Martina Iapichino

Yen Wen Wang

Savannah Gentry

Anne Pringle

Agnese Seminara

A precise

Keywords: ballistospory, fungi, morphometrics, biomechanics 30

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION 31

Molds, yeasts, and mushrooms are ubiquitous across Earth. Estimates of the number of 32 fungal species range from 1 to more than 5 million [START_REF] Blackwell | The fungi: 1,2,3, ... 5.1 million species?[END_REF], and fungi in 33 ecosystems function as decomposers, mutualists and pathogens. Emerging fungal 34 diseases endanger crops as well as wild plants and animals, threatening food security, 35 but fungal diseases also alter forest dynamics and contribute to the extinction of 36 animals. Losses cost millions of US dollars in damage [START_REF] Pennisi | 1000 genomes project gives new map of genetic diversity[END_REF][START_REF] Fisher | Emerging fungal threats to animal, plant and ecosystem health[END_REF]37 Kupferschmidt 2012). 38

Most fungal bodies (mycelia) are immobile, typically hidden within substrates. 39 Fungi use spores to reproduce and travel away from a natal habitat. Spores are carried in 40 air currents away from a source and when a spore lands in a favorable environment, it 41 germinates and begins or extends the life cycle. Basidiomycota are among the most 42 common fungi, encompassing pathogens like the honey mushroom as well as 43 charismatic mushrooms like matsutake and the fly agaric. The phylum is defined by the 44 production of sexually derived spores (basidiospores) on a basidium. Basidiospores are 45 launched via a surface tension catapult. Among species of mushroom-forming fungi, 46 e.g. agarics and boletes, spores typically form in groups of four from basidia arranged 47 along the gills or pores of a mushroom, each spore attached to a sterigma. A drop of water condenses on the hilar appendix of a spore. Buller's drop then collapses onto 50 another adaxial drop formed along the longitudinal axis of the spore itself (FIG. 1). 51 Upon coalescence, surface energy is converted into kinetic energy. The spore is ejected 52 horizontally away from the basidium and sterigma. The spore decelerates to rest after a 53 few milliseconds and then falls vertically between two gills or within the pore. 54 Ballistospore discharge was first observed by [START_REF] Schmitz | Beitrage zur Anatomie und Physiologie des Schwämme[END_REF]. In the 20th 55 century, Buller (1909) described the phenomenon in more detail, observing the 56 discharge of the spore and describing both the formation of the drop at the hilar 57 appendix and the subsequent launch of the spore together with the drop. The drop is 58 now referred to as Buller's drop and the discharge understood as a "surface tension 59 catapult". We also se he phrase spore-drop comple o mean he en ire spore 60 discharge complex, including the adaxial drop, Buller's drop and the spore. Progress in 61 understanding the anatomy and physics of the surface tension catapult was enabled by 62 the development of cameras. [START_REF] Webster | Ballistospore discharge in Itersonilia 464 perplexans[END_REF] provided photographic evidence of 63 Buller's drop forming at the hilar appendix just before discharge and proposed a two-64 phase mechanism for spore ejection: the first phase involved Buller's drop enveloping 65 the spore surface, acquiring momentum; the second entailed the sharing of momentum 66 and movement of the center of mass of the spore-drop complex, a result of rapid 67 wetting. Subsequent works modeled the conversion of surface energy into kinetic 68 energy with different degrees of complexity and imaged ballistospore launch with 69 progressively faster cameras [START_REF] Pringle | The captured launch of a 447 ballistospore[END_REF][START_REF] Noblin | Surface tension propulsion of fungal spores[END_REF]Stolze-Rybczynski 70 et al 2009;Fischer et al 2010b;[START_REF] Liu | Asymmetric drop 433 coalescence launches fungal ballistospores with directionality[END_REF]. [START_REF] Pringle | The captured launch of a 447 ballistospore[END_REF] observed 71 coalescence while [START_REF] Noblin | Surface tension propulsion of fungal spores[END_REF] described the process as encompassing four 72 stages and estimated approximately half of the total surface energy was dissipated balance to generate simulations of the fluid dynamics within the Buller's drop and the 75 adaxial drop during coalescence and described experiments with biomimetic drops. 76

These authors found that coalescence occurs in a regime where viscous dissipation in 77 the Buller's drop is negligible. Hence, energy is not dissipated to set Buller's drop in 78 motion, instead, it may be dissipated to break the spore from the sterigma. In addition, 79 [START_REF] Liu | Asymmetric drop 433 coalescence launches fungal ballistospores with directionality[END_REF] found that he phenomenon kno n in ph sics as pinning of the 80 contact line [START_REF] De Gennes | Wetting: statistics and dynamics[END_REF] provides directionality for the spore-drop complex as it 81 ejects away from the originating gill. 82

It has long been hypothesized that mushrooms form gills to increase the surface 83 area for spore production and pack the maximum number of spores into a minimal 84 investment of biomass (Buller 1909;[START_REF] Mcknight | Optimal gill packing in agaric sporocarps[END_REF]Fischer and Money 85 2010). To achieve an optimal morphology, the size of Buller's drop, the size of the 86 spore, and the distance between gills must be finely coordinated. While spore size and 87 intergill distance may be under genetic control [START_REF] Kues | Fruiting body production in basidiomycetes[END_REF]), Buller's drop forms 88 extracellularly [START_REF] Webster | Vapour as the source of water in B ller s 466 drop[END_REF]. Whether and how fungi control the size of Buller's 89 drop remains unknown, although data reporting characteristic sizes of Buller's drop for 90 different species suggest individual species do control size [START_REF] Pringle | The captured launch of a 447 ballistospore[END_REF]Stolze-91 Rybczynski 2009;[START_REF] Stolze-Rybczynski | Adaptation of the spore discharge mechanism in the Basidiomycota[END_REF]Fischer et al 2010b). 92

To explore whether the morphologies of gilled mushrooms enable the maximum 93 packing of spores within tissues, we first revisit the theory relating ejection velocity and 94 flight time to the horizontal distance travelled by a spore from the moment of launch to 95 the moment it begins settling underneath the gills (Buller 1909;[START_REF] Pringle | The captured launch of a 447 ballistospore[END_REF]96 Noblin et al 2009;[START_REF] Stolze-Rybczynski | Adaptation of the spore discharge mechanism in the Basidiomycota[END_REF]Fischer et al 2010b;[START_REF] Liu | Asymmetric drop 433 coalescence launches fungal ballistospores with directionality[END_REF]. 97

Using energy balance we obtain the ejection speed and highlight its dependence on the sizes and densities of the spore and Buller's drop. Combining expressions for ejection 99 speed and flight time, we predict the distance travelled ballistically by the spore-drop 100 complex before downward sedimentation starts. We then use our model to elucidate the 101 criteria enabling maximum packing of spores. In the phase space made up of the three 102 variables (i) drop radius, (ii) spore radius (defined as the radius of a sphere with the 103 same volume as the spore) and (iii) intergill distance, the criterion for maximum 104 packing is that spores must travel ballistically exactly midway between two facing gills. 105 Given two of the three variables (i)-(iii), the model predicts the third, assuming 106 maximum packing. To compare our model to empirical data, we collected mushrooms 107 of eight different species and measured spore size and intergill distance. By placing 108 these morphological data on the phase space generated from our model, we predict that 109 for collec ed species, he radi s of B ller s drop that maximizes spore packing ranges 110 between 23% and 50% of the radius of the spore (depending on the precise value of 111 spore density and efficiency of energy conversion). To validate the prediction, we 112 revisit previously published data for an additional 13 species [START_REF] Pringle | The captured launch of a 447 ballistospore[END_REF]Stolze-113 Rybczynski 2009;[START_REF] Stolze-Rybczynski | Adaptation of the spore discharge mechanism in the Basidiomycota[END_REF]Fischer et al. 2010b) and find B ller s 114 drop scales as 32% of spore size, consistent with our prediction. These results suggest 115 that B ller s drop radi s scales linearl i h spore radi s, and combined i h o r 116 model, generate a second prediction: to enable maximum packing, intergill distance 117 should be proportional to spore radius to the power 3/2. In the aggregate our work 118 s n hesi es hinking abo he morphologies of B ller s drop, spores, and gills, while 119 providing insights into the principles shaping ballistospory. 120

MATERIALS AND METHODS 121

Spore ejection speed. The reduction in surface energy following coalescence is 122 ~𝜋𝛾𝑅 2 where 𝛾 is surface tension of Buller's drop and 𝑅 is Buller's drop radius. By 123 balancing surface energy to kinetic energy of the spore-drop complex 1/2 𝑚 124 𝑚 𝑣 0 2 , we obtained: 125

𝑣 0 2 + (1) 126
where 𝑣 0 is the ejection velocity and and are the mass of the spore and of 127 Buller's drop. Note that we have neglected viscous dissipation because ballistospory 128 operates in a regime of low Onhesorge number [START_REF] Liu | Asymmetric drop 433 coalescence launches fungal ballistospores with directionality[END_REF]. By simple algebra we 129 can express in the following form 𝑣 0 𝑈 + where 𝑈 3𝛼𝛾/ 2𝜌 𝑅 is a 130 velocity scale independent of Buller's drop radius; 𝑅 is the radius of a sphere with the 131 same volume as the spore, this is the "equivalent radius" of the spore, and we will call it 132 "spore radius" for short; 𝑦 𝑅 /𝑅 is the normalized Buller's drop radius, i.e. the ratio 133 of Buller's drop to spore radii; 𝜌 and 𝜌 are densities of Buller's drop and spore 134 respectively and 𝛽 𝜌 /𝜌 is the ratio of Buller's drop to spore densities. The 135 parameter 𝛼 accounts for the fraction of available energy dissipated when the spore 136 breaks apart from the hilum, the point of attachment between the spore and the sterigma 137 (FIG. 1). 138

Relaxation time of the spore-drop complex. The complex's relaxation time is 139 determined by the air drag that causes rapid deceleration and is well approximated by 140 (2) 142 where 𝜌 , 𝜌 and 𝜌 are the densities of the air, of the spore and of Buller's drop 143 respectively; 𝛽 ̅ 𝜌 𝑅 3 𝑅 3 / 𝜌 𝑅 3 𝜌 𝑅 3 is the density of air divided by the 144 density of the spore-drop complex; 𝜐 is the kinematic viscosity of air and 𝑅 is the radius 145 of the spore-drop complex, which we consider as a sphere with volume equal to the sum of the spore and drop volumes 𝑅 𝑅 3 𝑅 3 . When Buller's drop is considerably 147 smaller than the spore which is often the case, 𝛽 ̅ ~𝜌 /𝜌 . Following simple algebra we 148 obtain the Stokes time of the spore-drop complex in the following non-dimensional 149 form 𝜏 𝑇 𝑦 3 1 where 𝑇 Preserve. We collected opportunistically, taking any mushroom that appeared in 154 good shape, but focusing on gilled (not pored) fungi. We collected specimens of 155 eight morphologically distinct species (TABLE 2). 156

We also retrieved published data from the literature; these data were generated 157 from a different group of species (TABLE 3) and provide measurements of Buller's 158 drops and spores, but do not provide information about intergill distances (Pringle et al 159 2005;[START_REF]Biomechanics of spore discharge in the basidiomycota[END_REF][START_REF] Stolze-Rybczynski | Adaptation of the spore discharge mechanism in the Basidiomycota[END_REF]Fischer et al 2010b). In 160 this series of papers, the authors captured ballistospory from high speed video 161 microscopy and obtained Buller's drop and spore size (but not intergill distance) from 162 image analysis. We discarded the species for which measures of Buller's drop size was 163 inferred indirectly. When spore volume was available, spore radius was calculated as 164 𝑅 3 4 1/3 . When spore volume was not available directly, we calculated it from 165 spore length and width, assuming spores are prolate spheroids,𝑉 4 3 𝜋𝐿 𝑊 2 . Values of 166 spore densities range from 0.8 to 3.8 𝑔/𝑐𝑚 3 [START_REF] Hussein | Species traits and inertial deposition of fungal spores[END_REF]. 167 Data from the literature were used to estimate the efficiency parameter 𝛼 as 168 follows. For two species of basidiomycota (Auricularia and Sporobolomyces), for which geometry of the spore and drop was observed directly, velocity of ejection was 170 73% and 68% of the theoretical maximum [START_REF] Noblin | Surface tension propulsion of fungal spores[END_REF], yielding a fraction of 171 usable energy 0.73 2 0.53 and 0.68 2 0.46, hence 𝛼 50 5 %, consistent with 172 estimates presented in (Stolze-Rybczynski 2009), based on a different hypothesis about 173 the geometry of the adaxial drop. 174

Preparing specimens for morphometrics. On the same day mushrooms were 175 collected, a scalpel was used to separate caps from stems (FIG. 2A). Caps were left face 176 down for 8 to 12 hr on a piece of paper covered with aluminum foil in order to create 177 spore prints (FIG. 2B). Spore prints are generated when spores fall from gills and settle 178 directly underneath the cap. Spore prints reflect the morphology of each collected 179 specimen, and the location of stems and patterns of gill spacing are easily seen from a 180 spore print. To image spores, three small pieces of foil, each measuring approximately 1 181 mm 1 mm, were cut (i) from close to each stem, (ii) equidistant between the stem and 182 the cap edge, and (iii) from near the edge of each cap. Spores were washed off each foil 183 piece and suspended in a 0.01%vol solution of Tween 80. 15 𝜇𝐿 of each spore 184 suspension were immediately spread onto a glass slide and spores imaged. Microscope 185 slides were sealed with nail polish to minimize evaporation of the Tween solution and 186 prevent the movement of spores during imaging. To measure distances between gills, a 187 photograph of each cap's underside, with a ruler included in the photograph, was taken 188 immediately after spore printing using a Canon EOS400D. 189 Identification of species using DNA barcoding.-Tentative field identifications were 190 confirmed by sequencing the nuc rDNA ITS1-5.8S-ITS region (ITS barcode; Schoch et 191 al. 2012) of each mushroom. To generate DNA barcodes for each collected mushroom, 192 we extracted DNA with an NaOH extraction method modified from Wang et al (1993). 193 First, the tissues of each sporocarp were ground finely with a pestle in 40 μL of 0.5 M transferred to 495 μL of 100 mM Tris-HCl (pH 8) and centrifuged at 13 000 rpm for 196 another min. Next, the resulting supernatant was used as template for PCR. We 197 amplified the ITS using primers ITS1F (Gardes and Bruns 1993) and ITS4 (White et al. 198 1990) one or more than one spore, using Zeiss Elyra LSM 780 and Zeiss LSM 710 confocal 209 microscopes. Spores were not stained as all collected species proved to be 210 autofluorescent. The laser wavelength used to excite autofluorescence was 405 nm. The 211 average area S and average radius of spores 𝑅 𝑆/𝜋 of each species were then 212 calculated from images of between 155 1180 spores using an image analysis tool 213 implemented in ImageJ v.1.51. A single pixel's dimension in 𝜇𝑚 was calculated from 214 the microscope and the images converted to greyscale (8-bit or 16-bit). ImageJ was then 215 used to threshold each image and convert the greyscale to a binary image, highlighting 216 all the spores to be counted and using the measurement of a single pixel to calculate the 217 area of each spore as shown in FIG. 2C-D. Spores touching other spores were not measured, nor were particles smaller than 2 μm 2 . Particles bigger than 2 μm 2 were 219 identified either as spores or not by eye. 220

Image analysis to measure intergill distances.-To measure distances between gills, 221 we first identified the center of each cap by eye (FIG. 2E). We then drew between 6 and 222 10 circles (depending on the size of each specimen's cap) concentrically around the 223 center of the cap (FIG. 2E). We then used ImageJ v1.51 to open each picture, set pixel 224 length in mm using the image of the ruler included in each photograph, and convert 225 images to greyscale (8-bit or 16-bit). The Oval Profile plugin was used to obtain 226 greyscale profiles traced along each of the concentric circles drawn onto an image. 227

Profiles were sampled at 3600 equally spaced points along each circle. Next, the area of 228 each circle was measured to calculate its average distance from the cap center, and these 229 measurements were later used to convert the distance between gills from radiants to 230 mm. The greyscale profile obtained from ImageJ along each circle was imported into 231 Matlab R2017b (one example in FIG. 2F) and analyzed with the function Findpeaks. 232 Peaks in the greyscale image identify the centers of gills, which appear white in 233 greyscale images. Peaks that were closer than 0.3 o were discarded as noise. We visually 234 inspected data to confirm that minor peaks did correspond to gills. Finally, we 235 quantified gill thickness as the width of the peak, defined as the distance where grey 236 value drops half way below peak prominence, which is a measure of peak height. The 237 distance between two gills, d, was defined as the distance between their centers minus 238 the half-width of each of the two gills (see close up of two peaks in FIG. 2G). 239

240

RESULTS

241

Ejection speed. We first focused on the velocity achieved by the surface tension catapult, which depends on various physical parameters as well as on the dimensions of 243 the spore and Buller's drop. Let us first recapitulate the physical processes that lead to 244 spore launch. Buller's drop coalesces with the adaxial drop to power the surface tension 245 catapult; both drops are made from condensed water vapor and appear following the 246 secretion of hygroscopic substances by the fungus. When Buller's drop coalesces with 247 the adaxial drop, the resulting reduction of surface area provides the surface energy to 248 accelerate the spore. Because the adaxial drop is pinned to the surface of the spore, 249

Buller's drop accelerates along the axis of the spore towards its distal tip. Once the 250 moving drop reaches the tip of the spore, capillarity and contact line pinning decelerate 251 water, and its momentum is transferred to the spore. Momentum transfer causes the 252 force that breaks the contact between spore and sterigma, resulting in spore ejection 253 away from the basidium. 254

In this physical process, surface energy is converted in kinetic energy and by 255 using the physical principle of energy conservation we obtained equation ( 1) 256 (MATERIALS and METHODS): this equation predicts there will be a radius of Buller's 257 drop that maximizes 𝑣 0 (FIG. 3A). By zeroing the derivative in (1) we obtained the size 258 of Buller's drop that maximizes ejection speed: 𝑦 2𝛽 1/3 and when considering 259 spores with densities once to twice the density of water [START_REF] Hussein | Species traits and inertial deposition of fungal spores[END_REF], 𝛽 = 1 to 2, 260 this equation implied that at 𝑦 Buller's drop radius is comparable to the equivalent 261 radius of the spore 𝑅 ~1.26𝑅 to 1.59𝑅 (the gray shade in FIG. 3A marks all values of 262 𝑦 , for 𝛽 ranging from 1 to 2). Note that at 𝑦 control of the ejection speed is 263 robust, i.e. ejection speed becomes insensitive to small deviations from the exact value 264 of Buller's drop size. Buller's drop is generally assumed to scale with spore length 265 (Fischer et al. 2010b), and this scaling appears to hold for at least 13 species of 266 basidiomycetes, as shown in [START_REF] Pringle | The captured launch of a 447 ballistospore[END_REF], Stolze-Rybczynski (2009), Stolze-to calculate spore radius 𝑅 pointing to 𝑦 𝑅 /𝑅 ~ 0.32 0.08 where we 269 report the best fit through the data and the mean residual. 𝑦 are represented 270 in cyan on the horizontal axis in FIG. 3A. Note that only three out of their 13 species 271 were gilled mushrooms (TABLE 3), but these three species lined up with the rest of the 272 data and showed no clear departure from the rest of the collected data (FIG. 3B). 273 Ballistic range. In order to understand how far a spore-drop complex travels after 274 launch, we analyzed the timescale 𝜏 over which the spore-drop complex decelerates to 275 rest, or relaxation time (see MATERIALS AND METHODS). We found that after 276 discharge, spores travel horizontally a distance 277 𝑥 𝑣 0 𝜏 278 with 𝑣 0 and 𝜏 from equations ( 1) and ( 2). Next, a spore stops abruptly and starts 279 to sediment vertically, out from beneath a pair of facing gills, following a trajectory 280 commonly known as a "sporabola" (FIG. 4A). 281 Maximum spore packing. In order to successfully escape the mushroom, a spore must 282 travel away from its basidium a distance 𝑥 into the clear space between gills before 283 settling downwards; it must travel far enough to avoid entrapment within the basidia 284 and spores underneath it. If 𝑥 is in fact dictated by this criterion that ensures spores 285 escape the gills without sticking to tissue underneath, and assuming all gills are 286 reproductive, then the distance between two facing gills, 𝑑, should be at least twice 𝑥, 287 hence 𝑑 2𝑥. To pack as many spores as possible within a mushroom and avoid 288 inefficient empty spaces, the distance between gills must be close to this minimum 289 value: 290

𝑑 2𝑥

where 𝑥 𝑣 0 𝜏. Maximum packing was first suggested by Buller (1909), and 292 this relationship appeared explicitly in more recent literature (Stolze-Rybczynski et al. 293 2009). Plugging in the values of 𝑣 0 and 𝜏 given by equations ( 1) and ( 2) we obtain: 294 295

𝑦 2 𝑦 3 𝛽 1/2 𝑦 3 1 2/3 𝑑 2𝑈𝑇 297 (3) 296
where U and T are the velocity and time scales derived in the previous two paragraphs 298 and summarized in TABLE 1 and𝑦 is the normalized Buller's drop radius, we add 299 the suffix .

to stress that this prediction is valid at maximum packing. Equation ( 3) 300 predicts the relationship between three variables: nondimensional Buller's drop radius 301

𝑦

, spore radius 𝑅 (𝑅 is contained in the expressions for U and T summarized in 302 TABLE 1) and intergill distance 𝑑 -at maximum packing. Hence given two of these 303 three variables, equation (3) predicts the third, assuming maximum packing and given 304 the values of the parameters (spore density, energy conversion efficiency). For example, 305 for any combination of spore radius and intergill distance, equation (3) predicts the 306 optimal radius of Buller's drop that achieves maximum packing. We solve Equation (3) 307 numerically and show the result for he normali ed radi s of B ller s drop, 𝑦 , in 308 FIG. 4 color-coded from 0 (cyan) to 5 (black) for different combinations of intergill 309 distances and spore radii using parameters listed in TABLE 1. At each point in the 310 phase space defined by spore radius and intergill distance, the color represents the value 311 of normalized Buller's drop radius that achieves maximum packing. Symbols and error 312 bars represent our own data, described, analysed, and discussed below. morphologies (see MATERIALS and METHODS) for eight wild mushroom species 315 (TABLE 2). While spore size varied from species to species, spores within a single 316 mushroom cap were considerably more similar and there were no consistent pattern of 317 variation, e.g. spores were not larger nearer the center or edge of a cap) (FIG. 5). 318 Moreover, the average intergill distance also remains constant with distance from the 319 center of the cap. The phenomenon is driven by the appearance of secondary gills 320 (lamellulae) towards the edge of a mushroom cap; as two gills with origins at the stipe 321 diverge, often a secondary gill will appear. The appearance of secondary and tertiary 322 gills keeps intergill distances constant. The only species in our dataset with variable 323 intergill distances was an unidentified species of Russula, which did not produce 324 secondary gills [START_REF] Phillips | Mushrooms of North America[END_REF]Fischer and Money 2010). Among measured species, 325 intergill distances varied from about 0.25 mm to 1.5 mm (FIG. 6), but there was no 326 obvious correlation between intergill distances and the size of the mushroom cap. 327

To ensure maximum packing in real species, we predicted Buller's drop radius 328

should be 23% to 50% of spore radius. Here we used equation ( 3) and data about 329 spore radius and intergill dis ance o predic B ller s drop radi s a ma im m packing. 330

We computed average and standard deviation of spore radius and intergill distance 331 across single individuals and placed these data on the phase space generated by our 332 hypothesis for maximum spore packing (symbols and error bars in FIG. 4B). While 333 most parameters of our model are known (TABLE 1), 𝛼 and 𝜌 are not. In order to 334 understand how the model depended on parameters, we derived an approximate formula 335 from equation (3), showing that in our model he op imal radi s of B ller s drop 336 depended only weakly on these free parameters. Indeed, when 𝑦 1, we could 337 expand equation (3) to leading order, i.e. neglecting the smallest terms in the equation. 338In equation ( 3 This compact formula explicitly connected the radius of Buller's drop at maximum 341 packing with the parameters of the problem. Particularly, it showed that Buller's drop 342 radius at maximum packing depended on 𝛼𝜌 1/2 . For the range of variation reported 343 in the literature (see TABLE 1), 𝛼𝜌 0.4 𝑔/𝑐𝑚 3 to 1.9 𝑔/𝑐𝑚 3 , and plugging these 344 values as well as data collected in this study in our model, we predicted that at 345 maximum packing 𝑦 50% to 23%. Importantly, the observed value of 𝑦 346 from the literature is within this range of variation: ⟨𝑦 ⟩ ~ 32%, which supports 347 the hypothesis of maximum packing. The value of the fitting parameters for which the 348 predicted Buller's drop radius for maximum packing averaged over our collected data 349 fits the observed value, 𝑦 ⟨𝑦 ⟩ = 32% is 𝛼𝜌 0.9 𝑔/𝑐𝑚 3 . For 350 the eight species analyzed in this study, we found a standard deviation of 12%. A 351 da ase of B ller s drop radi s, spore radi s, and intergill distance measured on the 352 same specimen is needed to further confirm the hypothesis of maximum packing. 353

Scaling of Buller's drop radius with intergill distance. Buller's drop is often 354

assumed to scale proportionally with spore size (Fischer et al. 2010b), and published 355 data shown in FIG. 3B corroborated this assumption (although note that only three out 356 of 13 species in these published data correspond to gilled mushrooms). However, the 357 assumption appears at odds with the prediction for maximum packing because equation 358 (4) implies that Buller's drop radius scales linearly with intergill distance 𝑑 and with the 359 inverse square root of spore radius 𝑅 . To resolve the paradox, intergill distance must 360 increase with increasing spore radius in the following way: 𝑑~0.32 8𝛾𝛼𝜌 27𝜌 2 𝜈 2 𝑅 3 365 where we have simply plugged 𝑅 ~0.32𝑅 into equation (4). In order to convincingly 362 prove or disprove this relationship, further data monitoring spore, drop, and gill 363 morphologies, as well as spore density and ejection velocity, are needed. 364 366 DISCUSSION 367

The intricate morphologies of gilled mushrooms are hypothesized to maximize 368 surface to volume ratios, an adaptation enabling the maximum packing of spores within 369 a minimal investment of biomass. The hypothesis requires intergill distances to be 370 exactly twice the horizontal range of an ejected spore: an ejected spore must both clear 371 its natal gill and avoid lodging into the gill across from it, and assuming gills are 372 crowded together as closely as possible (to efficiently use the space within a mushroom 373 cap), the spore will be finely tuned to reach midway between facing gills. Intergill 374 distances may be shaped by the reach of a spore, but because spore range is dictated by 375 he dimension of B ller s drop and i s densi y relative to the dimension and density of 376 the spore, the three parameters in ergill dis ance, spore si e, and B ller s drop si e 377 emerge as highly interdependent in the context of maximum packing. We find that 378 intergill distances and spore sizes from empirical observations populate a region of 379 phase space where our model predicts he radi s of B ller s drop enabling ma im m 380 spore packing ranges from 23% to 50% of spore radius. Previously published data 381 [START_REF] Pringle | The captured launch of a 447 ballistospore[END_REF][START_REF]Biomechanics of spore discharge in the basidiomycota[END_REF][START_REF] Stolze-Rybczynski | Adaptation of the spore discharge mechanism in the Basidiomycota[END_REF]Fischer et 382 al. 2010b) suggest B ller s drop radi s scales i h spore dimensions as 𝑅 ~32% 𝑅 383 and support the hypothesis of maximum packing. A further prediction of our model is that he linear scaling of B ller s drop and spore radii implies in ergill dis ance m s 385 scale as 𝑅 3/2 within an optimally packed mushroom. 386

The fungi used in our analyses (FIG. 3A) do not operate at maximum ejection 387 speed; the velocity of ejection for these spores remains well below the maximum. 388

Whether other species operate at maximum ejection speed remains to be elucidated. 389

This result may not be surprising because robust control of ejection speed may not 390 contribute an obvious selective advantage. By contrast, the maximum packing of spores 391 is expected to contribute significantly to the fitness of an individual. 392

More data are needed: our conclusions are based on a total of 21 species. 393 However, the data collected to date are consistent with the hypothesis of maximum 394 packing, confirming B ller s drop radi s is likely to be finely tuned to control range and 395 the spore away from its parent. The size of Buller's drop, together with material 480 parameters, determines the speed of spore discharge. Image adapted from 481 [START_REF] Webster | Ballistospore discharge in Itersonilia 464 perplexans[END_REF]. 482 483 1) predicts discharge speed. We represent the 499 normalized speed 𝑣 0 /𝑈 as a function of 𝑦 defined as the ratio of Buller's drop radius 𝑅 500 divided by spore radius 𝑅 (red line). The thickness of the red line represents predicted 501 speed for 𝛽 ranging from 1 to 2, where 𝛽 is the ratio of spore to drop density. B. Linear 502 scaling be een spore radi s and B ller s drop radi s from da a p blished in he 503 literature (see TABLE 3). Spore radius is calculated as described in the main text. 504 505 (Buller 1909). Maximum 510 packing implies that spores initially travel a distance 𝑥 𝑣 0 𝜏 to reach the midpoint 511 between two facing gills separated by 𝑑 2𝑣 0 𝜏 with 𝑣 0 and 𝜏 given by equations (1) 512 and (2). B. Prediction for normalized Buller's drop radius at maximum packing, 𝑦 , 513 obtained by numerically solving Equation (3) with the parameters listed in Table 1. 514

𝑦

is color coded from 0 (cyan) to 4.4 (black); white corresponds to the value 515 〈𝑦 〉 32%. Symbols correspond to data of intergill distances and spore radii 516 measured from 8 species collected by us in Michigan and Wisconsin (see Figure 4). 517

Each symbol is linked with a species as in Figure 5, 6 and Table 2. The optimal Buller's 518 drop radius for the 8 collected species is 𝑦 ~0.32 0.12, where we report average 519 standard deviation. The free parameters 𝛼 and 𝜌 are chosen within their range of 520 natural variation. The prediction varies little across the range of variation of 𝛼 and 𝜌 . 1 Webster and coauthors estimated that Buller's drop contain about 1% in mass of mannitol and sucrose [START_REF] Webster | 468 Manni ol and he oses are componen s of B ller s drop[END_REF], hence 𝛾 and 𝜌 are well approximated by the surface tension and density of water.

Ta 1

2 9,

 2 𝜈 is the air kinematic. 150 Data collection and published data.-Between 15 17 Sep 2017 we collected 151 mushrooms from the Upper Peninsula of Michigan. On 15 Oct 2017 we collected 152 mushrooms from the University of Wisconsin-Madison Lakeshore Natural 153

  follo ing PCR pro ocols o lined in Whi e e al. (1990) modifed for L cigen s 199 EconoTaq Plus Green 2x Master Mix (Lucigen Corp., Middleton, Wisconsin). 200 Amplified products were cleaned and then Sanger sequenced by Functional Biosciences 201 (Madison, Wisconsin). Sequences were deposited in GenBank under accession numbers 202 MK829236-MK829244. Two specimens, one Mycena (MK829242) and one Russula 203 (MK829243), could not be confidently identified to species despite their barcode data. 204 Microscopy and image analysis to measure spore geometry. 205 Images of spores were taken using microscopes housed at the Newcomb Image Center 206 at the University of Wisconsin-Madison. Spores were imaged either individually or in 207 groups (FIG. 2C), depending on whether a particular microscope's field of view housed 208

Figure 1 .

 1 FIGURE LEGENDS and FOOTNOTES 472

Figure 2 .Figure 3 .

 23 Figure 2. Collection and analysis of gill spacing in wild mushrooms. A. An image of 484

Figure 4 .

 4 Figure 4. Mushroom cap morphology and the maximum packing of spores. A. From 506

  522

Figure 5 .Figure 6 .

 56 Figure 5. Spore size does not vary across a single mushroom cap. Gray thin lines mark 523

  Figure 5 C ick he e acce /d ad;Fig e;fig e5. df

Table 1 .

 1 List of parameters and their estimated or measured values from the literature.

	Parameter		Symbol	Value	Reference	used in Fig 3
	Air density		𝜌	1 𝑘𝑔/𝑚		1 𝑘𝑔/𝑚
	Spore density		𝜌	0.8 to 3.8	(Hussein et	1.8 𝑔/𝑚
				𝑔/𝑚	al. 2013)
	Buller's drop		𝜌	1 𝑔/𝑚	same as	1 𝑔/𝑐𝑚
	density				water 1
	Buller's drop		𝛾	0.07 𝑁/𝑚	same as	0.07 𝑁/𝑚
	surface				water 1
	tension
	Efficiency of		𝛼	(50 ± 5)% (Noblin et al.	0.5
	energy				2009)
	conversion
	Spore to		𝛽 𝜌 𝜌 ⁄
	Buller's drop
	density ratio
	Air to Buller's		𝛽 ̅ 𝜌 𝜌 ⁄
	drop density
	ratio
	Reynolds	𝑅𝑒	𝑅 𝛾 𝜈 2 𝜌 ⁄
	number
	Reynolds	𝑅𝑒	𝑅 𝛾 𝜈 2 𝜌 ⁄
	number
	Intergill		𝑑
	distance
	Spore radius		𝑅
	Buller's drop		𝑅
	radius
	Normalized		𝑦 𝑅 𝑅 ⁄
	Buller's drop
	radius
	y at maximum		𝑦 𝑐𝑘
	packing
	Time scale	𝑇 2𝑅 2 9𝜈𝛽 ̅ ⁄
	Velocity scale

Table 2 .

 2 List of collected species, collection location, number of spores imaged and analyzed, corresponding symbol used in Figures.

	Ta	2
		Collected species	Location	No. spores analyzed	Symbol
		Camarophyllus borealis Marquette County,	231
			Michigan
		Cortinarius caperatus	Marquette County,	1180
			Michigan
		Amanita lavendula	Marquette County,	155
			Michigan
		Armillaria mellea	Marquette County,	301
		complex (A)	Michigan
		Armillaria mellea	Marquette County,	257
		complex (B)	Michigan
		Mycena sp.	UW-Madison	530
			Lakeshore Natural
			Preserve, Wisconsin
		Russula sp.	UW-Madison	1053
			Lakeshore Natural
			Preserve, Wisconsin
		Galerina marginata	UW-Madison	1159
			Lakeshore Natural
			Preserve, Wisconsin

Table 3 .

 3 Da a of B ller drop radi and pore radi from he li era re From left to right: Name of species, structure bearing spores B ller drop radi RB, spore length LS, spore width WS, spore volume VS, spore equivalent radius RS, reference; species that produce gilled mushrooms are in boldface.

	Ta	3
		Species (order or class)	Spore-	RB	LS	WS	VS	RS	Refernce
			bearing	(𝝁m)	(𝝁m)	(𝝁m)	(𝝁m)	(𝝁m)
			structure
		Trametes versicolor	Pores	0.6	-	-	6.4	1.15	Fischer et al. 2010
		(Polyporales)
		Aleurodiscus oakesii	Smooth,	5.2	23.2	17.0	-	18.86 Pringle et al. 2010
		(Russulales)	discoid
		Itersonilia perplexans	Yeast	4.9	15.1	12.8	-	13.52 Pringle et al. 2010
		(Tremellales)
		Tilletiopsis albescens	Yeast	3.7	13.4	4.4	-	6.38	Pringle et al. 2010
		(Exobasidiomycetes,
		incertae sedis)
		Laccaria amethystina	Gills	1.5	8.1	8.1	-	8.1	Stolze-Rybczynski
		(Agricales)							2009
		Stereum hirsutum	Pores	1.2	7.2	3.0	-	4.02	Stolze-Rybczynski
		(Russulales)							2009
		Xerula radicata	Gills	3.7	16.8	13.6	-	14.59 Stolze-Rybczynski
		(Agaricales)							2009
		Gymnosporangium juniperi-	Gelatinous	5.2	20.0	15.1	-	16.58 Stolze-Rybczynski
		virginianae	telial horns						2009
		(Pucciniales)
		Tilletia caries	Host tissue	5.2	21.4	7.6	-	10.73 Stolze-Rybczynski
		(Tilletiales)							2009
		Sporobolomyces	Yeast	3.8	11.5	7.9	-	8.95	Stolze-Rybczynski
		salmonicolor							2009
		(Sporidiobolales)
		Auricularia auricula	Smooth,	3.1	12.9	7.8	-	9.22	Stolze-Rybczynski
		(Auriculariales)	jelly-like						2009
		Polyporus squamosus	Pores	2.6	14.0	5.4	-	7.41	Stolze-Rybczynski
		(Polyporales)							2009
		Armillaria tabescens	Gills	1.5	6.8	6.1	-	6.32	Stolze-Rybczynski
		(Agaricales)							2009
		Clavicorona pyxidata	Smooth,	1.2	4.7	3.5	-	3.86	Stolze-Rybczynski
		(Russulales)	coral-like						2009


Laurent Le Gendre

Claire Le Paven-Thivet

Jacques Pinel

Didier Fasquelle

Jean-Claude Carru

François Cheviré

Franck Tessier

R Marchand

Lanthanum titanium oxynitride thin films

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