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For all supplemental material see 492_supplemental.zip.

SUPPLEMENTARY MOVIE

Visualization of the flow within the 3D printed model (Re = 0.376, see Table 1). Flow in the folds consists of horizontal bands of distinct red and blue color, indicating no adoral component to flow and no mixing within the folds, consistent with Hypothesis 2 (see text, Figure 2.2). The animation is sped up 16x. Figure 4 shows a single frame from the flow pattern observed, showing the steady-state flow pattern after nine minutes of flow. Click on image to activate animation.

figure 4 

SUPPLEMENTARY TABLE S1.

Full calculations for in vivo flow conditions.

What is the Re for blastoid Pentremites rusticus in vivo?
Assumed values shown in blue along with source
     
Environmental data    
nu, kinematic viscosity, m^2/s 1.00E-006 Use 1e-6 for 25C, 1.4 for 12C, 1.8 for 0C
     
Case 1: Cilia in pore canals    
     
At the pore    
Pore diameter, m 2.54E-005 Measured from Schmidtling and Marshall 2010
Pore area, m^2 0.0000000005  
Clia velocity, m/s 6.00E-004 Paul 1978
Volumetric flow rate m^3/s 0.0  
Re,D pore 0.01524 Implies laminar flow
     
In the folds    
Spacing between folds, m 4.26E-005  
Spacing between pores, m 2.65E-004 Schmidtling and Marshall 2010
Number of folds per pore 3 From Schmidtling and Marshall, figure 6
Fold area, m^2 0.0000000339 66.809793371
Volumetric flow rate m^3/s 0.0  
Velocity in fold, m/s 0.0000089807 m/s
Dh fold, 4*FA/WP 0.0000733744 for finite channels
Dh fold, 2*spacing 0.0000851648 for infinite array of closely spaced plates
Re,Dh fold 0.0006589551  
Re,Dh2 fold 0.0007648415 Implies laminar flow
     
     
Case 2: Cilia in folds    
     
At the pore    
Pore diameter, m 2.54E-005 Measured from Schmidtling and Marshall 2010
Pore area, m^2 0.0000000005  
Volumetric flow rate m^3/s 0.0  
Pore velocity, m/s 0.040085876  
Re,D pore 1.018181251 Implies laminar flow
     
In the folds    
Spacing between folds, m 4.26E-005 NOT measured in Schmidtling and Marshall; Tony measured in x-ray image
Spacing between pores, m 2.65E-004 7x pore diameter
Number of folds per pore 3 From Schmidtling and Marshall, figure 6
Fold area, m^2 0.0000000339  
Cilia velocity, m/s 6.00E-004 Paul 1978
Volumetric flow rate m^3/s 0.0  
Dh fold, 4*FA/WP 0.0000733744 For finite channels
Dh fold, 2*spacing 0.0000851648 For infinite array of closely spaced plates
Re,Dh fold 0.0440246517  
Re,Dh2 fold 0.0510989011 Implies laminar flow
     
Case 2 is the controlling case (higher Re, highest mixing expected there).    

 

SUPPLEMENTARY TABLE S2.

Full calculations for model flow conditions as tested.

What is the Re for the model as tested?
Assumed values shown in blue along with source
       
Environmental data      
nu, kinematic viscosity, m^2/s 1.00E-006 Use 1e-6 for freshwater at 25 C  
    Use 27e-6 for mineral oil  
       
At the faucet      
Time, s 4034 Tony Huynh measured  
Level change, m 0.0025 Tony Huynh measured  
Area, m^2 0.1114302763 Tony Huynh measured, for McMurdo portable tank with black screens  
Volumetric flow rate m^3/s 0.0000000691    
       
At the pore      
pPore diameter, m 1.85E-003 TH design, CM, DE measured  
Pore area, m^2 2.69E-006    
Number of pores connected 8    
Volumetric flow rate m^3/s 0.0000000086 1/8 of flow from tap  
Velocity at pore, m/s 0.0032113231    
Re,D pore 5.9409476461 transition, less than 10  
       
In the folds      
Spacing between folds, m 3.10E-003 TH design, CM DE measured  
Spacing between pores, m 0.0153 TH designed, DE meas STL  
Number of folds per pore 3 Tony Huynh designed into model, Fig 3  
Fold area, m^2 1.42E-004 52.9347713135  
Volumetric flow rate m^3/s 0.0000000086    
Velocity in fold, m/s 0.0000606657 m/s  
Dh fold, 4*FA/WP 0.0051554348 for finite channels  
Dh fold, 2*spacing 0.0062 for infinite array of closely spaced plates  
Re,Dh fold 0.3127578744    
Re,Dh2 fold 0.376127117 laminar  
  7.4 x in vivo Re  

SUPPLEMENTARY TABLE S3.

Full calculations for model operated to match lowest Re in vivo case.

What speed would we have to run at to match in vivo
Assumed values shown in blue along with source
     
Environmental data    
nu, kinematic viscosity, m^2/s 1.00E-006 Use 1e-6 for freshwater at 25 C
    Use 30e-6 for mineral oil
     
At the pore    
Pore diameter, m 1.85E-003 Tony Huynh designed
Pore area, m^2 0.000002688  
Volumetric flow rate m^3/s 0.0000000016  
Velocity at pore, m/s 5.85E-004  
Re,D pore 1.081961112 Transition, less than 10
     
In the folds    
Spacing between folds, m 3.10E-003 Tony Huynh designed
Spacing between pores, m 0.0153 Tony Huynh designed
Number of folds per pore 3 Tony Huynh designed into model, Fig 3
Fold area, m^2 0.00014229  
Volumetric flow rate m^3/s 0.0000000016  
Velocity in fold, m/s 0.0000110484 m/s
Dh fold, 4*FA/WP 0.0051554348 For finite channels
Dh fold, 2*spacing 0.0062 For infinite array of closely spaced plates
Re,Dh fold 0.0569592391  
Re,Dh2 fold 0.0685 Implies laminar flow
     
Re in vivo, case 1 0.0007648415  
Re in vivo, case 2 0.0510989011  

 

SUPPLEMENTARY FILE.

Stereolithography (STL) file of model of the distal end of the hydrospire in Pentremites rusticus (see palaeo-electronica.org/content/2015/1073-blastoid-hydrospire-fluid-flow) (see 492_supplemental.zip).

 
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Visualizing the fluid flow through the complex skeletonized respiratory structures of a blastoid echinoderm

Tony L. Huynh, Dennis Evangelista, and Charles R. Marshall

Plain Language Abstract

Our goal is to understand the functional effectiveness of the remarkably complex skeletonized respiratory structures in blastoids, an extinct group of bottom-dwelling stalked echinoderm. The complexity of the structures required a modeling approach, and we used a scaled-up 3D printed model of part of one of these structures to directly observe the pattern of water flow within it. Our results show that the path of water flow within the structures is consistent with a respiratory system able to function with high effectiveness. While it is notoriously difficult to determine the exact reasons for the evolutionary success of one group over other groups, it is possible that the hydrospires also helped the blastoids with some other function beyond respiration, for example, feeding.

Resumen en Español

Visualizando el flujo de fluido a través de las complejas estructuras esqueléticas respiratorias de un equinodermo blastoideo

Los blastoideos del grupo Spiraculata tienen extraordinarias estructuras respiratorias esqueléticas internas, las hidrospiras ("hydrospires"). Sin embargo, el patrón detallado del flujo de agua de mar dentro de las hidrospiras es desconocido, por lo que es difícil evaluar su eficacia respiratoria. Con el uso de un modelo físico impreso 3D de mayor escala (72x), para visualizar el flujo de agua a través de la parte más distal (aboral) de la hidrospira de Pentremites rusticus, mostramos que el flujo era coherente con el intercambio respiratorio eficaz en los pliegues de la hidrospira- el flujo continuaba horizontalmente dentro de los pliegues de la hidrospira pasando después a través de los canales de poros de la hidrospira y solo desarrollaba un componente adoral a su velocidad una vez que había entrado en los canales de la hidrospira. El flujo laminar ordenado observado es compatible con los valores de Reynolds que estimamos para un blastoideo vivo (Re = 0,0008-0,05). Aunque la mayoría de los análisis funcionales de hidrospiras espiraculadas se han centrado en su función respiratoria, también es posible que estas jugasen un papel importante en la alimentación, ayudando a sacar agua más allá de las braquiolas, lo cual constituye una hipótesis que puede ser objeto de futuras comprobaciones.

Palabras clave: Blastoidea; Echinodermata; Morfología funcional; Flujo de fluido; Hidrospira; Modelo impreso tridimensional

Traducción: Enrique Peñalver

Résumé en Français

Visualisation de l'écoulement du fluide à travers les structures respiratoires squelettisées complexes d'un échinoderme blastoïde

Les blastoïdes spiraculés ont des structures respiratoires squelettisées internes extraordinaires, les hydrospires. Cependant, le modèle détaillé de l'écoulement de l'eau de mer en leur sein n'est pas connu, ce qui rend difficile l'évaluation de l'efficacité respiratoire. En utilisant un modèle physique imprimé en 3D agrandis (72x) afin de visualiser le flux d'eau à travers la partie la plus distale (aborale) de l'hydrospire de Pentremites rusticus, nous montrons que le flux était consistent avec un échange respiratoire efficace dans les plis d'hydrospire - le flux a continué horizontalement dans les plis d'hydrospire après le passage à travers les canaux de pores d'hydrospire et a seulement développé un composant adorale à sa vitesse une fois qu'il était entré dans les canaux d'hydrospire. Le flux laminaire et ordonné observé est consistent avec les nombres de Reynolds que nous estimons pour un blastoïde vivant (Re = 0,0008 à 0,05). Alors que la plupart des analyses fonctionnelles des hydrospires spiraculés sont concentrées sur leur fonction respiratoire, il est également possible qu'ils ont joué un rôle dans l'alimentation, en aidant à puiser de l'eau au-delà des brachioles, ce qui est une hypothèse qui est prêtée à une analyse future.

Mots-clés: Blastoidea; Echinodermata; Morphologie fonctionnelle; Écoulement de fluide; Hydrospire; modèle imprimé en trois dimensions

Translator: Kenny J. Travouillon

Deutsche Zusammenfassung

Sichtbarmachung des Flüssigkeitsstroms durch die komplexen skelettierten Atemwegsstrukturen eines blastoiden Echinodermen

Spirakulate Blastoide haben außergewöhnliche interne skelettierte Atemwegsstrukturen, die Hydrospire. Jedoch ist nicht bekannt wie genau das Meerwasser hindurchströmt, was die Bestimmung der respiratorischen Wirksamkeit der Blastoide erschwert. Mit einem vergrößerten (72x) 3D-Druckmodell visualisierten wir den Wasserfluss durch den am weitesten distal (aboral) gelegenen Teil der Hydrospire von Pentremites rusticus. Wir zeigten, dass der Durchfluss bei effektivem respiratorischen Austausch in den Falten der Hydrospire gleich bleibend war – der Durchfluss setzte sich horizontal fort innerhalb der Falten der Hydrospire nachdem die Porenkanäle passiert waren und nahm erst eine adorale Geschwindigkeitskomponente an als die Hydrospiren-Kanäle passiert waren. Die beobachtete geordnete laminare Strömung stimmt mit der Reynoldszahl die bei einem lebenden Blastoiden (Re = 0.0008–0.05) zu erwarten ist überein. Während die meisten Funktionsanalysen über spirakulate Hydrospire auf die Atemwegsfunktion fokussiert sind, ist es ebenso möglich, dass diese eine Rolle bei der Nahrungsaufnahme spielten indem sie Wasser hinter die Brachiolen brachten, was eine Hypothese für weitere Untersuchungen ist.

Schlüsselwörter: Blastoidea; Echinodermata; Funktionsanalyse; Flüssigkeitsstrom; Hydrospire; dreidimensionales Druckmodell

Translator: Eva Gebauer

Arabic

in progress

Translator: Ashraf M.T. Elewa

 

 

FIGURE 1. Anatomy of the hydrospires of the blastoid Pentremites rusticus. 1.1, Location of one of the five radially distributed hydrospires within the calyx, showing incurrent hydrospire pores, and excurrent spiracle (inferred direction of water flow indicated by the arrows). 1.2, Oblique view of a section of a hydrospire and associated structures. Modified from Schmidtling and Marshall (2010).

figure1

FIGURE 2. Schematic showing hypothesized flow patterns within the hydrospire folds. 2.1, In Hypothesis 1, the flow has an adoral component representing respiratory leakage. 2.2, In Hypothesis 2, the flow is entirely radial, without leakage. See text for further discussion.

figure2

FIGURE 3. Digital and physical models use to visualize fluid flow. 3.1, Digital solid model of approximately the lower quarter of a hydrospire of Pentremites rusticus, using Blender (see text). 3.2, 3D-printed rendering of the digital model, shown with inlet headers connected.

figure3

FIGURE 4. Visualization of the flow within the 3D printed model (Re = 0.376, see Table 1). Flow in the folds consists of horizontal bands of distinct red and blue color, indicating no adoral component to flow and no mixing within the folds, consistent with Hypothesis 2 (see text, Figure 2.2). The still used in the print version of this paper is a single frame from the flow pattern observed, showing the steady-state flow pattern after nine minutes of flow. The animation is sped up 16x (for video see supplementary material).

figure4

 

TABLE 1. Calculation of the Re in the most distal (aboral) portion of the hydrospire, corresponding to the first eight hydrospire pores, in living Pentremites rusticus and in the 72x scale model.

 
  Pentremites rusticus Model
  If the cilia are in pore canals If the cilia are in the folds
Pore canal parameters      
       Pore diameter, m 25.4x10-6 25.4x10-6 1.85x10-3
       Pore area, m2 5.1x10-10 5.1x10-10 2.69x10-6
       Pore velocity, m/s 6.0x10-4 † 0.040 3.2x10-3
       Volumetric flow rate, m3/s 3.0x10-13 2.0x10-11 6.9x10-8
Hydrospire fold parameters      
       Fold spacing, m 4.3x10-5 4.3x10-5 0.0031
       Distance between adjacent pores, m 2.65x10-4 2.65x10-4 0.0153
       Area of folds between adjacent pores, m2 3.4x10-8 3.4x10-8 1.42x10-4
       Hydraulic diameter, m 8.6x10-5 8.6x10-5 0.0062
Velocity in folds, m/s 9.0x10-6 6.0x10-4 † 61x10-6
Kinematic viscosity, m2/s 1.8x10-6, 0°C seawater 1.8x10-6, 0°C seawater 1.0x10-6, 25°C freshwater
  9.4x10-7, 25°C seawater 9.4x10-7, 25°C seawater  
Reynolds number in folds 0.0004 at 0°C 0.03 at 0°C 0.376
0.0008 at 25°C 0.05 at 25°C
† Cilia velocity from (Paul, 1978)
 

author1Tony L. Huynh
Department of Integrative Biology
University of California, Berkeley
Berkeley, California 94720-3140 USA
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Tony L. Huynh received an undergraduate degree from the department of Integrative Biology at UC Berkeley in 2012. He likes using tools and approaches from mechanical engineering, computer-aided 3D design, and computed tomography (CT imaging) to answer evolutionary and ecological questions related to functional morphology. He is always on the hunt for another specimen to put into a CT scanner and has worked with vertebrates, invertebrates, and plants, along the entire range of geologic time. He is currently the lab manager for a metabolic physiology lab specializing in invertebrates at UC Berkeley.

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author2Dennis Evangelista
Department of Biology
University of North Carolina at Chapel Hill
Chapel Hill, North Carolina 27599-328
USA
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Dennis Evangelista is a postdoctoral researcher in comparative biomechanics. He studied mechanical and electrical engineering at MIT and Naval Postgraduate School before earning a PhD in Integrative Biology at UC Berkeley. He enjoys working with undergraduates, flight, swimming, maneuvering and control, behavior of aggregations, evolution, and biomechanics in extreme environments, and has worked on vertebrates, invertebrates, and plants, both extinct and extant. He is a good cook and is raising a guide dog puppy.

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author3Charles R. Marshall
University of California Museum of Paleontology and Department of Integrative Biology
University of California, Berkeley
Berkeley, California, 94720-478
USA
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Charles Marshall is currently the Director of the University of California Museum of Paleontology and a Professor in the Department of Integrative Biology at the University of California, Berkeley. His undergraduate degree from the Australian National University was in geology (especially in palaeontology), zoology, applied mathematics, physical chemistry, and history, and included the equivalent of a Masters thesis on lungfish. His Ph.D. (on sand dollars) was from the Committee on Evolutionary Biology at the University of Chicago, followed by an NIH postdoc in evo-devo with Rudy Raff at Indiana University (where he shifted to regular echinoids). His research interests have been described as being pathologically broad, which does not seem to be an unfair description.