Mixture model approach

Pan W, Lin J, Le G. 2002. How many replicates of arrays are required to detect gene expression changes in microarray experiments A mixture model approach. Genome Biology 3(5) research0022.1. 

The overall gain of the multiphase mixture model approach above is that the two-phase flow is still considered, but the simulations have only to solve pseudo-one-phase equations. Problems can arise if the equations are not averaged correctly. Also, the pseudo-one-phase treatment may not allow for pore-size distribution and mixed wettability effects to be considered. Furthermore, the multiphase mixture model predicts much lower saturations than those of Natarajan and Nguyen - and Weber and Newman even though the limiting current densities are comparable. However, without good experimental data on relative permeabilities and the like, one cannot say which approach is more valid. 

To use this mixture modeling approach, a priori studies must be made on all binary interactions existing in the mixture in order to obtain information on their mechanisms and the values of parameters for their description. This is a major limitation for the construction of mixture modeling since the number of binary interactions (AT) increases greatly with the number of mixture components (ri) (i.e., N = n(n - l)/2). Accordingly, for mixtures of 5 and 10 chemicals, 10 and 45 binary interactions must be previously studied, respectively. Considering the complexity of some of the mixtures to which humans or other species are exposed to, the characterization of all binary interactions is tedious or even impossible. Some alternative mixture modeling methods have therefore been proposed. 

We start with detailed definitions of the singlet and the pair distribution functions. We then introduce the pair correlation function, a function which is the cornerstone in any molecular theory of liquids. Some of the salient features of these functions are illustrated both for one- and for multicomponent systems. Also, we introduce the concepts of the generalized molecular distribution functions. These were found useful in the application of the mixture model approach to liquid water and aqueous solutions. 

In section 2.7, we introduced the generalized molecular distribution functions GMDFs. Of particular importance are the singlet GMDF, which may be re-interpreted as the quasi-component distribution function (QCDF). These functions were deemed very useful in the study of liquid water. They provided a firm basis for the so-called mixture model approach to liquids in general, and for liquid water in particular (see Ben-Naim 1972a, 1973a, 1974). 

Allison, D. B., et al. (2002). A mixture model approach for the analysis of micioarray gene expression data. 

Barthel J, Bachhuber K, Buchner R, Hetzenbauer H (1990) Dielectric spectra of some common solvents in the microwave region. Water and lower alcohols. Chem Phys Lett 165 369-373 Bates RG (1973) Determination of pH theory and practice. Wiley, New York Battino R, Clever HL (2007) The solubility of gases in water and seawater. In Letcher TM (ed) Developments and applications of solubihty, RSC Publishing, Cambridge, pp. 66-77 Ben-Naim A (1972) Mixture-model approach to the theory of classical fluids. II. Application to liquid water. J Chem Phys 57 3605-3612... 

The term mixture model is somewhat misleading when applied to the exact mixture-model approach to liquids (see Sec. 2.3). 

This section presents various aspects of the mixture-model approach to the theory of water. We start with some historical notes in Sec. 2.3.1, and then proceed with one very simple representative of the MM approach due to Wada (1961). Although this model was successful in reproducing some of the anomalies of liquid water, it suffers from two serious drawbacks. One is the question of the validity of the specific MM approach used in the theory, and the second is the validity of the assumption of the ideality of the mixture. We shall discuss these problems in Sec. 2.3.3, where an exact MM is developed. We show that the assumption of ideality, though inconsistent with the requirement of a successful MM, is not essential to the interpretation of the properties of water. 

In Sec. 2.3.5, we shall see that a central concept in the theories of water is that of the structure of water SOW). We shall devote Sec. 2.7.4 to discussing this concept. Here, we only point out that one way of defining the structure of water, is to use the mixture-model approach. The idea is very simple. We assume that liquid water may be viewed as a mixture of two components, say, an ice-like component and a close-packed one. We then identify the ice-like component with the more structured component, and hence, the degree of the structure can be measured simply by the concentration of this component. 

From (2.4.7), we can derive all the required thermodynamic quantities by standard relations. Before discussing applications of (2.4.7) it is worthwhile pausing to consider the equivalence of viewing the system as either a single component or as a mixture of two components. As in any mixture-model approach, we... 

This is the general expression for the heat capacity in the mixture-model approach. In our model, the first term on the right-hand side of (2.4.29) is zero and for the second term we have... 

A simpler version of the same principle uses the language of the mixture-model approach to liquid water. Within this approach, the principle states that there exists a range of temperatures and pressures at which there are non-negligible concentrations of two species one characterized by large partial molar volume and large absolute value of the partial molar enthalpy, and a second characterized by a smaller partial molar volume and smaller absolute value of the partial molar enthalpy. In order to obtain the outstanding properties of water, one also needs to assume that the concentrations of these two species are of comparable magnitude (see Sec. 2.3 for details). 

Such a splitting into four quasi-components can serve as a rigorous basis for a mixture-model approach for this liquid. This has direct relevance to the theory of real liquid water. 

We can now re-interpret the quantity Um)s — Um)o oti the right-hand side of (3.4.7) in terms of structural changes. A more appropriate term would be redistribution of quasi-components. We shall do it in two steps. First, we use the binding energy distribution function xbe introduced in Sec. 2.3. Second, we shall reformulate this quantity in terms of structure as defined in Sec. 2.7.4. Finally, we shall use the same quantity to apply to a two-structure mixture-model approach to water. 

Formulation of the problem within the mixture-model approach... 

We now briefly apply a continuous mixture-model approach to demonstrate the general aspect of the contribution of structural changes in the solvent. For simplicity, we refer to the energy change associated with the H(f>0 process. Following Sec. 3.5, we write... 

Appendix F Some Identities in the Mixture-Model Approach... 

Note that the temperature dependence of AG depends on the degree of structure of water, but it is not necessarily a monotonic dependence. We have seen that A5 depends on the structural changes induced by the solute, and the extent of the structural changes depends, in an ideal mixture model for water, on product XiXp. This means that in some regions, an increase in X can either increase or decrease the product Xi — xi). In a nonideal mixture-model approach, the dependence of the structural changes induced by the solute is not so simple as xi — Xi), but the general conclusion that A5 is non-linear or even monotonic in Xi is still valid. 

To the best of my knowledge, the formulation of the principle and its significance to the understanding of the anomalous properties of water was first published in 1972. This principle and its implications were repeatedly used later by many authors who rediscovered it. Similarly, the exact entropy-enthalpy compensation theorem and its implications for the theory of aqueous solutions were first discussed in terms of a mixture-model approach in 1965. Later, it was proved in a much more general form in Ben-Naim (1975b, 1978b). This theorem was reproved several times by several authors using different nomenclature and different notations. 

Generalized Molecular Distribution Functions and the Mixture-Model Approach to Liquids... 

Here, we have expressed Cy exphcitly in terms of the singlet and pair GMDF s. We recall that the heat capacity can also be expressed in terms of ordinary MDF s up to order four (see Section 3.8). In this respect, relation (5.57) for Cy is somewhat simpler than the previous one. A different interpretation of the term ACy, using the mixture-model approach, will be developed in Section 5.9. 

The procedure employed in this section will be generalized in the next section to obtain the basis of the so-called mixture-model approach to liquids. In fact, the treatment of a large number of problems in physical chemistry and especially in biophysics rests on arguments similar to those given above. 

Z THE MIXTURE-MODEL APPROACH TO LIQUIDS CLASSIFICATIONS BASED ON LOCAL PROPERTIES OF THE MOLECULES... 

Note that in (5.107), E stands for the energy, whereas in previous expressions in this section, we have used E for any extensive thermodynamic quantity. We also recall that this relation was derived in Section 5.5 on the basis of direct arguments which do not depend on the mixture-model approach. 

THERMODYNAMIC QUANTITIES USING THE MIXTURE-MODEL APPROACH... 

Because of the above difficulties, it is no wonder that many scientists have incessantly searched for other routes to studying liquid water and its solutions. The most successful approach has been the devising of various ad hoc models for water. In subsequent sections, we describe some of these theories and view them as approximate versions of the general mixture-model approach treated in Chapter 5. 

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