Analysis of the fluctuations about the steady state

In our scheme this ‘material’ aspect of the reaction function is emphasised, with the arrows describing relations between metabolites and reac- tions i.e. the properties of being either a substrate or a product. Very often metabolism is also described in terms of ‘pathways’. A common- sense definition of a pathway is quite obvious, namely it is a sequence of reactions sharing a common mass flow. An example of such a path- way is given by the thick line running through Fig. 1. The time evolution for the metabolic concentra- tions can be conveniently expressed in terms of a ‘time-evolution’ operator. This operator acts on the concentration vector for the initial conditions i.e. the vector comprising the values of the con- centrations at time t = 0 to produce the concen- tration vectors at a later time t. Our goal in developing the theory is to write the time-evolu- tion operator for the system of metabolic equa- tions as a superposition of operators representing the propagation along all the possible pathways of the given metabolic system. Then the weights of the pathways would be functions of time, rep- resenting the relative ‘importance’ of the path- ways at a given moment of time. We shall develop this into a systematic diagrammatic procedure for evaluating the variations in time of the metabo- lite concentrations.

2. Analysis of the fluctuations about the steady state

We start with the rate equation written in the following general form: dx dt = vx 1 where x denotes the vector of concentrations, while vx = {6 i {x j }} stands for the vector com- prising the rates of change of the concentrations. , we have by definition dx dt = 2 and hence vx = 0. Typically, a system of coupled equations can be efficiently characterised by the time evolutions of small fluctuations around the steady state. Such an approach, for example, forms the basis for the theory of fluctuations in statistical physics e.g. Landau and Lifschits, 1979, as well as the metabolic control analysis MCA method in bio- chemistry e.g. Kacser and Burns, 1973. The analysis of the time evolution of fluctuations, and the correlations between them, can help in deter- mining the stability of the steady or equilibrium state against changes in the external conditions. Indeed, provided the steady state x is stable, the fluctuations in concentration which may initially be present in the system at t = 0 would tend to vanish in the limit of t “ . By contrast, an unstable steady state would yield growth of the initially small fluctuations. Here we discuss the behaviour of small fluctua- tions in the concentrations around the steady state of a metabolic system. For this purpose we represent the state of the system as x t = x + n t vx : vx + D · n 3 where the operator D can be regarded as a con- stant in time N × N matrix with the elements D ij = 6 i x j x = x 4 Also n is an N-dimensional fluctuation vector defined by n = Æ Ã Ã Ã È n 1 n 2 … n N Ç Ã Ã Ã É = N i = 1 n i e i 5 1 œe 2 œ … œe N = I N the N × N identity matrix. The equation of mo- tion for n then reads: dn dt = D · n 6 which yields a formal operator solution n t = e D t · n0 7 for the time evolution of the fluctuations. It is useful also to introduce a correlation matrix C as the matrix whose elements are C ij = e i e D t e j 8 This quantity characterises the correlations be- tween the fluctuations of concentrations of the different metabolites. Specifically Eq. 8 charac- terises what would be the contribution to the concentration of the ith metabolite at time t pro- duced by a singular perturbation of the jth metabolite at zero time. The appropriately defined correlation function takes account of the interactions present in the system, and it yields the characteristic times associated with these interactions. We proceed next to the pathway representation as a means to carry out the evaluation of the above quantities. Eq. 7 may first be expanded as a power series with respect to time as n t = r = 0 t r r D r · n0 9 We may then decompose the vector v and the matrix D into sums of terms corresponding to the different elementary metabolic reactions, giving 6 i = M m = 1 h m i 6 m 10 Here 6 m denotes the rate of the mth reaction and h m i is the stoichiometric coefficient of metabolite i in this reaction. Also, we may write D = M m = 1 D m , with D ij m = h m i 6 m x j x = x = h m i o j m 6 m x j 11 where o n t = r = 0 t r r C r D m 1 · … · D m r · n0 12 Here the C r are ordered sets of indices {m 1 , m 2 , …, m r }, where each index labels one of the M different reactions. What we call here, for histori- cal reasons, the ‘Handscomb representation’ is, in fact, a series expansion that forms the foundation of any diagrammatic technique in statistical me- chanics see Rivers, 1987. The operator products in Eq. 12 represent the contributions from different reaction sequences. The expansion may now be used to rewrite Eq. 8 as C ij = r = 0 t r r C r {D m 1 · … · D m r } ij 13 In terms of the matrix elements this result becomes C ij = r = 0 t r r C r N k 2 , …, k r − 1 k = i, k r = j 5 r m = 1 D k m − 1 k m m m 14 In order to proceed further we now make some assumptions about the reaction rates in Eq. 1 and the corresponding matrix elements appearing in Eq. 14. We should note that these, or similar, assumptions are typical of previous MCA theories see Cornish-Bowden, 1995. 1. We assume that the reaction rate for any reaction m m does not depend on the concentra- tion of those metabolites that do not partici- pate in this reaction. Thus a necessary condition for any term in the expansion Eq. 14 to be nonzero is: Öm, k m , k m − 1 m m , 15 which means that k m and k m − 1 are each either a substrate or a product of the reaction m m. Fig. 2. A graphical representation of the terms contained in Eq. 18, a a D km m matrix element with k m an arrow; b a ‘diagonal’ D km m matrix element a loop. Using Eqs. 15 – 17 we now may rearrange the terms in Eq. 14 to give C ij = r = 0 t r r G r D ij G r − 1 n ss G r + n pp G r + n loop G r 5 r m = 1 D k m j m m m , 18 where it is now the absolute values of the D coefficients that appear. These terms can be repre- sented diagrammatically as shown in Fig. 2. In Eq. 18 we introduce new objects, ordered index sets, G r = {i 1 , …, i r } in which each index i m corre- sponds to a ‘path’ from a ‘pool’ vertex k i m through a ‘reaction’ vertex m i m to another ‘pool’ vertex, j i m , with a connectivity requirement that ÖG r , Öm, k i m = j i m − 1 19 Also the function D ij G r = 1 if G r , starts from the pool vertex i and ends at the pool vertex j otherwise D ij G r = 0. The other new notations and terminology are explained below. Graphically G r , can be represented as a ‘route’, or a pathway, on the metabolic graph. Each G r , is a connected ordered graph, i.e. it can be drawn without lifting a pen from the paper, which is a direct consequence of Eq. 19. It contains its own sequence of arrows, corresponding to the direc- tion, say, from a pool vertex k to pool vertex j so that a distinction is made between D kj m and D jk m , as well as ‘single index loops’ for the terms of the type D jj m . From now on we will use the term M-arrows for the arrows on a metabolic graph proper, that correspond to the substrate – product relations, and we will name G-arrows the arrows that specify the direction in a G r , graph. An example of a G r , graph with r = 4 for part of the metabolic graph in Fig. 1 is presented in Fig. 3, its contribution being Fig. 3. A sample G-graph comprising a part of the metabolic graph in Fig. 1. The M-arrows are represented by solid fines. 2. We assume that any reaction rate is inhibited by an accumulation of products and en- hanced by an accumulation of substrates see also a study on signs of control coefficients by Sen 1996. This implies the following sign rules: h m k 6 m x k x = x B 0. 17 It should be noted that, although the above assumptions are required for the version of dia- grammatic technique to be formulated here, some of the conditions allow straightforward generali- h m k 6 m x j x = x à à à \ 0, if k is a substrate of m and j is a product, or vice versa B 0, if j and k are both either substrates or products of m ; 16 Also n ss , and n pp in Eq. 18 stand for the total numbers of G-arrows in G r , which connect, re- spectively, substrate to substrate e.g. the G-ar- rows 3 “ 4 and 4 “ 5 in Fig. 3 and product to product, while n loop stands for the number of looped G-arrows in G r , e.g. single index loop 3 “ 3 in Fig. 3. The overall negative signs in Eq. 18 that are associated with each loop, substrate – substrate and product – product G-arrows are con- sequences of Eq. 16. Now we can formulate the rules for calculating the individual terms of Eq. 18 on any complete metabolism graph. The recipe is as follows: 1. Draw the complete metabolic scheme, with the reactions and pools depicted by vertices, as explained above i.e. like in Fig. 1. 2. Define an entry point i at a corresponding ‘pool’ vertex, and an exit point at jth ‘pool’ vertex. 3. Draw all possible pathways G r , that start at i and end at j, allowing for possible repetitions i.e. a bouncing back and forth between the reaction and pool vertices and loops around pool vertices. 4. For each pathway count the n ss , n pp and n loop quantities. 5. Finally, write the contribution for a given pathway as: p G r = t r r − 1 n ss G r + n pp G r + n loop G r 5 r m = 1 D k m j m m m 21

3. Monte-Carlo evaluation of the relative contributions of different metabolic pathways