A Natural O-Ring Optimizes the Dispersal of Fungal Spores

A natural O-ring optimizes the dispersal of fungal spores

Joerg A. Fritz1, Agnese Seminara1,3, Marcus Roper4, Anne Pringle2 rsif.royalsocietypublishing.org and Michael P. Brenner1

1School of Engineering and Applied Sciences, Kavli Institute for Bionano Science and Technology, and 2Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138, USA 3CNRS – Laboratoire de physique de la matie`re condense´e, UMR 7336, Parc Valrose, 06108 Nice, France 4Department of Mathematics, University of California, Los Angeles 90095, CA, USA Research The forcibly ejected spores of ascomycete fungi must penetrate several milli- Cite this article: Fritz JA, Seminara A, Roper metres of nearly still air surrounding sporocarps to reach dispersive airflows, M, Pringle A, Brenner MP. 2013 A natural and escape is facilitated when a spore is launched with large velocity. To O-ring optimizes the dispersal of fungal spores. launch, the spores of thousands of species are ejected through an apical ring, a small elastic pore. The startling diversity of apical ring and spore J R Soc Interface 10: 20130187. shapes and dimensions make them favoured characters for both species http://dx.doi.org/10.1098/rsif.2013.0187 descriptions and the subsequent inference of relationships among species. However, the physical constraints shaping this diversity and the adaptive benefits of specific morphologies are not understood. Here, we develop an elastohydrodynamic theory of the spore’s ejection through the apical ring Received: 26 February 2013 and demonstrate that to avoid enormous energy losses during spore ejection, Accepted: 28 May 2013 the four principal morphological dimensions of spore and apical ring must cluster within a nonlinear one-dimensional subspace. We test this prediction using morphological data for 45 fungal species from two different classes and 18 families. Our sampling encompasses multiple loss and gain events and potentially independent origins of this spore ejection mechanism. Subject Areas: Although the individual dimensions of the spore and apical ring are only biomechanics, biophysics, biomathematics weakly correlated with each other, they collapse into the predicted subspace with high accuracy. The launch velocity appears to be within 2 per cent of Keywords: the optimum for over 90 per cent of all forcibly ejected species. Although spore dispersal, fungi, morphological diversity, the morphological diversity of apical rings and spores appears startlingly fluid dynamics, elastohydrodynamics, diverse, a simple principle can be used to organize it. optimization 1. Introduction Author for correspondence: Spore dispersal is the primary determining factor for the range and distribution Joerg A. Fritz of fungi in nature. The importance of understanding this process in detail has been highlighted in recent years by an unprecedented number of fungal dis- e-mail: [email protected] eases, which have caused some of the most severe die-offs and extinctions ever witnessed in wild species [1] and are increasingly considered a worldwide threat to food security [2]. An effective control of these emerging diseases is possible only if we can understand and control how they propagate. The defining feature of the largest fungal phylum, Ascomycota, is the ascus, a fluid-filled sac from which spores are ejected. Ejection is powered by a build-up of osmotic pressure [3], which forces spores through a ring or hole at the tip of the ascus, after a critical pressure is reached [4]. Ascus and spore morphologies are highly variable and have been an essential element of species descriptions for more than 200 years [5,6]. Since spores are the primary agents of dispersal, these morphologies also play a critical role in the ascomycete life cycle: most fungi grow on highly heterogeneous landscapes, and to persist a fungus must move between disjoint patches of habitat [7], thus effective dispersal is critical Electronic supplementary material is available to the fitness of an individual. at http://dx.doi.org/10.1098/rsif.2013.0187 or To reach dispersive air currents, spores must be launched with enough speed to cross the stagnant air layer around the fungus, the fluid mechanical via http://rsif.royalsocietypublishing.org. boundary layer. Although typical boundary layer thicknesses are around 1 mm [8], a spore’s small size (approx. 10 mm) causes rapid deceleration after

U 20130187 10: Interface Soc R J rsif.royalsocietypublishing.org b l

spore ring x L W h(x)

Figure 1. The spore shooting apparatus. (a) Sporocarps on the stalk of a plant. (b) Flask-shaped sporocarp, containing five asci. (c) Upper part of an ascus with a mature spore close to the apical ring, which is still sealed. The length L and width W of the spore and the dimensions of the apical ring (‘, b, d) are indicated. (d) Spore moving at velocity U and deforming the apical ring at launch. A lubricating layer of fluid separates the spore from the ring. (e) The region where the spore first deforms the ring. Here, x measures the distance from the point where the spore starts to compress the ring; the gap thickness h varies with x and asymptotes to a constant value h0 at x . l as described in the text. Dashed line denotes dry contact deformation. launch, meaning that it must be launched at very high vectors, we test whether genetics are a constraint on mor- velocity even to travel a very small distance, and the likeli- phology. In fact, these species have very different apical hood of effective dispersal is directly correlated to the ring and spore shapes, suggesting natural selection is the thickness of boundary layer that the spore is able to cross [9]. force maintaining collapse into the one-dimensional subspace The critical role of the apical ring in spore dispersal caused for species with functional apical rings. speculation about whether the diverse morphologies of the spore ejection apparatus are tunedtoalloweffectivedispersal. Buller [10] proposed a relationship between the dimensions of the apical ring and the size of the spore, ostensibly to prevent 2. Results spores from tumbling during flight. Ingold [7] thought spores would be shaped to maximize the force used by apical rings to 2.1. Fluid mechanics of spore and apical ring coupling push on them. But, surprisingly, the individual geometric dimen- Figure 1 shows a representative context in which spore ejec- sions of apical rings and spores critical to these hypotheses are tion occurs. The sporocarps of a fungus are scattered on a either very weakly or not correlated. host (e.g. the stalk of a plant, figure 1a). These structures Here, we resolve this discrepancy by demonstrating a are produced by the fungus with the sole purpose of dis- strikingly tight coupling between the size of the spore and persing the spores. Within each sporocarp, there can be a nonlinear function of multiple dimensions of the apical hundreds of asci, each generally containing eight spores ring. The relationship is suggested by physical constraints (figure 1b). When the spores in an ascus are mature, osmo- on spore ejection: the requirement to efficiently convert the lytes are produced, leading to water influx into the highly potential energy stored in the ascus to kinetic energy of the elastic ascus, resulting in a significant increase in volume spore. The apical ring is an elastic seal, and distorts signifi- and pressure [4]. When the osmotic pressure p0 inside an cantly when the spore, which is lubricated by a thin fluid ascus is sufficiently high, the spores are singly ejected into layer, passes through it. The basic physical principles govern- the surrounding air. ing this kind of process were discovered 50 years ago, in the The speed U at which a spore is launched depends criti- study of elastomeric seals and O-rings used to control fluid cally on energy losses during ejection. If the osmotic flow in engines, pipes and other engineering applications pressure were entirely converted to kinetic energy, the [11]. By adapting these theories to the fluid mechanics of spore would be ejected at an ideal velocity spore ejection, we demonstrate that although there are at 2p0 least five independent dimensions to the morphological Uideal ; 2:1 ¼ sffiffiffiffiffiffiffirs ð Þ diversity of spores and apical rings, the need to minimize energy losses during ejection restricts spore and ascus where rs is the density of the spore and p0 is the overpressure morphologies to a one-dimensional subspace, where the in the ascus. dimensions of a spore and its apical ring are tightly coupled. However, the ideal launch velocity is necessarily degraded We test this theory using published electron micrographs by both friction and fluid loss as the spore moves through the of apical rings and spores [12–19] and a recently published apical ring (figure 1c,d). The apical ring consists of an elastic ascomycete phylogeny [20], which identifies two potentially material with thickness b and height ‘. The size of the opening independent groups of species with spores singly ejected of the apical ring before the spore starts to pass through it, d, is through apical rings. Quantitative descriptions of spores and much smaller than the width W of the spore. During the ejec- apical rings at a high resolution are available for 45 species, tion of the spore, the apical ring is strongly deformed, and with dimensions of the spore and apical ring characters varying there is a thin layer of fluid with viscosity m and density r, over one order of magnitude. Nonetheless, the observed vari- separating the apical ring from the spore. ation is confined to the predicted one-dimensional subspace Energy losses arise from two different processes occurring with surprising accuracy: energy losses are held within 2 in this lubricating fluid layer of thickness h0: first, there is fric- per cent of the theoretical optimum. tion between the spore and the apical ring, owing to the By assembling data on species where there is no selective viscous force in the fluid gap F W‘mU/h , opposing the ! 0 pressure to maximize ejection velocity, because spores are motion of the spore moving with velocity U. The total dispersed using different mechanisms, for example insect energy dissipated is then E FL, the product of this friction ¼ viscous force with the distance that the spore moves when the through it, we can neglect the deformation of the ascus 3 force is acting, which is the length L of the spore. The second wall for the elastohydrodynamic calculation. Under this sfryloitpbihn.r o nefc 0 20130187 10: Interface Soc R J rsif.royalsocietypublishing.org energy loss arises because the pressure in the ascus, and thus assumption, we can approximate the apical ring as a circu- the main accelerating force, decreases while the spore and lar cylinder with internal radius r d/2 and outer radius i ¼ lubricating fluid leave the ascus. If ascus pressure and r d/2 b, subject only to an internal pressure p . Owing o ¼ þ r volume are proportional, the energy lost due to fluid leaving to symmetry, the displacement of the ring depends only on the ascus is proportional to the kinetic energy, Efluid the radial distance r from the centre line. Classical elasticity 2 ¼ rWh0LU , up to a constant parametrizing the ratio of ascus theory [23] dictates that the deformation is given by volume before ejection to spore volume. If h is large, the 0 3 p r2 r2 u r i o : : energy loss is dominated by the fluid flow through the gap, 2 2 2 3 ¼ 2E ro ri r ð Þ while if h0 is small, the energy loss is dominated by friction. À The minimal total energy loss E E E occurs if l ¼ fluid þ friction With the spore passing through the apical ring, the defor- Efluid Efriction and thus if the physical gap thickness is mation of the inner surface of the ring is u(ri) W/2, close to the optimal value h h , with ¼ 0 ¼ * implying the elastic pressure m m 1=2 2 2 ‘ ‘ E ro ri b d b h a a ð Þ ; 2:2 pr W À E W ð þ Þ ; 2:4 Ã 1=4 2 2 ¼ sffiffiffiffiffiffiffirU ¼ 2rp0 ð Þ ¼ 3 rir ¼ Ã d d 2b ð Þ ð Þ o ð þ Þ where in the second equality, we have assumed that the where E 2/ 1 n2 E 8/3E: Here, we have assumed that Ã ¼ ð À Þ ¼ energy dissipation is sufficiently small that U Uideal, with the apical ring is incompressible (Poisson ratio n 1/2), as ¼ rs r. The proportionality factor a 0.45 can be found by are most biological materials. ¼ explicitly integrating the equations of motion for the spore, With a fluid gap separating the apical ring from the spore, as demonstrated in the electronic supplementary material. this purely elastic solution is modified. Dowson & Higginson Figure 2a shows a plot of the energy dissipated as a function [11] solved the coupled elastohydrodynamic problem by rea- of h0 following from this more complete analysis. lizing that the fluid gap thickness h0 itself only slightly increases the elastic distortion of the apical ring. The pressure distribution in the centre of the apical ring is thus still given 2.2. The fluid layer thickness h0 by the Hertzian solution, scaling as p Epx/j for x ! j. Simi- What physical mechanism determines h ? During spore ejec- ! 0 larly, away from the contact (negative x in figure 1e), the shape tion, the apical ring undergoes a strong deformation to allow ffiffiffiffiffiffiffiffi of the apical ring is mainly affected by the large elastic stresses the spore to pass, and this deformation causes a restoring within the contact, and so it is also given by the Hertzian sol- elastic pressure to push against the spore. On the other = ution, h j x/j 3 2. However, there will be deviations near hand, within the fluid gap there is viscous pressure caused ! j j the entry point (x 0), where the fluid pressures created by by the fluid motion itself. The fluid layer thickness h is deter- 0 the flow through the gap will significantly modify h(x). mined so that these two pressures exactly balance. The layer Solving for h(x) in this regime requires coupling the vis- thickness h thus depends in a non-trivial fashion on all of the 0 cous flow in the gap to the elastic deformation of the apical parameters of the problem outlined thus far: the dimensions ring. Viscous forces imply that the pressure gradient in the and elastic modulus E of the apical ring, the viscous forces gap is given by the Reynolds lubrication equation acting in the thin fluid layer and the size of the spore. dp h h Determining the dependence of the layer thickness h0 on 6mU À 0 : 2:5 these parameters is a classic problem in elastohydrodynamics, dx ¼ h3 ð Þ and it was examined in the 1960s to understand the properties A coupled solution to the elastohydrodynamic problem of engineering seals, for example, O-rings. The theoretical ideas requires that the pressure distribution p(x) and the gap worked out in this context are directly applicable to the present shape h(x) satisfy both the Reynolds equation (2.5) and the problem, and here we recapitulate the basic arguments elastic equations. [11,21,22] in the context of our system. Figure 1e shows the geo- The value of h0 is selected by the solution to this coupled metry of the contact, focusing on the edge of the apical ring elastohydrodynamic problem [11]. The dependence of h0 on where the spore enters from the ascus. The coordinate x para- parameters follows from a scaling argument at x 0 [21]. If metrizes distance from the entry point, located at x 0. ¼ l is the length scale over which the pressure varies in the Within the ascus, far from the spore entry point, the pressure fluid gap, P is the pressure scale and H is the scale of is p p and the shape of the apical ring is undeformed. 0 the gap thickness, equation (2.5) implies P/l mU/H2. First, it is convenient to consider what would happen ! The lubrication solution must match the Hertz solutions, if there were no fluid gap (h 0), and no flow through the 3=2 0 ¼ implying P Epl/j and H j l/j . Combining these contact. In this case, the elastic distortions and pressures ! ! ð Þ relations, we find that caused by the spore moving through the apical ring follow ffiffiffiffiffiffiffiffi 1=5 from Hertzian contact theory. The Hertz contact solution j m3p3=2‘2d d 2b 2 b b b 0 ð þ Þ ; : is completely specified by the local radius of curvature h0 H 3=5 3=2 3 2 6 ¼ ¼ K ¼ r E Wb d b !ð Þ R ‘ of the contact region,1 and the resulting elastic de- Ã ð þ Þ Ã ! formation, implying that when x , 0, h x j x/j 3=2, where K E j/ Uh is the ratio of the elastic modulus to the ð Þ¼ ð Þ ¼ Ã ð Þ whereas when x . 0, the pressure distribution for x/j ! 1 viscous pressure created in the gap. The proportionality con- obeys p x Epx/j; where j ‘ E/p is an elastic healing stant b 1.42 requires the complete elastohydrodynamic ð Þ¼ ¼ r ¼ length. Here, p is the elastic pressure exerted by the spore solution, outlined in the electronic supplementary material. r ffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi on the apical ring, well inside the contact x/j 1 . Since Figure 2b shows the exact gap height and pressure following ð Þ the ring deformation is dominated by the spore passing from this complete elastohydrodynamic analysis. Note that in (a) where g g(a, b) 371 (see the electronic supplementary 4 ¼ ¼ material for derivation).

0.9 20130187 10: Interface Soc R J rsif.royalsocietypublishing.org

t 2.4. Testing the prediction with morphological data /E l E 0.6 Equation (2.7) implies a strong constraint coupling spore and apical ring morphologies: the spore diameter W should be

linearly proportional to a single parameter Sr, capturing the different dimensions of the apical ring, energy loss, 0.3 d d 2b 2 Sr ð þ Þ ; 2:8 ¼ b d b p‘ ð Þ ð þ Þ

if the material parameters of all species,ﬃﬃ most notably p0/E , 1 2 3 * h* are reasonably conserved. While theoretical considerations gap thickness, h (µm) make it likely that both these values individually should be (b) roughly constant across different species (see the electronic supplementary material), no experiments determining the 2 10 3/5 elasticity of apical rings have been performed. The few avail- K 2/5 x K able measurements of p0 for different species indicate that this * h/ value might be roughly conserved [24]. p/E To test our prediction, we compiled a library of over 1000 1 5 papers from the mycological literature, and searched them for

pressure, high-resolution electron-micrograph images showing medial

gap thickness, cuts of mature apical rings (see the electronic supplementary

h0 material for search rules and example images). We found data −50 5for 45 species in two classes (18 families), with a good coverage position along ring, x/x K2/5 of the whole phylogeny of species whose spores are singly Figure 2. Simulations of the elastohydrodynamics of apical ring deformation ejected through an apical ring (figure 3, classes and families coupled with spore motion show the optimal thickness h (equation (2.2)). where data were found are shown in colour). The phylogeny * highlights the ubiquity of this trait in the ascomycetes (figure (a) Total energy dissipated through friction and fluid loss El Efluid Efriction, ¼ þ 3a), not only in two large classes (figure 3b,c) but also in more dis- as a function of the average gap thickness h. We normalize El with the final 2 2 tant families (e.g. Peltigeracea and Geoglossacea), potentially spore kinetic energy Et 2LW rU /3. Energy dissipation is minimized at the ¼ ej indicating multiple independent origins of this trait. It also optimal gap thickness h*, which is preserved as we vary the parameters of the model (G, C described in §4.1). Solid line and shading: results obtained shows several loss events (represented by dashed lines), where for a realistic set of parameters (G 0.05, C 2.3, D 0.79 mm1/2) and species ejecting spores through an apical ring evolved into ¼ ¼ ¼ niches where this trait conveyed no selective advantage and their expected variation, as described in the electronic supplementary material. (b) Normalized gap thickness (black) and pressure (grey) as a function of the was eventually lost, both on the class level (e.g. Laboulbeniales, a b,c distance from the point x 0, where the spore first starts to compress the figure 3 ) and family level (figure 3 ). ¼ From the images found with this search, we extracted the ring. The appropriate non-dimensionalization and numerical procedure are ‘ described in the electronic supplementary material. Dashed lines, solution of three independent dimensions of the apical ring (b, and d) W the dry contact problem; solid lines, solution of the full elastohydrodynamic relevant to our physical model, as well as the spore size . problem. When available, morphological data were taken from the same publication, to limit the influence of intraspecies varia- bility. If no spore size was reported or could be measured, it order to effectively minimize the fluid loss through the gap, the was taken as the average value reported in standard texts pressure in the gap (for x . 0) will be significantly higher than [25,26] (see the electronic supplementary material for details). in the ascus x 0 . This is consistent with our previous ð Þ Figure 4 shows the results of this analysis. Each data assumption that the pressure in the ascus is the main accelerat- point represents one species from the classes highlighted in ing force of the spore. The gap pressure acts nearly exactly figure 3. The individual dimensions of the apical rings are perpendicular to the spore motion during the ejection of the not strongly linked to the dimensions of the spores for the spore and can thus be neglected when calculating its accelera- same species. The spore width W does not correlate with ‘ or tion (see the electronic supplementary material). b (R2 0.11 and 0.10). The degree of correlation between W ¼ and d is higher (R2 0.64), indicating that species with larger ¼ 2.3. Optimality criterion spores have apical rings with slightly larger diameters. By con- We can now combine the results of §§2.1 and 2.2 to define an trast, figure 4b shows the correlation of the spore radius with Sr. 2 optimal spore shooting apparatus. To minimize energy losses The data collapse on a single straight line (R 0.84) is in excel- ¼ during spore ejection, the thickness of the fluid gap (h0, lent agreement with the theoretical expectation (equation (2.7)), 1=2 3 1=4 equation (2.6)) determined by the elastohydrodynamic sol- with only one free parameter D gm p0/E / p0r ¼ ð ÃÞ23ð Þ 2¼3 ution must be close to the optimal thickness of the fluid 0:79 + 0:06 mm1=2. If we assume r 1000 kg/m , m 10 ¼ ¼ gap (h , equation (2.2)). The equation h h implies the law Pa.s and p0 2 atm [27], the predicted elastic modulus of the * 0 ¼ * ¼ apical ring is E 1 MPa, consistent with the elastic moduli m1=2 p 3 d d 2b 2 * W g 0 ð þ Þ ; 2:7 of soft biological materials [28]. To quantify energy losses ¼ p r 1=4 E b d b p‘ ð Þ ð 0 Þ Ã ð þ Þ within this system, figure 4b also shows contours (grey ﬃﬃ Orbiliomycetes (a) Geoglossacea 5

Leotiomycetes 20130187 10: Interface Soc R J rsif.royalsocietypublishing.org Pezizomycetes

Laboulbeniales Saccharomycetes

Taphrinomycetes Lichinomycetes Sordariomycetes

Arthoniomycetes Lecanoromycetes Eurotiomycetes 0.1 Dothideomycetes (Peltigeracea)

(b) (d)

Nectriaceae (Hypocreales) Nectria epsiphaeria

Peltigera canina

Diaporthaceae (Diaporthales) Diaporthe eres Calosphaeriaceae (Calosphaeriales) Ceratostomella ampullasca Incertae sedis (Xylariales)

Sordariaceae (Sordariales) Sordaria macrospora Lasiosphaeriaceae (Sordariales) Lasiosphaeria spermoides

Xylariaceae (Xylariales)

Diatrypaceae (Xylariales) Xylaria longipes Quaternia 0.1 quaternata

Incertae sedis (Helotiales) (c) Dermateaceae (Helotiales)

Pezizella Hyaloscyphaceae (Helotiales) alniella Ombrophila Helotiaceae (Helotiales) hemiamyloidea Sclerotiniaceae (Helotiales)

Ciboria acerina Bulgaria Bulgariaceae (Leotiales) inqinans

Geoglossum nigritum 0.1

Figure 3. Phylogenetic tree highlighting the 45 species used in this study (adapted from [20]). Classes and families with functional apical rings are in colour, those with non-functional rings are represented by grey dashed lines, and classes with other dispersal mechanism are shown in solid grey. (a) Cladogram of the entire ascomycete phylum, delineating classes. Clades with functional apical rings are the Leotiomycetes (blue), Sordariomycetes (red), Geoglossaceae (orange) and the Peltigeracea in the Lecanoromycetes (green). More detailed phylogeny of the Sordariomycetes (b) and of the Leotiomycetes (c) delineating families. The families of the species used in this study are highlighted. (d) Examples of apical ring geometries (not to scale) to illustrate morphological diversity (adapted from [12–19]). Scale bars represents substitutions per site. (a) (b) 6 1 sfryloitpbihn.r o nefc 0 20130187 10: Interface Soc R J rsif.royalsocietypublishing.org (µm) b 0 10 2 ) 1/2 (µm)

1 (µm l r S

ring dimensions 0

3 5 ring shape, 2 (µm) d 1

0 5 10 15 0 5 10 15 spore diameter, W (µm) spore diameter, W (µm) Figure 4. Comparison between the theoretical prediction and real morphological data for 45 species represented in figure 3 by matching colours and symbols. (a) Correlation between the width of the spore W and individual dimensions of the ring. The R2 values of the correlations are 0.11, 0.10, 0.64 for ‘, b, d, respectively. (b) The non-trivial combination of lengthscales predicted theoretically correlates well with spore width (R2 0.84). The line represents equation (2.7) with fitting parameter D 0.79 mm1/2. ¼ ¼ Contour lines represent regions where spores attain 99% (light grey), 98% (grey) and 95% (dark grey) of the maximum ejecion velocity. shading) for spores attaining 99, 98 and 95 per cent of the maxi- 0.39), but in the right combination all morphologies are confined mum launch velocity, which can be obtained from a numerical to the one-dimensional subspace of the theoretical prediction integration of the equations of motion (see the electronic sup- (equation (2.7)). It is worth noting that our analysis explains plementary material). Nearly, all of the data fall within 2 per more of the variation in the dimensions W, d, ‘, b than traditional cent of the theoretical optimum. morphometric analysis using the first principal component (84 versus 64%). Principal component analysis finds the linear combination of parameters that best explains a wide variance. In 3. Discussion contrast, equation (2.7) depends nonlinearly on all of the parameters, in a fashion predicted by our mechanical analysis of The collapse of morphological data suggests that spore dissipation processes occurring during spore ejection (see the launching apparatuses have evolved to maximize dispersal electronic supplementary material). potential. A spore must escape its parent and if it can Morphologies may also be shaped by genetic constraints. penetrate through the fluid mechanical boundary layer To test whether genetics constrains fungi within the one- surrounding the sporocarp, it may be carried by the wind dimensional subspace, we explored the evolutionary trajectories and achieve long-distance dispersal. After launch from the of ascomycete species not subject to the selective force for range ascus, the velocity U(t) of a spore decelerates according to maximization. Several groups have evolved into niches where dU spore shooting is not critical to survival, because species use m zU; 3:1 dt ¼À ð Þ insects or other animals to disperse spores. Although nearly all of these species have completely lost the apical ring, the evol- where m is the spore mass and z is the drag coefficient. This utionary residue of spore ejection is seen in a few genera, for implies that the distance Z a spore ejected with initial velocity example, Geospora. Species of Geospora do forcibly eject spores, Uej will travel is given by but spores are ejected into a closed, subterranean sporocarp, m Z U : 3:2 where range maximization is irrelevant. ¼ ej z ð Þ Using the same methodology described for the forcibly The larger the range Z, the greater the variety of environ- ejecting species (see the electronic supplementary material for ments a spore can tolerate and still escape the boundary details), we collected morphological data for 13 species with layer. We have previously shown the shape of spores (the non-functional rings: seven are deliquescent, i.e. ascospores are ratio m/z) is tuned to within 1 per cent of the theoretical opti- not forciblyejected because the ascus wall dissolves; five are cleis- mum [29]; the present study demonstrates that the launch tothecial, i.e. spores are released within an enclosed sporocarp; velocity Uej is optimized to the same degree of precision by and one releases spores through a fissure in the ascus wall, and matching apical ring shape to spore size. not through the apical ring. The spore and ring morphologies Our theory shows that gradients away from the optimum of nine of 13 species are far from the subspace occupied by are steep—if a species moves off of the line in figure 3b, the spore shooting species (figure 5). So while over 90 per cent of energy dissipation penalty will be high, and the launch all species with functional apical rings have morphologies velocity Uej will plummet. within 2 per cent of the optimum, this is only the case for The most striking feature of the data collapse shown in about 30 per cent of species with non-functional apical rings. figure 4b is the large diversity of apical ring shapes captured These data confirm that the data collapse in figure 4b is by the model. Apical rings may be flat, thin, elongated or not the result of genetic constraints: alternate morphologies shallow, with only weak correlations between the different geo- are possible. In fact, the time of divergence from an ancestor metrical dimensions, as seen in figure 3d (R2 between 0.32 and with a functional apical ring is positively correlated with the and 7 0.6

90 dp 48 20130187 10: Interface Soc R J rsif.royalsocietypublishing.org a X 1 X bFGU3=5 X 1 X 4=10 ; 4:2 dX ¼ÀC ½ ð À ÞÀ ð ð À ÞÞ ð Þ

) 0.3

1/2 where X, U and pa are normalized by L, Uid and p0, respectively. The value of Uej U(X 1) after integration only depends on relative range loss (µm 60 ¼ ¼ r

S the three non-dimensional parameters G h /W, F H/h 0 0.2 0.4 0.6 ¼ * ¼ * substitutions per site and C, which is the ration of ascus to spore volume before ejection (see the electronic supplementary material for details). 30 In the physiologically relevant region of parameter space, the ring shape, solution has a sharp optimum in F, corresponding to the optimum in h shown in figure 2a.

0 5 10 15 4.2. Elastohydrodynamics spore diameter, W (µm) Near the entry point, elasticity theory [11] dictates that the gap Figure 5. Morphological analysis for 13 species with non-functional apical thickness is related to the pressure distribution by rings. Symbols represent classes (figure 3), species shown here are rep- 3 1 x s h x h x ds p s p s log j À j ; 4:3 resented by dashed grey lines in figure 3. Contours as in figure 4b. ð ÞÀ Hertzð Þ¼4pE ð f ð ÞÀ Hertzð ÞÞ j ð Þ Relaxation of the evolutionary constraint on the apical ring results in loss ðÀ1 where h x and p x are the Hertz solutions and p is of optimality. No signature of the linear relation between W and S , as pre- Hertzð Þ Hertzð Þ f r the solution to the Reynolds equation (2.5). We require p (x) dicted by equation (2.7) for functional apical rings, can be seen here f ! pHertz far from the entry point, i.e as x becomes large. For this (R2 0.076; p-value 0.32). The (inset) shows a positive correlation between ¼ to happen, the gap thickness must asymptote to a constant the phylogentic distance from the last ancestor with a functional apical ring value h h . We solve equation (4.13) iteratively at every point ! 0 (in substitutions per site) and the loss in range compared with an optimal x along the contact profile (for details, see the electronic sup- ring geometry for the same spore size (Zopt2Z)/Zopt. The distance from plementary material) to compute the pressure and hight profile the last common ancestor is measured on species level phylogenies shown in figure 2b. [31,32] using ancestral character reconstruction. The grey band corresponds to a 5% deviation from the optimum, which would contain all species This research was supported by the National Science Foundation with function apical rings. Note that three species are not shown in the through the Harvard Materials Research Science and Engineering Center (DMR-0820484) and the Division of Mathematical Sciences inset, since their phylogenetic status in unclear. (DMS-0907985), by the National Institute of General Medical Sciences (GM-068763), by a Marie Curie IO Fellowship within the loss of optimality of the apical ring (see inset of figure 5), 7th European Community Framework Programme to A.S., and by suggesting a role for genetic drift in shaping these mor- a fellowship from the Alfred P. Sloan Foundation to M.R. M.P.B. is an investigator of the Simons Foundation. phologies. In a phylum with almost no fossil record, and where molecular clock models remain problematic, morpho- We thank the Harvard Botany Libraries for their help and M. Mani, logical trait evolution may provide valuable additional data T. Schneider and D. Pfister for useful discussions and comments. We for dating species divergences. also thank three anonymous referees for their positive and extremely Our model highlights the key role of physics in genera- helpful feedback that significantly improved the quality and clarity of this paper. ting and shaping morphological diversity, which—even despite the emergence of molecular tools—remains a key to understanding the evolution of biodiversity. Endnotes 4. Material and methods 1The local radius of curvature is approximately ‘ as a result of the spore being much larger than the apical ring. We obtain the propor- 4.1. Integration of equations of motion tionality constant by data analysis as illustrated in the electronic 2 supplementary material. If ascus pressure pa and ascus volume covary linearly, then 2Experimental evidence shows that ascus pressure and volume do equation (2.6), the pressure evolution equation, and Newton’s covary during spore ejection [30] and linear covariance is most plaus- equation for the spore form a closed system of equations that ible given the material properties of the ascus in the relevant can be written in non-dimensional form as parameter regime (see the electronic supplementary material). For a different functional relationship, our model still predicts W Sr, dU 12 G 2=5 6=5 / U 3X 1 X pa U X 1 X 4:1 however, the interpretation of the constant of proportionality, and dX ¼ ð À Þ À b F ½ ð À Þ ð Þ thus our prediction for the elastic modulus of the ring, would change.

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Joerg Fritz⇤† Agnese Seminara⇤ Marcus Roper‡

Anne Pringle§ Michael P. Brenner⇤ May 25, 2013

Contents

1 Details of Theoretical Calculations 2 1.1 Basic Assumptions ...... 2 1.2 Elastic Deformation of Apical Ring ...... 4 1.3 Dynamics of Spore Motion ...... 6 1.4 Pressure Volume Relation for Ascus ...... 12 1.5 Elastohydrodynamics ...... 15 1.6 Derivation of prefactor for the apical ring law ...... 16

2 Morphological Analysis 17 2.1 SearchRules ...... 17 2.2 Selection Criteria and Measurements of Apical Ring Images . . . 18 2.3 Selection Criteria for Spore Dimension Measurements ...... 19 2.4 E↵ective Curvature of Apical Rings ...... 20 2.5 Data for Species with Functional Apical Rings ...... 21 2.6 Data for Species with Non-functional Apical Rings ...... 25

⇤School of Engineering and Applied Sciences and Kavli Institute for Bionano Science and Technology, Harvard University, Cambridge, MA 02138, USA †To whom correspondence should be addressed. E-mail:[email protected] ‡Department of Mathematics, University of California, Los Angeles, 90095, CA, USA §Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138, USA

1 1 Details of Theoretical Calculations 1.1 Basic Assumptions For clarity, we summarize here the assumptions that underly the theoretical model outlined in the main text. 1. The apical ring behaves like an elastic material under internal pressure, with Poisson ratio 1/2, so that the material is incompressible. While we are not aware of direct measurements of material properties for apical rings, owing to their large water content biological materials are generally nearly incompressible. Comparisons of ring micrographs right before and after ejection (see ﬁgure S1) indicate that the deformation is essentially purely elastic, with only a small plastic component, even after all (usually 8) spores have been ejected.

Figure S1: Comparison of rings before (A,C,E) and after (B,D,F) ejection for Neobulgaria pura (A,B) [1], Xylaria longipes (C,D) [2] and Mitrula paludosa (E,F) [3].

2. During ejection, the spore is separated from the ring by a thin layer of lubricating ﬂuid. Due to constraints in resolution in both time and space, this thin lubricating layer between spore and apical ring cannot be easily observed experimentally. However, from engineering applications at similar scales it is well known that this lubricating layer generally exists. Very complicated sealing geometries are needed to eliminate it, none of which occur in the species considered here. A complete removal of the lubricating layer would drastically increase friction between the ring and the spore and thus make spore ejection very dicult. 3. The shape of spores is well approximated by fore-aft symmetrical ellip- soids: the spores of all species considered in the paper are nearly perfectly

2 fore-aft symmetric. For species with aspect ratios below 10 the ellipsoid ﬁt is accurate to within the resolution of the available micrographs. For more ﬁliform spore shapes this approximation results in a small error, mostly in terms of e↵ective projection of forces, however numerical simulations (see section 1.3) show that these can be neglected.

4. The pressure in the ascus is proportional to its volume: while no pressure- volume relation has been determined experimentally for fungal asci, given the microscopic structure of the ascus wall and the extreme expansion of the ascus before ejection a hyperelastic model (think a balloon) is likely a good approximation (see section 1.4 for details). The main feature of such models is that pressure varies linearly with volume at small volumes and then becomes roughly independent of volume at high volumes (see figure S6). Experiments of Ingold [4] indicate that the spore range and thus pressure decrease significantly during the full ejection process. This means that a significant number of spores are ejected in the regime where pressure and volume are linearly related, and thus our model applies. Note that the model is robust with respect to details of the pressure to volume dependence, as long as they covary (i.e. pressure decreases as volume decreases). A di↵erent functional form of the pressure-volume relation only a↵ects our prediction for the Young’s Modulus of the ring, but not our prediction of W Sr,whereSr is a nonlinear combination of ring dimensions. /

5. The elastic deformation of the ring is not inﬂuenced by the ascus wall during ejection. The reason for this is two-fold. As seen in ﬁgure S1, the ascus wall is, even for the smallest rings, one order of magnitude thinner than the ring. In addition, the ascus wall strongly deforms for pressures on the order of the initial osmotic pressure p0 in the ascus, whereas the ring only deforms for the much larger lubrication pressure. This indicates that the elastic modulus of the ring is at least an order of magnitude higher than that of the ascus wall. These two factors together imply that the total force transmitted on the ring from the ascus wall (which is slightly deforming during ejection, since ascus volume is decreasing) can be neglected compared to the forces transmitted by the spore passing through the ring. Unfortunately no experimental data on the elastic modulus or other material parameters of either the apical ring or the ascus wall are available to make this argument more quantitative.

3 1.2 Elastic Deformation of Apical Ring We model the apical ring as a thick-walled short cylinder of an ideally elastic material in cylindrical coordinates (r, x, ✓). The ring has an internal radius ri, external radius ro and length ` in the x-direction. Our goal is to ﬁnd the deformation u of the ring as a function of an internal pressure pr. We will make the classical Lame assumptions: 1. axial-symmetry: u (in the r-direction) is the only displacement component 2. the radial displacement u only depends on the radius r: u(r, x, ✓)=u(r)

3. shear stresses on the elementary volume must be zero: ⌧xr = ⌧x✓ = ⌧r✓ =0 4. due to axial-symmetry and the constant thickness r r of the ring, the ra- o i dial and circumferential stresses only depend on the radius r: r(r, x, ✓)= r(r), ✓(r, x, ✓)=✓(r).

5. the axial stress x is constant along the length of the transverse section @x @x (independent of the coordinates x, r): @r = @x =0 We first consider the radial force equilibrium of an infinitesimal element of the ring with the dimensions dr, dx, d✓, keeping only the lowest order terms: d d✓ + r dr (r + dr)d✓dx rd✓dx 2 dxdr 0 r dr r ✓ 2 ⇡ ✓ ◆ d d drd✓dx + r dr2d✓dx + r rdrd✓dx drd✓dx 0 (1) , r dr dr ✓ ⇡ 2 Neglecting small products like (dr/dr)dr d✓dx and collecting terms, the above equation reduces to: d r + r ✓ = 0 (2) dr r We have recovered one equation with two variables r, ✓. If the complete stress field (r, x, ✓) in the ring is known, strains and the deformation can be determined easily. To derive two additional equations for these three variables we combine the strain-displacements equations: du ✏ = (3) r dr and 2⇡(r + u) 2⇡r u ✏ = = (4) ✓ 2⇡r r with Hooke’s generalized law for these strain components: 1 ✏ = [ ⌫( + )] (5) r E r x ✓ and 1 ✏ = [ ⌫( + )] . (6) ✓ E ✓ r x

4 Combining Eqns. (3) to (6) we ﬁnd

E u du ⌫ = + ⌫ + x (7) ✓ 1 ⌫2 r dr 1 ⌫ ✓ ◆ and E du u ⌫ = + ⌫ + x . (8) r 1 ⌫2 dr r 1 ⌫ ✓ ◆ Eqns. (2),(7),(8) give us three equations for the three stress components. By combining these equations we can derive an ODE for u(r)

d2u 1 du u + = 0 (9) dr2 r dr r2 Integrating twice we ﬁnd C u(r)=C r + 2 (10) 1 r

The constants C1 and C2 are determined by the boundary conditions of the problem. In our case we will require an overpressure (above atmospheric pressure) of pr on the inside of the ring. That is

(r = r )= p and (r = r )=0. (11) r i r r o Plugging Eq. (10) into (8) and comparison with Eq. (11) shows that for our case

1 ⌫ r2p 1 ⌫ r2r2p C = 0 r and C = i 0 r (12) 1 E r2 r2 2 E r2 r2 i o i o The most general expression for u(r)isthen

1+⌫ r2p r2 u(r)= i r 1 2⌫ + o r (13) E r2 r2 r2 0 i ✓ ◆ which for biomaterials with ⌫ 0.5 simpliﬁes to ⇡ 3 p r2r2 u(r)= r i o (14) 2E (r2 r2)r o i

5 1.3 Dynamics of Spore Motion To complement the scaling analysis presented in the paper, and exactly compute the energy dissipation during spore ejection we consider here the time evolution of the macroscopic motion of the spore. We introduce a closed system of equations for the velocity of the spore U(t), the gap thickness h(t) and the falling pressure in the ascus pa(t). The solution of these coupled equations with given initial condition gives the ﬁnal velocity Uej at which the spore is ejected.

Thickness of the lubrication layer. In the main text and section 1.5 we have outlined the method to solve the elastohydrodynamic equations to determine the constant gap thickness h0 at every instant in time. We summarize these results, highlighting the time dependence of the width of the spore r(t) and the velocity of the spore U(t):

µU(t) 3/5 h (t)= ⇠(t)2/5 (15) 0 E ✓ ◆ where =1.42 (see section 1.5), ⇠(t)=0.43` E/pr(t) (see section 2.4) and the dry contact pressure is p b(d + b) pr(t)=E r(t) (16) ⇤ d(d +2b)2

W For an ellipsoidal spore with width W and length L, r = L X(L X), where X is the Lagrangian coordinate of the spore (measured from the tip of the spore p to the apical ring, see ﬁgure S2).

Figure S2: Deﬁnition of the Langrangian coordinate X in this model of spore ejection

6 Note that this solution for h0 is only valid in the case where the ascus pressure is much smaller than the dry contact pressure, which holds in this system for most of the spore ejection, except during the very initial stages when the spore starts to compress the ring to pass through.

Force balance on the spore. We now proceed to write Newton’s law for the trajectory of the spore, subject to the ascus pressure pa pushing the spore out and viscous stress originating in the lubrication layer. If we approximate the pressure in the gap as the dry contact pressure, and the height of the gap as h0 from equation (15), the ﬂow in the lubrication layer is a simple shear with no pressure gradients: y u(y)=U (17) h0 so that the viscous stress on the spore wall is du U µ y=0 = µ (18) dy | h0 where y is the coordinate in the lubrication layer perpendicular to the spore wall. If U is the velocity of the spore and X its Lagrangian coordinate (measured from the tip of the spore to the apical ring), the equation of motion for the spore is:

l dU 2 U m = ⇡r p 2⇡rlµ cos[arctan r0]+2⇡ rpdx sin[arctan r0] (19) dt a h  0 Z0 where the r.h.s. terms, from left to right, originate from the ascus pressure, the viscous stresses and the pressure forces in the gap respectively. This last term is negligible, since the spores are approximately fore-aft symmetric. The cos and sin terms represent the projection of the forces along the direction of motion. Writing dU/dt = UdU/dX and using the mass of the spore m = ⇡/6⇢W 2L we have dU 6 12µl U = 3 X(L X)pa 2 U X(L X) (20) dX ⇢L ⇢WL h0 where we have neglected the projection terms, since theyp only introduce a weak dependence on the aspect ratio of the spore.

Pressure evolution. As discussed in the model assumptions and section 1.4 the pressure in the ascus depends linearly on ascus volume, so that p V a = (21) p0 V0 The volume V of the ascus decreases as the spore and the ﬂuid leave the ascus according to dV/dt = ⇡U(t)r(r + h0) where we used the linear velocity proﬁle (17) in the lubrication layer so that

dpa p0 = ⇡ Ur[r + h0] (22) dt V0

7 Nondimensional system. Equations (15), (20), (22) form a closed system of equations for the pressure in the ascus and the velocity of the spore. There are three length scales in this problem (H,h ,W ), we can thus define two non- dimensional parameters from them: F = H/h⇤ representing the optimum iden- tified in the main paper and G = h /W . We⇤ now non-dimensionalize with ⇤ X = XL˜ , U = 2p /⇢ U˜, r = W X˜(1 X˜) for ellipsoidal spores, p =˜p p 0 a a 0 and h = HU˜ 3/5/[X˜(1 X˜)]1/10qand obtain the two coupled equations: 0 p dU 12 G U =3X(1 X)p U 2/5[X(1 X)]6/5 (23) dX a F dp 48 a = [X(1 X) FGU3/5(X(1 X))4/10 (24) dX C where we have dropped the tildes. The solutions to these equations are completely determined by three non-dimensional parameters F and G introduced above and C representing the ratio of fluid volume in the ascus to the volume 2 of the spores V0 =8C⇡W L/6 (assuming 8 spores in the ascus, by far the most common number). As outlined in the main paper, we expect that to minimize dissipation, the morphology of the apical ring must be designed in such a way that the thickness of the lubrication layer matches the thickness at which minimum dissipation occurs, i.e. h0 = h , or F = Fopt. Note that equation (23) immediately shows that there is indeed⇤ an optimal F :ifF is too small, the spore rapidly loses momentum to viscous stress as the second term dominates the dynamics; if F is too large, the ascus pressure pushing the spore drops rapidly as a consequence of fluid loss (see eq (24)). We are interested in a numerical solution for Uej(F, G, C) in a defined region of the space spanned by the three independent parameters. We expect: (i) F = O(1), since at the optimum we must roughly have h0 = h (ii) 0.03

8 Figure S3: Final ejection speed as a function of F showing an optimal value, for G =0.05 and C =2.3. The color bar corresponds to the colors in the contour maps (Fig. S4 and Fig. S5).

9 Figure S4: Contour map of the ﬁnal ejection speed as a function of F and G, where the speed is normalized by the optimum value. The dark red denotes the parameter range where the optimum is attained. Note that there is a sharp optimum near F 0.32, and its location is insensitive to G over the physiological range of the species⇡ considered here.

10 Figure S5: Contour map of the ﬁnal ejection speed as a function of F and C, where the speed is normalized by the optimum value. The dark red denotes the parameter range where the optimum is attained. Note that there is a sharp optimum near F 0.32, and its location is insensitive to C over the physiological range of the species⇡ considered here.

11 1.4 Pressure Volume Relation for Ascus Here we outline a model calculation for the pressure volume relation in the ascus. As noted above, since the volume change in the ascus is small during the ejection of a single spore, for our calculation it is valid to approximate the pressure as depending linearly on the ascus volume. However, it is of interest to understand the full nonlinear relationship between pressure and volume in the ascus and here we outline a plausible model. Our model is based on the fact that the ascus walls are very thin, yet the ascus changes its volume by roughly an order of magnitude during its expansion before ejection. This makes it plausible that the ascus wall has material properties similar to polymer networks in a hyperelastic material (like e.g. in a balloon). The main feature of (ideal) polymer networks is that cross-links are evenly and randomly distributed in the bulk of the material. This leads to chains along which stress can be propagated that have random conﬁgurations. If we consider a deformation of this network, described by elongation factors i in dimensions xi we can calculate the change in entropy S associated with this deformation [5]

kn 3 3 S = e 2 3 ln (26) 2 i i i=1 i=1 ! X Y where ne is the number of chains. As a good approximation the volume of the 3 ascus wall stays constant and thus i=1 i = 1 which simpliﬁes things quite a bit. We want to consider a reversible process (dQ = TdS), where work is done Q by an external pressure, which creates reaction forces fi along the directions xi. The change in internal energy associated with this process is

dE = TdS + fidxi (27) i=1 X For an ideal polymer, we would expect (@E/@xi)T,V = 0 and thus @S f = T (28) i @x ✓ i ◆T,V The stress (with reference to the undeformed state) is l ⌧ = i f + ⌧ (29) i V i p where li is the length in direction xi, V is the volume of the bulk and ⌧p is an arbitrary constant (e.g. hydrostatic pressure). From this we can derive a stress-strain (or elongation ratio) relationship l @S ⌧ = i T i + ⌧ (30) i V l @ p ✓ i i ◆ or more concretely 2 ⌧i = Gi + ⌧p (31)

12 where G nekT/V is the shear modulus of the material. In the⇡ case of an elastic thin-walled cylinder, we have 3 major dimensions: z,r and t. We deﬁne the extension ratios along these dimensions as z = l/l0, r = r/r0 and t = t/t0. To impose conservation of volume we approximate the cylinder wall volume by Vw =2⇡rlt (32) ignoring the bottom and top and thus the volume is linear with regard to the major dimensions. Motivated by classical mechanics and the stresses in thin shells we introduce an anisotropy factor a to express the extension ratios as functions of each other. a (1+a) r = z ; t = z (33) where the last relation results from volume conservation. The arbitrary constant in the stress strain relation can be determined from ⌧t = 0 as 2(1+a) ⌧p = Gz (34) which gives us for the other two stresses

2 2(1+a) ⌧ = G (35) z z z ⇣ ⌘ and 2a 2(1+a) ⌧ = G (36) r z z The force balances in z and r direction⇣ give us a relation⌘ between the stresses and the pressure inside the ascus.

2 2⇡rt⌧z = p⇡r (37) and 2⇡l⌧r =2rlp (38) With this information we can eliminate the stresses and nondimensionalize with ⌧ = 2Gt0 sm r0 2 2(1 a) p z z = 1+2a (39) ⌧sm z and from equation (38) 2a 2(1 a) p z z = 1+2a (40) ⌧sm 2z ⇡r2l ⇡r2l ˜ 0 0 1+2a We can substitute z with a dimensionless volume V = ⇡r2l = z 1 0 0 and eliminate a by combining equations (39) and (40) to get a pressure volume relationship in dimensionless form

1/3 p (1 + V˜ )2 1 2 = (41) ⌧sm 2(1 + V˜ ) 0(1 + V˜ )2 (1 + V˜ 2)2 +1 1 @ ⇣ ⌘A 13 Figure S6 plots the predicted pressure volume relationship. The main point for the purposes of this manuscript is that during the ejection of a single spore, the volume does not change by very much, and in the region where pressure and volume covary (experimental evidence shows that they do [4]), their relation is indeed roughly linear. We therefore assume that fungal asci opperate in the small V˜ regime and consider the pressure volume relationship of the spore to be linearized around the initial ascus volume.

Figure S6: Expected pressure volume relation for a fungal ascus. In the region where pressure and volume covary (for small V˜ ), they are approximately linearly related. This is the regime where we expect asci to operate during spore ejection.

14 1.5 Elastohydrodynamics We now outline the solution to the elastohydrodynamic equations outlined in the main text. The structure of the solutions to these equations were discovered by Dowson and Higginson in the 1960’s; here we simply outline their original methodologies for solving the equations. More sophisticated variations of their methods have been extensively discussed. Our goal is to find a solution for the gap height profile h(x)intheen- try region of the apical ring that satisfies both the Reynolds equation and the elasticity equation of the ring. Guided by the scaling outlined in the main text we non-dimensionalize the system withp ¯ =(p/E )F 1/5,¯x =(x/⇠)F 2/5, h¯ =(3h/8⇠)F 3/5 with F = (32E ⇠)/(27µU). ⇤ Then the lubrication equation⇤ becomes

dp¯ h¯ h¯ f = 0 (42) dx¯ h¯3

1/2 3/2 subject to the conditionsp ¯c =¯x for x>>0 and h¯ =¯x for x<<0 arising from the Hertzian contact problem. This equation must be solved in conjunction with the elastic equation (Eqn. 6 of the main text). We solve the problem of matching the hydrodynamic to the elastic solution iteratively, following a method first outlined by Hooke [6]. The central idea of this method is to find the minimal deformation away from the dry contact gap profile that allows

the lubrication equation to be satisﬁed everywhere • the height and height gradient at a point far away from the contact to be • exactly the Hertz solution for the deformation and its gradient the pressure and pressure gradient at a point well into the contact to be • exactly the dry contact pressure and pressure gradient The basic steps in the algorithm are:

1. Guess a value for h¯0 2. At a point x well inside the contact region we calculate h¯ according 1 1 1/2 to (42) assuming the contact pressure gradient arising fromp ¯c =¯x 3. Add the constant h¯ to the height profile, so that in the first iteration h¯ = h¯ +¯x3/2 1 1 4. Given this height profile, perform a forward integration of the Reynolds equation from x to x , with the initial conditionp ¯ =0 1 1 5. Iteratively adjust the guess for h¯0 until the di↵erence between the pressure at x from the hydrodynamic solution and the elastic solution becomes negligible1 by repeating the steps above

15 6. Calculate a correction to the height proﬁle from this pressure di↵erence

as x 1 2 h¯(¯x)= R (¯pf p¯s)ln(¯x s) ds (43) x Z 1 where R is a relaxation factor 7. Add this correction to h¯ and repeat from step 1) until the norm of h¯ is suciently small

Following the method outlined above with R =0.01 and x = 10 yields a 1 value of h¯0 0.59 (see figure 2 in the main paper for a full profile of gap height and pressure).⇡ With this result we can also find the value of as defined in the main paper: 8 27 3/5 (µU)3/5⇠2/5 h0 =0.59 (44) 3 32 E3/5 ✓ ◆ ⇤ H | {z 1.}42 ) ⇡ | {z } 1.6 Derivation of prefactor for the apical ring law From the integration of the equations of motion (see Eq. (25)) we found that

H F = =0.32 opt h ⇤ Combining this with the results from the elastohydrodynamic solution (Eq. (44)) we ﬁnd (2⇢p )1/4(µU)3/5⇠2/5 0 =0.32, (45) (µ`)1/2E3/5 ⇤ which can be rewritten in terms of the dimensions of spore and apical ring using Eqns. (16) and (48) as

(2⇢p )1/4µ3/5(2p )3/10(0.43`)2/5d1/5(d +2b)2/5 0 0 =0.32 (46) ⇢3/10(µ`)1/2E3/5[Wb(d + b)]1/5 ⇤ or in terms of W = W (Sr)

211/4(0.43)2 p11/4µ1/2 d(d +2b)2 W = 0 . (47) 0.325 ⇢1/4E3 b(d + b)p` ⇤ Sr | {z } Thus we ﬁnd | {z } 211/4(0.43)2 = 371. 0.325 ⇡

16 2 Morphological Analysis 2.1 Search Rules We performed an in-depth search for publications that potentially contain high resolution images of apical ring cross sections to allow comparison with our theoretical predictions. We specifically targeted three main types of articles that are most likely to contain this type of data: 1. Studies on the actual structure of the ascus apex following Chadefaud’s initial observations [7]) 2. Original species descriptions 3. Comparative studies of ascus structure between genera and families. Our strategy was to first sample the literature in these three areas with strate- gically chosen keywords and then refine the results by searching references and author publications for articles that contain desired data. As a first step we queried both Google Scholar and Web of Science for any three of the following four criteria: 1 ”ultrastructure” or ”fine structure” or ”tem” or ”morphological studies” • ”inoperculate discomycetes” or ”sordariomycetes” or ”leotiomycetes” • ”ascus” or ”asci” • ”apical ring” or ”apical pore” or ”apical annulus” or ”ascus apex” or • ”apical thickening” This returns a library of over 500 potential publications that we checked by hand for appropriate data (see section 2.2). We found 26 papers that satisfy the criteria outlined there (hits). For each of these hits we compiled an additional list of all publications published by any of the authors • referenced in the hits • referencing these hits, based on both Google Scholar and Web of Science • resulting in an additional library of roughly 500 papers after screening for over- lap with the former list and subject matter (e.g. excluding publications in botany or about fungal classes that cannot have apical rings). We again searched this list by hand for images of apical rings according to section 2.2 and found an additional 12 hits. Repeating this last step for the new hits generated only a very

1An example of a valid search in Google Scholar notation would thus be ("inoperculate discomycetes" OR "sordariomycetes" OR "leotiomycetes") AND ("ascus" OR "asci") AND ("apical ring" OR "apical pore" OR "apical annulus" OR "ascus apex" OR "apical thickening").

17 small library of roughly 50 publications with no new results, indicating that we have thoroughly sampled the target subject areas. For a full list of papers used to generate the plots in ﬁgure 3 of the main paper (all hits) see supplementary table 1 and the associated ﬁgure S9.

2.2 Selection Criteria and Measurements of Apical Ring Images Our goal was to extract the key dimensions of apical rings (`, b, d) from published micrographs. To ensure accuracy we only used micrographs that 1. display a median or very nearly median longitudinal section or projection of an apical ring: this is the only geometric conﬁguration in which all 3 dimensions can be realiably measured.

2. have enough contrast and resolution such that the apical ring is clearly distinguishable 3. be in a developmental stage between advanced immature and mature in the notation of Verkley [1]: the individual dimensions of apical rings vary signiﬁcantly during development [3,8], however they essentially stop developing at some point before full maturity is reached. 4. the resolution is high enough such that potential measurement errors in ring shape are below 2% in terms of the estimated ejection velocity. ±

Figure S7: Examples for measured ring dimensions. Red is ring height `,blue is ring thickness b and green is pore diameter d for (A) Hypoxylon multiforme and (B) Xylaria longipes

Table 1 shows the results of these measurements, which were used to create the plots in ﬁgure 4 in the main paper.

18 2.3 Selection Criteria for Spore Dimension Measurements To determine whether a correlation between ring shape and spore size exists, we needed to determine the spore diameter for all species where accurate high- resolution apical ring cross sections are available. Variations in spore size can be quite pronounced in some species, so it is desirable to determine spore size from a source that as closely as possible matches the specimen for which the apical ring was measured. However most publications containing apical ring micrographs do not contain speciﬁc information on spore sizes, so we had to use average spore sizes reported elsewhere in the literature. We used the following order for spore size measurement sources, where the expected uncertainty mildly increases the further down an item is listed:

1. spore size given in the text of the publication, where the apical ring dimensions were measured 2. spore size measured from images in the publication where the apical ring dimensions were measured 3. in species with populations that vary in spore size, spore size reported in publications from the same or a similar population

4. average spore size from measurements of a representative number of spores (> 20) online or in a publication with explicitly reported median size 5. calculated average spore size from the lower and higher range of spore size in standard sources [10, 11]

Spore sizes and the source for the measurement are reported in supplementary table 1.

19 2.4 E↵ective Curvature of Apical Rings

Figure S8: Strong correlation between local curvature of the entry region of the apical ring and its length

The apical rings of ascomycetes function partly as seals that ensure the pressure loss in the ascus is not too high. In engineering applications of this type, the local radius of sealing rings at their inlet is generally strongly correlated with their length , because this shape is stable against wear. We tested this correlation for the biological sealing rings we study here by selecting three points along the inlet profile to determine the radius of curvature and find that this pattern is indeed preserved (figure S8). By linear regression we find that the radius of curvature is approximately R 0.195 ` where ` is the length of the apical ring. This give us an elastic healing⇤ ⇡ length [6] of the form

E E ⇠ = p8⇡R ⇤ 0.43` ⇤ (48) ⇤s pr ⇡ s pr

This correlation allows us to write Sr as a function of only three easily measured dimensions of the apical ring.

20 2.5 Data for Species with Functional Apical Rings

Table 1: Main Data

species name W (µm) ` b d (µm) S (µm1/2)# ⇥ ⇥ r Albotricha acutipila 1.50 [11] 0.85 0.30 0.42 [12] 2.16 1 ⇥ ⇥ Annulohypoxylon multiforme 4.50 [13] 1.16 0.57 0.80 [13] 3.57 2 ⇥ ⇥ Brunnipila clandestina 1.65 [11] 0.91 0.28 0.33 [12] 1.60 3 ⇥ ⇥ Bulgaria inquinans 6.75 [14] 1.20 0.87 1.30 [14] 5.79 4 ⇥ ⇥ Chlorociboria aeruginascens 1.75 [11] 0.58 0.40 0.26 [15] 1.43 5 ⇥ ⇥ Ciboria acerina 6.00 [16] 1.27 1.19 1.26 [16] 5.09 6 ⇥ ⇥ Ciboria batschiana 5.00 [10] 1.13 0.36 0.76 [17] 3.88 7 ⇥ ⇥ Cudoniella clavus 4.50 [11] 0.28 0.84 0.44 [15] 3.53 8 ⇥ ⇥ Dasyscyphella cassandrae 2.00 0.93 0.25 0.29 [12] 1.40 9 ⇥ ⇥ Dialonectria episphaeria 3.55 [13] 0.80 0.35 0.57 [13] 3.17 10 ⇥ ⇥ Diaporthe eres 2.57 [13] 1.41 0.71 0.54 [13] 1.95 11 ⇥ ⇥ Eutypella quaternata 3.50 [13] 0.34 0.14 0.52 [13] 6.11 12 ⇥ ⇥ Fasciatispora petrakii 6.00 [18] 0.77 0.26 0.65 [18] 4.30 13 ⇥ ⇥ Gelasinospora tetrasperma 14.50 1.77 1.02 2.33 [19] 9.75 14 ⇥ ⇥ Geoglossum nigritum 5.50 [11] 1.83 1.04 1.07 [20] 3.59 15 ⇥ ⇥ Hymenoscyphus herbarum 2.50 [11] 0.54 0.43 0.37 [15] 2.19 16 ⇥ ⇥ Hypoxylon fragiforme 5.70 0.91 0.57 0.98 [21] 5.19 17 ⇥ ⇥ Lachnum bicolor 1.75 [11] 1.38 0.38 0.31 [12] 1.15 18 ⇥ ⇥ Lachnum brevipilosum 1.70 0.83 0.21 0.26 [12] 1.30 19 ⇥ ⇥ Mitrula paludosa 2.75 [11] 1.22 0.52 0.72 [20] 3.13 20 ⇥ ⇥

21 Table 1: Main Data species name W (µm) ` b d (µm) S (µm1/2)# ⇥ ⇥ r Monilinia johnsonii 4.15 [17] 1.39 0.53 0.58 [17] 2.24 21 ⇥ ⇥ Natantiella ligneola 3.64 [13] 1.14 0.38 0.97 [13] 5.31 22 ⇥ ⇥ Neobulgaria pura 5.25 [10] 1.39 0.98 1.62 [14] 6.90 23 ⇥ ⇥ Neurospora crassa 16.58 [22] 0.61 0.28 1.27 [22] 12.60 24 ⇥ ⇥ Neurospora lineolata 12.00 0.77 0.38 1.26 [23] 9.40 25 ⇥ ⇥ Neurospora tetrasperma 17.00 1.04 0.68 1.90 [22] 11.36 26 ⇥ ⇥ Ombrophila hemiamyloidea 3.50 [24] 0.71 0.93 0.50 [24] 2.49 27 ⇥ ⇥ Ombrophila violacea 4.00 [11] 1.47 0.25 0.77 [14] 4.05 28 ⇥ ⇥ Peltigera canina 3.20 [25] 1.69 0.73 0.77 [25] 2.68 29 ⇥ ⇥ Pezicula ocellata 13.00 [26] 1.30 0.13 0.83 [26] 6.85 30 ⇥ ⇥ Pezizella alniella 2.75 [11] 0.73 0.32 0.43 [15] 2.37 31 ⇥ ⇥ Pezizella gemmarum 2.10 [11] 0.46 0.32 0.29 [15] 1.87 32 ⇥ ⇥ Phaeohelotium carneum 3.25 [11] 0.55 0.40 0.58 [15] 3.81 33 ⇥ ⇥ Podosordaria leporina 9.30 [27] 1.14 0.78 1.34 [27] 6.36 34 ⇥ ⇥ Poronia punctata 9.50 [10] 1.74 0.60 1.50 [28] 6.61 35 ⇥ ⇥ Rutstroemia lindaviana 1.75 [10] 0.53 0.26 0.23 [17] 1.36 36 ⇥ ⇥ Rutstroemia petiolorum 4.50 [10] 1.05 0.46 0.76 [17] 3.73 37 ⇥ ⇥ Ruzenia spermoides 4.94 [13] 1.06 0.36 1.05 [13] 6.24 38 ⇥ ⇥ Sclerotinia sclerotiorum 5.00 [11] 1.40 0.42 0.67 [17] 2.79 39 ⇥ ⇥ Sordaria ﬁmicola 16.08 [29] 0.72 0.41 1.57 [29] 12.93 40 ⇥ ⇥ Sordaria humana 15.00 [10] 2.28 1.08 2.92 [30] 11.54 41 ⇥ ⇥

22 Table 1: Main Data

species name W (µm) ` b d (µm) S (µm1/2)# ⇥ ⇥ r Spirodecospora bambusicola 13.00 2.59 1.18 2.98 [31] 10.75 42 ⇥ ⇥ Stromatinia rapulum 6.25 1.43 0.45 0.74 [17] 3.13 43 ⇥ ⇥ Xylaria hypoxylon 5.00 [11] 0.66 0.31 0.43 [32] 2.58 44 ⇥ ⇥ Xylaria longipes 6.00 [11] 1.85 0.47 0.89 [33] 3.44 45 ⇥ ⇥

Figure S9: Plot of ring shape Sr over spore size W with number labels corresponding to table 1

23 Figure S10: Plot of ring shape Sr over spore size W with number labels corresponding to table 2

24 2.6 Data for Species with Non-functional Apical Rings

Table 2: Main Data

species name W (µm) ` b d (µm) S (µm1/2) # reason for loss of function ⇥ ⇥ r Laboulbeniopsis termitarius 1.90 [34] 1.45 0.81 1.46 [34] 6.26 1 ascus wall dissolves [34] ⇥ ⇥ Thelebolus stercoreus 3.10 [35] 8.22 4.39 28.79 [35] 97.32 2 ejects spores into closed sporocarp [35] ⇥ ⇥ Thelebolus polysporus 3.75 [35] 3.25 4.39 11.06 [35] 72.92 3 ejects spores into closed sporocarp [35] ⇥ ⇥ Thelebolus crustaceus 4.50 [35] 5.21 2.35 18.42 [35] 88.28 4 ejects spores into closed sporocarp [35] ⇥ ⇥ Thelebolus microsporus 5.27 [35] 1.64 0.41 6.37 [35] 91.94 5 ejects spores into closed sporocarp [35] ⇥ ⇥ 25 Aniptodera salsuginosa 5.70 [36] 1.02 1.79 0.72 [36] 2.92 6 ejects spores through ﬁssure at the side of ascus [36] ⇥ ⇥ Magnaporthe salvinii 8.70 [37] 1.25 1.29 1.44 [37] 5.91 7 ascus wall dissolves [37] ⇥ ⇥ Prostratus cyclobalanopsidis 17.0 [38] 2.42 1.51 1.73 [38] 5.16 8 ascus detaches from sporocarp [38] ⇥ ⇥ Diaporthe phaseolorum 3.60 [39] 1.72 0.59 0.60 [39] 2.07 9 ascus detaches from sporocarp [39] ⇥ ⇥ Lollipopaia minuta 2.50 [40] 0.96 0.35 0.29 [40] 1.31 10 ascus detaches from sporocarp [40] ⇥ ⇥ Chaetosphaeria chaetosa 6.96 [41] 1.43 0.47 0.61 [41] 2.44 11 ascus wall dissolves [41] ⇥ ⇥ Aniptodera chesapeakensis 11.0 [42] 0.22 0.37 1.31 [42] 18.87 12 ascus wall dissolves [42] ⇥ ⇥ Halosarpheia marina 11.5 [42] 0.13 0.44 1.65 [10] 31.86 13 ascus wall dissolves [42] ⇥ ⇥ References

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