Kinetics fractal

H. Berry, Monte Carlo simulations of enzyme reactions in two dimensions Fractal kinetics and spatial segregation. Biophys. J. 83(4), 1891 1901 (2002). 

M. A. Savageau, Development of fractal kinetic theory for enzyme catalysed reactions and implications for the design of biochemical pathways. BioSystems 47(1 2), 9 36 (1998). 

Therefore, the estimation 0 im problem brings to the question of fractal dimension Df determination. At present two methods of indicated dimension determination one exist. First method consists of using of chemical reactions fractal kinetics general relationship [9] ... 

Before we close this section some major, unique kinetic features and conclusions for diffusion-limited reactions that are confined to low dimensions or fractal dimensions or both can now be derived from our previous discussion. First, a reaction medium does not have to be a geometric fractal in order to exhibit fractal kinetics. Second, the fundamental linear proportionality k oc V of classical kinetics between the rate constant k and the diffusion coefficient T> does not hold in fractal kinetics simply because both parameters are time-dependent. Third, diffusion is compact in low dimensions and therefore fractal kinetics is also called compact kinetics [23,24] since the particles (species) sweep the available volume compactly. For dimensions ds > 2, the volume swept by the diffusing species is no longer compact and species are constantly exploring mostly new territory. Finally, the initial conditions have no importance in classical kinetics due to the continuous re-randomization of species but they may be very important in fractal kinetics [16]. 

These properties are likely to have an important influence on the behavior of intact biochemical systems, e.g., within the living cell, enzymes do not function in dilute homogeneous conditions isolated from one another. The postulates of the Michaelis-Menten formalism are violated in these processes and other formalisms must be considered for the analysis of kinetics in situ. The intracellular environment is very heterogeneous indeed. Many enzymes are now known to be localized within 2-dimensional membranes or quasi 1-dimensional channels, and studies of enzyme organization in situ [26] have shown that essentially all enzymes are found in highly organized states. The mechanisms are more complex, but they are still composed of elementary steps governed by fractal kinetics. 

The power-law formalism was used by Savageau [27] to examine the implications of fractal kinetics in a simple pathway of reversible reactions. Starting with elementary chemical kinetics, that author proceeded to characterize the equilibrium behavior of a simple bimolecular reaction, then derived a generalized set of conditions for microscopic reversibility, and finally developed the fractal kinetic rate law for a reversible Michaelis-Menten mechanism. By means of this fractal kinetic framework, the results showed that the equilibrium ratio is a function of the amount of material in a closed system, and that the principle of microscopic reversibility has a more general manifestation that imposes new constraints on the set of fractal kinetic orders. So, Savageau concluded that fractal kinetics provide a novel means to achieve important features of pathway design. 

We thus expect a differential equation of the form of (4.13) to hold, where a is a proportionality constant, g (f) n (t) denotes the number of particles that are able to reach an exit in a time interval dt, and the negative sign denotes that n (f) decreases with time. This is a kinetic equation for an A + B - > B reaction. The constant trap concentration [5] has been absorbed in the proportionality constant a. The basic assumption of fractal kinetics [16] is that g(t) has the form g (t) oc I,, L. In this case, the solution is supplied by (4.14). 

It has been stated that heterogeneous reactions taking place at interfaces, membrane boundaries, or within a complex medium like a fractal, when the reactants are spatially constrained on the microscopic level, culminate in deviant reaction rate coefficients that appear to have a sort of temporal memory. Fractal kinetic theory suggested the adoption of a time-dependent rate constant , with power-law form, determined by the spectral dimension. This time-dependency could also be revealed from empirical models. 

Savageau, M., Michaelis-Menten mechanism reconsidered Implication of fractal kinetics, Journal of Theoretical Biology, Vol. 176, 1995, pp. 115— 124. 

Kosmidis, K., Argyrakis, P., and Macheras, P., Fractal kinetics in drug release from finite fractal matrices, Journal of Chemical Physics, Vol. 119, No. 12, 2003, pp. 6373-6377. 

Macheras, P., Argyrakis, P., and Polymilis, C., Fractal geometry, fractal kinetics and chaos en route to biopharmaceutical sciences, European Journal of Drug Metabolism and Pharmacokinetics, Vol. 21, No. 2, 1996, pp. 77-86. 

Macheras, P., Carrier-mediated transport can obey fractal kinetics, Pharmaceutical Research, Vol. 12, No. 4, 1995, pp. 541-548. 

Fractal kinetics was successfully shown to be a useful indicator for carrier-mediated transport (CMT) of substrates than classical Michaelis-Menten kinetics... 

Ogihara, T. Tamai, I. Tsuji, A. Application of fractal kinetics for carrier-mediated transport of drugs across intestinal epithelial membrane. Pharm. Res. 1998, 15 (4), 620-625. 

One of the most important assumptions in MM kinetics is that the reaction in question wiU proceed in a three-dimensional vessel filled with a well-stirred fluid that obeys Pick s law for diffusion. This is rarely the case in a living cell, where many reactions are localized to membranes (two dimensions) or to small regions somewhere within the cell, creating an effectively one-dimensional environment with little or no diffusion. To circumvent this limitation, fractal kinetics have been developed which allow for the approximation of enzymatic reaction velocities in vivo [7]. Fractal kinetics can utilize MM-type kinetic constants to create a model of events in a spatially restricted environment. Briefly, as the dimensionality of a reaction is reduced from three dimensions to one, the kinetic order of a bimolec-ular reaction, for example, increases from 2 in a three-dimensional case, to 2.46 in a two-dimensional environment (e.g., membrane), to 3 in a one-dimensional channel, up to 50 for the case where fractal dimensions are less than 1. In simple terms, the kinetic order is the sum of all stoichiometric coefficients of the reactants in a balanced chemical reaction equation. Rearranging the familiar equation for MM kinetics... 

Savageau, M. A. (1995). Michaelis-Menten mechanism reconsidered implications of fractal kinetics. J. Theor. Biol., 176, 115-124. 

As shown in Figure 4, for the channel in the corneal endothelium, we found (10) that the logarithm of the effective kinetic rate constant fceff as a function of the logarithm of the effective time scale feff is a straight line, which is consistent with eq 3. Thus, this channel has fractal kinetics. We also found a similar form for the currents recorded through channels in cultured hippocampal neurons (II). 

Thus, for a channel with fractal kinetics, equations 3 and 4 imply that the cumulative dwell time distribution has the form... 

We have encountered a number of difficult conceptual and practical problems in this chapter and throughout the entire book, and yet there are additional complications not treated. In a biological cell the concentrations of the various species may not be large, but many of the molecules (enzymes) may be large in size. Therefore molecular crowding [21] may occur because of this effect, mass action kinetics does not apply and a form of fractal kinetics applies [22]. These problems have not yet been considered in the interpretation of most experimental investigations. 

Kozlov, G. V Shustov, G. B. Dolbin, 1. V Zaikov, G. E. The physical significance of heterogeneity parameter in fractal kinetics of reactions. BuUetion of KBSC RAS,... 

In Fig. 12 the comparison of experimental and calculated within the frameworks of fractal kinetics (according to the Eq. (24) and (25)) kinetic curves for DMDAACh is showa As one can see, excellent corresporrdence between the theory and experiment is obtained the discrepancy does not exceed 6%, i.e., it is not higher than Q definition error [1],... 

---