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Fig. 5. Tie-lines across regions of L_d-L_o coexistence are evaluated by choosing one spectrum to lie along a given tie-line (1:1 DOPC/DPPC-d₆₂ + 15% Chol) and searching for a second spectrum such that both spectra can be represented as a superposition of the same two physical endpoint states. (a and b) This is accomplished by choosing a specific DOPC:DPPC-d₆₂ ratio (in this case along the red line of 1:2) and interpolating between acquired spectra at the red dots until a linear combination of the acquired spectrum (1:1 + 15% Chol) and the interpolated spectrum (1:2 + X% Chol) can be found that closely resembles a physical L_d phase endpoint spectrum. The best fit cholesterol composition, and therefore the tie-line slope, is determined through a least-squares minimization between the calculated L_d endpoint spectrum and an acquired reference spectrum from a composition expected to reside near the L_d tie-line endpoint (yellow square near the L_d label on the phase diagram). Errors in slope are determined directly from the covariance matrix produced by the least-squares minimization as described in Materials and Methods. Results from minimizations are shown in the third column of d for the best fit slope as well as slopes corresponding to ±1 and ±2 standard deviations away from the best fit solution. In the two right columns of d, the calculated endpoint spectrum is blue, the reference spectrum is green, and the residual is black. Once the slope is determined, we again perform a least-squares minimization between the calculated (L_o) endpoint and an acquired reference spectrum (yellow square near L_o label on phase diagram), and determine the location of tie-line endpoints based on the best fit parameter values as shown in c. Errors in tie-line endpoints are also determined from the covariance matrix at the best fit solutions and are propagated through to the tie-line calculation. Parameter errors are proportional to the residual at the solution and therefore depend on the availability of reference spectra in close proximity to the calculated endpoint. In this example, the L_d subtraction yields a good fit to a reference spectra that is very close to the calculated endpoint, whereas the L_o subtraction residuals remain large.

Fig. 6. (a) Within a three-phase region, each acquired spectrum is a linear superposition of endpoint spectra corresponding to DPPC-d₆₂ lipids in pure L_d, L_o, and S_o phases. Because the compositions and spectra of the endpoints are initially unknown, we first solve for three basis spectra by fitting each frequency component of the three-phase spectra to a plane as described in Materials and Methods. (b) The calculated basis spectra correspond to three single-component compositions outside of the three-phase triangle and are not themselves physical. (c) Linear combinations of these basis spectra are physical DPPC-d₆₂ spectra for compositions within the three-phase triangle. (d) The composition of each endpoint is determined by solving for the linear combination of basis spectra needed to resemble a reference spectrum acquired at a composition near the suspected triangle vertex. Errors in fitting parameters determine errors in the locations of vertices. (e) The relative intensities of endpoint spectra within a given acquired spectrum are given by a three-phase Lever rule. In our experiments, the intensity of spectral peaks from lipids in more ordered phases is attenuated due to different rates of spin relaxation in different phases [Hsueh YW, Gilbert K, Trandum C, Zuckermann M, Thewalt J (2005) Biophys J 88:1799-808]. All five acquired spectra within the three-phase triangle at 10°C can be constructed from endpoint spectra by using a Lever rule. Attenuation factors are applied consistently across all five spectra (here, S_o attenuation = 3 and L_o attenuation = 2) and are in good agreement with data from relaxation measurements (data not shown). Compositions shown are S₁ = 1:2 + 25%, S₂ = 1:2 + 20%, S₃ = 1:4 + 20%, S₄ = 1:4 + 30, and S₅ = 1:1 + 15%, where compositions are written as DOPC:DPPC-d₆₂ (mol:mol) + % Chol.