### EQ interpretations with regard to annotations

First of all, what appears unavoidable is a more complex provision for annotations, at least if complex phenotypes formalized in OWL/description logics (DL) [31] shall be composable in terms of the usual intersection. Implicitly, this has already been observed in [22], to some extent also in connection with the EQ formalism. The following adheres to the understanding of annotations as outlined in the Introduction and is inspired by the notion of *phenes* in [22]. Nevertheless, the subsequent variant differs in order to minimize changes to PATO and phenotype ontologies.

In order to solve especially the permutation problem of combined EQ descriptions, formally it suffices to have an "encapsulating" relation available. For instance, while (1) suffers from unwanted permutations, this is avoided in (2), where the encapsulating relation is termed hasPheno.

\begin{array}{c}\mathtt{\text{hasPheno}}\phantom{\rule{2.77695pt}{0ex}}\mathtt{\text{some}}\phantom{\rule{2.77695pt}{0ex}}\left(\mathtt{\text{red}}\phantom{\rule{2.77695pt}{0ex}}\mathtt{\text{that}}\phantom{\rule{2.77695pt}{0ex}}\mathtt{\text{inheresIn}}\phantom{\rule{2.77695pt}{0ex}}\mathtt{\text{some}}\phantom{\rule{2.77695pt}{0ex}}\mathtt{\text{eye}}\right)\phantom{\rule{2.77695pt}{0ex}}\mathtt{\text{and}}\\ \mathtt{\text{hasPheno}}\phantom{\rule{2.77695pt}{0ex}}\mathtt{\text{some}}\phantom{\rule{2.77695pt}{0ex}}\left(\mathtt{\text{short}}\phantom{\rule{2.77695pt}{0ex}}\mathtt{\text{that}}\phantom{\rule{2.77695pt}{0ex}}\mathtt{\text{inheresIn}}\phantom{\rule{2.77695pt}{0ex}}\mathtt{\text{some}}\phantom{\rule{2.77695pt}{0ex}}\mathtt{\text{tail}}\right)\end{array}

(2)

Naturally, the question arises which ontological reading applies to hasPheno. We interpret (2) as a concept for classifying organisms (by two phenotype descriptions). The hasPheno relation belongs to an interpretive view/pattern that overlays common interconnections of entities, centering on the organism. In terms of the example, one may consider an organism *O* that has an eye *E* as its part, while there is a red *R* that inheres in *E*. Thus *O* is indirectly related with *R* in terms of common relations like inherence and part-of. In the phenotype view, this allows us to view *O*, as *phenotype bearer*, to exhibit *R as a pheno* of *O*. The latter connection is reflected by the hasPheno link between *O* and *R*. We require that each hasPheno link is "justified" by a chain of basic relations like *inheres-in*, *part-of, has-function, participates-in*, etc., that connects the entity in the pheno role with the one in the phenotype bearer role (PB in Figure 2, 3, 4 below). This approach leaves existing ontologies intact, resolves the first two particular issues identified, and accounts for the fifth, as well.

### Enhancements for relational qualities

#### Purely formal extension

On the remaining issues of relational expressiveness and consistency of domain modeling, we first observe that the current relational EQ model forms a special case of reifying (only binary) relations with *fixed* auxiliary relations, cf. the structural part of [32]. The main uncommon feature is the naming of those auxiliary relations as inheresIn and towards. One should admit, though, that inheresIn is meant to link to the ontological notion of inherence, whereas towards is introduced for rather technical reasons in [17] (circumventing an inherence relation of higher arity). It remains to be explored in greater detail whether towards can be adequately reinterpreted in terms of the notion of *external dependence*, see [[33], esp. sect. 6.2.7]. The more common approach to name those auxiliary relations would have been the use of names counting arguments, like argument_{1} and argument_{2}. With the latter, an extension to *n*-ary relations is straightforward, which would solve the expressiveness issue. However, with fixed auxiliary relations there is no support for consistent domain modeling because the assignment of "values" to arguments is arbitrary. This may be the reason why all published variants of this pattern that we are aware of eventually suggest the *variable*, relation-specific naming of auxiliary relations [[29], sect. 5.1], [32, 34].

Therefore, we do not see that changing the interpretation of relational EQ statements could be sidestepped, if inter-modeler consistent domain modeling is to be supported any further. Striving at the same time for ontological adequacy somewhat systematically, we adopt the model of relations and (relational) roles from the General Formal Ontology (GFO) [35, 36], cf. also [37, 38]. As a side remark, we indicate that there are more types of roles in GFO, but for brevity we use roles and relational roles as synonyms herein. Note further that from here on 'role' is reserved for the ontological interpretation, whereas the meaning as set of pairs/as binary relation in the context of description logics and OWL is referred to as 'OWL property' or 'DL role'.

#### Ontological alternatives using relations

In brief, relations in GFO are considered as categories of relators. *Relators* are ontological individuals akin to qualities, but with the power to mediate/connect entities. A relator consists of *role* individuals (via hasRole/roleOf) and each role individual, besides depending on the relator, depends on a *player* (via playedBy/plays). The term 'player' is relative to this approach; in general, arbitrary entities can play a role within a relation. At the categorial/class level, each relation *R* is associated with a set of role categories that forms the *role base* for this relation. Basically, that means for each relator of type *R* that its roles must instantiate one of the role categories in that set, cf. [[38], sect. 3.3.3].

The GFO model of relations and roles can be encoded into an OWL representation in two obvious ways, termed *roles-as-properties* (Figure 2) and *roles-as-classes* (Figure 3). Common to both cases is to represent phenotype descriptions involving a relation *R* and (kinds of) entities *E*_{1}, . . . , *E*_{
n
} as argument restrictions. Either, corresponding to Figure 2, roles are left implicit in the OWL properties *o*_{1}, . . . , *o*_{
n
}, or, regarding Figure 3, role categories are explicated as OWL classes *O*_{1}, . . . , *O*_{
n
} (in between *R* and the *E*_{
i
}). Consider the example of *iron concentration in the spleen*, with the relation *concentration* and assuming that its two role categories are labeled *concentrated* and *concentrator*. The role of the *concentrated* is to be played by those entities that are concentrated in other entities, while the role of the *concentrator* is played by those (other) entities within which the first are concentrated. (Notably, in this technical reading *concentrator* must be understood merely like "container", which may be biologically misleading. In particular, anything in the *concentrator* role is *not* (necessarily) expected to cause or generate the concentration under consideration. A biological example of a concentrator in this non-intended sense could be a *kidney* that causes *urea* to be concentrated within the *bladder*. Despite this potential for confusion, *concentrator* is employed below in the "container"-sense in order to use a succinct role term that derives from the relation name.) The approach of roles-as-properties yields in OWL

\begin{array}{c}\mathtt{\text{hasPheno}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{some}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{(}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{concentration}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{and}}\\ \phantom{\rule{8em}{0ex}}\mathtt{\text{(concentrated}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{some}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{iron)}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{and}}\\ \phantom{\rule{8em}{0ex}}\mathtt{\text{(concentrator}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{some}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{spleen)}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{),}}\end{array}

(3)

whereas roles-as-classes leads to

\begin{array}{l}\mathtt{\text{hasPheno}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{some}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{(}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{concentration}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{and}}\\ \phantom{\rule{8em}{0ex}}\mathtt{\text{(hasRole}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{some}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{(concentrated}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{that}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{playedBy}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{some}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{iron)}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{)}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{and}}\\ \phantom{\rule{8em}{0ex}}\mathtt{\text{(hasRole}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{some}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{(concentrator}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{that}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{playedBy}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{some}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{spleen)}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{)}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{)}}\mathtt{\text{.}}\end{array}

(4)

The first of these cases equals the above approach of using variable, relation-specific names for the auxiliary relations [[29], sect. 5.1], [32, 34]. The second uses only two OWL properties hasRole and playedBy (and their inverses, possibly), but here this is unproblematic because the roles of the reified relation explicitly account for what is missing with fixed auxiliary relations without roles. Of course, both of these proposals will require a syntactic extension of the EQ model in order to capture the corresponding roles within EQ statements. Moreover, the roles-as-properties way may be simpler to reinterpret in other top-level ontological theories, because the roles presupposed by GFO are less explicit compared to roles-as-classes.

#### Ontological alternative using relations and qualities

The previous subsection suggests two ontologically inspired ways of formalizing relational qualities like *concentration* *of* (PATO:0000033, hereafter CO) in EQ statements that cure the immediate deficiencies previously described. We note that biological processes like concentrat*ing* anything in anything else are not taken into account here. Instead, both approaches are based on a purely relational reading of CO (and relational qualities, in general). CO is merely considered as a noun form of the phrase *is concentrated in* (CI). For example, '(a particular amount of) *iron* *I* is concentrated in a (particular) *spleen* *S*' is a "relational proposition", stating that *I* is concentrated in *S*. This proposition can be true or false, depending on whether the relation CI applies to *I* and *S* or not, but there is nothing to be measured, neither quantitatively nor qualitatively. (Pursuing this line of thought further in the example, one may wonder what remains as the actual difference between CI and relations such as 'is contained in' and 'is part of'.) In noun form, yet somewhat artificially, one may equivalently refer to 'there is concentration of *I* in *S*' for CI (note that *I* and *S* are particulars). However, we hold that CO comes in a second flavor, which is more amenable to specialization with notions like *increased concentration* *of* or to expressing specific values, e.g., 0.5_{g}/*l*. In phrases like 'the concentration of *X* in *Y* is 0.5_{g}/*l*', it appears more adequate to us to view CO as a proper quality which can be numerically quantified. Of course, immediately the question arises what that quality inheres in, which must be something that "includes" *X* and *Y*, not only one of the two. Here, computing the value of CO is instructive, which is based on values of qualities inhering solely in either *X* or *Y*, say, the weight of *X* and the volume of *Y*. The relationship between *X* and *Y* (of type CI, say) is characterized by the value within the CO phrase (in the second reading). Therefore, our current attempt of capturing relational qualities according to this analysis is to view them as inhering in particular relators, say a CI relator between *X* and *Y*. Admittedly, this is a deliberate, but no imperative choice among the possibilities within GFO. Other candidates for bearers of these qualities would be the overall relational fact, or one might consider the mereological sum of *X* and *Y*, in analogy to the inherence of relators in [[33], sect. 6.2.7]. If the latter option is to be followed, a more detailed analysis is required, though. Thinking of an amount of iron *I* concentrated in a spleen *S*, the question arises whether the mereological sum of *I* and *S* would differ from *S*. More generally, there may be interaction between the relation under consideration and forming a mereological sum of the relata. In any case, regarding implementation in OWL, we note that neither facts nor mereological sums are readily available on the basis of relators/relations and their arguments.

Eventually we arrive at a third approach, depicted in Figure 4, where the relation is characterized by a quality. In the example, that means that CI is distinguished from CO, the latter being understood as a quality that inheres in CI relators/instances. Accordingly, we refer to this approach as *relator-based-quality*. Note that the intuitive term 'relational quality' experiences a formal-ontological reinterpretation from relations in the previous cases of roles-as-properties and roles-as-classes to qualities proper (which are not relations) in the relator-based-quality approach. Looking again at *iron concentration in the spleen*, assuming the roles-as-properties approach for modeling a relation isConcentratedIn (with roles like above) and a relational quality concentration yields in OWL

\begin{array}{l}\mathtt{\text{hasPheno}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{some}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{(}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{concentration}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{that}}\\ \phantom{\rule{8em}{0ex}}\mathtt{\text{(}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{inheresIn}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{some}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{(}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{isConcentratedIn}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{and}}\\ \phantom{\rule{17em}{0ex}}\mathtt{\text{(concentrated}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{some}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{iron)}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{and}}\\ \phantom{\rule{17em}{0ex}}\mathtt{\text{(concentrator}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{some}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{spleen)}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{)}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{)}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathtt{\text{)}}\mathtt{\text{.}}\end{array}

(5)

This approach appears ontologically plausible to us currently, following the explanations above. Moreover, from the point of view of representation, it exhibits the beneficial property that CO is a "unary quality" like *color*, in the sense that it inheres in a single entity (a CI relator, which in turn accounts for the relational character of the quality). Any general account of representing measurements should thus be applicable to CO as it is to qualities like *color*. In these cases there is a single entity available - the quality - to which a measured value can be attached, where exploiting OWL datatype properties is one among several options that remain to be studied in future work. Furthermore, linking qualities to relators does not prescribe an overly specific relation model, but allows for adopting either of the approaches roles-as-properties and roles-as-classes in formalizing relations and roles, or even other theories (for which the quality bearer may require reinspection).