Downloaded by guest on September 30, 2021 a jce lwyadmypyial pntelyro tl i before air are still fungi of layer Sordariomycete the span some physically may of and whereas slowly spores 11), ejected thread-like (7, fragments long dispersed the easily small, dis- launch, into subsequently to associate during spores allowing cohesion but mass and projectile spore increasing appendages promote example, For sheathes species. mucilaginous fungal disparate in range a through pass thickness they (10). body of that fruiting air necessary fungal still is of it layer dispersive particular, enter boundary travel to In must body Spores flows. fruiting (8). originating air ejected hole the are a from spores each enough whereupon and far reached, within apex is pressure the pressure turgor at critical opens the a of maturity until species At climbs contain- many sac sacs (9). fluid-filled of spores asci, the spores from ing patches flows, ejected between air are dispersal fungi dispersive allow ascomycete reach and To flows 8). air spores (7, by Sexual carried habitat. suitable be of can patches disjoint readily between move be not can another selec- over of force optimum (6). the quantified physical of one strength for the 5), tion (4, allocation (3), behavior resource (2), and morphology physio- organism of be of that features can evolution diverse optimization clear in the of seen signatures also constrain although Similarly, will is shape. trade-offs body it ecological improve fitness, and may individual logical minimization of strong drag aspects in although leaves some tree However, tulip (1). streams, of flowing furling winds rapidly coordinated in precisely rocks the to or cleave that nymphs Mayfly of M spores fungal | optimization biological | hydrodynamics optima. physical the of toward of selection measurements group future of mega-diverse for forces framework a a in provides and shape species orga- explaining an for as shape principle optimization spore nizing biomechanical to uses study determinants Our shape evolution. other of and contributions a minimization relative within the drag measure shapes we spore family ascomycete of through single history prediction in evolutionary this the ejection confirm reconstructing of and imaging launch, spore high-speed predict of we shapes speed spore the the From min- size. the their of for 1% drag within possible remain imum to constrained are spores that shows of phylogeny real a from with we taken shapes shapes these spore hypothesis minimum compare volumes—and this of prescribed shapes—shapes for test drag spore To optimal bodies. nearly calculate the fruiting numerically in surrounding range their air maximize to still shaped hypoth- are We spores launch. after that rapidly esize decelerate to flows, spores air causes dispersive also of it out sedimentation can slowing drag microscopic by this their transport although aid and eject of drag, fluid Because great must flows. experience spores air size, fruiting fungi dispersive surrounding reach air ascomycete to still bodies nearly of of millimeters spores several through launched forcibly The Roper Marcus have shapes fungi drag-minimizing ascomycete of spores launched Explosively w.nsog/ci/di/1.03/pnas.0805017105 / 10.1073 / doi / cgi / www.pnas.org dtdb lxnr .Coi,Uiest fClfri,Bree,C,adapoe etme ,20 rcie o eiwMy2,2008) 23, May review for (received 2008 8, September approved and CA, Berkeley, California, of University Chorin, J. Alexandre by Edited 02138; MA Cambridge, colo niern n ple Sciences; Applied and Engineering of School utpeidpnetyeovdaattosehnespore enhance adaptations evolved independently Multiple must and landscapes heterogeneous highly on grow fungi Most nldn h temie hpso atsimn s and fish fast-swimming of shapes streamlined the drag, including minimizing for adaptations visible have organisms any a,b,1 ahlE Pepper E. Rachel , ersoatetrasperma.ByNeurospora c mta lnst the to clings that mm ∼1 0 pce.Oranalysis Our species. >100 ihe .Brenner P. Michael , b eateto raimcadEouinr ilg,and Biology, Evolutionary and Organismic of Department a n nePringle Anne and , hc eed ntedensity, the on depends which oigta uigeeto,tewr ftetro rsue(pV pressure turgor the of where work by the spore ejection, launched or during a that of size, noting speed shape, the estimate spore can We to identity. insensitive species itself is spores unappendaged iigtesoeavelocity a spore the giving togyafce yarflw u ieascn e fuprbounds m·s upper of 9–35 set second between a ranging give but flows similarly air are by affected distances strongly travel Spore walls. ascus and spore between rcinwti h su n hwta iiiaino frictional of minimization the that in show and shapes: ascus such the for within the friction selection within of (SI losses evidence information frictional supporting no minimizing find by We ejection ascus. of speed the (15). large projectiles a fungal for and data plant speed of supported launch range is of ejection collection of remarkable Vogel’s mode by this with species among conserved n ciiyo h soye rsn 1) eexpect we (14), U present osmolytes the of concentration the activity ascus by and the allowed overpressure that largest assuming the Furthermore, to size. inflated is or shape its on not but hc Eq. which ieacshsbe siae obe a to estimated within been pressure has turgor ascus ripe The measurements. indirect previous by constrained is but spore, ascomycete unappendaged ejected, singly ejection. siae ieol pe onson bounds upper only give estimates h au of value the 08b h ainlAaeyo cecso h USA the of Sciences of Academy National The by 2008 at © online information supporting 0805017105/DCSupplemental. contains article This hsatcei NSdrc submission. 1 direct PNAS a is article This interest. of conflict no declare A.P. authors and The M.P.B., M.R., and and data; M.P.B., paper. analyzed M.R., A.P. the and wrote research; M.P.B., designed M.R., A.P. research; and performed M.P.B., A.P. R.E.P., M.R., contributions: Author osiuu dpain a e esae omxmz range. maximize to shaped these be lacking yet spores may how adaptations consider We conspicuous (12). ascus the leaving fully jcin n hncnr hspeito hog high-speed through spore prediction of in this ejection speed of confirm the imaging predict then we real and shapes comparing ejection, optimal By these (13). with animals spores fast-swimming different and man-made very as projectiles such are macrobodies of forms shapes understood well drag-minimizing the from These spores. relevant sizes real and speeds to flow drag- of range construct the over to shapes optimization minimizing numerical use We minimization. owo orsodnesol eadesd -al [email protected] E-mail: addressed. be should correspondence whom To 0 ipepyia ruetsosta h pe feeto of ejection of speed the that shows argument physical simple A nod()hpteie htsoe r hpdt maximize to shaped are spores that hypothesized (7) Ingold h pe feeto a o endrcl esrdfrany for measured directly been not has ejection of speed The eaayea niepyoeyof phylogeny entire an analyze We ob osre cosalacmctswt xlsv spore explosive with ascomycetes all across conserved be to , V PNAS stevlm fasoe scnetdt iei energy, kinetic to converted is spore) a of volume the is 1 U ol give would b 0 sntkon h supinta anhsedis speed launch that assumption the known, not is eebr3 2008 , 30 December ersoatetrasperma. Neurospora U U 0 c −1 eateto hsc,HradUniversity, Harvard Physics, of Department ) 0 = pedxII Appendix = 1,1,1) oehls,although Nonetheless, 18). 17, (14, ρ 02 m·s 10–20  ≈ 2 p ,20kg·m 200 1, p ρ o.105 vol. U 0 uhseisfrdrag for species such >100 , ≈ 0 hyngettefriction the neglect they , −1 .803Ma(6;from (16); MPa 0.08–0.3 eaayeamdlfor model a analyze we www.pnas.org/cgi/content/full/ uhpressure-based Such . o 52 no. −3 7,o h spore the of (7), n thus and p, 20583–20588 [1] ,

EVOLUTION APPLIED MATHEMATICS Fig. 1. Real and drag-minimizing spore shapes. (A) Minimal drag shapes for Re = 0.1 (darkest), 1, 10 (lightest). The arrow points in the direction of flight and −1 spores are axisymmetric about this direction. (Left) Spore dimensions are given in physical units (assuming U0 = 2.1 m·s ). (Right) All spores are scaled to have equal volumes. The near fore-aft symmetry is not imposed (20). Comparison of minimal-drag shapes with Astrocystis cepiformis (B), N. crassa (C), islandica (D) spores. (Scale bar: 10 μm.) Like surface ornamentations (Fig. 2 and Table 1), rounding of spore apices only mildly increases drag. [B, reproduced with permission from ref. 38 (copyright 1998, British Mycological Society); C, reprinted from Experimental Mycology Vol 14, Glass NL, Metzenberg RL, Raju NB, Homothallic Sordariaceae from nature: The absence of strains containing only the a mating type sequence, 16 pp, 2008, with permission from Elsevier; D, reproduced with permission from ref. 39 (copyright 2006, British Society).]

losses favors oblate shapes. Such shapes are not seen in real spore adaptations such as appendages or septa, which may have different shape data (see Fig. 3A). ejection dynamics (see SI Appendix, Section III). When possible, With launch speed conserved, maximizing the spore range rejected species were replaced by species within the same genus requires minimizing the ratio of drag to mass. To see this, bal- lacking any such adaptations. To provide the greatest possible ance the fluid drag −ζu, where u is the speed and ζ is the Stokes spread of spore sizes, additional large-spored species were added drag coefficient, which is approximately independent of speed for to our analysis until every 10-μm interval of volumetric radii sufficiently small projectiles (10), against the inertia of the spore: contained at least five species. We find that the drag on a spore can be calculated by approxi- du mU mu =−ζ u ⇒ x ≈ 0 , [2] mating its shape by an ellipsoid of matching size and aspect ratio. dx max ζ This applies even to spores with pointed apices, or surface deco- rations such as warts, spines, or ridges, because at small Reynolds where x is the distance traveled and m is the spore mass. This numbers such shape features do not strongly contribute to drag. formula can only be used to infer distance traveled from spore Four of 102 species in our phylogenetic survey have decorated R shape for the smallest spores (for which the Reynolds number, e, surfaces. To show the validity of the approximation, we compare defined below, is <1). Although there is no formula relating dis- directly the computed drag on one smooth-spored species and R tance traveled to spore shape for larger spores ( e > 1), it is clear, three species of exceptionally rugose spores with the computed nevertheless, that minimization of the drag-to-mass ratio will max- fluid drag on the corresponding ellipsoids. Rugose spore silhou- imize the distance traveled by the spore during the early part of its ettes were traced from electron micrographs: an image of the trajectory. In the SI we describe the three assumptions necessary sculpted species Aleuria aurantia was provided by J. Dumais and for computing and then minimizing the drag on an ejected spore. images of the ridged species Peziza baddia and of the coarsely warty Peziza vacinii were taken from ref. 22, whereas the silhouette of Results and Discussion the smooth spored species Neurospora crassa was traced from a Shapes that minimize the ratio of drag to mass can be efficiently optical micrograph provded by N.B. Raju (see Fig. 2). The true computed by using algorithms developed for the design of airplane spore shape was approximated by revolving the silhouette to form wings (19): our implementation iteratively modifies the projec- an three-dimensional body. The computed drag on the spore is tile shape, at each iteration solving numerically for the flow field compared with the drag on its approximating ellipsoid in Table 1. around the shape and then for an adjoint flow field to determine the shape perturbation that gives the largest possible reduction in drag. The algorithm for computing minimum drag shapes is described in ref. 20 and summarized in the Materials and Meth- ods. Optimal shapes are parameterized by the Reynolds number Re ≡ U0a/ν, where ν is the kinematic viscosity of air, and a is the volumetric radius (the radius of a sphere with matching volume) of the spore. Assuming that U0 does not vary between species, the Reynolds number provides a dimensionless measure of the −1 spore size. For example, U0 = 2m·s implies Re = a/(9 μm). In Fig. 1, minimal drag shapes are displayed for Re ranging from 0.1 up to 10, and compared with real spore shapes. To determine whether the species shown in Fig. 1B are rep- resentative of ascomycetes with explosive ejection, we compared Fig. 2. Silhouettes of highly rugose spores (red curves), traced from optical or electron micrographs and approximating ellipsoids (black curves) con- real spores with optimal shapes in over 102 species by using a pre- structed to have the same volume and aspect ratio for N. crassa (A), A. viously created phylogeny of the phylum (21). Shape aurantia (B), P. baddia (C), and P. vacinii (D). (Scale bar: 5 μm.) Although data were gathered from 77 species whose spores are launched the approximating ellipsoids do not precisely capture the shape of the spore, forcibly and individually. We rejected species whose spores have they well-approximate the drag on the spore (see Table 1).

20584 www.pnas.org / cgi / doi / 10.1073 / pnas.0805017105 Roper et al. Downloaded by guest on September 30, 2021 Downloaded by guest on September 30, 2021 with melanchora , (Pertusaria spores 40 h pr weight spore the onaylyro erysilarcign otefutn body: fruiting the size to of clinging body air fruiting still a nearly for of Spores layer through them constraints. boundary carry to a inertia mechanical their for fluid size minimum a from exceed must follow bounds Both ieand size 0cm·s 20 m·s ftedseso fsoesaemyrsl rmcranspecies assumed certain this from than result larger speeds may Some for shape optimized 3B). spore being (Fig. of shapes dispersion optimal the and of real of agreement good give surface its and spore the encloses ornamentation. completely as approximated that accurately such ellipsoid more an is warts drag by the dense spores, less coarse, warty slightly these with is For but spores spores, for rough and accurate smooth high for with spore accuracy the (<1%) on drag the captures approximation shape The assuming ratio, aspect and speed volume launch drag same the the with having shapes ellipsoid spore an real on for drag the of Comparison 1. Table Roper ag.Frternet xedtecaatrsi onaylayer boundary characteristic setting on the obtain, exceed we to thickness range the For (Eq. range. formula Stokes the so ftesoe,i eywa 2) otearmyb rae sstill, as treated be may launch air of the spores, direction so smallest the (23), the weak in for very i.e., is boundary, spores, the the of to normal flow air umne yrno i ost esatrdoe h petri the over scattered be to that flows significantly Supposing been air dish.) have random ranges by whose augmented spores expect Fig. we (see piles contrast, by in dishes lying Petri spores the for 4C within only flows distances air travel by measuring controlled In range we of bodies. 18), enhancement (17, fruiting for measurements among range speed previous launch to contrast in variability the mated eiavacinii Peziza baddia Peziza aurantia Aleuria crassa Neurospora Species ae npesr rsoerne() eue ihsedimag- high-speed used we (8), measure directly range to spore ing or pressure on based si thickness istic (see etr oaSoe rgmdl(Eq. model drag Stokes tra- a spore the to Fitting jectory spore. launched the of trajectory the showing Eq. ihn2%o h h pia setrto hscomparison This ratio. velocity launch aspect a optimal the assume to of the to conforming half us the to requires and of corresponds optimum, which 25% the phy- 0.46%, within of the within 1% are from within species species drag- drags the 102 than have of drag logeny two more Seventy-three 1% the shape. than between more minimizing Species no shapes. suffer optimal curves dotted with phylogeny the in ieof size rpotoaplatystroma, (Graphostroma dataset the in spores yteotnqiewa i osbyn h onaylyr For layer. speed boundary of the wind upward beyond an flows supported from air be lift the to weak heavy quite too often are the large by too are that spores However, i.3A Fig. i.3A Fig. ymauigtedsacstaee yeetdsoe eesti- we spores ejected by traveled distances the measuring By eas u te jcinsedi esta rvosestimates previous than less is speed ejection fitted our Because ,wihaelkl ohv olwdietcltaetre.(By trajectories. identical followed have to likely are which ), ,wihmthstesz ftelretfril launched forcibly largest the of size the matches which μm, −1 2 .Oesqec a atrd(i.4A) (Fig. captured was One-sequence Methods). and Materials tal. et oifrluc eoiy ytaking by velocity, launch infer to ,wl-acigorpeitdeeto speed. ejection predicted our well-matching 4B), (Fig. a −1 anhvlcte nterne2.1(+1.4/ range the in velocities Launch Re. > lososcerupradlwrbud nsoesize. spore on bounds lower and upper clear shows also 2)w xetti onaylyrt aeacharacter- a have to layer boundary this expect we (24) oprsteapc ai and ratio aspect the compares 1.8 U √ .Ti scmaal otesz ftesmallest the of size the to comparable is This μm. 0 = ν 3 4 L/U π . m·s 2.1 pr rg NElpodda,n ro,% Error, nN drag, Ellipsoid nN drag, Spore a 3 wind ρ U U g Re 0 .122 3.7 0.6 0.3 0.3 2.23 2.04 1.73 4.04 2.31 2.05 1.73 4.03 0 ti eesr that necessary is it , −1 nlbrtr utrsof cultures laboratory in  ≈ L a ete eue odetermine to used be then be may 2)  . m ihntebudr layer boundary the Within mm. 0.1 2m ≈ ζ (if 1 m n idspeed wind and mm, 1 ≈ ·s −1 6π a ,w find we 2), a U othat so , = a = wind η 58 oe on nspore on bound lower a , ζ U Re 1 0 = μm). = ,then μm, ocnetbetween convert to , aafrtespecies the for data a 0cm ·s 20 6π Re < U aψμ 0  a .tetrasperma N.  = 9μU U − −1 = hr the where , 2 Re 1.24 ,w used we 1, 0 ρ wind .)m·s 1.1) . .vacinii. P. g oexceed to 0.7 U ≈ ±  wind 1/2 μm). 0.1), 0.01 −1 = ≈ mgna uoimcts ylo)(eaooyee)Ostropomycetidae, Leotiomycetes, Lecanoromycetidae. () (Lecanoromycetes) (black) (aqua) (yellow) Key Dothideomycetidae, Eurotiomycetes, drag. minimum (magenta) (Dothideomycetes) the (red) of Sordariomyceti- 1% (Sordariomycetes) dae, (green) within , species are (blue) ratio; symbols: curves aspect to phylo- dotted optimal by two the color-coded the displays species, curve between single black a The for (sub)class. size represents genetic and (length point ratio ratio Each number. aspect aspect Reynolds average against Spore the plotted spores. is real width) by with divided shapes optimal of Comparison r itd epciey nrf 6ad(stegenus the (as and 26 ref. groups in (under- these respectively, for hypogeous listed, genera are closed Representative cirrus, bodies. produce oily fruiting an that ground) as Pezizomycetes spores their on discharge and and with animals clades or dispersed within are insects nested that by of Sordariomycetes are on groups focus but We ejection. ejection, divergent forcible spore two lack by that demonstrated fungi well is maximization the within the lies in speeds dispersion launch large of limits. to range fitted leads the but launch ranges, 4 of measured Fig. angle in and shown speed are both perithecia velocities different launch 100 the of from counts Frequency substrate. the on area its 15 factor shape 3. Fig. rdcre ih xs steprcn rcino pce hs rgexceeds drag m·s whose 1-3.5 range species the of in fraction by cent possible per minimum the the is axis) right curve, (red osnu au rmmlil ult-ffi measures: quality-of-fit multiple from value consensus nvlmti ais n hnbtenbn tesldcrecrepnsto corresponds curve solid 10 (the width bins of between bins then bins and over radius, averaged ratios, volumetric aspect in spore real and optimal between differences esrso ult ffit. of quality of measures h feto eoigteslciecntan moe yrange by imposed constraint selective the removing of effect The 2) n h ubro prsi iewsetmtdfrom estimated was pile a in spores of number the and (25), μm ocbyeetdsoesae cosapyoeyo 0 species. 102 of phylogeny a across shapes spore ejected Forcibly ψ PNAS ,adtedse uv obn fwdh5 width of bins to curve dashed the and μm, ≈ −1 .5frasoeo egh31 length of spore a for 0.95 h pia setrtoin ratio aspect optimal The . % ohfisaecnitn ihaluc speed launch a with consistent are fits Both >1%. S eebr3 2008 , 30 December 1 bakcre,lf xs ie h u fsquared of sum the gives axis) left curves, (black (B o.105 vol. ) neec of Inference A sotie yuiga using by obtained is U 0 = aito in Variation D. n width and μm . m·s 2.1 o 52 no. Geopora U 0 )and μm) sn two using , −1 . 20585 and (A) S 2

EVOLUTION APPLIED MATHEMATICS Fig. 4. Experimental determination of spore ejection speed in N. tetrasperma. (A) Composite image of the flight of a N. tetrasperma spore at 63-μs intervals, including (slightly displaced) trajectory predicted from Eq. 2. The dotted curve shows the outline of originating perithecium. (B) Fit of spore trajectory (points) −1 to Eq. 2; giving launch speed U0 = 1.24 m·s . (C) Spore piles (outlined by solid curves) surrounding the perithecium (dotted curve). (Scale bar: 0.5 mm.) (D) Inferred launch speeds from spore prints of N = 100 fruiting bodies.

family Tuberales) in ref. 27. We collected spore-shape data for the where dt is any scalar increment of genetic change. The first terms type species of each of these genera, and added two additional in the left-hand sides of Eq. 3 represent the force of selection for species per genus, either drawn from the references or randomly shapes with an optimal aspect ratio α∗(u), and size u∗, whereas αi ui selected from the Cybertruffle taxonomic lists (28). dWt and dWt are independent Wiener processes, representing Spores from both groups of nonejected species have signifi- the fluctuating contributions of genetic drift and other shape con- cantly larger mean drag (one-tailed Wilcoxon signed rank test straints (33). We apply the model to a two-gene phylogeny of 44 P < 0.001) and fewer species within 1% of the drag minimum representative extant species (32). First, we find that differences in (ejected species, 73/102; insect dispersed group, 29/65; hypogeous spore size and shape between species are uncorrelated with genetic group 9/57) (see Fig. 5). The appearance of nonejected, spores with distances estimated from the phylogeny (r2 < 1.0 × 10−3, Mantel almost drag-minimizing shapes may result from the retention of test P = 0.09), suggestive of strong recent selection (see Materi- an ancestral shape in the absence of selection for a new optimal als and Methods). We therefore use shapes drawn independently shape; it is certainly for this reason that at least one species of from the invariant distribution of the stochastic process (Eq. 3) hypogeous ascomycete (Geopora) still ejects its spores forcibly, to fit the model to the real spore shape data and to obtain mea- albeit into the closed cavity of the fruiting body (29). sures of the relative contributions of drag minimization and drift 2 = By considering the coevolution of spore size and shape over to shape evolution. In particular, we estimate σα /2kα 0.267, so the reconstructed history of one family of fungi, we can deter- that in practical terms selection has dominated drift to constrain mine the extent to which drag minimization, as opposed to other spore aspect ratio to within ≈ 25% of the optimal value. Finally, constraints, has controlled evolution of spore shape. The Per- by comparing the fit of the strong selection model with the fit from tusariaceae are a cosmopolitan medium-sized family of lichen a drift-based reconstruction of the ancestral spore shapes (34) we species, most with forcibly ejected spores, and with the widest confirm that the covariation of aspect ratio with size in the family range of spore sizes of any ascomycete family (30–32). is much more likely to have arisen by selection than by chance (36) We model the evolution of spore shape (aspect ratio αi) and size (see Materials and Methods). (represented by ui = log Rei) within each Pertusariacean lineage by a stochastic process: Materials and Methods Calculation of Optimal Shapes. Shapes of projectiles that minimize drag for =− − ∗ + αi dαi(t) kα(αi α (ui))dt σαdWt prescribed Reynolds number and body volume were calculated by using a gradient descent algorithm, in which successive volume-preserving perturba- =− − ∗ + ui dui(t) ku(ui u )dt σudWt [3] tions were made to the shape of a suboptimal projectile (20). At each iteration

20586 www.pnas.org / cgi / doi / 10.1073 / pnas.0805017105 Roper et al. Downloaded by guest on September 30, 2021 Downloaded by guest on September 30, 2021 nequation: body an from the functions boundary of surface boundary the as the constructed to Deformations are multiplier Lagrange conservation. the volume and fluid, enforces momentum surrounding of the in conservation incompressibility enforce and respectively, that, multipliers Lagrange fasalprubto otesaeboundary, shape the to perturbation to small perturbation a arbitrary of Specifically, perturbation. an body, optimal to the a due inferred for then drag and fluid boundary, body in the change the calculated we window. plot the axes in The bound. rescaled this below been values have drag excess have spores hypogeous of 16% ( spores spores. ejected Sordariomycete forcibly the dispersed of for shape axes optimal logarithmic the and size—on shape same spore real the between drag in difference 5. Fig. Roper hp and shape oa rgo h oycnb ata igeitga vrteunperturbed the over integral single a as cast be can surface: body the on drag total h rttr fteo h ucinlrpeet h aeo okn fthe of velocity working adjoint of rate the the body, represents the functional on the force of the drag of term first The hn7%o ocbyeetdsoe,but spores, ejected More forcibly value. optimal of the 75% of than 1% within ( ratios drag-to-mass spores have region Pezizomycete shaded hypogeous and spores ejected with ∇· tal. et w jce n oeetdspores. nonejected and Ejected = w mesdi nifiiefli oan ecluaeteeffect the calculate we domain, fluid infinite an in immersed , ,sbett onayconditions boundary to subject 0, →− L[ ]= −∇ U + q ntefrfil,te h orsodn ml hnein change small corresponding the then far-field, the in  x ∂ δ + q L to B ∇· n η =− eoigteeetdspecies ejected the removing , ∇ · x σ u]dV 2 + w  · U ∂ =− α (x dS + α rvddta h don ed satisfy fields adjoint the that Provided )n. (x ρ (B + λ (u )  )    R ·∇ oprsno xesda o forcibly for drag excess of Comparison η

α 3 ∂ ∂ − (x dV 5 fisc ipre prsand spores dispersed insect of <45% )w (A) u n ) [w ∂ + ∂ iigtelclnra hf of shift normal local the giving −| w n oprsno xesdrag—the excess of Comparison · ρ (w) (∇· | (U + w ∂  = j λ n pressure and σ +  . ntedrag-functional: the on , 0 − dS w ntebudr fthe of boundary the on j ρ .Seiswti the within Species ×). . )∇ (u .platystroma G. u ·∇ j , insect- (×) and •) )u (q ) ed are fields from [4] [6] [5] λ ai aisapoiaeylnal with linearly approximately varies ratio n rf ssaedtriat yfitn h nain itiuino the of Pertusariaceae distribution the invariant by covered the sizes fitting (Eq. by process determinants stochastic shape as drift and Methods. Comparative a of images suf- of imaging sequence of focused h one 40 and However, spore. events focus. launched ejection dis- of 30 plane plane SZX-ILLB2-100 capture focal the to Olympus and leave ficed trajectory a to fps initial spores on 50,000 of the misalignment to mounted causes small up camera a of Even v7 rates scope. frame Phantom secting ejection and a day 40×, and using to tenth removed, up by the conidia of on righted, magnifications out were at imaged the carried dishes from Petri was light inoculation. imaging inocula- of after High-speed after deprived onward. were days imaging day 11 spore high-speed eighth taken for for were used used Photographs Cultures cultures remove tion. mycelia. to incubation applicator aerial of cotton-tipped and day illuminated a conidia and ninth with abraded light, the were incident On photography of day. print direction per the min control 10 to for container lid, a slotted in laid a stacks the with and sucrose stacked, 1% were dishes at Petri medium (37). crossing concentration in dishes Petri 10-cm-diameter in cultured ter), Speed. tetrasperma N. Launch of Measurement chord over search (see single-parameter ratios a width out to carrying and chord, associated the ecnte rt onteprubto ftesaebudr that boundary shape the shape: the of of perturbation volume the the fixing while down reduction drag write maximum achieves then can We iceiigtevlct n don eoiyfield velocity adjoint and Eq. velocity adjoint the and function discretizing equations Matlab flow (the Navier–Stokes routine step-size variable oyb h least-drag values the of set by small body a at shapes optimal from exact for interpolate of data we drag so and costly, ratio aspect computationally the is shapes Com- optimal Multiphysics. COMSOL of using putation by system discretized the solving and mesh, and elements, σ σ eeto nsae(hr oun (Eq. column) (third shape on (Eq. selection Eq. column) of (second selection distribution no stationary and column), (first selection Strong the from data shape criteria spore information to fitted Akaike Pertusariaceae models and evolution shape estimators for Parameter 2. Table nwhich in asinivratdsrbto 3) ota h pr hpso i species tip of matrices shapes covariance and spore expectation the with variables that random so independent are (35), distribution invariant Gaussian euneo hpsta h oypse hog orahteda minimum, drag the equation reach to an through passes giving body parameter the that A shapes preserved. of sequence is body the of hnb xrce rmteosre rtadscn oet ftespecies the of moments second and first observed the from extracted be then hr ehv defined have we where and k α u u 2 2 log −2 σ σ aiu ieiodetmtsfrec fteprmtr ntemdlcan model the in parameters the of each for estimates likelihood Maximum Re (bp/sub.) k (bp/sub.) (bp/sub.) u α Ic189 4.7143.30 141.07 128.95 AICc 2 2 u α α u u /2k /2k (0) (0) 1 /k ∗ oitreit ausof values intermediate to = α u α L λ .8 Eq. 0.38. a encoe oesr that ensure to chosen been has E

(p, Sri o 1,otie rmteFna eeisSokCen- Stock Genetics Fungal the from obtained 614, no. (Strain α α i i 2 PNAS u togslcinWa eeto ieselection Size selection Weak selection Strong q i

E ) 3 α IApni I Appendix SI α α ylna nt lmnso nusrcue triangular unstructured an on elements finite linear by (x 1.0118 128.50 131.81 119.70  hrfr ensa rsenUlnekpoeswith process Ornstein–Uhlenbeck an defines therefore n u i

u α u ) ora pr hp aa vrterneo spore of range the Over data. shape spore real to 3) i 2 − − − − −− spindle 0.127 0.278 0.267 1.17 eifrterltv tegh fda minimization drag of strengths relative the infer We ≡ i i

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EVOLUTION APPLIED MATHEMATICS data. Four dimensionless parameters must be estimated, and we choose these and the corresponding distribution for the pure Brownian process Eq. 9 can ∗ 2 2 = to be u , σα /2kα , σu /2ku and kα /ku (see SI Appendix, Fig. S4). be obtained by setting ku 0. A version of these formulae for the case The statistical support for the hypothesis of evolution of spores under when all lineages have the same ages (ti = tj for all i, j) is given in ref. 36. strong shape selection can be determined by comparing the fit to real spore Note that genetic and phenotypic distances between tip species will become shape data from model 3 with two other models for trait evolution: (i) a uncorrelated if and only if kut 1, where t is the smallest genetic dis- Brownian model (34), in which both u and α are unconstrained: tance between any pair of species, i.e., in the presence of strong selection. We measure t in units of the average number of substitutions per base pair, = 2 αi dαi (t) σα dW 2 t in the two genetic locii used in ref. 32. σu,α therefore have units of base pairs u du t = σ 2dW i [9] per substitution. Fitting these models to the observed species data requires i ( ) u t 2 estimation of (i) four or (ii) six parameters: the drift strengths, σα,u, the ances- and (ii) a Ornstein–Uhlenbeck process in which α is unconstrained, but there tral spore shape data [αi (0), ui (0)] and, in the case of process 10, the force of is selection on the size variable u: selection ku and optimal size u∗. The maximum likelihood estimators for all the parameters that feature = 2 αi dαi (t) σα dWt in the three models are given in Table 2. The Akaike Information Criterion 2 ui corrected for small sample sizes (AICc) allows the quality of fit of the three dui (t) =−ku(ui − u∗)dt + σ dW . [10] u t models to be compared (36), and shows that the strong selection hypothesis αi ,ui provides a significantly better fit to the real spore shapes data than either of dWt are independent Wiener processes, and t parameterizes the amount of genetic change from the common ancestor of the family. Distinct lineages the models in which there is no shape selection (last row of Table 2). co evolve from the ancestral state up until the time, t = tij of first diver- gence, so that shape and size data for the observed species have multivariate Gaussian distributions. Thus, if ti is the “age” of the the ith lineage then for ACKNOWLEDGMENTS. We thank Imke Schmitt, Jacques Dumais, and Eq. 10: François Lutzoni for making their data available to us, Jacques Dumais, L. Mahadevan, Nicholas Money, Andrew Murray, Sheila Patek, Frances Trail, Eαi (ti ) = αi (0) Dominic Vella, Howard Stone, and members of the Pringle lab for useful −k t −k t discussions and comments on the manuscript, and Ozymandias Agar for Eu (t ) = (1 − e u i )u∗ + e u i u(0) i i assistance with the spore launch imaging. This work was supported by fellow- = 2 cov(αi (ti ), αj (tj )) σα tij , ships from the Kodak-Eastman group and the Harvard University Herbaria 2 (M.R.) and by additional support from the Harvard Materials Research Sci- σ − + − − u ku(ti tj 2tij ) 2kutij ence and Engineering Center and National Science Foundation Division of cov(ui (ti ), uj (tj )) = e (1 − e ), [11] 2ku Mathematical Sciences.

1. Vogel S (1994) Life in Moving Fluids: The Physical Biology of Flow (Princeton Univ 21. Lutzoni F, et al. (2004) Assembling the Fungal Tree of Life: Progress, classification and Press, Princeton), 2nd Ed. evolution of sub-cellular traits. Am J Bot 91:1446–1480. 2. Parkhurst D, Loucks O (1972) Optimal leaf size in relation to environment. J Ecol 22. Norman JE, Egger KN (1996) Phylogeny of the genus Plicaria and its relation- 60:505–537. ship to Peziza inferred from Ribosomal DNA sequence analysis. Mycologia 88: 3. Taylor GK, Nudds RL, Thomas ALR (2003) Flying and swimming animals cruise 986–995. at a Strouhal number tuned for high power efficiency. Nature 425:707– 23. Schlicting H (1960) Boundary Layer Theory (McGraw–Hill, New York). 711. 24. Deering R, Dong F, Rambo D, Money N (2001) Airflow patterns around mushrooms 4. Pyke G, Pulliam H, Charnov E (1977) Optimal foraging: A selective review of theory and their relationship to spore dispersal. Mycologia 93:732–734. and tests. Q Rev Biol 52:137–154. 25. Brenner H, Happel J (1973) Low Reynolds Number Hydrodynamics with Special Appli- 5. Gilchrist M, Sulsky D, Pringle A (2006) Identifying fitness and optimal life-history cation to Particulate Media (Martinus Nijhoff Publishers, Leiden, the Netherlands), strategies for an asexual filamentous . Evolution 60:970–979. 2nd Ed. 6. Parker GA, Smith JM (1990) Optimality theory in evolutionary biology. Nature 26. von Arx J, Figueras M, Guarro J (1988) Sordariaceous ascomycetes without ascospore 348:27–33. ejaculation, Beih Nova Hedwigia 94:1–104. 7. Ingold C (1971) Fungal Spores: Their Liberation and Dispersal (Clarendon Press, 27. Ainsworth G, Sparrow F, Sussman A, eds (1973) The Fungi—An Advanced Treatise Oxford). (Academic Press, New York), Vol 4A. 8. Trail F (2007) Fungal cannons: Explosive spore discharge in the Ascomycota. FEMS 28. Cybernome: The Nomenclator for Fungi and Their Associated Organisms. Available Microbiol Lett 276:12–18. at: www.cybertruffle.org.uk/cybernome. Last accessed January 28, 2008. 9. Buller A (1934) Researches on Fungi (Longmans & Green, New York), Vol 6. 29. Burdsall HH, Jr (1965) Operculate asci and puffing of ascospores in geopora 10. Batchelor G (1967) Introduction to Fluid Dynamics (C.U. Press, Boulder, CO). (tuberales). Mycologia 57:485–488. 11. Jones E (2006) Form and function of fungal spore appendages. Mycoscience 47:167– 30. Archer A (1997) The Lichen genus Pertusaria in Australia. Bibl Lichen 69:1–249. 183. 31. Dibben M (1980) The chemosystematics of the Lichen genus Pertusaria in North Amer- 12. Ingold C (1939) Spore Discharge in Land Plants (Oxford Univ Press, London). ica north of Mexico. Publications in Biology and Geology (Milwaukee Public Museum, 13. Radhakrishnan V (1998) Locomotion: Dealing with friction. Proc Natl Acad Sci USA Milwaukee), Vol 5. 95:5448–5455. 32. Schmitt I, Lumbsch H (2004) Molecular phylogeny of the Pertusariaceae supports 14. Vogel S (2005) Living in a physical world II. The bio-ballistics of small projectiles. J secondary chemistry as an important systematic character set in lichen-forming Biosci 30:167–175. ascomycetes. Mol Phylogenet Evol 33:43–55. 15. Vogel S (2005) Living in a physical world III. Getting up to speed. J Biosci 30:303–312. 33. Hansen TF (1997) Stabilizing selection and the comparative analysis of adaptation. 16. Fischer M, et al. (2004) New information on the mechanism of forcible ascospore Evolution 51:1341–1351. discharge from immersus. Fung Gen Biol 41:698–707. 34. Felsenstein J (1985) Phylogenies and the comparative method. Am Nat 125:1–15. 17. Ingold CT, Hadland SA (1959) The ballistics of Sordaria. New Phytol 58:46–57. 35. Lemons DS (2002) Introduction to Stochastic Processes in Physics (John Hopkins Univ 18. Trail F, Gaffoor I, Vogel S (2005) Ejection mechanics and trajectory of the ascospores Press, Baltimore). of Gibberella zeae (anamorph Fuarium graminearum). Fung Gen Biol 42:528– 36. Butler M, King A (2004) Phylogenetic comparative analysis: A modeling approach for 533. adaptive evolution. Am Nat 164:683–695. 19. Mohammadi B, Pironneau O (2001) Applied Shape Optimization for Fluids (Oxford 37. Westergaard M, Mitchell H (1947) Neurospora V. A synthetic medium favoring sexual Univ Press, Oxford). reproduction. Am J Bot 34:573–577. 20. Roper M, Squires TM, Brenner MP (2008) Symmetry unbreaking in the shapes of 38. Dulymamode R, Cannon PF, Peerally A (1998) Fungi from Mauritus: Three Astrocystis perfect projectiles. Phys Fluids 20:093606. species from Pandanus. Mycol Res 102(11):1325–1330.

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