A mathematical technique for estimating the melting points of triacylglycerols from the component fatty acid melting points

Abstract

A method of correlating the value of the melting point of triacylglycerol (tag) with the values of the melting points of the three fatty acids constituting the tag as its acyl radicals was studied by applying the least squares method. The method was applied to the sets of values of the melting points of the a, b', and b forms of crystals of tag with saturated acyl radicals comprising decanoyl, lauroyl, myristoyl, palmitoyl, and stearoyl. The value of the melting point of the crystal form of the tag is expressed by prediction equations that comprise the values of the melting points of the three fatty acids above as variables. The best-fit equation can be fixed by satisfying the criterion of least squares to the prediction equations. The method enables us to estimate the values of the melting points of possible molecular species of tag consisting of the acyl radicals above. The correlation coefficient relevant to the correlation between the values of the melting points of the particular crystal form and the corresponding values of the best-fit equation is obtained. The more thermodynamically stable the crystal form, the larger is the value of the correlation coefficient.

A mathematical technique for approximating the values of melting points of diacylglycerols

“…The coefficients and constant fk p p 2 ð1; 2; 3Þg j that satisfy the criterion of least squares are denoted by fk Ãp p 2 ð1; 2; 3Þg j . As described in detail in a previous paper [10], the column vector of k Ãp or ðk Ãp Þ is expressed as …”

“…Now, until further notice, let t Y be an arbitrary value or t Y = 0 for numerical calculation. We postulate that each enantiomer and the mixtures of the pair of enantiomers have an identical melting point value, e.g., z sn-YOG = z sn-GOY = z b-YOG , and the physical property arising from the radical linked to sn-1 is identical to that arising from the radical linked to sn-3, provided the comparable conditions are the same [10]. The melting points of the b crystal form of DAG of which the acyl radicals are being selected from the elements of F are cited from [9] and are shown in the third column in Table 1.…”

“…These symbols will be used for representing the two acyl radicals and the one free hydroxyl radical in the molecule of DAG; e.g., 1,3-dilauroyl-glycerol is denoted by b-GYG, and b-GOY indicates sn-GOY and/or sn-YOG. A mixture of the pair of enantiomers is also expressed by b-GOY, irrespective of the mixing ratio of them [10,11].…”

“…Since the technique enables us to estimate the values of melting points of DAG, it will probably be useful for dealing with molecular species of DAG of which the physical properties are unknown, and will be helpful for designing oleo materials including DAG for industrial use. In a previous paper, we described a mathematical technique for estimating the melting points of TAG from the component fatty acid melting points, where the least squares method without accompanying variable parameters was employed [10]. Since the technique in the previous paper is considered to be a sort of prototype of that to be dealt with in this paper, that paper should be referred to for details of the technique.…”

“…The square of the correlation coefficient or r f 2 is called rsquared or contribution [10]. The expressions for Covðz j ; g j Þ, Varðg j Þ, and r g are obtained by replacing the symbol f j in Eqs.…”