INTRODUCTION

TOP ABSTRACT INTRODUCTION MATHEMATICAL MODEL THE LIMIT OF INSTANTANEOUS... PARAMETER ESTIMATION SCHEME COMPARISON TO EXPERIMENT DISCUSSION REFERENCES

Enzymatic degradation of fibrillar collagen networks is a fundamental process in connective tissue remodeling. An understanding of the mechanism of the degradation process is also of great importance in a host of biodegradable biomedical devices such as connective tissue analogs (Chamberlain et al., 1997; Sung et al., 1997; Compton et al., 1998; Riesle et al., 1998; Freyman et al., 2001), sutures (Hayashi et al., 1990; Okada et al., 1992), vascular grafts (Van Wachem, et al., 2001), drug-delivery devices (Gilbert and Kim, 1990; Freiss et al., 1996; Freiss, 1998; Fujioka et al., 1998; Wissink et al., 2000), etc. (Sabelman, 1985; Li, 1995; Nimni, 1995), as well as in processing of wood and cellulosic fibers (Viikari et al., 1991). Under in vivo and in vitro conditions, collagen aggregates into a network of approximately cylindrical fibrils with diameters varying between 20 and 500 nm, according to collagen age, type, composition, and tissue source (Stryer, 1988; Kadler et al., 1996; Riesle et al., 1998). Degradation of collagen monomers (i.e., tropocollagen) in solution by specific and nonspecific collagenases has been shown to follow Michaelis-Menten kinetics (Welgus et al., 1981a,b, 1982; Van Wart and Steinbrink, 1985; Mallya et al., 1992). In contrast, the degradation of fibrillar collagen depends on the age of the sample, with fresh samples behaving similarly to collagen molecules in solution, and older samples behaving anomalously with respect to enzyme concentration (Steven, 1976a,b). Namely, the erosion rate of mature fibrillar samples has been shown to depend hyperbolically on enzyme concentrations, even when the number of collagen monomers is greatly in excess of the number of enzyme molecules (Steven, 1976a,b; Welgus et al., 1980). Because this behavior is observed both with vertebrate collagenase and bacterial collagenase (and even with trypsin), it has been suggested that these anomalies are related to the microstructure of fibrillar collagen, namely to steric exclusion of the large enzyme molecules from the bulk of the fibrils due to very tight packing. Although this interpretation is widely accepted, and has been used to intuitively rationalize the experimental observations (Steven, 1976; Welgus et al., 1980), the implications of steric obstruction of enzyme binding sites on the transient degradation have not been explored quantitatively.

To date, except for the work of Suga et al. (1975), most mathematical modeling of enzymatic erosion of insoluble protein fibers has been of an ad hoc phenomenological nature. Hayashi and Ikada (1990) suggested that enzymatic erosion of insoluble polymer fibers is a pure surface erosion process and derived a simple model based on the assumption that the radius of the fiber decreases linearly with time. This model implies that the square root of the fiber mass decreases linearly with time, and seems to be consistent with most of the experimental data obtained by Okada et al. (1992) regarding the erosion of cross-linked collagen fibers by bacterial collagenase at 37°C. Although the assumption of surface erosion seems plausible for large enzymes, no justification was given for the assumption that the fiber radius decreases linearly with time.

Bailey and Ollis (1977) reanalyzed published data on the enzymatic erosion of insoluble proteins and demonstrated that the rate of erosion depends hyperbolically on enzyme concentration. They explained this hyperbolic dependence by suggesting that Langmuir surface adsorption is the rate-limiting step in the hydrolysis of insoluble enzyme. Although plausible, this explanation overlooks the fact that the Langmuir adsorption isotherm is used to describe adsorption of proteins onto noneroding surfaces (Letnam, 1951) and its applicability for eroding surfaces is questionable. Thus, although Sattler et al. (1989) were able to fit the initial rate of enzymatic hydrolysis of cellulose to a Langmuir adsorption isotherm, the resulting binding constant was found to be time dependent.

In this work, we derive a reaction diffusion model for the enzymatic erosion of insoluble fibrillar matrices that takes into account two inherent heterogeneities: 1) the macroscopic heterogeneity of the gel-solution system, which entails that we consider the diffusion of the enzyme into the sample and the diffusion of the degradation products out of the sample (Suga et al., 1975), and 2) the confinement of the binding sites available for the enzyme to the surface of the fibrils, which is translated into a novel kinetic scheme. The limit of instantaneous diffusion of this model is derived and studied using the quasi-steady-state approximation (QSSA). The latter approximation allows us to derive a closed-form approximation for the rate of degradation that is valid for a wide range of model parameters, and which can explain the success of the ad hoc correlations of previous researchers (Bailey and Ollis, 1977; Hayashi and Ikada, 1990). Moreover, the QSSA enables us to identify the model parameters (basic or composite) that govern the transient degradation under different experimental conditions and suggests an experimental-theoretical method of estimating these parameters. The limit of instantaneous diffusion is shown to be roughly valid for the experiments of Welgus et al. (1980) on the in vitro erosion of fibrillar collagen by matrix fibroblast collagenase at 37°C. Although those experiments are only partially consistent with the theoretical-experimental method proposed in the current paper, the QSSA is shown to be roughly valid for them and enables us to estimate the Michaelis-Menten constant of that system, which could not be assessed before. Using this estimate to simulate additional erosion experiments (Welgus et al., 1980), elucidates the role of diffusion. Moreover, the consistency between simulation and experiment reinforces the validity of the proposed model and the parameter estimates obtained in this work.

	MATHEMATICAL MODEL

TOP ABSTRACT INTRODUCTION MATHEMATICAL MODEL THE LIMIT OF INSTANTANEOUS... PARAMETER ESTIMATION SCHEME COMPARISON TO EXPERIMENT DISCUSSION REFERENCES

Overview

Here we derive a model for the erosion of an insoluble fibrillar matrix (e.g., collagen gel) by a specific enzyme that has a single cleavage site on the monomer of which the fibril is composed (e.g., skin fibroblast collagenase). The fibrillar gel is modeled as a solid porous network immersed in a buffered enzyme solution. The fibrils are idealized as perfect cylinders of tightly packed monomeric rods (see Fig. 1). This idealization is a good approximation as long as the fibril diameter, d_f, is much larger than the diameter of the monomer, d_m. When this network comes in contact with the enzyme solution, the enzyme diffuses into the gel where it binds to specific sites on monomers located at the surface of the fibrils. Due to their size, the enzyme molecules cannot penetrate the tightly packed (cross-linked) monomers that make up an individual fibril. This problem is inherently heterogeneous, because the reaction is confined to the gel.

View larger version (13K): [in this window] [in a new window]   FIGURE 1   Schematics of an idealized cylindrical fibril segment composed of many tightly packed cross-linked monomers that are modeled as rigid rods. The blowup depicts a single monomeric subunit. The diameter of the fibril, df, is much larger than the diameter of its cylindrical monomeric subunits, dm.

A crucial simplifying assumption in the subsequent derivation is that cleavage is the rate-limiting step of fibril erosion (e.g., as in fibrillar collagen at T > 35°C (Sakai and Gross, 1967; Welgus et al. 1980)). Thus, once a monomer at the surface of a fibril is cleaved, it is assumed to spontaneously detach and go into the solution, where it diffuses, eventually reaching the gel-liquid interface. This assumption will enable us to use standard enzyme kinetics for the degradation process at the surface of the fibrils (Lin et al., 1999), and moreover implies that fibril erosion is confined to its surface.

Basic kinetic scheme

The reaction between the enzyme and monomer substrate is assumed to be of the common Michaelis-Menten type (Stryer, 1988; Suga et al., 1975; Lin et al., 1999)

(1)

Here E and C denote the free and bound enzyme (inside the matrix), respectively,  denotes the substrate (monomer) and P denotes the degradation products (i.e., the cleared monomers), _f is the rate constant of formation (per unit substrate) of the enzyme-substrate complex, k_r is the rate constant of dissociation of the enzyme substrate complex, and k_cat is the catalysis rate. The final step in kinetic scheme 1 is irreversible because the enzyme only catalyzes the degradation of the monomers. Moreover, because we assume that cleaved monomers detach spontaneously and go into solution, association of cleaved monomers is expected to be negligible.

Kinetic scheme 1 implies the following kinetic equations for the immobilized species in the gel matrix:

(2)

(3)

and the following reaction-diffusion equations for the mobile species in the gel matrix (Crank, 1975):

(4)

(5)

where D_e,g and D_p,g are, respectively, the diffusion coefficient of the free enzyme and the degradation products inside the gel matrix. Denoting the concentration (per unit volume) of available (unoccupied) binding sites (i.e., attached monomers) on the surface of the fibrils by S, we note that _f has to satisfy the relation,

(6)

where k_f is the rate of complex formation per available substrate molecule. Eq. 6 is a manifestation of the fact that the available binding sites are all located at the surface of the fibrils. Thus, whereas k_f is a basic (constant) parameter of the system, the effective reaction parameter _f is a variable proportional to the ratio S/. Substituting Eq. 6 into Eqs. 2 and 3, we obtain

(7)

(8)

where

(9)

denotes the total concentration of surface binding sites, both free and bound. To close this system of equations, we have to relate between and

(10)

the total concentration of collagen monomers. Such a relation is derived in the following section.

Concentration of surface binding sites

Consider a (constant) reference volume V_g(r) centered at a point labeled by the vector r inside the gel. This reference volume is chosen such that it is small compared to the total volume of the gel matrix, but large compared to the typical network diameter, so that it contains many fibril segments (see Fig. 2). Let N(r, t) and n(r, t), respectively, denote the total number of rod-shaped monomers of length L_m and fibril segments of length L_m in V_g(r) at time t. As degradation proceeds, the number of fibril segments n and the monomer diameter d_m are constant, but the total number of monomers N and the fibril diameter d_f decrease.

View larger version (13K): [in this window] [in a new window]   FIGURE 2   Multiscale geometry of the problem. (A) Schematics of the (macroscopic) gel matrix (g) and the surrounding solution layer (s). (B) Blowup of a (mesoscopic) region of volume Vg around a point r  g. (C) Blowup of a typical fibril segment in Vg revealing its microscopic structure.

The total molar concentration of (undegraded) monomers, , therefore satisfies the relations

(11)

(12)

where ₀ and N₀ denote the initial values of and N, respectively. Figure 3 shows a cross section of a typical fibril segment. The total concentration of surface binding sites, , is the product of the concentration of fibril segments of length L_m, n₀/V_g, with the total number of monomers (e.g., circles of diameter d_m) on the circumference of such a cross section, d_f/d_m, namely,

(13)

The area of the cross section depicted in Fig. 3 is d/4 and it is covered by tightly packed circles (e.g., monomer cross sections) of diameter d_m. Because the monomers are incompressible, the circles do not overlap and are equivalent to squares of side d_m in terms of coverage. Equating the total volume of fibril segments in V_g(r), n₀L_m(d/4), to the total equivalent volume of monomers in V_g(r), NL_md, we obtain

(14)

and

(15)

Sequential application of Eqs. 11 and 14 yields

(16)

Substituting the latter result into Eq. 13, and using Eq. 15 to further simplify, yields

(17)

where the proportionality constant  is defined as

(18)

Initially we have

(19)

where

(20)

Recall that the results of this section were obtained by approximating the fibril as a cylinder with a smooth surface and are only valid provided that d_m d_f. For fibrillar collagen, d_m  1.5 nm and 22 nm d_f(0)  500 nm (Hulmes et al., 1995). This implies that the approximations of this section should always be roughly valid at least during the initial stages of fibrillar collagen degradation.

View larger version (27K): [in this window] [in a new window]   FIGURE 3   Schematics of a cross section perpendicular to the axis of the fibril. The area of such a cross section is d/4 and is covered by many tightly packed circles (e.g., monomer cross sections) of diameter dm. Because the monomers are incompressible, the circles do not overlap and are equivalent to squares of side dm in terms of coverage.

The reaction diffusion equations

Incorporating result 17 into Eqs. 7 and 8, we obtain

(21)

(22)

(23)

(24)

Here, _g denotes the gel matrix, and we remind the reader that  =  + C. Because appears naturally in Eq. 22, it is convenient to replace Eq. 21 by a rate equation for . This is achieved by adding Eqs. 21 and 22, to obtain

(25)

In this work, we consider the uniform initial conditions,

(26)

(27)

As (standard) boundary conditions, we impose continuity of the fluxes across the gel-liquid interface (denoted as _g)

(28)

(29)

where E_s and P_s denote the concentrations of free enzyme and degradation products in the surrounding solution layer, respectively, and D_e,s, D_p,s are the corresponding diffusion coefficients. Because the enzymatic reaction is confined to the gel, the dynamics of the free enzyme and the degradation products in the surrounding liquid layer are described by the equations,

(30)

(31)

(32)

(33)

where _s denotes the volume of the external solution layer.

The low-temperature limit

The activation energies associated with the enzymatic degradation of fibrillar collagen are significantly higher than the corresponding values for degradation of tropocollagen monomers in solution (Welgus et al., 1981). The fact that enzymatic degradation of fibrillar collagen becomes negligible at 25°C and 4°C for skin fibroblast collagenase and bacterial collagenase, respectively, has enabled researchers to measure the binding of these enzymes to fibrillar collagen in the absence of significant degradation (Welgus et al., 1980; Matsushita et al., 1998). Such experiments correspond to substituting k_cat  0 in Eqs. 22-33 to obtain

(34)

(35)

(36)

(37)

(38)

(39)

It is noteworthy that this is the standard reaction-diffusion formulation of absorption and binding of a solute by a porous matrix, with  = ₀ playing the role of the maximal binding capacity.

	THE LIMIT OF INSTANTANEOUS DIFFUSION

TOP ABSTRACT INTRODUCTION MATHEMATICAL MODEL THE LIMIT OF INSTANTANEOUS... PARAMETER ESTIMATION SCHEME COMPARISON TO EXPERIMENT DISCUSSION REFERENCES

The above model of enzymatic erosion of insoluble fibrillar gels is much more complex than the Michaelis-Menten kinetic scheme that is commonly used to study enzymatic processes in solution. Namely, the proposed model depends on seven more model parameters than the standard Michaelis-Menten scheme and involves the solution of a set of coupled partial differential equations, as opposed to a set of coupled ordinary differential equations in the Michaelis-Menten model. According to Buckingham's  theorem (Buckingham, 1914), the number of independent dimensionless variables is equal to the number of physical quantities (e.g., model parameters) minus the number of independent physical dimensions (e.g., length and time). This implies that Eqs. 22-33 depend on ten dimensionless variables, whereas the standard Michaelis-Menten scheme depends on only three dimensionless variables. Thus, the inclusion of diffusion complicates the model significantly.

However, whenever the characteristic time scales of the diffusion of the free enzyme and degradation products in the gel and in the external solution layer are much shorter than the characteristic time for product formation, diffusion can be assumed to be instantaneous. In this case, the concentrations of the reactants and products can be assumed uniform within the gel, subject to the global enzyme conservation relation,

(40)

where |_g| and |_s| are the volumes of gel and the external solution layer, respectively. This conservation relation is derived by noting that, whereas the bound enzyme is confined to the gel, the free enzyme distributes uniformly in the aqueous phase both inside the gel and in the surrounding fluid layer. For sparse gels, we can safely neglect partition and substitute E = E_s in Eq. 40 to obtain

(41)

where V_t = |_{g  s}| is the total volume of the system. Introducing the simplifying definitions

(42)

(43)

into relation 41, we obtain

(44)

To summarize, in the limit of instantaneous diffusion, our model reduces to the set of nonlinear ordinary differential equations,

(45)

(46)

subject to the initial conditions

(47)

(48)

and the substrate conservation relation

(49)

Here,

(50)

is the Michaelis-Menten constant of the system. These equations are analogous to the equations describing enzymatic reactions in solution. This analogy will be exploited to derive approximate analytical solutions and parameter estimation methods.

With the exception of the low-temperature limit, Eqs. 45-48 are nonintegrable. In the low-temperature limit, k_cat  0 and  = ₀, so that Eq. 46 takes on the form,

(51)

where

(52)

is the Langmuir binding constant of the enzyme, and

(53)

are the roots of the quadratic equation,

(54)

Integration of Eq. 51 yields

(55)

where we introduced the simplifying notation,

(56)

Note that _±(₀)  0 and that (₀) corresponds to the steady-state concentration of bound enzyme, C_eq, as can be inferred by taking the t   limit of result 55. Hence,

(57)

regardless of the rate of diffusion. Namely, because this is an equilibrium result, its validity transcends that of the limit of instantaneous diffusion.

The quasi-steady-state approximation

As already mentioned, Eqs. 45-48 are nonintegrable for k_cat > 0. However, these equations are no more complex than the equations corresponding to the standard Michaelis-Menten scheme (Stryer, 1988). The latter are also nonintegrable but have been successfully analyzed using the QSSA (Segel and Slemrod, 1989; Borghans et al., 1996; Schnell and Mendoza, 1997; Schnell and Maini, 2000). Below, we derive the QSSA corresponding to Eqs. 45-48 and analyze its validity using a modification of the procedure described by Borghans et al. (1996).

The total QSSA

(58)

(59)

(60)

(61)

(62)

(63)

(64)

(65)

(66)

(67)

(68)

(69)

(70)

(71)

(72)

(73)

(74)

The first-order tQSSA

(75)

(76)

(77)

(78)

(79)

(80)

(81)

(82)

(83)

(84)

(85)

(86)

(87)

(88)

(89)

(90)

(91)

(92)