Logarithmic gain

Figure 7. Logarithmic gain for the repressible threonine biosynthetic operon of Escherichia coli. An auxotrophic mutant of E. coli B unable to synthesize homoserine was grown for more than four generations in glucose minimal medium supplemented with various concentrations of homoserine (Savageau and Steward, 1970). At low concentrations of homoserine, a correspondingly low amount of threonyl-tRNA is produced, which leads to derepression of the threonine biosynthetic operon at high concentrations homoserine (and threonyl-tRNA) the operon is repressed (Savageau, unpublished data).

Figure 8. Logarithmic gain for the second messenger cAMP in the fat pad of the rat responding to perfusion with various concentrations of the hormone epinephrine (replotted from the data of Butcher et al., 1965).

The logarithmic gain factors are characterized by the percentage change of a dependent concentration X, in going from one steady-state to another as the result of a 1% increase in an independent concentration X, while all other independent concentrations and parameters are held constant. These factors also can be determined mathematically by differentiation of the explicit steady-state solution. 

In a similar fashion, one can obtain the logarithmic gains in the dependent fluxes through the pools of the system. 

The logarithmic gain between a given independent and dependent variable is not determined by any one component, or even by a small number of components, in the system. Rather it is a systemic property that in general depends upon all of the components and their specific organization or design. These systemic factors. 

LOG Logarithmic amplifier with 60/100 dB dynamic range without gain setting 0.1 to 10 MHz (-3 dB)... 

Experimentally accessible is D by means of scattering methods [144], The corresponding fractal analysis of scattering data is gaining special attractivity from its intriguing simplicity. In a double-logarithmic plot of I (s) v.v. s the fractal dimension is directly obtained from the slope of the linearized scattering curve. It follows from the theory of fractals that... 

The molecular binding energies obtained from chemical conceptions are lower than the activation energies gained from the slope in the logarithmic representation of the Arrhenius formula. 

The thorough study of PEA by Fytas et al.30 obtained qualitatively identical results. The parameter /3 was independent of pressure and temperature. The logarithm of the average relaxation time increased linearly with pressure. The parameter B was found to be 37.5 cm3/mol. This is substantially smaller than for polystyrene and reflects the much larger volume swept out by the anisotropy relaxation in polystyrene as compared to the longitudinal density relaxation in PEA. Although these types of experiments are very difficult, the information gained is well worth the effort. 

The Hammett equation (26-6) states that the relative reactivity (expressed in logarithmic form) of a substituted benzene derivative is proportional to the substituent constant a. For a given reaction, a plot of log (/c//c0) or of log (K/Ka) versus cr should be linear with slope p. Some idea of the validity of the Hammett equation can be gained from Figure 26-4, which shows plots of log kjko) or of log (KjKa) against a for several different reactions. For the examples given, the fit to the Hammett equation is fair. A number of p values (slopes) are listed separately in Table 26-7. It can be seen that p values vary... 

Fig. 3.9. The effect of changes in the photomultiplier tube voltage and amplifier gain on the appearance of six signals with intensities in the relationship of 1 2 10 20 100 200 to each other. A Linear amplification. B Logarithmic amplification, (continued on next page)...

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