Sigmoid behaviour

Allosteric enzymes do not follow the Michaelis-Menten kinetic relationships between substrate concentration Fmax and Km because their kinetic behaviour is greatly altered by variations in the concentration of the allosteric modulator. Generally, homotrophic enzymes show sigmoidal behaviour with reference to the substrate concentration, rather than the rectangular hyperbolae shown in classical Michaelis-Menten kinetics. Thus, to increase the rate of reaction from 10 per cent to 90 per cent of maximum requires an 81-fold increase in substrate concentration, as shown in Fig. 5.34a. Positive cooperativity is the term used to describe the substrate concentration-activity curve which is sigmoidal an increase in the rate from 10 to 90 per cent requires only a nine-fold increase in substrate concentration (Fig. 5.346). Negative cooperativity is used to describe the flattening of the plot (Fig. 5.34c) and requires requires over 6000-fold increase to increase the rate from 10 to 90 per cent of maximum rate. 

A decreasing reduction current due to peroxide consumption could be observed when increasing amounts of octane were injected into the peroxide containing solution. The graph shows a sigmoidal behaviour with a levelling off at about 40 mM octane. A decrease of about 5 /aM peroxide per mM octane is obtained at up to 20 mM octane (Fig. 2.13) [106]. 

A detailed examination of the reaction products of 2MP on both USHY catalysts indicates the modes of reaction are 1) isomerization, monomolecular and dimerization-cracking on pristine sites and 2) biinolecular chain processes between an adsorbed product carbocations and a gas phase reactant molecule. Figure 1 presents the experimental average conversion and the corresponding predicted conversion, with respect to time on stream, along lines of constant catalyst to reactant ratio. The sigmoidal behaviour exhibited at low conversions and short times on stream is consistent with the presence of the second type of reaction, one mediated by chain processes. Such behaviour contrasts sharply with that previously reported for linear paraffins on USHY (9) where 1 was observed to increase monotonically with reaction time. A kinetic model has been proposed (10) which accounts for all the mechanisms active in this system. The model assumes that the surface reaction is rate controlling in all cases and is ... 

In this experiment jmax exhibited sigmoidal behaviour, increasing upon addition of polymer at low P/E ratios, and then becoming constant at P/E values near 10. If one assumes that the current is proportional to the rate of electron transfer between the enzyme and the polymer, i.e. that the electrodes are kinetically limited by the rate... 

It is appropriate to start with BaN6 since this compound has been studied particularly intensively and has been regarded as a model in the development of the theory of kinetics of decompositions of solids. The sigmoid a—time curves for BaN6 pyrolyses, Fig. 15, are typical examples of solid state autocatalytic behaviour. 

The behaviour [821] of AgN02 is closer to that expected of a nitro compound than a nitrite. Decomposition (308—363 K) yields Ag metal and N02. a—Time curves are sigmoid with a prominent linear region (0.15 < a < 0.45) but the Arrhenius plot was curved at >333 K. This was attributed to inadequate gaseous product removal. In contrast to the behaviour observed for most other solids, pre-irradiation with 7-rays inhibits subsequent thermal decomposition [829]. 

The a—time curves for the vacuum decomposition at 593—693 K of lanthanum oxalate [1098] are sigmoid. Following a short induction period (E = 164 kJ mole-1), the inflexion point occurred at a 0.15 and the Prout—Tompkins equation [eqn. (9)] was applied (E = 133 kJ mole-1). Young [29] has suggested, however, that a more appropriate analysis is that exponential behaviour [eqn. (8)] is followed by obedience to the contracting volume equation [eqn. (7), n = 3]. Similar kinetic characteristics were found [1098] for several other lanthanide oxalates and the sequence of relative stabilities established was Gd > Sm > Nd > La > Pr > Ce. The behaviour of europium(III) oxalate [1100] is exceptional in that Eu3+ is readily reduced... 

The more usual pattern found experimentally is that shown by B, which is called a sigmoid curve. Here the graph is indicative of a slow initial rate of kill, followed by a faster, approximately linear rate of kill where there is some adherence to first-order reaction kinetics this is followed again by a slower rate of kill. This behaviour is compatible with the idea of a population of bacteria which contains a portion of susceptible members which die quite rapidly, an aliquot of average resistance, and a residue of more resistant members which die at a slower rate. When high concentrations of disinfectant are used, i.e. when the rate of death is rapid, a curve ofthe type shown by C is obtained here the bacteria are dying more quickly than predicted by first-order kinetics and the rate constant diminishes in value continuously during the disinfection process. 

The temperature record showed a more variable pattern. For the relatively slowly polymerising styrene and indene, the initial reaction rate was always so much less than the rate of mixing-in of the initiator solution, even with high initiator concentrations, that the reaction curves showed good first-order behaviour from the start. For the more reactive monomers, all the reaction curves were sigmoid, with a marked but variable acceleration lasting rarely more than 3 seconds. The behaviour of the conductivity mentioned above indicates that this is not due to the slowness of the initiation reaction,... 

The range of frequencies used to calculate the moduli are typically available on many instruments. The important feature that these calculations illustrate is that as the breadth of the distributions is increased the original sigmoidal and bell shaped curves of the Maxwell model are progressively lost. A distribution of Maxwell models can produce a wide range of experimental behaviour depending upon the relaxation times and the elastic responses present in the material. The relaxation spectrum can be composed of more than one peak or could contain a simple Maxwell process represented by a spike in the distribution. This results in complex forms for all the elastic moduli. 

The sorption behaviour of a number of dairy products is known (Kinsella and Fox, 1986). Generally, whey powders exhibit sigmoidal sorption isotherms, although the characteristics of the isotherm are influenced by the composition and history of the sample. Examples of sorption isotherms for whey protein concentrate (WPC), dialysed WPC and its dialysate (principally lactose) are shown in Figure 7.13. At low aw values, sorption is due mainly to the proteins present. A sharp decrease is observed in the sorption isotherm of lactose at aw values between 0.35 and 0.50 (e.g. Figure 7.13). This sudden decrease in water sorption can be explained by the crystallization of amorphous lactose in the a-form, which contains one mole of water of crystallization per mole. Above aw values of about 0.6, water sorption is principally influenced by small molecular weight components (Figure 7.13). 

Metallic lithium in the form of a suspension has been used to polymerize isoprene (97) but the system is not too suitable for an exact analysis of the mechanism. The conversion-time curves are sigmoidal in shape. Minoux (66) has shown that the overall rate is not very dependent on the amount of lithium dispersion used as expected if the organo-lithium intermediates are highly associated. The molecular weight of the polymer is more dependent on quantity of lithium used. The observed kinetic behaviour is very similar to that shown in lithium alkyl initiation. This suggests that apart from differences in the initiation step, the mechanisms are quite similar. 

The temperature dependence of the yield stress, ay, of PMMA obtained at a strain rate, s = 2 x 10-3 s-1, is shown in Fig. 18. A sigmoidal curve is observed, which looks like the temperature dependence of the Young s modulus, E. When increasing the strain rate a similar behaviour is observed. 

There are exceptions to Michaelis-Menten behaviour. For example allosteric enzymes which instead of a hyperbolic curve in a Lversus [S] graph yield a sigmoidal plot (the behaviour is rather like non-catalytic allosteric proteins, such as haemoglobin, Section 2.5. This type of curve can indicate cooperative binding of the substrate to the enzyme. We have discussed cooperativity in Section 1.5 (see also Section 10.4.3). In addition, regulatory molecules can further alter the activity of allosteric enzymes. 

The behaviour of S as a function of time t can be crudely classified into two cases. The first corresponds to formation of nuclei of the product and their growth. As shown schematically in Fig. 1, the reaction interface increases until growing nuclei overlap extensively and then decreases, resulting in a sigmoidal dependence of the extent of reaction, a, on time t. When diffusion is not significant, the early stage of the nucleation is described by a power law dependence of a on t... 

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