Polymerization nucleated equilibrium

Nucleated supramolecular polymerization, on the other hand, is a much more sensitive function of the external conditions. Indeed, a sharp polymerization point can be identified below which almost no material is in the polymerizated state and above which the self-assembled polymers exhibit a strong variation of their mean size with varying concentration, temperature, and so on. Nucleated equilibrium polymerization requires the existence of... 

In effect this is a self-organized system that maintains a steady state, although not at chemical equilibrium, by a balance of reaction rates against reactant fluxes, a condition commonly seen in real-world systems. The titration experiments permit a rather direct measurement of the rate of nucleation and polymerization reactions that transform Ala monomers to Alb polymers. The titrations at pH 4.75 and 4.90 yielded only Ala and Alb. Although some Ale was formed in titrations at pH 5.00, the polymeric product was about 90% Alb. Hence, the data obtained in these experiments should be useful for determining the effect of pH on the rate of the polymerization process leading from Ala to Alb. At least a part of the information from titrations made at pH 5.20 should also be useful because during much of those experiments the principal product also was Alb. 

Consider a situation in which a concentrated polymeric solution enters the extraction zone of, say, an extruder in circumstances when the pressure in the extraction zone. Pa, is less than the equilibrium partial pressure of the volatile component in the feed solution. Under these conditions the solution will be supersaturated at the extraction pressure, flashing of the volatile component will occur, gas bubbles of radius Rq will be formed, and the concentration will immediately fall from Wi to wq. If bubble formation occurs by homogeneous nucleation, the rate at which these bubbles will be formed per unit volume of solution should depend on the difference between the equilibrium partial pressure of the volatile component and the devolatilization pressure. Since this pressure difference is greatest when the solution first enters the extraction zone, the rate of formation of bubbles will at first be high but as devolatilization pro-... 

The exact form of 7p. the number of bubbles nucleated per unit volume per unit time, is not known since there appears to be no published scientific studies which deal with nucleation rates in concentrated polymeric solutions at pressures below the equilibrium partial pressure. Blander and Katz (1975) have studied nucleation rates of pure liquids as a function of temperature at atmospheric pressure and Prud homme (1982) has studied nucleation rates of polymeric solutions containing up to 0.60 weight fraction of polymer as a function of temperature, concentration, and polymer molecular weight at atmospheric pressure. Both sets of investigators concluded that nucleation rates seem to follow the equation... 

The threshold concentration of monomer that must be exceeded for any observable polymer formation in a self-assembling system. In the context of Oosawa s condensation-equilibrium model for protein polymerization, the cooperativity of nucleation and the intrinsic thermodynamic instability of nuclei contribute to the sudden onset of polymer formation as the monomer concentration reaches and exceeds the critical concentration. Condensation-equilibrium processes that exhibit critical concentration behavior in vitro include F-actin formation from G-actin, microtubule self-assembly from tubulin, and fibril formation from amyloid P protein. Critical concentration behavior will also occur in indefinite isodesmic polymerization reactions that involve a stable template. One example is the elongation of microtubules from centrosomes, basal bodies, or axonemes. 

Zeolite crystallization represents one of the most complex structural chemical problems in crystallization phenomena. Formation under conditions of high metastability leads to a dependence of the specific zeolite phase crystallizing on a large number of variables in addition to the classical ones of reactant composition, temperature, and pressure found under equilibrium phase conditions. These variables (e.g., pH, nature of reactant materials, agitation during reaction, time of reaction, etc.) have been enumerated by previous reviewers (1,2, 22). Crystallization of admixtures of several zeolite phases is common. Reactions involved in zeolite crystallization include polymerization-depolymerization, solution-precipitation, nucleation-crystallization, and complex phenomena encountered in aqueous colloidal dispersions. The large number of known and hypo-... 

Electron microscopy of the final latex of the experiments given in Table I showed almost no new nucleation. The particle size distributions were narrow and indicated no noticeable coagulation as well. New nucleation would lead to increased rates whereas coagulation would have the opposite effect. Any decrease in the rate therefore must be due to a decrease in [m], if we assume n to be constant. We therefore determined the tofuene/polymer ratio in the seed latex in the absence and presence of the various additives. Toluene was chosen as the solvent, because it is similar to styrene and allows the measurement of equilibrium solubilities without the risk of polymerization. Table II gives the experimental values of the toluene solubility in the seed as a function of time. The results indicate that the swelling is nearly complete within 5 to 10 min. 

If the nucleated assembly is self-catalyzed, then this modifies only the role of the equilibrium constant K in the non-self-catalyzed model and has to be replaced by the ratio K/Ka. This means that we again obtain for the fraction polymerized material f=KaNa/(1 + KaNa) but that the degree of polymerization averaged over the active material only now obeys (Ciferri, 2005)... 

Figure 7 The fraction polymerized material fas a function of the dimensionless time Ty according the kinetic Landau model discussed in the main text, with h the nucleation rate. Shown are results valid in the limit where the nucleation reaction is rate limiting, for a quench to X/Xp = 2 where in equilibrium f— 0.5. Depolymerization is much faster than polymerization.

The first interval is the interval of particle nucleation (interval I) and describes the process to reach an equilibrium radical concentration within every droplet formed during emulsification. The initiation process becomes more transparent when the rate of polymerization is transferred into the number of active radicals per particle n, which slowly increases to n 0.5. Therefore the start of the polymerization in each miniemulsion droplet is not simultaneous, so that the evolution of conversion in each droplet is different. Every miniemulsion droplet can be perceived as a separate nanoreactor, which does not interact with others. After having reached this averaged radical number, the polymerization kinetics is slowing down again and follows nicely an exponential kinetics as known for interval III in emulsion polymerization or for suspension polymer-... 

Figure 10.5 The equilibrium monomer concentration as a function of the total monomers for a nucleation-elongation protein polymerization process described by Equations (10.7) and (10.13). The constant K = k+ /K is the association constant of a monomer to the polymer [A ]eq is the equilibrium free monomer concentration and A is the total monomer concentration. The variable a = 1 — K /K depends on the difference in equilibrium constants between elongation and nucleation steps. At a = 0 there is no difference in equilibrium constants at er = l nucleation is highly unfavorable compared to elongation and there is infinite cooperativity in the polymerization process.

Microtubule assembly in cells differs in some ways from assembly in vitro. In cells, nucleation of microtubules requires a third type of tubulin, which is called y-tubulin, that functions in concert with other proteins in the form of a y-tubulin ring complex. In most animal cells, the y-tubuIin ring complex is located at the pericentriolar region of the microtubule organizing center (or centrosome) where it nucleates microtubule assembly at the minus ends (7). The y-tubulin does not become incorporated into the microtubule, but rather it only localizes to the minus ends. Assembly of tubulin to form microtubules during the early stages of polymerization in vitro can be considered a pseudo first-order reaction. A steady state is eventually attained in which both the soluble tubulin concentration and the microtubule polymer mass attain stable plateaus (8). The critical concentration at apparent equilibrium (actually a steady state, see below) is the concentration of soluble tubulin in apparent equilibrium with the microtubule polymers. 

If polymerization is carried out under conditions of high nucleation, polymer chains are deposited rapidly and indiscriminately and the active sites necessary for growth are occluded. If growth is carried out near the equilibrium concentration, depolymerization becomes important and again the active sites become occluded. 

The rate-limiting step in actin polymerization appears to be nucleation, the formation of an actin cluster large enough (typically three or four G-actins) for the rate of monomer association to exceed the rate of dissociation. Once filaments of this size form, they continue to grow, and the concentration of G-actin monomers decreases until it is in equilibrium with F-actin. The concentration of monomeric actin at equilibrium is called the critical concentration, Cc. In vitro, Cc is 0.1 mM. The value in vivo is variable, depending in part on the concentration of ATP. In the presence of ADP, instead of ATP, both ends of the filament grow at the characteristic slow rate. ATP speeds up the rate of polymerization and lowers the effective Cc. 

The third model, nucleated polymerization (NP) (Jarrett and Lans-bury, 1993), predicts that the S- and A-states are in equilibrium in solution, but the Estate is predominant. Soluble A-state protein is stabilized by association with assembled A-state complexes. Thus, the rate-limiting step is formation of a polymerization-competent surface or nucleus rather than conformational conversion. 

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