Minimal modeling

The first study in which a full CASSCE treatment was used for the non-adiabatic dynamics of a polyatomic system was a study on a model of the retinal chromophore [86]. The cis-trans photoisomerization of retinal is the primary event in vision, but despite much study the mechanism for this process is still unclear. The minimal model for retinal is l-cis-CjH NHj, which had been studied in an earlier quantum chemisti7 study [230]. There, it had been established that a conical intersection exists between the Si and So states with the cis-trans defining torsion angle at approximately a = 80° (cis is at 0°). Two... 

We employ the general scheme presented above as a starting point in our discussion of various approaches for handling the R-T effect in triatomic molecules. We And it reasonable to classify these approaches into three categories according to the level of sophistication at which various aspects of the problem are handled. We call them (1) minimal models (2) pragmatic models (3) benchmark treatments. The criterions for such a classification are given in Table I. 

The present perturbative beatment is carried out in the framework of the minimal model we defined above. All effects that do not cincially influence the vibronic and fine (spin-orbit) stracture of spectra are neglected. The kinetic energy operator for infinitesimal vibrations [Eq. (49)] is employed and the bending potential curves are represented by the lowest order (quadratic) polynomial expansions in the bending coordinates. The spin-orbit operator is taken in the phenomenological form [Eq. (16)]. We employ as basis functions... 

There are other, more complex, variants of this minimal model. FHP-II, for example, is a 7-bit variant of FHP-I, adding a zero-velocity rest particle and a few collision extra rules involving that rest particle [frischS ]. FHP-III is a collision-saturated version of FHP-II ([frischST], [d HumSTc]). There are also 8-bit variants that include up to three rest particles per site and use particles of different mass ([d Hum87c], [wolf86c]). For simplicity, we will not consider models with rest particles. 

De Gaetano A, Mingrone G, Castagneto M. NONMEM improves group parameter estimation for the minimal model of glucose kinetics. Am J Physiol 1996 271 E932-7. 

Garavelli M, Gelani P, Bernardi F, Robb MA, Olivucci M (1997) The CyHgNH protonated shiff base an ab initio minimal model for retinal photoisomerization. J Am Chem Soc 119 6891... 

At the genomic level, an RBN models genomic regulation. The state of a cell is described via its state vector. A state vector may be viewed as a set of the proteins that exist in a cell at some moment in time. However, even in this minimal model, two other properties of real cells must be included cell division and spatial patterning. 

This minimal model can be used in many situations where one clearly defined factor is crucial for the quality of the product. It can be organised as a voluntary agreement between independent enterprises, or it can be imposed by a dominant enterprise such as a wholesaler or retailer, as a condition for obtaining a contract to sell products to this enterprise. 

Hamiltonian equations, 627-628 II electronic states, 632-633 triatomic molecules, 587-598 minimal models, 615-618 Hartree-Fock calculations ... 

Minimal models, Renner-Teller effect, triatomic molecules, 615-618... 

Hamiltonian equations, 610—615 minimal models, 615-618 multi-state effects, 624 pragmatic models, 618—621 spectroscopic properties, 598-610 linear molecules ... 

All three of these predictions from this minimal model are manifest in the etiology of prion disease an inversely proportional relationship between PrPc expression and prion incubation period in transgenic mice predisposition by relatively subtle mutations in the protein sequence and a requirement for molecular homogeneity between PrPSc and PrPc for efficient prion propagation [4, 5, 20]. It is clear that a full understanding of prion propagation will require knowledge both of the structure of PrPc and PrPSc and of the mechanism of conversion between them. 

This is a (minimal) model including the formation of the complex R1-R2, the active precursor complex APC that interconverts to those states belonging to the active successor complex ASC, as discussed in the previous section. The chemical reaction, in this model, ends up with the formation of the products PI and P2. The kinetic parameters k+ and k- hide the effects of quantum interconversions via the intermediate Hamiltonian Hc(ij). Let us introduce this feature in the kinetic model, so that... 

In addition, good use of packages facilitates full separation of interfaces from implementations. As an extreme example, a single class might be implemented in a package that imports the packages with the type definitions of all types that class must implement each such package contains the minimal model of any other types that it must interact with. 

Figure 5. A minimal model of glycolysis One unit of glucose (G) is converted into two units of pyruvate (P), generating a net yield of 2 units of ATP for each unit of glucose. Gx, Px, and Glx are considered external and are not included into the stoichiometric matrix. A A graphical depiction of the network. B The stoichiometric matrix. Rows correspond to metabolites, columns correspond to reactions. C A list of individual reactions. D The corresponding system of differential equations. Abbreviations G, glucose (Glc) TP, triosephosphate, P, pyruvate.

Probably the most well-known pathway to exemplify the occurrence of complex dynamics in metabolic networks is the glycolytic pathway of yeast. Arguably one of the most modeled pathways ever, minimal models of yeast glycolysis were studied since the 1960s [94, 273, 305 308] and give rise to a rich spectrum of... 

Figure 21. The nullclines of the minimal model of glycolysis (schematic). The graphic analysis allows to deduce the qualitative dynamics of the system. Each area in the phasespace is characterized by the signs of the local derivatives, corresponding to increasing or decreasing concentration of the respective variable. The gray arrows indicate the direction a trajectory will go. Note that the trajectories may only intersect vertically or horizontally with the nullclines. For simplicity, the nullclines are depicted schematically only, for the actual nullclines corresponding to the rate equations see Fig. 22C.

Figure 22. The nullclines corresponding to the minimal model of glycolysis. Depending on the value of the maximal ATP utilization Vm3, the pathway either exhibits a unique steady state or allows for a bistable solution. Note that the nullcline for TP does not depend on VThe corresponding steady states are shown in Fig. 23. Parameters are Vm 3.1, K 0.57, ki 4.0, K i 0.06, and n 4 (the values do not correspond to a specific biological situation).

Figure 23. The steady state ATP concentration as a function of maximal ATP utilization Vmi for the minimal model of glycolysis. The letters denoted on the x axis correspond to the different scenarios shown in Fig. 22A D. Bold lines indicate stable steady states. Note that the physiologically feasible region is confined to the interval ATP0 e [0,Ar]. For low ATP usage (Vm3 small), there are three steady states, two of which are stable. However, both stable states are outside the feasible interval.

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