Connectivity Theorem

The Connectivity Theorem states that the flux control coefficient and elasticity are related, that is,... 

The flow or concentration control coefficients are related to elasticity coefficients through the conservation relations and connectivity theorems. 

Connectivity theorems allow to relate the control coefficients (systemic properties) to the elasticity coefficients (properties of the network s enzymes individually as if in isolation) (Westerhoff and Van Dam 1987 Heinrich and Schuster 1996 Fell 1997). The connectivity theorems have given us a strong insight into the functioning of metabolic pathways. For example, it follows directly from these theorems that enzymes that are very sensitive to the concentrations of metabolites, such as substrates, products and allosteric effectors, tend to have little control over the flux. This is illustrated by overproduction of phosphofructokinase in bakers yeast, an enzyme often referred to textbooks as rate-limiting. Yet, overproduction of phosphofructokinase does not lead to a significant flux increase, since the cell compensates by lowering the level of its allosteric effector fructose 2,6-bisphosphate (Schaaff et al. 1989 Davies and Brindle 1992). 

Kholodenko, B.N., Sauro, H.M. and Westerhoff, H.V. (1994b) Control by enzymes, coenzymes and conserved moieties. A generalisation of the connectivity theorem of metabolic control analysis. Eur. J. Biochem. 225, 179-186. 

The connectivity theorems are another important feature of MCA. Through these theorems, one can relate the control coefficients to the elasticity coefficients. The connectivity theorem for flux-control coefficients states that, for a common metabolite S, the sum of the products of the flux-control coefficient of all (i) steps affected by S and its elasticity coefficients toward S, is zero [Kacser, Bums, and Davies 1973],... 

The connectivity theorems describe how perturbations on metabolites of a pathway propagate through the chain of enzymes. The summation theorems, together with the connectivity relations and enzyme elasticities, and possibly some additional relations in the case of branched pathways [e g., Fell and Sauro 1985],... 

The developed control theory for metabolic systems allows inferring of, for example, the effects of local changes, like the properties of an enzyme on global properties as the flux through the system. Furthermore, general global properties of the systems were captured by summation and connectivity theorems, see [5] for a comprehensive review. 

The flux control connectivity theorem relates the flux control coefficients to the elasticity coefficients ... 

Since an actual crystal will be polyhedral in shape and may well expose faces of different surface tension, the question is what value of y and of r should be used. As noted in connection with Fig. VII-2, the Wulff theorem states that 7,/r,- is invariant for all faces of an equilibrium crystal. In Fig. VII-2, rio is the... 

Recalling yls), the adiabatic-to-diabatic transfoiniation angle [see Eqs. (74) and (75)] it is expected that the two angles are related. The connection is formed by the Flellmann-Feynman theorem, which yields the relation between the s component of the non-adiabatic coupling term, x, namely, x, and the characteristic diabatic magnitudes [13]... 

We refer to this equation as to the time-dependent Bom-Oppenheimer (BO) model of adiabatic motion. Notice that Assumption (A) does not exclude energy level crossings along the limit solution q o- Using a density matrix formulation of QCMD and the technique of weak convergence one can prove the following theorem about the connection between the QCMD and the BO model ... 

The compactness properties are closely connected with the reflexivity of spaces. On that score we formulate two theorems widely used in this book (Vainberg, 1972). 

TT-theorem) or from the governing equations of the flow. The latter is to be preferred because this method will give a sufficient amount of dimensionless numbers. Furthermore, it will connect the numbers to the physical process via the equations and give important information in cases where it is necessary to make approximations. 

The use of Polya s Theorem in the enumeration of rooted trees is amply described in Polya s paper and needs little comment here. We shall note an important point in connection with the enumeration of alkyl radicals. A radical is a portion of a molecule that is regarded as a unit that is, it will be treated much the same as if it were a... 

A series of four papers by G. W. Ford and others [ForG56,56a,56b, 57] amplified this work by using Polya s Theorem to enumerate a variety of graphs on both labelled and unlabelled vertices. These included connected graphs, stars (blocks) of given homeomorphic type, and star trees. In addition many asymptotic results were derived. The enumeration of series-parallel graphs followed in 1956 [CarL56], and in that and subsequent years Harary produced... 

The light shed by Redfield s paper on the close connection between Polya theory and symmetric function theory is well illustrated by a particularly simple way of looking at Polya s Theorem -- one that shows the way to further developments. Suppose the store of figures consists of n distinct figures, as for example with necklace problems using n kinds of beads. The figure generating function is then... 

If a trial function 9 leads to a kinetic energy 1 and a potential energy Vx which do not fulfill the virial theorem (Eq. 11.15), the total energy (7 +Ei) is usually far from the correct result. Fortunately, there exists a very simple scaling procedure by means of which one can construct a new trial function which not only satisfies the virial theorem but also leads to a considerably better total energy. The scaling idea goes back to a classical paper by Hylleraas (1929), but the connection with the virial theorem was first pointed out by Fock.5 It is remarkable how many times this idea has been rediscovered and published in the modern literature. 

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