Summation relationship

For a lucid account of the kinetics of multi-enzyme systems, the reader should consult Cornish-Bowden who defines such related parameters as flux control coefficients, summation relationships, and response coefficients. 

As mentioned, there are other summation relationships that can be derived in a similar way. For example, in arriving at the result above, we deduced that a 1% increase in the activity of every enzyme would produce not only a 1% change in flux but also a zero change in the concentration of any metabolite. It follows from this sort of consideration that if we define concentration control coefficients for any particular metabolite, they will sum to zero. These other relationships have, in general, received less attention than the flux summation theorem, which is now regarded as being fundamental to the understanding of metabolic control. 

Due to the unity summation relationship we only need ( c 1) DPE equations to fully define the system. The resulting Jacobian system of equations is a matrix of size ( c— 1) X ( c— 1) where entry ij is the first partial derivative of ft with respect to component xf. 

Savageau M A, Volt E O and Irvine D H 1987 Bioohemioal systems theory and metabolio oontrol theory 2. The role of summation and oonneotivity relationships Math. Biosci. 86 147-69... 

Under most circumstances the equations given in Table 10.4 accurately calculate the thermodynamic properties of the ideal gas. The most serious approximations involve the replacement of the summation with an integral [equations (10.94) and (10.95)] in calculating the partition function for the rigid rotator, and the approximation that the rotational and vibrational partition functions for a gas can be represented by those for a rigid rotator and harmonic oscillator. In general, the errors introduced by these approximations are most serious for the diatomic molecule." Fortunately, it is for the diatomic molecule that corrections are most easily calculated. It is also for these molecules that spectroscopic information is often available to make the corrections for anharmonicity and nonrigid rotator effects. We will summarize the relationships... 

For a given geometry, view factors are related to each other, one example being the reciprocity relationship given in equation 9.126. Another important relationship is the summation rule which may be applied to the surfaces of a complete enclosure. In this case, all the radiation leaving one surface, say i, must arrive at all other surfaces in the enclosure so that, for n surfaces ... 

Some difference formulae. In the sequel, when dealing with various difference expressions, we shall need the formulae for difference differentiating of a product, for summation by parts and difference Green s formulae. In this section we derive these formulae within the framework similar to the appropriate apparatus of the differential calculus. Similar expressions were obtained in Section 2 of Chapter 1 in studying second-order difference operators, but there other notations have been used. It performs no difficulty to establish a relationship between formulae from Section 2 of Chapter 1 and those of the present section. 

Upon expressing from the equilibrium condition the complex concentration M12 through the concentrations of monomers, and substituting the expression found into relationship (21) we obtain, invoking the formalism of the Markov chains, final formulas enabling us to calculate instantaneous statistical characteristics of the ensemble of macromolecules with colored units. A subsequent color erasing procedure is carried out in the manner described above. For example, when calculating instantaneous copolymer composition, this procedure corresponds to the summation of the appropriate components of the stationary vector jt of the extended Markov chain ... 

In addition to the equations arising from the material and energy balances over a stage, and the equilibrium relationships, there will be a fourth relationship, the summation equation for the liquid and vapour compositions ... 

Describe the factors that influence the strength of skeletal muscle contraction including multiple motor unit summation, asynchronous motor unit summation, frequency of nerve stimulation, length-tension relationship, and diameter of the muscle fiber... 

For the following relationships the sign = means approximately equal to, when X is small. These equations are derived by using a Taylor s series (see Series Summation and Identities ). 

This expression demonstrates use of the Einstein summation convention 6. The significance of r is made clear by examining a particle momentarily at rest in a Lorentz system. The components of the vector, transformed dx = (0, 0, 0, icdt ) and dr2 = —(1 /c dx dx = [dt )2. Thus dr is the time interval on a clock travelling with the particle and is therefore referred to as the interval of the particle s proper time or world time. The relationship between dr and an interval of time as measured in a given Lorentz system can be derived directly by expanding the equation... 

The utility and success of Metabolic Control Analysis is mostly due to a number of simple relationships that interconnect the various coefficients and that bridge between local and global properties of the network. First, the summation theorems relate to the structural properties of the network and are independent of kinetic parameters [96]. Using Eq. (90) and (91), it is straightforward to verify that... 

Various methods by which the Kow of PAHs could be calculated are based on their molecular structures, i. e., a quantitative structure-property relationship (QSPR) approach [ 1,199,200]. One of the most famous techniques involves summation of hydrophobic fragmental constants (or f-values) for all groups in a molecule of a specific compound. On the other hand, Aboul-Kassim [1] and Aboul-Kassim et al. [202, 203] introduced a modeling technique based on the molecular connectivity indices of various PAHs, ranging from low- to high-molecular weight compounds. More details are given in Chap. 4 of this volume. 

The relationship among heat capacity, entropy, and temperature in crystalline solids may be understood on the basis of two fundamental concepts the Boltzmann factor and the partition function (or summation over the states, from the German term Zustandsumme). Consider a system in which energy levels Eq,... 

To demonstrate the truth of this statement, note if any one orbital occurs more than once in Eq. (22) (i.e., if iq = ij, then using the relationship hj = will reduce the degree of this term in the polynomial. Because of this, the summation in Eq. (22) includes monomials of every degree between one (when i = o = = ig) and g (when h ig)- Because the eigenvalues of... 

Like the potential, other electrostatic functions can be expressed as Fourier summations over the structure factors (Stewart 1979). The electric field, being the (negative) gradient of the potential, is a Fourier series in which the power of the magnitude of H increases from —2 to —1, as expected from the reciprocal relationship between direct space and Fourier space. Starting with... 

Besides having a convenient measure of the interrelationship between two variables, it is also useful to develop procedures to describe the relationships betweeen samples so that subsequently the samples can be grouped according to how similar or dissimilar they are to one another. One set of possible functions to describe the relationship between samples is the correlation and covariance function defined in the equations 6 to 9 with the meaning of the indexes changed so that j and k refer to different samples and the summations are taken over the n variables in the system. The use of such functions will be explored later in this chapter. 

Ultrafast laser excitation gives excited systems prepared coherently, as a coherent superposition of states. The state wave function (aprobabihty wave) is a coherent sum of matter wave functions for each molecule excited. The exponential terms in the relevant time-dependent equation, the phase factors, define phase relationships between constituent wave functions in the summation. 

The 8 value of a polymer may also be calculated from Small s relationship, in which 2G is equal to the summation of the molar attraction constants of the individual repeating units constituting the polymer ... 

What we really are is a web of interconnections, the summation of all of our relationships, all the people we know and those we are still to meet. It s not that we are in the web, the web is what we are. Vowel sounds change the colors pitch and tone alter the shape of the enclosing... 

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