Skeletal structures for geometric objects are an important concept in diverse areas of science. Motivated by applicational needs, different types of skeletons and their geometric and algorithmic properties have been studied, in computational geometry and also in more practically oriented fields.medial axis. It represents the set of centers of all multi-tangential circles (or spheres) inscribed to the object. Defined by distances to the boundary, the medial axis keeps a strong correspondence to the shape of the object; we refer the reader to [The most prominent and widely used skeletal structure is the eader to , 8, 29 fstraight skeleton of a polygon (or polytope) offers a potential alternative. This structure is not defined via distances but rather in a procedural way, by means of a mitered boundary offsetting process which shrinks the object in a self-parallel way till it vanishes. In two dimensions, the shrinking process is combinatorially trivial, and the geometry of the straight skeleton is well understood [The so-called derstood , 4. It iThe planar straight skeleton has proved useful in various areas, including CAD , image processing (shape comparison and manipulation), architecture (automatic roof design), GIS (terrain and city modeling), and others; see e.g. , 21, 32 Offset calculation (a standard operation in CAD), the construction of a skeletal structure (that encodes the shape of the polytope in a suitable manner), and decomposition . However, the precise definition of the shrinking process in 3D bears difficulties, and even the existence of a mitered offset surface which is continuous and not self-intersecting is no longer trivial.It is desirable to find appropriate generalizations of the straight skeleton to three dimensions, as this would provide piecewise linear solutions for three main operations on nonconvex 3D polytopes which the medial axis cannot offer: shrinking a given polytope means offsetting its boundary surface in inward direction, in a self-parallel way and at unit speed. Thereby the polytope facets ‘sweep out’ the cells of the 3D straight skeleton. The polytope edges (which move in angle bisector planes), and the polytope vertices , trace out the sheets and the spokes of the skeleton, respectively, which border its cells. The shrinking polytope undergoes changes of various kinds, purely geometrical, or of combinatorial nature (changes in the boundary structure), or topological .Surprisingly, not much attention has been paid in computational geometry to these basic questions. Intuitively speaking, Erickson (see ) observeThe present paper, which contains and extends material from the conference papers , 10, is thinning .convex polytopes, where the straight skeleton coincides with the medial axis, and can be interpreted as the (projected) lower envelope of n hyperplanes in 4-space that correspond to the n polytope facets. The skeleton therefore consists of convex cells whose overall size  is axis-aligned  polytopes is the medial axis in the There are certain special cases where 3D straight skeleton algorithms have been known. First of all for see e.g. . Similarmentclasspt{minimac factor . In thesmentclasspt{minimad-space, including their positively weighted versions. This improves the trivial bound by an order of magnitude. The argument is based on a unique layer partition for straight skeletons and related cell complexes, introduced in Sect. We give an Here we define the type of polytope we would like to work with, and give some related definitions and explanations.convex polytope is the finite intersection of closed halfspaces of Euclidean three-space polytope is a bounded subset of vertices (points of intersection of three linearly independent supporting planes), edges . Note that polytope facets are connected, but are not required to be simply connected. Moreover, there might exist so-called touching faces, for example, a vertex or an edge touching the interior of a non-incident facet. That is, the boundary complex of a polytope needs not be face-to-face. Polytopes with boundary singularities of various kinds arise naturally during the shrinking process, even for the simple setting where the input polytope is homeomorphic to a ball and all its facets are triangles.A entclass1pt{minimav of touching vertex if there exists some v and having a positive radius of at most v is of the touching type, but the vertices u and w are not.) A vertex is called non-touching, otherwise. Among the latter vertices, certain types are particularly relevant. A vertex v is pointed if there exists an open (geometric) disk whose intersection with v. A saddle vertex is incident to edges that positively span 3-space.A polytope, entclass1pt{minimae of a polytope convex, reflex, or flat respectively, if the dihedral interior angle spanned by e’s incident facets is smaller, larger, or equal to e is part of a boundary singularity of v of convex if all edges incident to v are convex. Likewise, v is called reflex if all its edges are reflex. Note that convex vertices are pointed, but reflex vertices are neither pointed nor saddle vertices. A saddle vertex necessarily has edges of both convexity types. The degree of a vertex v of v. Vertex degrees have to be at least 3, because vertices come from intersecting three or more planes.An edge entclass1pt{minimaWe impose no general position assumption on a polytope, nor is this needed for our structural and algorithmic results. For the ease of exposition, a generic behavior of the facet offset planes will be assumed at certain places, though only temporarily.In this section, we give a characterizing definition of mitered offset surfaces and a proof of their existence, along with a discussion of some of their relevant properties.v, of m. We have v can be arbitrarily large, n counts the number of facets of v is a non-touching vertex; we will come back to the general situation (which is similar) in Sect. Let entclass1pt{minimav. Each such facet arrangement defined by the offset planes v. We denote this arrangement by Let U, at the vertex v, sufficiently small to intersect only faces incident to v. The intersection of U with the polytope v is a non-touching vertex. Note that U . In the following definition, the planes v to Now center a sphere, valid offset surface for v is (the graph of) a radial function i.For every x on the boundary of A v thus is a radially visiblev, because then all planes v. These properties are sufficient to have an offset polytope, v according to A valid offset surface v of v, then exterior of Let r be an infinite ray originating at v and with r can intersect v. Let r does not intersect any edge of r infinitesimally, otherwise.) For increasing v, such that the facet v, then (and only then) a second point, r with Let Facets that shift in the ‘wrong’ direction contradict the polytope shrinking process, in the sense that a sequence of polytopes v which are not radially visible but do not self-intersect; see Fig. There exist surfaces for The question of the existence of valid offset surfaces arises. An affirmative answer follows from the results in , 10 descA valid offset surface U lies in the function domain v. Assume that some ray r emanating from v intersects the boundary of a (convex) cell C in C has to be adjacent there to another cell in C into a bounded and an unbounded part—a contradiction.Let To get rid of the bounded facets of Theorem C shown in Fig. u, of degree 6. Adding the cell D instead of C leads to a temporary violation of radial visibility (middle). We have the peculiar situation that one offset plane (marked with H) supports the surface with both sides now, such that one of the two facets defined by H (the hidden triangular facet of D) would shift toward the exterior of the polytope; cf. Lemma C next (right). In the resulting valid surface, all facets are unbounded again. No more cells can be added in this example without destroying radial visibility beyond repair.Valid offset surfaces are not unique in general; see . The smav gets resolved into 8 surface vertices, rather than only 5 as in the other two cases.  trace out extraneous skeleton cells, such that the volume swept over by an individual offset plane is not interior-connected any more; see Sect. convex vertex v of v splits into vertices of degree 3 in the generic case, i.e., when no 4 offset planes pass through the same point. Indeed, we can have similar positive features for arbitrary polytope vertices v.  offset surface for v whose edge graph is a forest with at most Let v such that all facets of v’s polytope facets. To see that w have degree w is incident to at least 4 facets and, because exactly 3 planes in w, at least 2 facets for w stem from the same plane, say unbounded surface facets incident to w  envelope of two valid offset surfaces for inner offset of its boundary. In fact, the quest for an outer offset for The shrinking process for the polytope For algorithmic purposes, it is of advantage to reduce the vertex resolution problem to one dimension less. We will distinguish two cases, depending on the type of the polytope vertex considered. More specifically, we discuss a two-dimensional reduction for pointed polytope vertices first, and then proceed to a generalization for arbitrary vertices, including the touching vertex type.v of the polytope v needs not be a convex vertex, that is, v can have reflex incident edges. However, there exists a plane E that intersects all edges of v. Moreover, as v is a non-touching vertex by assumption, E intersects v) in a simple polygon, which will be denoted by v is of degree m then m edges, Let us consider any pointed vertex entclass1pt{minimaE and the facet(s) of E influences both u shared by edges E). Therefore, u traces out a skeleton arc along the line E are radial projections of each other .In the edge events that occur in the construction of u, is created in an edge event; see Fig. e has to shrink to length zero, and the offsets u, forming an interior angle of at least e define a convex interior angle (as drawn in the figure). We fix the sectional plane E orthogonal the line segment u. Note that we have u at time u at the same time, which is necessary for the occurrence of u as an offset vertex. A similar argument for the nonexistence of u applies when the offsets reflex interior angle (this case is not shown in the figure). It remains to recall that the vertex structure of E. Suppose that a nonconvex vertex, u is created in an edge event, then E chosen above; see Fig. e, the vertex u may even be flat in order to have u actually is not created as a polygon vertex.  vertex In summary, there is a unique way to proceed in the construction of ted case .monotone polygons, in particular, the intersection of a face The inner arcs of skeleton , but doeskeleton , 13.v). Moreover, its faces are monotone in the direction of their defining polygon edges.The weighted straight skeleton sticking events; see e.g. [u of a starting from u, in a way such that the two faces which share a are monotone in the required directions.To see the monotonicity property, we observe that non-monotonicity always arises in the neighborhood of skeleton nodes created by so-called see e.g. . These am is the degree of v. The next assertion, which has been stated informally in [v.By Lemma mentclasspt{minimav that radially projects to There is a valid offset surface for U centered at v.  Consider an arbitrary point v and fulfills (1) and (2) by construction. To see that a of a. Arc a is contained in an angle bisector plane v. This implies that arc a radially projects to the line a.) We conclude that for each Clearly, the domain for such a surface v. In particular, each polytope facet v gives rise to a single and unbounded facet Lemma v (left) could be resolved in two different ways, then the two surfaces  were obtained. The former surface cannot come from f is not monotone there. So it must be the latter one, which (unlike the former surface) does not result from a self-parallel polygon offsetting process, not even for arbitrary edge weights. We conclude that there is a unique solution, which is the one shown in Fig. We remark that Lemma skeleton roof; see e.g. [x to the supporting line There is another and more well-known projection surface for weighted straight skeletons of a polygon see e.g. , 21. Thiv in Lemma valleys of this roof  have to start at the boundary of The skeleton roof corresponding to t}P; see . As a cov and base v from E, because the planes v.We observe finally that when the edge weights In this subsection, we discuss a two-dimensional reduction of the vertex resolution problem for arbitrary polytope vertices.v of the polytope v. Using two sectional planes leads to one or more unbounded polygons of intersection with v is high, these polygons can be arbitrarily complex, having a large number of convex and reflex vertices. Though the weighted straight skeleton inside each such polygon can be defined and computed much like in the (bounded) case before, we now face the task of combining several skeletons that stem from two different planes. This inherits the problem of merging straight skeletons, which is unsolved so far. In fact, a solution would imply a novel divide & conquer method for computing straight skeletons.Non-pointed polytope vertices are more complicated to deal with. For example, if a vertex U as in Sect. v be an arbitrary vertex of the polytope v is non-touching , but it necessarily disconnects if v is a touching vertex, and This situation can be circumvented when a sectional sphere v always exists. In fact, Definition v concerns the facet planes for v. In addition to the facets of v as a vertex, there exist facets v, and whose offset planes v may touch a facet v is a vertex may touch Still, a valid offset surface for v. We observe that bisector graph, v and regions) where the function Let now v of the offset planes not considered to be part of leaves. The inner nodes of The arcs of v with degree 10. The domain v is of sufficiently high degree m. This is already true for surfaces having the forest structure as in Corollary g and v can either be a touching vertex, being the common apex of v splits into Figure outerplanar  if all regions of Our next aim is an extension of Corollary v be an arbitrary vertex of the polytope m be the number of facets that contain v. There exist (orphan-free) valid offset surfaces v such that Let Cycles in outerplanar bisector graphs stem from the shape of the domain v, the minimum surface maximum surface v, which is always outerplanar, is the so-called spherical skeleton, U, introduced in [m is the number of facets of v.To obtain a canonical bisector graph for a polytope vertex Observe that the existence of v, the offset arrangement To select more than one bisector graph for m3) time . This isv under consideration. Being easy to implement, this method is plausible when the degree m is a constant, independent of the number n of facets of the input polytope. Indeed, most solids can be approximated accurately by (boundary-meshed) polytopes with vertices of small constant degree. Also, in the generic case, the shrinking process can only create offset polytope vertices of degree A computationally simpler and still flexible approach, and the one we have implemented for the 3D straight skeleton construction, is the following. By Corollary v to be resolved. We can treat each component v is a non-touching vertex, where we also have only one connected component. Let now U with the angle bisector planes The graph enumeration is facilitated by the following observations. Let G be an outerplanar graph for G is a bisector graph for v if and only if (1) all inner nodes of G have degree G can be embedded on the respective components of the circle arrangement Let G fulfills conditions (1) and (2) then a valid offset surface with facets from the offset planes G does not embed inside G that is not radially visible from v, implying that G cannot be a bisector graph: Let two arcs in the embedding of G cross at the point v and through a suitable point in the neighborhood of x intersects two surface facets (rather than only one) in their interiors. If G provides us with extra information, such that only connected graphs with inner nodes of degree exactly 3 need to be generated: Arcs of G missing in the corresponding bisector graph, and thus causing its disconnectedness, reveal themselves by the identity of the respective offset planes.  in Lemma see e.g. . As a usk leaves .multiple event can be processed, i.e., a combination of events of possibly different types, which take place at the same point in space, and which can be arbitrarily complex.  events in Sect. In summary, a universal engine for vertex resolution is obtained, which works for arbitrary polytope vertices and in all degenerate cases. This ‘vertex splitter’ is useful not only for initially splitting the higher-degree vertices of the input polytope entclass1pt{minimaAfter having settled some basic questions about mitered offset surfaces for general nonconvex polytopes, we can now return to the main concern of this paper: the construction of 3D straight skeletons for such input polytopes. We will maintain full generality in this section, to point out the general validity of the results. In Sect. events. These are combinatorial changes in the boundary structure of the offset polytope v, then four or more offset planes participate, which when shifted further, constitute the respective offset arrangement Like in two-dimensions, the skeleton construction complies with the shrinking process for the given polytope entclass1pt{minimainitial events, to distinguish them from the non-initial events that occur later in the offsetting process for cell, or sheet, or spoke, respectively, as we shall name these skeleton components. The edges of Initially, for infinitesimally small v of v undergoes a change. The vertex v becomes part of the skeleton in the event, being an endpoint of certain skeleton spokes. We will call such endpoints the corners of the skeleton. Note that events may happen simultaneously also for a fixed value Every event is associated with a vertex By the results in Sects. Let entclass1pt{minimanonconvex sets in general. Studying their structure is therefore a nontrivial task.In the decomposition a straight skeleton defines for a polytope skeleton cell of a facet connected set, and the reason is that We start by arguing that skeleton cells are always bordered by some polytope facet. The C, then can split into several interior-connected orphan cells, though C still stays connected through the vertices that have been resolved.  facets of C is monotone, in the sense that the intersection of C with any line normal to its defining facet(s) of voids in skeleton cells, even when the polytope In any case, the cells share another property which is helpful when using 3D straight skeletons as a partitioning structure: Each cell facial structure present in a straight skeleton for We now take a closer look at the s, and consider an endpoint v of s which is an inner corner of the skeleton , one plane in the triple for mentclasspt{minimav is not a touching vertex for the cells Geometrically, when a skeleton cell is considered as a separate polytope, it can have several kinds of vertices, including the reflex and the saddle type. Touching vertices occur as well, namely, as corners that concatenate orphan parts of the same skeleton cell. However, in the geometric degeneracy described above, the vertex spoke graph of a 3D skeleton next, i.e., the graph formed by its spokes and corners. Whereas skeleton cells are always connected, this is not true for the spoke graph. Two examples can be found in Sect. Returning to topological issues, let us study the Consider any straight skeleton of the polytope I be some isolated component of the spoke graph that is not connected to the boundary of I from I, and the cells therein (whose existence is witnessed by the spokes in I) are not bordered by the boundary of Suppose the contrary, and let medial axis of By Lemma see e.g. . This cocombinatorial complexity of a 3D skeleton is an interesting quantity. It is defined as the total number of cells, sheets, spokes, and corners of the skeleton. Let the polytope n facets. The number of cells is at most n, as each facet borders a unique connected cell. However, a cell can consist of various orphan cells connected via corners. The number of corners is trivially bounded from above by Naturally, the i, j, k. By a similar index count, a single bisector plane Let now n facets. The combinatorial complexity of any straight skeleton for Let Clearly, the result also holds when ial axis  is \documentclasspt{minimaVarious different straight skeletons for a polytope The information about all possible mitered offsets and straight skeletons of a polytope is comprised in the structure of the individual events. It is therefore of interest to provide a description and categorization of these events, which is the purpose of this section. Though rich in detail and space consuming, this task leads to several insights into the geometry and combinatorics of the offsetting process, and its occasional peculiarities. Also, the presented material will be helpful when an implementation different from ours is intended.  of events.When constructing a straight skeleton in the plane, one of the two possible (generic) events is the edge event . A polygedge (vanish) events. Such events are totally local and can be detected by the fact that some edge of O(1) time.A generalization to three dimensions entails several events of this kind. One up to six edges of the offset polytope e disappears (left) and is replaced by the edge e ‘flips’ to convex vertex v of v into two vertices of degree 3 is a unique binary tree. Concerning the 3D skeleton construction, two spokes and a sheet are completedv, and the construction of two spokes spanning a new sheet in a different bisector plane starts at v. Also, four already partially constructed sheets get extended further.We start with the simplest edge event, which we denote with v is pointed but nonconvex, or is a reflex vertex. However, there are events where exactly one edge vanishes but with different characteristics, for instance when a saddle vertex is generated. Such edge events correspond to the sticking event in the polygon case [The situation stays similar when the generated degree-4 vertex gon case , and canv is incident to two convex and two reflex edges (left). Remarkably, v can be resolved in two different ways (middle left/right), where the vanished edge is either flipped or not. This is the example of smallest vertex degree where the offset surface is ambiguous; in fact, this is the only ambiguous event among the non-initial events (in the generic and orphan-free case). The bisector graph a and b so as to cut off the arcs c and d. Equivalently, in the three-dimensional setting, See Fig. v, the construction may continue either with a new sheet in a different, or in the same bisector plane. In the former instance, the skeleton continues like for a convex vertex v. In the latter instance, the constellation of the involved sheets and cells deserves particular attention. The completed sheet and the starting sheet are coplanar and touch at v after the event. The four extending sheets also have v as a corner, but v sits between two collinear spokes on their common boundary, being a ‘flat’ corner there. This is exactly the situation depicted in Fig. v ‘wrap around’ this corner, such that each of them is supported by both sides of a single bisector plane. These cells are therefore adjacent in two sheets. Observe that the offset polytope retains the incidence structure of its boundary in this event instance.The actions for the 3D skeleton dictated by event v, which is the event v is incident to a facet containing a reflex interior angle at v (right). In the shown polytope, the edge e of the kite-shaped facet vanishes (left), and the edge e and same two facet offsets (shown shaded). Therefore the polytope retains its boundary incidence structure in the event, though there is a switch in the orientation of the edges e and e and Figure v, which henceforth stays a vertex of the offset polytope for a while. The bisector graph is a 3-star. Three sheets, three spokes, and also one cell of the skeleton get completed at corner v, and three already partially constructed sheets get adjacent at a new spoke. This spoke has v as a starting point.We now turn to the description of edge events where more than one edge vanishes. Let us treat the case of (exactly) 3 edges next. The respective event, denoted by v. Such a tetrahedron can sit on top of some facet f of f may collapse in the former case. In both cases, four spokes and six sheets get completed at v, that is, all that can be incident to a corner in the generic case. Three cells are completed in the former case, and four in the latter. After that, v disappears as a vertex of v, though with the extension of the cell swept out by f in the former case. As a remarkable fact, the skeleton spoke graph disconnects then, at least locally: In the neighborhood of v, the current part of the spoke graph loses contact to the shrinking boundary of the offset polytope. Note further that the bisector graph is empty in both cases, because the spherical polygon is either void or a hemisphere.Going up by one dimension, 6 edges vanish in the event collinear edges of a facet of v the tetrahedron shrinks into. The spherical polygon is a 2-gon in this event, formed by two semicircular arcs. The bisector graph consists of a single arc a (right). Four spokes, five sheets, and two skeleton cells get completed at v, which then disappears from v, that is, the spoke graph construction does not continue there again (in spite of the fact that the bisector graph is connected now). The skeleton continues with the extension of the partial sheet that corresponds to the arc a, and the two incident cells.In a similar event, triangle collapse event in the polygon case. Notably, in three dimensions some edges of topologically change the polytope or its non-vanishing facets. There are reasons for not including such events into the class of edge events, which will become apparent in Sects. The last three events are the generalizations of the An example where one edge vanishes is the event in Fig. This completes the enumeration of edge events. In retrospect, these events can create all types of polytope vertices but the touching type, though only with degree at most 4. The offset polytope does not change its topology in edge events, nor does any of its facets, unless polytope pieces, facets, or holes in facets collapse entirely. The spoke graph of the skeleton locally disconnects when 5 and possibly when 6 edges vanish  tetrahedron T. Clearly, if v lies inside T, then T cannot influence the offset surfaces at all, and v then also lies inside the reflected tetrahedron v sees a single facet of T, then T defines one or three orphan facets in the inner offset surface, and T nor v cannot see three facets of T, because all planes in v. So let v see exactly two facets of T. If the presence (or the absence) of T defines two orphan facets in the inner surface or violates its radial visibility, then T and The arity of events now depends on how unique events  are unary, but not conversely. For example, the inverse We conjecture that the arities of self-inverse, because they preserve the boundary structure of On the other hand, it will turn out that generic non-initial events always define unary inverse pairs v uniquely determines the structure of the 3D straight skeleton, in the entire neighborhood of the skeleton corner v. A canonical way of defining such pairs is obtained, for example, by utilizing the spherical skeleton  events are characterized by the property that either the boundaries of two coplanar facets of So-called keletons , where tf of f (possibly) gets into self-contact can be calculated directly in f has k edges. As an alternative, a weighted straight skeleton for f can be used. Negative edge weights [f can have edges that shift toward its exterior. A simple and usually fast implementation is by triangulating all the facets of O(h) overall time for all facets per increase of h facets and each of them is of constant size.Detection of events in this class can still be done locally, either directly by observing the collapse of edges in  weights  may occuriangles . This wovertex/edge contact, v of v. Observe that v is non-touching, but is neither a pointed vertex nor a saddle vertex. We may call v an almost pointed vertex, because its spherical polygon U, but not in any open one (as is required for being pointed); v gets split into three degree-3 polytope vertices.Facets events show the richest forms, and they can be systematized by the dimension of the facet boundary pieces that get into contact. Most similar to the planar case is the type fextends now locally at v after the event. In the 3D skeleton, a spoke ends at the corner v and three spokes start there. Also, four sheets are extended, and two sheets are created.The same anatomy arises for a similar case of a vertex/edge contact, v of degree 5 is created intermediately (right), which is of the touching type now: The boundary of the spherical polygon consists of two parts, a triangle and a 2-gon. On the other hand, the spherical polygon is still connected, and the bisector graph even has the same tree structure as in the events There is another case of a vertex/edge contact, the facet event vertex/vertex contact. The two short edges of the shaded facet simultaneously shrink to length zero (left). At the same moment, the object on top of the adjacent facet, f, shrinks so as to touch the boundary of f at a vertex v of f (middle). The vertex v has degree 5 and is of the almost-pointed type. The spherical polygon is connected, but the bisector graph is disconnected, being a forest consisting of a 3-star and a single arc. This graph has only one inner node; accordingly, v is ‘resolved’ into only one vertex of the offset polytope. That is, v does not split at all, but rather loses two collinear edges.Consider Fig. complementary setting . Therefore, the respective reverse actions take place in the skeleton. Note at this point that the inverse of a facet event has to be a facet event again.Actually, this event is the inverse of the facet event v of v resolves into only two offset vertices, because two collinear edges of the shaded facet merge into one. Thereby, a tunnel is generated in the offset polytope. The spherical polygon is connected but contains two holes , into two 3-stars and two spherical triangles, respectively. This event is the inverse of the vertex/vertex contact event Finally, there also exist facet events of mixed type, and a vertex/edge/edge contact. Such an event, v of v is of degree 8 and has two pairs of collinear edges (right). The degree of the constructed skeleton corner v is still 4 , because v splits into only two vertices in the offset polytope (middle). These vertices belong to two objects sitting on the two holes born in the event. The spherical polygon is boundary-connected, but the bisector graph is a forest with three components, two 3-stars and an isolated arc. Once more, two skeleton spokes get completed and two created. Six already existing sheets concur at v and extend, like in the event v is of the almost-pointed type in A facet event of the mixed type can also involve a For all v be some vertex of v is at most 4, and each plane can contribute at most 2 facets of v. But the number of facets and edges incident to v must be the same, because these faces correspond bijectively to the arcs and nodes of the spherical polygon for v.Let vertex/vertex/edge/edge contact has already been covered by the event To convince ourselves that the enumeration of facet event types is complete, we observe that the In summary, all facet events are unary, as are their inverses by Lemma global (collision) events, where combinatorially unrelated faces of It remains now to consider the class of see e.g. . Events unique, because the resolved vertex v always lies inside the tetrahedron T defined by the arrangement By convention, we assign the inverse of a global event to the same class. All such events are vertex/facet collision. The front vertex of a tetrahedral dent of the polytope approaches the shaded facet (left) and pierces a triangular hole into its interior (middle), which suggests the name piercing event, v of v resolves into 3 vertices that span a hole of the pierced facet. This constructs a tunnel in the offset polytope. In the 3D skeleton, three sheets get v as a corner and extend, but their common spoke is completed there. Three spokes are born at v, along with three sheets spanned by them.See Fig. The inverse of the piercing event, edge/edge collision or kissing event, v of v then resolves into 4 vertices spanning a nonplanar quadrangle (middle), which opens a tunnel in the offset polytope. We recall from the beginning of Sect. isolated corner v with four incident spokes is generated. Four sheets are initiated too, and the two sheets traced out by the front edges of the colliding wedges are extended. This leads to a (temporarily) isolated part of the spoke graph. However, this part stays connected to the offset polytope boundary from the exterior, and will therefore connect to some other part of the spoke graph in later events.The The inverse of the kissing event, vertex/vertex collision leads to 6 or 5 offset planes passing through the same point, unless two pairs of identical planes are present. In that case, the event shapes into the facet event vertex/edge collision yields 5 concurring offset planes, unless an identical pair is involved. The resulting event is the facet event vertex/facet collision is the piercing event Combinatorially, there are six possibilities to pair up polytope faces of dimensions 0 to 2. Let us go over them systematically now. Two possibilities are  ruled out by our generic assumption: The type edge/edge collision is only partially covered by the kissing event, because the contacting edges may be collinear. One or two pairs of coinciding offset planes are involved then, but in the latter case the planes have differently-oriented normal vectors in each pair. Such a collinear contact of edges, and also the next type, the edge/facet collision, can be interpreted as limit cases of the type facet/facet collision, the last possible combination. This event necessarily requires two coinciding offset planes of different orientation.By contrast, the type not excluded by the generic assumption. We observe that, in the moment of collision, more than one new vertex can be created in the polytope v of These last three event types contain geometric degeneracies which are The enumeration of the generic events is now complete. In spite of their sometimes quite complex anatomy, all events but one are unary. In the less obvious cases, this can be verified easily by using bisector graphs as a tool. Another general observation is that events do not change class when being inverted .contacting events, as opposed to the contact-free edge events. This distinction will turn out important in Sect. One can give a gross classification of events, based on whether or not the offset polytope On the other hand, only the contacting events have the ability to alter the 3D topology of the polytope. Tunnels may be created or destroyed, either locally like in the facet event Let v in the offset polytope X the new void in the polytope v lies on the boundary of v which intersects the boundary of X at least twice. This contradicts the radial visibility of the created offset surface. Otherwise, the void did appear in the interior of X have collectively swept over X during the interval v. But prior to the resolution of v, these planes must have arrived at v. Therefore, they have swept over points in X already earlier, for offset values smaller than Assuming the contrary, let a corresponding event happen at the point interior of boundary of Notice that, nevertheless, an event capable of creating a void in the input dimension 2 (for a polygon) and 3 (for a polytope) is that an additional dimension is needed to enable the generating boundary singularities.This situation parallels the two-dimensional case. When the input polygon is shrunk, no holes can be generated, neither in its interior nor at its boundary. It is instructive to visualize that such planar hole generations do take place for the polytope facets—in the piercing event and, for example, in the facet events of type vertex/vertex contact, respectively. The intuitive explanation for the lack of the corresponding events in the The variety of the skeleton construction events and their properties is admittedly confusing, and an overview may be helpful. The following table summarizes the (generic and orphan-free) events that can take place in the shrinking process for a nonconvex polytope. We respect the order of classes and the notations established in the preceding subsections: edge events contact, if any (indicated as vertex/edge/facet); the degree and the type of the offset polytope vertex v created at this moment; the structure of the bisector graph ; the connectivity behavior of the spherical polygon boundary-connected, boundary-disconnected, or disconnected); the number of spokes/sheets/cells completed and created in the straight skeleton being constructed (the number of extending skeleton faces is the rest to 4 / 6 / 4); the local behavior in the spoke graph construction ; the 3D topological changes in the offset polytope, if any; and finally the inverse event  and of bisector graph structures (Column 6).Our main results on mitered offset surfaces and straight skeletons are based on arrangements of planes in 3-space, and therefore can be generalized naturally. We will address extensions in three different directions—which can be combined—in the present section and in Sects. roof surface in three dimensions [It is well known that the straight skeleton of a polygon in the plane defines a mensions , 21. We t stands for the 4th coordinate. Now consider the hyperplanesskeleton roof for Let We can also define more general roofs for x on i. We call the graph of roof for the polytope roof complex for Let Roof complexes are cell complexes inside offset surfaces in Definition U in U then have bisector graphs as their spoke graphs, and among them the spherical skeleton [Note the similarity between roof surfaces and One difference to straight skeletons is that a roof complex can contain additional cells, namely, more than one connected cell for a given facet The union of cells in a roof complex for X be the point set in question, and let X is not monotone such that X in a disconnected set, then the function Let proof in  to one mBy the monotonicity property expressed in Lemma O(n) connected components.The spoke graph of any roof complex for Although not every roof corresponds to a straight skeleton, any fixed roof  case in . Let \dou of Figure v above.Note that a tetrahedron collapse without explicit reference to The event Let Being orphan-free is not a sufficient condition for a roof complex to be a straight skeleton, as we recall from Fig. For certain roof complexes, a canonical partition of the corners into so-called layers can be specified. This structural property, which we will describe now, leads to the first non-trivial upper bound on the size of straight skeletons in Consider a polytope ty Lemma . Moreoveenumeration method for the sheets in There exists an contacting events from Sect. vblue if its event is contact-free, and red, otherwise. The time stamp of v.Each corner of layers of The greedy process above can be guided by corner colors. In a first round, we complete a maximal set of sheets by blue corners. We start with the triangular sheets, and collect all accessed blue corners in a set shrinking order, i.e., for ascending time stamps. This explains why we can continue with the next blue layer after collecting The following interpretation of layers implies the correctness of the claim. greedy order for We obtain a linear extension of the partial order in each layer Visualizing the situation for straight skeletons in two dimensions Fig. , left, wred layer. We then scan through this modified greedy orderOur next aim is to realize the greedy order of events in a different geometric way, with the intention of analyzing their overall number. To this end, in the layer partition for blue corner ignored. That is, the parts of Initially, red layer, a self-intersection of For each corner The philosophy behind this construction is that, because events are processed in independent portions, they can be performed consistently on Note that n facets, and let Let n, and we obtain v in e on e. We study the interplay on e. The three lines v at the same time, causing e to vanish. Edge e may reappear immediately, for example when v is a saddle vertex, but this can happen only once. Otherwise, as no contacting events can hinder their influence, at least one of For the cardinalities of layers we give the following arguments. While staying within a fixed layer O(n) edges and facets. If From Theorem near-convex if it allows for some straight skeleton with a single (blue) layer. That is, no contacting events occur, hence the greedy order is identical to the shrinking order. For example, convex polytopes are near-convex. Also, the surface The number of layers depends on the shape of the polytope If For general roof complexes for From the algorithmic point of view, roof complexes provide an alternative way of computing monotone decompositions of a polytope n roof hyperplanes The surface of a roof for zone in To construct a roof for ing step , and theDepending on the kind of cell complex we are aiming at, we can continue the process. For simplicity, let us assume that n cells is obtained, where the requirement that Still, the complex n facets. A decomposition n monotone polytopal cells and inner corner degree 4 can be computed in Let The cell complex minimal or the maximal orphan-free roof  , is dooment}$$On^$$\end{domentclass2pt{minimtracing algorithm, based on the enumeration method for sheets in Sect. The cell adding method is conceptually simple and flexible, and may construct polytope decompositions of small size in many cases. For example, the size of orphan-free 3D straight skeletons is bounded by  Theorem , and ten Theorem . This raweighted mitered offset surfaces might be of particular interest, as they offer additional flexibility for adapting to given needs.A concept already considered in , 21 is tpolygon, As a warm-up, we reconsider the polygon case in the light of Definition entclass1pt{minimav of valid offsets are now given by the (dimension-independent) conditions in Definition v. The spherical polygon becomes a possibly disconnected circular domain; see Figs. For a given vertex t. It is easy to verify that t either must, or cannot, contribute to an inner offset which is radially visible. That is, edge events and split events are unique.If no degeneracies occur in the shrinking process for multiple events, Definition vertex event in the literature [switching event, as a point-symmetric exchange of edges takes place. Switching events have the advantage that they keep the straight skeleton of For terature . Edge evinterior-connected when orphan edges are allowed in events (but remain connected via the resolved vertex). In fact, a fixed edge n edges; see Fig. On the other hand, skeleton cells do not stay v. See Fig. sliding event, its three radially visible offsets, and the different skeletons generated by them. The sliding event does not disconnect the weighted straight skeleton. In the offset line arrangement t degenerates to two slabs separated by the third offset line. Therefore, this type of ambiguity cannot be fixed by ruling out orphan edges owing to the more general definition of an offset boundary; the arity of the sliding event is 2.In the weighted case, ambiguity arises in a generic way in a new edge event: The offsets of two parallel edges We now study the effect of weighting on a polytope Most importantly, Definition same orientation may identify for some value The generic condition  is constructed in one go at time v becomes a skeleton corner of degree 4, like for unweighted skeletons. The spoke graph does not continue at v. This event is generic, belongs to the class of edge events because one edge vanishes, and is of arity 2. Notice that Similarly, when the polygon sliding facet event. It stems from two parallel planes in There also exists a non-initial generic non-complementary setting).We recall from Sect. No other new weighted events exist in the generic case. initial events can change when weighted, but generic non-initial events do not, except possibly for facet events, albeit new event types are enabled. The incidence structure of the produced weighted straight skeleton, n interior-connected cells, when the construction obeys orphan-free vertex resolution. Also, the property is needed for Lemma In summary, the When orphan facets may be produced in the resolution of vertices, then a cubic lower bound exists. The construction in Fig. voids—but only if without orphan cells.The only larger difference to the unweighted case is the lack of monotonicity of the skeleton cells Lemma . For genNo other proofs in earlier sections are based on the monotonicity property, such that the respective results carry over to the weighted setting. For instance, if ds Lemma , so thatconvex polytopes n facets and volume 1. For any positive numbers In this context, let us mention an optimality property of weighted roofs for roved in . Let \dod-space The concepts of polytope, radial visibility, and arrangement of planes generalize to Euclidean v, being based on its offset arrangement d. This also includes the orphan-free case, and positively weighted surfaces. Straight skeletons of nonconvex polytopes in j incident to each inner corner v. In particular, exactly v. Geometric—but no combinatorial—degeneracies can occur in the cell structure, due to doubly-adjacent faces of various dimensions; cf. Sect. The proof of the existence of mitered offset surfaces for a polytope vertex n facets can have n, and the number of skeleton corners is trivially bounded by The regular structure of a straight skeleton stands in contrast to the incidence structure of the medial axis of entclass1pt{minimav of d-simplex as the only bounded cell. This implies that, interestingly, such events are either unary or binary; there exist at most two resolution surfaces for v, like in the three-dimensional case  quadratically many facet events and global events, because contacting faces of different dimensions have to be paired up. This is probably also the most suitable method for detecting these events, as higher-dimensional data structures tend to lose their efficiency in practice. However, all edge events  can still be detected directly.Clearly, the number of event types within each class increases with the dimension. There are v can be found reasonably quickly for small dimension d. Also for initial events, where As bisector graphs are not appropriate any more, the algorithm of choice for event processing is the cell adding procedure using the arrangement n roof hyperplanes in The concept of roofs and their complexes inside a polytope j-sheets, for n for (weighted) straight skeletons, because each blue/red pair of layers can be charged to the construction of some skeleton cell again. The count for corners per blue layer can be done with respect to the angle Finally, the sheet enumeration method from Sect. n facets, for arbitrary constant dimension d. Every (weighted or unweighted) straight skeleton for Let monotone terrains in  hyperplane arrangements in monotone paths in line arrangements [k-level in an arrangement of planes in unbounded terrain facets are present, then obviously the terrain size in O(n) in n faces. It remains unclear what happens in higher dimensions. However, from the straight skeleton bound in Theorem n unbounded facets behave like zones, having a combinatorial complexity of Roof surfaces can be viewed as ngements . They raIn this section, we report on some experimental results and potential applications. Many more details concerning implementation aspects and experiments are provided in .We have implemented an algorithm that computes (weighted) straight skeletons in The detection of events, on the other hand, is a notoriously difficult and time-consuming task, as is well known from the planar case. We have mentioned some possibilities for speed-up along with the event description. The problem is in finding the contacting events, especially the global ones. It seems hard to capture the interplay of the reflex faces of the polytope le graph , 32, whiFigure Pizza Box  in the table below) and Fig. Spiky), and related ones. One of our larger boundary-triangulated input polytopes (ID: Verwortakelt) is displayed in Figs. L of layers if they exist. Except for extreme polytopes, the number of corners always was a small multiple (less that 10) of the number of polytope vertices. The  runtime is manageable for moderate polytope sizes. A brief excerpt of our experiments is given in the following table.We have tested various polytopes of different sizes, mostly boundary-triangulated and modeling common shapes, but also polytopes of extreme shape, like the ones in Fig. convex edges, or reflex edges respectively, at each event. As can be seen from the table (Columns 2 and 3), the former strategy led to fewer events, and thus to straight skeletons of smaller combinatorial size, apart from the first four polytopes whose skeletons are unique. This choice also tends to keep the volume large, whereas the polytope is ‘slimmed down’ more quickly when reflex edges are maximized. However, extreme volumes are not achieved; there exist counterexamples for initial events.Events with arity larger than 1 give choice among different (orphan-free) offset surfaces, especially for initial events. For example, we can maximize the number of all values of Of course, the volume can be maximized (or minimized) directly in each event. Unfortunately, this local optimization strategy does not guarantee that the volume of the offset polytope The effect of weighting on the straight skeleton can be seen in Column 4. In the respective polytope, all facets carry unit weight but one, whose weight was put to 10. This can lead to a drastic reduction of the skeleton size (Rows 1 to 3), when the heavy-weighted facet defines a large cell and extrudes many other facets which had larger cells in the unweighted skeleton. The effect gets slightly stronger when convex offset edges are maximized, wherefore we used this construction rule.L in Column 5 shows that usually only a few layers are present, unless the polytope has an extreme shape. Convex polytopes, like the one in Row 2, have a single layer; see Corollary The layer count surface area. Denote with v of v is the sum of the volumes of the respective pyramids based on v, the  volume difference of the offset polytopes Concerning optimization criteria, let us mention a relation between the volume of the offset polytope construction events. This question is also practically relevant, as such skeletons realize a minimum number of corners.An interesting though probably hard problem is to find (or even characterize) straight skeletons that require a minimal number of Planar straight skeletons enjoy various applications, and the same can be expected for their generalizations in three dimensions. We conclude this paper with mentioning a few promising candidates.offset calculation for triangular, or more generally, polygonal meshes. This operation is important in CAD, for example, for the thinning of solid objects and tool path generation. A suitable offset definition for the meshed surface is usually problematic. Surfaces derived from the medial axis [The most direct application is ial axis  are wellial axis , 34.convert a triangulated input surface into a polygonal surface where all vertices have degree 3.Our method automatically produces a topologically correct mitered offset mesh (which can be re-triangulated if desired), and also determines the largest allowable offset threshold flattened if We say that a polytope  surface . In otheHaving agreed on a particular skeleton for The two-dimensional case is much simpler. Exactly the near-convex polygons can be flattened, and this property can be recognized by constructing a unique straight skeleton. However, if weighting is allowed then no efficient method is known even in the planar case. A discussion of several generalized notions of convexity for polygons (being different from near-convexity) can be found in .decomposition of refine the cells into simple shape: We can use the boundary surface of Every straight skeleton of a polytope se Lemma , but can methods , can be D for the polytope D is much easier than that for any straight skeleton: The arising events only need to be processed in increasing order of time stamps, rather than also be detected, which is by far the more costly task. The time stamp v of D to the 4 supporting planes of v.We recall that a suitable roof complex convergence result similar to the one in [Like in two dimensions, a straight skeleton in rtcoming , which hrtcoming , is to ‘e one in  can be esee e.g. .


Circular RNAs (circRNAs) are emerging as a novel class of noncoding RNAs which potential role as gene regulators is quickly gaining interest. circRNAs have been studied in different tissues and cell types across several animal species. However, a thorough characterization of the circRNAome in ejaculated sperm remains unexplored. In this study, we profiled the sperm circRNA catalogue using 40 porcine ejaculates. A complex population of 1,598 circRNAs was shared in at least 30 of the 40 samples. Generally speaking, the predicted circRNAs presented low abundances and were tissue-specific. Around 80% of the circRNAs identified in the boar sperm were reported as novel. Results from abundance correlation between circRNAs and miRNAs together with the prediction of microRNA (miRNA) target sites in circRNAs suggested that circRNAs may act as miRNA sponges. Moreover, we found significant correlations between the abundance of 148 exonic circRNAs and sperm motility parameters. Two of these correlations, involving ssc_circ_1458 and ssc_circ_1321, were confirmed by RT-qPCR using 36 additional samples with extreme and opposite sperm motility values. Our study provides a thorough characterization of circRNAs in sperm and suggests that circRNAs hold potential as noninvasive biomarkers for sperm quality and male fertility. In humans, infertility is an increasing problem in contemporary society, affecting one in twenty males2, and has become a subject of research in swine4. While in humans, men attend In Vitro Fertilization (IVF) clinics to address infertility issues, the boars at artificial insemination (AI) studs have been previously selected on the basis of their semen quality and consequently, infertility is rare. However, these studs contain subfertile boars and also present phenotypic variation on the semen quality of the animals. Men’s infertility and these phenotypes in the boar studs are expected to share a common molecular basis. Unlike in humans, where fertility data is based on few records per person, the swine industry is currently recording semen quality and fertility phenotypes each time a sire is used for AI. In pig intensive production systems, an AI boar typically inseminates around 1,500 sows during its productive life . Thus, large datasets of the reproductive ability of these boars is becoming available.Swine, due to its similarity to humans in its genome sequence, anatomy and physiology, is quickly becoming an important model for biomedical research5, horse6 and swine7 where significant correlations between sperm motility and fertility have been found. In humans, this technique is also applied to estimate the in vitro fertilizing potential of the ejaculates used in assisted reproductive treatments8.Sperm motility and kinetic parameters provide an objective and reproducible measurement of semen quality that is automatically assessed by the computer-assisted semen analysis (CASA) system. CASA measures among other parameters, the percentage of total motile spermatozoa, the average curvilinear velocity (VCL), the average straight line velocity (VSL) and average path velocity of the sperm cells (VAP) based on trajectories of motile sperm. This approach has been commonly used to assess semen quality in animal breeding prior to AI in cattle9. These studies have focused their research not only on messenger RNAs (mRNAs) but also on different classes of non-coding RNAs, including microRNAs (miRNAs) and transfer RNAs (tRNAs)10.Multiple research efforts have demonstrated that sperm quality parameters and fertility outcomes are related to the presence or absence of sperm RNAs. Different studies have provided evidence that the absence or deregulation of certain RNAs is associated to infertility and/or sperm motility in human, mice and cattle11. circRNAs are highly stable in vivo in comparison to their mRNA linear counterparts, because of their circular structure, which confers protection against exonuclease-mediated degradation12. circRNAs mostly have an exonic origin. However, intronic and intergenic circRNAs have also been described13. While the first ones are preferentially located in the cytoplasm, the other two are found in the nucleus. Expanding views on their function suggest that circRNAs can directly regulate the abundance of their cognate mRNA and can also act as miRNA sponges14, thus impeding the post-transcriptional inhibitory roles of these miRNAs on their target mRNAs. circRNAs have been identified across tissues in several species including the fruitfly, human, mice and swine, following tissue-specific expression patterns12.Circular RNAs (circRNAs) are a novel class of non-coding RNAs with a closed loop structure, mainly formed through pre-mRNA back splicing event15. There are few reports characterizing circRNAs in reproductive organs and cell types including oocyte, embryo, placenta, granulosa cells, immature spermatogenic cells, seminal plasma, testis and more recently, spermatozoa18. Up to date, a small number of studies have assessed the circRNA predictive potential for reproduction outcomes. Chang and colleagues identified circRNAs associated to embryo quality and suggested their role as potential predictors of live birth19. A novel study carried by Chioccarelli et al., has characterized the human sperm circRNAome using a circRNA microarray in three normozoospermic donors18. In this study, the authors compared, within each sample, two sperm sub-populations defined by their good and bad quality in terms of motility and morphology. This comparison yielded 148 differentially abundant circRNAs between the two semen quality groups18.During the last few years, there has been a considerable interest in the potential of circRNAs as health biomarkers. Several studies have identified circRNAs which abundances were associated to cancer, aneurysms, hypertension, heart failure, diabetes or arthritis20 and the data, in line with previous studies9, showed that the majority of the transcripts are highly fragmented and with low abundances. Considering their stability and abundance, circRNAs could hold an important potential as reliable biomarkers for sperm quality and fertility. To shed light into the potential relevance of circRNAs on semen quality, we have carried a de novo characterization of the circRNA repertoire in 40 porcine spermatozoa samples and assessed their potential role as miRNA sponges using total and small RNA-seq. Furthermore, we have investigated the correlation between their abundance and sperm motility traits.Our group has recently carried a thorough description of the porcine sperm transcriptomeThe average percentage of motile spermatozoa was 75.5 with a standard deviation (SD) of 17.4, VCL , VSL  and VAP .RNA extraction from 40 mature sperm samples free from somatic cells yielded an average of 2.2 fg per spermatozoa (range between 0.8 and 3.7 fg) , histone H3-K36 methylation  and chromatin organization  (Supplementary Table −4), cilium assembly  and developmental process   analysis of the circRNA host genes revealed an enrichment for epigenetic functions including histone modification 22 and swine 23. Twenty-four % and 11.3% of the boar sperm circRNAs had potential orthologs in human and mouse, respectively 24, and rat and mouse brains25. All the boars from our dataset were sexually mature with ages ranging between 9 and 54 months of age. Sexual maturity in boars is a process that starts at the age of 8 months and finalizes when animals reach 2 years. Thus, we divided the samples in those coming from boars approaching sexual maturity with ages below 2 years old and those produced by mature pigs with ages above 2 years old. There was no significant difference in the number of circRNAs identified  nor in their RNA abundance  between the two groups. We repeated the analysis considering only extreme ages: young  and mature . Again, there was no difference in the number of circRNAs identified  or in their abundance .We assessed whether sperm circRNAs depicted an age-accumulating profile as it has been previously observed in rat testesin silico prediction of miRNA target sites involving the miRNAs and exonic circRNAs identified in our data. We assessed miRNA target sites in the exonic circRNA sequences based on sequence complementarity using the tool miRanda26. The analysis yielded 4,987 potential targets involving all the 95 miRNAs and 1,103 circRNAs. To reduce the proportion of false-positives, only the 81 interactions  that were identified in both approaches were used for network visualization  and the 95 miRNAs (from short RNA-seq libraries). The analysis identified 1,882 significant correlations that included 95 miRNAs and 458 exonic circRNAs. In parallel, we also performed an A total of 148 exonic circRNAs from 142 genes showed significant correlations between their abundance and sperm motility traits (Supplementary Table PTGES3) with 75.0 CPM, and ssc_circ_0670 (from BAZ2B) with 10.6 CPM, both also detected in human22  and in swine testes23. The PCR amplification of the two circRNAs resulted in a single electrophoretic band of the expected size  and a Sanger Sequencing approach. First, we randomly selected two circRNAs with different RNA abundance levels to test whether we could confirm the bioinformatically predicted circRNAs. The chosen circRNAs were: ssc_circ_1141  and ssc_circ_0118 from PDE10A  displayed 2 amplification bands , VCL , VSL  and VAP . The RT-qPCR assays presented efficiencies between 99.6% and 105.2%.The RT-qPCR levels of the 6 circRNAs were measured in 36 animals presenting extreme and opposite values of sperm motility (N = 18 for each phenotypic distribution tail) from a dataset of 300 boar ejaculates with phenotype records. None of the 36 samples was included in the RNA-seq study. The two sample groups displayed significant phenotypic differences for all the traits: percentage of total motile spermatozoa  and ssc_circ_1321 from PAPOLA . A third circRNA, ssc_circ_1132, from LIN7A, showed significant differences between the two motility groups   Fig.  but in tups Fig. .18. circRNAs have been reported as promising potential biomarkers for human health including cancer, diabetes, cardiovascular diseases or the pathogenesis of pre-eclampsia15. Here, we provide novel data supporting the involvement of circRNAs on the male’s reproductive function as reflected by their association with several sperm motility parameters.This study provides the first RNA-seq based circRNA repertoire of mature spermatozoa in a mammalian species. We have also assessed the sperm circRNA role as miRNA sponges and evaluated their associations with sperm motility as a parameter of semen quality. Our results provide further evidence and expand the relevance of spermatozoa RNAs in sperm biology and quality, as recently described in other studies23, human17 or rat24. Nonetheless, the list of boar sperm circRNAs showed a modest overlap with these other datasets . The sperm circRNA repertoire showed similar circular genomic characteristics including the proportion of genomic overlap with protein coding and intergenic regions, the number of exons and the circRNA length  harboring the 15 most abundant exonic circRNAs 24, all our samples were from adult pigs with less dissimilar ages (between 9 and 54 months) taking into account that swine has a longer life cycle. Second, the study in rats assessed the whole testis payload, which includes cells from the spermatogenic lineage and also, leukocytes, endothelial, Sertoli and Leydig cells. Our study in pigs exclusively queried mature-selected ejaculated spermatozoa. Cells with high proliferation rates34 such as spermatogonial stem cells (SSC) have been suggested to accumulate less circRNAs due to passive thinning out during their continuous self-renewal and proliferation until they become spermatozoa. Spermatozoa originates from SCC through a highly constant and orchestrated process and upon maturation, they enter a silent state with no apparent transcriptional activity. This would imply that in spermatozoa, circRNAs do not accumulate through age. Interestingly, our results indicate that circRNAs in mature sperm are stable and thus, provide further support for the potential of circRNAs as noninvasive biomarkers for male reproductive conditions.A previous study in rat testes identified a dynamic circRNA age-dependent pattern of expression and suggested a relation between their abundance and function with the male’s sexual maturity and spermatogenesisin silico prediction of miRNA target sites to increase the reliability of the circRNA-miRNA relationship predictions. This network of 81 interactions contained some circRNA genes and miRNAs previously related to semen quality and male fertility. Remarkably, 2 circRNAs, ssc_circ_0954 and ssc_circ_1454 were linked to 4 different miRNAs each. The first, ssc_circ_0954 arises from DCDC2C, a gene identified in the human sperm flagellum end-piece with a suggested role on microtubule dynamics by acting as a depolymerization/polymerization balancing system35. This circRNA was predicted to interact, among others, with miR-361-3p which was found to be dysregulated in subfertile men36, miR-423-5p, which was altered in oligozoospermic men37 and miR-28-5p, a miRNA that was dysregulated in normozoospermic infertile individuals38. The other cirRNA, ssc_circ_1454, is transcribed from MTHFD2L, a mitochondrial isozyme from the folate cycle metabolic pathway, a vitamin that has been also related to semen quality and fertility in men39. One of the miRNAs targeted by ssc_circ_1454 was miR-16, which was dysregulated in subfertile men36. The network also showed 10 circRNAs that may be regulating miR-28, a miRNA (miR-28-5p) that was dysregulated in normozoospermic infertile individuals38. The potential miR-28 regulators included ssc_circ_1370, arisen from FAM92A, whose protein may play a role in ciliogenesis40, and ssc_circ_0002 from WDR7, which is associated to sperm quality in cattle41. Likewise, 9 circRNAs were predicted to regulate miR-26a, which has been in turn, linked to VCL, VSL and VAP motility parameters in a previous study in swine42. miR-26a was also targeted by ssc_circ_0361, a circRNA from ACTL6A, which is a gene with a crucial function for embryo development43 and ssc_circ_1352 from CAGE1, an acrosomal protein with proposed roles in fertility44. Additional interesting interactions included ssc_circ_0345 from the hotspot gene SLC5A10 (sodium-dependent mannose and fructose transporter) , is a circRNA hosted by CAMK4, a gene that has been implicated in sperm motility in humans46. ssc_circ_0780, correlated with the percentage of motile cells , is hosted by LRGUK, a gene that is required for sperm assembly including the growth of the axonome, a key structure for the flagellar beating in sperm47. We successfully tested by RT-qPCR 6 of the 148 exonic circRNAs that correlated to sperm motility. We validated their correlation with motility traits for 2 of these 6 circRNAs, ssc_circ_1458 from LRBA and ssc_circ_1321 from PAPOLA. Both were significantly down-regulated in the ejaculates with low sperm motility  and the host gene is implicated in RNA and ATP binding. Thus, none of the two host genes have been previously related to sperm function or fertility. However, the association is confirmed by RT-qPCR and we cannot exclude unidentified relevant functions on sperm motility. Two other circRNAs were also of high interest and tested as they were within the top 15 most abundant   of the 12 hotspot genes hosted a circRNA with significant abundance correlation with sperm motility. Thus, one third of the hotspot genes included circRNAs correlated with sperm motility whilst only 15.0% (148) of the 984 genes hosting the 1,598 circRNAs were correlated with motility (Supplementary Table 23 (data not shown). Altogether, this indicates that circRNA hotspot genes may have relevant tissue-specific functions.Remarkably, 4 , which includes colloidal silica particles coated with silane in an isotonic salt solution in a 15 ml RNAse-free tube. Then, the samples were subjected to centrifugation at 300 × g for 20 min at 20 °C. After centrifugation, all the upper phases were removed and the cell pellets were transferred to a new RNase-free 15 mL tube, washed with 10 mL of RNase-free PBS and centrifuged at 1,500 × g for 10 min at 20 °C. The supernatants were then removed and the pellets optically inspected using a microscope to confirm the removal of somatic cells. RNA was extracted from purified sperm cells, treated with TURBO DNA-free™ Kit (Invitrogen) and quantified using QubitTM RNA HS Assay kit (Invitrogen). We assessed RNA integrity with the 2100 Bioanalyzer and yield using the Agilent RNA 6000 Pico kit (Agilent Technologies). We then performed RT-qPCR assays for PRM1 and PTPRC mRNAs as well as for intergenic/genomic DNA to verify proper sperm purification.Ejaculates were purified to remove somatic cells and immature sperm cells and purified sperm was stored at −96 °C with TrizolForty sperm RNA samples were subjected to total RNA-seq. Thirty-four of these samples were also used for small RNA-seq. For total RNA-seq libraries, ribosomal RNA (rRNA) was depleted with the Ribo-Zero Gold rRNA Removal Kit (Illumina) and libraries were constructed with the SMARTer Universal Low Input RNA library Prep kit v2 (Clontech). Resulting libraries were sequenced in a HiSeq2000/2500 system (Illumina) to generate 75 bp long paired-end reads. For small non coding RNA-seq libraries, extracted RNA without rRNA depletion was directly subjected to library preparation with the NEBNext Small RNA Library Prep Set kit (New England Biolabs) and sequenced on a HiSeq2000 (Illumina) to generate 50 bp single-end reads.53. The identification of circRNAs was carried on these reads with the find_circ pipeline14 with stricter filter stringency. To reduce the false positive rate in the discovery of circRNAs, we selected circular splice transcripts with at least two unique supporting reads in the anchor segment and with Phred quality scores of 35 or more. Moreover, only circRNAs predicted in at least 30 samples were kept. The RNA abundance of the predicted circRNAs were normalized as the number of BSJ spanning reads per million raw reads (CPM)21. The functional regions of circRNAs were identified based on their co-location with genomic features  from the Ensembl database (release 91) with BEDtools54. Our catalogue of boar sperm circRNAs was contrasted with other publically available porcine circRNA databases including heart, liver, spleen, lung, kidney, ovarium, testis, skeletal muscle, fat and fetal brains23. We also queried several human tissues  and murine (cell lines and brain sections) available at the circBase database22. Genomic coordinates from the human and mouse circRNAs were liftover to Sscrofa11.1 using the UCSC liftover tool55.For the total RNA fraction, raw reads were filtered by removing adaptor sequences and low-quality reads with Trimmomatic v.0.3656. The mapping of sncRNAs was performed with the sRNAtoolbox v.6.1757 with default settings and providing miRBase58 release 21 as library dataset. Multi-adjusted read counts were then normalized by sequencing depth as CPM. We only considered the miRNAs that were detected >1 CPM in all the samples.For the small RNA-seq analysis, trimming of adaptors and low quality bases was performed with Cutadapt v1.059 using the RNA abundance levels of the of exonic circRNAs and miRNAs after stabilization with log2 transformation. Only negative RNA abundance correlations between circRNAs and miRNAs were kept. We further assessed circRNAs-miRNAs interactions using miRanda v.3.3a26 to predict miRNA target sites in the circRNAs sequences. The potential interactions identified with both approaches were kept and visualized with Cytoscape v.3.7.060.To identify circRNA:miRNA interactions, we carried a Partial Correlation with Information Theory (PCIT) analysis61 with the overrepresentation test and P-values corrected with FDR. Annotation Data Set was “GO biological process complete”. Pearson correlation was used to determine associations between circRNA abundance levels and sperm motility parameters. P-values < 0.05 were considered statistically significant.GO analysis was carried with PANTHER v.13.1circRNAs were validated by Sanger Sequencing and quantified by RT-qPCR using divergent primers. Primers were designed using the Primer Express software (Applied Biosystems). Primer sequences are shown in Supplementary Table 62.The abundance level of 6 circRNAs, correlated to sperm motility parameters in the RNA-seq study, was analyzed by RT-qPCR in 36 samples, none of them included in the RNA-seq. These 36 samples belong to two groups with extreme and divergent values for sperm motility from a bank of 300 ejaculates with phenotypic records. Isolated RNA could not be treated with RNAse R due to extremely low RNA yield  applying the 2-ΔΔCt method. The same software was used to compare the biological groups. Significance was set at a P-value < 0.05.Quantitative PCR reactions were performed in triplicate following the manufacturer’s instructions. The final reaction (15 μL) included 7.5 μL SYBR Select Master Mix (Life Technologies -Thermo Fisher Scientific), 300 nM of each primer and 3.75 μL of cDNA 1:4 diluted on a QuantStudio 12K Flex Real-Time PCR System (Applied Biosystems). To evaluate the efficiency of the RT-qPCR assays, standard curves with 6 serial dilutions from a pool of sperm cDNA were generated. Thermal profile was set as follows: 50 °C for 2 min, 95 °C for 10 min and 40 cycles at 95 °C for 15 sec and 60 °C for 60 sec. Moreover, a melting profile  was programmed at the end of the RT-qPCR to assess the specificity of the reactions. The genes Supplementary Information.Supplementary Tables.


ALK, ROS1, RET and NTRK gene fusions and MET splicing events in advanced-stage NSCLC patients. Overall, our RNA fusion panel was able to detect all fusions and a splicing event harbored in a RNA pool diluted up to 2 ng/µL. In addition, It also successfully analyzed 46 (95.8%) out of 48 routine samples previously characterized by conventional non - NGS technology, representing a robust tool for routine setting.Gene fusions represent novel predictive biomarkers for advanced Non Small Cell Lung Cancer (NSCLC) patients. In this study, we developed and validated a narrow Next Generation Sequencing gene panel able to cover Gene fusions represent novel predictive biomarkers for advanced non-small cell lung cancer (NSCLC). In this study, we validated a narrow NGS gene panel able to cover therapeutically-relevant gene fusions and splicing events in advanced-stage NSCLC patients. To this aim, we first assessed minimal complementary DNA (cDNA) input and the limit of detection (LoD) in different cell lines. Then, to evaluate the feasibility of applying our panel to routine clinical samples, we retrospectively selected archived lung adenocarcinoma histological and cytological (cell blocks) samples. Overall, our SiRe RNA fusion panel was able to detect all fusions and a splicing event harbored in a RNA pool diluted up to 2 ng/µL. It also successfully analyzed 46 (95.8%) out of 48 samples. Among these, 43 (93.5%) out of 46 samples reproduced the same results as those obtained with conventional techniques. Intriguingly, the three discordant results were confirmed by a CE-IVD automated real-time polymerase chain reaction (RT-PCR) analysis . Based on these findings, we conclude that our new SiRe RNA fusion panel is a valid and robust tool for the detection of clinically relevant gene fusions and splicing events in advanced NSCLC. ALK) rearrangement (3–6%), which can predict marked sensitivity to crizotinib, alectinib, and brigatinib treatments (10)-ALK(20), Solute Carrier Family 34 Member 2 [SLC34A2](4)-ROS1(32), Coiled-Coil Domain Containing 6 [CCDC6](1)-RET(12) and Tropomyosin 3 [TPM3](8)-NTRK1(10)] were purchased from Horizon Diagnostics . In addition, several cell lines, each carrying a specific gene fusion, were purchased from the American Type Culture Collection. RNA was extracted from of the following cell lines: H3122 [EML4(13)-ALK(20)], H2228 [EML4(6)-ALK(20)], Hs746T (MET exon 14 skipping), H596 (MET exon 14 skipping), HCC-78, [SLC34A2(4)-ROS1(32)]; LC2ad [CCDC6(1)-RET(12)], EBC-1, (MET Amplification), SUDHL-1, -NTRK1(10)] ] . FinallyRK1(10)] .ALK, ROS1, RET, and NTRK gene fusions, as well as the MET exon 14 skipping mutation.For molecular reference standards, RNA was extracted with the All Prep DNA- RNA Mini Kit , according to the manufacturer’s instructions, and re-suspended in 50 µL of RNAsi/DNAsi free water . RNA was directly extracted from the above mentioned cell lines  with the High Pure RNA Isolation kit  according to the manufacturer’s instructions. TapeStation 4200 , a microfluidic based technology, evaluated the RNA concentration (ng/µL) and RNA integrity number (RIN). The RNA was eluted in 30 μL of RNAsi/DNAsi free water . Retro-transcription was carried out with SuperScript IV VILO Master Mix (ThermoFisher Scientifics) according to the manufacturer’s instructions. The cDNA was then quantified by using the D1000 Genomic assay on TapeStation 4200 (Agilent Technologies) . Our cusALK, ROS1, RET, and NTRK gene fusions, as well as the MET exon 14 skipping mutation. This retrospective analysis was based on previous diagnoses of gene rearrangements involving ALK, ROS1, and RET made by various Italian institutions by immunohistochemistry (IHC), fluorescent in situ hybridization (FISH), and NGS analyses  S. Sample characteristics and gene alterations are reported in both To evaluate the performance of our SiRe fusion panel in a clinical setting, we retrospectively selected archived lung adenocarcinoma histological and cytological (cell blocks) samples harboring The fusion primer pool for SiRe was designed with the AmpliSeq designer software v.7.4.1., which uses hg19 as a reference sequence. Overall, 16 samples in 520 chips on a S5/S5XL Ion Torrent platform (ThermoFisher Scientifics), were run simultaneously. Remarkably, the pool of 91 primer pairs covered all the targeted gene fusions, reaching a 5000× coverage for each amplicon. Moreover, since the fusion primer pool was fully compatible with the custom NSCLC DNA primer pool, a DNA/RNA pool was generated in a single NGS run. The DNA libraries can be processed in parallel with RNA libraries, under the same thermal and sequencing conditions. The RNA primer pool, as reported in the method section, was developed to be fully integrable with SiRe primers pool . The SiRALK, ROS1, RET, and NTRK, as well as MET exon 14 skipping mutations. Taking into account all the mutant cases, we obtained, on average, 99.70% (ranging from 99.13% to 99.98%) on-target reads, 103.55 bp read lengths (ranging from 93.00 to 114.00), 311,798.44 mapped reads , and 318415.89 coverage uniformity . The SiRe RNA fusion panel was able to detect ALK, ROS1, RET, and NTRK rearrangements, as well as MET exon 14 skipping mutations. All the results, including the fusion partner genes, are reported in All samples processed for the feasibility test passed the quality filters. The NGS run parameters ensured an optimal analytical performance of the RNA fusion panel, as evidenced by its ability to reliably detect clinically relevant rearrangements in ALK, ROS1, RET samples, as well as the MET exon 14 skipping kit and the NTRK fusion kit  fusion positive samples of advanced stage NSCLC patients  retrieved from the archives. Sample characteristics, mutational status, and technologies are reported in ALK, ROS1, RET and NTRK) and MET exon 14 skipping mutations is pivotal in choosing correct management of advanced-stage NSCLC [MET exon 14 skipping mutations. To this end, an integrated RNA- and DNA-based NGS approach should be preferred [Advanced lung cancer, like many other malignancies, is driven by various genetic changes, including gene fusions and splicing events. Thus, the identification of therapeutically relevant gene fusions  the results obtained with conventional techniques. Considering the three discrepant cases between NGS and FISH analysis, it should be considered that in all instances a “borderline” FISH positive results was obtained, featuring a percentage of rearranged nuclei around 15%. In addition, our approach was able to identify all gene fusions and the splicing event harbored in the RNA pool extracted from cell lines, even when the RNA concentration was as low as 2 ng/µL. Taking into account this parameter, it is conceivable that the inadequate NGS results may be related to low quality RNA. This underlines its applicability to cytological samples and small histological biopsies, which represent 80% of all advanced stage NSCLC sample types.In our experience, we demonstrated that our narrow SiRe NGS fusion panel is a reliable tool to detect clinically relevant gene fusions and ®) and RNA (SiRe fusion) primer pool in an Ampliseq based approach to cover all the clinically relevant NSCLC alterations  simultaneously in a single run, regardless of the small sample size of available tissue. A further study is ongoing to evaluate in addition to the specificity also the sensitivity of our panel on a wide cohort of patients, including not only mutated but also wild type cases. In conclusion, the study fully validates the feasibility of integrating a DNA (SiRe


The highly conserved HIV‐1 transactivation response element (TAR) binds to the trans‐activator protein Tat and facilitates viral replication in its latent state. The inhibition of Tat–TAR interactions by selectively targeting TAR RNA has been used as a strategy to develop potent antiviral agents. Therefore, HIV‐1 TAR RNA represents a paradigmatic system for therapeutic intervention. Herein, we have employed biotin‐tagged TAR RNA to assemble its own ligands from a pool of reactive azide and alkyne building blocks. To identify the binding sites and selectivity of the ligands, the in situ cycloaddition has been further performed using control nucleotide (TAR DNA and TAR RNA without bulge) templates. The hit triazole‐linked thiazole peptidomimetic products have been isolated from the biotin‐tagged target templates using streptavidin beads. The major triazole lead generated by the TAR RNA presumably binds in the bulge region, shows specificity for TAR RNA over TAR DNA, and inhibits Tat–TAR interactions. Taking the lead: The highly dynamic secondary TAR RNA structure of HIV‐1 is used to template cycloaddition reactions leading to the generation of triazole‐containing thiazole peptidomimetics as selective leads. The triazole ligand with flexible methylene units selectively binds to TAR RNA over TAR RNA without a bulge and TAR DNA. The ligand presumably binds in the bulge region of the TAR RNA and inhibits its interaction with the Tat peptide. Human immunodeficiency virus type‐1 (HIV‐1)Herein we demonstrate the rapid discovery of HIV‐1 TAR‐Tat inhibitors by TAR RNA guided in situ cycloaddition of azide and alkyne fragments. The in situ cycloaddition was also performed with a TAR RNA without the UCU bulge  and a TAR DNA  and azide (2 a–k) building blocks , a carboxylic acid (2 b), a nitro (2 d), an ester (2 g), an amino acid (2 h), a carboxamide (2 j), and a guanosine azide (2 k). The combinations of these building blocks could lead to 88 possible regioisomers ‐ and syn‐‐triazoles).For the in situ cycloaddition, we designed and synthesized a library of alkyne (nd azide –k buildim) in 20 mm sodium cacodylate, 180 mm NaCl, and 10 mm MgCl2 buffer at pH 7.4 and 25 °C for 72 h. To identify the selective leads for the TAR RNA, control experiments were performed with biotinylated TAR RNA w/b and biotinylated TAR DNA  and 3 ba  in a ratio of 41:59 by the TAR RNA  was selectively formed using the control TAR RNA w/b . HPLC and ESI‐MS analysis showed the reactions yielded similar lead compounds .anti‐ regioisomers. Comparison of the HPLC traces of the RNA‐templated products with the authentic 1,4‐triazole samples  revealed that the in situ reaction generates predominantly the anti‐ isomers.Hit triazole products anti‐triazoles were further determined by time‐dependent cycloaddition of the corresponding alkyne  and azide (2 a) fragments in the presence of the biotin‐tagged target and control DNA and RNA targets . Alkyne 1 b reacted with azide 2 a in the presence of TAR RNA to form anti‐3 ba in 62 % yield . Similarly, lead anti‐3 ca (from 1 c and 2 a) was obtained in 54 % yield with TAR RNA . However, in the presence of TAR DNA, anti‐3 ca was formed in 42 % yield. Lead anti‐3 aa (from 1 a and 2 a) was formed in 51 % yield in the presence of the control TAR RNA w/b.The yields of the lead 3 ba showed an exothermic binding interaction with TAR RNA with a dissociation constant (Kd) of 0.49 μm. In comparison, ligand 3 ca showed an approximately 15‐fold lower binding affinity, with Kd=7.2 μm for the TAR RNA  studies . Ligands 3 ba and 3 ca show an increase in their fluorescence intensity of approximately 5.1‐ and 5.3‐fold respectively, upon titration with TAR RNA . In comparison, 3 ba didn't result in a significant increase in the fluorescence intensity in the presence of TAR DNA and TAR RNA w/b .The binding specificity of the hit triazoles was further analyzed using a fluorescence titration assay. Upon excitation at A Figure . Howeveranti‐triazole ligands to displace the Tat peptide from a preformed complex between fluoresceinated carboxytetramethylrhodamine (TAMRA) labeled Tat peptide and fluorescein amidite (FAM) labeled TAR RNA by a Förster resonance energy transfer (FRET) based assay , which indicates 3 ba has a higher affinity for TAR RNA. However, ligand 3 aa wasn't able to displace 50 % of the bound Tat peptide from the TAR RNA  was performed with the AutoDock program v4.1 to elucidate the structure–activity relationship of the ligands with the RNA structure . Anti‐3 ca, on the other hand, exhibited a planar crescent‐shaped structure which stacks between the major groove of the TAR RNA  and TAR DNA. The bulge region further widens the groove (in TAR RNA and DNA) and makes the stacking interactions more favorable but nonspecific for anti‐3 ca.Molecular docking of In summary, in situ click chemistry has been successfully applied to discover novel thiazole‐based peptidomimetics as HIV‐1 TAR‐Tat inhibitors. The TAR RNA itself acts as a template for the cycloaddition and promotes the formation of a thiazole peptidomimetic 1,4‐triazole lead that shows high selectivity for TAR RNA. Isothermal calorimetric and fluorescence studies reveal that a thiazole peptidomimetic with two thiazole rings and flexible methylene units strongly interacts with the bulge region of TAR RNA and effectively inhibits the Tat‐TAR RNA interactions. Further studies are currently underway to apply this method for the discovery of novel TAR RNA inhibitors and to investigate their antiviral activities.The authors declare no conflict of interest.As a service to our authors and readers, this journal provides supporting information supplied by the authors. Such materials are peer reviewed and may be re‐organized for online delivery, but are not copy‐edited or typeset. Technical support issues arising from supporting information (other than missing files) should be addressed to the authors.SupplementaryClick here for additional data file.