Single-species accumulation kinetics

As earlier in the case of diffusion-controlled concentration decay, one-species reactions, A -l- A —) A and A 4- A 0, in one-dimension, could be used as a proving ground for the testing theory of particle accumulation, when random particle creation is added. For the former reaction the master equation could be easily derived in the form [95] 

The general solution of equation (7.3.25) may be expressed as a sum of the terms e w x) with the eigenfunctions w x) satisfying 

This eigenvalue problem is solved by inspection - this is just Airy s equation. The properly normalized stationary (A = 0) solution is 

The stationary concentration ns p,D) of the single species coagulation process with random particle production is, from equation (7.3.27) 

These results agree well with what was said above about the A + B — 0 reaction (see Fig. 7.5) - the larger reactant diffusities and/or smaller irradiation intensity, the smaller saturation concentrations ris. The Monte Carlo simulations [95] very well confirm these results. These simulations were performed on a lattice of 10 sites, by the direct simulation method. The interparticle probability density was also measured in the simulations, and the results are compared with theory the agreement is excellent. 

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This is the simplest explanation for the observation that when L and M have come to an equilibrium which contains these species in comparable amounts, the concentration of L decreases to near zero even while M remains at its maximal accumulation. Recent measurements of the quasi-equilibrium which develops in asp96asn bacteriorhodopsin before the delayed reprotonation of the Schiff base confirm this kinetic paradox [115]. Two M states have been suggested also on the basis that the rise of N did not correlate with the decay of M [117]. In monomeric bacteriorhodopsin the two proposed M states in series have been distinguished spectroscopically as well [115]. It is well known, however, that kinetic data of the complexity exhibited by this system do not necessarily have a single mathematical solution. Thus, assurance that a numerically correct model represents the true behavior of the reaction must come from testing it for consistencies with physical principles. It is encouraging therefore that the model in Fig. 5 predicts spectra for the intermediates much as expected from other, independent measurements, and the rate constants produce linear Arrhenius plots and a self-consistent thermodynamic description [116]. 

Kinetic studies are important for interpreting the results of the toxicity studies. Pharmaco- and toxicokinetics should be evaluated in single- and, in some cases, multiple-dose studies. Multi-dose pharmacokinetic studies are important to assess accumulation between doses, the impact of antibody formation on systemic exposure and whether inducible clearance exists therefore, these studies should be done in relevant species - preferably in conjunction with the toxicity studies. Information on the disposition of interferons and interleukins, particularly the clearance rate and clearance organs, is desirable. 

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