Metabolic control analysis in a nutshell

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1 Metabolic control analysis in a nutshell Jan-Henrik S. Hofmeyr Det. of Biochemistry University of Stellenbosch Private Bag X, Matielan 76 Stellenbosch, South Africa jhsh@maties.sun.ac.za ABSRAC Metabolic control analysis is a owerful quantitative framework for unerstaning the relationshi between the steay-state roerties of a (bio)chemical reaction network as a whole an the roerties of its comonent reactions. Although in essence it is a tyical sensitivity analysis of a ynamical system, the stoichiometric structure of reaction networks gives it a character of its own. It has rove very useful for both theoretical an exerimental analysis of cellular systems, leaing to ee insights into matters of control an regulation. his aer attemts to cature in a nutshell the etaile erivation from first rinciles of all of the imortant theorems of control analysis starting with the general kinetic moel of a reaction network.. INRODUCION Like the evolution of life, the eveloment of metabolic control analysis can be likene to a rocess of tinkering. What now stans as the theoretical boy of control analysis is the result of a iecemeal aition to an refinement of theorems resente in the original aers of Kacser an Burns, an Heinrich an Raoort. A lanmark aer on the formalisation of control analysis is by Reer, although others have rovie some formal escrition from first rinciles,, 8, 9,,. At resent the most comlete formalisation can be foun in. One may therefore rightly question the nee for another treatment. However, reinventing the metahorical wheel often yiels new insights, an it is in this sirit that this aer is offere as a journey of iscovery through the algebraical lanscae of metabolic control analysis. We start at the very beginning with the general kinetic moel for a network of chemical reactions, an then rocee ste by ste, oing our best to avoi those unexlaine jums which, although seemingly obvious to exerience mathematicians, leaves us lesser mortals feeling woefully inaequate. Nevertheless, the reaer is at least assume to be acquainte with introuctory ifferential calculus. he great strength an elegance of symbolic matrix algebra is utilise throughout, but there is nothing mysterious about it. he rules of matrix algebra are similar to those of orinary algebra 7, but be minful of two things: (i) two matrices can only be multilie if the number of columns of the first matrix equals the number of rows of the secon (an m n matrix can only be multilie by a n k matrix; the rouct will have imensions m k),an (ii)matrix multilication is not commutative, i.e., the rouct AB is usually not equal to BA. For those that feel more comfortable with exlicit matrix equations, the examle rovie in Aenix B serves as a starting oint. It must, however, be mae clear from the outset that this is neither a literature review, nor a iscussion of the theoretical an exerimental alications of control analysis or the hysical interretation of control roerties. For this the reaer is referre to the original literature, erhas with the excellent monograhs 5 an as oints of earture. Here only selecte key references are sulie.. HE KINEIC MODEL he kinetic moel for any (metabolic) network of coule chemical reactions an transort rocesses can be written as a set of nonlinear ifferential equations (see e.g., ): s Nvs, () where, for a system of n coule reactions that inter-convert m substances (from here on calle metabolites ), s is an m-imensional column vector of metabolite concentrations, N is an m n-imensional matrix of stoichiometric coefficients (the stoichiometric matrix), v is an n-imensional column vector of reaction rates, an is a -imensional column vector of arameters. Only variable metabolite concentrations are inclue in s; metabolite concentrations which are buffere externally an can therefore be regare as constant are consiere to be inclue in the arameter vector. In any systemic state the reaction rates v are functions of both metabolite concentrations s an arameters such as kinetic constants an fixe external concentrations. his is exresse in eqn. by the functional relationshi v vs,. he structure or toology of the reaction network is emboie in the stoichiometric matrix N. Anelementc ij of N is the stoichiometry, usually an integer, with which metabolite S i articiates in reaction j (if S i is a reactant, c ij < ; if a rouct, c ij > ; otherwise, c ij ). wo invariant roerties can be extracte from N, namely (i) the conservation relationshis that arise when the ifferential equations are not all linearly ineenent, an (ii) the steay-state flux relationshis. Here we iscuss the first; the secon will follow once the steay state has been treate. By Gaussian elimination to row echelon form (see, e.g., 7) we can etermine whether the rows of N (an, therefore, the ifferential equations themselves) are linearly ineenent (see Aenix B for an examle). If they are ineenent then r m, wherer is the rank of N (the number of ineenent equations). If r < m then there arem r eenencies among the ifferential equations. Eliminating m r eenent rows of N leaves a reuce stoichiometric matrix, N R, with r ineenent rows. N an N R can be relate by constructing a link matrix L with imensions m r so that N LN R. If N is re-arrange so that the ineenent rows 9

2 come first, then L an the concentration vector s that corresons to the rows of the re-arrange N have the structure Ir si L an s () L s where I r is an r-imensional ientity matrix an L an (m r) r- imensional matrix that exresses the eenent time erivatives in terms of the ineenent time erivatives (see eqn. 6 below); s i refers to ineenent an s to eenent concentrations. Using these relationshis the kinetic moel in eqn. can be written as s LN Rvs i, s, () where the functional relationshi v vs i, s, emhasises the fact that the eenencies among the ifferential equations allows the artitioning of s into r ineenent concentrations s i an m r eenent concentrations s. Now the kinetic moel can be exane into si Ir LN s R vs i, s, N L R vs i, s, () which can be slit into two equations: s i N R vs i, s, (5) s L N R vs i, s, L s i which, when combine, give s L s i (7) It is clear that if L is known we nee only consier the kinetics as exresse by eqn. 5, as eqn. 6 allows the exression of the linear eenencies between the rates of change of metabolite concentrations: (s L s i ) (8) where is a null vector (a vector of zeros). his imlies that s L s i (9) where is an (m r)-imensional vector of constant (conserve) sums of concentrations. he full concentration vector s can therefore be exresse as a function of s i an : si Ir s s s L i Ls i () where reresents an r-imensional subvector of zeros.. Functional relationshis in the steay state In the steay state the kinetic moel s/, an the equations simlify to a system of non-linear equations of the form N R vs i, s, () When there are no conservation relationshis (when r m), the equation system reuces to a slightly simler form Nvs, () but we shall only consier eqn. as it is more general. he solution to eqn. is a vector of ineenent concentrations s i s i, () (6) Note that the concentrations are now steay-state concentrations (as the context is clear we eem it wise not to confuse things by altering the symbol; from here on s, s i,ans only refer to steaystate concentrations). Furthermore, the solution can be exane to a vector of eenent steay-state concentrations, which eens through eqn. 9 on s i an, s s s i, () an a steay-state reaction rate vector J vs i, s, (5) for which we reserve the secial name flux vector. Usually, we cannot solve for the steay-state concentrations an fluxes analytically, although the owerful symbolic maniulators available toay (e.g., Mathematica, Male, Reuce) enlarge the scoe of what is ossible. Excet for the simlest cases, analytical solutions are in any case extremely ifficult to interret. he central question aske by metabolic control analysis is how the steaystate variables change when the steay-state changes in resonse to a erturbation in one or more arameters. In orer to answer this question it is necessary to ifferentiate the steay-state equations with resect to the arameters, an for this we must have an accurate icture of the functional relationshis in these equations. A iagrammatic reresentation makes these neste functional relationshis more transarent: s i v s It is clear that the steay-state concentrations s i an s as well as the steay-state fluxes J, ultimately een only on the arameters an the conservation sums. Nevertheless, the intermeiary levels of functional eenencies are imortant when the steay-state equations are ifferentiate with resect to an. However, before we turn to this toic we comlete our structural analysis of N by consiering the relationshis that exist between fluxes in the steay state.. Flux-relationshis in the steay state We showe above that linear eenencies among the rows of N can be cature in the link matrix L. Similarly, in the steay state when N R v (or Nv, if there are no conservation relationshis) their exist eenencies among the columns of N (or N R ) that can be exresse as NK or N R K (6) where K is the kernel (or nullsace) of N. Each column of K is a articular solution to eqn. 6, an the set of columns are linearly ineenent an therefore san the nullsace. Because each column reresents an ineenent flux, it follows that: J KJ i (7) where J is an n-imensional column vector of all the steay-state fluxes, an J i is an (n r)-imensional column vector of ineenent fluxes (recall that r isthe rank ofthe stoichiometricmatrix). K 9

3 therefore has imensions n (n r). If K is re-arrange so that the n r rows that correson to ineenent fluxes come first, then the flux vector J is artitione into n r ineenent fluxes J i an r eenent fluxes J, an eqn. 7 becomes Ji In r J J K i (8) where I is an (n r)-imensional ientity matrix an K an r (n r)- imensional matrix that exresses the eenent fluxes in terms of the ineenent fluxes, J K J i.. DIFFERENIAION OF HE SEADY- SAE EQUAION Because the arameters an etermine the steay-state, any change in these arameters can otentially change the steay state. If the arameter erturbation is small enough, the change from steay state (s, J ) to steay state (s, J ) is aroximate by s s s ( ) (9) an J J J ( ) () the first two terms in the so-calle multivariate aylor exansion. Similar equations for erturbations in can be obtaine by relacing by. he matrices of artial erivatives s/, J/, s/, an J/ are thus of great interest an will be obtaine next. Reaers will be familiar with the ifferentiation of an exlicit mathematical equation of the form y f (x) with resect to x,orif,asis the case here, y is a function of more than one variable y f (x, z), artial ifferentiation with resect to either x or z or both at the same time. However, we are working not with simle scalar variables, but with vector variables. In aition, equations such as eqn. are imlicit functions. Fortunately, the extension to vector variables an imlicit functions is not ifficult at all. Let us first get a feel for this rocess by ifferentiating eqn., the exlicit function s L s i,. Weusetheartoftheiagram of functional relationshis given above that starts at s as a guie. First, let us artially ifferentiate s with resect to s i (at constant an ): ( ) s L () s i, When we ifferentiate with resect to at constant there are two routes from s to, one via s i an one irect: ( ) ( ) ( ) s s si s i, Inserting eqn. we obtain ( ) s ( s ) s i, () ( ) si L () Note how we use the chain rule to follow the ifferent branches of the tree structure of the iagram of functional relationshis. Because we are ifferentiating with resect to vectors, all the imensions have to be consistent; this is why an ientity matrix aears rather than a. Finally, we ifferentiate with resect to at constant : ( ) s ( ) s s i, ( ) ( ) si si L () We shall nee all three of these equations when we next rocee to ifferentiate the kinetic moel in steay state. We first ifferentiate with resect to at constant, an then with resect to at constant. Case :, Eqn. is an imlicit equation of the form f (x, y, z). In general, to obtain, say, y/ x one coul in rincile solve for y an artially ifferentiate with resect to x while keeing z constant. his is often imossible, but there is, fortunately, a much simler way aroun this, namely imlicit ifferentiation ( ) ( ) ( ) f f f f x y z (5) x y,z y x,z z x,y where f is calle the total ifferential. If only one variable, say x, is consiere to change at constant y an z, then one obtains ( ) ( ) ( ) ( ) ( ) f f y f z (6) x y x z x y,z x,z o ifferentiate Eqn. with resect to we use this technique in combination with the chain rule to traverse the three routes from v to on the iagram of functional relationshis: ( ) ( ) ( ) ( ) ( ) v si v s si N R s i s,, s s i,, s i, ( ) v (7) where is a null matrix. Inserting eqn. an collecting the first two terms in the square brackets we obtain ( ) ( ) v v I N r si v R s i s N L R (8) By efinition the column vector of matrices is L. he artitione matrix v v s i s is the matrix ( v/ s), of artial erivatives of reaction rate functions with resect to the iniviual concentrations ins. In control analysis these artial erivatives are calle elasticity coefficients, efineas ε v k s j v k / s j. he above matrix of elasticity coefficients is symbolise with ε s. Similarly, ( v/) is a matrix of elasticity coefficients with resect to, symbolise by ε. he bar in ε remins us that these are orinary artial erivatives, not the normalise (scale) artial erivatives which are usually use in control analysis an which we shall encountere further on (an for which we shall use the unbarre symbol; this istinction between barre (non-normalise) an unbarre (normalise) symbols is mae throughout the aer for all the coefficients of control analysis an their matrices). Using this symbolism eqn. 8 is written as ( ) si N R ε s L N R ε (9) z he steay-state eqn. 5 for fluxes is exlicit. Differentiation with resect to yiels: ( ) ( ) ( ) J v si s i s,, ( ) ( ) ( ) ( ) v s si v () s s i,, s i, his exression is ientical to the sum of terms within the square brackets of eqn. 7. Substituting an collecting terms as before x,y y 9

4 gives ( ) ( ) J si ε s L ε () which simlifies to s ( LM N R ) ε () Case :, Now we imlicitly ifferentiate eqn. with resect to at constant. On the iagram of functional relationshis there are two routes from v to ; one of them branches at s into two subroutes to. We ifferentiate in two stes to avoi getting things mixe u first we ifferentiate to the level of s i an s : ( ) ( ) ( ) ( ) v si v s N R () s i s,, s s i,, Now we must take care of the two routes from s to. In fact, we have alreay one this in eqn. so we just substitute: ( ) ( ) ( ) ( ) v si v si N R L () s i s,, s s i,, Multilying out an collecting the first two terms as before gives ( ) si N R ε s L ε s () Exlicit ifferentiation of eqn. 5 with resect to at constant gives: ( ) ( ) ( ) ( ) ( ) J v si v s (5) s i s,, s s i,, We rocee exactly as in eqns. an to obtain ( ) ( ) J si ε s L ε s (6) his conclues the ifferentiation of the steay-state equations.. MEABOLIC CONROL ANALYSIS We are now in a osition to erive the basic efinitions of an relationshis between all the ifferent matrices of the coefficients of metabolic control analysis (the boxe equations in the rest of this section).. Concentration resonse with resect to he matrix rouct N R ε s L, which aears in eqn. 9, is the socalle Jacobian matrix, which we symbolise with M. he nature an significance of the Jacobian matrix is exlaine in Aenix A. Here we just note that if a steay state exists the Jacobian matrix is invertible. From here on, for the sake of brevity, we leave out the subscrits that inicate which vectors remain constant uring ifferentiation. In this sirit, eqn. 9 is now written as s i M N R ε (7) From eqn. it follows that s L s i L M N R ε (8) Combining the eqns. 7 an 8 gives s i s Ir L M N R ε (9) We have therefore obtaine the first of the matrices of artial erivatives that we seek. he elements of this matrix are concentrationresonse coefficients, efineasr s j k s j / k ; they quantify the steay-state resonse in a metabolite concentration s j to a erturbation in arameter k. he matrix s/ will be symbolise by R s.. Concentration-control coefficients What are the elements of the matrix LM N R, or in more exlicit form, L(N R ε s L) N R? Besies L an N R, which are integer matrices, this matrix contains only artial erivatives of the rates with resect to the steay-state concentrations, i.e., elasticity coefficients; nothing in the matrix eens exlicitly on or. Now, consier a set of arameters such that each uniquely affects a single reaction, i.e., in the elasticity matrix ε v k k an v k l for k l () his means that ε is now a iagonal matrix (only the iagonal elements v k / k are non-zero). he inverse of a iagonal matrix is again a iagonal matrix but with all iagonal elements in recirocal form. herefore, by multilying both sies of eqn. with this inverse leas to a matrix exression for LM N R in which an element in row j an column k is s j / v k () k k his is, in fact, the funamental efinition of a concentration-control coefficient: C s j k s j k / v k k () he matrix LM N R is therefore the matrix of concentration-control coefficients C s LM N R () he matrix s/ can be recognise as a matrix of concentrationresonse coefficients R s. Eqn. is therefore a statement of the artitione concentration-resonse roerty of metabolic systems: R s C s ε (5). Flux-resonse an control coefficients Differentiation of the steay-state flux equation with resect to le to eqn.. Inserting eqn. 7 we obtain J ε ( ) s LM N R ε ε (6) From eqn. we recognise the brackete term as C s,sothat J ( ε ) s C s I n ε (7) Similar to the revious section, an element in the ith row an kth column of ε s C s I n can be seen to be a flux-control coefficient of reaction k: C J i k J i k / v k k (8) 9

5 so that the matrix of flux-control coefficients is efine as C J ε s C s I n (9) he matrix J/ can be recognise as a matrix R J of flux-resonse coefficients, iniviually efine as R J i k J i / k, that quantify the steay-state resonse in a flux J i to a erturbation in arameter k. Eqn. 7 is therefore a statement of the artitione flux-resonse roerty of metabolic systems: R J C J ε (5). Concentration-resonse with resect to Differentiation of eqn. with resect to le to eqn., which, if s i / containing concentration-resonse coefficients with resect to the conservation sums is symbolise by R s i, can be written as N R ε s L R s i N R ε s (5) which, using N R ε s L M an re-arranging gives R s i M N R ε s (5) Eqn. can be written as R s L R s i (5) which, inserting eqn. 5, is R s L M N R ε s (5) Combining eqns. 5 an 5 gives R s i Ir R s M L N R ε s (55) which reuces to R s LM N R ε s As LM N R C s we get R s C s ε s (56) (57) Finally, we refer to write this equation in terms of the full elasticity matrix. his can be one if we realise that herefore ε s ε si ε s ε s R s ( C s ε s I m ) (58) (59) he righthan matrix singles out the eenent metabolites that are each unique to a ifferent conservation equation, ensuring that each conservation sum is erturbe ineenently..5 Flux-resonse with resect to Differentiation of eqn. 5 with resect to le to eqn. 6, which, if J/ containing flux-resonse coefficients with resect to the conservation sums is symbolise by R J, can be re-written as R J ε sl R s i ε s (6) We can now insert the exression for R s i in eqn. 5 to give R J ε s( LM N R ) ε s ε s (6) As C s LM N R we obtain R J ( ε C s s I n ) ε s (6) he brackete term is the efining exression for C J (eqn. 9). herefore, R J C J ε s (6) As before, we rather write this in terms of the full elasticity matrix R J C J ε s (6) It is ossible to exress R J in terms of R s. Inserting eqn. 9 into eqn. 6 gives R J ( ε C s s I n ) ε s (65) Multilying the first RHS rouct out an recollecting terms gives R J ε s( C s ε s I m ) (66) Using eqn. 59 we finally get R J ε R s s (67).6 Normalising the central equations In control analysis the use of the imensionless normalise form of the control an elasticity coefficients is generally referre,. With one trivial excetion, the basic equations eveloe in the revious sections look exactly the same in normalise form, rovie that the K, L, N R an M-matrices an are scale aroriately. o o this we efine the iagonal matrices D J an D s which resectively have the steay-state fluxes an concentrations on their iagonal (justas with the coefficient matrices the flux an concentration vectors are arrange so that the ineenent variables come first, the eenent variables secon). heir inverses (D J ) an (D s ) have inverse fluxes an inverse steay-state concentrations on their iagonals. Similarly, we efine D s i, D,anD. Using these iagonal matrices, the matrices that occur in the control-matrix equation are scale as follows (note that the absence of a bar enotes normalise matrices): C J (D J ) C J D J (68) C s (D s ) C s D J (69) ε s (D J ) ε s D s (7) R J (D J ) R J D (7) R s (D s ) R s D (7) R J (D J ) R J D (7) R s (D s ) R s D (7) L (D s ) L D s i (75) K (D J ) K D J i (76) N R (D s i ) N R D J (77) M (D s i ) M D s i (78) he equations central to control analysis erive above (the boxe equations) are now summarise in the normalise format: 95

6 Matrix efinition of concentration-control coefficients C s LM N R (79) Matrix efinition of flux-control coefficients C J ε s C s I n (8) Partitione concentration-resonse roerty with resect to arameters : R s Cs ε (8) Partitione flux-resonse roerty with resect to arameters : R J CJ ε (8) he artitione resonse roerties with resect to iffer slightly from the non-normalise eqns. 59 an 6 in that the ientity submatrix in the righthan matrix is relace by a matrix containing inverse mole fractions of the eenent metabolites on its iagonal. he artitione concentration-resonse roerty with resect to is: R s (Cs ε s I m ) D s (8) Partitione flux-resonse roerty with resect to : R J CJ ε s (8) Relationshi between R J an Rs : D s R J ε sr s (85).7 Summation theorems he summation equations for flux an concentration control coefficients follow irectly from the efinitions of C s (eqn. 79) an C J (eqn. 8) an the relationshi N R K (the normalise form of N R K ). he first is calle the summation theorem for concentration-control coefficients: C s K L(N R ε s L) N R K (86) an the secon the summation theorem for flux-control coefficients: C J K (ε s C s I n )K K (87).8 Connectivity theorems he flux an concentration connectivity equations follows from the invertibility of the Jacobian matrix N R ε s L (the normalise form of N R ε s L). Multilying C s an C J by ε s L gives, first, the connectivity theorem for flux-control coefficients: C s ε s L L(N R ε s L) N R ε s L L (88) an, secon, the connectivity theorem for flux-control coefficients: C J ε s L (ε s C s I n )ε s L (89) ogether, the summation an connectivity theorems allows the exression of control coefficients in terms of elasticity coefficients. his is arguably the most owerful feature of metabolic control analysis an is treate next..9 he control-matrix equation It is ossible to combine the summation an connectivity theorems into a generalise matrix form, which we call the control-matrix equation. Quite a few ermutations of such an equation have been suggeste,, 6, 7, 8,,,,, 6, 9,, but the one that follows arises naturally from the formalism eveloe in this aer 6, 8. Furthermore, it simlifies to the form C i E I (see below), which shows exlicitly how the matrix exressing ineenent systemic roerties, C i, an the matrix exressing structural an local roerties, E, are inverses of each other (if the rouct of two square matrices equals the ientity matrix, then they are inverses of each other). his means that control coefficients can be calculate from elasticity coefficients, C i E, an vice versa, E (C i ) (the last case being strictly true only if there are no conservation equations; see next section). he result will once again stress the funamental role of the K an L-matrices in control analysis. he control-matrix equation is forme by combining eqns as 6, 8: C J K C s εs L K (9) L he matrices can be artitione in terms of ineenent an eenent variables to give C J i C J C s i K εs L I n r K (9) I r C s L Extracting the equations for the ineenent variables J i an s i gives: C J i C s i K εs L In r (9) I r which, if C i C J i C s i an E K ε s L, reuces to the articularly elegant form: C i E I n (9) Both C i an E are square invertible n n matrices, 6, i.e., the equation can also be written as EC i I, which exresses fluxcontrol coefficients in terms of concentration-control an elasticity coefficients. hese equations are comletely general an hol for any network of reactions.. he inverse roblem We have seen that C i E : if all the elasticity coefficients have been etermine (either exerimentally or by calculation as the normalise artial erivatives of the rate equations), the control coefficients with resect to the ineenent concentrations an fluxes can be calculate by inverting E. he control coefficients with resect to the eenent variables are calculate using the relationshis: C s L C s i (9) C J K C J i (95) which follow from eqns. an 8. However, consier the inverse roblem, i.e., calculating the elasticity coefficients from exerimentally etermine control coefficients. Using E (C i ) we can calculate E by inverting C i. If L I, i.e., if there are no conservation constraints, the task is accomlishe the righthan r columns of E form the elasticity matrix ε s an therefore contain the values of the elasticity coefficients. However, if L I, some elements in the righthan r columns of E contain linear functions of elasticity coefficients, an more information is neee to solve for the iniviual elasticities. 96

7 his extra information can only be obtaine by erturbing the conservations sums in the column vector an measuring the resulting steay-state changes in all the fluxes an concentrations. We augment on the left both sies of eqn. 85: R J ε sr s (96) with the matrix ε s L to give ε s L R J ε sl ε s R s (97) which can be re-arrange to solve for ε s : ε s ε s L R J L Rs (98) he n n matrix L R s has been rove to be invertible. 5. DISCUSSION his aer set out to rovie, in a nutshell, the comlete formal basis for metabolic control analysis in a way that leaves as little as ossible unexlaine. In articular, care has been taken to show how the functional relationshis in the steay-state equations hang together, thereby roscribing how the steay-state equations shoul be ifferentiate. For a more extensive exosition of much of the material covere in this aer the reaer is referre to the excellent monograh by Heinrich an Schuster, which is a treasure trove of information on biochemical moelling an control analysis. However, there are asects covere here which are either absent from their treatment (the resonse to in Sections. an.5, an the inverse roblem in.) or ifferent (Sections.7,.8, an.9, where full scaling is use instea of the artial scaling use in ). Metabolic control analysis has been alie to many tyes of systems, which has le to interesting extensions of the theory, for examle, multi-level or hierarchical systems,, moular systems, 5, signal transuction athways 6, time-eenent henomena,, transition times 8, oscillating systems, channelle systems 5, an grou-transfer athways 7. Co-resonse analysis 6 is an extension built on the control-matrix equation escribe in this aer. It not only has useful exerimental imlications (control analysis requiring neither kinetic knowlege of the comonent reactions nor quantitative information about the magnitues of the effects of erturbations on iniviual enzyme activities), but also forms the basis for the analysis of regulatory asects of metabolism (for examle, suly-eman analysis, 5, 7). 6. ACKNOWLEDGEMENS First I must thank the many stuents who serve as critical eitors of this material in a number of workshos hel in the ast ecae, secon my colleagues Johann Rohwer, Jacky Snoe, Athel Cornish-Bowen an Hans Westerhoff for many iscussions. 7. REFERENCES L. Acerenza. Metabolic control esign. J. theor. Biol., 65:6 85, 99. M. Cascante, R. Franco, an E. I. Canela. Use of imlicit methos from general sensitivity theory to evelo a systematic aroach to metabolic control. I. Unbranche athways. Math. Biosci., 9:7 88, 989. M. Cascante, R. Franco, an E. I. Canela. 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9 If the number of ineenent concentrations is less than the number of metabolites then by efinition N LN R an s Ls i,or equivalently, δs Lδs i. Using the argument of eqns. 6 it follows that (δs i) N R ε s ol δs i (6) From the general efinition of the Jacobian matrix given in the first aragrah of this section we see that M N R ε s ol (7) is the Jacobian matrix. he Jacobian matrix can be normalise as follows: Define the iagonal matrices D vo, D so an D so i which resectively have the reaction rates, concentrations an ineenent concentrations obtaining at state s o on their iagonal (the rate an concentration vectors are arrange so that the ineenent variables come first, the eenent variables secon). heir inverses (D vo ),(D so ) an (D so i ) have the inverse rates an inverse concentrations on their iagonals. Using the ientities δ ln s i (D so i ) δs i (8) N R (D so i ) N R D vo (9) ε s o (D vo ) ε s o D so () L (D so ) L D so i () eqn. 6 can be written as (δ ln s i) N R ε s ol δ ln s i () he kinetic moel for erturbations from state s o has therefore been transforme to logarithmic sace. From this formulation the normalise Jacobian matrix is seen to be M N R ε s ol () which, if there are no eenent metabolites (L I) simlifies to M Nε s o () B. AN EXPLICI EXAMPLE Fig. reresents a simle reaction network containing both a branche flux an a moiety-conserve cycle 9. Here we show how the K an L-matrices can be constructe from an analysis of its stoichiometric matrix. Once these matrices are available it is a simle matter to formulate the control matrix equation C i E I exlicitly (for a numerical solution of this examle see 6). X X7 S S S X6 X X 5 Figure : A examle reaction network. he first ste is to write own the stoichiometric matrix N for this system, labelling the rows an columns (the left-han matrix in eqn. 5). N is then augmente with an ientity matrix in which each column reresents a time erivative (the right-han matrix in eqn. 5). Note that only variable metabolites S i are reresente. he terminal X-metabolite concentrations must be constant (at non-equilibrium values) for a steay state toexist,an are therefore consiere art of the arameter set. R R R R ṡ ṡ ṡ S S S (5) Next the augmente matrix is subjecte to Gaussian elimination to row echelon form : R R R R ṡ ṡ ṡ S (6) S S he rank of the stoichiometric matrix is an there is one conservation relationshi s s,where is the conserve sum. here are two ineenent fluxes an two ineenent metabolites. Choosing J an J (the R an R -columns without ivots in the reuce stoichiometric matrix N R in eqn. 6) as the ineenent fluxes, the K-matrix follows from the flux relationshis J KJ i : J J J J J J J J J J J J K is scale to K (D J ) KD J i as follows: J J J J J J J J J J J J J J J J (7) (8) Either S or S can be chosen as the eenent metabolite. We choose S.heL-matrix follows from the relationshis in s/ Ls i /, which are rea off from the last row of the righthan matrix in eqn. 6: ṡ ṡ ṡ ṡ ṡ ṡ ṡ ṡ L is scale to L (D s ) LD s i as follows: s s s s s he matrix rouct ε s L for this system is ε v s ε v s ε v s ε v s ε v s s s ε v s ε v s s s ε v s ε v s ε v ṡ s (ε v s s s ε v (ε v s s s ε v s ) s ) ṡ (9) () () Simle systems can be analyse by han, more comlicate systems with one of the numerous comuter tools that o this automatically, such as Mathematica, Matlab, etc. or eicate metabolic simulators (e.g., 9,, 8, Metatool ft://mushark.brookes.ac.uk/ub/software/ibmc/metatool) 99

10 Note that the re-orering of fluxes in the K-matrix is reflecte in ε s. Finally, the C i E I control-matrix equation is constructe using K an ε s L: C J C J C J C J ε v s C J C J C J C J ε v s C s C s C s C s J J C s C s C s C s J J ε v s (ε v s s s ε v s ) J J J J (ε v s s s ε v s ) () o solve the inverse roblem (ε s from (C i ),R s,anrj ) we nee to construct eqn. 98. here is only one conserve sum s s. he matrix rouct ε s L is known from (C i ),sothat ε s ε v s R J ε v s R J ε v s (ε v s ε v s s s ) R J (ε v s ε v s s s ) R J R s R s s s R s ()