This subproject is one of many research subprojects utilizing the resources provided by a Center grant funded by NIH/NCRR. Primary support for the subproject and the subproject's principal investigator may have been provided by other sources, including other NIH sources. The Total Cost listed for the subproject likely represents the estimated amount of Center infrastructure utilized by the subproject, not direct funding provided by the NCRR grant to the subproject or subproject staff. A novel method has been developed to determine the tie-lines of any three-component two-phase coexistence region by fitting electron-spin resonance (ESR) spectra obtained from compositions on the coexistence curve and within the coexistence region. The method was applied to the liquid-ordered (Lo) and liquid-disordered (Ld) phase coexistence region of the lipid system brain-sphingomyelin/dioleoylphosphatidylcholine/cholesterol (SPM/DOPC/cholesterol). The essential aspects of the method were a unique parameterization of the coexistence region called a "ruled surface" and knowledge of the coexistence curve. Least-squares fitting of the ESR spectra with the parameterized model resulted in a tie-line field consistent with other known information on this lipid system. The use of the ruled surface allows one to guarantee that tie lines do not cross. The best-fit tie-line field consisted of the set of tie lines which were not exactly parallel;they exhibited a gradual change in slope with the largest slope within the coexistence region connecting the coexistence curve compositions with the highest and lowest cholesterol concentrations. The results were compared with those from more constrained methods of representing the tie-line fields. An accurate determination of the tie-line field of phase coexistence regions in lipid systems is a necessary step in determining coexisting lipid compositions to serve as models of cell plasma membranes.