Unmixing linear

If reference emission spectra of a set of pure fluorophores are available, and if an emission spectrum of an unknown mixture of any combination of these fluorophores is acquired under the same conditions, this equation can be used to determine the abundance of the different fluorophores in the mixture. The use of this equation to determine the abundance of the fluorophores present is called linear unmixing. To illustrate the basis of linear unmixing, we will first use this equation to analyze the emission spectra of the mix capillary containing an unknown mixture of Cerulean and Venus depicted in Fig. 8.1. The unmixing approach we describe will utilize reasonable guesses for the values of x1 (representing the abundance of Cerulean) and x2 (representing the abundance of... 

Fig. 8.3. The basis of linear unmixing. Unnormalized emission spectra of the three capillaries are shown in panel A. The linear unmixing algorithm is based on the hypothesis that a complex emission spectrum (an emission spectrum of a sample containing 2 or more fluorophores) can be modeled as a weighted sum of the emission spectra of the individual fluorophores present. Thus, the Mix spectrum should be the sum of the Cerulean and Venus spectra after each is multiplied by an abundance factor. In panel B the abundance factor for Venus is held at a value of 1, while the value of the Cerulean abundance factor is varied from 0.6 to 1.4. Because the Cerulean and Venus capillaries each contained 10 /iM of fluorophore, an abundance range of 0.6-1.4 corresponds to a concentration range of 6-14 /iM. In panel C the Cerulean abundance factor is held at a value of 1 (10 /rM) while the abundance factor for Venus was altered from 0.2 to 1 (2-10 /rM). Note that when the Cerulean spectrum was multiplied by 1 (corresponding to 10 /rM) and added to the Venus spectrum multiplied by 0.6 (corresponding to 6 /rM), the linear unmixing model matched the complex spectrum measured for the mix capillary.

This is the same equation as the standard linear unmixing equation for two fluorophores. 

Standard linear unmixing of a spectral image of a sample composed of two fluorophores yields a measure of the concentration of each fluorophore present for each pixel. If FRET is occurring, linear unmixing will produce an apparent donor concentration ( apparent) that underestimates the true donor concentration (d) by a factor of 1 -ED ... 

Linear unmixing will also produce an apparent acceptor concentration ( apparent) that will over-estimate the true abundance of the acceptor (a) ... 

Linear unmixing is applied to each spectral image to produce four measurements at each pixel, Apparent an[Pg.384]

Linear unmixing of the spectral image acquired at will also yield two observables ... 

Multi-spectral imaging and linear unmixing add a whole new dimension to laser scanning fluorescence microscopy. Biotechniques 31, 1272-8. 

Zimmermann, T. (2005). Spectral imaging and linear unmixing in light microscopy. Adv. Biochem. Eng. Biotechnol. 95, 245-65. 

Thaler, C. and Vogel, S. S. (2006). Quantitative linear unmixing of CFP and YFP from spectral images acquired with two-photon excitation. Cytometry A 69, 904—11. 

The beta PDF is widely used in commercial CFD codes to approximate the mixture-fraction PDF for binary mixing. This choice is motivated by the fact that in many of the canonical turbulent mixing configurations (Fig. 5.8) the experimentally observed mixture-fraction PDF is well approximated by a beta PDF. However, it is important to note that all of these flows are stationary with Nmf = A m — 1 = 1, i.e., no linear mixture exists between the inlet conditions. The unmixed PDF is thus well represented by two peaks one located at % = 0 and the other at % = 1, which is exactly the type of behavior exhibited... 

The most simple diblock copolymers are linear chains, in which one part of the chain consists of one type of monomer, say polystyrene (PS), and the other one of another type, say polybutadiene (PB), as illustrated in Figure 14. PS and PB usually phase separate at low temperatures however, because of their chemical connectivity, block copolymers cannot unmix on a macroscopic scale. They can only phase separate on a microscopic scale, the size of which is determined by the length of the polymers. 

In diffusion combustion of unmixed gases the combustion intensity is limited by the supply of fuel and oxidizer to the reaction zone. The basic task of a theory of diffusion combustion is the determination of the location of the reaction zone and of the flow of fuel and oxidizer into it for a given gas flow field. Following V. A. Schvab, Ya.B. considered (22) the diffusion equation for an appropriately selected linear combination of fuel and oxidizer concentrations such that the chemical reaction rate is excluded from the equation, so that it may be solved throughout the desired region. The location of the reaction zone and the combustion intensity are determined using simple algebraic relations. This convenient method, which is universally used for calculations of diffusion flames, has been named the Schvab-Zeldovich method. 

For a such case, linear arrays of electrodes may be used however, this will lead to extended structures when using, e.g., more than four electrodes. Instead of using such unidirectional motion, the droplet may be moved in circular fashion by a square-like 2x2 array of electrodes. Indeed, faster mixing times compared with simple droplet merging can be achieved, albeit not faster than for the respective four-electrode linear structure. Actually, a small portion of the droplet remains unmixed. This was explained as due to the droplet pivoting around the array center. For this reason, a non-symmetric array (2 x 3) was developed as mentioned below. 

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