives: d.r,/dr=f,(.r pdz,/dr ,.___._ r_.-_ ,.___. 2.) = F,(x ,.___._ r,. -,.._.. Z, ) (;=I (I= ____. II) (Dla) I.....s) (Dlb) where (n +s)=p. Eq. Dla is called the clegetlet-ate sy.ster?z. eq. D 1b the adjoined system. 02. Tihonov ‘s Theorem The solution of the whole (original) system (eqs. D 1) tends to the solution of the degenerate system when p + 0 if the following conditions are fulfilled: (a) the solution -7, = +,(_x,, _ _ _ , x,,. =,. - _ . _ =,).___._,=~~(s,.___._Y,,.T,,___.=_~) is an isolated root of the algebraic system F,(X ,..._._r,.-_ I.‘.‘. =,)=cl (I=1 . . . . . s) (D2) Acknowledgements The author is greatly indebted to the MaxPlanck-Gesellschaft zur Fiirderung der Wissenschaften e-V. for the scholarship for staying in the Max-Planck-Institut fiir Biophysikalische Chemie in Giittingen and would like to express his gratitude to Professors L. De Mayer, M. Eigen and H. Strehlow for the kind hospitality he received during his stay. References I H. Kijima and S. Kijima, Biophys.Chem. 16 (1982) 181. 2 A .N. Tihonov. M at. Sb. N.S. 31 (73) (1952) 575 fin Russian). 3 J.L. M assera. M ath. Rev. 14 (1953) 1085. 4 Y.A. M itropolskir Problemes de la theorie oscillations 5 Yu.M . non-stationnaires Romanowski,N.V. ski, M athematical (i.e.. in the small neighborhood of this root there 1975) (in Russian). modetling asymptotique (Gautiers-V illars, St’epnanova Paris. des 1966). and D.S. Tchemav- in biophysics (Nauka. M oscow. 6 7 8 9 10 11 12 13 A.M. Zhabotinsky, Koncentracionnye kolebania (Nauka, Moscow, 1974) (in Russian). W. Klonowski: Thermodynamics of dissipative structures in enzymatic systems as an example of process modelling in biophysics. IFTR Reports l/1978, Warsaw (in Polish). W. Klonowski, Probl. Contemp. Biophys.. 5 (1980) 199. (PWN. Warsaw) (in Poli -5). J.D. Mur r ay: Nonlinear-differential-equation models in biology (Clarendon Press. 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