A new mathematical model for the interpretation of translational research evaluating six CTLA-4 polymorphisms in high-risk melanoma patients receiving adjuvant interferon

Abstract

Adjuvant therapy of stage IIB/III melanoma with interferon reduces relapse and mortality by up to 33% but is accompanied by toxicity-related complications. Polymorphisms of the CTLA-4 gene associated with autoimmune diseases could help in identifying interferon treatment benefits. We previously genotyped 286 melanoma patients and 288 healthy (unrelated) individuals for six CTLA-4 polymorphisms (SNP). Previous analyses found no significant differences between the distributions of CTLA-4 polymorphisms in the melanoma population vs. controls, no significant difference in relapse free and overall survivals among patients and no correlation between autoimmunity and specific alleles. We report new analysis of these CTLA-4 genetic profiles, using Network Phenotyping Strategy (NPS). It is graph-theory based method, analyzing the SNP patterns. Application of NPS on CTLA-4 polymorphism captures allele relationship pattern for every patient into 6-partite mathematical graph P. Graphs P are combined into weighted 6-partite graph S, which subsequently decomposed into reference relationship profiles (RRP). Finally, every individual CTLA-4 genotype pattern is characterized by the graph distances of P from eight identified RRP's. RRP's are subgraphs of S, collecting equally frequent binary allele co-occurrences in all studied loci. If S topology represents the genetic "dominant model", the RRP's and their characteristic frequencies are identical to expectation-maximization derived haplotypes and maximal likelihood estimates of their frequencies. The graph-representation allows showing that patient CTLA-4 haplotypes are uniquely different from the controls by absence of specific SNP combinations. New function-related insight is derived when the 6-partite graph reflects allelic state of CTLA-4. We found that we can use differences between individual P and specific RRPs to identify patient subpopulations with clearly different polymorphic patterns relatively to controls as well as to identify patients with significantly different survival.

Figure 1. Example how experimentally determined CTLA-4 genotype (top panel) for a patient (id = 55) is transformed into a) prp graph and b) PRP graph.

a-major allele, b-minor allele, ab-heterozygous allele status vertices. Each SNP is represented by a graph partition (rectangles), identified by the SNP code. Lines – graph edges, representing the co-occurrences of all alleles in the patient's CTLA-4 genotype.

Figure 2. Study graphs g(a) and G (b) constructed as union of all prp's (g) or PRP's (G).

Symbols as in Fig. 1, thickness of edges in g and G are proportional to co-occurrence frequencies of respective SNP pairs, connected by the edge.

Figure 3. Three examples showing how elements of distance vectors are computed for the same patient #55.

In all figures, prp ( RRP in c)) for this patient  =  dashed lines, rrp's (or RRP in c))  =  solid lines. Double arrows indicate mismatch in SNP co-occurrences. Elements of are sums of these mismatches (in computations, we add negative sign to make identity (zero mismatches) mathematically largest). a,b) Comparison of patient's genotype to the second and third reference SNP relationship patterns rrp₃ and rrp_{2 .} c) Comparison of patient's genotype to the 4^th reference SNP relationship pattern RRP₄ .

Figure 4. Decomposition of study graphs g (picture represents both cases and control subcohorts) into rrp's 1–8.

Case rrp's are shown by solid, control by dashed edges. Coefficients show the multiplicities of respective rrp's in the g-decompositions (top  =  case graph, bottom  =  control graph). Symbols as in Fig. 1.

Figure 5. Case-control discrimination by “missing” CTLA-4 genotype reference profile rrp₈ (dashed lines in all figures).

Solid lines in schemes a) – e) show five prp CTLA-4 genotype profiles, found exclusively for 219 (77%) patients identified from the complete case cohort by condition that their prp have maximal possible distance from the rrp₈. Symbols as in Fig. 1.

Figure 6. Selection of maximally case-control survival discriminating combination of distances from all RRP's.

Points are defined by the coordinates (see text) computed by averaging the distance differences over all patients separately in case and control sub-cohorts for all 190 possible RRP pairs. In the neighborhood of diagonal line are non-discriminatory combinations. The two lines are used to identify the combinations, with maximal case – control and control-case bias in PRP-RRP distances. The optimal selection is shown by boxes.

Figure 7. Histograms showing heterogeneity of distributions of individuals shown in the CTLA-4 genotype landscape, defined by the inter-personal differences in prp's for the five most discriminating RRP combinations.

Two selected combination of distance differences are plotted on x and y axes, on the z axis are numbers of subjects having a given combination of the distance differences. Blue-controls, red-cases.

Figure 8. Comparison of CTLA-4 genotype relationship profiles of five most case-control discriminating RRP's.

RRP₂ (dashed edges) is shown in both panels for reference. Symbols as in Fig. 1.

Figure 9. Comparison of RRP's, distances from which are most significantly different in the two survival groups (overall survival longer or shorter than 5 years).

RRP₁₀ is shown by solid edges in both panels (a,b) for reference.