Temporal model of fluid-feeding mechanisms in a long proboscid orchid bee compared to the short proboscid honey bee

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Temporal model of fluid-feeding mechanisms in a long proboscid orchid bee compared to the short proboscid honey bee Lianhui Shi , Jianing Wu , Harald W Krenn , Yunqiang Yang , Shaoze Yan PII: DOI: Reference:

S0022-5193(19)30387-X https://doi.org/10.1016/j.jtbi.2019.110017 YJTBI 110017

To appear in:

Journal of Theoretical Biology

Received date: Revised date: Accepted date:

6 May 2019 16 September 2019 18 September 2019

Please cite this article as: Lianhui Shi , Jianing Wu , Harald W Krenn , Yunqiang Yang , Shaoze Yan , Temporal model of fluid-feeding mechanisms in a long proboscid orchid bee compared to the short proboscid honey bee, Journal of Theoretical Biology (2019), doi: https://doi.org/10.1016/j.jtbi.2019.110017

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Highlights  The erection angles of microtrichia were added in the modified model of honey bees.  A model was proposed to evaluate the volumetric and energetic intake rate of orchid bees.  The honey bees have higher volumetric and energetic intake rate compared with orchid bees.

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Temporal model of fluid-feeding mechanisms in a long proboscid orchid bee compared to the short proboscid honey bee Lianhui Shia,†, Jianing Wub,c,†, Harald W Krennd, Yunqiang Yanga,*, and Shaoze Yanc,*

a

School of Engineering and Technology, China University of Geosciences (Beijing),

Beijing, 100083, People‟s Republic of China b

School of Aeronautics and Astronautics, Sun Yat-Sen University, Guangzhou,

510006, People‟s Republic of China c

Division of Intelligent and Biomechanical Systems, State Key Laboratory of

Tribology, Department of Mechanical Engineering, Tsinghua University, Beijing, 100084, People‟s Republic of China d

Department of Integrative Zoology, University of Vienna, Vienna, 1090, Austria

*Author to whom correspondence should be addressed. E-mail: [email protected] (Y. Yang), [email protected] (S. Yan). †

These authors contributed equally to this study.

Notes: 1. Lian-Hui Shi: one of the first authors who wrote the manuscript, performed the experiments and analyzed the data. His email address: [email protected] 2. Jia-Ning Wu: one of the first authors who found the special compensation mechanism, designed the experiments and revised the manuscript. His email address: [email protected] 3. Harald W Krenn: the general author who performed the analysis with constructive discussions about the orchid bees. His email address: [email protected] 2

4. Yun-Qiang Yang: a corresponding author who conceived the project and designed the experiments. His email address: [email protected] 5. Shao-Ze Yan: a corresponding author who conceived the project and designed the experiments. His email address: [email protected]

Abstract Bees (Apidae) are flower-visiting insects that possess highly efficient mouthparts for the ingestion of nectar and other sucrose fluids. Their mouthparts are composed of mandibles and a tube-like proboscis. The proboscis forms a food canal, which encompasses a protrusible and hairy tongue to load and imbibe nectar, representing a fluid-feeding technique with a low Reynolds number. The western honey bee, Apis mellifera ligustica, can rhythmically erect the tongue microtrichia to regulate the glossal shape, achieving a tradeoff between nectar intake rate and viscous drag. Neotropical orchid bees (Euglossa imperialis) possess a proboscis longer than the body and combines this lapping-sucking mode of fluid-feeding with suction feeding. This additional technique of nectar uptake may have different biophysics. In order to reveal the effect of special structures of mouthparts in terms of feeding efficiency, we build a temporal model for orchid bees considering fluid transport in multi-states including active suction, tongue protraction and viscous dipping. Our model indicates that the dipping technique employed by honey bees can contribute to more than seven times the volumetric and energetic intake rate at a certain nectar concentration compared with the combined mode used by orchid bees. The high capability of the honey bee‟s proboscis to ingest nectar may inspire micropumps for transporting viscous liquid with higher efficiency.

Keywords: honey bee; orchid bee; nectar feeding; micropump

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1. Introduction The word „biofluiddynamics‟ was coined to describe fluid mechanics problems in biology (Lighthill, 1975). Many flower-visiting insects gained most of their energy from floral nectar which contains various carbohydrates in rather high concentrations (Cundall, 2000; Kim et al., 2012). The drinking strategies of animals were researched for a long time and adaptations of feeding organs to nectar feeding have been studied extensively (Daniel and Kingsolver, 1983; Krenn et al., 2005; Wu and Shi et al., 2018). According to the principal forces involved, the drinking techniques of a broad range of animals could be generally classified into dipping (Kim et al., 2011), licking (McClung and Goldberg, 2000), lapping (Reis et al., 2010; Crompton and Musinsky, 2011; Gart et al., 2015), and suction (Tamm and Gass, 1986). Licking and suction were considered two principal mechanisms for the uptake of surface liquids or nectar from flowers (Kingsolver and Daniel, 1995). Mouthparts which function primarily in conformity to the properties of adhesion and capillarity possess a wettable apical surface and perform lapping, licking, dipping or sponging movements to pull fluids adhering to surface structures into the mouthparts. Mouthparts which function in accordance with a suction mode often had long slender and tubular food canals to load nectar from flowers with more or less, long or narrow corolla tubes (Krenn et al., 2005). A pressure gradient is created between the proboscis tip and the proboscis base by a muscular pumping organ inside the head. The morphology of mouthparts and the nectar drinking behavior of honey bees have been investigated (Snodgrass, 1956; Snodgrass 1984; Dade 2009; Wu and Zhu et al., 2015). The mouthparts of bees (Apidae) include the mandibles and an elongated proboscis for fluid-feeding. The proboscis is composed of the paired elongated galeae and labial palpi which together form a food canal that encloses the central glossa. Advances in micro-imaging techniques have resulted in novel insights of the drinking mechanism of Apis mellifera (Kim et al., 2011; Yang et al., 2014). The fluid-feeding technique of the mouthpart of a honey bee was described as “partly suctorial and 4

rather lapping” (Huxley, 1877). However, it was found that these two principal mechanisms of fluid uptake were involved simultaneously in honey bees in the later studies (Kingsolver and Daniel, 1995; Krenn et al., 2005; Wu and Zhu et al., 2015). During lapping, (1) nectar is loaded onto the hydrophilic distal part of the extended glossa by capillary action; (2) the glossa retracts whereby adhering nectar is transported into the food canal. (3) Afterwards, the muscular pump in the head takes action to suck nectar from the food canal into the mouth (Wu and Zhu et al., 2015). When a honey bee drinks nectar, the glossal microtrichia erect rhythmically to save energy for fluid uptake (Zhao C. and Wu et al., 2015). Theoretical predictions of volumetric flux derived from the modified viscous dipping model and compared with experimental data (Kim and Bush, 2012; Yang et al., 2014). The predictions considered the erection of glossal microtrichia for loading nectar, and concluded that a honey bee augmented its working power in protracting glossa against an increased nectar viscosity. This assumption then estimated the theoretically optimal sucrose concentration as 33% (wt. / wt.) which was in accordance with the nature-preferred nectar concentration, namely 35% (wt. / wt.) (Yang et al., 2014; Wu et al., 2015). Apidae are a species-rich group of insects and their mouthparts exhibit a range of diversity in morphology and size (Michener, 2000). For instance, the Neotropical Euglossini, commonly known as orchid bees, contain approximately 190 species and are widely distributed from Mexico to central Argentina (Cameron, 2004). The orchid bees are an interesting group of insects: some species have the brilliant metallic coloration, and extraordinarily long proboscises, which may exceed the body length in many species (Fig. 1, Williams and Whitten, 1983; Borrell, 2004; Roubik and Hanson, 2004; Borrell, 2006; Düster et al., 2018). The similarity of the mouthparts of the honey bee and orchid bees in morphology is that the mouthparts all comprise of three main parts, a pair of galeae, a pair of labial palpi and a glossa (Fig. 1(g), (i)). The major differences are the greater length of the proboscis and the number, length and 5

shape of structures on the glossa surface where nectar is taken up first (Düster et al., 2018). The functional morphology of the proboscis and the nectar-feeding mechanism were studied in various Euglossa species in context with feeding ecology and flower-visiting behavior. For instance, feeding experiments after removing the glossa, suggested that this tongue-like organ was not the essential mouthpart structure for liquid uptake, indicating a mode of merely sucking (Borrell, 2004). Similar to honey bees, the proboscis of an orchid bee consists of the elongated galeae and slender labial palpi, which together form the feeding tube that encircles the glossa (Stell, 2012). Recent observations under semi-natural conditions showed that orchid bees consumed large amounts of fluid by this suction technique without glossal movements, but small residual amount of fluids was lapped using repeating glossal protraction-retraction movements, which gave a rise to the feeding efficiency (Düster et al., 2018).

Fig. 1 Fluid transport by proboscis of honey bees and orchid bees. (a)-(f) Six typical frames of a honey bee‟s tongue in a feeding cycle in which the red arrows indicate the direction of the glossa‟s movement. (g) Mouthpart of an orchid bee (Euglossa imperialis) (Gruber 2013). (h) Image of an orchid bee (Euglossa sp.). (i) Mouthparts of a honey bee (Apis mellifera) which comprise of three main parts a pair of galeae, a pair of labial palpi and a glossa. (j) Image of a feeding honey bee. 6

A simple model of the honey bee feeding was presented in which the fluid was entrained by the outer surface of the tongue through the combined action of viscosity and capillarity according to the Landau-Levich-Derjaguin theory (Kim and Bush, 2012). Viscous dipping is generally characterized by an extensible glossa being submerged into nectar, loaded, then extracted in a cyclic fashion. One expects the volume entrained to be proportional to the area of the immersed tongue surface area S and the thickness e of the nectar layer. If T0 represents the time needed for drinking nectar each cycle, then the volumetric flux is given by Q = Se / T0. The active suction model was proposed by Newton‟s second law and Poiseuille flow in which effect of gravity and inertial were negligible (Kim et al., 2011). The volumetric flux can be written as Q = πa2u, where a is the radius of the feeding tube and u represents the mean speed of the fluid (Kim and Bush, 2012). Notably, the liquid transport mechanism of orchid bees may be related to its special biophysical characteristics. However, the modeling of sucking-lapping-combined drinking strategy remains unexplored, as well as the connections between the feeding pattern and habitat specificity. In this paper, we will propose a mathematical model to describe the multi-phased feeding fashion of orchid bees, and analyze the key parameters that determine the liquid intake rate, energy consumption rate and energy reward, bridging a connection between the feeding techniques, behavior and flowers preference.

2. Materials and methods 2.1 Bee sample preparation We used the western honey bees (Apis mellifera ligustica) from Xiangshan, Beijing, China (40.00°N, 116.33°E) and fed them in an indoor beehive, which was equipped with a ventilation to maintain the temperature at 25°C and the humidity at 50%, respectively (Li et al., 2016). Males of the orchid bee Euglossa imperialis were 7

attracted by fragrance baits of three different aromas, cineol, eugenol and methyl salicylate, because they are known to be attractive to E. imperialis (Roubik and Hanson, 2004). The orchid bees were allured and captured in the tropical research station La Gamba (8.72°N, 83.23°W; average air temperature 27.3°C) adjacent to the Piedras Blancas National Park in Costa Rica (Weissenhofer and Huber, 2001; Düster et al., 2018). We confirmed that there were no specific permissions required for the experiments with honey bees, and a collection and research permit was given to M. Gruber in February 2010 by the Costa Rican Ministerio del Ambiente y Energía.

2.2 Observation of feeding cycles The experimental setup, which we used to observe the feeding process and the erection angle of glossa, is comprised of a cold light source, a high-speed camera (Phantom M110, USA) with a microscope (Axiostar Plus, Zeiss, Germany), a glass feeder filled with sucrose solution and a positioner (Wu and Zhu et al., 2015; Wu and Shi et al., 2018). To ensure the honey bees‟ mouthparts remained in one fixed position and the freedom of nectar drinking, the thoraxes of honey bees were tightly fixed on the special fixture by resin glue. Notably, the position of the honey bees was adjusted through the positioner. The glass feeder contained 35% (wt. / wt.) sucrose solution, and the temperature was 25°C (Wu and Shi et al., 2018). When the honey bees started to drink the sucrose solution, a series of photographs were captured by the high-speed camera at a frequency of 500 frames per second with the enlargement ratio of 5× in vertical view. The feeding behavior of the orchid bees (n = 11) was captured inside a mosquito net in the garden of the La Gambe station in Costa Rica by using a Sony V50 video camera (Sony, Japan). Droplets of artificial nectar (approximately 10 µL with 30% sugar) were offered to orchid bees. The movements of the single component of the proboscis could be analyzed by micro videography (Düster et al. 2018). Video footage of the nectar feeding trials was used to compare the proboscis movements of orchid bees with honey bees. This material was filmed by H. Gruber and was 8

previously used for studying the mouthpart morphology of Euglossa bees (Düster et al. 2018).

2.3 Measurement on volumetric feeding rate To obtain the experimental data referring to the volumetric intake rate, five worker honey bees were captured from the beehive. To record the feeding duration from the beginning of sucrose solution dipping until a honey bee had completely consumed a droplet, a digital video camera (Canon 80D, Japan) was used in this experiment. A plastic dish was placed at the bottom of the bottle before putting the honey bee into the glass bottle. Then the droplets, of 35% (wt. / wt.) sucrose solution, with a volume of 20 µL were placed at the plastic dish by a micropipette (Top Pipette 20 ~ 200 µL, DragonLab, China). The freely moving E. imperialis orchid bees (n = 11) visited a yellow rubber plate where several small drops (each 10 µL) of artificial nectar (30% wt. / wt.) had been placed using a micropipette.

2.4 Statistical Analyses of the hair erection angles We filmed independent protraction events of five honey bees and averaged the erection angles when the microtrichia on the glossa reached the maximum angle. To obtain the average erection angles of glossal microtrichia of the worker honey bees (n = 5), we selected 15 videos of the typical dipping cycles, and a vector method was used in our experiments. When honey bees were in the maximum angle phase (Fig. 1(d)), the glossal parts where protracted out of the feeding tube were measured. We chose two microtrichia strands adhered to the same membrane. Then we named two vector x and y which can be obtained by connecting two pairs of key points A1 and A2, B1 and B2 (supplementary material Fig. S1). The four endpoints represent the direction between the microtrichia and the glossal long axis, respectively. The angle θ between two microtrichia was defined as   arccos

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x y (Wu and Zhu et al., 2015). The x y

data statistical software SPSS (Statistics 20.0, IBM, Armonk, NY, USA) was used to calculate the erection angles. The mathematical software Matlab (R2016a, MathWorks, Natick, MA, USA) was used to analyse and simulate the proposed physical models of honey bees and orchid bees.

3. Results and discussion 3.1 Comparison of nectar feeding of honey bees and orchid bees The average nectar feeding cycle (n = 15) of five honey bees lasts about 156 ± 1 ms (Fig. 1(a) - (f)). During the drinking process the glossal microtrichia erect and nectar is trapped between the rows of the unfolded microtrichia. A typical feeding cycle of a honey bee (Fig. 2(a)) can be separated into three phases, namely the glossa protraction, microtrichia unfolding and glossa retraction phases (Zhao et al., 2015; Li et al., 2015; Yang et al., 2017). In the initial drinking stage, the glossa of a honey bee is encompassed by the feeding tube which is formed by the galeae and labial palpi, and the glossa protracts with the microtrichia flatten. At 54 ms, the bushy microtrichia start to erect when the glossa reaches the maximum extension. The glossa remains still for a short time (about 2 ms) as the microtrichia stay erect. The bushy microtrichia on the glossa reach the maximum angle at about 75 ms. In the final phase, when the glossa retracts into the feeding tube with the nectar adhering to it, the glossal microtrichia remain extended until the end. The sugar water is trapped between the rows of microtrichia until it is sucked by the cibarial pump (Düster et al., 2018). A feeding cycle of Euglossa can be summarized into three phases, namely the tongue protraction, motionless suctorial feeding and the lapping phase (Düster et al., 2018). Fig. 2(b) shows feeding of an orchid bee that takes much longer time than a honey bee. In the tongue protraction phase (supplementary material Movie 2), which is timed from 0 s to 1 s, the glossa stretches out of the tube for a short distance, with microtrichia adhered to the tongue body. During the time interval [2 s, 15 s], the proboscis tip is soaked in the liquid, then the glossa keeps motionless and the liquid is 10

sucked into the feeding tube. At the time of 15 s, when little nectar remains, the tip of glossa further extends out of the tube with the microtrichia starting to erect immediately. The nectar fills the gaps between glossal microtrichia at the time of 16 s. During the lapping mode phase from 16 s to 20 s, similar to the feeding model for honey bees (Fig. 2(a)), the glossa slowly protracts and retracts several times with the unfolded microtrichia and transports the adhering fluid back into the feeding tube.

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Fig. 2 Comparison of two typical nectar drinking mechanisms. (a) A typical feeding cycle of a honey bee (Li et al., 2016). We film the feeding cycle of honey bees about 156 ms (supplementary material Movie 1). (b) The Euglossa first use a suctorial mode of feeding (2-15 s) which is characterized by the motionless glossa and after 16 s the glossa further protracts and retracts to transport fluid in the feeding tube. The duration of one feeding cycle of an orchid bee is about 20 s (supplementary material Movie 2).

There are some morphological differences of the mouthparts between honey bees and orchid bees on the length of the proboscis and the protracted mouthpart (Table 1). The lapping feeding is the major technique of fluid uptake in honey bees, and the sucking feeding is the primary mode of nectar up-take in orchid bees.

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Table 1 Morphology of honey bees and orchid bees. Species

Body Length

Average length

Protracted mouthpart

of proboscis Honey bees (Apis mellifera) Orchid bees (Euglossa

7.00 mm 16.00 ± 1.21 mm

3.00 ± 0.01 mm

(Waddington and Herbst, 1987)

ca. 15.00 mm

22.22 ± 1.45 mm

28.11 ± 0.66 mm

(Roubik, 2004)

(Düster et al., 2018)

(Düster et al., 2018)

imperialis)

3.2 Modeling of viscous dipping of a honey bee The model developed by Yang et al to estimate the average volumetric flux suggests the microtrichia contribute to the nectar ingesting, and the erection angle in their viscous dipping model is 90 ° (Yang et al., 2014). However, with the observation of the microstructures on the proboscis of the honey bees, we find that the honey bees average erection angle of glossal microtrichia is 38.54 ± 0.41° in a feeding cycle. The erection angles in different parts of the glossa from five samples are shown in supplementary material Table S1. We propose one modified model for accurately estimating the volumetric flux of the honey bees. As shown in Fig. 3, the glossa can be treated as a cylinder that is covered by cuticle microtrichia, and the overlapping galeae and labial palpi can be regarded as a tube. Fig. 3 elucidates the temporal model of one feeding cycle of a honey bee, which is divided into three steps. Namely, the duration of the glossal protraction is denoted by T1, and T2 is the time during which the microtrichia unfold. The phases of glossal retraction costs a duration of T3. In this model, the average radius of the glossa is a (Fig. 3). The average length of the microtrichia is h, and the average erection angle of microtrichia is θ. The nectar solution has a viscosity μ, a sucrose mass concentration s, and a density ρ. The parameter T = T1 + T2 + T3 is the period of a dipping cycle. As the glossa performs reciprocating movement in the sucrose solution, the honey bee need to overcome the inertial force and viscous resistance. We use the 35% (wt. / wt.) 13

sucrose solution at 25 ℃, so the density is ρ = 1.15 g mL-1 (Pieter, 1953). The dynamic viscosity μ (Pa · s) rises sharply when the concentration of the nectar increases, namely s increases which can be calculated (Pivnick and McNeil, 1985)

100.8752 s /(100 s )  s  ( s)  1097

2

/9901

(1) .

According to the Eq.1, we can get the viscosity of μ = 3.60 mPa · s at s = 35% (wt. / wt.) sucrose solution. We measure the tongue length as L = 1.60 mm, and the average retraction speed of glossa is v = L / ( T2 + T3 ) = 29.60 mm / s (Fig. 1 (a)-(f)). We measure the diameter of the cylinder is A = 425 μm. The Reynolds number can be calculated by Re = Aρv / μ ≈ 4 (Pan et al., 2016). Namely, the power to overcome the viscous drag is Pv ~ μLv2. The power required for tongue acceleration can be written as Pt ~ ρtA2v3, where the glossa density of ρt = 1 × 10-6 g mm−3 (Wu and Shi et al., 2018). We can get the ratio R (s) = Pt / Pv ~ ρtvA2 / μL and plot the ratio with the nectar concentration s. Fig. 4 shows that the R (s) gradually decreases against the nectar concentration and is less than 2 × 10-3, so the inertial effect can be ignored (Kim et al., 2011).

Fig. 3 Fluid transport model elucidating the feeding cycle of a honey bee. The circulation is divided into three phases, namely glossa protraction, microtrichia unfolding, glossa retraction. Galeae and labial palpi overlap and form the tubular food 14

canal.

The average volume of the liquid acquired in a cycle can be calculated by regarding it as the volume of the nectar attached on the cylinder V1 minus that of glossal microtrichia immersed in the nectar V2 (Yang et al., 2014). The volume V1 and V2 can be calculated as

V    a  h sin  2   a 2  vT  1  1  .  2 V2   2 nad 2 hvT1 3 

(2)

The average volumetric flux can be estimated as

V V Q1  1 2  T

2  h sin   2a  h sin   vT1   2 nad 2 hvT1 3 , T

(3)

in which n is density of the microtrichia and d is the diameter of glossal microtrichia on the base. The microtrichia can be simplified as cone and we assume that the power rate applied in viscous dipping of honey bees, W , remains constant with respect to nectar concentration (Pivnick and McNeil, 1985). This assumption leads to

v ~ (W / L)1/2  1/2  k  1/2 , where the coefficient k indicates the relative value of working power. Following the reasoning of Kim et al. 2011, we assume that the coefficient T1 / T will not change with the nectar viscosity (Kim et al., 2011). The density of nectar solution is proportional to the mass concentration at 25 ℃ (Daniel et al., 1989). To calculate the energetic intake rate of honey bees at different mass concentrations, we fit the relationship between density and mass concentration as ρ = 0.9896 + 0.0047 s (g mL-1). The energetic intake rate E of honey bee can be given by the product of the energy content per unit mass of sugar c = 1.54 × 107 J / kg and the sucrose concentration s (Daniel et al., 1989). We then arrive at E1 (Kingsolver and Daniel, 1979)

E1   scQ1 /100

15

.

(4)

Fig. 4 The ratio of inertial to viscous scales against the nectar concentration under the two feeding models. The black curve shows the ratio of inertial of the viscous dipping model, and the blue curve shows the ratio of inertial to viscous scales.

Parameters and values used to estimate the volumetric flux and energetic intake rate of honey bees are listed in Table 2.

Table 2 Parameters for estimating the volumetric flux and energetic intake rate of Apis mellifera. Parameter T1 T = T1 + T2 + T3 L A a

Description Protraction time Period of a dipping cycle The glossal length out of the Diameter of tube the feeding tube Radius of glossa Distribution density of

Value 54 ± 0.5 ms 156 ± 1 ms 1.60 ± 0.01 mm 425 ± 2 μm 48.30 ± 0.19 μm 2500 mm-2

d

the microtrichia Diameter of glossal microtrichia

h

Length of the microtrichia

(Wu and Zhu et al., 2015) 6 ± 0.1 μm (Zhu et al., 2016) 170 ± 5 μm

θ

Average erection angle of

n

(Li and Wu et al., 2016) 38.54 ± 0.41 deg.

microtrichia

3.3 Modeling of suction behavior of an orchid bee As shown in Fig. 5, we simplify the glossa as a cylinder, and the surrounded galeae and labial palpi are treated as a tube. Fig. 5 illustrates the nectar feeding cycle of an 16

orchid bee, which can be divided into three steps, protraction, suction and lapping, respectively. We define that, the cylinder has an average radius b, the length of the feeding tube as H and the average radius as r. We also state that, the average rising speed of nectar is u, the viscosity is μ, and the density of the source solution is ρ. The nectar can be described by the Newton‟s second law written as (Kim and Bush, 2012)

mu   (r 2  b2 )P  mg  2 H (r  b) .

(5)

Here m is the mass of the liquid in the tube, △P is the pressure difference generated at the height H of the flow, τ is the shear stress along the outer wall and g is the gravitational acceleration. The variables, namely, m and τ can be estimated as

4u    r b  m   (r 2  b 2 ) H  

(6) .

So we can simplify equation (5) as

H

du P 8 Hu   Hg  2 2 . dt  (r  b ) 

(7)

Fig. 5 Fluid transport model illustrating the drinking cycle of an orchid bee. The circulation is composed of three phases. (a)-(b) In the protraction phase, the glossa extends to get the nectar with the microtrichia flatten. (c)-(d) In the suction phase, the

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glossa remains motionless. (e)-(h) In the lapping phase, the glossa moves and microtrichia unfold to load nectar.

Suction can be classified according to what produces the driving pressure and whether the flow is counteracted primarily by fluid inertia or viscosity, such as inertial suction, viscous suction and capillary suction (Kim et al., 2012; Kim and Bush, 2012). We introduce a reduced condition of the suction feeding model that active suction with tube of characteristic length H ~ 1 cm, consequently ρgH∕△P < 0.1, and the effect of gravity on the dynamics behaviors of the system is relatively minor which can be negligible (Kim et al., 2011). The ratio of inertial to viscous scales can be written as ζ (s) = 4r2ρf∕μ (Lee et al., 2009; Ewald and Williams, 1982). Fig. 4 shows that the ratio ζ (s) gradually decreases against the nectar concentration and is far less than 0.1 with different nectar solution, so the inertial effect can be ignored. In the suction model, we note an assumption that the nectar motion is described by Poiseuille flow (Kim et al., 2011), in which the volumetric flux is written as Qs   (r 2  b 2 )u 

 (r 2  b2 )2 P 8 H .

(8)

By measuring the dependence of flow rate on sugar concentration, we speculate the suction power in drinking of orchid bees as constant values (Pivnick and McNeil, 1985). The power rate required to overcome the viscous friction on the feeding tube is given by W  Qs P , which remains constant with respect to nectar concentration. So equation (8) may be expressed as Qs  (r 2  b 2 )

W 8 H .

(9)

In fact, the orchid bee drinks larger amount of nectar by suction phase than lapping phase, and the residual volume of nectar is negligible (Düster et al., 2018). That means Ql

Qs which Ql is the average volumetric flux in lapping phase. So, we can

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assume that Q2  Qs  Ql  Qs , which Q2 is the total volumetric flux on average. The energetic intake rate E2 of orchid bees can yield

E2   scQ2 /100 .

(10)

3.4 Comparison of energetic intake rates The mathematical model of fluid-feeding shows the volumetric flux and the energetic intake rate of both orchid bee and honey bee as a function of nectar concentration, respectively (Fig. 6). The black curve (circle) and the magenta curve (square) represent the volumetric flux and energetic intake rate of honey bees, respectively. The energetic intake rate of honey bees arrives at a peak at the 35% (wt. / wt.) sucrose concentration, which matches the results provided by Yang (Yang et al., 2014). With the parameter s increases, the volumetric flux gradually decreases. The red and blue curves express the volumetric flux and energetic intake rate of orchid bees, respectively. Similar to the feeding maps of a honey bee (Yang et al., 2014), the values of volumetric flux of orchid bees also gradually reduce with the increase of s. The energetic intake rate of orchid bees arrives at the maximum value under a same concentration of s = 35% (wt. / wt.) as a honey bee. Furthermore, theoretical prediction in Fig. 6 indicates the following results. (1) The volumetric flux of honey bees in a dipping cycle is 0.91 μL s-1 at real nature-preferred concentration s = 35% (wt. / wt.). The average volumetric flux of orchid bees can reach 0.13 μL s-1. The maximum energetic intake rate of the honey bees is 5.68 J s-1, and the peak of the orchid bees is 0.81 J s-1. In conclusion, the volumetric flux and the energy intake rate of honey bees in our model are more than 7 times larger than the results of orchid bees. When drinking the same volume of nectar in a continuous feeding cycle, for instance 5 μL, a honey bees cost only about 6 second and an orchid bee will consume more than 38 second (Fig. 6). Compared to the lower flow flux and energy rewards of orchid bees, the honey bees can ingest 19

more liquid in the same duration, which insinuates the honey bees drink nectar much quicker than orchid bees, which benefits the competition with other flower-visiting insects and probably reduce the risk of predation. (2) We plot two curves of energetic intake rates under the assumption of constant working power of the studied bee species that are shown as a blue curve and a magenta curve (Kim et al., 2011). Both the theoretically optimal sucrose concentration of honey bees and orchid bees calculated by our model are 35% (wt. / wt.), which is consistent with the real nature-preferred concentration (Kim et al., 2011; Kim and Bush, 2012; Yang et al., 2014).

Fig. 6 Volumetric flux and energetic intake rate against the nectar concentration of honey bees and orchid bees. The black and magenta curves show the volumetric flux and energetic intake rate of honey bees (Apis mellifera). The blue and red lines show the volumetric flux and energetic intake rate of orchid bees (Euglossa imperialis).

(3) In addition, increasing working power against nectar viscosity, we suppose that 20

W   m , where the parameter m indicates the energetic intake rate that working power increases with viscosity (Kim et al., 2011; Yang et al., 2014). We plot energetic intake rate of orchid bees against nectar concentration, and m is equal to 0 which means the working power keeps constant (Fig. 6). Then we vary m from 0 to 1 and find that the optimal concentration and maximizing energetic rewards also shift to the bigger values shown as blue lines in Fig. 6. We can suggest that, when m varies, we will get different theoretically optimal concentrations for orchid bees. For instance, the optimal value is 35% (wt. / wt.) with m = 0, which is the same value as a honey bee that uses the viscous dipping mode. Both of the suction pattern and viscous dipping pattern reach a peak at the sucrose concentration 35% (wt. / wt.) under the assumption of constant working power. When m rises to m = 0.3, the optimal concentration would be 42% (wt. / wt.), which has an increase of 7% (wt. / wt.) compared with the condition under which m = 0. Meanwhile the peak of the energetic intake rate rises from 0.81 J s-1 to 1.01 J s-1. From the assumption that the working power remains constant (Kim and Bush, 2012), we then represent Qs in terms of μ as Qs ~ (W / H )1/2  1/2   1/2 , and write the energetic intake rate

E2

as

E2 ~ Qs  s . When dipping nectar with higher viscosity, we know that the orchid bees will continuously promote pumping power and the volumetric flux, both of which may get higher energetic intake rate and optimal nectar concentration. However, the sucrose concentration s rises rapidly signifies the rapid growth of the viscosity μ (Eq. 1). According to the Newton‟s inner friction law in the laminar flow, we know that the viscous drag is written as Ff   S

du . Here the du/dy is the fluid velocity gradient dy

and S is the contact area. We can set the viscous drag Ff   , then arrive at the frictional force Ff  s . The principle between the sucrose concentration and volumetric flux is shown in Fig. 6. When orchid bees drink with 42% (wt. / wt.) 21

nectar solution or higher optimal concentration, the orchid bees must spend more energy to balance the viscous drag and take more time to drink nectar, which is detrimental to the survival of orchid bees (Fig. 6).

4. Conclusion We discovered that the lapping feeding behavior of honey bees has a great impact on the efficiency of nectar uptake. For the feeding patterns of the studied bee species, we proposed two hydrodynamic models of the honey bee and the orchid bee to compare the volumetric flux and the energetic intake rate against the nectar concentration. Numerical analysis shows that the energetic intake rate of the viscous dipping pattern of the honey bee rivals the suction pattern of the orchid bee, because the dipping mechanism is beneficial for higher nectar intake rate. The orchid bees can collect nectar through their characteristic long proboscises from flowers with various flower lengths (Pokorny et al., 2014; Kimsey, 1980). The advantage of long-proboscid Euglossa bees is that they are able to reach nectar even in long-spurred flowers which supply larger amount of nectar but are inaccessible for short-tongued bees, like a honey bee. However, the honey bees ingest liquid with a viscous dipping mode, which is a more efficient drinking strategy compared with the sucking mode. In this work, our physical model indicates the feeding efficiency of honey bees is higher than the orchid bees. The orchid bees live in the Neotropics and collect nectar all the year to supply themselves with energy from mainly long-spurred flowers. In contrast, honey bees are eusocial insects, and a cast of worker bees are specialized in collecting nectar and can transform it into honey as a source of carbohydrates to support the entire colony (Karasov and Carlos, 2007). Honey bees are generalized flower-visitors which are able to exploit a wide range of various types of flowers in various concentrations. They collect nectar in a short time to build up honey reserves for unsuitable periods of time in the year (Winston, 1991). Generally, the honey bees will stop foraging activities when the ambient temperature drops 22

below about 10°C. Then they consume their stored honey to produce body heat to warm up the colony during the winter. Honey bee workers temporally exhibit a wide range of behaviors, such as feeding the brood, building the wax combs, sharing and distributing nectar among nest mates, cleaning the hive and doing the guarding duty. The mouthparts can be seen as minute tools usable for collection of nectar as well as for the distribution of nectar among the nest mates and its storage in a honey comb inside the hive. The high evolutionary pressure to supply the whole colony with nectar from easily accessible flowers might be the reason for the high efficiency of the feeding technique since many other nectar-feeding insects are also able to use this type of blossoms.

Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant no. 51475258); the Fundamental Research Funds for the Central Universities (Grant no. 2652017063); the Research Project of the State Key Laboratory of Tribology under Contract SKLT2016B03; and the Research Grant of Sun Yat-Sen University for Bairen Plan with a contract number of 76200-18841223.

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