Saturation parameter

In equation (Cl.4.14) the saturation parameter essentially defines a criterion to compare the time required for stimulated and spontaneous processes. If I then spontaneous coupling of the atom to the vacuum modes of the field is fast compared to the stimulated Rabi coupling and the field is considered weak. If s" 1 then the Rabi oscillation is fast compared to spontaneous emission and the field is said to be strong. Setting s equal to unity defines the saturation condition... 

Note that negative Acoj (red detuning) produces a force attracting the atom to the intensity maximum while positive (blue detuning) repels the atom away from the intensity maximum. The spontaneous force or cooling force can also be written in tenns of the saturation parameter and the spontaneous emission rate. 

At large Euler numbers when AP < 1, the vapor essure may be calculated by the Clausius-Clapeyron equation. In this case Ps and ft in Eq. (9.38) correspond to the saturation parameters. 

Since the coolant and its vapor are conductive fluids. Toy = TLy = 7, where the subscripts s and f correspond to the saturation parameters and the interface surface, respectively. The saturation pressure and temperature are weakly connected (Sect. 10.9.1), so that Ts is determined practically by the external pressure Pg,oo-... 

Figure 39, Chapter 3. Bifurcation diagrams for the model of the Calvin cycle for selected parameters. All saturation parameters are fixed to specific values, and two parameters are varied. Shown is the number of real parts of eigenvalues larger than zero (color coded), with blank corresponding to the stable region. The stability of the steady state is either lost via a Hopf (HO), or via saddle node (SN) bifurcations, with either two or one eigenvalue crossing the imaginary axis, respectively. Intersections point to complex (quasiperiodic or chaotic) dynamics. See text for details.

Importantly, the discussion given above holds for a large class of possible rate functions making the interpretation of the saturation matrix independent of a specific functional form. We note that for any rate equation that is consistent with the generic form given in Eq. (47), we can specify an interval for the saturation parameter. Specifically, for an irreversible rate equation of the form... 

Figure 27. Interpretation of the saturation parameter. Shown is a Michaelis Menten rate equation (solid line) and the corresponding saturation parameter d (dashed line). For small substrate concentration S Km the reaction acts in the linear regime. For increasing concentrations the saturation parameter d ... 

The limiting cases are limvo 0 a = 1 and limy. x a = 0. To evaluate the saturation matrix we restrict each element to a well-defined interval, specified in the following way As for most biochemical rate laws na nt 1, the saturation parameter of substrates usually takes a value between zero and unity that determines the degree of saturation of the respective reaction. In the case of cooperative behavior with a Hill coefficient = = ,> 1, the saturation parameter is restricted to the interval [0, n] and, analogously, to the interval [0, n] for inhibitory interaction with na = 0 and n = , > 1. Note that the sigmoidality of the rate equation is not specifically taken into account, rather the intervals for hyperbolic and sigmoidal functions overlap. 

To highlight the relationship of the matrices A and to the quantities discussed in Section VILA (Dynamics of Metabolic Systems) and Section VII.B (Metabolic Control Analysis), we briefly outline an alternative approach to the parameterization of the Jacobian matrix. Note the correspondence between the saturation parameter and the scaled elasticity ... 

Despite the obvious correspondence between scaled elasticities and saturation parameters, significant differences arise in the interpretation of these quantities. Within MCA, the elasticities are derived from specific rate functions and measure the local sensitivity with respect to substrate concentrations [96], Within the approach considered here, the saturation parameters, hence the scaled elasticities, are bona fide parameters of the system without recourse to any specific functional form of the rate equations. Likewise, SKM makes no distinction between scaled elasticities and the kinetic exponents within the power-law formalism. In fact, the power-law formalism can be regarded as the simplest possible way to specify a set of explicit nonlinear functions that is consistent with a given Jacobian. Nonetheless, SKM seeks to provide an evaluation of parametric representation directly, without going the loop way via auxiliary ad hoc functions. 

Rate Equation Saturation Parameter Kinetic Parameter... 

An analysis is terms of the normalized partial derivative (saturation parameter) is invariant with respect to the specific rate equation. Note that not all choices are necessarily equally plausible. 

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... 

The overall influence of ATP on the rate V (ATP) is measured by a saturation parameter C (—oo, 1]. Note that, when using Eq. (139) as an explicit rate equation, the saturation parameter implicitly specifies a minimal Hill coefficient min > C necessary to allow for the reverse transformation of the parameters. The interval 6 [0,1] corresponds to conventional Michaelis Menten kinetics. For = 0, ATP has no net influence on the reactions, either due to complete saturation of a Michaelis Menten term or, equivalently, due to an exact compensation of the activation by ATP as a substrate by its simultaneous effect as an inhibitor. For < 0, the inhibition by ATP supersedes the activation of the reaction by its substrate ATP. 

The parameterization of the remaining reactions is less complicated. For simplicity, the rate v2(TP,ADP) is assumed to follow mass-action kinetics, giving rise to saturation parameters equal to one. Finally, the ATPase represents the overall ATP consumption within the cell and is modeled with a simple Michaelis Menten equation, corresponding to a saturation parameter 6 e [0,1], The saturation matrix is thus specified by four nonzero entries ... 

Figure 29 shows the bifurcation diagram for different values of the saturation parameter 6 of the ATPase reaction. Qualitatively, the plot shows the same... 

As the second step, the matrix of saturation parameters has to be specified. For simplicity, and following the model of Wolf et al. [126], all reactions are assumed to be irreversible and dependent on their substrates only. The matrix is then specified by 12 free parameters ... 

The dependence (1 TP of v, on ATP is modeled as in the previous section, using an interval C [—00,1] that reflects the dual role of the cofactor ATP as substrate and as inhibitor of the reaction. All other reactions are assumed to follow Michaelis Menten kinetics with ()rs E [0, 1], No further assumption about the detailed functional form of the rate equations is necessary. Given the stoichiometry, the metabolic state and the matrix of saturation parameter, the structural kinetic model is fully defined. An explicit implementation of the model is provided in Ref. [84],... 

Evaluating the structural kinetic model, we first consider the possibility of sustained oscillations. Starting with the simplest scenario, all saturation parameters are set to unity, corresponding to bilinear mass-action kinetics and... 

Figure 31. Dynamics of glycolysis. Upper panel The eigenvalue with the largest real part as a function of the feedback strength fcf j., of ATP on the combined PFK HK reaction. All other saturation parameters are unity 6% 1. Shown is (solid line) together with the imaginary...

In this case, the saturation parameter (/( of the overall reaction is simply the weighted sum of the individual saturation parameters 6% ... 

Note that this expression also provides a conceptual foundation to approximate complex processes, like ATP utilization, by a single reaction with a single saturation parameter sampled randomly from a specified interval. 

Slightly more complex are constraints with respect to the feasible intervals that are induced by interactions between metabolites. Until now, all saturation parameters were chosen independently, using a uniform distribution on a given interval We emphasize that this choice indeed samples the comprehensive parameter spaces, and for all samples, there exists a system of explicit differential equations that are consistent with the sampled Jacobian. However, obviously, not all rate equations can reproduce all sampled values. In particular, competition between substrates for a single binding site will prohibit certain combinations of saturation values to occur. For example, consider an irreversible monosubstrate reaction with competitive inhibition (see Table II) ... 

Estimating the saturation parameter for both reactants, we obtain... 

Since both reactants compete for the same binding site, both saturation parameters are interrelated 0J < 0J. A similar situation occurs for two substrates that compete for the same binding site. 

Nonetheless, note that such constraints hinge upon detailed knowledge of the functional form of the rate equation. For example, for noncompetitive inhibition, no restriction occurs Both saturation parameters may attain any value with in their assigned interval, independent of the saturation of the other reactant. We thus emphasize that choosing all saturation parameters... 

Alternatively, if such restrictions need to be incorporated, it is suggested to sample from the explicit rate equation. For example, the generic rate law in Eq. (Ill), discussed in Section VII.C.3, allows to generate an ensemble of saturation parameters that obey all relevant inequalities. 

The construction of the structural kinetic model proceeds as described in Section VIII.E. Note that in contrast to previous work [84], no simplifying assumptions were used the model is a full implementation of the model described in Refs. [113, 331]. The model consists of m = 18 metabolites and r = 20 reactions. The rank of the stoichiometric matrix is rank (N) = 16, owing to the conservation of ATP and total inorganic phosphate. The steady-state flux distribution is fully characterized by four parameters, chosen to be triosephosphate export reactions and starch synthesis. Following the models of Petterson and Ryde-Petterson [113] and Poolman et al. [124, 125, 331], 11 of the 20 reactions were modeled as rapid equilibrium reactions assuming bilinear mass-action kinetics (see Table VIII) and saturation parameters O1 1. 

For the irreversible reactions, we assume Michaelis Menten kinetics, giving rise to 15 saturation parameters O1. C [0, 1] for substrates and products, respectively. In addition, the triosephospate translocator is modeled with four saturation parameters, corresponding to the model of Petterson and Ryde-Petterson [113]. Furthermore, allosteric regulation gives rise to 10 additional parameters 7 parameters 9" e [0, — n for inhibitory interactions and 3 parameters 0" [0, n] for the activation of starch synthesis by the metabolites PGA, F6P, and FBP. We assume n = 4 as an upper bound for the Hill coefficient. 

Figure 41. Evaluating the stability of the simple example pathway shown in Fig. 40 Metabolic states and the corresponding saturation parameters are sampled randomly and their stability is evaluated. For each sampled model, the largest positive part within the spectrum of eigenvalues is recorded. Shown is the probability of unstable models, as a function of (A) The size of the system. Here the number of regulatory interactions increases proportional to the length of pathway (number of metabolites). Other parameters are maximal reversibility ymax 1 and p 0.5. (B) An increasing number of regulatory interactions. The number of metabolites m 100 is constant. Maximal reversibility ymax 1 and p 0.5. (C) Maximal reversibility v /v . and p 0.5. Other...

Figure 43. Measuring the importance of parameters. A The distribution of the saturation parameter 0pga (saturation of TPT with respect to PGA) for the full ensemble of models, chosen randomly from the unit interval. B The distribution of the saturation parameter 0pga after restricting the ensemble to stable states only. Within the ensemble of stable models, instances with 0pga close to the linear regime are overrepresented.

Figure 44. The correlation coefficient of the saturation parameters with stability, here identified with the largest real part within the spectrum of eigenvalues. Models (Jacobians) of the Calvin cycle are iteratively sampled and the correlation coefficient between each saturation parameter and the largest real part of the eigenvalues are evaluated. Negative values imply a negative correlation, that is, small saturation parameters correspond to a higher probability of instability.

The Kolmogorov Smirnov test Closely related to the visual approach employed in Fig. 42, the KS test evaluates the equality of two distributions. Under the assumption that a saturation parameter has no impact on stability, its distribution within the stable subset is identical (in a statistical sense) to its initial distribution. Deviations between the two distributions, such as those shown in Fig. 42, thus indicate a dependency between the parameter and dynamic stability. 

To investigate these two questions, a parametric model of the Jacobian of human erythrocytes was constructed, based on the earlier explicit kinetic model of Schuster and Holzhiitter [119]. The model consists of 30 metabolites and 31 reactions, thus representing a metabolic network of reasonable complexity. Parameters and intervals were defined as described in Section VIII, with approximately 90 saturation parameters encoding the (unknown) dependencies on substrates and products and 10 additional saturation parameters encoding the (unknown) allosteric regulation. The metabolic state is described by the concentration and fluxes given in Ref. [119] for standard conditions and is consistent with thermodynamic constraints. 

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