Granularity dependence of <i>bona fide</i> boundaries of some object clusters.

<p><b>Left:</b> An object cluster consisting of six object entities. This cluster is exclusively demarcated by <i>bona fide</i> boundaries, and so is any of its sub-clusters (e.g. the two sub-clusters, each consisting of three object entities (in light-grey and in dark-grey). <b>Right:</b> The object cluster consisting of six object entities constitutes an object at a coarser granularity level. This object is demarcated from its surrounding complement by a <i>bona fide</i> boundary. Contrary to the finer granularity level, however, within this coarser level the two sub-clusters cannot be demarcated by <i>bona fide</i> boundaries anymore: The adherence relation between the objects involved (light-grey and dark-grey) at the finer level maps to a coherence relation at the coarser granularity level. Therefore, the respective parts are demarcated by a <i>fiat</i> boundary<i>^mat</i>. <b><i>Fiat</i></b><b> boundary</b><b><i>^mat</i></b><b>:</b> demarcates <i>fiat</i> parts of a material entity.</p

Three Foundational Types of Spatio-Topological Relations between Material Entities (from [4]).

<p>Three Foundational Types of Spatio-Topological Relations between Material Entities (from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030004#pone.0030004-Vogt1" target="_blank">[4]</a>).</p

Taxonomy of top-level types of constitutively organized material entities.

<p>A taxonomy of top-level types of material entity and important subtypes that can be distinguished in constitutively organized material entities. types that are presently distinguished in the Basic Formal Ontology (BFO) are in dark grey.</p

Sortation by Type using Categories based on Granular Representation.

<p>Left: Following the rule of sortation-by-type and applying a more complicated granularity scheme (Vogt 2010) one infers this type granularity tree from the instance granularity tree shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030004#pone-0030004-g007" target="_blank">Fig. 7</a>. This results in types of entities belonging to type granularity levels for which BFO provides no respective categories as templates for granularity specific ontologies (here marked ‘???’): BFO (version 1.1) provides no category for molecules that are not part of any cell, since they are neither a cell (i.e. BFO's ‘object’), nor a cell aggregate (i.e. BFO's ‘object aggregate’) or fiat cell part (i.e. BFO's ‘fiat object part’). Right: By applying the notion of <i>granular representation</i> and the here proposed categories of <i>portion of matter</i> and <i>portion of matter aggregate</i>, the transformation of the instance granularity tree into a type granularity tree can be completed. With these additional categories and the notion of granular representation one can account for the effects of trans-granular multiple instantiation in cumulative-constitutively organized material entities.</p

Integrated and Simplified Taxonomy of Top-Level Categories of Material Entity.

<p>A taxonomy of top-level categories of material entity and important subcategories that can be distinguished in constitutively and cumulative-constitutively organized material entities. By introducing the category ‘material entity aggregate’ one can reduce the number of <i>foundational categories</i> that exhaustively cover all possible types of constitutively and cumulative-constitutively organized material entities to four. All other categories represent <i>additional categories</i> that are organized into three hierarchical levels and subsumed under the foundational category ‘material entity aggregate’. All clusters of material entities are subsumed under ‘material entity cluster’ and all groups under ‘material entity group’ respectively. <i>Black boxes indicate Basic Formal Ontology (BFO) categories of material entity, dark grey boxes the additional categories suggested by </i><a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030004#pone.0030004-Vogt1" target="_blank">[4]</a>, <i>and white boxes additional categories required for accommodating all cumulative-constitutively organized material entities.</i></p

Trans-Granular Multiple Instantiation and the Distinction of Instance Granularity and Type Granularity.

<p>A) Left: Compositional partitions of a <i>constitutively</i> organized idealized multicellular organism into its constitutive object parts. Four partitions are shown: (i) into organs (<i>f</i>); (ii) into organ cells (<i>e</i>); (iii) into organelles (<i>c,d</i>) of organ cells; and (iv) into organelle molecules (<i>a,b</i>). Right: Compositional partitions of a <i>cumulative-constitutively</i> organized idealized multicellular organism into its constitutive object parts. The same corresponding four partitions are shown: (i) into organs (<i>j</i>) alongside with cells (<i>i</i>) and extracellular molecules (<i>g,h</i>), both of which are not part of any organ; (ii) into organ cells (<i>q</i>) and extracellular molecules (<i>o,p</i>) that are part of some organ, organelles (<i>m</i>,<i>n</i>) that are part of cells which are not part of any organ, and cellular molecules (<i>k,l</i>) that are neither part of any organ nor any organelle; (iii) into organelles (<i>v,w</i>) of organ cells and molecules (<i>t,u</i>) that are part of these organ cells but not part of any organelle, as well as molecules (<i>r,s</i>) of organelles of cells which are not part of any organ; and (iv) into organelle molecules (<i>x,y</i>) that are part of organ cells. B) Left: The <i>instance granularity tree</i> of <i>constitutively</i> organized <i>bona fide</i> objects based on the corresponding four partitions. Each partition constitutes a <i>cut</i> in the instance granularity tree (<i>Cut I–IV</i>) and thus an <i>instance granularity level</i>. Contrary to cumulative-constitutively organized material entities, particular instances of the same type of material entity do <i>not</i> belong to different cuts and thus are restricted to a single level of instance granularity. The types' extensions do <i><u>not</u></i> transcend the boundaries between instance granularity levels. Right: The <i>instance granularity tree</i> of <i>cumulative-constitutively</i> organized <i>bona fide</i> objects based on the corresponding four partitions. Particular instances of the same type of material entity, like for instance of the type <i>MOLECULE</i>, belong to different cuts and thus different levels of the respective instance granularity. In other words, the extension of the type <i>MOLECULE</i> transcends the boundaries between instance granularity levels. C) Left: The constitutive instance granularity tree that results from the corresponding four partitions <i>can</i> be directly transformed into the respective <i>type granularity tree</i> – no sortation required since they are topologically identical. Right: The cumulative constitutive instance granularity tree that results from the corresponding four partitions <i>cannot</i> be directly transformed into or mapped upon the respective <i>type granularity tree</i>. However, by (i) following the simple and intuitive rule of sortation-by-type (i.e. a type occupies the same granularity level as its finest grained instance) and by (ii) applying a more complicated granularity scheme (Vogt 2010), one can infer a type granularity tree. Unfortunately, this results in types of entities belonging to type granularity levels for which BFO provides no respective categories as templates for granularity specific ontologies (here marked ‘???’). For instance when looking at the cellular level, a cellular ontology that is based on BFO provides no category for molecules that are not part of any cell, since they are neither a cell (i.e. BFO's ‘object’), nor a cell aggregate (i.e. BFO's ‘object aggregate’) or fiat cell part (i.e. BFO's ‘fiat object part’), nor any of the additional categories of the extended BFO. (modified from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030004#pone.0030004-Vogt2" target="_blank">[19]</a>).</p

Second order basic types of material entity.

<p>Possible subcategories of the three basic types of aggregates that can be differentiated on grounds of distinguishing two types of relation between the objects within the aggregate (i.e. metric proximity and adherence) and the presence or lack of <i>fiat</i> boundaries<i>^immat</i>: (i) clusters <i>are not</i> demarcated by <i>fiat</i> boundaries<i>^immat</i> and are further characterized by topological adherence between the entities of the aggregate (through chemical bonds or physical junctions); (ii) groups <i>are</i> demarcated by <i>fiat</i> boundaries<i>^immat</i> and are further characterized merely by metric proximity of the entities of the aggregate – they lack adherence. Since also clusters can spatially relate to one another on grounds of metric proximity, clusters can also be part of groups. <b>A–D</b>. The four basic types of object aggregate – one object cluster and three types of object group, all of which either consist of objects, object clusters, or both. Note that an object cluster is only demarcated by <i>bona fide</i> boundaries and thus does not represent a <i>fiat</i> whole. <b>E–H</b>. The four basic types of <i>fiat</i> object part aggregate – one <i>fiat</i> object part cluster and three types of <i>fiat</i> object part group, all of which either consist of <i>fiat</i> object parts, <i>fiat</i> object part clusters, or both. <b>I–K</b>. Four out of 26 basic types of object with <i>fiat</i> object part aggregate – one object with <i>fiat</i> object part cluster and three out of 25 possible types of object with <i>fiat</i> object part group. <b><i>Fiat</i></b><b> boundary</b><b><i>^mat</i></b><b>:</b> demarcates <i>fiat</i> parts of a material entity; <b><i>fiat</i></b><b> boundary</b><b><i>^immat</i></b><b>:</b> demarcates <i>fiat</i> parts of an immaterial entity (i.e. a hole).</p

Definitions of additional Top-Level Categories of Material Entity for the Basic Formal Ontology (from [4]).

<p>Definitions of additional Top-Level Categories of Material Entity for the Basic Formal Ontology (from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030004#pone.0030004-Vogt1" target="_blank">[4]</a>).</p

A Cluster of Cells at the Molecular and the Cellular Level of Granularity.

<p>Left - A material entity that is a cluster of molecules at the molecular level of granularity. The molecule cluster constitutes two cells embedded in extracellular matrix (ECM). The two cells are connected with each other via a cell-cell junction. The molecules form a molecule cluster since they are connected with each other via molecular bonds. Right - The same material entity at the cellular level of granularity in which it is a cell cluster with portion of ECM cluster. The cell molecules constitute two cells that are connected via a cell-cell junction, thus forming a cell cluster. The ECM molecules form a portion of ECM that is a fiat entity.</p

Definitions of the basic types of material entity of the Basic Formal Ontology (BFO version 1.1).

<p>Definitions of the basic types of material entity of the Basic Formal Ontology (BFO version 1.1).</p