Computing the probabilities of HLA-like matching

Abstract

We present several alternative computation schemes, accompanied with appropriate software, to compute the probabilities of the (2n+1) possible match levels between the alleles in n genetic sites of a given individual, and the alleles in the same n sites of an individual who is drawn randomly from a given population. The modeling generalizes the asymmetric HLA-criterion which defines the donor-recipient immunological compatibility in kidney or bone-marrow transplantation. We discuss our algorithms by order of their run-time complexity with respect to n. We show the advantage of using computational schemes over explicit expressions even for the HLA present-day count of n=3.

Appendices

Appendix A: The mismatch equations for three HLA sites

Barnes and Miettinen (1972) consider a recipient who has antigens corresponding to alleles a _i and a _j from HLA site A, and b _k and b _l from site B. (In those days the DR locus was not considered relevant to graft survival). Thus, they deal with an HLA system with two sites only and with the 5 possible match levels considering two sites (a total of zero mismatches, 1 mismatch, down to 4 mismatches). In what follows we generalize their results to three sites, to account for mismatches in site DR as well. We retain the assumption on statistical independence, and the indifference between combinations of mismatches in the specific three sites which amount to a fixed given value of I. Thus, for example, the three outcomes “one single foreign antigen occurs at donor’s site A”, and the same with site B, and the same with site DR—are all equivalent (though not equal-probable), and they are contained in one event I=1.

Define:

a _j and a _i :

alleles of recipient’s site A.

b _k and b _l :

alleles of recipient’s site B.

c _o and c _p :

alleles of recipient’s site DR.

I :

total number of HLA mismatches.

f _u :

P{a _i ∪a _j } as in (1), pertaining to site A.

g _u :

P{b _k ∪b _l } as in (1), pertaining to site B.

h _u :

P{c _o ∪c _p } as in (1), pertaining to site DR.

S(−u,f):

probability of homozygosity at the donor’s, as in (2), pertaining to site A.

S(−u,g):

probability of homozygosity at the donor’s, as in (2), pertaining to site B.

S(−u,h):

probability of homozygosity at the donor’s, as in (2), pertaining to site DR.

The probability of perfect match is clearly

$$ P(I = 0) = f_{u}^{2}\cdot g_{u}^{2}\cdot h_{u}^{2}.$$

(A.1)

Since {I=1} is the set union of three disjoint events “exactly one mismatch, and it occurs in A”, “exactly one mismatch, and it occurs in B” and “exactly one mismatch, and it occurs in DR”,

(A.2)

where the terms in brackets signify exactly one mismatch in site A, B, or DR, respectively. See (4) in the main text. Now, a total of two mismatches occur when there are two single mismatches in two sites, and zero mismatches in the other, or when there are two mismatches in one single site. Invoking (1)–(3) again we get:

(A.3)

In a similar manner the equations for 3, 4 and 5 mismatches follow as well:

(A.4)

(A.5)

and

(A.6)

For an absolute mismatch we have the obvious:

(A.7)

Appendix B: A guide to the Software

Web address: The codes for Algorithm 2 (MS-Excel), Algorithm 3 (MATLAB) and Algorithm 4 (MS-Excel) can be found at http://www.bgu.ac.il/~idavid. (See “Kidney Matching Computation”). All of the codes (except for one variant of Algorithm 4, see below) now contain the data per Example 2.

Algorithm 2

To see the VB code which underlies the calculation button in “main”, one may press Alt+f11 (editor), then “tools” → “macros” → “step inside”. The current file is set or n=5 HLA sites, and one may change the alleles in the “patient” sheet, and the genetic distribution per site in the “sites” sheet, as he/she pleases. Anyone who is proficient with Excel would also know how to handle a different number of sites.

Algorithm 3

In the folder “Algorithm 3” one may recognize four MATLAB files. To operate them one should store them in a designated folder, then to open MATLAB7 and to change the working directory to that folder. Then press “load hla_ws”, and then run the program by “calc_pj(patient)”. The answer is given in the familiar MATLAB format. To view the data (the same data as in the Excel workbook) one may go to “window” → “workspace” and then to examine/modify the matrices “patient” (equivalent to the above Excel-sheet “patient”) and “hla_probs” (equivalent to the above Excel-sheet “sites”). To remove/add sites just erase/add adjacent columns in these two matrices!

Algorithm 4

In this folder there are two Excel files. “Algorithm 4—five sites.xls” runs poly_mult of Fig. 4, which uses power2_mult as explained in Sect. 5 above. Both functions use a procedure, 2poly_mult, which multiplies two polynomials of arbitrary degree. (Our 2poly_mult appears in the VB code of this workbook, currently it does not employ the FFT algorithm). Running the program is made by the appropriate button in the sheet “main”. The sheets “patient” and “sites” are the same as in the workbook for Algorithm 2, so everything is embedded in the framework of HLA computation as before. Alternatively, “Algorithm 4—fifty quadratics” contains only two sheets, “BigPoly—main” and “quadratics”. The second sheet holds the coefficients of fifty random quadratics (array pij of Fig. 4). The workbook as a whole demonstrates the algorithm for a large value of n.