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Those describe the genotype of an animal (or plant, virus etc.).

The nomenclature can be very varied and domain-specific, but for those two examples:

$dt^{sz}$ is a Syrian golden hamster model with a spontaneous mutation (i.e. occurred during breeding, without specific human intervention) which gives predisposition to seizures.
They are described in this paper:

Spontaneous seizures: a new mutation in Syrian golden hamsters. - Yoon et al. - J Hered 1976

SPONTANEOUS motor seizures with tonic muscle spasms and severe paralytic states have been observed in inbred line BIO® 86.93 of Syrian hamsters (Mesocricetus auratus auratus). These animals had been inbred for 23 generations and maintained by Bio-Research Consultants. The seizures, which last from 2 to 5 hours, can be precipitated by mild stress, and occur in animals between the ages of 30 days and 60 days. Breeding tests have shown that the condition is transmitted as a simple recessive; accordingly, the gene symbol sz is proposed for this condition.

The lowercase indicates that the mutation is recessive that is, you need to have two copies of the mutation to see its effect (its phenotype).

dt comes from dystopia, another symptom of the mutation.

As for Rgs9-Cre/+;gtROSA/+ mouse, the genotype is a bit more complex.

Whenever you see /+ it means that the animal is heterozygous, that is, it only carries one copy of that specific gene.

For instance these are transgenic animals producing the bacterial protein Cre under the Rgs9 promoter. A promoter is a sequence of DNA that controls the expression of a certain gene. In this case the DNA sequence for the promoter of the gene Rgs9 was attached to the DNA sequence for Cre. This new construct was then inserted into the genome of the mouse. The result is a mouse in which the cells that normally transcribe the Rgs9 gene also produce Cre.

This mouse line was then crossed with a reporter mouse line called gtROSA26, which expresses a certain protein called $\beta$-galactosidase in a Cre-dependent manner.

The gene for the galactosidase is present in all the cells, but only where Cre is present (in our case the Rgs9 positive cells) it will be transcribed. Eventually the crossing leads to the expression of $\beta$-galactosidase in Rgs9 positive cells. $\beta$-galactosidase has the nice property that it can change the colour of certain chemicals to blue, which allows to easily visualize the cells that produce it. This can help you answer questions like: "where is the Rgs9 gene transcribed?".

Wikipedia has a neat page on Gene nomenclature with lots of links to nomenclature guidelines.

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These strings are specific genotypes of these animals. They often denominate mutations or transgenes. For the hamster $dt^{sz}$ stands for dystopic, the sz comes from the first denomination of this symptoms as seizure. See here for a brief explanation.

For the mouse you have both alleles named: Rgs9-cre is a transgene, + in this context always stands for the wildtype. So this mouse has one allele of the transgene Rgs9-cre while the other allele is wildtype for it. Transgene constructs with the cre-recombinase can be inserted into the genome upon induction, which is a useful tool to modify genes. This needs to be done in mice which harbour the gtROSA transgene, which denominates a specific genetic background for the mice. See here for more details:

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Thank you! Do you have any suggested books/articles/material for me to read as a primer to this material? I need to be familiar with this stuff to be working in a research lab. –  Dissenter Jun 13 at 20:28

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