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

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

We now rewrite the preceding formalism in a way that will be suitable for applying Monte- Carlo numerical simulations. First Eq. 20 can be re-expressed in terms of a ratio of two mean values: C ij k,l C kl = r = 0 G r D ij G r jG r pG r r = 0 G r j G r pG r r pG r j G r = − 1 n ss G r + n pp G r + n loop G r ; pG r = pG r r, G r pG r ; 23 We note that the set of functions pC r has the properties: 1. ÖG r , pG r ] 0 24 2. ÖtB, r = 0 C r pG r = 1 25 These justify our usage of the functions pG r as probability distribution in Eq. 22, where the angular brackets denote an average value taken with respect to the distribution pG r . A general Monte-Carlo procedure to calculate the ratio in Eq. 22 may now be implemented as follows: 1. We set up a Markov chain in the space of all possible pathways G r with a limit distribution given by pG r . 2. At each step we generate a certain pathway G r . 3. We calculate its overall sign contribution j G r . 4. We add jG r to one summator, B. 5. If G r starts at a pool vertex i and ends at a pool vertex j, then we add jG r to another summator, A this accounts for theD ij G r factor. 6. After a predefined and substantial number of steps we calculate Eq. 22 as: C ij k, l C kl = A B 26 As regards methods for a practical implementa- tion of the Markov chain, the most obvious can- didates may be either some sort of ‘percolation’ procedure see Hinrichsen and Koduvely, 1998 or a Handscomb-type procedure Handscomb, 1962; Rojdestvenski and Cottam, 2000. The latter im- plies sampling the space of all G r graphs via Markov process consisting of steps that increase r, G r “ G r D k r + 1 j r + 1 m r + 1 G r = G r − 1 D k r j r m r “ G r − 1 Unfortunately, these straightforward ap- proaches are impractical in many cases. The rea- son is the well known ‘sign problem’, which is frequently encountered in theoretical physics ap- plications that deal with diagrammatic expansions see Rozhdestvensky and Favorsky, 1992. In- deed, the numerator and denominator in Eq. 23 may be infinite sums comprising terms with alter- ing signs. Different signs arise here because of the rules Eqs. 16 and 17. The numerical values of such sums are typically determined with consider- able uncertainty and their ratios may be com- pletely ill defined. Such a situation in the quantum Monte-Carlo simulations of antiferromagnetic systems is described, for example, by Rojdestven- ski 1995. Some possible ways to deal with this problem are either to concentrate on specific cases where this sign problem does not appear as in Handscomb, 1964 or to choose a different repre- sentation of variables as in Rozhdestvensky and Favorsky, 1992. We will discuss, in fact, a combi- nation of these two approaches in application to the current problem.

4. Two types of branching in metabolic graphs