In silico investigation of ion transport and water flux in cystic
fibrosis epithelia
Claire Walsh
Supervisors: Guy Moss, Paola Vergani and Vivek Dua
March 7, 2012
Abstract
An in silico investigation into how the permeability of apical and basolateral membranes
a↵ects water flux through epithelial cells, was carried out. Using a model of epithelial ion
transport constructed by [O’Donoghue, 2011], univariate sensitivity analysis was carried out.
It has shown that increasing basolateral potassium permeability increases water flux into the
luminal compartment, in agreement with other modelling results from
[Novotny and Jakobsson, 1996b]. However the results from other parameter variations do not
agree with that work. This discrepancy points to key di↵erences between the models of
[Novotny and Jakobsson, 1996a] and [O’Donoghue, 2011], in particular the role of a finite ASL
compartment and water permeable paracellular pathway. The analysis also suggests a
monotonic correlation between membrane permeability and transepithilial potential di↵erence
V(t).
1
1
1.1
Introduction
1.2
Physiology of CF
CFTR is expressed in the apical membrane
of epithelial cells and is responsible for anion
transport across the membrane. An absence of
CFTR leads to decreased chloride permeability in epithelia tissues where it is expressed,
which include the pancreas, pulmonary system, intestine, liver, sweat glands and reproductive system. This causes di↵ering responses
in each of these systems or organs: pancreatic insufficiency due to decreased enzyme secretion, distal intestinal obstruction syndrome,
increased sweat chloride levels, blocking of vas
deferens leading to infertility and chronic airway infection. The chronic lung infection accounts for approximately 90% of all CF mortality and consequently treatment and prevention are a major area of research; however,
the pathogenesis is still not fully understood
[Boucher, 2007].
It is well accepted that loss of CFTR function causes highly viscous mucus to be produced in the airways which is not well mobilised by normal mucociliary clearance. This
mucus is colonised with bacteria in particular Pseudomonas aeruginosa [Boucher, 2007]
and leads to repeated pulmonary exacerbation
with highly inflammatory response. This inflammatory response leads scarring and permanent damage of the lungs. As this permanent damage only occurs later in life (CF
newborns are found to have normal lung
capacity [Linnane et al., 2008]) an opportunity to dramatically improve life expectancy
and quality through good prenatal diagnosis
and e↵ect preventative treatment, is present
[Linnane et al., 2008]. There are several hypotheses as to what form this treatment should
take depending on what theory of disease
pathogenesis is accepted. This lack of consensus and the potential benefits that could be
gained, makes the CF epithelium an excellent
candidate for a modelling approach.
Historical Backround
Cystic Fibrosis (CF) is an autosomal recessive
disorder, leading to a dysfunctional or absent cystic fibrosis transmembrane regulator
(CFTR). The disease is thought to a↵ect
one in every 2500 births in the caucasian
population, making it the most prevalent
lethal, genetic disorder for this demographic
[Cohen-Cymberknoh et al., 2011]. The disease a↵ects a range of systems in the body,
characteristic phenotypes include: pancreatic
insufficiency, elevated sweat chloride level and
development of chronic lung infection, which
is normal the cause of mortality. The disease
was first recognised as distinct from celiac’s
disease in 1938 [Davis, 2006] at which point
the chronic lung infection was thought to arise
from malnutrition caused by the pancreatic
insufficiency [Andersen, 1938]. In 1948 the
elevated sweat chloride levels were first noticed in the New York heat wave by Paul
di Sant’Agnese [Di sant’agnese et al., 1953].
This observation provided the insight that
CF was a systems disorder with a single underlying cause, and provided the first simple
diagnostic test. The sweat test, in conjunction with family history, genetic screening
and nasal potential di↵erence measurements
(NPD), is still used today. At around the
same time the foundation of CF centres and
establishment of a policy of aggressive antibiotic treatment, lead to significant increases in
life expectancy. The discovery of the CF gene
in 1989 by genetic cloning [Kerem et al., 1989,
Riordan et al., 1989, Rommens et al., 1989],
provided a host of possible new drug targets
and genetic therapy options, into which much
current research is focused.
2
1.3
on the properties of the airway surface liquid
(ASL), the thin liquid layer composed of the
periciliary liquid layer (PCL) and mucus layer
covering the apical surface of the epithelium
(see Figure (1)). The first of the two theories, the so called compositional hypothesis,
was first purported by [Zabner et al., 1998,
Smith et al., 1996,
Goldman et al., 1997].
It states that the loss of CFTR function
in CF prevents Cl absorption leading to
a hypertonic ASL, similar to the form of
the disease in the sweat glands. This high
salt concentration inactivates the -defensin
antimicrobial agent, thereby eliminating a
key line of defence against inhaled pathogens
[Goldman et al., 1997].
[Smith et al., 1996]
showed that cultured CF human epithelial
cells were unable to kill an addition of P.
aeruginosa applied to the apical face. When
wild type CFTR was expressed in these cells
they were able to reduce or in some cases
eliminate the bacterium after a 24hr period.
Crucially this work also showed that the ASL,
when removed by water washing from both
CF and non-CF cells, has similar bacterial
killing properties. Only ASL in situ on the
apical surface of the CF epithelium showed
the deficiency. The authors concluded that
although CF and non-CF ASL contain the
same broad spectrum antimicrobial agents,
the hypertonic composition of the ASL,
caused by the deficient CFTR, renders certain
antimicrobials inactive. Measurements of ASL
composition by both [Zabner et al., 1998] and
[Smith et al., 1996] showed a hypertonic ASL
for CF compared to non-CF epithelium.
Since these experiments however, many
other groups using more advanced methods have found no evidence for hypertonic
ASL[Tarran et al., 2001, Matsui et al., 1998,
Boucher, 2007].
Additionally the concept
that airway epithelial cells could support an
osmotic gradient is not supported by the
known permeability of the tissue. As a result
Aims
This work aims to use just such an approach
to investigate the relationship between water
and ion flux through the epithelial sheet. Using a model of airway epithelial cells created
by [O’Donoghue, 2011] a univariate sensitivity
analysis will be used to identify the a↵ects of
varying membrane density of modelled transport channels, on water flux through the cell.
The following sections will consist of a review
of the role of water flux in CF lung disease
pathogenesis, a summary of the modelling attempts to date along with a more detailed explanation of the model used here, and a univariate sensitivity analysis using the model.
2
The role of water flux in CF
lung disease pathogenesis
ASL
ASL
airway epitheliumairway epithelium
- Na+ absorption (ENaC,
Na+K+ATPase)
- Na+ absorption
(ENaC, Na+K+ATPase)
Figure
1:
Diagram
of a ciliated epithe- Cl- secretion (CFTR,
NKCC1)
- Cl secretion (CFTR, NKCC1)
lium
showing the ASL composed of the
- H2O movement
(AQP3,4,5)
- Hcell
2O movement (AQP3,4,5)
PCL and mucus layer, image taken from
[Dua et al., 2011]
There is much controversy over the pathogenesis of CF lung disease, which recently
has focused on two opposed mechanisms in
particular
[Donaldson and Boucher, 2003].
Both of the proposed mechanisms are centred
3
amiloride may help prolong the a↵ect, but
again clinical trials have produced inconclusive
results [Wark et al., 2009]. In addition recent
work on newborn piglet trachea have shown
no evidence for decreased ASL depth as well
as no evidence for a hyperabsorption of N a+
in the more distal airways [Chen et al., 2010].
These findings are highly significant as they
question many of the beliefs currently held
with regards to the pathology of CF lung
disease. In particular this study highlights the
difficulties in generalising from results in the
proximal airways to the more distal, where
transport channel expression di↵ers.
the majority of groups have abandoned this
hypothesis in favour of the volume depletion
hypothesis.
The volume hypothesis, originally put forward
by [Boucher, 1994] and [Matsui et al., 1998],
has received a much wider level of support in
recent years with results from in vitro, in silico
and some in vivo studies supporting the hypothesis [Donaldson and Boucher, 2003]. The
hypothesis states that epithelial ion transport
is the main mechanism for regulation of ASL
volume via regulation of osmotic gradients.
Contrary to the composition hypothesis, it
is thought that hyperabsorption of N a+ ,
associated with CF [Hummler et al., 1996,
Stutts et al., 1997, Chen et al., 2010],
increases water flux across the epithelial sheet
from luminal to serosal, thus depleting the
ASL [Boucher, 1994]. In particular certain
groups maintain that it is the PCL specifically
that is reduced in depth [Matsui et al., 1998].
The PCL provides a relatively non-viscous
fluid in which the cilia are able to beat
e↵ectively, and acts as a lubrication layer on
which the overlying mucus gel may slide more
easily. Depletion of this layer was shown by
[Matsui et al., 1998] to cause direct contact of
the cilia the mucus gel.
Although this theory is more widely accepted,
there is still a doubt over whether it is a true
representation of the pathology. Treatments
which target the ASL depletion such as isotonic and hypertonic saline nebulisation have
not shown a significant impact during clinical
trials [Wark et al., 2009]. However this failing
in clinical situations is argued by some not to
be as a result of incorrect pathology, but due
to the treatments not e↵ectively hydrating
the ASL long term. The in vitro studies of
[Tarran et al., 2001] showed that immediate
a↵ect of hypertonic saline treatment did
increase ASL volume but that within 12mins
this had returned to its basal levels. It is
thought that addition of mannitol and or
3
3.1
Approaches to
the epithelium
modelling
Physiology
Epithelial tissue separates and regulates transport between the external environment and internal compartments of an organism. Epithelial cells are polarised from the apical membrane to basolateral membrane with di↵ering
transmembrane proteins in each (see Figure
2). Airway epithelium tissue consists of several types of epithelial cell most commonly the
columnar ciliated epithelial cells. These are arranged in single layer close packed sheet. Cells
are bound together by tight junction binding
proteins which seal the paracellular pathways
to macromolecules and prevent apical and basolateral membrane proteins from migrating
[Alberts et al., 2002]. Transport across the epithelium occurs via a variety of ion channels,
pumps, transporters and passively along the
paracellular pathway. The tight binding proteins can vary the permeability of the paracellular pathway to small molecules and in this
way control how ’leaky’ the epithelial sheet is.
4
3.2
Past Modelling of Airway Ep- basal levels soon after increasing due to a
dynamic calcium response.
ithelium
There have been various models of the
epithelium and epithelial cells which include paracellular pathways over the
past two decades, the first of which
are
[Hartmann and Verkman, 1990]
and
[Duszyk and French, 1991]. The former used
data from canine trachea to provided ion
channel permeabilities and saturabilities.
The model was able to investigate the effect ion transport permeabilities had on
short circuit current and membrane potentials.
[Duszyk and French, 1991] produced
the first human airway epithelial model
and used it to analyse the steady state
responses of the cell parameter changes.
[Horisberger, 2003] used their model to
show that the coupling between ENaC and
CFTR proteins, seen experimentally, could
be achieved through electrical coupling without any form of direct protein interaction.
Particularly relevant to this work are the
models of [Novotny and Jakobsson, 1996a]
and [Warren et al., 2009] which take a more
rigorous approach to modelling the water
transport as well as ion transports in the
airway epithelium. The former of the two
was originally used in a univariate parameter
investigation similar to this work and will
be discussed further in the context of the
results. It has also recently been extended
to include intracellular pH which introduces
the bicarbonate and hydrogen ions, to the
model
[Falkenberg and Jakobsson, 2010].
[Warren et al., 2009] focus particularly on the
e↵ects of calcium signalling on PCL volume.
Their model, which includes a finite PCL or
ASL, was used by them to investigate the
volume hypothesis and specifically to simulate
the experiment of [Tarran et al., 2001]. The
model was able to show that, similar to the
experimental data, PCL depth returned to
Epithilial sheet
Tight binding
protein and
paracellular
pathway
Apical Membrane
Basolateral membrane
Figure 2: Diagram of the epithelium sheet with
the ion channels of the model depicted, also
showing the paracellular pathway, basolateral
and apical membranes.
The
model
used
in
this
work
[O’Donoghue, 2011], has a focus from those
mentioned above. It primarily aims to simulate human nasal epithelial cells (HNE) and
so far has been used to simulate the NPD
diagnostic test. Focusing the model on HNE
is advantageous due to greater availability
of data on HNE from NPD testing. This
data is used by O’Donoghue for optimisation
of the parameters sets for both CF and
non-CF cases. A disadvantage of this is that
conclusions as to how water flux in the distal
airways might behave based on simulation by
5
this model must be treated with caution.
3.3
absorption of N a+ is considered a phenotype
of the disease by most [Hummler et al., 1996,
Stutts et al., 1997, Chen et al., 2010].
There are a variety of distinct basolateral
Cl and K + channels known to exist including the calcium activated K + channel
(KCa3.1) and a voltage gated K + channel
(KvLQT1)[Bardou et al., 2009]. There is no
conclusive evidence as to what the basolateral
Cl channel might be but the I-V relation and
response to cAMP, pH and Ca+ have been investigated [Itani et al., 2007]. In the model basolateral K + and Cl transport occur via a
single ion channel each, which can be considered to encapsulate flux from all channel types.
The NaK ATPase pump is a well characterised protein, located in the basolateral membrane of epithelial tissue. In a single cycle
it pumps 3 N a+ ions out and 2 K + into
the cell. This pump is responsible for setting
the N a+ electrochemical gradient of the cell
[Alberts et al., 2002].
The NKCC co-transporter is a large protein
located basolaterally that transports two anions and two cations (N a+ and K + alongside
2Cl ). The transporter uses the N a+ electrochemical gradient to provide the energy for this
transport [Alberts et al., 2002].
Aquaporins are plasma membrane proteins
which transport water down the osmotic gradient across the apical and basolateral membranes [Matthay et al., 1996].
Of the channels mentioned above notable exceptions include calcium activated apical chloride channels as well as apical K + . These are
at present not included the in model predominately to increase simplicity but future work
will aim to extend the model to incorporate
them.
Each transport channel above provides an ion
flux which is a function of the cellular and extracellular solutions, as well as the abundance
or density of the channel in the membrane.
The NKCC co-transporter ion flux is given by
Model description
This section highlights the key points of the
model; for a more detailed explanation see
[O’Donoghue, 2011]. The model used in this
work is a single epithelial cell model, which
aims to simulate the NPD tests. To do this
it is constructed as a system of 6 variables:
cell volume Wi (t), moles of N a+ , Cl and K + ,
(N a+ (t), Cl (t) and K + (t)) and apical and
basolateral membrane potentials Vmap (t) and
Vmba (t) respectively. The luminal and serosal
solutions are modelled as having infinite volume and fixed concentration, in accordance
with NPD simulation. The transport channels
included or not in the model are done so with
dual criteria of significance to CF pathology,
and parsimony of the model.
In the apical membrane the modelled channels
are, CFTR, ENaC and aquaporins; in the basolateral membrane they are Cl and K + ion
channels, the NaK ATPase pump, aquaporins and the NKCCl co-transporter. As well as
these the paracellular conductance of K + , Cl
and N a+ are also modelled. (For a diagrammatic representation see Figure 2).
CFTR is a cAMP-dependent, protein kinase A
regulated, anion channel. The chloride channel
is activated by phosphorylation of several sites
in the R domain of the protein and shows a
linear I-V relation for symmetric Cl distribution but show rectifying for asymmetric distributions [Cao, 2005]. CFTR is expressed in the
apical membrane of airways, sweat ducts, pancreas, and digestive system epithelial tissue.
ENaC is a sodium selective ion channel found
in the apical membrane of the kidney, colon,
lung and sweat gland epithelium. It is particularly characterised by a sensitivity to amiloride
[Cao, 2005]. ENaC activity has been shown to
be regulated by CFTR in many instances making it of particular interest in CF where hyper6
+
JN KCC (t) =
max
JN
KCC
+
+
+
a aK (aCl )2
a aK (aCl )2
aN
aN
s
s
s
i
i
i
a+ /k
K + /k + + 1)(aCl /k
2
(aN
+
1)(a
+
Na
K
Cl + 1)
i
i
i
max
eqn.(1). Where JN
KCC is a parameter defining membrane density of the transporter, the
constant kN a+ , kK + and kCl are the binding
affinities for the respective molecules.
For the four ion channels the flux can be modelled in terms of current using the GoldmanHodgkin-Katz flux equation, this gives rise to
ap
ba , I ap and
four ion channel currents IN
, IK
+
a+
Cl
ba
ICl respectively [Schultz, 1980].
(1)
each compartment will have an osmolarity of
290mOsm/L. The water flux is then simply directly proportional to the osmolarity di↵erence
between the relevant compartments.
Jwap (t) = Lw vw ([S]lum
Jwba (t) = Lw vw ([S]i
[S]i )
[S]ser )
(5)
(6)
where Lw is the hydraulic conductivity of the
plasma membrane to H2 O and vw is the partial
n
n
x
molar volume of water. Positive apical water
a
ae exp( zn ' )
Inx = Pnx 'x F i
(2)
x
flux denotes water flow out of the cell, posi1 exp( zn ' )
tive basolateral water flux denotes water flowVmx F
ing into the cell. Each of the above equations
x
' =
(3)
RT
is used to construct the ODEs for the six varix
x
where In is the current per unit area, Pn is the ables of the model.
permeability of membrane x per unit area to
dWi
= Jwba Jwap
(7)
ion n, Vmx is the electric PD across membrane
dt
x, F is the Faraday constant, R is the ideal gas
constant, T is the absolute temperature, ani is
ap
IN
dN a+
i
a+
the thermodynamic activity of ion n in the cell
= JN KCC 3JN aK
(8)
dt
F zN a+
(ane is the same for extracellular activity), and
ap
ba
ICl
+ ICl
zn is the valence of ion n.
dCli
= 2JN KCC
(9)
The flux of NaK through the basolateral memdt
F zCl
brane is given by:
ba
IK
dKi+
+
= JN KCC 2JN aK
(10)
dt
F zK +
J
(t) = mJ max V
(t)
(4)
N aK
N aK N aK
and the equivalent electric circuit to the model
is used to write down the equations for membrane potentials
where m is a factor of 3 for N a+ and 2
max is a parameter determining
for K + , JN
aK
the maximum pump flux and is proportional
to the density of pumps in the membrane
and VN aK (t) is turnover rate, modelled using the existing the Smith and Crampin model
[Smith and Crampin, 2004].
The water flux acts to preserve an iso-osmolar
intracellular compartment, the osmolarity [S]
of the compartments is the sum of the [N a+ ],
[K + ], [Cl ] and impermeable ions [ ] in the
respective compartment.
is chosen so that
dVmap
=
dt
dVmba
=
dt
1 X ap
(Ii (t) + Iipa (t))
Cm
(11)
i
1 X ba
(Ii (t) + Iipa (t))
Cm
(12)
i
Where Iipa is the current of the paracellular
pathway and Cm is the capacitance of the
plasma membrane.
7
The model as described above has several assumptions and omits some features of other
models described in the previous section. The
most significant to this work are:
noted by the vector P was set for the CF and
non-CF case by previous optimisation work
in [O’Donoghue, 2011] (for a full list see appendix). In this work the sensitivity of waap
ter flux to the model parameters PNapa+ , PCl
+,
ba
ba
max
max
PCl+ , PK + , JN KCC and JN aK was investigated. These parameters are all proportional
to the membrane density of the respective
transport channel. The analysis was done using simulation over 40000 seconds, in which one
of the parameters was varied to a new value for
5000seconds, then returned to the baseline. A
continuous top hat function of the form of eqn.
(13) was introduced to the model to perform
this change.
• The luminal and serosal compartments are
assumed to be infinite and have constant
concentration. Other models which consider water flux included an ASL compartment which can alter in both concentration and volume. This means that this
work can only use apical water flux as
a means of considering airway hydration.
Additionally fixed osmolarity of the luminal compartment will alter relation of water flux due to ion flux over the apical
membrane, by comparison to these other
models.
✓(t, T 1, T 2) =
• The paracellular pathways are modelled
as permeable only to N a+ , Cl and K +
not to water molecules. This means that
other than flux through the apical and
basolateral membranes there can be no
water movement. (At present there is
no osmolarity gradient between luminal
and serosal compartments and hence there
would be no paracellular flux of water even
if it was water permeable but when an
ASL compartment is added the role of this
pathway might need to be considered.)
4
1
(1 + e
(t T 1)
0.1
)(1 + e
(t T 2)
0.1
)
(13)
where T 1 is the time until the step up and T 2
is the time till the step down. T 1 and T 2 were
set at 20000 and 25000 seconds respectively for
the entirety of the analysis. The parameter
values were altered from their baseline in percentage steps of 5%, 10%, 20%, 50% and the
corresponding negative steps (see appendix for
absolute values). This range was chosen as for
the majority of parameters, this represented
a wide spread of values within the physiologically realistic range set by [O’Donoghue, 2011],
as well as providing a good means of comparison. The simulations produced a general trend
in all six variables and the water fluxes that
can be seen in Figure 3. All six variables can
be seen to move from the initial steady state
to a new steady state and then return in line
with the applied top hat function. For the
water fluxes the steady state must always be
zero with only a transitive response to parameter changes. The area under the curve during these transitive changes provides the total moles of water per unit area moved across
the given membrane, this area was calculated
Methods
All the simulations done in this work use MatLab version 17.13 (R2011b). The simulation protocol was as per previous investigations on this model by O’Donoghue, for full
details see [O’Donoghue, 2011]. In brief, the
system of ODEs was numerically solved using odes15, with initial conditions given by
the steady state solutions obtained using the
fmincon constrained minimisation solver with
sqp algorithm. The full set of parameters de8
Figure 3: Showing the general trend of all 6 variables as well as apical and basolateral fluxes.
This figure is the product of an increase in PNapa by 20% from baseline
using the trapz function and was used as the varied. It should also be noted that the time
comparative measure for the various parameter scale over which the water flux returns to a
changes.
steady state value of zero is the same for all
Na
parameter alterations. Comparison of Pap
max
and JN aK from Figure 4. also shows that the
5 Results and Discussion
directionality of the water flux response varied
depending on the parameter. The results of
The change in apical water flux with variation Figure 4. can be more clearly expressed as a
ap
ba
ba
of parameter values PNapa+ , PCl
+ , PCl+ , PK + ,
function of area under the curve for each case.
max
max
JN
KCC and JN aK was of particular interest in Integration of each of the curves in Figure
the context of this work. As there is no ASL 4. provides the total moles of water per unit
compartment to the model, positive apical area, moved across the apical membrane in
water flux (water flow out of the cell) phys- response to the parameter alteration. This
iologically implies water flow into the ASL. was plotted against the normalised parameter
Figure 4. shows how the apical water flux value for each of the parameter changes for
varied upon the step up to the new parameter both CF and non-CF parameter sets. Figure
value for all transport channels. As can be 5. shows the plots.
seen all parameters a↵ect the apical water As can be seen all ion channels show a positive
flux and all can both increase or decrease it gradient of varying magnitude whilst the codepending on the new value. It can be seen transporter and the pump both have negative
that all simulation began at a steady state gradients. In order to discuss these results in
and that in response to the parameter change context I have compared them to the results
implemented at 20000 seconds the water flux
9
Figure 4: Graph showing how varying the parameters representing the density of a particular ion
channel in the membrane a↵ects water flux through the apical membrane in non-CF individuals.
The y axis shows the area under the curve of Jwap vs T curve and hence represents the total
water volume which moves across the apical membrane. Positive Jwap represents water flowing
out of the cell into the luminal compartment.
of [Novotny and Jakobsson, 1996b] who also
carried out a univariate sensitivity analysis on
their similar model.
In the case of the ion channels, Figure 5 shows
that increasing the density of any ion channel
in the model, increases the water flow from
the intracellular compartment through the
apical membrane and into the luminal compartment where the ASL is located. For the
+
two basolateral channels Kba
and Clba this is
intuitive. Increasing the density of basolateral
channels increases flux from the intracellular
to the serosal compartment; this decreases
intracellular osmolarity and water flows out of
the cell. In the case of the apical ion channels
CFTR and ENaC this a↵ect is somewhat
less intuitive. In the simulation, increasing
the density of ENaC channels in the apical
membrane increases the intracellular [N a+ ]
as expected but decreases the intracellular
chloride and potassium concentrations as well
as the cell volume see Figure 3. The sum
of the ion concentration produces a decrease
in intracellular osmolarity and hence water
flows out of the cell. This is contrary to the
findings of [Novotny and Jakobsson, 1996b]
who find an increase in the same parameter in
their model, causes apical volume reduction
and cell volume increase. This discrepancy
highlights several di↵erences between the
models, and the significance of these di↵erences. As mentioned the infinite volume and
concentration invariant luminal and serosal
compartments of this model, mean that there
can only be equal net fluxes either into or out
of the intracellular compartment and no net
flux from luminal to serosal. Additionally in
the model of [Novotny and Jakobsson, 1996a]
the ASL layer is depleted not only by flux
into the cell but also by evaporation from
the mucosal layer at a rate of 4.38 ⇥ 10 8
l.m2 s 1 . This di↵erence means that a direct
comparison of the results is not justified,
however no other models with infinite luminal
10
(a)
(b)
Figure 5: Graph showing how varying the parameters representing the density of a particular
ion channel in the membrane a↵ects water flux through the apical membrane in (a) CF and (b)
non-CF cases. The y axis shows the area under the curve of Jwap vs T - the total water volume
per unit area which moves across the apical membrane. Positive Jwap represents water flowing
out of the cell into the luminal compartment.
and serosal compartments report on water
fluxes, and as such investigation of the source
of the di↵erences between these two models is
of value.
In the case of the N aK pump again the same
discrepancy occurs, an increase in pump density in this model reduces apical water flux. In
[Novotny and Jakobsson, 1996b] simulations
airway surface dehydration is reported in
response to decreased pump density. In this
model an increase in pump activity leads to a
decrease in intracellular N a+ and a increase
intracellular Cl and K + , this can be seen in
Figure (6). This net ion concentration change
leads to an increased intracellular osmolarity
and a decrease in the flux out of the apical
membrane. It could be understood that in
this model, the increase in the pumps density
increases the sodium electrochemical gradient,
thereby increasing the activity of the NKCC
co-transporter which leads to the increase in
Cl and contributes to the increase in K + .
As
two
of
the
results
discussed
above are opposed to the work of
[Novotny and Jakobsson, 1996b] it seems
necessary that further investigation of
water flux will require the addition of a
finite ASL. This will allow a more direct comparison not only with the models
of
[Novotny and Jakobsson, 1996b]
and
[Warren et al., 2009] but also with experimental evidence which report ASL measurements.
Figure 7. shows the maximum gradients of
Figures 5. The gradients are a measure of the
maximal e↵ect of the parameters on the apical
water flux and provides a comparison between
the CF and non-CF cases.
CF and non-CF show the same trends for all
parameter changes. In all cases other than
apical sodium variation, the non-CF case has
a greater magnitude response.
ba
The gradient of PK
is the second
largest for both the CF and non-CF
11
Figure 6: Graph showing how the intracellular ion concentrations vary as a result of increasing
max by 20%, as can be seen N a decreases while both Cl and K increase.
JN
i
i
i
aK
Figure 7: Chart showing the maximum gradients of Figure 5
cases, and its direction agrees both
with the [Novotny and Jakobsson, 1996b]
model and the in vitro
work of
[Cowley and Linsdell, 2002]. It is know that
basolateral K + channels play important roles
in the regulation of fluid movements in the cell
[Wang, 2009] and it is thought that increasing
basoalteral K + channels could stimulate
CaCC channels [Cowley and Linsdell, 2002].
If this is the case, and it is possible to find a
targeted therapy for the appropriate channel,
this could o↵er an interesting prospect for
treatment.
Another area of interest is in understanding
whether measurements that can be made
relatively easily in vivo correlate with water
flux and can hence be used as a biomarker
for airway hydration.
A commonly used
measurement is the transepithilial PD (V (t))
measured during nasal PD diagnostic tests.
Investigating the relation of this to water flux
in the model was done via similar methods
to those above. Figure 8 shows a plot of
moles of water per unit area moved over the
apical membrane against the steady state
value of V(t) after the parameter change was
made. The graphs show that regardless of
the parameter varied there is a monotonic
correlation between V(t) and apical water flux
over the range of parameters investigated.
This correlation could be highly useful as V(t)
measurements could be used as a biomarker
of water flux changes providing a more quantitative measurement for studies investigating
the a↵ect of therapies designed to target
water flux. The work here only constitutes a
cursory investigation into the relation and a
far more detailed study would be required to
ascertain whether this relation is seen when
12
Figure 8: Graphs showing that transepithilial PD is correlated with apical water flux. The
graphs show the relation for the di↵ering parameter changes on the CF parameter set
the modelling environment is expanded to
include other ion channels and finite ASL
compartment. Nevertheless it is not unphysiological to think that correlation between
V(t) and apical water flux would be present
as both are determined by compartmental ion
compositions.
6
Conclusion
In conclusion the analysis done here suggests
that increasing basolateral potassium channel
density may help to increase water flux onto
the apical surface of the epithelium and in doing so increase ASL volume. The analysis has
also highlighted some key di↵erences between
this and other models in the field, these need
closer investigation and it may that in order
to e↵ectively use this model to simulate water flux it is necessary to include an finite ASL
compartment and water permeable paracellular pathways. This work also suggests that
there may be monotonic correlation between
the parameters analysed and V(t).
7
Further Work
The most important further work in terms of
water flux modelling is to expand the model to
include an additional finite variable composition ASL. The framework for this has already
been set out in [O’Donoghue, 2011], and would
include 4 additional ODEs. For parameterisation of this work it is would be important
to have good experimetnal data on ASL variation as there are many methods of measurement and it is not clear whether it is the entire
ASL or merely the PCL that is considered in
some works.
Another extension to this work would be
in the form of calcium excitation modelling. As mentioned in previous sections
[Warren et al., 2009] have developed a model
in which calcium activated channels are also
included. Simulation with the model could be
possible via a multi-variate parameter analysis. Increasing the apical chloride permeability and the basolateral potassium would
mimic the result of a calcium excitation,
and a comparison of the outcome with that
13
of [Warren et al., 2009] would provide insight [Cohen-Cymberknoh et al., 2011] Coheninto the two di↵ering model environments.
Cymberknoh, M., Shoseyov, D., and
Kerem, E. (2011). Managing cystic fibrosis:
strategies that increase life expectancy and
References
improve quality of life. American journal
of respiratory and critical care medicine,
[Alberts et al., 2002] Alberts, B., Johnson, A.,
183(11):1463.
Lewis, J., Ra↵, M., Roberts, K., and Walter, P. (2002). Molecular biology of the cell. [Cowley and Linsdell, 2002] Cowley, E. and
Garland Science, 4 edition.
Linsdell, P. (2002). Characterization of basolateral k+ channels underlying anion se[Andersen, 1938] Andersen, D. (1938). Cyscretion in the human airway cell line calu-3.
tic fibrosis of the pancreas and its relation
J Physio, l(538):747–57.
to celiac disease: a clinical and pathologic
study. Archives of Pediatrics and Adolescent [Davis, 2006] Davis, P. (2006). Cystic fibrosis
Medicine, 56(2):344.
since 1938. American journal of respiratory
and critical care medicine, 173(5):475.
[Bardou et al., 2009] Bardou, O., Trinh, N.,
and Brochiero, E. (2009). Molecular diver- [Di sant’agnese et al., 1953] Di sant’agnese,
sity and function of k+ channels in airway
P., Darling, R., Perera, G., and Shea, E.
and alveolar epithelial cells. American Jour(1953). Abnormal electrolyte composition
nal of Physiology-Lung Cellular and Molecof sweat in cystic fibrosis of the pancreas.
ular Physiology, 296(2):L145–L155.
Pediatrics, 12(5):549.
[Boucher, 2007] Boucher, R. (2007). Airway [Donaldson and Boucher, 2003] Donaldson, S.
surface dehydration in cystic fibrosis: pathoand Boucher, R. (2003).
Update on
genesis and therapy. Annu. Rev. Med.,
pathogenesis of cystic fibrosis lung disease.
58:157–170.
Current opinion in pulmonary medicine,
9(6):486.
[Boucher, 1994] Boucher, R. C. (1994). Human airway ion transport. part one. Ameri- [Dua et al., 2011] Dua, V., Moss, G., and Vercan Journal of Respiratory and Critical Care
gani, P. (2011). Cystic fibrosis and tranepMedicine, 150(1):271–81.
ithelial ion and water transport. CP3 Presentation.
[Cao, 2005] Cao, L. (2005). Modulation of
CFTR and ENaC Channel Function by In- [Duszyk and French, 1991] Duszyk, M. and
teracting Proteins and Trafficking. Acta
French, A. S. (1991). An analytical model
Biomedica Lovaniensia. Coronet Books Inc.
of ionic movements in airway epithelial cells.
Journal of Theoretical Biology, 151(2):231 –
[Chen et al., 2010] Chen, J., Stoltz, D., Karp,
247.
P., Ernst, S., Pezzulo, A., Moninger, T.,
Rector, M., Reznikov, L., Launspach, J., [Falkenberg and Jakobsson, 2010] Falkenberg,
Chaloner, K., et al. (2010). Loss of anion
C. V. and Jakobsson, E. (2010). A biotransport without increased sodium absorpphysical model for integration of electrical,
tion characterizes newborn porcine cystic fiosmotic, and ph regulation in the human
brosis airway epithelia. Cell, 143(6):911–
bronchial epithelium. Biophysical Journal,
923.
98(8):1476 – 1485.
14
[Goldman et al., 1997] Goldman, M., Anderson, G., Stolzenberg, E., Kari, U., Zaslo↵,
M., and Wilson, J. (1997). Human defensin-1 is a salt-sensitive antibiotic in
lung that is inactivated in cystic fibrosis.
Cell, 88(4):553–560.
[Hartmann and Verkman, 1990] Hartmann,
T. and Verkman, A. (1990). Model of ion
transport regulation in chloride-secreting
airway epithelial cells. integrated description of electrical, chemical, and fluorescence
measurements.
Biophysical Journal,
58(2):391 – 401.
[Horisberger, 2003] Horisberger, J.-D. (2003).
Enac-cftr interactions: the role of electrical coupling of ion fluxes explored in an
epithelial cell model. Pflügers Archiv European Journal of Physiology, 445:522–528.
10.1007/s00424-002-0956-0.
P., Turner, S., et al. (2008). Lung function in infants with cystic fibrosis diagnosed
by newborn screening. American journal
of respiratory and critical care medicine,
178(12):1238.
[Matsui et al., 1998] Matsui, H., Grubb, B.,
Tarran, R., Randell, S., Gatzy, J., Davis, C.,
and Boucher, R. (1998). Evidence for periciliary liquid layer depletion, not abnormal
ion composition, in the pathogenesis of cystic fibrosis airways disease. Cell, 95(7):1005–
1015.
[Matthay et al., 1996] Matthay,
M.,
Folkesson, H., and Verkman, A. (1996).
Salt and water transport across alveolar
and distal airway epithelia in the adult
lung.
American Journal of PhysiologyLung Cellular and Molecular Physiology,
270(4):L487–L503.
[Hummler et al., 1996] Hummler, E., Barker,
[Novotny and Jakobsson, 1996a] Novotny, J.
P., Gatzy, J., Beermann, F., Verdumo, C.,
and Jakobsson, E. (1996a). Computational
Schmidt, A., Boucher, R., and Rossier, B.
studies of ion-water flux coupling in the air(1996). Early death due to defective neonaway epithelium. i. construction of model.
tal lung liquid clearance in ↵enac-deficient
American Journal of Physiology-Cell Physimice. Nature genetics, 12(3):325–328.
ology, 270(6):C1751–C1763.
[Itani et al., 2007] Itani, O., Lamb, F.,
Melvin, J., and Welsh, M. (2007). Baso- [Novotny and Jakobsson, 1996b] Novotny, J.
and Jakobsson, E. (1996b). Computational
lateral chloride current in human airway
studies of ion-water flux coupling in the airepithelia. American Journal of Physiologyway epithelium. ii. role of specific transLung Cellular and Molecular Physiology,
port mechanisms. American Journal of
293(4):L991–L999.
Physiology-Cell Physiology, 270(6):C1764–
[Kerem et al., 1989] Kerem, B., Rommens, J.,
C1772.
Buchanan, J., Markiewicz, D., Cox, T.,
Chakravarti, A., Buchwald, M., and Tsui, [O’Donoghue, 2011] O’Donoghue, D. (2011).
Quantitative model of ion transport in huL. (1989). Identification of the cystic fiman airway epithelial cells.
brosis gene: genetic analysis.
Science,
245(4922):1073.
[Riordan et al., 1989] Riordan, J., Rommens,
[Linnane et al., 2008] Linnane, B., Hall, G.,
J., Kerem, B., Alon, N., Rozmahel, R.,
Nolan, G., Brennan, S., Stick, S., Sly,
Grzelczak, Z., Zielenski, J., Lok, S., Plavsic,
P., Robertson, C., Robinson, P., Franklin,
N., Chou, J., et al. (1989). Identification of
15
the cystic fibrosis gene: cloning and characterization of complementary dna. Science,
245(4922):1066.
cells. PhD thesis, Hong Kong University of
Science and Technology.
[Wark et al., 2009] Wark, P., McDonald, V.,
and Jones, A. (2009). Nebulised hypertonic
[Rommens et al., 1989] Rommens, J., Iansaline for cystic fibrosis. Cochrane Database
nuzzi, M., Kerem, B., Drumm, M., Melmer,
Syst Rev, 2.
G., Dean, M., Rozmahel, R., Cole, J.,
Kennedy, D., Hidaka, N., et al. (1989). Identification of the cystic fibrosis gene: chro- [Warren et al., 2009] Warren, N., Tawhai, M.,
and Crampin, E. (2009). A mathematical
mosome walking and jumping. Science,
model of calcium-induced fluid secretion in
245(4922):1059.
airway epithelium. Journal of Theoretical
Biology, 259(4):837 – 849.
[Schultz, 1980] Schultz, S. (1980). Basic principles of membrane transport. IUPAB bio[Zabner et al., 1998] Zabner, J., Smith, J. J.,
physics series. Cambridge University Press.
Karp, P. H., Widdicombe, J. H., and Welsh,
M. J. (1998). Loss of cftr chloride chan[Smith et al., 1996] Smith, J. J., Travis, S. M.,
nels alters salt absorption by cystic fibroGreenberg, E., and Welsh, M. J. (1996).
sis airway epithelia in vitro. Molecular Cell,
Cystic fibrosis airway epithelia fail to kill
2(3):397 – 403.
bacteria because of abnormal airway surface
fluid. Cell, 85(2):229 – 236.
[Smith and Crampin, 2004] Smith, N. and
Crampin, E. (2004). Development of models
of active ion transport for whole-cell modelling: cardiac sodium-potassium pump as
a case study. Progress in biophysics and
molecular biology, 85(2-3):387–405.
[Stutts et al., 1997] Stutts, M., Rossier, B.,
and Boucher, R. (1997). Cystic fibrosis
transmembrane conductance regulator inverts protein kinase a-mediated regulation
of epithelial sodium channel single channel
kinetics. Journal of Biological Chemistry,
272(22):14037.
[Tarran et al., 2001] Tarran, R., Grubb, B.,
Parsons, D., Picher, M., Hirsh, A., Davis,
C., and Boucher, R. (2001). The cf salt controversy:: In vivo observations and therapeutic approaches. Molecular cell, 8(1):149–
158.
[Wang, 2009] Wang, D. (2009). Regulation of
anion secretion in human airway epithelial
16