Fluid and solute transport across the retinal pigment epithelium: a theoretical model

Abstract

The retina is composed of two main layers—the neuroretina and the retinal pigment epithelium (RPE)—that are separated by a potential gap termed the sub-retinal space (SRS). Accumulation of fluid in the SRS may result in a retinal detachment. A key function of the RPE is to prevent fluid accumulation in the SRS by actively pumping fluid from this space to the choroid. We have developed a mathematical model of this process that incorporates the transport of seven chemical species: Na⁺, K⁺, Cl⁻, ${\text{HCO}}_3^{-}$, H⁺, CO₂ and H₂CO₃. This allows us to estimate solute and water fluxes and to understand the role of the different membrane ion channels. We have performed a global sensitivity analysis using the extended Fourier amplitude sensitivity test to investigate the relative importance of parameters in generating the model outputs. The model predicts that flow across the RPE is driven by an osmotic gradient in the cleft gap between adjacent cells. Moreover, the model estimates how water flux is modified in response to inhibition of membrane ion channels and carbonic anhydrase (CA). It provides a possible explanation for how CA inhibitors, which are used clinically to prevent fluid accumulation in the SRS, may be acting.

1. Introduction

The retina is composed of two main layers—the neuroretina and the retinal pigment epithelium (RPE)—that are separated by a potential space termed the sub-retinal space (SRS). The RPE sits on a connective tissue layer called Bruch’s membrane, below which is the choroid. The RPE is bounded above (the apical surface) by the SRS and below by the choroidal space, as shown in figure 1. The RPE cells themselves are coupled by tight junctions and this layer constitutes the outer blood–retinal barrier.

Figure 1.

Sketch of the retinal pigment epithelium cell. Drawing by Prof. Federica Grillo (University of Genoa, Italy). (Online version in colour.)

The photoreceptors face the apical surface of the RPE. The functions of the RPE include pumping fluid from the SRS to the choroid and transporting the carbon dioxide produced by the retina, and particularly by the photoreceptors, to the choroid, where it can be eliminated [1].

In a healthy retina, the RPE actively pumps fluid from the SRS to the choroid at a rate of about 4–10 μl h⁻¹ cm⁻² [2,3]. It is a failure of this pumping that can lead to fluid accumulation in the SRS, which is associated with a number of pathological conditions, including serous retinal detachment, age-related macular degeneration and diabetic retinopathy [4]. The aim of this model is to understand how this fluid transport occurs.

The fluid transport is isotonic (i.e. without a significant difference in osmolarity between the apical and basal regions) and the standard explanation is that ‘water follows the ions’, i.e. water transport occurs in the main direction of ion transport [5]. However, the physical mechanisms behind this coupling are obscure.

A number of mathematical models have been developed to study isotonic fluid transport by epithelia. Diamond & Bossert [6] first proposed what is known as the standing gradient osmotic flow (or local osmosis) model. This model postulates the presence of an osmolarity gradient that is generated in the cleft gap between two adjacent cells. This gradient is generated by the active pumping of ions and it drives fluid into and along the cleft, resulting in transepithelial water transport. This model has been extended and applied to study isotonic water transport in a number of epithelia (e.g. [7–10]).

Electro-osmosis has been proposed as an alternative mechanism. It occurs as a result of the interaction of an electric field with non-zero spatial charge density that forms in a thin region close to the cell membrane. We previously developed a mathematical model that couples water and ion flow across the RPE that incorporates both local and electro-osmotic effects [11], allowing the investigation of the relative importance of these two possible mechanisms for fluid transport. This model looked at the presence of three different species: sodium (Na⁺), potassium (K⁺) and chloride (Cl⁻). The model predicted that local osmosis, driven by the osmolarity gradient in the cleft gap, is comparable to the experimentally measured values of the flow and extensively dominates electro-osmosis.

However, that model had a number of limitations and the objective of this work is to extend the work in [11] to include the role of bicarbonate (${\text{HCO}}_3^{-}$), carbon dioxide (CO₂), carbonic acid (H₂CO₃) and protons (H⁺) along with the relevant ion channels, as they are thought to play an important role in fluid transport [12], with the aim of generating a more accurate quantitative estimate of transepithelial water and ion fluxes. Adding ${\text{HCO}}_3^{-}$ to our model is essential if one wants to describe physiological ion fluxes in detail. In our previous model, the Na⁺ flux across the RPE was imposed to be zero. The addition of these extra species now allows the sodium flux to be modelled.

The importance of CO₂ and bicarbonate in fluid transport is shown by the enhancement of fluid flow by carbonic anhydrase (CA) inhibitors [13,14]. The enzyme CA catalyses the conversion of CO₂ and H₂O to carbonic acid (H₂CO₃) and it is well known that some patients with macular oedema will respond to CA inhibition [15,16]. Adijanto et al. [1] conducted in vitro experiments on human fetal RPE, suggesting that an increase in apical and basal CO₂ enhances water transport. The motivation behind developing the current model is to better understand such processes.

The main objectives of the present modelling work are thus as follows. First, we examine the coupling between water and ion transport in the context of the RPE. Second, we consider the impact of CO₂, ${\text{HCO}}_3^{-}$, pH and H₂CO₃ on water transport, including alterations of the balance between the species via CA inhibition. Finally, we explore the effects of ion channel regulation on water transport.

2. Methods

2.1. Model set-up

We represent the RPE cell layer as a row of rectangles with length L and width H in a two-dimensional space (figure 2). The cells (denoted with superscript i for intracellular; see table 1) are separated by a region of thickness 2h, representing the cleft gap (g) and connected with tight junctions (tj). Above the cell layer there is the apical region (a), which is the SRS. The RPE sits on Bruch’s membrane, and below this is the basal region (b), which is the choroid.

Figure 2.

Left cell: Diagram of ion channels in the RPE cell membranes, with a sketch of the domain and coordinate system. Right cell: Typical values for species concentrations (in mM) and electrical potential (mV) under the open-circuit condition. The prescribed values are in black and the values predicted by the model are in blue for the reference parameters in table 2, with an identical concentration of CO₂ in the apical and basal regions. (Online version in colour.)

Table 1.

List of notations and variables used in the model. Superscripts s and m denote generic regions from the set {a, b, g, i}. We highlight with a ‘*’ the unknowns computed by the model in the regions reported in parenthesis. With ‘**’ we denote the variables that are functions of the unknowns.

a, b, g, i	values of s labelling apical, basal, cleft gap and intracellular regions, respectively
$n_k^s$, *(i, g)	concentration of species k in region s (mM) such that $n_0^s=$[Na+]s, $n_1^s=$[K+]s, $n_2^s=$[Cl−]s, $n_3^s=$[${\text{HCO}}_3^{-}$]s, $n_4^s=$[H+]s, $n_5^s=$[CO2]s and $n_6^s=$[H2CO3]s
X, *(i)	concentration of non-diffusible anions in the cell (mM)
B, *(i) and HB, **	concentration of cytosolic and protonated buffer (mM)
zk	valence and charge of species k
$f_k^s$, **	kinetic description of reactions of species k in region s
ϕs, *(a, g, i)	electrical potential in region s (V)
${\mathbf{j}}_k^s$, **	flux of species k per unit area in region s (mol m−2 s−1)
$J_k^{sm}$, **	membrane flux of solute k per unit area from region s to region m (mol m−2 s−1)
Qsm, **	water flux per unit area from region s to region m (m s−1)
u, **	fluid velocity (m s−1)
p, *(g)	fluid pressure in the cleft gap (Pa)

The apical membrane is assumed to be folded, leading to an area increase of a factor of 10 [1], which we neglect in the geometry but account for in the values of membrane permeability to water and uncharged solute. We introduce a Cartesian system of coordinates (x, y), so that x spans the length of the cleft and x = 0 corresponds to Bruch’s membrane (figure 2).

We make the assumption that the solutions are well mixed in the apical and basal regions. The solute concentrations in these spaces are fixed and we formulate a mathematical model for water and ion transport across the cell and the cleft gap. We consider the presence of seven species: Na⁺, K⁺, Cl⁻, ${\text{HCO}}_3^{-}$, CO₂, H₂CO₃ and H⁺.

CO₂ and ${\text{HCO}}_3^{-}$ are subject to chemical rearrangement via the following two reactions:

$${\text{HCO}}_3^{-}+{\text{H}}^{+}\hspace{0.17em}\underset{k_{-1}}{\overset{k_1}{\rightleftharpoons }}\hspace{0.17em}{\text{H}}_2{\text{CO}}_3\hspace{0.17em}\underset{k_h}{\overset{k_d}{\rightleftharpoons }}\hspace{0.17em}{\text{CO}}_2+{\text{H}}_2\text{O}.$$ 2.1

The first reaction is the dissociation of H₂CO₃ into the ions ${\text{HCO}}_3^{-}$ and H⁺; it is rapid and happens almost instantaneously. The second reaction is normally slow, but can be catalysed up to six orders of magnitude in speed by CA [8]. There are a number of isoforms of CA: CA II is found in the cytosol of the RPE [17], whereas CA IV and XIV are attached to the cell membranes and are active in the extracellular membranes [18,19]. In effect, the presence of CA activity can be considered to be ubiquitous. The rates of this reaction are reported in table 2.

Table 2.

Model reference parameters; please refer to the text for further details on their motivation. The detailed choice of the parameters is described in the electronic supplementary material.

Although the second step of reaction (2.1) involves the production and consumption of water, the resulting changes are insignificant compared with the total water present. Accordingly, the amount of water is considered to be fixed.

The concentrations of anions and cations do not balance in the cell; there is an excess of cations. This deficiency of measured anions is called the anion gap and is due to some species not included in the model; in particular, both anions (such as lactate and phosphate) and non-diffusible proteins, which carry a net negative charge.

We model the anion gap by assuming the presence of a combination of two non-diffusible species. The first one, a buffer termed B⁻, will react with H⁺ within the physiological pH range and the reaction can be modelled as follows:

$${\text{B}}^{-}+{\text{H}}^{+}\hspace{0.17em}\underset{k_{-3}}{\overset{k_3}{\rightleftharpoons }}\hspace{0.17em}\text{HB},$$ 2.2

where HB is a protonated buffer, a product of the reaction (2.2), and is not charged. The total buffer concentration is fixed and conserved, and we denote it by C_B := B + HB, where B and HB denote the concentrations of B⁻ and HB, respectively. We note that this representation of cytosolic buffering is highly simplified, representing a large number of different reactions.

The second group consists of non-diffusible molecules that are strong acids and do not react with H⁺ in the physiological pH range. We denote their concentration by X and the corresponding fixed negative charge by $\tilde{X}:=z_{X}X$, where z_X is the mean valence of these molecules. Therefore, the total concentration of fixed negative charge in the cell is $\tilde{X}+B$ and the total concentration of non-diffusible molecules is X + C_B.

The pumps, co-transporters and ion channels that we incorporate in the model are adapted from Adijanto et al. [1] and Reichhart & Strauß [20] (figure 2). A peculiar feature of the RPE cell is that the Na⁺/K⁺ ATPase is located in the apical membrane. It pumps three Na⁺ out of the cell and two K⁺ in (denoted in the following with subscript ATP). The apical membrane also has the NKCC co-transporter that transports Na⁺, K⁺ and two Cl⁻ (NKCC), the K⁺ channels, the electrogenic channel NBC1 that co-transports two ${\text{HCO}}_3^{-}$ and one Na⁺ (aNBC_1:2) and the Na⁺/H⁺ exchanger (NHE) [21,22]. Following Adijanto et al. [1], we also account for the possible presence of the electroneutral ${\text{Na}}^{+}\text{–}{\text{HCO}}_3^{-}$ co-transporter (aNBC_1:1), where the a stands for apical.

The basolateral membrane has K⁺ channels, Cl⁻ channels, the ${\text{HCO}}_3^{-}$/Cl⁻ exchanger AE2 (AE) and the NBC channel that co-transports Na⁺ and ${\text{HCO}}_3^{-}$. The stoichiometry of the NBC channel is not clear, with estimates varying from 1 : 2 (Na⁺ : 2${\text{HCO}}_3^{-}$) to 1 : 3 [1]. We have modelled this NBC channel as a combination of channels with stochiometry 1 : 2 (bNBC_1:2) and 1 : 3 (bNBC_1:3) in the basolateral membrane.

The tight junction is considered to be impermeable to anions [1].

We assume that both cell membranes are permeable to CO₂ and H₂CO₃, since these molecules are not polarized and can pass through the lipid bilayer. Water is transported across the membranes by aquaporins and we have attributed a given hydraulic conductivity to each membrane.

Throughout this paper, we adopt the notation presented in table 1 and the values reported in table 2 as the baseline. Unless stated otherwise, we use superscripts to label regions and subscripts to label species, which we number in the following way:

$$\begin{array}{ccccccc}{\text{Na}}^{+}& {\text{K}}^{+}& {\text{Cl}}^{-}& {\text{HCO}}_3^{-}& {\text{H}}^{+}& {\text{CO}}_2& {\text{H}}_2{\text{CO}}_3\\ 0\hspace{0.17em}& 1\hspace{0.17em}& 2\hspace{0.17em}& 3\hspace{0.17em}& 4\hspace{0.17em}& 5\hspace{0.17em}& 6\hspace{0.17em}\end{array}$$

We first present a spatially resolved model of solute transport with reactions. We then simplify the model using asymptotic expansions in terms of the aspect ratio of the cleft, δ = h/L, and exploiting the fact that different time scales characterize reaction and diffusion processes. We then model fluid transport in the cleft. This model does not include electro-osmosis in the cleft, as it has been demonstrated to be subdominant [11], and instead focuses on water transport driven by osmosis. Finally, we impose water balance in the cell and describe the solution procedure.

2.2. Model of solute transport and water

Each ion k ∈ {0, 1, …, 6} is conserved in region s ∈ {i, g}

$$\frac{\mathrm{\partial }{n}_k^s}{\mathrm{\partial }t}=-\mathrm{\nabla }\cdot {\mathbf{j}}_k^s+f_k^s,$$ 2.3a

where $\mathrm{\nabla }=(\mathrm{\partial }/\mathrm{\partial }x,\mathrm{\partial }/\mathrm{\partial }y)$, and is transported by electrodiffusion and advection [26]

$${\mathbf{j}}_k^s=-D_k(\mathrm{\nabla }{n}_k^s+\frac{z_{k}F}{RT}{n}_k^s\mathrm{\nabla }{\varphi }^s)+\mathbf{u}{n}_k^s.$$ 2.3b

The anion gap is modelled to ensure electroneutrality,

$$\sum _{k=0}^4{z}_k{n}_k^s=X^s,s\in \{i,g\},$$ 2.3c

where Xⁱ = z_XX + B and X^g = 0.

In the cell, we have an additional equation for the negatively charged cytosolic buffer (which we assume cannot diffuse),

$$\frac{\mathrm{\partial }B}{\mathrm{\partial }t}=f_{\mathrm{B}}^i.$$ 2.3d

The concentration of HB can be found from the relation HB=C_B − B. The $f_k^s$ are the reaction terms, modelled via the law of mass action [27] as follows:

$$\begin{array}{ccc}\hspace{0.17em}& f_3^s=k_{-1}{n}_6^s-k_1{n}_3^s{n}_4^s,& f_{\mathrm{B}}^i=k_{-3}HB-k_3{n}_4^{i}B,\hspace{0.17em}{f}_4^s=f_3^s+f_{\mathrm{B}}^s,\\ \hspace{0.17em}& f_5^s=-k_h{n}_5^s+k_d{n}_6^s,\hspace{0.17em}& f_6^s=-f_5^s-f_3^s,\hspace{0.17em}s\in \{i,g\}\hspace{0.17em}\end{array}$$ 2.4

with all other $f_k^s$ equal to zero.

We start simplifying our equations by comparing the time scales for reaction and diffusion processes. The time scale for diffusion in the cell is about L²/D ≈ 0.06 s, where D = 1.7 × 10^{− 9} m² s^{− 1} (K⁺ in water, [28]) is a representative diffusion coefficient.

The time required for the first step of the reaction (2.1) and for the buffer reaction (2.2) is about 10^{− 7} s. It thus makes sense to assume that the first step of the reaction (2.1) and the reaction (2.3d) are both effectively in equilibrium. This leads to the following relations:

$$n_6^s=K_d{n}_3^s{n}_4^s,K_d=\frac{k_1}{k_{-1}},\hspace{0.17em}s\in \{i,g\}$$ 2.5a

and

$$B=\frac{K_b{C}_{\mathrm{B}}}{K_b+n_4^i}\text{and}{K}_b=\frac{k_{-3}}{k_3}.$$ 2.5b

We also note that the time scale for lateral diffusion in the cleft is much faster than in the cell, owing to its small width. Nevertheless, the relation (2.5a) still holds, as we prove in the electronic supplementary material.

In the second step of the reaction (2.1), catalysed hydration of CO₂ takes about 6 × 10^{− 4} s and dehydration 2 × 10^{− 6} s. These reaction rates are up to 10⁶ times slower in the absence of CA. As we are interested in studying the effect of CA inhibition, we do not assume that this step of the reaction (2.1) is in equilibrium [24].

We obtain an equation for total CO₂ conservation by considering (2.5a) and adding equations (2.3a) for k = 3, 5, 6 (bicarbonate, carbon dioxide and carbonic acid, respectively), resulting in

$$\begin{array}{rl}\hspace{0.17em}& \frac{\mathrm{\partial }{n}_3^s}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{n}_5^s}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{n}_6^s}{\mathrm{\partial }t}=-(\mathrm{\nabla }\cdot {\mathbf{j}}_3^s+\mathrm{\nabla }\cdot {\mathbf{j}}_5^s+\mathrm{\nabla }\cdot {\mathbf{j}}_6^s),s\in \{i,g\}.\end{array}$$ 2.6

We can further manipulate equations (2.3a) by subtracting equation (2.3a) for k = 4 from equation (2.3a) for k = 3 and adding (2.3d), to obtain the following relationship for conservation of charge:

$$\frac{\mathrm{\partial }{n}_3^s}{\mathrm{\partial }t}-\frac{\mathrm{\partial }{n}_4^s}{\mathrm{\partial }t}+\frac{\mathrm{\partial }B}{\mathrm{\partial }t}=-\mathrm{\nabla }\cdot {\mathbf{j}}_3^s+\mathrm{\nabla }\cdot {\mathbf{j}}_4^s,s\in \{i,g\}.$$ 2.7

We will be using equations (2.6) and (2.7) along with (2.5a) and (2.5b) instead of equations (2.3a) for k = 3, 4, 6 (${\text{HCO}}_3^{-}$, H⁺ and H₂CO₃, respectively) and (2.3d).

We do not explore transients (for example, the effects of transitioning from light to dark) and only focus on the steady-state behaviour of the system. Therefore, we have dropped all time derivatives.

2.2.1. Boundary conditions

We prescribe concentrations of solute in apical and basal regions $n_k^s,\hspace{0.17em}k\in \{0,\dots ,6\},\hspace{0.17em}s\in \{a,b\}$, as in table 2.

We impose at the membrane separating the generic regions s and m (either a cell membrane or a tight junction) the condition that the flux on the s side of the membrane of species k is equal to the function $J_k^{sm}$,

$${\mathbf{n}}^{sm}\cdot {\mathbf{j}}_k^s=J_k^{sm},k\in \{0,\dots ,6\}.\hspace{0.17em}$$ 2.8

This describes both the passive and active flows of species k across this membrane and the choice of the function is described in §2.2.2. Here, n^sm is the unit normal pointing from region s to region m.

Without loss of generality, we impose the potential in the basal region (at the level of Bruch’s membrane) to be zero. We prescribe Dirichlet conditions for all variables at the bottom of the cleft gap, x = 0 (which is also at the level of Bruch’s membrane), so that

$$n_k^g=n_k^b,{\varphi }^g=0,\hspace{0.17em}k\in \{0,\dots ,6\}.$$ 2.9a

We also have symmetry conditions, at y = 0 and y = H/2 + h, which read

$$\frac{\mathrm{\partial }{\varphi }^s}{\mathrm{\partial }y}=\frac{\mathrm{\partial }{n}_k^s}{\mathrm{\partial }y}=0,k\in \{0,\dots ,6\},\hspace{0.17em}s\in \{i,g\}.$$ 2.9b

Finally, we impose zero current at the apical region to simulate the open-circuit condition and to be able to compute the potential in the apical region (ϕ^a)

$$\begin{array}{r}{\int }_h^{H/2+h}\sum _{k=0}^4{z}_k{\mathbf{j}}_k^i\cdot {\mathbf{n}}^{ia}\hspace{0.17em}\text{d}y+{\int }_0^h\sum _{k=0}^4{z}_k{\mathbf{j}}_k^g\cdot {\mathbf{n}}^{ga}\hspace{0.17em}\text{d}y=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{at}\hspace{0.17em}x=L.\end{array}$$ 2.9c

A schematic of boundary conditions is shown in figure 3. The problem in the cell and cleft is simplified to a set of ODEs along the x-axis in both cell and cleft (see the electronic supplementary material for the derivation) with an error of approximation of order δ = h/L ≈ 10^{− 3}. The resulting system of equations is summarized in electronic supplementary material, §§1 and 2.

Figure 3.

Sketch of the imposed boundary conditions. (Online version in colour.)

2.2.2. Ion fluxes through cell membranes and tight junctions

There are two main pathways for fluid and solutes to cross the RPE. These are the transcellular pathway (where the transported material passes through the cell) and the paracellular pathway (where the material crosses the tight junctions and then passes down the cleft). The flux through cell membranes and the tight junctions of solute k that accounts for all active and passive transport from region m to region s is in general given by the following expression:

$$J_k^{ms}={\mathcal{J}}_k^{ms}+{\mathcal{A}}_k^{ms}+\sum _{l=0}^6\sum _{\alpha \in {\mathcal{C}}^{ms}}{s}_{lk}^{\alpha }{P}_{\alpha }\mathrm{\Delta }{\overline{\mu }}_l^{ms}.$$ 2.10

We note that $J_k^{ms}=-J_k^{sm}$. The first term in (2.10) accounts for passive flux of solute through the membrane. In the case of ions, the passive flux through ion channels is modelled with the Goldman–Hodgkin–Katz expression [27],

$${\mathcal{J}}_k^{ms}=P_k^s{V}_k^{ms}\frac{n_k^m-n_k^s\exp (-V_k^{ms})}{1-\exp (-V_k^{ms})},k\in \{0,\dots ,4\},$$ 2.11

where $V_k^{ms}=(z_{k}F)/(RT)({\varphi }^m-{\varphi }^s)$ and $P_k^s$ is the permeability of membrane s to ion k. We refer the reader to tables 1 and 2 for the meaning of other symbols and values of the constants in this expression. We model the flux of CO₂ and H₂CO₃ across the cell membranes with the following expressions:

$${\mathcal{J}}_k^{ms}=P_k^s(n_k^m-n_k^s),k\in \{5,6\}.$$ 2.12

The second term in (2.10), ${\mathcal{A}}_k^{ms}$, accounts for the active transport, which in the case of RPE is due to the presence of Na⁺/K⁺ ATPase in the apical membrane. We take the flux through this channel to be represented by the following expression [29]:

$${\mathcal{J}}_{\mathrm{ATP}}=P_{\mathrm{ATP}}{\left(\frac{n_0^i}{K_{\mathrm{Na}}+n_0^i}\right)}^3{\left(\frac{n_1^a}{K_{\mathrm{K}}+n_1^a}\right)}^2,$$ 2.13

where P_ATP is the parameter that determines maximum pump capacity and $K_{\mathrm{Na}}=0.2(1+n_1^i/(8.33\hspace{0.17em}\text{mM}))\hspace{0.17em}\text{mM}$ and $K_{\mathrm{K}}=0.1(1+n_0^a/(18.5\hspace{0.17em}\text{mM}))\hspace{0.17em}\text{mM}$ are apparent Na⁺ and K⁺ dissociation constants. Therefore, the terms representing active transport for Na⁺ and K⁺ are ${\mathcal{A}}_0^{ia}=3{\mathcal{J}}_{\mathrm{ATP}}$ and ${\mathcal{A}}_1^{ia}=-2{\mathcal{J}}_{\mathrm{ATP}}$, with the other ${\mathcal{A}}_k^{ms}$ equal to zero.

Finally, the last term in (2.10) represents fluxes through co-transporters and exchangers (excluding ATP), which are modelled using a linear non-equilibrium thermodynamics approach (similar to [8]). The set of such channels in the apical membrane is denoted by ${\mathcal{C}}^{ia}=\{a{\text{NBC}}_{1:1},a{\text{NBC}}_{1:2},\text{NKCC},\text{NHE}\}$ and in the basolateral one by ${\mathcal{C}}^{ib}={\mathcal{C}}^{ig}=\{\text{AE},b{\text{NBC}}_{1:2},b{\text{NBC}}_{1:3}\}$ (with no such channels in the tight junctions). The inner sum in (2.10) is performed for all channels in ${\mathcal{C}}^{ms}$ for given regions m and s. The factor $s_{lk}^{\alpha }$ is the product of the number of ions l and k in the stochiometry of channel α, $\alpha \in {\mathcal{C}}^{ms}$. For each channel α, this approach requires only one parameter to describe flux through this channel, P_α, $\alpha \in {\mathcal{C}}^{ms}$, with the values reported in table 2. $\mathrm{\Delta }{\overline{\mu }}_k^{sm}$ is the dimensionless chemical potential difference of ion l between regions m and s, $\mathrm{\Delta }{\overline{\mu }}_l^{ms}=\ln (n_l^m/n_l^s)+V_l^{ms}$.

2.3. Fluid flow

Fluid flow in the cleft is governed by Stokes equations for incompressible flow,

$$-\mathrm{\nabla }p+\mu {\mathrm{\nabla }}^2\mathbf{u}=0$$ 2.14a

and

$$\mathrm{\nabla }\cdot \mathbf{u}=0,$$ 2.14b

where u = (u, v) is fluid velocity, p is pressure and μ is fluid dynamic viscosity. We use the fact that the equations have a gauge freedom in the pressure to assign zero pressure in the cell, pⁱ = 0. Following the experimental evidence that the pressure jump across the RPE is negligible [30], we further assume it has values of zero in the apical and basal regions, p^a = p^b = 0, treating these regions as zero-dimensional compartments. Across cell membranes and tight junctions, there are osmotic and pressure driven fluxes, given by expressions

$$Q^{ag}=k^{\mathrm{tj}}(RT\sum _{k=0}^6(n_k^a-n_k^g(L))+p(L))$$ 2.15

and

$$\begin{array}{r}{Q}^{is}=k^s[RT(\sum _{k=0}^6(n_k^i-n_k^s)+X+C_{\mathrm{B}})-p^s],s\in \{a,b,g\},\end{array}$$ 2.16

where k^b = k^g = k, k^a = A_f k and p^g = p(x). Therefore, boundary conditions on the cell walls and tight junction of the cleft are $v(x,\pm h)=\mp Q^{ig}$ and u(L, y) = Q^ag. We also assume non-slip conditions for the longitudinal velocity at y = ±h, u(x, ±h) = 0 and impose zero pressure at the boundary with the basal region. We use lubrication theory to simplify the problem to an ODE for the pressure in the cleft (see electronic supplementary material, §3, for the derivation).

Finally, we impose water balance in the cell using the following relation:

$${\int }_0^H{Q}^{ia}\hspace{0.17em}\text{d}y+{\int }_0^H{Q}^{ib}\hspace{0.17em}\text{d}y+2{\int }_0^L{Q}^{ig}\hspace{0.17em}\text{d}x=0.$$ 2.17

This balance is required for the volume of the cell to remain constant and it provides the equation to determine the concentration of large negatively charged molecules in the cell, X. We note that X is related to concentration of ions in the cell through the electroneutrality condition (2.3c), providing an additional coupling between fluid and ion transport.

2.4. Summary of the model

The mathematical model described above has been formulated in two dimensions. We use asymptotic expansions with respect to the small aspect ratio of the cleft δ = h/L to simplify it to a one-dimensional problem in the x-direction (see electronic supplementary material for the derivation). The model has 20 variables, denoted with a ‘*’ in table 1: concentration of all species in the cell (s = i) and the cleft (s = g) ($n_k^s(x)$, $k\in \{0,\dots ,6\}$), electrical potential (ϕ^s(x)), concentration of negatively charged molecules in the cell (X), intracellular buffer (B(x)), fluid pressure (p(x)) and transepithelial potential (TEP) (ϕ^a). These variables are found from solving numerically (see electronic supplementary material for the details) the nonlinear system of seven algebraic equations along with the 13 ODEs summarized in table 3. All the simplified equations along with the boundary conditions, the solution procedure and methods are presented in electronic supplementary material, §4.

Table 3.

Summary of the model with references to equations in the main text in two-dimensional form (second column) and their simplified one-dimensional version in the electronic supplementary material (third column). The numbers in parentheses in the first column indicate the number of equations.

equations	full model	simplified model
conservation of solute in the cell and the cleft (8)	(2.3a) for k = 0, 1, 2, 5	(4-2a), (4-2b)
conservation of charge (2)	(2.7)	(4-2c)
conservation of total CO2 (2)	(2.6)	(4-2d)
equilibrium of the reaction (2.2) and first part of (2.1) (3)	(2.5a), (2.5b)	(4-2e), (4-2f)
electroneutrality (2)	(2.3c)	(4-2g)
fluid transport (1)	derived from (2.14)	(4-2h)
water balance in the cell (1)	(2.17)	(4-2i)
open-circuit condition (1)	(2.9c)	(4-3g)
boundary conditions at x = 0 and x = L
imposed values at the exit of the cleft	(2.9) and p(0) = 0	(4-3a)
membrane conditions	(2.8)	(4-3b)–(4-3e)
water flux across the tight junction	derived from u(L, y) = Qag	(4-3f)

2.5. Sensitivity analysis

Parameter estimation is always challenging as some of the necessary values are reported in the literature while others have a range of values reported. Our chosen set of baseline parameters are presented in table 2, and we describe how the choices were made in electronic supplementary material, §6.

Owing to the large number of parameters, we wish to determine their relative importance on the model output, focusing on their effect on water and ion fluxes. We accordingly perform global sensitivity analysis (GSA) using a variance-based method, known as the extended Fourier amplitude sensitivity test (eFAST), proposed in Saltelli et al. [31] and adapted to biological models in Marino et al. [32]. We consider the 12 parameters listed in table 4 and assume that their logarithms are uniformly distributed within given ranges, with the purpose of spanning several orders of magnitude. We further note that the qualitative results of the GSA are valid and, in conclusions reported below, are unchanged if the parameters are instead uniformly distributed across their range.

Table 4.

Parameters for the sensitivity analysis and their allowed ranges in the sensitivity study.

membrane permeability	range, 108 m s− 1
$P_1^b$ - permeability of basal membrane to K+	[1, 10]
$P_1^a$ - permeability of apical membrane to K+	[10, 50]
$P_2^b$ - permeability of basal membrane to Cl−	[0.5, 3.16]
amplitudes of ion channels	range, 106 mol m− 2 s− 1
PATP - amplitude of the pump	[1, 3.16]
PNKCC - amplitude of co-transporter	[0.5, 5]
$P_{{a\mathrm{NBC}}_{1:2}}$ - amplitude of apical ${\text{Na}}^{+}-{\text{HCO}}_3^{-}$ co-transporter	[0.1, 3.16]
$P_{{a\mathrm{NBC}}_{1:1}}$ - amplitude of apical ${\text{Na}}^{+}-{\text{HCO}}_3^{-}$ electroneutral co-transporter	[0.01, 1]
$P_{{b\mathrm{NBC}}_{1:2}}$ - amplitude of basal ${\text{Na}}^{+}-{\text{HCO}}_3^{-}$ co-transporter 1 : 2	[0.1, 3.16]
$P_{{b\mathrm{NBC}}_{1:3}}$ - amplitude of basal ${\text{Na}}^{+}-{\text{HCO}}_3^{-}$ co-transporter 1 : 3	[0.01, 1]
PAE - amplitude of ${\text{Cl}}^{-}-{\text{HCO}}_3^{-}$ exchanger	[0.1, 3.16]
PNHE - amplitude of Na+−H+ exchanger	[0.01, 10]
tight junction permeability to cations	range, 106 m s− 1
Ptj - permeability of tight junction to Na+ and K+	[1, 100]

eFAST produces two sensitivity indices for each parameter: the first-order sensitivity index S_i and the total sensitivity index S_Ti. S_i describes the ‘main effect’ of each parameter and is a measure of the expected reduction in variance of the model output if that parameter were kept fixed. S_Ti captures both this first-order effect and the interaction of a given parameter with other parameters. In other words, it measures the expected variance that would be left if all but a given parameter were fixed. These indices allow one to determine parameters which affect the model output most. A detailed meaning and definition of these indices can be found in Saltelli et al. [31]. In electronic supplementary material, §5, we report the particular choices we made for the implementation of the method.

3. Results

Photoreceptors have different levels of metabolic activity, varying with exposure to light: it is much higher in the dark. This means that in dark conditions CO₂ is produced in the SRS at a higher rate. We, therefore, focus on two physiologically relevant cases. In the first one, we impose the same values of CO₂ concentration in the apical and basal regions, representing retina exposed to light. We first show the results for the baseline parameter values reported in table 2 and then we provide the results of the sensitivity analysis. In the second case, we consider a higher CO₂ concentration in the apical region than in the basal, representing retina in the dark.

3.1. Equal CO₂ concentration in the sub-retinal space and the choroid

In the absence of concentration jumps across the RPE the concentration of all species in the cell is constant. In figure 2 (right cell), we report the values of the concentrations and the electrical potential computed by the model. The concentrations predicted for Na⁺, K⁺ and TEP (ϕ^a) compare well to experimentally measured values [12,20].

The measured values of the Cl⁻ concentration vary across the literature, from 62 mM in the bovine RPE [33] to about 20 mM in the toad and chick RPE [34,35]. The value computed by the model is within this range, but we note that it is high compared with some other mammalian cell types. Nevertheless, the sensitivity analysis shows that this value does not significantly affect the behaviour of the system (results not reported).

${\text{HCO}}_3^{-}$ concentration and pH predicted values are comparable to those reported in Gallemore et al. [36]. We note that, owing to the large permeability of cell membranes to CO₂ and H₂CO₃, their concentrations in the cell are predicted to be almost exactly the same as the values imposed in the apical and basal regions, i.e. $n_5^i=1.7\hspace{0.17em}\text{mM}$ and $n_6^i=0.49\times {10}^{-2}\hspace{0.17em}\text{mM}$.

With the reference parameter set from table 2, the model predicts the following ion fluxes:

$$\begin{array}{ccccc}\hspace{0.17em}& {\text{Na}}^{+}\hspace{0.17em}& {\text{K}}^{+}\hspace{0.17em}& {\text{Cl}}^{-}\hspace{0.17em}& {\text{HCO}}_3^{-}\hspace{0.17em}\\ \text{ion flux},\text{mol}\hspace{0.17em}{\text{m}}^{-2}\hspace{0.17em}{\text{s}}^{-1}& -1.44\times {10}^{-6}& -1.69\times {10}^{-6}& -2.34\times {10}^{-6}& -7.79\times {10}^{-7}\end{array}$$

directed from the SRS to the choroid (corresponding to a negative value). Measurements of ion fluxes across the RPE are limited and incomplete, with different authors even finding fluxes in opposite directions. Our model predicts an Na⁺ flux in agreement with the value of − 1.5 × 10^{− 6} mol m^{− 2} s^{− 1} measured in an isolated bullfrog RPE-choroid preparation [2]. The calculated K⁺ and Cl⁻ fluxes are comparable to those in Dvoriashyna et al. [11] and the predicted ${\text{HCO}}_3^{-}$ flux is smaller than that estimated by Hughes et al. [2], − 3.13 × 10^{− 6} mol m^{− 2} s^{− 1}.

We also find that concentration gradients do form in the thin cleft gaps that separate adjacent cells, owing to the presence of active ion pumping, as shown in figure 4a. In this figure, we also show the electrical potential along the cleft gap. The potential gradient is positive, opposite to the prediction of Dvoriashyna et al. [11]. We attribute this to using a different value for the tight junction cation permeability. This change in value was required in order to obtain the experimentally observed values of TEP. This implies that the value of ϕ^g on approaching the apical region is closer to the TEP.

Figure 4.

(a) Left axis: deviation of the ion concentrations along the cleft from those in the basal region for the reference parameters in table 2. The x-axis goes along the symmetry mid-line (y = 0) of the cleft, from the basal region to the tight junction. Right axis: electrical potential along the x-axis in the cleft. (b) Osmolarity distribution along the cell (y = H/2 + h) and cleft (y = 0). The osmolarities in the apical and basal regions are shown with a circle and a triangle, respectively. Inset: schematic of the direction and magnitude of water fluxes across cell membranes. (Online version in colour.)

Figure 4b shows the osmolarity in the cell and the cleft predicted by the model. The osmolarity in the cleft is higher than that in the cell and this induces a local osmotic flux. The overall effect is the generation of a net water flux across the RPE, directed from the apical to the basal region. A schematic of water fluxes through the cell and the cleft is shown in the inset of figure 4b. For the reference parameters in table 2, we find a value of the water flux per unit surface across the RPE equal to Q ≈ − 9.36 × 10^{− 9} m s^{− 1}, which is comparable to measured values of −[1.1, 2.8] × 10^{− 8} m s⁻¹ [3].

3.1.1. Global sensitivity analysis

We investigate the sensitivity of water and ion fluxes to the model parameters reported in table 4 with a GSA algorithm, eFAST. The first-order and total sensitivity indices of model parameters for water flux are shown in the histograms in figure 5. The height of each bar is a measure of how much the corresponding parameter affects water flux. Inspection of figure 5 shows that first-order and total indices lead to qualitatively similar results, identify the same parameters as being important for modifying water flux (the same is true for ion fluxes).

Figure 5.

First and total sensitivity indices for water flux. For the large bars, arrows indicate how the magnitude of water flux is modified, up or down, by an increase in that particular parameter. The dummy parameter is introduced to verify the results: its deviation from zero can be interpreted as the uncertainty in the method, e.g. due to an insufficient number of points. (Online version in colour.)

For the most significant parameters, we show in figure 6 scatter plots with the values of water flux, Q, taken in each numerical experiment versus the corresponding value of the parameter. In the plots, we also report in black a curve corresponding to a second-order polynomial function fitted to the data with the least-squares method.

Figure 6.

Scatter plots for water flux Q versus four parameters with largest sensitivity indices: (a) P_ATP, (b) $P_{{b\mathrm{NBC}}_{1:2}}$, (c) P_NHE and (d) P^tj. Each point corresponds to a numerical experiment. Different colours correspond to different searching curves. (Online version in colour.)

In figure 7, we report total sensitivity indices for ion fluxes. In the figure indices corresponding to different ions are reported with different colours. The sensitivity analysis for ion fluxes highlights the great importance that the Na⁺–H⁺ exchanger and the tight junction permeability have on the system.

Figure 7.

Total sensitivity index for ion fluxes. The arrows indicate the direction of ion flux change with the increase in a given parameter. (Online version in colour.)

It is believed that ion and water fluxes through epithelia are tightly linked to each other and that ‘water follows the ions’. We examine this statement in figure 8, where we plot water flux versus each of the ion fluxes. Each point corresponds to a numerical experiment and we identify a general trend in each scatter plot by reporting (in black) a line fitted with the least-squares method. The figure shows that a strong link exists between water flux and Na⁺ and ${\text{HCO}}_3^{-}$ fluxes. To a smaller extent, Cl⁻ flux is also correlated with water flux. On the other hand, a clear correlation is not found for K⁺ flux. The model does confirm that there is a link between ion and water fluxes, though it is not always conclusive and the concept that water follows the ions is not a general principle.

Figure 8.

Scatter plots for water flux Q versus (a) Na⁺, (b) K⁺, (c) Cl⁻ and (d) ${\text{HCO}}_3^{-}$ flux. Each point corresponds to a numerical experiment. (Online version in colour.)

We simulated varying the CO₂ concentration on both sides of the epithelium from 1% to 13% and determined the predicted responses of water flux, following the experiments performed by Adijanto et al. [1]. Unlike in the experimental observations, our model does not predict any significant change in water transport with variation of CO₂ (results not shown).

3.2. Different CO₂ concentration in the sub-retinal space and the choroid

CO₂ is produced by photoreceptors at high rates in dark conditions. This results in a higher CO₂ concentration in the SRS than in the choroid. To model this condition, we assume that CO₂ concentration in the basal region is fixed at 5% and we increase the concentration in the apical region to 7%. This creates a jump in CO₂ concentration across the RPE, which induces spatial variability of all variables in the x-direction in the cell. These induced departures of ion concentrations from the corresponding average values ${\hat{n}}_k^i$ along the cell are shown in figure 9a. While the average values are approximately the same as reported in figure 2, there is a concentration gradient, with the concentrations of Na⁺ and K⁺ decreasing towards the apical membrane, whereas concentrations of Cl⁻ and ${\text{HCO}}_3^{-}$ increase. Moreover, figure 9a also shows that a potential gradient became established across the cell in a basal-to-apical direction.

Figure 9.

The case of apical CO₂ at 7% and basal CO₂ at 5%. (a) Left axis: deviation of ion concentrations from their average value in the cell along the x-axis, ${\hat{n}}_k^i={\int }_0^L{n}_k^i\hspace{0.17em}\text{d}x/L$. Right axis: deviation of the electrical potential along the x-axis in the cell from its average value. (b) Osmolarity distribution in the cell, cleft, apical and basal regions. Inset: schematic of the direction and magnitude of water fluxes across cell membranes. (Online version in colour.)

Interestingly, despite these changes in the ion concentration and the potential gradients that were generated in the cell, there was little change in the net ion fluxes. This is because the concentration and the potential gradient act in opposite directions on the ion transport. With the established CO₂ gradient, there was a transepithelial CO₂ flux equal to −9.05 × 10⁻⁵ mol m⁻² s⁻¹, and directed from the apical to the basal region.

The osmolarity in the cell and a schematic of water fluxes across the cell membranes are shown in figure 9b. The decrease in osmolarity towards the apical membrane reduces the magnitude of the transepithelial water flux to the value of −0.43 × 10⁻⁸ m s⁻¹.

We also performed a GSA for the case with a CO₂ gradient but do not report these results since the conclusions are similar to those discussed above.

3.2.1. Impact of carbonic anhydrase inhibition

In order to test the effect of CA inhibition, we introduce a factor α that multiplies the reaction rates (k_d and k_h) of equation (2.1) and their reduction simulates inhibition of this reaction. The response of water flux Q to a progressive decrease of this factor is shown in figure 10a. The three curves correspond to three different CO₂ concentrations (5%, 7% and 9%) imposed in the apical region (the basal CO₂ concentration is kept fixed to 5%). The figure shows that inhibition of CA leads to a significant change in water flux towards the choroid. When a CO₂ drop across the RPE is imposed, water flux towards the choroid reduces. However, water flux increases significantly if reaction (2.1) is inhibited, especially when a CO₂ jump across the RPE is imposed. When CO₂ concentration is the same in the apical and basal regions the water flux reaches a maximum value for α ≈ 10⁻⁴. The scatter plot in figure 10b shows the effect of CA inhibition on water flux Q for various model parameters. The average behaviour suggests that Q increases in magnitude with CA inhibition.

Figure 10.

Water flux as a function of the CA activity. The values of α on the x-axes are factors that multiply the reaction rates k_d and k_h of the reaction (2.1). The value α = 1 corresponds to full catalysis (the reaction rates from table 2); as the value of α decreases the catalyser is progressively inhibited. In (a), the typical solution, with other parameter values from table 2, is considered. Different curves correspond to different values of apical CO₂ concentration: 5%, 7% and 9%, while basal CO₂ is kept at 5%. In (b), we vary nine model parameters from table 4, keeping membrane permeabilities to ions ($P_1^b$, $P_1^a$ and $P_2^b$) fixed, while inhibiting CA with apical CO₂ at 7% and basal CO₂ at 5%. Different colours correspond to different searching curves. (Online version in colour.)

4. Discussion

Fluid accumulation in the SRS remains a relatively poorly understood problem, despite its clinical relevance for the formation and development of various sight-threatening pathological conditions, such as macular oedema, retinal detachment and age-related macular degeneration [37], and so a good understanding of this process is of considerable importance. Mathematical modelling has the potential to significantly contribute in this field by developing a framework that can help clarify underlying physical mechanisms and direct and interpret further experimental work.

In the present work, we develop a fully coupled, spatially resolved model of fluid and solute dynamics across the RPE and aim to study mechanisms of their interaction, extending the study of Dvoriashyna et al. [11]. We also perform a GSA to help identify which processes have the largest impact on transport.

The model predicts that ion concentrations in the cell and TEP are within physiological range. We also find that the concentration gradient in the cleft drives a water flux from the SRS to the choroid of the order of 10⁻⁸ m s⁻¹, which is compatible with the experimentally measured values of [1.1, 2.8] × 10⁻⁸ m s⁻¹ [2,3]. The predicted ion fluxes also have magnitudes comparable to those found experimentally [2,38,39].

The sensitivity results, from GSA, suggest that the tight junction plays a key role in Na⁺ transport, and large tight junction permeability to cations is required for the Na⁺ flux to be directed from the SRS to the choroid. This is in agreement with the hypothesis of Gallemore et al. [37], who argued that the passive flux through the tight junction towards the choroid should exceed the active flux of Na⁺ across the cell towards the SRS and establishes a net Na⁺ transport towards the choroid. Our model also predicts that water flux is roughly proportional to Na⁺ flux, with a proportionality constant of 0.0026 m³ mol⁻¹.

The ${\text{HCO}}_3^{-}$ flux is closely related to Na⁺ transport, owing to their coupling through ${\text{Na}}^{+}-{\text{HCO}}_3^{-}$ co-transporters in cell membranes. Hughes et al. [2] performed experiments on isolated bullfrog RPE-choroid preparations that suggested that fluid absorption is strongly dependent on the active absorption of ${\text{HCO}}_3^{-}$. The results of our model agree with this, since our model predicts that water flux is roughly proportional to ${\text{HCO}}_3^{-}$ flux, and provides a proportionality constant of 0.0016 m³ mol⁻¹.

The GSA suggests that Cl⁻ transport is directed in most cases towards the choroid. We note that the direction of this flux is obtained from a balance between basal ${\text{Cl}}^{-}-{\text{HCO}}_3^{-}$ exchanger and basolateral permeability to Cl⁻. Our model confirms the hypothesis that fluid transport is enhanced by a transepithelial Cl⁻ flux (e.g. [33]), although the effect is less evident than that for Na⁺ and ${\text{HCO}}_3^{-}$.

The GSA indicates which manipulations to perform in order to adjust the flow. In particular, the model predicts that increasing the tight junction permeability to cations will result in an increase of the transepithelial Na⁺ flux, and thus of water flux, towards the choroid. To a smaller extent, inhibition of basal ${\text{Na}}^{+}\text{–}{\text{HCO}}_3^{-}$ (1 : 2) co-transporter and anion exchanger also increases the rate of fluid removal from the SRS. The effect of the anion exchanger on water flux has been confirmed experimentally by Iserovich et al. [40].

The model predicts that the water flux is also sensitive to the amplitudes of Na⁺−H⁺ exchanger (NHE) and Na⁺−K⁺ ATPase, inhibition of which results in a decrease in water flux towards the choroid. Therefore, we anticipate that a side effect of NHE inhibitors, such as amiloride, or reduced activity of Na⁺−K⁺ ATPase, e.g. due to low metabolic activity, may result in fluid accumulation.

The model also predicts that if CO₂ is being excessively produced in the SRS, then the water flux in the direction of the choroid reduces. This is because of the presence of spatial variability in ion concentration induced by a CO₂ gradient in the cell. This in turn implies a change in osmolarity from the basal to the apical region, such that the osmolarity in the cell decreases towards the apical membrane. As a result, the osmotic water flux across the apical membrane is reduced (or even reversed), resulting in an overall reduction in water flux towards the choroid. In other words, the model suggests that CO₂ accumulation in the SRS tends to decrease water transport. The implication is that acidosis and fluid accumulation in the SRS are predicted to be linked to each other, which is experimentally testable.

CA inhibitors are used to facilitate fluid transport [13,14] or to treat macular oedema [15,16]. This behaviour is captured by the model since we find that, in the case of a CO₂ gradient across the RPE, inhibition of CA smooths the gradients of all species in the cell (except for CO₂), resulting in an increase in water transport towards the choroid. Thus the model provides a possible explanation for the physical mechanism behind the functioning of CA inhibitors.

Adijanto et al. [1] studied the dependency of water flux on CO₂ concentration in the apical and basal solutions. The authors found that an increase in CO₂ concentration on both sides of the epithelium from 5% to 13% results in a doubling of water flux, suggesting that water transport is facilitated by CO₂. Our model suggests that water flux is independent of equal CO₂ concentration changes on both sides of the membrane, in disagreement with Adijanto et al. [1], which may also indicate that the model is incomplete and additional factors may contribute for large uniform increases in CO₂. However, owing to the high level of coupling among numerous contributions to overall transport, the addition of further factors would make it difficult to identify and assess their potential impact.

To summarize, we present a mathematical model that explains some of the features of fluid and solute transport across the RPE. It couples fluid, ion and CO₂ dynamics across the epithelium and shows that fluid transport can be driven by local osmosis in the cleft. The model provides possible explanations of the mechanisms behind the functioning of CA inhibitors used to treat fluid accumulation in the SRS. The results of the model, and particularly of the GSA, may be used in providing mechanism-based insight for targeting inhibition of other channels to treat pathological eye states associated with fluid transport across the RPE. We finally remark that the model is based on factors that are common to all absorptive or secretive epithelia and, therefore, can constitute a basic framework for the study of fluid flow across other tissues.

Data accessibility

The codes written to produce the results of this research are available at the Oxford University Research Archive (ORA) with the doi:10.5287/bodleian:6g4oazYNE.

Authors' contributions

All authors contributed to formulating the mathematical model. M.D. wrote the codes and carried out the calculations. All authors edited the manuscript and gave final approval for publication.

Competing interests

We declare we have no competing interests.

Funding

The work has been supported by the Macular Society and UK Fluid Networks, funded by EPSRC grant no. EP/NO32861/1. M.D. acknowledges the support of the Italian National Group for Mathematical Physics.