What you are describing is a typical situation in any mapping of biological annotations from one database to another. There are two main categories of problems here:
Difference between Gene and Transcript and Probe
You have mentioned that you are specifically interested in gene symbols, whereas "The GeoMx DSP looks at the expression of a panel of mRNA transcripts". Genes are different from transcripts.
It makes sense for probes to be transcript-specific, because that is what is represented at a single molecule level. I would guess (but don't know) that this is the reason for the first number after the Probe DisplayName, representing a transcript probe, rather than a gene probe (e.g. IFNA7/17, rather than IFNA7).
Given that different genes can share sequence - and I'm aware that IFNA is a particular issue in this regard - it is not surprising that a single transcript probe matches multiple different genes (or multiple different full-length transcripts, for that matter).
I would expect that nanoString already has software to carry out appropriate deconvolution of probe-level counts to create transcript-level counts. That should be your first priority: communicating with nanoString to get transcript-level counts for your data. You shouldn't be trying to process the data from the raw probe-level counts; these are too fine-grained to be useful for what you want to do.
Once you have transcript-level counts, you can then decide how to map the transcript to genes. While some transcripts can be components of multiple different genes, it is usually the case that each transcript can be uniquely mapped to an isoform of a single gene. It is a project-specific decision whether you take the counts from the most abundant transcript for each gene, or aggregate them together for a total count, or choose the counts for the longest transcript, or something else.
Database Mapping / Conversion
In almost all cases where annotations from one database need to be converted to another database, there is a many-to-many mapping (i.e. many things from the first database can be mapped to many things from the second database). More verbosely, relating to your particular situation:
- There will be some gene symbols that have no associated Entrez IDs
- There will be some Entrez IDs that have no associated gene symbols
- There will be some gene symbols that have multiple associated Entrez IDs (as in your example)
- There will be some Entrez IDs that have multiple associated gene symbols
Deciding on what to do for each of these situations is also a project-specific decision; there is no perfect answer that will work in all situations.
Because you say you need Entrez IDs (not a strict requirement for GSEA, but you have claimed this in your question), the gene symbols with no associated Entrez IDs will need to be discarded, which will be a loss of data. Due to the bulk / aggregate nature of GSEA, this is unlikely to substantially influence results.
Where there are multiple associated Entrez IDs, it would not be surprising if those IDs were almost always present together in the same gene sets. To avoid loss of data in this situation (with some small risk of false-positive association), the multiple mappings can be duplicated. For example, a count that is assigned to 5 different genes could be treated as either a single count for each of those 5 genes, or as 0.2 counts for each of those 5 genes (i.e. one count distributed equally among all the possible genes), or assigned randomly to one of the 5 different genes - all of these approaches have been proposed and implemented in the past for gene counting. The other option is, as you say, dropping counts entirely, but that seems like an approach which would lead to bias away from well-annotated genes.
In any case, your general problem needs a project-specific decision. It's up to you to discuss this with your collaborators to find the best approach that answers the biological question your group has about the data.
