Drosophila segmentation is a well-established paradigm for developmental pattern formation. However, the later stages of segment patterning, regulated by the “pair-rule” genes, are still not well understood at the system level. Building on established genetic interactions, I construct a logical model of the Drosophila pair-rule system that takes into account the demonstrated stage-specific architecture of the pair-rule gene network. Simulation of this model can accurately recapitulate the observed spatiotemporal expression of the pair-rule genes, but only when the system is provided with dynamic “gap” inputs. This result suggests that dynamic shifts of pair-rule stripes are essential for segment patterning in the trunk and provides a functional role for observed posterior-to-anterior gap domain shifts that occur during cellularisation. The model also suggests revised patterning mechanisms for the parasegment boundaries and explains the aetiology of the even-skipped null mutant phenotype. Strikingly, a slightly modified version of the model is able to pattern segments in either simultaneous or sequential modes, depending only on initial conditions. This suggests that fundamentally similar mechanisms may underlie segmentation in short-germ and long-germ arthropods. Drosophila embryos, all segments are patterned rapidly and simultaneously during early development, in a process known as “long-germ” embryogenesis. In contrast, many insect embryos retain an ancestral or “short-germ” mode of development, in which segments are patterned sequentially, from head to tail, over a period of time. In both types of embryo, the patterning of segment boundaries is regulated by a network of so-called “pair-rule” genes. These networks are thought to be quite divergent due to the different expression patterns observed for the pair-rule genes in each case: regularly spaced arrays of transient stripes in Drosophila, and dynamic expression within a posterior “segment addition zone” in short-germ insects. However, even in Drosophila, a clear understanding of pair-rule patterning has been lacking. Here, I make a computational model of the Drosophila pair-rule network and use simulations to explore how segmentation works. Surprisingly, I find that Drosophila segment patterning relies on pair-rule gene expression moving across cells over time. This conclusion differs from older models of pair-rule patterning but is consistent with the subtly dynamic nature of pair-rule stripes in real embryos, previously described in quantitative studies. I conclude that long-germ and short-germ segmentation involve similar expression dynamics at the level of individual cells, even though they look very different at the level of whole tissues. This suggests that the gene networks involved may be much more conserved than previously thought.Segmentation in insects involves the division of the body into several repetitive units. In  Drosophila melanogaster has a segmented body plan. This segmental pattern is laid down in the embryo during the first 3 hours of development. During this time, the anteroposterior (AP) axis of the blastoderm is progressively patterned down to cellular-level resolution by an elaborate, multi-tiered network of genes and their encoded transcription factors . Co. Coen, oftz expression is directly patterned by Eve; runt is patterned by Odd, which is itself patterned by Eve; and slp is patterned by Runt, which, as just mentioned, is indirectly downstream of Eve. In the first simulation    and thene mutant , simulat off see .eve mutants [odd and ftz primary stripes are broader than usual  are derepressed  The loss of Runt activity then contributes to the misexpression of the slp primary stripes, which turn on at the end of cellularisation and would normally be patterned by both Eve and Runt  and replaced it with early repression from Runt and Odd, both of which repress the eve late element in wild-type embryos (hairy by gap inputs (G1 in the model) and replaced it with autorepression, which is a common phenomenon for her/hes family genes [opa expression to the decay of Cad by making Cad repress opa. These changes allowed the gap inputs G1 and G2 to be removed from the model and simplified the temporal control of the patterning process, putting everything downstream of Cad and Hairy.The only explicit regulatory changes that I made to the simulation were to the control logic of  embryos , right. ly genes ,108. ThiDrosophila network and explored how it behaved at the tissue level, given various sets of initial conditions. I found that if I set up a repeating initial phase gradient of Hairy expression  and left initial Cad expression unchanged, the simulation output was essentially identical to before , thDrosophila-like pair-rule gene expression and with “clock-and-wave-front” pair-rule gene expression similar to that seen in, e.g., Tribolium. This finding suggests (1) that many of the regulatory interactions within the Drosophila pair-rule network might be conserved in short-germ arthropod species and (2) that long-germ and short-germ segmentation involve essentially similar patterning mechanisms and dynamics. Specifically, segment-polarity fate appears to be determined in both cases by the intersection of 2 sets of temporal signals, those carried by the pair-rule genes and those carried by broadly expressed extrinsic timing factors. In long-germ embryos, a periodic pattern of segment-polarity fates is achieved by directly imparting spatial information to the pair-rule genes using the gap system and stripe-specific elements. In short-germ embryos, however, retracting wave fronts of the timing factors provide this spatial information, and the gap genes play other roles.The modified network , right iDrosophila pair-rule network using a combination of simulated and experimental data to reveal how segment patterning is achieved. I have discovered a functional role for dynamic gap inputs in correctly phasing the pair-rule stripes and propose revised mechanisms for the patterning of the odd-numbered and even-numbered parasegment boundaries. In contrast to previous models based around the principle of static morphogen gradients, these mechanisms involve a coordinated interplay between intrinsic network dynamics and extrinsic spatiotemporal signals and do not necessarily require graded pair-rule activity.In this manuscript, I have analysed the structure and dynamics of the en stripes and hence both sets of parasegment boundaries [en. Later, it was recognised that Eve does not regulate en directly but instead represses several other pair-rule factors that themselves repress en [en stripes [eve mutant embryos. Instead, I propose that Eve conveys positional information largely via its expression dynamics, which are decoded downstream by the rest of the pair-rule gene network.  represent the relictual components of some ancestral segmentation clock that drove oscillations of these genes with double-segment periodicity [The findings also clarify the relationship between long-germ and short-germ segmentation. I have shown that pair-rule patterning is fundamentally a temporal process and that it is therefore straightforward to convert the ion zone . This hyosophila . Strikiniodicity .Drosophila.The evolutionary transition from short-germ segmentation to long-germ segmentation is thought to have been driven largely by selection for faster development. A segmentation clock can only tick so fast, and therefore represents a tight production bottleneck when generating segments: a longer body requires a longer time .  a heterochronic shift in the deployment of the pair-rule network from posterior segment addition zone to blastoderm and (2) the elaboration of the posterior gap network and associated stripe-specific elements. These latter processes are likely to have occurred progressively along the AP axis, with intermediate forms exhibiting a composite mode of development: simultaneous for more anterior segments, sequential for more posterior segments . The proDrosophila pair-rule network, 2 of the primary pair-rule genes are patterned by gap inputs while 3 are patterned by cross-regulation. However, the system works equally well if only 1 of these genes is explicitly patterned and the rest are cross-regulated  Conseqng modes –122. Theng modes .Drosophila blastoderm, the regulatory situation appears to be partially redundant: at least 3 of the primary pair-rule genes possess stripe-specific elements in any given pair-rule repeat, but only 2 of these genes lack zebra elements [runt retain a zebra element when they already have a full set of stripe-specific elements?Above, I stated that stripe-specific elements and cross-regulation are theoretically interchangeable for positioning pair-rule stripes. In the elements . Why theDrosophila trunk, based on comparisons with the scuttle fly Megaselia abdita [runt, ftz, and odd all possess a stripe-specific element for stripe 3 [The first question has 2 likely answers. Using gap inputs to directly pattern multiple genes speeds up the first emergence of a correctly phased pair-rule repeat and hence reduces the total time required for segmentation as well as the minimum magnitude of the stripe shifts  Movies. a abdita ,124.) Thstripe 3 , correspstripe 3 .runt, ftz, and odd may therefore increase the speed of patterning, their output patterns are not as regular as those of hairy and eve, probably answering the second question. Irregular gap-driven pair-rule stripes, if not later refined by cross-regulation, would result in aberrant segment-polarity patterns. For example, cross-regulation of the runt pair-rule stripes appears to be absent in Dichaete mutant embryos, and this leads to severe and variable segmentation defects [While the stripe-specific elements of  defects .runt stripes were as regular as those of eve, cross-regulation might still be required. The eve stripes are patterned with a precision of about 1% AP length, i.e., differing in position from embryo to embryo by roughly 1 nucleus [eve and runt stripes both showed positional variation of this magnitude, and this noise was uncorrelated, segmentation defects would likely occur. The benefit of the dynamic patterning carried out by the pair-rule network is that the influence of Eve activity on gene expression is not restricted to the cells currently within each Eve stripe but also extends to cells posterior to these stripes, like wakes stretching out behind moving ships. This means that entire pair-rule repeats become coordinated with the Eve stripes, regardless of where exactly these stripes are located in the blastoderm. Thus, spending extra time to cross-regulate is likely to minimise the consequences of noise in the original spatial signals, a trade-off that is characteristic of many developmental patterning systems [However, even if these  nucleus . This po systems .hairy and eve patterned by gap inputs) requires a minimum shift of 3 nuclei, while versions including greater control of pair-rule gene expression by gap inputs reduce this minimum to 1 or 2 nuclei  Careful experimental characterisation of the quantitative relationship between the protein concentrations of input factors and the transcription rate of output factors could form the basis for future, more sophisticated models of angement –131 and/angement –134) andangement ,136) of Drosophila undergoes an extensive topology change at gastrulation. I then presented evidence that dynamic Eve expression is integral to segment patterning and also that the Drosophila pair-rule network is broadly compatible with both simultaneous and sequential modes of segmentation.In the introduction to this paper, I highlighted 3 recent findings related to pair-rule patterning: (1) that pair-rule orthologs exhibit oscillating expression in the segment addition zones of short-germ arthropods, (2) that pair-rule gene expression in long-germ insects is patterned by dynamic gap gene expression, and (3) that the pair-rule network in Drosophila, the “early” pair-rule network is derived from the “clock” part of an ancestral short-germ segmentation mechanism, while the “late” network is derived from the stabilising genetic interactions that would have originally established parasegment boundaries anterior to the segment addition zone.I therefore propose the following evolutionary hypothesis, which ties together and potentially explains the 3 original findings. (1) Long-germ and short-germ segmentation are not dichotomous modes of patterning but rather alternative behaviours of a largely conserved pair-rule network. (2) In long-germ insects, ad hoc patterning of stripes by gap inputs stands in for the expression of particular pair-rule gene “clock enhancers”, meaning that gap gene networks are under strong selection to preserve dynamic behaviour. (3) In These predictions can be tested in the future by comparative studies in emerging arthropod model species.http://dx.doi.org/10.5061/dryad.cg35k [eve mutation used was eve3 (gift of Bénédicte Sanson) and was balanced over CyO hb::lacZ (Boomington stock no. 6650) in order to easily distinguish homozygous mutant embryos. Whole mount double FISH, microscopy, and image analysis were carried out as described previously [eve/Eve doubles in eve. Images were analysed using Fiji [www.python.org) using the libraries NumPy [http://ginsim.org) [Wild-type in situ images are from a previously published data set deposited in the Dryad repository: ad.cg35k . The eveeviously . The eveing Fiji ,140: intes NumPy  and Matpes NumPy . SBML-qusim.org) . All modS1 Fighairy, eve, runt, ftz, odd, and slp. (A) hairy and eve pair-rule stripes partially overlap during cellularisation. At gastrulation, hairy expression fades away, while the eve stripes narrow from the posterior and then also fade. (B) eve and runt pair-rule stripes partially overlap during cellularisation. The eve stripes become increasingly narrow at gastrulation, and the runt secondary stripes emerge anterior to the eve stripes. By early GBE, eve expression has faded away and runt expression has resolved into a regular, segmental pattern. The refinement of the eve stripes occurs more gradually in real embryos than in the simulation. Arrowheads in B”‘ indicate new runt expression related to the developing nervous system. (C) eve and ftz pair-rule stripes are at first expressed in complementary patterns. Starting from late cellularisation, they both narrow from the posterior (eve more than ftz). eve expression later fades away, while ftz persists. (D) eve and odd pair-rule stripes are at first expressed in complementary patterns, before both narrowing. odd secondary stripes emerge at the posterior of the narrowing eve domains, which then fade away, leaving segmental stripes of odd. (E) hairy and runt pair-rule stripes are expressed in complementary patterns during cellularisation. hairy expression then fades away, while runt transitions to segmental stripes. (F) hairy and ftz pair-rule stripes slightly overlap during cellularisation. hairy expression then fades away, while the ftz stripes narrow. (G) hairy and odd pair-rule stripes slightly overlap during cellularisation. hairy expression then fades away, while odd transitions to narrow segmental stripes. (H) runt and ftz pair-rule stripes partially overlap throughout cellularisation. At gastrulation, runt secondary stripes emerge to the posterior of the narrowing ftz stripes. Later, the runt primary stripes refine from the posterior, and the overlaps with ftz are lost. (I) runt and odd pair-rule stripes slightly overlap during cellularisation. These overlaps resolve at late cellularisation (slightly earlier than in the simulation). At gastrulation, new odd expression emerges just anterior to the runt primary stripes, while new runt expression emerges just posterior to the refining odd primary stripes. By early GBE there is a regular segmental pattern of abutting odd and runt stripes, separated by gaps. Arrowheads in I”‘ indicate new runt expression related to the developing nervous system. (J) The ftz and odd stripes are fairly congruent during cellularisation. At gastrulation, both narrow from the posterior, and the odd secondary stripes intercalate between them. Over the course of patterning, their anterior boundaries also become offset from one another. (K) The slp primary stripes emerge later than the hairy stripes, but share an anterior border with them. (The slp stripes in the simulation are too broad at this stage—they should be narrower than the hairy stripes.) hairy expression then fades away, while slp transitions to a segmental pattern. (L) The slp primary stripes emerge later than the eve stripes, and abut their anterior borders.  At gastrulation, the eve stripes narrow from the posterior and then fade, while slp transitions to segmental stripes. (M) The slp primary stripes emerge later than the runt primary stripes, and are offset slightly from their posterior boundaries.  At gastrulation, secondary stripes of both genes emerge between the primary stripes (the widths of the simulated slp stripes are now appropriate). The expression patterns become largely congruent, except at the posteriors of the runt primary stripes. These differences resolve later, when the runt primary stripes narrow. (N) The slp primary stripes emerge later than the ftz primary stripes, and partially overlap with them. At gastrulation, the secondary slp stripes emerge just anterior to the ftz domains, which narrow from the posterior, losing the overlaps with the slp primary stripes. (O) As for (N), the slp primary stripes partially overlap the odd primary stripes, and these overlaps are later lost by the odd stripes narrowing from the posterior. The secondary stripes of odd and slp intercalate between the primary stripes, and abut one other.Expanded version of (DOCX)Click here for additional data file.S2 FigEach panel shows an uncropped version of the corresponding double fluorescent in situ image in (DOCX)Click here for additional data file.S3 Figprd transcripts (magenta) are shown relative to eve (green), odd (blue) or slp (cyan) transcripts, at four different embryo ages (left) or simulation timepoints (right). Note that the figure presents a different set of developmental ages / simulated timepoints from those in prd and slp are not; at “mid cellularisation”, prd is expressed but slp is not; at “late cellularisation”, slp is additionally expressed. While the simulated expression of prd is broadly appropriate early on (e.g. compare prd and odd expression at mid-cellularisation / T26), several aspects of prd expression are not recapitulated by the model. First, the changing phasing of prd posterior borders and eve anterior borders between mid-cellularisation and late cellularisation  and the consequent emergence of single-segment periodicity. (B) Uncropped views of the embryos shown in (A). Scale bars 100 μm.This figure compares double FISH data of pair-rule gene expression with simulated transcript expression, as in see also . Second,(DOCX)Click here for additional data file.S4 Figeve (magenta), prd (blue), slp (green), en (red), odd (yellow), and wg (cyan). From left to right, the panels in each row show: 1) a whole embryo view; 2) an enlarged view of pair-rule repeats 2–6; 3) the individual channel for the first gene listed in the row label; 4) the individual channel for the second gene listed in the row label. Arrowheads mark the locations of prospective odd-numbered parasegment boundaries. Scale bars 100 μm. (B) Summary schematic showing the relative expression of the Eve, Prd, and Slp protein domains, and the en, wg, and odd transcript domains, at the odd-numbered parasegment boundaries. (C) The same schematic as in B, with the addition of the regulatory interactions between Eve, Prd, and Slp and their target genes. Aside from the colours, which are changed to match the in situ images, this schematic is identical to (A) Double FISH images of segmentation gene expression in gastrulation stage embryos . Each row shows a different pairwise comparison of the transcripts of the genes (DOCX)Click here for additional data file.S5 Figrunt (green) and slp (blue) along a narrow ventral strip of the trunk of the two embryos pictured. In the wild-type embryo, the runt stripes are all (except stripe 7) roughly symmetrical and strongly expressed. In the eve mutant embryo, runt stripes 1–6 , have much lower intensity than runt stripe 7  and exhibit a sawtooth pattern, in which expression intensity decreases from anterior to posterior. The slp expression in the eve mutant embryo, while broad, does display a pair-rule modulation, which is in opposite phase to the downregulated runt stripes. Therefore, the same two regulatory interactions  are evident in both the in situ data and the simulated data Click here for additional data file.S6 FigSimulation output showing the expression patterns generated by the early pair-rule network, assuming various anterior-to-posterior shifts speeds of the gap inputs (shown in black). See supplementary movies for full simulation output  = 3x. Ga(DOCX)Click here for additional data file.S7 Fighairy receives gap inputs (1), a minimum of time 10x the delay is required (network model as in simulation 12). If both hairy and eve receive gap inputs (2), the minimum is 7x the delay (network model as in simulation 10). If runt or ftz/odd additionally receive gap inputs (3), the minimum is 6x the delay . Finally, if all the primary pair-rule genes receive gap inputs (4), the minimum is 5x the delay . Note that the stripes of ftz and odd are considered a single pattern, because of their identical regulation in the simulations. Minimum duration required for generating a correct segment pattern , under various simulation scenarios. Minimum simulation duration (Y-axis) is given as a multiple of the synthesis / decay delay parameter value of the pair-rule gene products . See (DOCX)Click here for additional data file.S1 Text(PDF)Click here for additional data file.S2 Text(PDF)Click here for additional data file.S1 File(PY)Click here for additional data file.S2 FileCorresponds to the network in Simulation 10 and (PY)Click here for additional data file.S3 FileCorresponds to the network in Simulation 10 and (SBML)Click here for additional data file.S4 FileCorresponds to the network in Simulations 12 and 13, and (SBML)Click here for additional data file.S1 MovieSimulated expression of the primary pair-rule genes, assuming static gap inputs.(MP4)Click here for additional data file.S2 MovieSimulated expression of the primary pair-rule genes, assuming dynamic gap inputs.(MP4)Click here for additional data file.S3 MovieSimulated expression of the primary pair-rule genes, assuming slow gap shifts.(MP4)Click here for additional data file.S4 MovieSimulated expression of the primary pair-rule genes, assuming fast gap shifts.(MP4)Click here for additional data file.S5 MovieSimulated expression of the primary pair-rule genes, assuming very fast gap shifts.(MP4)Click here for additional data file.S6 Moviehairy, eve, and runt take gap inputs.Pattern establishment when (MP4)Click here for additional data file.S7 Moviehairy, eve, and ftz/odd take gap inputs.Pattern establishment when (MP4)Click here for additional data file.S8 MoviePattern establishment when all primary pair-rule genes take gap inputs.(MP4)Click here for additional data file.S9 MovieSimulated behaviour of the whole pair-rule system, assuming static gap inputs.(MP4)Click here for additional data file.S10 MovieSimulated behaviour of the whole pair-rule system, assuming dynamic gap inputs.(MP4)Click here for additional data file.S11 Movieeve mutant.Pair-rule gene expression in a simulated (MP4)Click here for additional data file.S12 MovieSimultaneous patterning by the modified pair-rule network.(MP4)Click here for additional data file.S13 MovieSequential patterning by the modified pair-rule network.(MP4)Click here for additional data file.


Ab initio calculations combined with statistical thermodynamics approaches reveal that the site occupation of the carbides may be incorrectly predicted when only the commonly used approach of full sublattice occupation is considered. We found that the γ-M23C6 structure can be understood as a familiar sodium chloride structure with positively charged rhombic dodecahedron (M(4a) M12(48h)) and negatively charged cubo-octahedron (M8(32f) C6(24e)) super-ion clusters, together with interstitial metal atoms at the 8c sites. The stability of the partially occupied phase can be easily rationalized on the basis of a super-ion analysis of the carbide phase. This new understanding of γ-M23C6 carbides may facilitate further development of high-chromium heat-resistant steels.The effect of mixed partial occupation of metal sites on the phase stability of the γ-Cr For instance, a high density of γ-M23C6 carbides , increases the creep rupture strength, as grain boundary sliding and surface cracking are reduced. Although the chemical compositions of some of these carbides have been determined, experimental studies on their mechanism of formation, atomic site occupation, and chemical interactions between species in multicomponent phases have seldom been reported4.The phase stability of carbide precipitates plays a crucial role in improving the creep lifetime of steels19, in particular with regard to Frank-Kasper structures20. When mixed occupancy of sublattice sites was considered it was only to the extent as necessitated by the cluster expansion method, say when nearest neighbor interactions between sites of the same type needed to be evaluated, such as is the case in the σ structure24. Mixed occupancy was considered in the specific context of the coherent potential approximation25, and for several configurations in the Fe-Cr σ structure26 but an in-depth analysis including associated effects such as due to lattice vibrations remains lacking. Here, we address this issue by considering many partially occupied configurations of a non-trivial technically important carbide phase, Cr23C6, while including vibrational effects.The partial/full occupation of a sublattice site refers to the fraction of the site that is occupied by a particular atomic species, averaged over many unit cells. To date, the vast majority of ab initio total energy studies have considered only configurations in which sublattice sites were fully occupied by a single atomic species23C6 crystal has a face-centered cubic (fcc) structure with the 23C6) unit cells and consists of 92 metal atoms occupying four inequivalent metallic sites, namely, the 4a, 8c, 32f, and 48h sites, and 24 non-metallic atoms occupying the 24e sites, as represented using Wyckoff notation, according to the results of X-ray powder diffraction27. Figure 23C6, and Fig. 27. In this structure, each 4a site is surrounded by 12 metal atoms in 48h sites, forming a cubo-octahedron with the composition Cr(4a)Cr12(48h). On the other hand, the neighboring 32f sites form a cube, and protruding from the faces of the cube are the 24e sites that together form a concentric octahedron. The closely positioned 32f and 24e sites together form a rhombic dodecahedron with the composition Cr8(32f)C6(24e) and eight Cr(8c) atoms lie at the interstitial positions. In this work, we treat only the mixed, i.e. partial, occupation of the metallic sites, and not the C occupied sites. We suppose that C atoms are fully occupying the 24e sites. Partial occupancy of the carbon sublattice can be important in some carbides29, but in this carbide we do not expect it to play a role of significance because the carbon vacancy formation enthalpy is very large (~1.05 eV/C vacancy)30.The γ-Mx23−FexC6 in the range of 0 ≤ x ≤ 7.3632. At a low Fe fraction, the Fe atoms occupy the 4a and 8c sites, and they simultaneously begin to occupy the 48h and 32f sites at higher Fe fractions. The occupancy sequence was found for 4a and 8c sites from atomistic simulations using pair potentials obtained through the lattice inversion technique33. However, simple pair potentials generally favor topologically close packed structures and generally cannot yield complex structures such as the present carbide as a true ground state. Therefore, in an MD simulation at finite temperatures, these pair potentials quickly lead to an unrealistic complete collapse of the crystal structure33. Moreover, the pair potentials do not account for changes in magnetic order, cannot reproduce elastic anisotropy, and fail to reproduce non-isotropic structural relaxations. Clearly, ab initio approaches which properly account for many-body interactions are much preferred for studies of the occupancy sequence in M23C6.Experimentally, according to single-crystal X-ray diffraction data, there is evidence of the selective metallic site occupation of Fe in γ-Cr23C6. Medvedeva et al.34 considered Cr22Fe1C6, where Fe either fully occupies the 4a site, or partially occupies an 8c, 32f, or 48h site. It was found that Fe prefers to occupy the 4a site and that Cr22Fe1(4a)C6 is more stable than a mixture of Cr23C6 and Fe23C636, whereas other substitutions (Fe1(8c), Fe1(32f) and Fe1(48h)) decrease the stability of Cr22Fe1C6. A similar result was reported by Fang et al.37. However, in contrast to the present work, previous work neglects important thermal excitations such as the effect of vibrational entropy on stability. This severely limits a comparison between previous ab initio results and experiment.A few ab initio studies considered partial site occupation in Mx23–FexC6 (x = 0–3) carbide phase in detail using the cluster-expansion method (CEM) and the cluster-variation method (CVM) based on first-principles as function of composition and temperature. We found that if the partial site occupation is more properly taken into account in the cluster expansion, the preferred site occupancy is substantially different from the case when the cluster expansion mostly relies on full occupation configurations.In this paper, we investigated the impact of site occupancy of Fe in the γ-Cr38 already recognized that the Cr23C6 structure can be viewed as composed of Cr(4a)Cr12(48h) and Cr8(32f)C6(24e) clusters in a NaCl arrangement with Cr(8c) atoms as interstitials, see Fig. 40 coupled with an analysis of the electron levels of the isolated clusters supports the ionic NaCl interpretation of the structure, as evidenced by Figs xFe = x/23), shown in Fig. (4a)Cr12(48h) is more than 6 electrons deficient, rather independent of Fe content, and Cr8(32f)C6(24e) has more than 7 additional electrons, while the interstitial 8c sites balance the charges. As is typical of ionic configurations we expect to see a significant gap between the highest occupied and the lowest unoccupied molecular orbitals (HOMO–LUMO gap) for this charge state. Figure (4a)Cr12(48h) features a large HOMO–LUMO gap for 70, 74, and 80 valence electrons, implying preferred charges of 8+, 4+, and 2−, the former corresponding nicely to the Bader charge of more than 6 electron deficient. The isolated Cr8(32f)C6(24e) cluster has a large HOMO-LUMO gap for 72, 75, and 80 valence electrons, implying preferred charges of 0, 3−, and 8−, the latter agreeing well with the Bader charge of more than seven excess electrons.Although subsequently forgotten, Westgren(4a)Cr12(48h) and Cr8(32f)C6(24e) super-ions. When some Fe atoms substitute for Cr atoms on the 48h sites in the Cr(4a)Cr12(48h) or on the 32f sites in the Cr8(32f)C6(24e) super-ions, the high symmetry of the super-ions is lost. Such symmetry breaking lowers the degeneracies of electron levels and therefore also reduces the HOMO-LUMO gap. Large HOMO-LUMO gaps indicate favorable electronic configurations. Therefore, Fe substitutions on 48h or 32f sites destabilize the super-ions and thus the whole carbide structure. When Fe substitutes for Cr on the 4a site in Cr(4a)Cr12(48h), or Fe substitutes Cr on the 8c sites, the symmetry remains exactly the same, the favorable large HOMO-LUMO gap of the super-ions is maintained, and the structure remains stable. Therefore, incorporation of Fe in Cr23C6 is energetically preferred on the 4a and 8c sites.The Fe-Cr substitution behavior can be understood in terms of the HOMO-LUMO gaps of the CrT = 0 K. Indeed, the vibrational free energy contributions can cause a re-ranking of the most stable structures , to the ab initio computed ground state enthalpy, H,T is the absolute temperature. The tilde superscript is a reminder that the configurational entropy term has not been accounted for. The Debye vibrational free energy45 is computed withkB is Boltzmann’s constant, θD is the Debye temperature, and D3 is the Debye integral.Consideration of the vibrational effect in the free energy calculations can significantly alter the predicted carbide phase stability obtained at es Table  in the lThe first term on the right-hand side denotes the zero-point vibrational contribution and the second indicates the temperature dependent vibrational contribution.The formation free energies of the perfectly periodic supercells, Gf(T), with full inclusion of configurational entropy and configurational enthalpy associated with temperature dependent partial site occupancy. The thermodynamically favored site occupation is determined by minimizing ΔGf(T). In our calculations, we have neglected the effects of thermal volume expansion.The free energies, xFe for the cases of both full and partial occupation at various temperatures. It is apparent that many Fe substituted carbides have lower energy than mixtures of the terminal carbides (Cr23C6 and Fe23C6) at the same composition. As a function of temperature, it is apparent that even when configurational entropy effects are ignored, the vibrational free energy contribution by itself causes shifts in the most favorable carbide structures: at T = 0 K  configuration becomes preferred. At xFe ≈ 0.087, (Fe – Fe1/8 –) and (Fe – – Fe1/12) configurations have similar low free energies at T = 0 K. Vibrational free energy contributions increasingly favor the (Fe – – Fe1/12) configuration over the (Fe – Fe1/8 –) configuration as temperature is raised. When even more Fe atoms substitute for Cr (xFe ≈ 0.130), even more partial occupied configurations (Fe – Fe1/8 Fe1/12), (Fe – Fe2/8 –), and (Fe – – Fe2/12), have free energies very close to the fully occupied (Fe Fe – –) configuration at 0 K. However, at higher temperatures, the vibrational excitations cause a gradual preference for the (Fe – Fe2/8 –) configuration. This is not an isolated case, at higher temperatures several partial occupancy structures sink below the convex hull formed by full occupancy structures only. So a general trend emerges where vibrational excitations increasingly favor partial occupancy configurations over full occupancy configurations as temperature is raised. Clearly the vibrational effects by themselves already can produce shifts in site occupation behavior. The volume per atom decreases monotonically with xFe, as shown in Fig. S1a. The atomic volumes of the most stable configurations were linearly fitted as (10.045 − 0.549 xFe [Å3]), where the negative slope is in agreement with the reported linear fits to experimental results of (10.439 − 0.574 xFe [Å3]) by Yakel32, (10.385 − 0.390 xFe [Å3]) by Shaw46, and (10.443 − 0.536 xFe [Å3]) by Westgren47. The computed atomic volumes are smaller than the reported experimental values. This is a well-known deficiency of the generalized gradient approximation (GGA) of the exchange-correlation functional48, especially for systems containing early 3d-elements50. For very Cr-rich carbides there is no net magnetization, however, already at an iron concentration xFe ≈ 0.043 a transition to the ferromagnetic state occurs, see Fig. S1b.Figure 0 K Fig.  the  carbide structures, and the relevant changes with increased Fe content, we examined the electron localization function (ELF)51. ELF contour plots in the (110) plane of the carbide phase are displayed for four different Fe occupations, ranging from xFe = 0.0 to 0.130, see Fig. S2b. It is readily apparent that with increasing Fe content the character of bonding locally changes. While the ELF usually indicates degree of covalency, here it nicely illustrates changes in spin polarization. The arrows indicate the area in the bonding region between nearest neighbor 48h-48h Cr atoms perpendicular to the cross section plane where the spin polarization is strongly increased due to the presence of extra Fe atoms.To understand the nature of chemical bonding in the most stable γ-CrT = 1200 K, xFe ≈ 0.087 (Fe – – Fe1/12), was calculated as approximately −4.09 eV/supercell, quite comparable with the experimental enthalpy of −3.34 eV/supercell52 and higher than the value of −12.12 eV/supercell reported using a thermodynamic Calphad assessment of the Fe–Cr–C system53.At 32. In the results without the vibrational contribution, we found that inclusion of partial occupation structures in the cluster expansion gave improvement with experiment for the 4a and 48 h sites at both 600 K and 1200 K. However, for the 8c and 32f sites no such clear improvement could be seen. The results with vibrational contributions show more often that including the partial occupation structures in the cluster expansion does not improve the agreement with experimentally measured site occupations, the only exception to this being the 4a site at 600 K. The inclusion of vibrational effects seems to improve the agreement with experiment in particular for the results for the 4a site, for the other sites no such clear-cut conclusion can be drawn. Overall, we see a fair agreement with the experimental results, in particular for the 4a, 32f and 48h sites theoretical results are found within the experimental error bars. For the 8c sites agreement, in all theoretical treatments we considered is less good.Finally, Fig. 37. With increasing iron content, in contrast to the experimental findings, we find that the 4a sites saturate before the 8c sites take Fe occupancy. The Fe uptake in the 32f and 48h sites increases gradually with iron content, and in the 48h sites a little more rapidly than in the 32f sites, in good agreement with experiment32.Fe is found to preferentially substitute for Cr at the 4a site, which is consistent with earlier work23C6 structure as an NaCl arrangement of super-ions. We show that such super-ions can be recognized with approximate constant charges irrespective of composition, at least within the range 0 ≤ x ≤ 3 in Crx23–FexC6. The super-ion description not only gives a rationalization for the crystal structure, but also gives an understanding for the site preference of Fe in Cr23C6. As this reasoning does not rely on the specifics of Fe, but merely assumes that elastic strain effects play an subordinate role, we expect that other alloying elements, with an atomic size comparable to Cr, can be predicted and understood as well.In this work, we revived an old interpretation of the CrIt is also shown that including vibrational effects changes which structures populate the convex hull. With increasing temperature, it is seen that the vibrational free energy causes structures with partial site occupancy to sink below the convex hull formed by the full occupancy structures, even when configurational entropy effects are neglected. Of course, configurational free energy terms would further favor partial occupancy structures over full occupancy structures. This implies that vibrational effects cannot be ignored in the prediction of site preference.Including partial occupancy structures in the cluster expansion gives significant changes in the site occupation. However, these changes did not give a univocal improvement with the experimental data. Inclusion of vibrational effects however, did give a systematic improvement of computed site occupancies with experimental results.54 to study the effect of the partial occupation of inequivalent metallic sites in the γ-Crx23−FexC6 (x = 0–3) carbide phase, where the interactions were used to compute the entropy and Gibbs energy of the phase as function of the temperature and composition within the tetrahedron approximation21. The cluster expansion was carried out using a pool of clusters generated by the condition that no two sites in a cluster are separated more than 0.31 times the M23C6 lattice parameter. As the lattice parameter is about 1 nm, this means no sites are further than about 0.3 nm apart. Using this criterion, five distinct types of 4-site clusters are the largest clusters that can be generated55. These clusters are shown in Fig. In this work, we have used CEM and CVM57. The number of valence electrons are 4, 6, and 8 for C:(2s22p2), Cr:(3d54s1), and Fe:(3d74s1), respectively. We considered collinear spin-polarization for all structures within the generalized gradient approximation (GGA) for the exchange–correlation functional as formulated by Perdew, Burke, and Ernzerhof (PBE)48. All of the calculations were performed using the supercell technique, where periodic boundary conditions and full relaxation of internal coordinates and lattice parameters were considered. The adopted supercell contains one conventional cubic unit cell consisting of 92 metal atoms and 24 carbon atoms. The cutoff energy for plane wave expansion was 520 eV and the k-point sampling grid was 7 × 7 × 7. The calculations were considered converged when the change of the total energy between successive self-consistency iterations reached 10−5 eV or less and simultaneously the magnitude of the greatest force on any atom was less than 10−4 eV/Å.As inputs for the CEM, we performed total energy calculations using pseudopotentials generated by the projected augmented wave (PAW) method as implemented in the Vienna Ab initio Simulation Package (VASP)Supplementary information


Furthermore, the FE approach gives access to more information such as the von-Mises stresses. It is found that the influence of the abutment screw preload has only a minor effect on the microgap formation and local stress distribution. The congruence between FE simulations and in-vitro measurements at the micrometer scale underlines the validity and relevance of the simple FE method applied to study the micromovement of the abutment and the abutment screw preload in conical implant-abutment connections under load.State-of-the art, two-piece dental implants made from titanium alloys exhibit a complex micromechanical behavior under dynamical load. Its understanding, especially the formation of microgaps, is of crucial importance in order to predict and improve the long-term performance of such implants. Microgap formation in a loaded dental implant with a conical implant-abutment connection can be studied and quantified by synchrotron radiography with micrometer accuracy. Due to the high costs and limited access to synchrotron radiation sources, alternative approaches are needed in order to depict the microgap formation. Therefore, synchrotron radiography is used in this article to validate a simple finite element model of an experimental conical implant design. Once validated, the model is in turn employed to systematically study the microgap formation developed in a variety of static load scenarios and the influence of the preload of abutment screw on the microgap formation. The size of the microgap in finite element analysis (FEA) simulations is consistent with that found in  Today, a two-piece screw implant is the most common, comprising; I. implant body, II. abutment and III. abutment screw, which connects the abutment axially on the implant  testing , 5. Off  testing , 8, 9. C testing , 3. In a testing , 11, 12. testing , 14. Forin vitro synchrotron-based PCR results. The latter is a unique property of this study, i.e. the experimental validation of a model predicting microscopic structures inside a macroscopic, fully opaque object.The aim of this study is simply to validate whether FEA based on linearly elastic deformation is sufficient to predict microgap formation in dental implants. It is explicitly noted that this study does not aim to establish an FE model which describes perfectly the mechanical behavior of a two-piece dental implant design. This is the task for a later, continuing work. Again, the model introduced in this work is simply intended to predict the formation of microgaps. Once validated, it can be extended to quantify the influences of force and application angle, as well as screw mounting torque, on the stability of the abutment in a conical IAC. Furthermore, it would reduce the need for complex characterization methods such as PCR, which have only limited availability. Hence, a simple CAD model of an experimental conical implant was developed, virtually loaded and the resulting microgap at the IAC compared to 22.1in vitro at the BESSY-II light source. The implant and abutment were assembled and screw tightened with a torque of 15 Ncm. Each implant was rigidly mounted in a brass cylinder (15 × 15 mm2) with methylmethacrylate-based adhesive  up to 3 mm below the shoulder of the implant according to ISO standard 14801:2008 with a 10 mm steel ball connected to the abutment (X60). The center of the steel ball was 8 mm above the shoulder of the implant ; its geometric parameters are listed in lant cf. b.Fig. 1 on the BAMline . The X-ray images were recorded as described previously 2.2® 3D . Based on radiographic observations of the inner and outer geometry, a CAD model was created for the ECI implant as detailed in ascr is not simulated by a torque, but by a downward force (assembly preload) of the screw ) applies to the interface between steel ball and abutment.Finite element analysis calculation employed Autodesk InventorM was estimated from Eq. A = 15 Ncm, d2 = d1 − 0.64952 · P = (1.5–0.23) mm = 1.27 mm the pitch diameter, thread pitch P = 0.35 mm, and DKm the average of outer and inner diameters of the friction ring between screw-head and abutment DKm = (dW + DKi)/2 = 1.95 mm. The last term is multiplied with a cosine to account for a tapered screw head in the ECI (α = 32.5°). The friction coefficients μG (thread) and μK (head) were both set to 0.5 M = const. = 120 N (which corresponds to 10 Ncm torque) [Since the screw mounting force is approximated by an elastic downward force the value for the assembly preload F torque) , 25.® 3D software were applied in order to obtain a generic FE approach that enables estimation of the microgap formation. According to the software description, all of the mesh elements are solid tetra 10 type . These are volumetric elements with the mesh size at the implant shoulder set to 0.1 mm. At less critical positions the meshes were reduced up to 0.3 mm element size. The FE analysis was calculated with approximately 151 000 nodes and 96 000 elements. The contact condition of the different implant components were defined to be separated [Standard settings of the Autodesk Inventoreparated , 25 , “lower left“ (LL), “lower right“ (LR) and “upper right“ (UR), corresponding to the four edges of the IAC trapezoid  for loosening of the abutment screw is a cause for clinical revision/intervention, the interest and necessity for studying the mechanics of dental implants will most likely increase with the complexity of their design.The experimental results support the use of FE modeling which is already widely used to assess the biomechanical behavior of dental implants , 22. So 5in-vitro synchrotron radiography. Compared to the use of laboratory X-ray sources the use of synchrotron radiation allows one to probe the interior of the implant with high spatial resolution [i.e. it full-filled the needs of our descriptive study, a finer mesh is recommended to reliably determine the stresses generated around stress concentration sites such as threads. A more detailed mesh sensitivity study would be required in order to allow for the model to predict the behavior of a two-piece implant design in a truly realistic manner.FE analysis has been successfully validated using solution , 26. As Wolfram Wiest and Alexander Rack: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Wrote the paper.Simon Zabler: Conceived and designed the experiments; Performed the experiments; Wrote the paper.Alex Schaer: Analyzed and interpreted the data; Contributed reagents materials, analysis tools or data.Michael Swain and Katja Nelson: Analyzed and interpreted the data.Alex Schaer was an employee of Camlog Biotechnologies AG and Katja Nelson receives lecture honoraria from Camlog Biotechnologies AG.German Research Foundation (DFG grant: Ne 1646/ Za 656).This work was supported by the No additional information is available for this paper.