Q:

What makes rice gluten-free?

Rice does NOT trigger symptoms in gluten-related disorders and is commonly called gluten-free. In its broadest definition covering several species of cereal grains, gluten is composed of two families of storage proteins, prolamins and glutelins, which are ALSO present in rice. So what makes rice gluten-free? Is there a defining sequence-related or structural aspect of gluten missing from rice?

A:

tl;dr:  The terminology used to compare plant proteins is used on a level of abstraction too high to give meaningful medical results. Prolamins are too broad of a chemical category to assay their biology and qualities in disease.
Gluten-related disorders is a less than ideal term for a wide variety of disorders involving grains. While coeliac disease is triggered by wheat gluten, a wheat allergy is less well placed into this analytical category. For the consumer's sake of simplicity the equation of gluten=wheat like grains makes some sense, as a gluten-free product usually is also free of wheat.
But all storage proteins in grasses are classified into four groups, called Osborne fractions:

Albumin: water soluble, sometimes called leukosin in wheat (most likely allergen)
Globulin: soluble in salt-water, called edestin in wheat
Prolamin: soluble in strong ethanol, called gliadine in wheat and oryzin in rice (not to be confused with the enzyme, but because of the name double often called just prolamin)
Glutelin: alkali/acid-soluble to non-soluble, called glutenin in wheat and oryzenin in rice

That is a very broad classification system for a mixture of several proteins in one group (or "family") based on very basic properties. The actual contents, configurations and other chemical, biological or medical qualities of the various proteins are not really covered with this scheme. But it is already apparent that although the proteins from rice and wheat may be classified similarly but are different enough to get their different names early on.
For StackExchange-simplicity's sake let's concentrate on the example of coeliac disease:

Celiac disease (CD) is more than just an “allergy” or “sensitivity” to wheat and gluten. It is a lifelong, permanent intolerance to the gliadin fraction of wheat protein and its related alcohol-soluble proteins (prolamins) found in rye and barley. In patients with the genetic susceptibility to CD, ingesting these proteins leads to an autoimmune enteropathy that will self-perpetuate as long as these foods remain in the diet. The good news is that, unlike most autoimmune conditions, removal of the environmental trigger (gluten) from the diet of a biopsy-proven celiac results in complete symptomatic and histologic resolution of the disease in the majority of patients.
Differentiating CD from wheat allergy, gluten sensitivity, and other autoimmune gastrointestinal (GI) diseases (such as Crohn’s disease) can be challenging.
From: Michelle Maria Pietzak: "Dietary Supplements in Celiac Disease", in: S. Devi Rampertab & Gerard E. Mullin (Eds.): "Celiac Disease", Humana Press: New York, Heidelberg, 2014, p137.)

If is only the gliadin fraction and the closely related proteins in rye and barley that trigger this disease, then how does gliadin compare to oryzin?

A singular feature of rice is that prolamin, which represents the major endosperm storage protein in other cereals except, oats (Shewry and Halford, 2002), is a minor protein in all rice grain milling fractions, whereas glutelin is the dominant protein in brown and milled rice. The proportion of albumin, globulin, glutelin and prolamin has been reported to be 5–10, 7–17, 75–81 and 3–6%, respectively, in brown rice, 4–6, 6–13, 79–83 and 2–7%, respectively, in milled rice, and 24–43, 13–36, 22–45 and 1– 5%, respectively, in rice bran (Adebiyi et al., 2009; Agboola et al., 2005; Cao et al., 2009; Ju et al., 2001; Juliano, 1985; Zhao et al., 2012).
Rice prolamin has been reported to be composed of three polypeptide groups having MW of 10, 13 and 16 kDa, with the 13 kDa prolamin being predominant, as determined by SDS-PAGE (Hibino et al., 1989; Ogawa et al., 1987). In the current study, prolamin showed one major band with MW of about 10 kDa. Also, two minor subunits of about 18 kDa and 31–32 kDa were present, most likely due to cross-contamination with glutelin. The method used to extract the rice protein fractions from RF coupled with their solubilisation in the strong reducing buffer prior to SDS-PAGE analysis, provided good resolution of the proteins characterising the different fractions, which allowed the identification of the protein subunits of the intact rice protein ingredients.
Luca Amagliania et al.: "Composition and protein profile analysis of rice protein ingredients", Journal of Food Composition and Analysis
Volume 59, June 2017, Pages 18-26, [DOI].(https://doi.org/10.1016/j.jfca.2016.12.026)

The exact structure of all the components making up a fraction of these proteins is not known yet. But it is clear that rice oryzin/prolamin does not trigger the following:

Dietary gluten storage proteins from wheat, rye, and barley contain protein sequences that elicit a diverse array of immunological response. Oats do not typi- cally elicit an immunological response unless there is sufficient cross-contamination from milling and handling of gluten-rich grains (i.e., wheat). Alpha-2 gliadin (α2-gliadin) contains a 33 amino acid sequence that is resistant to digestion by human gut and pancreatic enzymes and is a classic CD antigen.
In order to mount an immunological response to gluten proteins, a number of events need to take place. The antigen must breach the protective barrier of the small intestine to be presented to the B and T cells of the mucosal immune system by major histocompatibility complex molecules (MHCs) present on antigen- presenting cells (APCs) such as dendritic cells. Gluten proteins appear to traverse the cells and leak between cells due to defective regulation of tight junction proteins such as zonulin-1, providing a target for therapy 4. A number of agents can initiate a breach in barrier function (i.e., infections, nonsteroidal medications, bacterial overgrowth); thus, defective permeability may be an antecedent to disease develop- ment as proposed by Fasano. The resultant processing of indigestible gluten anti- gens by the mucosal immune system leads to active small intestine inflammation whose inflammatory cytokines can further loosen the tight junctions and promote further entry of more gluten peptides to perpetuate the vicious cycle.
The enzyme tissue transglutaminase (TTG) removes the amide group from glutamine of gluten peptides such α2-gliadin, leaving it in a highly negatively charged state, which increases its affinity and binding to MHC HLA-DQ2.5 or DQ8. The aforementioned antibodies against TTG and deamidated gliadin become an important screening tool for CD.
Gerard E. Mullin 2014 (above), chap. Pathobiology, p2. 

