44 DETERMINATION OF COMPLEX REACTION MECHANISMS

Fig. 4.14 Plot of the results of a calculation of the steady-state concentration of fructose 6-phosphate for the system shown in fig. 4.13. The enzyme models are either based on Michaelis– Menten formalisms or modifications of multiple allosteric effector equations. The gate exhibits a function with both AND and OR properties. At low concentrations of both inputs, the mechanism functions similarly to an OR gate, while at simultaneously high concentrations of the input species (citrate and cAMP), the output behavior more closely resembles a fuzzy logic AND gate. The mechanism satisfies the requirements for a fuzzy aggregation function. (From [7].)

chemical signals of cellular energy states, such as the adenosine phosphates, cAMP, citrate, and NAD.

For another implementation of electronic devices by means of macroscopic kinetics we mention the construction of frequency filters and signal processing [14], but do not explore this subject further.

We have seen that computations can be achieved by chemical and biochemical reaction mechanisms, and have located computational functions in biological reaction systems. This identification helps in understanding functions and control in such systems. It also helps in suggesting new approaches to the determination of causal connectivities of reacting species, of reaction pathways, and reaction mechanisms, by exploring analogs of investigations in electronics, system analysis [15–17], multivariate statistics [18–20], and other related disciplines. We begin with a study of the response of chemical kinetic systems to pulse perturbations of the concentration of one or more chemical species.

References

[1]Hjelmfelt, A.; Weinberger, E. D.; Ross, J. Chemical implementation of neural networks and Turing machines. Proc. Natl. Acad. Sci. USA 1991, 88, 10983–10987.

[2]Hjelmfelt, A; Weinberger, E. D.; Ross, J. Chemical implementation of finite-state machines.

Proc. Natl. Acad. Sci. USA 1992, 89, 383–387.

[3]Hjelmfelt, A.; Ross, J. Chemical implementation and thermodynamics of collective neural networks. Proc. Natl. Acad. Sci. USA 1992, 89, 388–391.

[4]Hjelmfelt, A.; Schneider, F. W.; Ross, J. Pattern recognition in coupled chemical kinetic systems. Science 1993, 260, 335–337.

[5]Hjelmfelt, A.; Ross, J. Implementation of logic functions and computations by chemical kinetics. J. Phys. D. 1995, 84, 180–193.

[6]Hjelmfelt, A.; Ross, J. Pattern recognition, chaos, and multiplicity in neural networks of excitable systems. Proc. Natl. Acad. Sci. USA 1994, 91, 63–67.

[7]Arkin, A.; Ross, J. Computational functions in biochemical reaction networks. Biophys. J. 1994, 67, 560–578.

[8]Laplante, J. P.; Pemberton, M.; Hjelmfelt, A.; Ross, J. Experiments on pattern recognition by chemical kinetics. J. Phys. Chem. 1995, 99, 10063–10065.

[9]Hauri, D. C.; Shen, P.; Arkin, A. P.; Ross, J. Steady-state measurements on fructose 6-phosphate/fructose 1,6-bisphosphate interconversion cycle. J. Phys. Chem. B 1997, 101, 3872–3876.

[10]Okamoto, M.; Sakai, T.; Hayashi, K. Switching mechanism of a cyclic enzyme system: role as a “chemical diode.” BioSystems 1987, 21, 1–11.

[11]Okamoto, M.; Hayashi, K. Dynamic behavior of cyclic enzyme systems. J. Theor. Biol. 1983, 104, 591–598.

[12]Okamoto, M.; Katsurayama, A.; Tsukiji, M.; Aso, Y.; Hayashi, K. Dynamic behavior of system realizing two-factor model. J. Theor. Biol. 1980, 83, 1–16.

[13]Okamoto, M.; Hayashi, K. Control mechanism for a bacterial sugar-transport system: theoretical hypothesis. J. Theor. Biol. 1985, 113, 785–790.

[14]Samoilov, M.; Arkin, A.; Ross, J. Signal processing by simple chemical systems. J. Phys. Chem. A 2002, 106, 10205–10221.

[15]Klir, G. J. Reconstructability analysis: overview and bibliography. Int. J. Gen. Sys. 1981, 7, 1–6.

[16]Conant, R. C. Extended dependency analysis of large systems, Part II: Static analysis.

Int. J. Gen. Sys. 1988, 14, 125–141.

[17]Conant, R. C. Extended dependency analysis of large systems, Part I: Dynamic analysis.

Int. J. Gen. Sys. 1988, 14, 97–123.

[18]Marriot, F. H. C. The Interpretation of Multiple Observations; Academic Press: New York, 1974.

[19] Mardia, K. V.; Kent, J. T.; Bibby, J. M. Multivariate Analysis; Academic Press: San Francisco, 1979.

[20]Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes in C; Cambridge University Press: New York, 1988.

[21]Ganapathisubramanian, N.; Showalter, K. Bistability, mushrooms, and isolas. J. Chem. Phys. 1984, 80, 4177–4184.

[22]Schacter, E.; Chock, P. B.; Stadtman, E. R. Regulation through phosphorylation/ dephosphorylation in cascade systems. J. Biol. Chem. 1984, 259, 12252–12259.

2.The initial response of the relative concentration of a species in time decreases as the number of reaction steps, as described in point 1, increases.

3.Species directly (not directly) connected by reaction to the initially perturbed species have nonzero (zero) slope.

4.All responses of species in the relaxation back to the stationary state are positive deviations from that state unless there is feedback, or feedforward, or there are higher order (>1) kinetic steps.

5.For short times after a pulse, prior to the exit of the material from the pulse, the concentration change of the pulse is conserved; the sum of deviations of concentrations, weighted by stoichiometric coefficients, must be constant and equal to the change in concentration due to the initial pulse. This property is useful in confirming that all species produced from the pulse due to reactions have been detected, and in determining the correct stoichiometric coefficients of reactants and products.

6.When it is possible to identify rate expressions for reaction steps, then rate coefficients may be estimated from the concentration measurements.

7.Equation (5.8) shows that a maximum in u2 occurs between the relaxation curves of the preceding and succeeding species.

8.A pulse applied to a species in the middle of an unbranched chain, such as species 5 in fig. 5.3(b), propagates well downhill in Gibbs free energy to species 6, 7, and 8 but weekly uphill to species 4 and not past that (for the particular set of rate coefficients chosen for this example).

We are beginning to see from figs. 5.2 and 5.3 and the deductions (points 1–8) how the causal connectivity, at least in this simplest of reaction mechanisms, can be deduced from pulse perturbations.

Further regularities appear for the case of irreversible reactions. Suppose we have the first-order reactions

Division of eq. (5.10) by eq. (5.12) and addition of 1 to both sides (see eq. (5.3)) yields

The relative concentration of X at its maximum value in the relaxation to the stationary state equals the concentration of X1 at that instant of time. This is illustrated for the system in fig. 5.1 for irreversible reactions, that is, the case for which all the reverse rate coefficients are set to zero (see fig. 5.4).

Of course most reactions are not simple first order, but it is important to note that for appropriate isotope experiments the responses to pulses of isotopes are always first order, regardless of the order of the kinetics, second or higher, that is, regardless of the

50 DETERMINATION OF COMPLEX REACTION MECHANISMS

Fig. 5.4 Plot of the relative deviation of concentration in the sequence of reactions in fig. 5.1 for the case of irreversible reactions. (From [1].)

nonlinearities of the kinetics. To show this, we study the kinetics of a species A; the system is in a stationary state, equilibrium or not, and we apply a pulse of radioactive A, labeled A , in such a way that the concentration of the sum of A and A is constant. We also require that the kinetic isotope effect of this radioactive tracer be negligible, that is, the rate coefficients for the reactions of A and A are essentially the same.

We assume the rate of disappearence of A to be given by

We write the stoichiometric equation XA + (ν − x)A = products, x = 1, 2, . . . , ν. Hence the mass action law kinetic equation for the time variation of the isotope A , the rate R , is for the case ν = 2:

Since the sum of the concentrations of A and A are constant, we see that the rate of change of A , that is, R , is linear, first order, in A regardless of the value of ν. This is an important and widely applicable result [6].

Many reaction sequences consist of converging and diverging chains; an example of converging irreversible reactions is shown in fig. 5.5. Calculations of the response of this system to a pulse perturbation of X1 and X8 are plotted in figs. 5.6(a) and (b), respectively. The plots indicate the convergence of two chains at the species X3.

We proceed with the consideration of a linear chain of coupled firstand second-order reactions (fig. 5.7). If species X1 is pulsed, then the relaxation of the various species is shown in fig. 5.8. There are interesting approximate relations for such systems among the amplitudes of changes of relative concentrations. Consider the variation of X2 upon a pulse of X1 administered to the system; the deterministic rate equations are

Fig. 5.5 Chemical reaction mechanism for converging chains of irreversible first-order reactions. The rates of production of species X1 and X8 are held constant at 0.1 and 0.5, respectively. [Xi ]0 denotes the stationary state concentration of species Xi . (From [1].)

Fig. 5.6 Plots of relative deviation in concentration versus time for species of the mechanism in fig. 5.5. In (a), a perturbation of the concentration of species X1 causes a pulse to propagate through the sequence X2, X3, X4, X5; in (b), perturbation of species X8 causes the pulse to branch species X3, and the maximum value (of the relative deviation) of a species occurs on the curve for the species that produces it. For the branch species, the maximum value of u3 is equal to αi ui , where i is 2 or 6, depending on whether the pulse originates from species X2 or X6. The coefficients α2 and α6 are the relative contributions of the branch fluxes to the total flux: α2 = j1/( j1 + j2) and α6 = 1 − α2. (From [1].)

We carry out a linearization by substituting X2 = X2s + δX2, and similarly for X1, to obtain

The terms in Xs cancel due the stationarity condition. At the maximum of the relaxation of X2 after a pulse perturbation, eq. (5.17) is zero; the ratio of the rate coefficients can be related to the ratios of the concentrations at the stationary state, and hence with

52 DETERMINATION OF COMPLEX REACTION MECHANISMS

Fig. 5.7 Chemical reaction mechanism of a linear chain of coupled firstand second-order reactions. The order of a reaction is given by the stoichiometry. (From [1].)

Fig. 5.8 Plots of relative deviation in concentration versus time for species of the mechanism in fig. 5.7. A pulse perturbation of the concentration of species X1 results in the responses shown. The pulse propagates through the chain with maxima of relative deviations of species ordered according to the positions of species in the chain. From the plot we

find the relations: u ≈ (1/2)u1, u ≈≈ 2 ≈ 3

2u2, u6 (1/2)u5, and u7 2u6; the values of these coefficients may be

related to the orders of the reactions through eq. (5.18). (From [1].)

eq. (5.3) we have

This approximate relation, and others, can also be seen in fig. 5.8, and it yields the stoichiometric coefficient 2 of the reaction 2X2 → X3. Thus pulse methods can provide information on the stoichiometry of elementary reactions in a reaction mechanism.

Suppose we have a reaction mechanism with feedback, as shown in fig. 5.9. Here species X7 acts as a catalyst on the reaction X3 → X4. If we perturb species X1, then the relaxation of all species is shown in fig. 5.10(a), in a plot of concentration deviationsX as a function of time; at longer times X3 becomes negative, which indicates the possibility of feedback. This possibility is confirmed in fig. 5.10(b) in which the catalytic species X7 is perturbed. The effect of the catalysis makes X3 negative andX4 positive at once, thus confirming the catalytic effect of species X7 on this reaction.

Other geometries of reaction pathway have been investigated: branched chains, that is, several reaction chains converging or diverging at some species, and cycles of reactions.