Sigmoidal curve

Fig. 4. Typical sigmoid curve for the response of a biological system to chemical injury.

Figure 6-9 shows Fhs and Fs plotted against pH, according to Eqs. (6-61) and (6-62), for a weak acid of p/c = 4.0. Because of their appearance such curves are called S-shaped or sigmoid curves. 

This property of the sigmoid curve permits to be easily estimated. This is an advantage of the k-pH plot. If the inflection point cannot be accurately located, the dissociation constant may still be estimated. Let [H j = Ka in Eq. (6-64) then Eq. (6-66) results. 

A frequently encountered pH-rate profile exhibits a bell-like shape or hump, with two inflection points. This graphical feature is essentially two sigmoid curves back-to-back. By analogy with the earlier analysis of the sigmoid pH-rate curve, where the shape was ascribed to an acid-base equilibrium of the substrate, we find that the bell-shaped curve can usually be accounted for in terms of two acid-base dissociations of the substrate. The substrate can be regarded, for this analysis, as a dibasic acid H2S, where the charge type is irrelevant we take the neutral molecule as an example. The acid dissociation constants are... 

As noted previously, in all cases these various functions describe an inverse sigmoidal curve between the displacing ligand and the signal. Therefore, the mechanism of interaction cannot be determined from a single displacement curve. However, observation of a pattern of such curves obtained at different tracer ligand concentrations (range of [A ] values) may indicate whether the displacements are due to a competitive, noncompetitive, or allosteric mechanism. 

Displacement experiments yield an inverse sigmoidal curve for nearly all modes of antagonism. 

All regions of the function should be defined with real data. In cases of sigmoidal curves it is especially important to have data define the baseline, maximal asymptote, and mid-region of the curve. 

FIGURE 11.10 Removal of outliers points to achieve curve fits, (a) The least squares fitting procedure cannot fit a sigmoidal curve to the data points due to the ordinate value at 20j.iM. Removal of this point allows an estimate of the curve, (b) The outlier point at 2jiM causes a capricious and obviously errant fit to the complete data set. Removal of this point indicates a clearer view of the relationship between concentration and response. 

FIGURE 11.13 A collection of 10 responses (ordinates) to a compound resulting from exposure of a biological preparation to 10 concentrations of the compound (abscissae, log scale). The dotted line indicates the mean total response of all of the concentrations. The sigmoidal curve indicates the best fit of a four-parameter logistic function to the data points. The data were fit to Emax = 5.2, n = 1, EC5o = 0.4 pM, and basal = 0.3. The value for F is 9.1, df=6, 10. This shows that the fit to the complex model is statistically preferred (the fit to the sigmoidal curve is indicated). 

Sigmoid curves, attributable to nucleation and growth reactions, were observed for the decompositions of cobalt phthalate and silver mellitate these are marked in Table 16. The decomposition of nickel terephthalate [88] obeys the Avrami—Erofe ev equation [eqn. (6)], for which n is 1.0— 1.5 and E = 226 8 kJ mole-1. Decompositions of Co—Ni mixed mellitates are discussed in Sect. 7. 

The focus on productivity in growing systems requires a time component in the study of ecosystem responses. The response of productivity to stress must therefore be considered in three dimensions (Fig. 6). This figure illustrates the effects of a stress at any particular time on the classic sigmoid curve of growth (productivity). Positive production will occur only if the stress is less than the ultimate stress and the residual strain (permanent productivity reduction) will be seen as a lowering of the growth curve below the upper boundary (the z dimension in Fig. 6). 

Figure 2.19. Intersection of two linear regression lines (schematic). In the intersection zone (gray area), at a given c-value two PD-curves of equal area exist that at a specific y-value yield the densities zi and Z2 depicted by the dashed and the full lines. The product zi Z2 is added over the whole y-range, giving the probability-of-intersection value for that x. The cumulative sum of such probabilities is displayed as a sigmoidal curve the r-values at which 5, respectively 95% of Z2) s reached are indicated by vertical arrows. These can be...

This latter transform is used to linearize the usual sigmoid curve produced in plotting B/Bq vs log dose. This transform may be accomplished by using a computer, logit paper or a table of logit values. 

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. 

A, obtained if the disinfection process obeyed the first-order kinetic law. B, sigmoid curve. This shows a slow initial rate of kill, a steady rate and finally a slower rate of kill. This is the form of curve most usually encountered. C, obtained if bacteria are dying more quickly than first-order kinetics would predict. The constant, K, diminishes in value continuously during the process. 

Poorly absorbed compounds have been identified as those with a PSA>140Af Considering more compounds, considerable more scatter was found around the sigmoidal curve observed for a smaller set of compounds [74]. This is partly due to the fact that many compounds do not show simple passive diffusion only, but are affected by active carriers, efflux mechanisms involving P-glycoprotein (P-gp) and other transporter proteins, and gut wall metabohsm. These factors also con-... 

Permeability-pH profiles, log Pe - pH curves in arhficial membrane models (log Pjpp - pH in cehular models), generally have sigmoidal shape, similar to that of log Dod - pH cf. Fig. 3.1). However, one feature is unique to permeabihty profiles the upper horizontal part of the sigmoidal curves may be verhcally depressed, due to the drug transport resistance arising from the aqueous boundary layer (ABL) adjacent to the two sides of the membrane barrier. Hence, the true membrane contribution to transport may be obscured when water is the rate-limiting resistance to transport. This is especially true if sparingly soluble molecules are considered and if the solutions on either or both sides of the membrane barrier are poorly stirred (often a problem with 96-well microhter plate formats). 

Actual response curves often follow a sigmoidal curve as shown in Fig. 2.10. This is characteristic of systems having a series of multiple lags and hence of systems which are characterised by several time constants. 

The above equation then represents the balanced conditions for steady-state reactor operation. The rate of heat loss, Hl, and the rate of heat gain, Hq, terms may be calculated as functions of the reactor temperature. The rate of heat loss, Hl, plots as a linear function of temperature and the rate of heat gain, Hq, owing to the exponential dependence of the rate coefficient on temperature, plots as a sigmoidal curve, as shown in Fig. 3.14. The points of intersection of the rate of heat lost and the rate of heat gain curves thus represent potential steady-state operating conditions that satisfy the above steady-state heat balance criterion. 

The absorbance values obtained are plotted on the ordinate (linear scale) against the concentration of the standards on the abscissa (logarithmic scale), which produces a sigmoidal dose-response curve (Figure 5). The sigmoidal curve is constructed by... 

Figure 5 An example calibration curve. Absorbance is plotted against log (concentration of analyte). The competitive equilibrium binding process results in a sigmoidal curve that is fitted using a four-parameter fit. The IC50 is defined as the concentration of analyte that results in a 50% inhibition of the absorbance...

Some compounds exhibit pH behavior in which a bell-shaped curve is obtained with maximum instability at the peak [107]. The peak corresponds to the intersection of two sigmoidal curves that are mirror images. The two inflection points imply two acid and base dissociations responsible for the reaction. For a dibasic acid (H2A) for which the monobasic species (HA-) is most reactive, the rate will rise with pH as [HA-] increases. The maximum rate occurs at pH = (pA) + pK2)/2 (the mean of the two acid dissociation constants). Where an acid and base react, the two inflections arise from the two different molecules. The hydrolysis of penicillin G catalyzed by 3,6-bis(di-methylaminomethyl)catechol [108], is a typical example. For a systematic interpretation of pH-degradation profiles, see the review papers by van der Houwen et al. [109] and Connors [110]. 

The real time data (the process reaction curve) in most processing unit operations take the form of a sigmoidal curve, which is fitted to a first order with dead time function (Fig. 6.2) 1... 

These sigmoidal curve shifts can be described by shifts in the inflection point pH as a function of the aqueous and membrane permeability [19]. 

In the first example, the predicted oral absorption for a series of ACE inhibitors has been compared with published values of human bioavailability. For the generation of calculated absorption, a sigmoidal curve between observed human absorption and PSA for a series of reference compounds was used [25], The predicted oral absorption for ACE inhibitors is plotted against the calculated PSA values is shown in Fig. 19.6 however, as expected, only a partial correlation existed between predicted absorption and observed in vivo bioavailability. 

Cooperative enzymes show sigmoid or sigmoidal kinetics because the dependence of the initial velocity on the concentration of the substrate is not Michaelis-Menten-like but gives a sigmoid curve (Fig. 8-7). 

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