Table Of ContentIon mediated crosslink driven mucous swelling kinetics
S. Sircara and A. J. Roberts
SchoolofMathematicalSciences,UniversityofAdelaide,SA5005,Australia
a CorrespondingAuthor(email:[email protected])
5
1
0
2 Abstract
n Wepresentanexperimentallyguided,multi-phasic,multi-speciesionicgelmodeltocom-
a
J pareandmakequalitativepredictionson therheologyofmucusofhealthyindividuals(Wild
0 Type) versus those infected with Cystic Fibrosis. The mixture theory consists of the mucus
2 (polymerphase)andwater(solventphase)aswellasseveraldifferentions:H+,Na+ andCa2+.
The model is linearized to study the hydration of spherically symmetric mucus gels and cal-
]
O ibratedagainstthe experimentaldata ofmucus diffusivities. Nearequilibrium,the linearized
T formoftheequationdescribingtheradialsizeofthegel,reducestothewell-knownexpression
o. used in the kinetic theory of swelling hydrogels. Numerical studies reveal that the Donnan
i potentialisthedominatingmechanismdrivingthemucusswelling/deswellingtransition.How-
b
- ever,thealteredswellingkineticsoftheCysticFibrosisinfectedmucusisnotmerelygoverned
q
bythehydroelectriccompositionoftheswellingmedia,butalsoduetothealteredmovement
[
ofelectrolytesaswellasduetothedefectivepropertiesofthemucinpolymernetwork.
1
v
3 Keywords:Donnanpotential,CysticFibrosis,mucusdiffusivity,polyelectrolytegel
0
0
5
1 Introduction
0
.
1
0 Mucus is a polyelectrolyte biogel that plays a critical role as a protective, exchange and transport
5
mediuminthedigestive,respiratoryandreproductivesystemsofhumansandothervertebrates[1,2].
1
: Its swelling mechanism are of special interest because of its role in understanding a variety of
v
i diseases including cystic fibrosis (CF) [3, 4, 5]. The conformation of the long-chain, negatively
X
chargedmucus glycoproteinsdepends stronglyon factorssuchas pH,ionicstrength andionic bath
r
a composition [6]. Mucin is present in secretory vesicles at very high concentrations where they
are shielded primarily by a combination of divalent ions (e.g., Ca2+) [7]. Experiments show that
the mucus gel may swell explosively, up to 600-fold, in times that are the order of a few seconds,
a process not observed in hydrogel swelling [8, 9]. Experiments also confirm that this rapid and
massive expansion of the mucus gel is driven by an exchange of calcium in the vesicle for a
monovalent ion (e.g., Na+) in the extracellular environment [9]. This is because, calcium being
divalent, must balance two negative charges rather than one. Hence, a divalent Ca2+ ion can act
as a ‘cross-linker’ between two polymer strands, allowing much tighter condensation than when
thenegativechargesofthenetworkareshieldedbymonovalentions(Fig.1).Further,experiments
demonstrate that the exocytosed mucin is recondensed if the calcium concentration of the ionic
mediumisincreasedsufficiently[10].Theseobservationsindicatethattheamountandthenature
1
ofthesaltdissolvedinthesolventdeterminestheinitialandtheequilibriumconfigurationofthese
gels.
Experimental findings have reported that (unlike hy-
drogelswelling)asimpleosmoticpressuredifferenceof
solute particles across the mucus gel cannot explain the
massiveandtheexplosivepost-exocytoticswellingobser-
vations [11]. Mucushydration results froma balancebe-
tweenosmoticforces,diffusionalhindrance(oftheensem-
ble of entangled polymer mesh of the mucus), polyionic
charges of the mucin and the other fixed polyions en-
trapped within the gel’s matrix,the last two factors being
themainideabehindtheforcesviaDonnanpotential[12].
Figure 1: Schematics showing how the
The Donnan potential is not constant, but modulated by
ion-displacementchangesbetweencal-
the concentration offree cationsand polycationsandthe
cium ion (Ca2+) and a monovalent ion
pH of the hydrating fluid [13]. Marriott et al. observed
(e.g.,Na+)leadingtochangesinthenet-
that a decreased monovalent ion concentration, or an in-
workstructureofthemucusmatrixand
creased Ca2+ concentration, in the airway surface liquid
itseventualexpansion.
was a limiting factor in mucus hydration in CF-infected
mucus[14].Lisleindicatedtheroleofdefectiveprocess-
ing of mucin polymers, via increased sulfation, in the abnormal mucus hydration in CF [15]. In-
creasedsulfationandsialilationfoundinCFmucins[16]areimportantformucusswelling,since
the high affinity of sulfate residues for calcium drastically shifts the normal monovalent/divalent
ion-exchange properties of mucins and their swelling characteristics following the exocytotic re-
lease [17]. In summary, besides the ionic composition of water on the surface of the airway, the
defective processing of mucins and its abnormal ion exchange properties are essential factors to
explainthecharacteristicallydeficienthydrationandrheologyofmucusinCF.However,theprecise
quantitativelinkbetweentheseelementsandmucushydrationstillremainsunknown[18].
Thetheorytounderstandtheswellinganddeswellingofionicgelshasalonghistorybeginning
with the classical work of Flory [19, 20, 21] and Katchalsky [22] (see also [23, 24]). An early
study to understand the kinetics (and not simply the equilibria) of swelling and deswelling, was
done by Tanaka and colleagues [25, 26] who developed a kinetic theory of swelling gels viewing
a gel as a linear elastic solid immersed in a viscous fluid. Although they neglected the motion of
the fluid solvent, the model reasonably explained the swelling of a gel to its equilibrium volume
fraction.Theirworkgaverisetotheconceptofgeldiffusivityasawaytocharacterizethekinetics
of swelling. The gel diffusivity is defined as D = L2/τ where L is the equilibrium size (length
or radius) of a gel and τ is the time constant of exponential swelling toward the equilibrium size.
They found that expansion of the gel was governed approximately by a diffusion equation with
diffusion coefficient D. However,how the diffusivity of the gel,or more generally the kinetics of
swelling (perhaps with large changes in volume fraction) is affected by the movement of the ions
anddefectivepropertiesofthemucuspolymernetworkispoorlyunderstood.
Subsequent studies relaxed the assumption of linear elasticity by defining the force on the gel
to be the functional derivative of the free energy function for the polymer mesh [27, 28, 29, 30,
31]. However, most of these works neglected the fluid flow that must accompany swelling. Wang
et al. [32] added fluid flow by application of two-phase flow theory but considered only small
polymer volume fractions and small gradients in the volume fraction. Durning and Morman [27]
2
also used continuity equations to describe the flow of solvent and solution in the gel, but used a
diffusionapproximationwithaconstantdiffusioncoefficienttodeterminethefluidmotion.More
recently,Wolgemuthetal.[13]extendedthesetheoriestostudytheswellingofapolyelectrolytegel.
However, these current state of the art models do not couple the binding/unbinding of the ions to
the network micro-structure,an idea which is critical in describing the bulk mechanical properties
ofthepolymer[33,34].
The purpose of this paper is to provide a new comprehensive model detailing the swelling
kineticsofmucin-likeionicgelsanduseittocalibratetherheologicaldataofsphericallysymmetric
mucin granules, exocytosed from the goblet cells of CF-patients versus those released from the
non-CF individuals. The novel feature of this model is the coupling of the dynamical motion of
the swelling gel with the ionic binding to the polymer network. We use this model to show how
the altered kinetics of the CF-infected mucus depends on the electro-chemical environment of
theswellingmediaaswellastherheologicalpropertiesofthepolymernetwork.Thenextsection
presentsthedetailsofthismodel(§2),includingequationsofmotion(§2.1)andchemistryofbinding
reactions (§2.2). Sections §2.3 and §2.4 outline the linearized analysis of spherically symmetric
swelling gels and methods to estimate model parameters, respectively. The results pertaining to
theswellingkineticsandtheequilibriumconfigurationoftheseionicgelsunderdifferentchemical
stimuliarepresentedin§3.Weconcludewithabriefdiscussionoftheimplicationoftheseresults
onthepathophysiologyoftheinfectedmucus.
2 Multi-species, multi-phase mucus-gel model
Thepolymergelismodeledasamulti-componentmaterial,consistingofk differenttypesofparti-
cles.Specifically,thismaterialconsistsofsolventparticles,polymers,andseveralsmallmolecular
ion species. The polymer is assumed to be made up of monomers (i.e., charge units), denoted
asM,eachofwhichcarriesasinglenegativecharge.Thepositivelychargedionsinthesolventare
Hydrogen (H+), Sodium (Na+) and Calcium (Ca2+). The negatively charged ions could include
Hydronium(OH−)andChloride(Cl−).Becausethenegativelychargedionsareassumedtobenot
involved in any binding reactions with the gel, acting only as counterions to positive charges, we
identify these ions by the name chloride. The binding reactions of the positively charged ions with
themonomersare
M− +H+ −(cid:41)−k−h(cid:42)− HM, M− +Na+ −(cid:41)−k−n(cid:42)− NaM, M− +Ca2+ −(cid:41)−k−(cid:42)c− MCa+, M− +MCa+ −(cid:41)−k−(cid:42)x− M Ca,
2
k−h k−n k−c k−x
(1)
where k and k , for C = h,n,c,x, are the binding and the unbinding rates, respectively. We
C −C
assume that all the binding sites/charge sites are identical although the binding affinities for the
different ions are different. The species M Ca are cross-linked monomer pairs, and the species
2
M−,MCa+,NaMandHMaredifferentmonomerspecies,allofwhichmovewiththesamepoly-
mer velocity. The ion species are freely diffusible, but because they are ions, their movement is
restrictedbytherequirementtomaintainelectroneutrality.Finally,becauseasmallamountofwater
dissociates into hydrogen and hydronium, we are guaranteed that there are always some positive
andnegativeionsinthesolvent.
3
2.1 Equations of motion and interface conditions
SupposewehavesomevolumeV ofamixturecomprisedofktypesofparticles(includingpolymer,
solvent and ion species) each with particle density (number of particles per unit volume), n (x,t)
j
andparticlevolume,ν ,movingwithvelocity,v (x,t),j = 1,...,k.Thesubscripts(1,2)denote
j j
thepolymerandthesolventphase,respectively.Wedenotethesephaseswithsubcripts(p,s)respec-
tively.Thevolumefractionsforpolymerandsolventareθ = ν n andθ = ν n .Conservationof
p p p s s s
polymerimplies
∂θ
p
+∇·(v θ ) = 0. (2)
p p
∂t
Weassumethattheothermolecularspeciesdonotcontributesignificantlytothevolume.Therefore,
the volume fractions, θ + θ = 1. It follows from (2) and a similar conservation argument for
p s
solventthat
∇·(θ v +θ v ) = 0. (3)
s s p p
Themotionofthepolymerandsolventphaseofthismulti-componentmixtureisgovernedbythe
StokesequationforNewtonianfluid,whichare
θ θ
s p
∇·(θ σ (v ))−ξ φ (v −v )− ∇µ = 0, (4)
p p p p p s p
ν ν
s p
and
θ θ (cid:88) θ
s s ˆ s
∇·(θ σ (v ))−ξ φ (v −v )− φ ∇µ − ∇µ = 0, (5)
s s s p s p j j s
ν ν ν
s s s
j≥3
whereσ (v) = ηj(∇v+∇vT)+λ I∇·v,aretheviscousstresses,η > 0andλ aretheviscosities,
j 2 j j j
ξ is the drag coefficients. φp = np/np+ns, φs = ns/np+ns, φj = nj/(cid:80)i(cid:54)=1,2ni for j ≥ 3 (assuming that
the ions are dissolved in the solvent), are the polymer, solvent and ion species per total solvent
particle fractions, respectively. The first and second terms in PDE (4, 5) represents the viscous
forces and drag forces due to the friction between the two phases, respectively. The third term in
PDE (5)representstheforceduetothedissolvedcounter-ionsinthesolvent(osmoticeffect),while
the last terms in PDE (4, 5) are the chemical forces due to the respective phases. The chemical
potentialsaregivenby
µ = k TM +z Φ +ν P,
p B p m e p
µ = k TM +ν P, (6)
s B s s
where M and M represent the entropic contribution to the respective chemical potentials (de-
p s
scribed later in this section), k is the Boltzmann constant, T is temperature, P is pressure, z is
B m
theaveragechargepermonomer(definedlaterin§2.2),Φ istheelectricpotentialarisingfromthe
e
unbalancedcharges.Fortheionspecies,theparticlevolume,ν ,iseffectivelyzerosothat
j
µ = k T (lnφ +1−2σ )+z Φ , j ≥ 3, (7)
j B j I j e
wherez isthechargeonthejthionicspecies.Incalculatingparticlefractions,φ ,weassumethat
j j
theparticledensityoftheionsisinsignificantlysmallcomparedtotheparticledensityofpolymer
(cid:80)
andsolvent, n (cid:28) n +n .Theionspeciessatisfytheforcebalance
j≥3 j s p
ξ n (v −v )−n ∇µ = 0, j ≥ 3. (8)
j j s j j j
4
Finally,sincethereisafreemoving-edgetothegel,ononesideofwhich(insidethegel)θ− = θ ,
p p
andontheothersideofwhich(outsidethegel)θ+ = 0,θ+ = 1,implyingthatthereisnopolymer
p s
outsidethegel.Thesuperscripts(+,−)denoteregionsinsideandoutsidethegel,respectively.The
interfaceconditionsare
(cid:18) (cid:19)
k T P
σ (v−)n = B M− +z Ψ +ν n, (9)
p p ν p m e pk T
m B
and
(cid:18) (cid:19)
k T P
(cid:0)σ (v+)−σ (v−)(cid:1)n = B M+ −M− −σ+ +σ− −ν n, (10)
s s s s ν s s I I sk T
s B
where
(cid:18) (cid:19)
1 1 T (cid:16)χ (cid:17)
M = lnφ + −1 φ + 0 φ2 +µp ,
p N p N s T 2 s 0
(cid:18) (cid:19)
1 T (cid:16)χ (cid:17)
M = lnφ + 1− φ −σ + 0 φ2 +µs . (11)
s s N p I T 2 p 0
N isthenumberofmonomersperpolymer-chainandT isareferencetemperature.Thehydrostatic
0
pressureisP− = P,P+ = 0andthenormalizedelectrostaticpotential(Ψe = Φe/kBT)is,Ψ−e = Ψe,
Ψ+ = 0. The normal to the free surface is denoted by n. The term σ in the solvent chemical
e I
(cid:80)
potential (Eq. (11)) is the total ion particle fraction (σI = j≥3nj/ns) and represents osmotic
pressure as characterized by van’t Hoff’s law. The quantities χ,µp,µs are the Flory interaction
0 0
parameter,standardfree energies forpure polymerandpure solventrespectively. These arerelated
tothecross-linkfraction,α (thefractionofmonomersboundwithcalcium),
(cid:18) (cid:19)
1
χ = z((cid:15) +(cid:15) )−2 1− (cid:15) −(cid:15) α,
1 2 1 1
N
(cid:18) (cid:19)
z 1 1
µp = −(cid:15) +(cid:15) 1− + (cid:15) α,
0 12 4 N 2 3
z
µs = −(cid:15) . (12)
0 22
(cid:15) (i = 1,...,4)arethenearestneighborinteractionenergiesparametersofthevariousmonomer-
i
monomer and monomer-solvent pairs and z is the number of interaction sites on a lattice (i.e.,
coordinationnumber)[35].EliminatingP fromEqns.(9,10)wefindasingleinterfacecondition
(cid:0) (cid:1)
σ (v−)−σ (v−)+σ (v+) n = Σ n, (13)
p p s s s s net
whereΣ isthenetswellingpressureattheinterface,
net
Σ M− M− T µ0 z σ− σ+
net = p − s + 0 s + mΨ + I − I . (14)
e
k T ν ν T ν ν ν ν
B m s s m s s
Intheaboveequation,thefourthtermrepresentsswellingpressurecomingfromanyelectriccharge
on the monomers, referred as the Donnan pressure (z Ψ is the corresponding Donnan potential)
p e
whereas the last two terms represents the osmotic swelling pressure coming from the difference
betweentheconcentrationsofionsdissolvedinthegelandthosedissolvedinthebath.Sircaretal.
5
[35] derivedthese equations of motion and the interface conditions using the standard variational
arguments to minimize the rate of work dissipated within the polymer and the solvent. Unlike
previoustheories,thismodelaccuratelycaptureshowthebinding/unbindingofthedissolvedions
influences the motion of the swelling gel. The chemistry of the dissolved ions and the charged
polymerspeciesisdescribednext.
2.2 Ionization chemistry
This section formulates a mathematical model to represent how the chemical species move and
react. Let the concentrations per total volume of the polymer species be denoted by x = [M Ca],
2
m = [M−],v = [NaM],w = [MCa+]andy = [HM],withthetotalmonomerconcentration
m = m+2x+w+v +y. (15)
T
The concentrations per solvent volume of the ion species are denoted as c = [Ca2+], n = [Na+],
h = [H+],andc = [Cl−].Withconcentrationsexpressedinunitsofmolesperliter,therelationship
l
between ion particle fractions φ and concentrations c is φ = ν N c , where N is Avagadro’s
j j j s A j A
number. To describe thechemicalreactions, weuse thelaw ofmass action. Since allthe monomer
speciesareadvectedwiththepolymervelocityv ,themonomerspeciesevolveaccordingto
p
∂j
+∇·(v j) = R+ −R−, j = x,v,w,y (16)
∂t p j j
whereR+,R− aretheforwardandbackwardratesforthemonomerbindingreactionsinEqn.(1),
j j
respectively.Themonomerconcentration,m,isobtainedfromEqn.(15)andmT = θp/νpNA.Under
theassumptionoffastchemistry,wesettherighthandsideofthe PDE (16)tozero,whichreduces
intothefollowingsetofequationsforeachofthemonomerspecies
θ θ θ θ
w = s mc, v = s mn, y = s mh, x = s m2c, (17)
K φ2 K φ2 K φ2 4K2φ4
c s n s h s c s
where Kc = k−c/kc, Kn = k−n/kn, Kh = k−h/kh, Kx = k−x/kx = 4Kc. The above expressions (17),
assumethattheunbinding (dissociation) reactionsareionizationreactionsthatrequiretwo “units”
ofsolvent.Wetaketheunbindingreactionratestobek φ2,forC = c,x,n,handbecausecalcium
−C s
isadivalention,2k = k andk = 2k .
x c −x −c
Similarly,undertheassumptionoffastdiffusionandchemistry,thelawofmassactionforthe
ionspeciesreducesto
C = C e−zCΨe (18)
b
with symbol C = c,h,n,c , z = z = 1,z = 2 and z = −1. The subscript ‘b’ denotes
l n h c c
l
the corresponding bath concentrations [35]. The electrostatic potential, Ψ , is determined by the
e
electroneutralityconstraintinsidethegel,namely,
(2c+n+h−c )θ +z m = 0, (19)
l s m T
where z is the average residual charge of the unbound monomers which depends on the amount
m
ofbindingwithions,
z m = w−m. (20)
m T
6
Sinceboththeelectrostaticpotentialandpolymerparticlefractionareassumedtobezerooutside
thegel,electroneutralityinthebathrequiresthat
2c +n +h −c = 0. (21)
b b b lb
Finally,thecrosslinkfraction(orthefractionofmonomersboundwithcalcium)isα = x ,where
mT
Eqns.(15,17)definetheconcentrations,m ,x,respectively.
T
Insummary,thedynamicalmotionofafreelyswellingmucus-gelismodeledbythesystemof
equations including the mass conservation PDE (2), total volume conservation PDE (3) together
withforcebalance PDES (4-5)andinterfaceconditionEqn.(13),subjecttotheconstraintsEqn.(15)
(monomerconservation),Eqn.(18)(ionmotion)andEqn.(19)(electroneutrality).
2.3 Linearized analysis of spherically symmetric swelling gels
Wenowconsidertheswellingkineticsofaradiallysymmetricmucusgelusingalinearizedanalysis
ofthegoverningequations,describedintheprevioustwosections.Thevolumefractionθ isnon-
p
zeroonthedomain0 ≤ r < R(t),withR(maximumradiusofthesphere)afunctionoftime.Since
onlythefinitesolutionsofthegoverningequationsareofinterest(inparticular,finitesolutionsof
the force balance, PDES (4, 5)) inside this domain, we assume that vp = vs = 0 at r = 0. The
volumeconservation, PDE (3),impliestheconstraintθpvp+θsvs = 0throughoutthedomain.Using
this constraint, we reduce the two force balance equations into a single equation (by multiplying
PDE (4)byvolumefractionθs and PDE (5)byvolumefractionθp andsubtracting)
(cid:20) (cid:18) ∂v λ ∂ (cid:19)(cid:21) (cid:26) (cid:20) ∂ (cid:18)θ v (cid:19) λ ∂ (cid:18)r2θ v (cid:19)(cid:21)(cid:27)
θ ∇· θ η p + p r2v +θ ∇· θ η p p + s p p
s p p ∂r r2 ∂r p p s s∂r θ r2 ∂r θ
s s
(cid:18) (cid:19)
φ φ
p s
+ξv θ + −θ θ ∇(Σ ) = 0, (22)
p p p s net
ν ν
s p
whereastheinterfacecondition(Eqn.(13))reducesto
∂v ∂ (cid:18)θ v (cid:19) λ ∂ λ ∂ (cid:18)r2θ v (cid:19)
η p +η p p + p r2v + s p p = Σ at r = R(t). (23)
p ∂r s∂r θ r2 ∂r p r2 ∂r θ net
s s
Notethat∇(µp − µs) = ∇(Σ ),whereΣ istheinterfaceswellingpressuredefinedinEqn.(14).
νm νs net net
Thevelocitiesinsidethegelarev− = v andoutsidethegelarev+ = 0.Next,weassumethat
p/s p/s p/s
thelinearizedvelocityv issmall(theequilibriumvelocityv∗ = 0)andthatθ = θ∗ +δθ ,where
p p p p p
θ∗ is the equilibrium polymer volume fraction. Sircar et al. [35] gives details on the equilibrium
p
solution.Sincethisisamovingboundaryproblem,itisconvenienttomapthedomain0 ≤ r < R(t)
ontothefixeddomain0 ≤ y < 1bymakingthechangeofvariablesr = R(τ)y andt = τ.Further,
we seek space-time variable separated solutions of the form v = a(τ)f (y) and δθ = a(τ)f (y).
p 1 p 2
Undertheseassumptionsthereducedforcebalance, PDE (22)is
2(η +2λ )R (cid:20) 2λ ξR (cid:18)φ∗ φ∗(cid:19)(cid:21)
f(cid:48)(cid:48) + e e f(cid:48) + e + s + p f = −θ∗RΣθ (θ∗)f(cid:48), (24)
1 (η +λ )y 1 (η +λ )y2 η +λ ν ν 1 s net p 2
e e e e e e p s
andtheinterfacecondition,Eqn.(23),is
2λ
(η +λ )f(cid:48) + ef = θ∗Σθ (θ∗)f at y = 1. (25)
e e 1 y 1 s net p 2
7
Thesupercripts((cid:48))and(θ)denotethederivativeofthefunctionswithrespecttothevariablesyandθ,
respectively.Thenetshearandbulkviscositiesaregivenbyη = θ∗η +θ∗η andλ = θ∗λ +θ∗λ .
e s p p s e s p p s
φ∗,φ∗ are the equilibrium particle fractions for the polymer and solvent, respectively. For a finite
p s
solutionofEqn.(24),itisassumedthatf (0) = 0.Further,weassumethatf(cid:48) = f .Theseassump-
1 2 1
tionsreduceEqn.(24)intothehomogeneous,sphericalBesseldifferentialequationwhosesolutions
(cid:113)
have the form f = yγJ (βy), where γ = 1 − (ηe+2λe)R, β = ξR (φ∗s + φ∗p)+θ∗RΣθ (θ∗)
1 n 2 ηe+λe ηe+λe νp νs s net p
(cid:113)
and n = |γ2 − 2λe |. The Bessel functions, J , are of the first kind of order n. Since there are
ηe+λe n
several solutions satisfying the condition f (0) = 0, we choose the solution with the lowest order
1
n = n ,whichdeterminestheequilibriumradiusoftheswellinggel,R = R(τ → ∞),
min f
(cid:32) (cid:115) (cid:33)
(cid:18) (cid:19)
1 2λ η +λ
R = − n2 + e e e . (26)
f 2 min η +λ η +2λ
e e e e
Thevariable-separable,linearizedformofsolution(linearizednearequilibrium)formassconserva-
tion,Eqn.(2),gives
1∂a 1 θ∗ 1
= − p (y2f )(cid:48) = − , (27)
a∂τ f R y2 1 τ
2 f ch
whereτ isaconstant(tobedeterminedlater).ThefirstandthelastpartofEqn.(27)impliesthat
ch
a(τ) = e−τ/τch. Because the polymer is conserved (neither created nor destroyed) the velocity of
themovingboundarymustbethesameasthegelvelocityattheboundary,v (y = 1),
p
∂R
= v (y = 1) = e−τ/τchf (1). (28)
p 1
∂τ
The solution to the above equation is numerically computed via Matlab ODE solver ode15, with
trivial initial conditions. In particular, we note that if R ≈ R , then f (1) ≈ constant and in this
f 1
caseEqn.(28)canbesolvedexactly,i.e.,R(τ) = R (1−e−τ/τch).Thisexpressionisidenticalto
f
the radial expansion of the linearized form given by Tanaka et. al [25] in their kinetic theory of
swelling hydrogels. Finally, the time constant of gel-swelling towards the equilibrium size (Eqn.
(27)),is
f R y2 (cid:12)
τ = 2 f (cid:12) , (29)
ch θ∗(y2f )(cid:48)(cid:12)
p 1 y=1
whichdefinesthediffusivityforsphericalgels,D = Rf2/τch.
2.4 Parameter Estimation
Twosetsofdataareusedtocalibratethemodelforthekineticsofsphericallysymmetricswelling
mucus gels. The first experiment, conducted by Verdugo et al., measures the diffusivity data (R2
f
versus τ values) of the swelling kinetics of exocytotic mucin granules from four (human) CF-
ch
patients and three healthy individuals at pH = 7.2, T = 37◦C and [Ca2+] = 1mM as well as
b
[Ca2+] = 2.5mM [8]. In the second experiment, performed by Kuver et al., kinetic data was
b
collected from culturedgallbladder epithelial cells from wild-type andCF-infected mice at pH =
7.0, T = 37◦C, [Ca2+] = 4mM and [Na+] = 140mM [36]. The microscopic composition of the
b b
8
mucusinbothoftheseexperimentswerereportedidentical.Fig.2aandFig.2bpresentstheresults
fromthesetwoexperiments,respectively.
Table 1 lists the values of the parameters used in our numerical calculations. The constants in
the model are the monomer particle volume, ν , the solvent particle volume, ν , the coordination
p s
numberofthepolymerlattice,z,andthenearestneighborinteractionenergies,(cid:15) (Eqn.(12)),shear
i
andbulk viscositycoefficientsof solvent(i.e.,water),η ,λ ,respectively,at referencetemperature.
s s
Theundeterminedconstantsarethebindingaffinitiesofthevariouscationswiththegel,K ,K ,K
h c n
(introducedinEqn.(17))andthepolymerviscositycoefficients,µ ,λ ,andthedragcoefficient,ξ
p p
(Eqn.(4)).
Table1:Constantparameterscommontoallthenumericalsimulations.Thereferencetemperature
fortheviscositycoefficientsandsolubilityparametersisfixedatT = 25◦C.
0
Constants Value Units Source
Repeatunitperchain(N) 266 – [7]
Molecularvolumeofmucus(ν ) 5×10−20 m3 [7]
p
Molecularvolumeofwater(ν ) 2×10−23 m3 [7]
s
Shearviscosityofwater(η ) 8.88×10−4 Pas [37]
s
Bulkviscosityofwater(λ ) 2.47×10−3 Pas [37]
s
Hildebrandsolubility(δ ) 1.0928(α = 0),1.3258(α = 1) MPa1/2 [38]
p
Hildebrandsolubilityforwater(δ ) 48.07 MPa1/2 [39]
s
Theradiallysymmetricallyswellingmucusgelhasa3-Dconfiguration,whichsuggeststhatwe
choose the coordination number, z = 6, mimicking the 3-D structure of the polymer lattice [19].
Thestandardfreeenergies,k T µ0andk T µ0andtheenergyinteractionparameters,(cid:15) (Eqn.(12))
B 0 p B 0 s i
arefoundfromtheHildebrandsolubilitydata,δ [39].Thevaluesforthesolubilitydataformaterials
i
mimicking mucus glycoproteins are given in a study by Mimura [38]. The fully un-crosslinked
(no Ca2+ binding) and fully crosslinked (Ca2+ bound) states are denoted by α = 0 and α = 1,
respectively. The standard free energy is the energy of all the interactions between the molecule
and its neighbors in a pure state that have to be disrupted to remove the molecule from the pure
state.Therelationbetweenthestandardfreeenergiesandthesolubilityparameters(Table1)are
−k T µ0 = ν δ2
B 0 s s s
−k T µ0 = ν δ2, (30)
B 0 p m p
where ν = 2 × 10−23 cm3 is the volume of one molecule of water at reference temperature,
s
T = 298K. The negative sign in Eqn. (30) indicates that k T µ0,k T µ0 < 0, since they are the
0 B 0 p B 0 s
interaction energies. Using the relations in Eqn. (30), these values are fixed at (cid:15) = 4.84, (cid:15) =
1 2
3.74, (cid:15) = −13.70, (cid:15) = 0.ThereferencetemperatureoftheexperimentswasfixedatT = 25◦C.
3 4 0
The volume of a mucin oligomer/multimer chain is calculated by mathematically modeling a
chaincomprisingN freelyjointedcylindricalsegmentsoflengthl andwidthd,whereN = L/l is
thenumberofKuhnsegments(oreffectiverigidcylindricalsegmentswhichdeterminesthedifferent
conformations a chain can have), and L is the end-to-end length of the chain. For a gel forming
mucin(e.g.,human MUC5AC,usedintheexperimentaldatatocalibrateourmodel),l ≈ 0.03µm,
L ≈ 8µm and N = 8/0.03 ≈ 266 [6, 40]. The radius of gyration, R , (defined as the average
g
9
distance from all Kuhn segments to the center of mass of the chain) and the pervade volume, V,
(i.e.,theapproximatesphericalvolumeofaspherewithradiusR )is[23]
g
l0.8 ×d0.2 ×N0.588 π
R ≈ , V ≈ 4× ×R3. (31)
g 2.45 3 g
Substitutingvalues,thevolumeV(= ν ) ≈ 0.05µm3.
p
The undetermined parameters, namely the binding affinities K ,K ,K and the rheological
h n c
coefficients η ,λ ,ξ are computed by minimizing a nonlinear least-square difference function be-
p p
tween the experimental values of diffusivity data (R2 versus τ ) and the corresponding model
f ch
output(Eqns. (26, 29)),implementedvia the MATLAB non-linearleast-square minimizationfunc-
tion lsqnonlin. These values are found as log (K ) = −2.27,log (K ) = −3.65,log (K ) =
10 n 10 h 10 c
−3.12,η = 0.11,λ = 0.31,ξ = 1.97(wild-type mucus) and log (K ) = −2.55,log (K ) =
p p 10 n 10 h
−3.98,log (K ) = −7.12,η = 1.02,λ = 2.87,ξ = 0.21(CF-infectedmucus).Theclosenessof
10 c p p
fit between the model (highlighted by the solid lines) and the experimental data points is shown
in Fig 2. The error bars represent the maximum and minimum deviation from the sample points
andsetat 5%marginoferror.Noticethelinear relationship betweenR2 andτ ,predictedbythe
f ch
experimentsandaccuratelycapturedbythemodel.
3 Results and discussion
Themainideabehindthisworkistoprovideanobjectivecomparisonoftheswellingpropertiesof
normalversusunhealthymucus,immersedinanextracellularmediumwithchemicallycontrolled
composition. Well established results exist in literature which detail the relationship between the
swelling rate, final size of the gel and the swelling time [25, 26]. However, these results do not
explicitlyoutlinehowtheseandotherphysiochemicalelementsinfluencingthemucushydrationand
rheology, depend onthe nature of the swellingmedia. Hence, in§3.1,we explore the relationship
betweentheradialsizeoftheswellingmucusgelandtherheologyofthemucuspolymer,namethe
dragandtheviscositycoefficients.Theeffectofthecalciumandthesodiumionsinthesolventon
theequilibriumsizeofthemucusblobaredetailedin§3.2and§3.3,respectively.
3.1 Swelling kinetics
Numericalsimulationswereperformedtooutlinethedifferencesbetweentheswellingkineticsfor
WT and CF-infected mucus, by altering the electro-chemical composition of the swelling media.
Fig. 3 presents the radius of the swelling gel, R(τ), versus time, τ, for WT mucus with calcium
bathconcentrations, C = 1mM andC = 2.5mM (the‘dash-dot’ and‘solid’ curve,respectively),
b b
as well as for CF-infected mucus (highlighted by the ‘dotted’ and ‘dashed’ curve, respectively).
The sodium ion concentration and the pH in the bath are fixed at N = 0 and pH = 7.2. These
b
concentrationscorrespondtotheinvitroconditionsofVerdugo’sexperiments[8].
In particular, note that the expansion of the CF-infected mucus (‘dashed’ and ‘dotted’ curves,
Fig.3)isappreciablyslowerthanWTmucus(‘dash-dot’and‘solid’curves,Fig.3).Thisdifference
inswellingprofilesisaconsequenceofsmalldragcoefficientoftheCF-infectedmucusgels,relative
totheviscositycoefficients(i.e.,theratioδ = ξ issmallforCF-infectedmucus).
θs∗(ηp+λp)+θp∗(ηs+λs)
Usingtherheologicalparameters,estimatedin§2.4(i.e.,η ,λ ,ξ),thesolventparameterslistedin
p p
10
