Geometric Approach to Orthogonality in 2DLC

The retention maps summarized in Fig. 12.2 allow one to visually compare the degree of separation orthogonality. Several promising 2DLC setups for the separation of tryptic peptides have been identified. However, to quantitatively compare the data orthogonality and estimate an achievable 2DLC peak capacity, more rigorous mathematical tools are needed. 

The description of the degree of retention data correlation is more complicated than it appears. For example, the 2D retention maps cannot be characterized by a simple correlation coefficient (Slonecker et al., 1996) since it fails to describe the datasets with apparent clustering (Fig. 12.2f). Several mathematical approaches have been developed to define the data spread in 2D separation space (Gray et al., 2002 Liu et al., 1995 Slonecker et al., 1996), but they are nonintuitive, complex, and use multiple descriptors to define the degree of orthogonality. 

In their seminal work from 1983, Davis and Giddings used a statistical theory to define the number of peaks observable in 1DLC separation upon the injection of a sample of different complexity on a column of a given peak capacity (Davis and Giddings, 1983). The theory was later extended into 2D separation space (Davis, 2005 Shi and Davis, 1993), also discussed in Chapter 2 of this book. The theory implies that when the 1D or 2D separation space is randomly covered with the number of peaks equal to the separation space peak capacity (area), the normalized surface coverage is 

Similar mathematical solution can be derived from a Poisson distribution of random events in 2D space. The probability that 2D separation space will be covered by peaks in ideally orthogonal separation is analogical to an example where balls are randomly thrown in 2D space divided into uniform bins. The general relationship between the number of events K (number of balls, peaks, etc.) and the number of bins occupied F (bins containing one or more balls, peaks, etc.) is described by Equation 12.3, where N is the number of available bins (peak capacity in 2DLC). 

In the special case considered here, where the number of events K is equal to number of bins (N=K peak capacity is equal to the number of separated components), this formula can be rearranged into Equation 12.3. 

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