Generalized compensation (n-colors)

(Go back one page to three-color compensation, or back to the introduction)

As we cram more and more colors on each laser, which are closer and closer together, we have to compensate between every pair of channels. Furthermore, we are starting to use fluorophores which can be excited by multiple lasers, which generates the need for cross-laser compensation. (For instance, Cy5PE can be efficiently excited not only by the 488 nm Argon laser line, but also a 633 nm HeNe laser line--and emit very similarly to APC. APC, however, is not excited by the 488 nm line. Therefore, we can compensate from the first laser Cy5PE channel to the APC channel to remove the contribution of Cy5PE fluorescence on the second laser, leaving only APC fluorescence).

Multi-color compensation is a simple extension of two-color compensation, through the use of linear algebra. Let's assume that we are measuring n different fluorescent molecules, all of which may contribute fluorescence to each of the other channels. Therefore, the measured signal in any channel is given by:

M(1) = A(11) x F(1) + A(21) x F(2) + ... A(n1) x F(n)
M(2) = A(12) x F(1) + A(22) x F(2) + ... A(n2) x F(n)
M(n) = A(1n) x F(1) + A(2n) x F(2) + ... A(nn) x F(n)
where M(i) is the measured fluorescence in channel i; F(i) is the amount of fluorescent molecule (i) present on the cell of interest, and A(ij) is the ratio of the fluorescence of molecule (i) in channel (i) to the fluorescence of molecule (i) in channel (j).

Thus, the coefficients A are the spillover coefficients. These values are determined by running singly stained samples: if we are compensate fluorescent molecule i, then F(j) will be zero for every j that is not equal to i. Therefore, the coefficient A(ij) is equal to the measured value M(j) divided by the measured value M(i) (therefore, A(ii) = 1).

Once we have determined the coefficients A, we can exactly determine the amount of each fluorescent molecule by solving the above set of equations: there are exactly n unknowns in n equations. In matrix algebra, these equations are written as:
M = A x F
If we premultiply both sides by the inverse of the coefficent matrix (A'), then we have:
A' x M = A' x A x F = F
Computationally, therefore, we invert the coefficient matrix A, multiply by the measured values M, and the result F gives the concentration of each fluorescent molecule present in the measured cell. F is the completely compensated set of signals. Note that the coefficients in the matrix A' contains the compensation values.

In the case of three-color compensation commonly done on today's flow cytometers (first laser), only two pairs of compensations are done (for a description, click here). In this case, the coefficients A(31) and A(13) are assumed to be zero (i.e., no spillover between fluorescein and Cy5PE and vice versa).

Finally, it's time to deal with autofluorescence.