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Effects of airway surface liquid height on the kinetics of extracellular nucleotides in airway epithelia Tauanne D. Amarante a, Jafferson K.L. da Silva a, Guilherme J.M. Garcia b,1,n a b

Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte, Minas Gerais, Brazil Departamento de Matemática e Física, Universidade Federal de São João Del Rei, Ouro Branco, Minas Gerais, Brazil

H I G H L I G H T S

A mathematical model is proposed for the biochemical network of extracellular nucleotides in airway epithelia, which includes diffusion of nucleotides in the airway surface liquid. The model predicts the kinetics of ATP, ADP, AMP, ADO and INO after hypotonic challenge. Due to a sharp concentration gradient, nucleotide concentrations near the epithelial surface are much higher than volume-averaged concentrations. CNT3 inhibition leads to the greatest availability of ADO for A2B activation. NTPDase1/highTNAP inhibition leads to the greatest availability of ATP for P2Y2 activation.

art ic l e i nf o

a b s t r a c t

Article history: Received 13 December 2013 Received in revised form 7 July 2014 Accepted 18 August 2014 Available online 24 August 2014

Experimental techniques aimed at measuring the concentration of signaling molecules in the airway surface liquid (ASL) often require an unrealistically large ASL volume to facilitate sampling. This experimental limitation, prompted by the difﬁculty of pipetting liquid from a very shallow layer ( 15 μm), leads to dilution and the under-prediction of physiologic concentrations of signaling molecules that are vital to the regulation of mucociliary clearance. Here, we use a computational model to describe the effect of liquid height on the kinetics of extracellular nucleotides in the airway surface liquid coating respiratory epithelia. The model consists of a reaction–diffusion equation with boundary conditions that represent the enzymatic reactions occurring on the epithelial surface. The simulations reproduce successfully the kinetics of extracellular ATP following hypotonic challenge for ASL volumes ranging from 25 μl to 500 μl in a 12-mm diameter cell culture. The model reveals that [ATP] and [ADO] reach 1200 nM and 2200 nM at the epithelial surface, respectively, while their volumetric averages remain less than 200 nM at all times in experiments with a large ASL volume (500 μl). These ﬁndings imply that activation of P2Y2 and A2B receptors is robust after hypotonic challenge, in contrast to what could be concluded based on experimental measurements of volumetric concentrations in large ASL volumes. Finally, given the central role that ATP and ADO play in regulating mucociliary clearance, we investigated which enzymes, when inhibited, provide the greatest increase in ATP and ADO concentrations. Our ﬁndings suggest that inhibition of NTPDase1/highTNAP would cause the greatest increase in [ATP] after hypotonic challenge, while inhibition of the transporter CNT3 would provide the greatest increase in [ADO]. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Purinergic signaling Mucociliary clearance Bronchial epithelium Mathematical model Enzyme inhibition

1. Introduction Extracellular nucleotides are essential in protecting the respiratory tract against inhaled pathogens due to their key role in regulation of mucociliary clearance. By activating the membrane receptors n

Corresponding author. E-mail address: [email protected] (G.J.M. Garcia). 1 Present address: Biotechnology & Bioengineering Center, Medical College of Wisconsin, Milwaukee, WI, USA. http://dx.doi.org/10.1016/j.jtbi.2014.08.030 0022-5193/& 2014 Elsevier Ltd. All rights reserved.

A2B and P2Y2, the concentrations of extracellular adenosine (ADO) and adenosine triphosphate (ATP) regulate mucus secretion rate, ciliary beating frequency, mucus hydration, and mucus viscosity (Schmid et al., 2011). A quantitative understanding of the concentrations of ADO and ATP may be important for the development of new therapies for respiratory pathologies, especially cystic ﬁbrosis (Zuo et al., 2008; Garcia et al., 2011, 2013; Herschlag et al., 2013). Individuals with this genetic disease have chronic bacterial lung infection due to impaired mucociliary clearance, that is caused by mucus dehydration and excessive mucus secretion (Boucher, 2007, 2007).

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The biochemical network that regulates the concentrations of extracellular ATP and ADO was recently described quantitatively by a computer model (Zuo et al., 2008; Garcia et al., 2011) based on experimental data collected in cultures of human respiratory epithelia with airway surface liquid (ASL) at a height of 1500 μm. Although this model reproduced its target experimental data, it cannot be used to investigate the regulation of extracellular nucleotides in vivo, because in vivo the ASL height is approximately 15 μm (Harvey and Schlosser, 2009). Aiming to describe the behavior of the system in vivo, we improved the computer model by including diffusion and using experimental data from the literature where the extracellular ATP concentration was investigated for ASL heights ranging from 14 μm to 2730 μm (Okada et al., 2006). The new version of the model, proposed in this work, consists in the solution of the reaction–diffusion equation in cylindrical coordinates, with boundary conditions that express nucleotide secretion, nucleoside absorption, and the reactions catalyzed by ectonucleotidases on the epithelial surface. The model successfully predicts the behavior of ATP concentration at the epithelial surface after hypotonic challenge. On the cell surface, where the P2Y2 and A2B receptors are located, the model shows that ATP concentration reaches the range for P2Y2 receptor activation ( 1 μM Mason et al., 1991) in response to hypotonicshock induced ATP secretion. In addition, the model predicts the kinetics of adenosine diphosphate (ADP), adenosine monophosphate (AMP), ADO and inosine (INO), for which no experimental data is available after hypotonic challenge. Finally, and most importantly, the model was used to predict which enzymes when inhibited provide the greatest increase in ATP and ADO concentrations, thus identifying the best targets for therapeutic modulation of mucociliary clearance.

2. Materials and methods 2.1. Motivation Zuo and coworkers (Zuo et al., 2008; Garcia et al., 2011) proposed a mathematical model that describes the kinetics of nucleotide and nucleoside ASL concentrations by coupled differential equations. It takes into account that ATP, ADP and AMP are secreted by the respiratory epithelium to the ASL, where they are metabolized by ecto-enzymes and converted into the nucleosides ADO and INO which in turn are absorbed by the respiratory epithelium (Fig. 1). The original model was based on experimental data obtained by off-line measurements after pipette sampling of human bronchial epithelial cell cultures with ASL volume of 350 μl, which corresponds to a liquid height of 1500 μm (Zuo et al., 2008). To construct the model, the authors assumed that the inﬂuence of diffusion could be neglected in determining the concentration of extracellular purines. There was a logical inconsistency in the original model as it relied on the ASL height in vivo ( 15 μm Harvey and Schlosser, 2009) to neglect the diffusion, but aimed to reproduce and analyze the kinetics of cell culture experiments with a large ASL volume (350 μl), where the assumption of no vertical gradients fails. We performed some analytical calculations (see Supporting Material B) that show that diffusion and ATP degradation occur at similar time scales, and therefore it is essential to take diffusion into account in order to accurately reproduce the kinetics of extracellular nucleotides in cell culture experiments involving large ASL volumes. Thus, in spite of reproducing the in vitro data for volumetric-average concentrations at ASL height of 1500 μm, it was unclear whether the original model reproduced the kinetic behavior in vivo, where ASL height is 15 μm.

Fig. 1. Nucleotide transport and metabolism in the airway surface liquid (ASL). Respiratory epithelia release ATP, ADP and AMP at rates JATP, JADP and J AMP , respectively. The nucleotides are dephosphorylated by ecto-enzymes into ADO and INO, which are absorbed by the transporter CNT3 at rates jADO e jINO, respectively. ATP and ADO are signaling molecules that regulate mucociliary clearance by activating the receptors P2Y2 and A2B. The abbreviations of the enzymes included in the model are: NTPDase (Nucleoside triphosphate diphosphohydrolases), NSAP (tissue non-speciﬁc alkaline phosphatase), E-NPPs (EctoNucleotide pirophosphatase / phosphodiesterases), ecto 5'-NT or CD73 (ecto-5'nucleotidase), ecto-AK (ecto adenilase kinase), ADA1 (adenosine deaminase 1).

2.2. Experimental data Okada and coworkers (Okada et al., 2006) performed a series of in vitro experiments to investigate the extracellular ATP concentration at an ASL height similar to the ASL height in vivo. To simulate the native human airway epithelia they used a well differentiated primary human bronchial epithelial cell culture. They accessed ATP concentration after hypotonic challenge for ASL volumes ranging from 25 μl to 500 μl using three different techniques, namely (1) pipette or micro-sampling followed by offline luminometry; (2) real-time luminometry with soluble luciferase; (3) real-time luminometry with cell-attached SPA-luc. The last technique measures ATP concentration on the cell surface, while the other two techniques provide an average ATP concentration throughout the ASL volume. Since the P2Y2 receptors are located at the cell surfaces, physiologically the most important measurement is the concentration at the cell surface. Their experimental data show that the volumetric-average ATP concentration after hypotonic challenge depends on ASL height (Fig. 2). Their data also show that the volumetric-average ATP concentration is much lower than the concentration near the cell surface. Therefore, the assumption by Zuo et al. (2008) that nucleotide concentration is uniform in the ASL is not valid. 2.3. Model equations To reproduce the experimental data by Okada et al. (2006) and solve the inconsistency of the original model, diffusion was added to the mathematical model of Zuo and collaborators (Zuo et al., 2008) (Fig. 1). Our model consists in solving the diffusion equation ∂c ¼ D∇2 c; ∂t

ð1Þ

with boundary conditions that take into account the reactions that occur at the cell surface. Here c is the ASL concentration of a chemical specie ([ATP], [ADP], [AMP], [ADO] or [INO]; units of mol/ m3), t is the time (unit of s) and D ¼ 4:6 10 10 m2 =s is the diffusion constant of ATP (Hubley et al., 1996). Experimental data (Bowen and Martin, 1964) suggest that the diffusion coefﬁcient is nearly identical for all nucleotides and nucleosides. Bowen and Martin reported that the diffusion coefﬁcient is D ¼ 4:3 10 10 m2 =s for ATP, D ¼ 4:0 10 10 m2 =s for ADP, D ¼ 4:3 10 10 m2 =s for AMP, D ¼ 5:2 10 10 m2 =s for ADO, and D ¼ 5:2

T.D. Amarante et al. / Journal of Theoretical Biology 363 (2014) 427–435

-1

1500

1000

ATP(nM)

ATP(nM)

1500

500

0

429

0

hypotonic

1

2

3

-1

Time (min)

V=25µl V=50µl V=100µl V=300µl V=500µl

1000

500

0

0

hypotonic

1

2

3

Time (min)

ATP(nM)

1500

-1

1000

500

0

0

hypotonic

1

2

3

Time (min)

Fig. 2. Kinetics of ASL ATP concentration after hypotonic challenge in primary cell cultures of human bronchial epithelium as reported by Okada et al. (2006). ATP concentrations were measured by different techniques and several ASL volumes. (A) Luminometry off-line samples, (B) soluble luciferase and (C) SPA-luc attached cells. Data from Okada et al. (2006).

^ The ﬂux of particles through a boundary is given by F ¼ D∇c n, where n^ is a unit vector normal to the boundary. Both the secretion of ATP, ADP, and AMP, and the uptake of INO and ADO occur on cell surfaces (z¼0 boundary). As enzymes are also located at the cell membrane, the boundary conditions at z¼ 0 are ∂c D ∂z

z¼0

Fig. 3. Cylindrical geometry of the cell cultures used in the mathematical model. Nucleotide release, nucleoside uptake, and enzymatic reactions occur only at the epithelial surface (z ¼ 0) as indicated by the arrows.

ð2Þ

Thus, the model equations are D

∂½ATP ¼ J ATP νAK ∑ νi ; ∂z z ¼ 0 i ¼ 1;2;3;10

ð3Þ

D

∂½ADP ¼ J ADP þ2νAK þ ∑ νi ∑ νi ; ∂z z ¼ 0 i ¼ 1;2;3 i ¼ 4;5

ð4Þ

D

∂½AMP ¼ J AMP νAK ∑ νi þ ∑ νi ; ∂z z ¼ 0 i ¼ 6;7;8 i ¼ 4;5;10

ð5Þ

D

∂½ADO ¼ jADO þ ∑ νi ν9 ; ∂z z ¼ 0 i ¼ 6;7;8

ð6Þ

D

∂½INO ¼ jINO þ ν9 ; ∂z z ¼ 0

ð7Þ

10

10 m2 =s for INO. Therefore, we assume that D ¼ 4:6 10 10 m2 =s for all chemical species in the model. Since the cell cultures used by Okada et al. (2006) were cylindrical with a radius R ¼6.0 mm, the diffusion equation was investigated in cylindrical coordinates (Fig. 3). The liquid height H depends on the amount of ASL in the experiments, ranging from 25 to 500 μl. To determine the liquid height in the model, it was assumed that the upper surface of ASL is ﬂat, so that H ¼V/A where V is the liquid volume and A ¼ π R2 is the surface area of the cell cultures. For the ASL volume of 350 μl used by Picher and collaborators (Picher and Boucher, 2001), this formula predicts an ASL height of 3100 μm which is larger than the experimental value of 1500 μm. This discrepancy is due to the meniscus that forms along the border of the cell culture, which we have neglected in this analysis. Although the meniscus was neglected due to the greater complexity that its consideration would add to this analysis, a good agreement was found between model predictions and experimental data.

¼ release rate uptake rate þ reaction rates:

where J is the nucleotide secretion rate and j is the nucleoside uptake rate, and νi is the rate of the chemical reaction i described in Table 1.

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Table 1 Kinetic parameters from Zuo et al. (2008). The parameters J and Vmax are in units of nmol/min/ml and KM has unity of μM. Reaction

Enzyme/Transporter

Parameters

Experimental

Simulation

ATP-ADP þ Pi

NTPDase 1/highTNAP

ð1Þ V ð1Þ max , K M

2.1, 12

6.3, 16.3

NTPDase 3

V ð2Þ max , V ð3Þ max , V ð4Þ max , V ð5Þ max , V ð6Þ max , V ð7Þ max , V ð8Þ max , K ðinÞ ATP K ðinÞ ADP V ð9Þ max , V ð10Þ max , Þ V ðfmax , ðbÞ V max ,

8.1, 136

15.5, 114.9

14.1, 451

20.0, 418.0

lowTNAP ADP-AMP þ Pi

NTPDase 1/highTNAP NTPDase 3/lowTNAP Ecto 50 -NT

AMP-ADO þ Pi

highTNAP lowTNAP Feed forward inhibition Feed forward inhibition

K ð2Þ M K ð3Þ M K ð4Þ M K ð5Þ M K ð6Þ M K ð7Þ M K ð8Þ M

7

28.4

10

20.4

K ð9Þ M

0.6, 25

0.3, 17.0

K ð10Þ M

0.9, 22

1.2, 28.3

Þ Þ K ðfATP , K ðfAMP

3.8, 23, N/A

2.2, 30.4, 24.7

K ðbÞ ADP

1.8, 43

2.2, 61.8

0.0012 – – 0.12, 2.2

0.0011 0.0131 0.0125 0.2, 1.2

–

0.2, 1.2

ATP þ AMP-2ADP

EctoAK (direta)

ATP þ AMP’2ADP

EctoAK (inversa)

ATP ﬂux ADP ﬂux AMP ﬂux ADO uptake

– – – CNT3

JATP JADP JAMP

INO uptake

CNT3

ðu;2Þ V ðu;2Þ max , K M

ðu;1Þ V ðu;1Þ max , K M

The equations that model the enzymatic reaction rates and uptake rates have the same form as in the original model (Zuo et al., 2008; Garcia et al., 2011). So it was considered that, except for the AMP hydrolysis and the ecto-AK reversible reaction, all reaction rates can be modeled by Michaelis–Menten kinetics,

νj ¼

V jmax ½S K jM þ ½S

;

ð10Þ

where the index j refers to the enzymes j ¼ 1–5, 9–10 listed in Table 1, [S] is the substrate concentration, Vmax is the maximum rate, and KM is the Michaelis constant. Since the ecto-enzymes are attached to the apical membrane of epithelial cells, the reactions occur on the cell surface, which means that [S] is the concentration at z ¼0. It was assumed that the reversible reaction ATP þ AMP⇄2ADP catalyzed by the enzyme ecto-AK follows a Bi Bi mechanism (Valero et al., 2006; Voet and Voet, 2010) described by V fmax

νAK ¼ 1þ

inhibition V jmax ½AMP

νj ¼

ð9Þ

K fATP

½ATP

þ

K fAMP

½AMP

þ

K fATP K fAMP

½ATP½AMP

V bmax Kb K bADP 1 þ 2 ADP þ ½ADP ½ADP

!2 :

ð11Þ where Vfmax is the maximum rate of the forward reaction ATP þ AMP-2ADP, Vbmax is the maximum rate of the backward reaction 2ADP-ATP þ AMP and KfATP, KbADP, KfAMP are constants. It was experimentally observed that ATP and ADP inhibit AMP hydrolysis because they compete for binding the enzymes that catalyze this reaction. Therefore, the hydrolysis rate of AMP was described by Michaelis–Menten kinetics with competitive

1.7, 13.0 6.2, 27.2

E-NPPs

∂c jz ¼ H ¼ 0: ∂z

1.8, 14

11.9, 694.9

ATP-AMP þ PPi

ð8Þ

10.7, 83.9

9.8, 717

ADA1

∂c jr ¼ R ¼ 0; ∂r

0.5, 2.8

9.1, 103 4.2, 36

ADO-INO

The boundary condition is reﬂective around the cylinder wall and at the top of the liquid

1.4, 5

K jM

1þ

½ATP K in ATP

þ

½ADP K in ADP

!

;

ð12Þ

þ ½AMP

in where the index j¼6,7,8 and Kin ATP and KADP are inhibition constants (Table 1). The uptake rates of ADO and INO must consider the competition of these nucleosides for the same transporter, so they are given by

jADO ¼ K u;1 M

jINO ¼ K u;2 M

V u;1 max ½ADO ! ; ½INO 1 þ u;2 þ ½ADO KM V u;2 max ½INO ! ; ½ADO 1 þ u;2 þ½INO KM

ð13Þ

ð14Þ

u;2 where V u;1 max , V max are the maximum uptake rates of [ADO] and u;2 [INO] respectively and K u;1 M , K M are constants. To implement the model, we used the kinetic parameters (Vmax's and K M 's) obtained by Zuo et al. (2008) (see Table 1) because it is a complete set that reproduces the experimental steady-state concentrations as well as the transient after addition of 100 μM ATP. Although the nucleotide ﬂuxes and V max have units of mol/s/m2 in our model, these parameters were written in the original model in units of nmol/min/ml under the assumption that the ASL volume was ﬁxed at V¼ 0.35 ml. Thus, to use the kinetic parameters of Table 1, we multiplied J and V max by (V/A), where V¼0.35 ml and A¼1.12 cm2 were respectively the ASL volume and area used in the experiments to which the model parameters were ﬁtted (Picher and Boucher, 2001; Zuo et al., 2008).

2.4. ATP secretion after hypotonic challenge To reproduce the experimental data by Okada and coworkers (Okada et al., 2006), the rate of ATP release ðJ ATP ðtÞÞ in response to hypotonic challenge must be included in the model. Unfortunately, the mechanism of ATP release by the epithelial cells is still

T.D. Amarante et al. / Journal of Theoretical Biology 363 (2014) 427–435 2000

[ATP](nM)

1500

1000

500

0

0

0.5

1

1.5

Time (min) Fig. 4. ATP concentration at the cell surface (z¼ 0) after hypotonic challenge in the presence of enzyme inhibitors. Symbols: experimental data for an ASL volume of 50 μl; curve: exponential ﬁt (Eq. (15)).

unknown, thus, to determine J ATP ðtÞ we used the experimental data for [ATP](t) after hypotonic challenge in the presence of an inhibitor mixture, as measured by Okada et al. (2006) (Fig. 4). In the presence of enzyme inhibitors, there is no ATP hydrolysis so that dN ATP =dt ¼ AJ ATP ðtÞ, where N ATP ¼ V½ATP is the number of moles of ATP in the ASL, so that J ATP ðtÞ ¼ ðV=AÞd½ATP=dt. We ﬁtted the experimental data on Fig. 4 using the following exponential ﬁt ½ATP ¼ cðtÞ ¼ c1 ð1 e λt Þ;

ð15Þ

where the values of the ﬁtting constants are c1 ¼ 1701 nM and λ ¼4.487 min 1. Therefore, the ATP secretion rate following hypotonic challenge is J ATP ðtÞ ¼ V=Aλc1 e λt where V¼50 μl and A¼ 1.12 cm2 are the ASL volume and surface area used in these experiments. This equation is used until the ﬂux becomes lower than the baseline ﬂux J basal ¼ 6:25 10 11 mol=s=m2 (Zuo et al., 2008) (Table 1), when the baseline ﬂux kicks in. The set of coupled differential equations was solved numerically using the ﬁnite volume method, which is a discretization method used to transform a set of partial differential equations into a set of algebraic equations (Patankar, 1980). In this method, continuous functions are only determined at a ﬁnite number of points, dubbed the mesh or grid. The domain is divided into control volumes such that there is one control volume around each point of the grid. The differential equations are then integrated over each control-volume to derive the algebraic equations. Its advantage compared to other numerical methods for solving differential equations is that, for any grid size, the ﬂux from a control volume to another is conserved. In other words, the laws of physics are obeyed for each control volume (Patankar, 1980). Two codes were written independently in MATLAB and C þ þ by two different investigators. The codes were benchmarked by comparing the numerical prediction to an analytical solution of the time-dependent diffusion equation with time-independent boundary conditions (see Supporting Material A).

3. Results One of the assumptions made by Zuo and collaborators (Zuo et al., 2008) to built their model was to consider that the nucleotide concentration is uniform in the ASL. To illustrate the importance of taking diffusion into account, we used our model to investigate how ATP and ADO concentrations vary along the centerline of the cell culture for an ASL volume of 350 μl (Fig. 5). This was the ASL volume of the experimental data used by Zuo et al. (2008) to ﬁt their model parameters. Fig. 5 demonstrates that the assumption of an uniform ASL is invalid.

431

The qualitative behavior of the volumetric-average ATP concentration predicted by the model is in agreement with the experiments, i.e, both the model prediction (Fig. 6(a)) and experimental data (panels A and B in Fig. 2) indicate that ATP concentration after hypotonic challenge is inversely related to ASL volume. This behavior is expected because ATP becomes more diluted as the ASL volume increases. However, the curves predicted by the model reach their peak at time t 13 s which is sooner than the experimental peak which occurs at t 35 s. Furthermore, the peak of the volumetric-average ATP concentration is smaller than measured experimentally. The model prediction for the peak volumeaveraged ATP concentration for an ASL volume of 50 μl is 180 nM, while the experimental value obtained with soluble luciferase is 960 nM and the value measured with pipetting sample is 300 nM (Fig. 7(a)). Thus the model prediction value is closer to measurements by pipetting sample. The large difference between the two experimental techniques is noteworthy and may reﬂect differences in the accuracy of each experimental technique. The predicted [ATP] at z ¼0 (Fig. 6(b)) is in good agreement with the experimental values measured with cell-attached SPA-luc (Fig. 2(c)). Both the predicted values and experimental curves suggest that the average concentration at z ¼0 does not depend as strongly on ASL volume (Fig. 7(a)). The model prediction approaches the experimental behavior quantitatively, namely the peak [ATP] at z ¼0 after hypotonic challenge is 1200 nM both in the model and in the experiments for all ASL heights studied. The P2Y2 receptor is situated on the apical membrane surface and its ED50 is 1000 nM of ATP (Mason et al., 1991), thus a robust activation is expected after hypotonic challenge. However, there is a discrepancy between the estimated time for ATP concentration on the surface to reach its maximum, t 3 s in the model (Fig. 6 (b)), and the time experimentally measured t 25 s (Fig. 2(c)). The radial proﬁle of ATP concentration is displayed in Fig. 8. Although our model takes into account the cylindrical geometry of the cell cultures, no radial concentration gradient was predicted by the model. This quasi-uniform concentration was observed at all time points (Fig. 8(a)), at all heights from the epithelial surface (Fig. 8(b)), and for all nucleotides. This behavior can be explained by the fact that both nucleotide release and the enzymatic reactions occur on the epithelial surface, with no radial dependence. As previously discussed, ATP and ADO have important biological roles in regulating mucociliary clearance. Therefore, it is important to understand the kinetics of both of these purines in the ASL. The experimental measurements by Okada et al. (2006) provide information only for ATP concentration. Given that our model reproduces the experimental behavior of ATP, it becomes an useful tool to investigate the behavior of ADO in ASL in vivo. The predicted behavior for ADO when the ASL is subjected to hypotonic challenge is depicted in Fig. 6(c) and (d). As in the case of ATP, the maximum volumetric-average ADO is inversely dependent on liquid height (Fig. 6(c)), but the peak [ADO] at z ¼0 is almost independent of liquid height (Figs. 6(d) and 7(b)). As expected, the peak ADO concentration occurs after the peak ATP concentration, because ADO is a metabolite of ATP. Our results suggest that the peak ½ADOz ¼ 0 occurs 25 s after the peak ½ATPz ¼ 0 . The steady state ADO concentration ( 160 nM Zuo et al., 2008; Garcia et al., 2011) is enough to partially activate the A2B receptor. However, at steady state, only 14% of A2B receptors are active. This estimate was obtained assuming that the activity of the receptor A2B obeys Michaelis–Menten kinetics (Alon, 2007) activation ½ADO ¼ ; maximum activation ED50 þ ½ADO

ð16Þ

where ED50 ¼ 1 μM (Huang et al., 2001; Lazarowski et al., 2004) is the ADO concentration required for 50% of the maximum

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1000

2500

[ATP] (nM)

800

[ADO] (nM)

t=7.5s t=15s t=30s t=45s

600 400 200 0

0

1

2

1500 1000 500 0

3

t=15s t=34s t=45s t=60s

2000

0

1

2

z (mm)

3

z (mm)

Fig. 5. Concentration proﬁle at the centerline axis of the cell culture (r¼ 0) at different times after hypotonic challenge, including the time of peak concentration (t ¼7.5 s for [ATP] and t¼ 34.0 for [ADO]) for an ASL volume of 350 ul. (A) [ATP] as a function of the distance from the cell surface (z); (B) [ADO] as a function of the distance from the cell surface (z).

VOLUMETRIC-AVERAGE

1000

500

0

0

1

2

AT CELL SURFACE

1500

ATP (nM)

ATP (nM)

1500

1000

500

0

3

V=25µl V=50µl V=100µl V=300µl V=500µl

0

1

2500

2500

2000

2000

1500 1000 500 0

0

1

2

3

Time (min)

ADO (nM)

ADO (nM)

Time (min)

2

1500 1000 500 0

3

0

1

Time (min)

2

3

Time (min)

Fig. 6. Concentration of ATP and ADO after hypotonic challenge predicted by the model for different volumes (500 μl (open triangles), 300 μl (gray diamonds), 100 μl (solid triangles), 50 μl (open circles), and 25 μl (solid circles)). (A) and (C) volumetric-average concentration; (B) and (D) concentration on the cell surface (z ¼0).

3000

Peak [ADO] (nM)

Peak [ATP] (nM)

2000 1500 1000 500 0

0

100

200

300

400

ASL volume (µl)

500

2500 2000 1500 1000 500 0

0

100

200

300

400

500

ASL volume (µl)

Fig. 7. Peak ATP and ADO concentrations after hypotonic challenge as a function of ASL volume in a cylindrical cell culture. Model predictions for the concentration at the cell surface (solid line) and volumetric-average concentrations (dashed line) are compared to the experimental measurements (available for ATP only) at the cell surface (solid diamonds) and volumetric-average concentrations (gray squares and open circles). The experimental data for volumetric-average concentration were obtained with two different techniques, (1) pipette or micro-sampling followed by off-line luminometry (gray squares); (2) real-time luminometry with soluble luciferase (open circles).

T.D. Amarante et al. / Journal of Theoretical Biology 363 (2014) 427–435

1000

1500 t=5s t=15s t=30s t=45s

1000

z=0µm z=18.6µm z=37.2µm z=55.7µm z=74.3µm z=111µm z=223µm

800

[ATP] (nM)

[ATP] (nM)

433

500

600 400 200

0

0

1

2

3

4

5

0

6

0

1

2

r (mm)

3

4

5

6

r (mm)

Fig. 8. Radial proﬁle of ATP concentration. (A) At the cell surface (z ¼ 0), for an ASL volume of 350 μl, [ATP] as a function of the distance from the centerline axis (r) at different times; (B) at the time of peak concentration (t¼ 7.5), for an ASL Volume of 25 μl, [ATP] as a function of the distance from the centerline axis (r) at different distances from the cell surface (z).

VOLUMETRIC-AVERAGE

2000 1500 1000 500 0

0

1

2

AT CELL SURFACE

2500

Concentration (nM)

Concentration (nM)

2500

3

ATP ADP AMP ADO INO

2000 1500 1000 500 0

0

Time (min)

1

2

3

Time (min)

Fig. 9. Model prediction for the concentrations of ATP, ADP, AMP, ADO, and INO for an ASL volume of 25 μl after hypotonic challenge.

activation. In Fig. 6(d) we see that hypotonic challenge produces a large increase in the concentration of adenosine in the ASL due to the stimulated ATP secretion. The model predicts that, after hypotonic challenge, ADO concentration on the surface remains above the ED50 of A2B for a time of approximately 50 s. In other words, for 50 s the activation of A2B is greater than 50% (Fig. 6(d)). Using our model, we also predict the behavior of the other nucleotides for an ASL height similar to the liquid height in vivo as shown in Fig. 9. We used the model to study the effect of inhibiting each enzyme that catalyzes the reactions that involve ATP and ADO. In particular, we ask which enzymes when inhibited lead to the greatest increase in the concentrations of ATP and ADO available for activation of P2Y2 and A2B receptors (i.e., concentrations at z ¼0). We also investigated how enzyme inhibition affects the activation period of P2Y2 and A2B , which we deﬁned as the period of time during which ATP and ADO concentrations were greater than the ED50 of the P2Y2 and A2B receptors, respectively. The results, depicted in Fig. 10, show that NTPDase 1/highTNAP inhibition provides a 2-fold increase in the peak ATP concentration and a 3-fold increase in the activation period of P2Y2. On the other hand we observe that the inhibition of the transporter CNT3 provides the greatest increase in ADO concentration and causes a permanent activation of A2B receptors. To highlight the importance of including diffusion in order to reproduce the experimental measurements of ATP concentration after hypotonic challenge in cultures of airway epithelium a comparison was performed between the original mathematical model (Zuo et al., 2008; Garcia et al., 2011) (model without diffusion) with the revised model introduced in this manuscript

(model with diffusion) (see Supporting Material B). Since the original model assumed a uniform nucleotide concentration throughout the ASL, the ATP concentration at the cell surface was assumed to be identical to the volumetric-average [ATP]. The original model was not able to access the actual concentration at the surface, underpredicting the concentration of greatest physiological importance, the one available for the receptors.

4. Discussion Our computational results reinforce the view that cell signaling in respiratory epithelia is highly localized (Okada et al., 2006; Ribeiro et al., 2003). This localized signaling is accomplished via a source-and-sink paradigm, in which the signaling molecule originates very close to the location where it is metabolized (e.g., extracellular ATP is released and degraded at the epithelial surface). This strategy leads to a very sharp concentration gradient (Fig. 5(a)) whose spatial scale is determined by the diffusion coefﬁcient of the signaling molecule and its secretion/degradation rates. The existence of these sharp concentration gradients creates a challenge for biochemists who investigate the signaling network that regulates mucociliary clearance. Conventional pipette sampling combined with off-line measurements require an unrealistically large ASL volume, and thus are likely to underpredict the true physiological concentrations near the epithelial surface, possibly leading to incorrect conclusions. For example, based on volumetric measurements of ATP concentration alone, one could incorrectly conclude that ATP never reaches the concentration needed for P2Y2 activation after hypotonic challenge (Okada et al., 2006). In contrast, our simulations conﬁrm that hypotonic-

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Fig. 10. Effect of enzyme inhibition on (A) peak [ATP] and peak [ADO] and (B) activation period of ATP and ADO.

challenge induced ATP secretion leads to a robust activation of P2Y2 receptors by increasing the ATP concentration locally (Fig. 5(a)). This activation, however, is short-lived because ATP is quickly metabolized, so that its concentration drops below the ED50 for P2Y2 activation in less than 1 min. Although model simulations reproduced quite well the experimental behavior of ATP near the cell surface, two important discrepancies were noted when comparing model predictions and experiments for the volumetric-average ATP concentration. First, the time scale of events was much faster in the model where ATP reached its peak concentration in 13 s, in contrast to the 35 s observed experimentally. Second, the volumetric-average ATP concentration was much lower than experimental measurements with soluble luciferase (Fig. 7(a)). In this regard, one should note that there were signiﬁcant differences between the experimental data obtained with off-line luminometry (Fig. 2(a)) and with soluble luciferase (Fig. 2(b)). This inconsistency between the two experimental techniques was not discussed by Okada et al. (2006). Our simulations raise the possibility that the soluble luciferase technique does not provide a mere average concentration in the ASL volume, but perhaps there is a tendency to give more weight to molecules near the epithelial surface. We speculate that a bias for near-surface ATP concentration may be explained by a recent report that mucins form a dense brush near the epithelial surface which may reduce the diffusion coefﬁcient near the cell surface and thus increase the residence time of large soluble molecules near the surface (Button et al., 2012). In addition, we also speculate that the discrepancies between model simulations and experimental measurements are partially due to neglecting the meniscus that forms along the cell culture border. As seen in Figs. 2 and 6, ATP kinetics depends on liquid height and the existence of the meniscus changes the vertical distance z between the air–liquid interface and the epithelial surface. A future version of this model should include the meniscus to test whether this would lead to a better agreement between simulations and experiments. In summary, we developed a model aimed at describing the effects of liquid height on the kinetics of extracellular nucleotides in the airway surface liquid that coats respiratory epithelia. The model was used to investigate the role of each enzyme on ATP and ADO concentrations. We partially reproduced the experimental data for ATP concentration at the cell surface obtained by Okada et al. (2006). The volumetric-average ATP concentration prediction reproduces only the qualitative behavior of the experimental data. As the receptors are localized on the apical surface we are more interested on the concentration at the surface. Due to the good agreement between model predictions and the experimental data for ATP concentration at z¼0, it was possible to make predictions for ADO concentration at the cell surface and to simulate the effect of enzyme inhibition. According to the model, peak [ATP] is the greatest and P2Y2 activation period is the longest with NTPDase 1/highTNAP inhibition, while peak [ADO] is the greatest and A2B

activation period is the longest with CNT3 inhibition in bronchial epithelium. It is expected that an increase on ATP and ADO concentrations will lead to an improvement on mucociliary clearance.

Supporting citations Reference Butkov (1968) appears in the Supporting Material A.

Acknowledgments This research was funded by the Brazilian science agencies Conselho Nacional de Desenvolvimento Cientíﬁco e Tecnológico (CNPq) and Fundação de Amparo à Pesquisa do estado de Minas Gerais (Fapemig). We are also grateful to Dr. Richard Boucher and Dr. Timothy Elston (University of North Carolina at Chapel Hill) and to Dr. Maryse Picher for some discussions on this project.

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