The Minimal Model

In looking for global properties of the interaction of competing macrosocieties it is not the intention here to introduce variables describing microscopic details of this interaction. Instead a rather coarse-grained approach will be used (other authors [6.2-7] have proposed interesting and much more detailed models of world-wide economic interactions). 

There is, however, one aspect in which the constructed model will go further (to the authors knowledge) than hitherto proposed models In the spirit of the concepts developed in the preceding chapters the effect of political psychology on the interaction of competing macrosocieties will be quantitatively included. 

The minimal requirement of such a model is that it should describe - in a coarse-grained manner - the properties of the quasi-stationary states peace and tension , the stability or lability of these states and the possibility of eventual transitions between them. Furthermore, the model cannot avoid including a consideration of the possibility of a transition to the state war . Only in this way can an estimation be made as to which trend parameters in the model are responsible for an increase or decrease of the threat of this transition. 

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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... 

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... 

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. 

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 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.

In addition to bistability and hysteresis, the minimal model of glycolysis also allows nonstationary solutions. Indeed, as noted above, one of the main rationales for the construction of kinetic models of yeast glycolysis is to account for metabolic oscillations observed experimentally for several decades [297, 305] and probably the model system for metabolic rhythms. In the minimal model considered here, oscillations arise due to the inhibition of the first reaction by its substrate ATP (a negative feedback). Figure 24 shows the time courses of oscillatory solutions for the minimal model of glycolysis. Note that for a large... 

Figure 24. The nullclines (upper panels, gray lines) and time courses (lower panels) for oscillatory solutions of the minimal model of glycolysis. Left panels Damped oscillations. The...

The metabolic state of the minimal model is specified by one independent flux value v° and 3 steady-state metabolite concentrations,... 

To specify the matrix 0we take into account the minimal model discussed in Section VII.A.4 The first reaction vj (ATP), including the lumped PFK reaction, depends on ATP only (with glucose assumed to constant). The cofactor ATP may activate, as well as inhibit, the rate (substrate inhibition). To specify the interval of the corresponding saturation parameter, we use Eq. (79) as a proxy and obtain... 

Figure 29 Bifurcation diagram of the minimal model of glycolysis as a function of feedback strength and saturation 6 of the ATPase reaction. Shown are the transitions to instability via a saddle node (SN) and a Hopf (HO) bifurcation (solid lines). In the regions (i) and (iv), the largest real part with in the spectrum of eigenvalues is positive > 0. Within region (ii), the metabolic state is a stable node, within region (iii) a stable focus, corresponding to damped transient oscillations.

Figure 30. A medium complexity model of yeast glycolysis [342], The model consists of nine metabolites and nine reactions. The main regulatory step is the phosphofructokinase (PFK), combined with the hexokinase (HK) reaction into a single reaction vi. As in the minimal model, we only consider the inhibition by its substrate ATP, although PFK is known to have several effectors. External glucose (Glc ) and ethanol (EtOH) are assumed to be constant. Additional abbreviations Glucose (Glc), fructose 1,6 biphosphate (FBP), pool of triosephosphates (TP), 1,3 biphosphogly cerate (BPG), and the pool of pyruvate and acetaldehyde (Pyr).

Figure 31 shows the largest eigenvalue of the Jacobian at the experimentally observed metabolic state as a function of the parameter 0 TP. Similar to Fig. 28 obtained for the minimal model, several dynamic regimes can be distinguished. In particular, for sufficient strength of the inhibition parameter, the system undergoes a Hopf bifurcation and the pathway indeed facilitates sustained oscillations at the observed state. 

The minimal models used here and elsewhere in the book are models that can be described and solved to demonstrate a phenomenon with a minimal number of molecular parameters—sometimes, this is the same as die maximal number of simplifying assumptions. 

Perhaps the first step in this direction consisted of the realization that the minimal models usually employed for mechanistic investigations (which fold into the Y-structure) are suboptimal These truncated motifs require high (millimolar) concentrations of Mg + ions for proper folding and subsequent catalytic activity (typical rates of ca 1 min ) °. In contrast to this, natural hammerhead ribozymes are active at submillimolar (physiological) concentrations and exhibit much higher cleavage rates ca 870 min in the case of HHRzs isolated from schistosomes) . ... 

The real power in the multi-coefficient models, however, derives from the potential for the coefficients to make up for more severe approximations in the quantities used for (/) in Eq. (7.62). At present, Truhlar and co-workers have codified some 20 different multicoefficient models, some of which they term minimal , meaning that relatively few terms enter into analogs of Eq. (7.62), and in particular the optimized coefficients absorb the spin-orbit and core-correlation terms, so they are not separately estimated. Different models can thus be chosen for an individual problem based on error tolerance, resource constraints, need to optimize TS geometries at levels beyond MP2, etc. Moreover, for some of the minimal models, analytic derivatives are available on a term-by-term basis, meaning that analytic derivatives for the composite energy can be computed simply as the sum over tenns. 

The minimal model comprises an uncatalyzed and unspecific direct formation of R and S (ko), a simple and unspecified description of the autocatalytic steps assuming monomers as catalytic species (k ), and the monomer-dimer equilibria (k2, M and (M M in which different rates of dimer formation for homochiral and heterochiral species is allowed. This model translates into the following set of differential equations ... 

While it is commonly accepted that the required mutual inhibition originates from dimerization as expressed by the minimal model, the exact nature... 

In the following, only the properties of the minimal model are discussed since the alternative model gives rise to very similar results. 

Fig. 6 Simulated time-evolution of heterochiral and homochiral dimers in the Soai reaction by using the minimal model, giving rise to a nearly 1 1 distribution of both types of species. Same initial conditions and parameters used as in Fig. 5...

For the insulin example in Section 5.3, the information in the available in-vivo data is concerned primarily with the importance and speed of the internalization. The minimal modeling shows that both the internalization, the subsequent dephosphorylation and the eventual recycling of the free receptor are essential to explain the given data. The time-constants for these processes may also be estimated with a relatively high accuracy. On the other hand, information about the reversal of these processes, and the details of the interconversions between the various forms of phosphorylated receptors, are virtually non-existent in the given data. 

Several models of different complexity - e.g. the minimal model - describe the IVGTT [122-125]. Typically, beta cells respond with a large first phase, followed by a much smaller second phase as the glucose concentration rapidly declines. 

Equations (8.46) and (8.47) describe the Sangren-Sheppard model. While the equations are straightforward and can be thought of as the minimal model that captures the important biophysical phenomena of solute exchange along a capillary, this model represents nearly the maximal level of complexity that can be effectively analyzed without invoking numerical approximations to simulate it.6... 

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