Mechanisms of self-organization for the collagen fibril lattice in the human cornea

Abstract

The transparency of the human cornea depends on the regular lattice arrangement of the collagen fibrils and on the maintenance of an optimal hydration—the achievement of both depends on the presence of stromal proteoglycans (PGs) and their linear sidechains of negatively charged glycosaminoglycans (GAGs). Although the GAGs produce osmotic pressure by the Donnan effect, the means by which they exert positional control of the lattice is less clear. In this study, a theoretical model based on equilibrium thermodynamics is used to describe restoring force mechanisms that may control and maintain the fibril lattice and underlie corneal transparency. Electrostatic-based restoring forces that result from local charge density changes induced by fibril motion, and entropic elastic restoring forces that arise from duplexed GAG structures that bridge neighbouring fibrils, are described. The model allows for the possibility that fibrils have a GAG-dense coating that adds an additional fibril force mechanism preventing fibril aggregation. Swelling pressure predictions are used to validate the model with results showing excellent agreement with experimental data over a range of hydration from 30 to 200% of normal. The model suggests that the electrostatic restoring force is dominant, with the entropic forces from GAG duplexes being an order or more smaller. The effect of a random GAG organization, as observed in recent imaging, is considered in a dynamic model of the lattice that incorporates randomness in both the spatial distribution of GAG charge and the topology of the GAG duplexes. A striking result is that the electrostatic restoring forces alone are able to reproduce the image-based lattice distribution function for the human cornea, and thus dynamically maintain the short-range order of the lattice.

1. Introduction

The almost perfect transparency of the human cornea depends on the regular lattice arrangement of the collagen fibrils within the stromal lamellae and on the precise maintenance of an optimal hydration [1]. The achievement of both conditions depends on the presence of stromal proteoglycans (PGs) that associate with the collagen fibrils. The stroma, which comprises about 90% of the corneal thickness, is organized into lamellae (fibres organized in flat sheets) that contain collagen types I and V in the form of 25–30 nm diameter fibrils and small leucine-rich repeat PGs. The collagen fibrils within a lamella appear in parallel arrays following the direction of the lamella. In a lamella cross-section, the fibrils assume a quasi-regular packing arrangement with centre-to-centre interfibrillar distance of 40–60 nm; this arrangement is referred to as the fibril lattice. The short-range order of this lattice produces optical interference effects that render the tissue transparent [1–4].

The stromal PGs are sulfated and have linear sidechains of negatively charged disaccharide units called glycosaminoglycans (GAGs) that are bound covalently at one end to a PG core protein, which in turn is attached to a collagen fibril [5]. The charged GAG chains interact electrostatically with ions in the interfibrillar fluid to produce osmotic pressure. The resulting propensity of the stroma to swell by absorbing water is counteracted in the in vivo cornea by an active ion transport mechanism located in the endothelial cellular layer [6]. It is through the interplay of the GAG-based osmotic pressure and the active ion pumping mechanisms that hydration in the in vivo cornea is controlled. A measure of the overall swelling tendency of stroma in vitro, which includes osmotic and other effects, is provided by the tissue swelling pressure which has been measured by numerous investigators [7,8].

The GAGs also have an important but less-understood role in the maintenance of the fibril lattice. Keratan sulfate (KS), the predominant stromal GAG component, has been shown to be involved in modulating the fibril lattice by the knockout of Chst5¹ in the mouse [9]. Scott [10] proposed that two or more GAG chains, originating at different core proteins on neighbouring fibrils, may form an antiparallel duplexed association that appears as a bridge-like structure spanning the interfibrillar distance in electron microscopy after staining, and that such structures may be capable of controlling the interfibrillar spacing. The open question of how negatively charged (and mutually repulsive) GAG chains might form durable antiparallel associations is discussed by Knupp et al. [11]. The idea of an entropic elastic interconnected fibril–GAG network was employed in the theoretical study on lattice forces and spatial order by Farrell & Hart [12] who proposed a model based on a sixfold symmetric elastic network. A sixfold symmetric structural model was also proposed by Müller et al. [13] in which GAG bridges (presumably formed by duplexing) were proposed to link next-nearest-neighbouring fibrils. However, recent three-dimensional electron tomography imaging by Lewis et al. [5] and Knupp et al. [11] shows that while GAG chains do appear to form bridging structures between two or more fibrils, these structures do not exhibit any approximation to a regular sixfold symmetric network or indeed any kind of regularity in spatial organization. Further, many GAG chains are not acting as bridges and these linear structures reach into the interfibrillar fluid at various orientations. All of this suggests that fibrillar bridges alone cannot mediate the lattice [5].

Maurice [2,14] speculated that repulsive forces must exist between adjacent fibrils to keep them separated and that the origin of these forces was related to the swelling force within the extracellular matrix. A conceptual model for the lattice forces has recently been described by Lewis et al. [5] in which an attractive force system between a fibril pair resulting from thermal fluctuations of antiparallel GAG duplexes is balanced by a repulsive force system which has an origin in osmotic pressure. In this model, the lattice dimensions will adjust to reflect a state of equilibration between the two opposing force systems, which depends on factors such as the number and type of GAG duplexes. The contractive entropic force from GAG duplexes remains an essential element of this model.

In the present work, a theoretical model for the forces that allow stable self-organization of the lattice is presented. An electrostatic-based fibril restoring force is introduced which results from a hypothesized local change in charge density induced by motion of a fibril relative to its neighbours in the lattice. It is also assumed that some GAG chains form antiparallel duplexes bridging between fibrils and these have been modelled by assuming that they possess the entropic elasticity of a worm-like chain. In developing the mathematical description, a conventional equilibrium thermodynamics approach [15] is adopted and Poisson–Boltzmann (PB) mean-field theory is used to describe interactions between GAG fixed charges and mobile ions. The tissue swelling pressure and lattice restoring forces are found from suitable derivatives of the free energy. An approximation for the electrostatic forces is also derived using Donnan theory and used in a coarse-grained dynamic model to describe the self-organization of the fibril lattice. The dynamic system, solved using a molecular dynamics technique, is used to explore the effects of randomness in the GAG chain organization by comparing the computed lattice radial distribution functions with an image-based estimate [16].

In a theoretical study on corneal transparency, Twersky [17] conjectured that collagen fibrils must be centred in a ‘tough adherent transparent coating’ that prevents fibril aggregation, an idea that has experimental support in the work of Fratzl & Daxer [18] whose measurements suggest that collagen fibrils are surrounded by a cylindrical coating consisting of PGs. The current model is used to explore the possibility that the collagen fibrils have a PG-dense coating. It is shown that such a coating will relatively have small influence on the tissue osmotic pressure and fibrillar restoring forces, except when the coatings of adjacent fibrils become close, at which point an additional and rapidly increasing electrostatic repulsive force is generated.

2. Ionization and charge properties of the glycosaminoglycans in corneal stroma

2.1. Predominant glycosaminoglycans

The predominant corneal GAGs are KS, dermatan sulfate (DS) and chondroitin sulfate (CS). These are linear unbranched polymers that are composed by repeating disaccharide units. In order to characterize the fixed charge carried by the corneal GAGs, we must estimate the average charge per disaccharide unit. The molecular structure of each GAG chain is shown in figure 1. The KS disaccharides are sulfated at both Gal and GlcNAc at the 6-carbon position. The CS disaccharides are sulfated at either the 4- or 6-carbon of the GalNAc residue, and are designated CS4 or CS6, respectively (figure 1). A detailed FACE² [19] analysis discloses that only a portion of the available GAG monosaccharides is sulfated in the human cornea. In KS, the fraction of sulfated Gal and GlcNAc is 54% and 95%, respectively. For CS, the sulfation fraction is significantly lower. For CS4, the fraction of sulfated GalNAc is 28%, whereas for CS6 the fraction is only 8% (table 1).

Figure 1.

Disaccharide repeat units of (a) KS, (b) DS, (c) CS4 and (d) CS6.

Table 1.

Sulfation fraction (%) of monosaccharides in KS and CS/DS for normal corneas, data taken from [19].

KS		CS/DS
GlcNAc sulfation	Gal sulfation	GalNAc4-sulfation	GalNAc6-sulfation
95 ± 5	54 ± 2	28 ± 2.5	8 ± 0.7

2.2. Estimation of glycosaminoglycan ionization fractions

As sulfate groups carry one negative charge per group, the average charge per disaccharide unit for KS can be found from table 1 as z_KS = 0.95 + 0.54 = 1.49. Similarly, for CS/DS³ the value is z_CS/DS = 1.0 + 0.28 + 0.08 = 1.36. Introducing the GAG ionization fraction f, defined as the ratio of the average charge per disaccharide unit to the number of groups that are ionizable, it is found that f_KS = z_KS/2 = 74.5% for KS, and f_CS/DS = z_CS/DS/2 = 68.0% for CS/DS. The average ionization fraction of all corneal GAGs may then be estimated as

2.1

where x_KS and x_CS/DS are the molar fractions of KS and CS/DS, respectively. Given the percent weight ratio between the two GAG species, w_KS = 65% and w_CS/DS = 30% [19,20] and the molecular weight (g/mol per disaccharide unit), M_KS = 464 g mol⁻¹ and M_CS = 513 g mol⁻¹ [21], the molar ratio of KS : CS/DS is estimated as x_KS : x_CS/DS = w_KS/M_KS : w_CS/M_CS = 2.40, which leads to x_KS = 70.5% and x_CS/DS = 29.5%. Substitution of these values in equation (2.1) produces f_avg = 72.6%.

2.3. Stromal glycosaminoglycan fixed charge

If a representative volume within the normally hydrated stroma contains N_g GAG chains with average contour length L_c and if b is the length of a monosaccharide unit, the total GAG fixed charge in the volume is

2.2

where e is the unit charge. The average stromal fixed charge density is then

2.3

where is a measure of the GAG number density. However, in the current study, the charge density distribution must be characterized at the nanometre scale where it will be spatially non-uniform.

X-ray scattering studies have been used to reveal structural transformation of collagen fibrils under varying tissue hydration [18,22]. Fratzl & Daxer [18] performed x-ray scattering studies of collagen fibrils under varying hydration produced by drying the tissue. Their results suggest that the stromal PGs appear in relatively high density in the vicinity of the fibrils where they may form a charge-rich and water-binding PG-coating surrounding each fibril. The radius of the coating was measured to be r_c = 18.25 nm. They showed that the coating radius is insensitive to hydration over a wide range, suggesting that the water bound by the coating is released only at very low levels of tissue hydration. The existence of such a surface ultrastructure on the collagen fibrils is corroborated by image studies from Miyagawa et al. [23] and, to some extent, by Müller et al. [13]. In particular, Miyagawa et al. [23] had suggested that the PG-coating is primarily composed of KS. A theoretical study by Twersky [17] proposed that collagen fibrils must be centred in a transparent coating and the coated fibrils occupy approximately 60% of the matrix volume, giving r_c = 21.56 nm.

To accommodate the possibility that the collagen fibrils have coatings consisting of PGs, two classes of GAGs are introduced for modelling purposes (figure 2c): (i) an interstitial class, which spans the interfibrillar electrolyte volume and (ii) a fibril coating class, which spans a fibril coating volume. The stromal fixed charge Q is partitioned with a parameter such that the fixed charge Q_inter = λQ is associated with the interstitial GAGs, and the charge Q_coat = (1 − λ)Q is associated with the coating GAGs. The parameter λ facilitates modelling of the case with no coating GAGs, λ = 1, coating and interstitial GAGs, 0 < λ < 1, and the extreme (non-physiological) case of coating GAGs only, λ = 0.

Figure 2.

(a) The corneal stroma tends to swell by imbibing water, which is described by the swelling pressure P_s. In healthy cornea, the swelling is countered by the continuous removal of bicarbonate ions from the stroma by pumps located in the endothelium and which results in a stable stromal hydration. (b) An illustration of the organization of the corneal stromal showing several lamellae and a keratocyte cell. Collagen fibrils within lamellae are found in parallel arrays following the fibril direction; each lamella makes a large angle with its immediate neighbours and keratocyte cells are found interspersed between adjacent lamellae. (c) An idealized cross section through a lamella showing a hexagonal collagen fibril lattice; the irregular lines between the fibrils are illustrative of the GAG chains; a typical unit cell employed for analysis is indicated by the inset triangle. (d) The unit cell domain is partitioned into fibril, coating and interstitial regions, denoted Ω_fibril, Ω_coat and Ω_interstitial, respectively. (Online version in colour.)

The coating GAGs are not explicitly identified with KS or CS/DS. However, such identification can provide a lower bound on the physiological value of λ. The charge fraction contributed by KS and CS/DS is given as Q_KS/Q_CS/DS = z_KSx_KS/z_CS/DSx_CS/DS = 2.62. This implies that Q_KS/Q = 72.3% and Q_CS/DS/Q = 27.7%, which provides an estimation for the lower bound of λ_l = 0.28 based on the assumption that all the KS is contained in the PG coating [23].

3. Unit cell model

3.1. Thermodynamic equilibrium

The simplest possible geometric representation of a unit cell within a stromal lamella is that of an equilateral triangular prism with vertices at the centre of three fibrils (figure 2c,d). The cell contains a parallel array of collagen fibrils aligned with the lamella direction and arranged with hexagonal packing. The centre-to-centre interfibrillar spacing at normal hydration is denoted by , the fibril radius by r_f, and the PG-coating radius by r_c (figure 2d). The antiparallel GAG duplexes, which are formed by two or more GAG chains, bridge the neighbouring fibrils and are assumed to repeat continuously in the fibril axial direction with an average periodicity h (figure 3). The unit cell length is taken to coincide with h. Measurements of for the human cornea are summarized in table 2 and values for other unit cell parameters are summarized in table 3, which indicates that we have assumed and h = 25 nm as representative values from measurements.

Figure 3.

(a) Transverse view of a fibril pair bridged by an antiparallel GAG duplex. Each GAG chain is attached at one end to a collagen fibril via its PG core protein. The end-to-end distance L_i of the antiparallel GAG duplex depends on the centre-to-centre distance of the bridged fibril pair d_i, fibril radius r_f, and the protein binding sites on the fibril surfaces described by θ₁ and θ₂. (b) Axial view of the fibril pair. The antiparallel GAG duplex structures are assumed to lie in a plane perpendicular to the fibril axis and occur with an average axial periodicity h.

Table 2.

Summary of interfibrillar spacing measurements for human cornea at physiological conditions.

measured (nm)	method	references
43.0 ± 2.5	electron microscope	[13]
43.4 ± 3.4	transmission electron microscope	[16]
61.9 ± 4.5	synchrotron x-ray diffraction	[24,25]
60.8 ± 2.0	synchrotron x-ray diffraction	[26]

Table 3.

Parameter values employed for the unit cell.

parameter	value
ϕcol, volume fraction of the collagen fibrils (%)	24.9a
ϕker, volume fraction of the keratocytes (%)	11.2a
ϕelectrolyte, volume fraction of the electrolyte fluid (%)	63.9a
, average charge density of the corneal stroma (mEq l–1)	38.6
, interfibrillar distance at normal hydration (nm)	53.0b
rf, radius of the collagen fibril (nm)	14.73c
rc, radius of the PG coating (nm)	18.25, 21.56d
λ, the charge partition parameter	0.3–1
Lp, persistent length of GAG duplex (nm)	8.72e
Lc, contour length of GAG duplex (nm)	90.88, 329.09f
Nb, number of bridging GAG duplexes in the unit cell	3/2
h, average axial periodicity of the antiparallel GAG duplexes (nm)	25g
t0, corneal thickness at normal hydration (mm)	0.5h
C0, physiological bath concentration (M)	0.15
T, temperature (K)	298

^aValues derived from [24,27].

^bRepresentative of the mean value from the measurements shown in table 2.

^cValue derived from the data of [24].

^dValues taken from [18] and derived from [17].

^eValue taken from [28].

^fCalculated from equation (3.8) for the nearest-neighbour and next-nearest-neighbour bridging connections.

^gRepresentative of the mean value from measurements in [13,29].

^hValue taken from [8].

For isolated corneal stroma immersed in an ionic bath, the total Gibbs free energy of the unit cell can be defined as

3.1

where is the Helmholtz free energy, P and V are atmospheric pressure and total system volume (including the bath solution), respectively, and is the external work applied to the unit cell. At equilibrium, stationarity of the Gibbs energy implies:

3.2

The PV term is essentially constant [30] and equation (3.2) reduces to . From the standard theory for electrolyte gels [15], the Helmholtz free energy has electrostatic, entropic elastic and mixing components, i.e.

3.3

For the cornea, it has been demonstrated that the free energy of mixing is very small [21,28,31] and this component is therefore neglected in the present study.

3.2. Electrostatic free energy

The unit cell domain is denoted and contains fibril and electrolyte regions The electrolyte domain is further partitioned into coating and interstitial regions (figure 2d). For a binary electrolyte (e.g. NaCl), the electrostatic potential φ over Ω_electrolyte is governed by the PB equation [32]

3.4

where ρ_GAG is the non-uniform GAG fixed charge density, ε_w is the dielectric permittivity of the electrolyte solvent, F is the Faraday constant, C₀ is the bath concentration at which the electrical potential is zero, R and T are the ideal gas constant and temperature, respectively. When λ ≠ 1 the charge density ρ_GAG(x) will be discontinuous at the Ω_interstitial − Ω_coating interface, whereas the potential φ and electric field E = −∇φ are required to be continuous across the interface.⁴

The mean-field approximation of the free energy for a binary electrolyte can be expressed as a functional of φ, which is stationary at the solution to the PB equation [33] in the form:

3.5

The first and second terms in equation (3.5) measure the electrostatic free energy of the GAG fixed charges and ionic charges, respectively, and the last term measures the contribution from the electric field.

3.3. Entropic elastic free energy

As noted in §1, Scott [10] proposed that two or more GAG chains, each attached to a fibril through its PG core protein, may form a duplexed association in which the linear chains are 180° apart, creating a bridge-like structure spanning the two fibrils (figure 3). Other imaging studies have supported this proposition, e.g. [5,13,34]. It is assumed that the antiparallel, duplexed GAG structures possess the free energy of a worm-like chain (WLC) [35], providing an entropic elastic behaviour arising from thermal fluctuations that will induce a contractive force between the bridged collagen fibrils. In this case, the free energy takes the form:

3.6

where N_b is the number of GAG duplexes in Ω_cell, k_b is the Boltzmann constant, L_p is the persistence length of the GAG duplex, L_c is the contour length representing the length of the fully stretched duplexed chains, and L_i is the end-to-end distance of the ith chain.

In order to characterize L_i, it is necessary to specify the locations of the PG binding sites where each protein core attaches to each fibril. Although it is known that the binding sites in the fibril axial direction occur at the a, c and d/e bands within the fibril D-period [36], the current model assumes that the antiparallel GAG duplexes are perpendicular to the collagen fibrils [36] with an axial periodicity h (figure 3b). However, the circumferential location of each of the PG binding sites on the two bridged fibrils can produce significantly different end-to-end bridging distances (figure 3a). By assuming that each circumferential PG binding site is random for each fibril, the end-to-end distance L_i may be replaced by the root mean square (r.m.s.) distance . For the nearest-neighbour fibril bridging

3.7

where d_i is the centre-to-centre distance of the bridged fibril pair. The distance function L_i(θ₁, θ₂) is illustrated in figure 3a.

The r.m.s. end-to-end distance of a free WLC at normal hydration is denoted by ℓ⁰ and is assumed to be given by equation (3.7) with d_i replaced by the normal hydration distance Using the relationship between ℓ⁰ and L_p and L_c [37], it follows that

3.8

This expression allows the contour length L_c to be determined in terms of L_p, r_f, and . Table 3 gives the values used in this study. For the nearest-neighbour bridging, which leads to ℓ⁰ = 37.83 nm and L_c = 90.88 nm. This implies that duplexed GAG chains are approximately 42% stretched at normal hydration.

4. Stromal swelling pressure

4.1. Thermodynamic definition

Stromal swelling pressure P_s is defined as the pressure applied on a sample of the tissue in an ionic bath that is required to prevent it from swelling. The dependence of P_s on hydration, bath ionic strength, temperature and pH have been measured in numerous species including human, and by various means. The application of the current model to the corneal swelling pressure provides an important validation case. The results for P_s are also used directly in the lattice force model presented in §§5 and 6.

Consider a representative volume of stromal tissue at normal hydration. This volume will contain keratocyte cells and is distinct from the unit cell Ω_cell previously introduced. Let the tissue volume be subject to a dilation where V_stroma will be referred to as the current volume. If it is assumed that the total collagen, keratocyte and PG content are preserved under the dilation while the incompressible electrolyte fluid is free to enter or leave the volume during dilation,⁵ then at fixed temperature and bath ionic concentration, the external work needed to achieve the volume V_stroma is

4.1

At equilibrium, it follows from equation (3.2) that

4.2

where and are the electrostatic and entropic elastic swelling pressure components, respectively. The swelling of the corneal stroma has been observed to be predominantly unidirectional in the thickness direction [31,38] and experimental measurements of the interfibrillar spacing of the swollen cornea suggest that the fibril lattice repacks itself bidirectionally [38]. In the swollen stroma it is therefore reasonable to assume that the fibrils will assume a lattice with equal interfibrillar spacing. Then the interfibrillar centre-to-centre distance d_c under volume dilation J can be determined as

4.3

4.2. Stromal charge density

In the stroma, collagen fibrils and keratocyte cells occupy volume that is excluded from the electrolyte fluid. Table 4 lists the approximate weight fraction of each constituent of the stroma [27]. Denoting the collagen and keratocyte volume fractions as ϕ_col and ϕ_ker, respectively, and denoting the volume fraction of the remaining constituents as ϕ_electrolyte = 1 − ϕ_col − ϕ_ker (primarily water), and using the data of Meek & Leonard [24], we have estimated ϕ_col = 24.9%, ϕ_ker = 11.2% and ϕ_electrolyte = 63.9%.

Table 4.

Composition of the corneal stroma [27].

constituent	wet weight (%)
collagen	14.6
cellular water	11.4
proteoglycans	1.0
matrix water	64.8
other proteins including cellular components	8.2

Assuming that keratocyte cells dilate with the tissue dilation [39] and assuming collagen fibrils are non-swelling [22,31,38], we find two charge densities associated with Q_inter and Q_coat under macroscopic dilation J to be as follows:

4.4

and

4.5

where is given by equation (2.3). The above expression has used the coating volume fraction ϕ_coat defined by For consistency with the findings of Fratzl & Daxer [18], the coating volume does not change with dilation J such that

The non-uniform (piecewise constant) GAG fixed charge density ρ_GAG appearing in the PB equation (3.4) can now be defined as

4.6

where the integer-valued n(x) is given by

4.7

4.3. Poisson–Boltzmann-based numerical solution for P_el

A sequence of dilated unit cells, parametrized by the dilation J, is constructed with interfibrillar centre-to-centre distance given by equation (4.3). The numerical method proceeds by solving the nonlinear PB equation (3.4) with equation (4.6) for the electrostatic potential φ(J) by the finite-element method. The electrostatic free energy is then evaluated using equation (4.2) for each member in the unit cell sequence. The electrostatic contribution to the swelling pressure is determined for any dilation J from equation (4.2) and where is approximated numerically by central difference. As J is reduced, the coating domains will overlap and the charge density in the overlap region is doubled as indicated by n(x) above. Electrostatic potential contours for J ∈ [0.42,1.0] are illustrated in figure 9a–d for typical stromal parameters, including the case of coating overlap.

Figure 9.

(a–d) The contour plots of the PB-based electrostatic potential (φ) within the unit cell with volume dilations J = 1.0, 0.6, 0.5 and 0.42; (e–h) the comparison between the φ and Donnan-based approximation along the fibril centre-to-centre line. In these calculations, the PG-coating radius r_c = 18.25 nm and charge partition λ = 0.65.

4.4. Donnan-based approximation for P_el

The Debye length is the characteristic distance in an electrolyte over which the potential of a charged surface decays by 1/e. For the human stroma, this is estimated to be

4.8

Because this value is small compared to the unit cell dimensions, it is reasonable to consider an approximation based on Donnan potentials associated with the ρ_inter and ρ_coat charge densities. The piecewise constant Donnan potential is given by [32]

4.9

where ρ_GAG is given by equation (4.6). Although satisfies the PB equation (3.4), it will not satisfy the necessary continuity conditions at the Ω_coat − Ω_interstitial interface. Introducing equation (4.9) into (3.5) and using equation (4.2), the Donnan-based electrostatic contribution to the swelling pressure is given as (appendix A)

4.10

The second term on the right-hand side of equation (4.10) provides the correction for the effect of the coating. If the coating charge is set to zero (λ = 1) and the keratocyte volume is neglected (ϕ_ker = 0), the above swelling pressure reduces to the classical Donnan form with the collagen volume exclusion effect included [40]. On the other hand, if all the GAG fixed charge is in the coating (λ = 0), then ρ_inter = 0 and is zero for all dilation J, consistent with the assumption that the coating volume does not change with tissue dilation. The accuracy of equations (4.9) and (4.10) is examined in §7.

4.5. Entropic elastic contribution

The GAG duplex r.m.s. end-to-end distance ℓ_i, given by equation (3.7), may be expressed as a function of J by substituting equation (4.3) in equation (3.7) with d_i = d_c resulting in

4.11

Use of the chain rule allows P_str(J) to be determined from equations (4.2), (3.6) and (4.11) as

4.12

where is a measure of duplexed GAG bridging density in the corneal stroma. For the parameter values given in table 3, the entropic elastic swelling pressure component P_str remains negative (contractive) for all J > 0.146.

5. Interfibrillar lattice restoring forces in an ideal hexagonal cell

5.1. Rationale and thermodynamic definition

Consider an individual fibril displaced to a new position in the lattice while all other fibrils maintain their positions. Linear GAG disaccharide chains are associated with each fibril and attach to that fibril at one end only via the PG core protein. It is hypothesized that the attached GAG chains will displace with their supporting fibril. Because the longer chains reach into the interfibrillar fluid, a local change in GAG fixed charge density is created with an increase in advance of the displacement and a reduction behind (figure 4). In our model, the interstitial GAG class introduced in §2 are assumed to be involved in this process, whereas the shorter coating class GAGs will not, until the coatings become close. Additionally, GAG chains that have formed interfibrillar antiparallel, duplexed structures will experience changed entropic forces resulting from changes in end-to-end distances. For the purpose of fully characterizing the two restoring forces, the analysis in this section is based on an ideal hexagonal unit cell with perfect duplexed GAG bridging based on nearest-neighbour connectivity. In §6, the important effects of random GAG chain topology is considered.

Figure 4.

The electrostatic restoring force mechanism which acts on fibrils to maintain the lattice organization is illustrated for the case where all fibrils in the lattice are held fixed in their lattice positions and a single fibril is displaced relative to its lattice position. (a) Shows a set of fibrils in a regular lattice arrangement with their associated GAG chains and a distribution of mobile ions. In this situation, which corresponds to normal physiological conditions, the osmotic pressure is essentially uniform. (b) Depicts the osmotic pressure in each subcell around the undisturbed fibril. The osmotic pressure in each subcell exerts a force on the fibril (see equation (5.5)) but because the osmotic pressure is uniform, all six forces balance and the net force is zero, as indicated by the red dot. (c) Shows a fibril displaced from its lattice position. The GAG chains attached to the fibril move with the fibril and the GAG fixed charge density and mobile ion concentration increase in advance of the fibril displacement and reduce behind it. (d) Indicates the resulting osmotic pressure in each of the subcells, which is higher where GAG fixed charge density has increased and lower where it has reduced. The forces exerted by each of the subcells now has changed magnitude with the result that a net force acts on the fibril with a direction oriented towards the original lattice position. This restoring force is shown as a red arrow. This mechanism, which exploits the stromal collagen–GAG organization, operates regardless of the direction of the perturbation and regardless of whether GAG chains are part of a duplex or are independent. When any fibril undergoes a dynamic excursion from its lattice position, the electrostatic restoring forces act to restore the fibril.

The interfibrillar lattice restoring force f_l(r) is defined as the force that must be imposed on a fibril to maintain its current position r. The external work required to move the fibril from its lattice position r₀ to displaced position r, while fixing the position of other fibrils in the lattice is given by

5.1

From equations (3.2) and (3.3), f_l is given as

5.2

where and are defined as the electrostatic and entropic elastic components of the restoring force.

5.2. Electrostatic restoring force

The region between the target (central) fibril and its six neighbours in the lattice is divided into triangular prism subcells with initial volume The central fibril is then displaced to position r causing a change ΔV_k in the volume of each subcell (figure 4b). Let measure the kth subcell change in volume due to the fibril displacement to r and set

The GAG fixed charges within subcells are assumed to be conserved during fibril displacement. The fixed charge density ρ_inter given by equation (4.4) is modified to for the kth subcell as

5.3

Since the volume of the hexagonal cell is fixed (J is independent of r), it follows that Σ_kΔJ_K = 0. Figure 5 shows for each subcell when the central fibril is displaced vertically and indicates that charge density increases in advance of the displacement and reduces behind it. Note that ρ_coat and ρ_GAG and are still given by equations (4.5) and (4.6), respectively.

Figure 5.

The charge density ratio (k = 1–6) in each subcell as a function of displacement r when the fibril is vertically perturbed in an ideal hexagonal cell. (Online version in colour.)

The electrostatic free energy of the cell can be evaluated by summing over subcells

5.4

where takes the precise form of equation (3.5), but with the kth subcell volume as the integral domain. An energy sequence i = 1, … ,n is determined numerically using equation (3.5), after solving the PB equation on the subcells. The force f_el(r) is then obtained from equation (5.2) using central difference. This scheme considers all electrostatic effects, including coating overlap.

5.3. Donnan approximation

We make the approximation , and an analytical formulation of f_el is found by

5.5

where is the kth subcell Donnan osmotic pressure component from equation (5.5). This result shows clearly how the osmotic pressure in the subcell serves as the source for the interfibrillar restoring force. The vector ∂J_k/∂r is directed towards the centre of the lattice for all k (see appendix B for details); thus equation (5.5) describes a set of centrally oriented forces with differing magnitudes depending on the subcell osmotic pressure. Clearly, when the fibril is in its lattice position, the osmotic pressure is uniform and the electrostatic force vanishes identically. Away from the lattice position, the electrostatic force is always a restoring force. An interpretation of the electrostatic restoring force as given by equation (5.5) is provided in figure 4. Equation (5.5), which evaluates the electrostatic restoring force without solving the PB equation, is employed in a fibril dynamics model described in the next section.

5.4. Entropic elastic restoring force

Consider the centre of the displaced fibril at location r, and the centre of a neighbouring fibril in the hexagonal cell at location at r_i, with centre-to-centre distance Observing from equation (3.6) that a GAG duplex between two fibrils will have entropic free energy

The relationship between the bridging duplex end-to-end r.m.s. distance ℓ_i and d_i is given by equation (3.7). Then for all six nearest-neighbour bridges, we have, from equation (5.2)

5.6

In the above expression, −∂d_i/∂r_i = v_i, a unit vector pointing from r to r_i, indicating the direction of the contractive force on the displaced fibril.

6. Lattice dynamics model with an imperfect glycosaminoglycan topology

A dynamic model is introduced to describe fibril movements and lattice self-organization. Consider a realistic fibril lattice, with many collagen fibrils, in a transient state; in analogy with molecular dynamics theory, any imbalance in the interfibrillar forces on fibril i will induce a fibril displacement

6.1

where m_i is the fibril mass and a_i is the acceleration. To incorporate the force computations from the previous sections, the lattice is divided into equilateral triangular subcells defined by the nearest-neighbouring fibrils. In order to efficiently compute the electrostatic force, equation (5.5) is used to avoid solving the nonlinear PB equation at each time step. In addition, a soft-sphere potential [41] is introduced to prevent fibril–fibril interpenetration as fibrils have a finite size. The soft-sphere potential can be described by a generalized inverse power law (GIPL)

6.2

where r_ij refers to the distance between fibril pairs and σ_r = 2r_f that corresponds to the fibril diameter. A force field based on a value of n = 32 was used in this study and gives an extremely short-range force field such that the force magnitude drops by more than 95% after two fibrils are separated by 3 nm.

To describe a realistic (i.e. random) GAG chain system [5], two types of imperfection are introduced: (i) the stromal charges are not equally allocated to each subcell and (ii) not all bridging sites are occupied. In this approach, the reference charge densities in subcells have been randomly assigned which obeys the Gaussian distribution , where σ_p is the standard deviation. To describe a defective GAG bridging network, each bridging site has been assigned a probability p_b for the existence of GAG duplex.

The dynamic system has been solved using a typical molecular dynamics routine implemented in MATLAB. As is customary, all the physical parameters have been reduced to dimensionless (table 5). The velocity-Verlet algorithm is used for the time integrator, and the standard cell subdivision and neighbour-list algorithms [42] have been implemented to reduce the interaction computations. The results have been interpreted by computing the radial distribution function (RDF), which is a measure of the average number density of fibril centres at a given radius r from every other fibril normalized by the bulk fibril number density. A structure with a long-range order will have many sharp peaks in its RDF corresponding to the distances between the particles with high probability, whereas the RDF for a purely random structure will be constant with a value of unity. For structures with short-range order, such as the fibril lattice in the corneal stroma, the RDF has multiple peaks with decaying magnitude over a fixed interval and then quickly converges to unity thereafter [16]. The classical approach of evaluating the two-dimensional RDF is given as [42]

6.3

where A is the system area and h_n is the number of fibril pairs (i, j) for which (n − 1)Δr ≤ r_ij ≤ nΔr and r_n = (n − 1/2)Δr. N_m is the total number of the fibrils in the system.

Table 5.

Dimensionless units for the fibril dynamics model in two dimensions.

parameter	expression	value
m0, fibril mass (kg m–1)	a	9.26 × 10−13
L0, length unit (m)		10−9
τ0, time unit (s)		10−9
f0, force unit (N m–1)		9.26 × 10−4
D0, energy unit (J m–1)		9.26 × 10−13

^aThe density of collagen fibril ρ_col = 1.36 × 10³ kg m⁻³ [39].

7. Results

7.1. Swelling pressure

7.1.1. Dilation and hydration

For convenience in interpreting results, stromal hydration H, defined as water weight per unit dry weight, may be approximately⁶ related to the tissue volume dilation J by

7.1

such that normal (physiological) hydration corresponds to J = 1 and H⁰ = 3.2 [4]. The linearity is consistent with measurements [8,43].

7.1.2. Keratocyte and collagen electrolyte volume exclusion effects

Predicted electrostatic swelling pressure component P_el, given by equation (4.10), is plotted against volume dilation J in figure 6a for three cases: no volume exclusion, only collagen volume exclusion and collagen and keratocyte volume exclusion. The volume exclusion effects arising from these two stromal components are important for all J < 1 corresponding to hydration levels lower than normal. For dilations J < 0.5 (hydration less than 33% normal), the volume exclusion effect is significant, increasing P_el by almost an order. Collagen volume exclusion is the most important because the keratocyte volume is smaller and also because it dilates with the tissue.

Figure 6.

(a) The predicted electrostatic swelling pressure component P_el as a function of volume dilation J for three sets of collagen and keratocytes volume fractions (ϕ_col, ϕ_ker). The volume exclusion effects increase P_el significantly when J < 1. (b) Variation of P_el against charge partition λ on three selected values of J. P_el is relatively insensitive to λ when λ > 0.5 and reduces significantly when λ < 0.5. (Online version in colour.)

7.1.3. Collagen proteoglycan-coating effects

The effect of the coating charge partition λ on P_el is illustrated for three values of J in figure 6b. P_el is relatively insensitive to λ for 0.5 < λ ≤ 1, and reduces significantly in the range 0 ≤ λ < 0.5 as progressively more of the total charge is allocated to the coating region and less to the interfibrillar region. In the extreme case where all charge is located in the coating region, λ = 0, the predictions converge to zero swelling pressure, reflecting the condition that the coating volume does not change with hydration [18].

7.1.4. Swelling pressure prediction versus experimental measurements

Predicted swelling pressure P_s, given by equation (4.2), is compared with experimental measurements [7,8] in figure 7. The average stromal charge density used in these calculations was found by requiring the predicted P_s to exactly match the measured swelling pressure at J = 1 (normal hydration) and λ = 0.65. Olsen & Sperling [8] provide a power-law fit of their data where t is the stromal thickness, and this was used to determine the physiological swelling pressure as 84.35 mmHg.

Figure 7.

(a) The predicted swelling pressure P_s versus volume dilation J for three representative λ and its comparison with experimental measurements [7,8] over volume dilation range 0.55 ≤ J ≤ 1.5. The swelling data have been extracted from the original papers by the free graphics software Plot Digitizer, and the hydration data H from [7] have been transformed to thickness t by the observed linear relation H = 7.09t – 0.44 [8,43] (similar to equation (2.4)). (b) The predicted swelling pressure P_s versus volume dilation J for three values of The charge partition is fixed at λ = 0.65. (c,d) Predicted and measured swelling pressure in the low hydration range 0.4 ≤ J ≤ 1 for r_c = 18.25 nm (c) and r_c = 21.56 nm (d). The coating overlap introduces significant increase in the swelling pressure. At extremely low hydration, P_s for each λ value converges to a single curve, as the overlap domain is pervasive and the charge distribution approaches a uniform state. (Online version in colour.)

In figure 7a, computed and measured swelling pressure are compared over a range from swollen J = 1.5 (168% normal hydration) to compressed J = 0.55 (39% normal hydration) for three values of λ. The curve for λ = 0.65 agrees by design with the experimental data at J = 1, and shows excellent agreement with the data over the entire range of hydration. The other two curves corresponding to λ = 1 and 0.3 show the effect of changing the charge partition; the solution for λ = 0.3 deviates significantly from the experimental data. In figure 7b, the predicted swelling pressure is shown for three values of interfibrillar spacing (from 43 to 60 nm) with other unit cell parameters being fixed.

To illustrate the effect of coating overlap on swelling pressure, two coating radii r_c = 18.25 nm and 21.56 nm [17,18] have been modelled at very low hydration and the results are summarized in figure 7c,d. For both radii, the swelling pressure exhibits a very rapid increase with reducing hydration, owing to fibril coating overlap, with values that are in good agreement with the experimental data. The range of hydration for each coating is terminated when the coating reaches the collagen core.

In figure 8, the ratio between the electrostatic P_el and entropic P_str components of P_s is given and indicates that at normal hydration the magnitude of P_el is at least an order greater than the magnitude of P_str. At lower hydration, the ratio increases quickly, exceeding two orders at 60% normal hydration. When swollen to about 300% normal hydration, they become comparable.

Figure 8.

Ratio of magnitude of P_el and P_str versus J at λ = 0.65. P_el exceeds P_str by two orders of magnitude at J = 0.7 (hydration about 60% normal) and the ratio between these two components decreases to 1 as the tissue is swollen to J = 2.5 (hydration about 300% normal). The GAG elastic network is assumed to be ideal, wherein all the bridging sites are occupied. The bridging density is ρ_b = 4.38 × 10²² m⁻³ in these calculations and is comparable to the value estimated by [12]. (Online version in colour.)

7.1.5. Assessment of the Donnan potential-based approximate solution

Contour plots of the electrostatic potential φ, obtained by finite-element solution of the PB equation (3.4) are shown in figure 9a–d for four hydration values. In order to compare φ with the Donnan approximation given by equation (3.4), the profile of the two potentials is plotted along the fibril centre-to-centre line in figure 9e–h. At normal hydration, figure 9e shows that agrees reasonably well with φ. As the hydration is reduced, the coating domains approach figure 9f,g and then overlap figure 9h. As can be anticipated, the four representative hydration values illustrated in figure 9e–h clearly indicate that the Donnan approximation cannot accurately represent φ when the coatings are close or overlapping. A comparison of the value of P_el computed from the PB solution φ using equation (4.2) and from the Donnan potential using (4.10) is shown in figure 10a,b. The Donnan solution, which takes into account the coatings but not their interactions, is very accurate until the coatings become close, at which point the Donnan approximation does not share the rapid increase in swelling pressure predicted by the PB solution.

Figure 10.

(a) The effect of the φ and on the electrostatic free energy F_el and (b) electrostatic swelling pressure P_el for three representative values of λ. The PB-based and Donnan-based solutions for P_el agree well over the dilation range where PG-coatings are well separated. The Donnan-based solution cannot capture the effect of PG-coating overlap and thus loses accuracy at low hydration. (Online version in colour.)

7.2. Interfibrillar restoring forces in an ideal hexagonal cell

In order to characterize and compare the electrostatic and entropic restoring forces f_el and f_str, developed in §5, we consider an ideal hexagonal cell with fibrils at each vertex (figure 4). The hydration is taken to be normal with J = 1, and GAG fixed charges are allocated equally in each subcell. The resulting electrostatic restoring force magnitude f_el is computed from equation (5.2) and depicted in figure 11a,b for two fibril coating radius values r_c of 18.25 and 21.56 nm. The force–displacement relationship exhibits two quasi-linear regimes with a transition where the coatings begin to interact. This result quantifies the effect of the PG-coatings on the fibrils as a mechanism preventing fibril aggregation. For example, with r_c = 18.25 nm and λ = 0.65, the restoring force on the fibril at a displacement of 10 nm is less than 1 mN m⁻¹; when the displacement is extended to 20 nm, the force increases to approximately 3 mN m⁻¹.

Figure 11.

(a,b) Electrostatic restoring force magnitude f_el on a horizontally displaced fibril in the ideal hexagonal cell for two coating radius values (a) r_c = 18.25 nm and (b) 21.56 nm and three representative λ values. (c,d) Force–angle relationship for the electrostatic restoring force when the central fibril is displaced around small and extended circles. The force field is almost isotropic on the small circle before the PG-coating overlap. The overlap occurs on the extended circle and produces a degree of anisotropy. (Online version in colour.)

In order to characterize the isotropy of this force, the central fibril was displaced around a circle while the vertex fibrils remained fixed. The results for the magnitude of the restoring force are shown in figure 11c,d with two circles selected for each r_c. The smaller circle produces almost perfectly isotropic results as indicated by the flat force–angle curves in figure 11c,d. On the extended circle, the coatings interact and produce a degree of anisotropy.

For the purpose of characterizing the entropic restoring force associated with the antiparallel GAG duplexes, it is assumed in this ideal situation that all six nearest-neighbour bridging sites are occupied producing a perfect network with hexagonal symmetry. The results, shown in figure 12, indicate an essentially linear force–displacement relationship. At a fibril displacement of 10 nm, the entropic elastic restoring force is approximately 0.03 mN m⁻¹ and is thus two orders lower than the electrostatic restoring force. Note, however, that at lattice location r = r₀, we have f_el = −f_str = 0, which reflects the ideal symmetry of this case.

Figure 12.

Entropic elastic restoring force f_str versus the fibril displacement in an ideal hexagonal cell. The force–distance relationship is nearly linear and isotropic. (Online version in colour.)

Table 6 summarizes the electrostatic restoring stiffness, defined as the fibril restoring force per metre of fibril divided by the fibril displacement. The two quasi-linear regimes were considered and the electrostatic restoring stiffness in each regime was computed by least-squares regression. For example, with r_c = 18.25 nm and λ = 0.65, the electrostatic stiffness was 69 kPa for a fibril displacement less than 10 nm and 483 kPa for a fibril displacement greater than 16 nm (a sevenfold increase). In contrast, the entropic restoring stiffness is only 3 kPa.

Table 6.

Electrostatic restoring stiffness (kPa) in the two regimes (before and after the PG-coating overlap) obtained from the computed f_el in figure 11.

rc = 18.25 nm				rc = 21.56 nm
	λ = 0.3	λ = 0.65	λ = 1.0		λ = 0.3	λ = 0.65	λ = 1.0
r < 10 nm	39.12	68.98	79.04	r < 8 nm	40.98	67.05	75.43
r > 16 nm	1133	483.3	232.7	r > 12 nm	228.3	167.8	146.0

7.3. Fibril dynamics model for lattice self-organization

The proposed lattice restoring force mechanisms are studied further using the fibril dynamics model which features an imperfect GAG chain topology modelled by introduction of random variability in both fixed charge density and GAG chain elasticity. The fibril dynamics model is based on a two-dimensional periodic box containing 256 (16 × 16) collagen fibrils. The system is initialized with a perfect hexagonal lattice and zero initial velocities. The hydration is taken to be normal (J = 1) and charge partition λ = 1. The simulation has been run long enough to ensure equilibrium and the results are found independent to the system size and choice of time step. The RDF g(r) is obtained from equation (6.3) at each time step and averaged with respect to time.

Figure 13a shows the result of g(r) for the case where only the electrostatic restoring force f_el is considered, using three standard deviation values of 0.1 and 0.2. The charge samplings in the subcells for these three cases are summarized in the bar histogram in figure 13b. When and 0.1, the computed g(r) closely matches measurements on electron micrographs of human corneas [16] and indicates short-range order in the lattice such that correlations between any two fibrils becomes nearly random after r = 200 nm. As σ_ρ increases to , a more irregular lattice with a short-range order up to 80 nm results. The average interfibrillar distance, which corresponds to the location of the first peak in g(r), is maintained at in all three cases.

Figure 13.

Comparison between the predicted RDF g(r) from the fibril dynamics model with the measured g(r) from the electron micrograph of human cornea [16]. (a) The effect of electrostatic restoring force f_el on g(r) for three values of σ_ρ. (b) The bar histogram showing the Gaussian sampling of the initial charge density on 512 subcells under three values of σ_ρ; the curves are the ideal values computed from the Gaussian distribution. (c) The effect of entropic elastic restoring force f_str on g(r) for four imperfect GAG networks. (d) The predicted g(r) obtained by combining f_el and f_str for five GAG networks. The standard deviation for the charge density is fixed at . The interfibrillar spacing is preserved by the electrostatic restoring force regardless of the presence of duplexed GAG bridging.

The influence of duplexed GAG bridging elasticity on the fibril lattice order has been studied by computing g(r) for four imperfect networks with f_el excluded. The results are summarized in figure 13c. For a nearly perfect network (i.e. p_b = 0.9), the average interfibrillar distance is maintained as but the peaks of the computed g(r) do not decay as fast as the measurement with increasing r, indicating that a long-range order lattice is predicted in this case [12]. For the networks with more than 20% bridges missing (i.e p_b < 0.8), the location of the first peak in g(r) is immediately reduced to 31 nm, which is close to the value of fibril diameter and implies that fibril aggregation is occurring in the lattice. The results indicate that the duplexed GAG structures alone cannot maintain the interfibrillar distance when the network is imperfect.

Figure 13d shows the computed g(r) when both f_el and f_str are present, with and p_b varying from 0 to 0.9. The peak locations are maintained in all these cases and clearly the lattice structure becomes insensitive to the duplexed GAGs bridging topology when f_el is present.

8. Discussion

The proposed collagen–PG interaction model governing the fibrillar lattice was validated by predicting the stromal swelling pressure. The results, which showed good agreement with experimental data over the entire tested range of hydration, were used to establish a reasonable value for the coating charge parameter of λ = 0.65, although there is not a high sensitivity to this parameter with regard to swelling pressure. This value lies midway between the limit of no coating charge, λ = 1 and the lower limit λ = 0.28 established in §2. It may be possible to explore this parameter more directly by using three-dimensional electron tomography images such as those in [5,11]. Experimental results for swelling pressure at extremely low hydration, which are on the order of 2500 mmHg at a hydration of around 20% of normal, cannot be matched by classical Donnan theory. The proposed model modifies the Donnan result with two additional inputs: the electrolyte volume exclusion provided by the collagen and keratocyte stromal constituents, and the effect of fibril PG-coating interaction.

In the present work, we have used a theoretical model to describe mechanisms by which movement of a fibril out of its lattice position produces electrostatic and entropic elastic restoring forces. In this model, the GAG chains generate the underlying tissue osmotic pressure and they are also responsible for a local perturbation in that pressure when a fibril, along with its GAG cargo, is displaced from its lattice position. All GAG chains can contribute to such a perturbation, regardless of whether they are individual chains or are part of an antiparallel, duplexed association bridging between adjacent fibrils. It has been shown that in the dynamic situation where a fibril is out of its lattice position, the electrostatic and entropic elastic restoring forces: (i) are both restoring forces, (ii) essentially operate independently and not as an antagonistic mechanism in which they are required to oppose each other and (iii) the electrostatic forces alone can restore the lattice. In addition, according to our calculations, the resulting electrostatic-based restoring forces subordinate the entropic elastic forces of the antiparallel GAG duplexes and are relatively insensitive to the random organization of the GAGs.

As described in §4, stromal swelling pressure has both osmotic (positive) and GAG entropic elastic (negative) components; it depends strongly on hydration and has a considerable physiological value of approximately 60 mmHg. However, common experience informs us that healthy living corneas do not swell; the hydration (and thus swelling) of the stroma is precisely controlled to be 78% water weight by means of an active ion transport mechanism located in the bounding endothelial layer (figure 2a) [14]. Hydration values exceeding optimal have been shown to result in corneal opacity. The active mechanism that counters the swelling pressure in the living cornea is implicit in the presented model because the stromal volume (for any hydration level) is maintained by forces that precisely balance the swelling pressure. Further, the fibril lattice theory presented is valid for the living system when the local perturbations in the osmotic pressure are based on the physiological hydration (as was done for the lattice dynamics model in §§6 and 7).

It has also been shown that, according to the model, if the fibrils do have a GAG-dense coating, and our results do not preclude that, then when fibril pairs approach the point where the coatings are close or overlapping, an additional restoring force mechanism is activated and the restoring stiffness (force per unit displacement) increases sixfold over the value before the coatings engaged. In this way, the model gives a quantitative assessment of the coating effectiveness with regard to preventing fibril aggregation.

In our modelling of randomness of the GAG organization, it is always assumed that the duplexed GAG chains are perpendicular to the fibril axis [13]. In the light of recent findings, this is not realistic because many are tilted with respect to the fibril axis [5]. Such considerations could be included in the theory, but presumably would tend to further reduce force estimates because only a component of the force vector will act as a restoring force. In regard to the relatively small bridging forces, it is interesting to note that in recent studies of other connective tissues such as tendon, cartilage and ligament, interfibrillar PG-based filaments have been found to be unlikely to have any mechanical role [44].

The dynamic analysis was based on a lattice with a random GAG organization. The randomness of the GAGs was reflected in both the spatial distribution of charge and the topology of the bridges. A striking result is that the electrostatic restoring force alone was able to reproduce the radial distribution function of the fibrils for the human cornea, confirming the short-range order of the lattice; no input from the GAG duplexes was needed. The converse did not hold. The dynamic case with bridging entropic forces alone actually predicted fibril aggregation on a defective network because the entropic force is essentially contractive.

The lattice dynamics study presented was performed at normal hydration. At a swollen state, the cornea can become opaque and manifest stromal ‘lakes’ [1,27], which are regions in which the lattice is profoundly disturbed with reduced numbers of collagen fibrils. Although results have not been presented here, the proposed model is applicable to modelling these conditions. In the current model, PG/GAGs have been assumed to be distributed uniformly throughout the stroma. Biochemical studies have shown that the anterior corneal stroma has a higher ratio of DS to KS [45]. The proposed theory could be adapted to model how the corneal hydration profile may be affected by this distribution. Another potential application concerns the loss of KS which occurs in corneas with macular corneal dystrophy [5,26] or to genetically mutated corneas [9], which leads to a reduction in interfibrillar distance and a more chaotic lattice due to the lack of PG regulation.

Appendix A. Derivation of the Donnan-based electrostatic swelling pressure component P̂_el

The piecewise constant Donnan potential is written by combining equations (4.6)–(4.8) as

A1

where the charge densities ρ_inter(λ,J) and ρ_coat(λ) associated with the two GAG classes are given by equations (4.4) and (4.5), respectively. Substitute equation (A 1) into equation (3.5), the Donnan electrostatic free energy approximation is given as

A2

where I_inter and I_coat are the two integrands in the interstitial and coating regions:

A3

and

A4

The Donnan-based electrostatic contribution to the swelling pressure is then given by equation (4.2) with replaced by the Donann approximation , equation (A 2). Using equations (A 1), (A 3) and (A 4), it is observed that

A5

and

A6

therefore can be derived as

A7

Appendix B. Derivation of the Donnan-based electrostatic interfibrillar restoring force

From equation (5.5), the electrostatic interfibrillar force f_el is expressed under Donnan approximation as

B1

The volume dilation on the kth subcell J_k is expressed as

B2

and

B3

where is the current volume of the subcell and measures the subcell cross-section area and is a function of r,r_k1,r_k2, which are the coordinates of fibrils forming the kth subcell. The analytical form of and its derivative are given as

B4

and

B5

where r = (x,y) and refers to the sign of the term in the absolute operator of equation (B 4). Combining equations (B 1), (B 3) and (B 5) gives

B6

Funding statement

This research was supported by the Stanford University Bio-X Interdisciplinary Initiatives Program (IIP), which is gratefully acknowledged.