INTRODUCTION

TOP ABSTRACT INTRODUCTION RESULTS DISCUSSION APPENDIX REFERENCES

In analyzing the binding of ligands in solution to receptors on the surface of a cell or biosensor, it is common to model the receptors as fixed or diffusing particles on a two-dimensional (2D) surface. In many experimental systems, however, receptors are distributed in a layer above a surface. In the case of cells, specific ligands bind to sites within the extracellular matrix (glycocalyx). In the Biacore (Biacore AB, Uppsala, Sweden) an optical biosensor used widely for quantitative analysis of interactions between biomolecules, receptors are often attached to polymers that form a layer on a sensor chip (Rich and Myszka, 2000).

The extent to which a binding layer affects the interpretation of Biacore data has been investigated experimentally (Karlsson and Fält, 1997; Parsons and Stockley, 1997), numerically (Schuck, 1996), and analytically (Edwards, 2001). The results we present here facilitate the analysis of Biacore experiments, resolve an apparent contradiction between earlier numerical predictions and experimental results, and provide tools for assessing and including the effect of a binding layer when ligands bind to sites on spherical cells or beads.

Our approach is based on effective rate models (reviewed in Goldstein et al., 1999). When ligands in solution bind to receptors on a surface, whether or not the receptors are distributed in a three-dimensional (3D) layer, a complete model for binding and dissociation includes transport of the ligand to the surface and reaction at the surface. However, the resulting partial differential equation (PDE) models, including diffusion and possibly convection of the ligand, are impractical as the basis for extracting reaction rate constants from binding data. Approximate models, consisting of an ordinary differential equation (ODE) with effective rate coefficients that are explicit functions of both the transport and reaction parameters, provide the basis for simple and accurate estimation of reaction rates, under a wide range of experimental conditions (e.g., Goldstein and Dembo, 1995; Myszka et al., 1998; Mason et al., 1999; Edwards et al., 1999). Here we derive effective rate coefficients for ligands binding to receptors distributed in a layer above the surface of a spherical cell or the sensor surface of a Biacore.

Effective rate coefficients for binding and dissociation within a layer modify, in a simple way, the analogous expressions for the case where binding occurs on an impenetrable surface. The modification is that the intrinsic reaction parameters, i.e., the association and dissociation rate constants, are reduced by a factor that depends on the thickness of the layer and the mean free path of the ligand in the layer (roughly, the distance a ligand travels within the layer before it binds to a receptor). If the mean free path is large relative to the height of the binding layer, then the layer can be modeled as a 2D surface. This makes sense because, in this limit, concentrations of free and bound ligand are essentially uniform in the layer. The 3D structure of the binding layer would only be important if, in the course of an experiment, gradients developed in the ligand concentration in the layer, for example, if binding occurred to a significantly greater extent at the top than at the bottom of the layer.

The main conclusion for the Biacore is that, for typical ligands and experimental design, effective rate coefficients derived from a model where receptors are distributed in the dextran layer are very close to those derived under the assumption that binding takes place on an impenetrable 2D surface. The dextran layer affects the interpretation of binding data only if diffusion of the ligand within the layer is much slower than the diffusion of a typical biological molecule in the aqueous medium outside the layer (where the diffusion coefficient is expected to be between 10⁷ and 10⁶ cm²/s). This result is consistent with simulations by Schuck (1996), who used a diffusion coefficient of 10⁸ cm²/s for a ligand inside the dextran layer and predicted gradients in the concentrations of free and bound ligand within the layer. If such gradients are pronounced, they may lead to misinterpretation of binding data. However, our result is also consistent with Biacore experiments of Karlsson and Fält (1997) and Parsons and Stockley (1997). Both groups did not detect differences in binding or dissociation between sensor chips with receptors distributed in a dextran layer and chips with receptors bound directly to the surface. Karlsson and Fält argued that parameter values used in the Schuck calculations were not characteristic of Biacore conditions.

	RESULTS

TOP ABSTRACT INTRODUCTION RESULTS DISCUSSION APPENDIX REFERENCES

General form of effective rate coefficients for binding in a layer

The role of effective rate coefficients is to provide a good approximate description of ligand/receptor binding kinetics at a surface, using an ODE of the form,

(1)

where B and R are concentrations of bound and free receptors on the surface, C_T is the bulk concentration of ligand far from the surface, k is the effective forward (association) rate coefficient, and k is the effective reverse (dissociation) rate coefficient. The concentrations of bound and free receptors, B and R, satisfy the conservation law R + B = R_T, where R_T is the total surface concentration of receptors, assumed to be constant for the period of the experiment being modeled. Typical units for the concentrations of reactants are cm³ or nM for the ligand and cm² or nM-cm for receptors. The formulation in Eq. 1 is for monovalent receptors and ligands.

If transport of the ligand is rapid relative to the reaction at the surface, the system is in the "reaction limit," and Eq. 1 holds with the effective rate coefficients equal to the intrinsic association and dissociation rate constants, denoted by k_a and k_d. In general, effective rate coefficients are not constants; they depend on R (equivalently, on B).

For experiments where ligands bind to receptors distributed in a polymeric layer attached to the surface of a spherical cell or the sensor surface of a Biacore flow cell, we derive effective rate coefficients of the form

(2)

which differ from those derived previously for binding to a 2D surface only in the "thickness factor" T(), defined below, that multiplies the intrinsic rate constants k_a and k_d. The components of Eq. 2 are defined in the next section, where we summarize the PDE model that underlies the derivations. In brief, R is the effective surface concentration of free receptors if receptors are considered to be confined to a surface of area A, and, therefore, RA is the total number of receptors in the layer; k₊ is the transport-limited forward rate constant, i.e., the rate constant for binding to the surface, as the receptor concentration tends to  and the surface becomes a perfect absorber;  is the ratio of the height of the layer to the mean free path of the ligand in the layer, when the surface concentration of free receptors is R; and

(3)

The graph of T is shown in Fig. 1. In the limit of small , or equivalently, when the binding layer is thin relative to the mean free path of the ligand, T()  1 and the effective rate coefficients approach the form expected when binding takes place on an impenetrable surface. In this case, the layer can be ignored, and binding can be modeled as occurring on a surface. When   1, T() acts on the intrinsic reaction rates k_a and k_d as a reduction factor. In this case, if the effective rate model without the correction for the layer is used to analyze binding data, the intrinsic rate constants will be underestimated by the factor T().

View larger version (0K): [in this window] [in a new window]   FIGURE 1   Graph of the thickness factor T() = tanh()/ that reduces effective rate coefficients when ligand/receptor binding takes place in a 3D layer rather than on a 2D surface. When  (Eq. 5b) is small, which occurs when the layer is thin relative to the mean free path of the ligand in the layer, then T()  1 and the layer has a negligible effect on binding.

Overview of the PDE model for ligand concentrations

Figure 2 sketches (a) a spherical cell with radius a and (b) a cross section of a Biacore flow cell, with height h and length l. In both cases, the height of the polymeric layer is d. The diffusion coefficient for the ligand is D_i in the layer and D_o outside. In the model for binding to a cell, transport of the ligand to the cell is by diffusion alone. For the Biacore, there is also convection, with maximum velocity v_c. In both cases, a partition coefficient  gives the effective fraction of the volume of the layer in which the ligand is free to diffuse. If the pore size in the polymer matrix is comparable to the size of the ligand, then the excluded volume will be greater than the volume that the polymer matrix occupies (Deen, 1987). We assume that receptors and ligands bind monovalently with intrinsic forward and reverse rate constants k_a and k_d.

View larger version (0K): [in this window] [in a new window]   FIGURE 2   Diagrams of (a) a spherical cell and (b) the flow chamber of a Biacore biosensor, each with receptors distributed in a polymeric layer of height d.

In the next section, we present methods for calculating effective rate coefficients from the steady-state flux of ligands to receptors. In the model, a steady state is maintained by, in effect, replenishing ligand and receptor. This is accomplished by holding the ligand concentration, C_T, constant as the distance from the surface approaches , and holding the free receptor concentration in the layer, R^3D, constant and uniform. (Note that we are not assuming that the free receptor concentration remains constant over time. Rather, we have a continuum of models, one for each free receptor concentration R^3D, that allows us to calculate effective rate coefficients for association and dissociation at each R^3D.) The uniformity assumption for free receptors is a substantive simplification, as discussed further below.

For the rectangular geometry of the Biacore, the effective surface concentration of free receptors satisfies

(4)

For a spherical cell, Eq. 4 holds approximately, in the limit where the height of the layer is small relative to the radius of the cell, i.e., d/a 1. Also, the form of the function T that gives the thickness factor T() in Eq. 2 has the simple form given by Eq. 3 only when d/a 1. A typical lymphocytic cell has radius a  5 × 10⁴ cm. The extracellular matrix has height d  10⁶ cm (Bongrand, 1988). Therefore, at least for these cells, d/a  2 × 10³ and it is reasonable to use the approximations given by Eqs. 3 and 4. (The general forms of T and R for a sphere are given in the Appendix, Eqs. A6 and A7.)

For both the spherical cell and the Biacore, the mean distance a ligand travels in the polymer layer before binding occurs (i.e., the mean free path) is

(5a)

at the start of a binding experiment when all receptors are free and uniformly distributed in the layer (see Appendix). When Eq. 4 holds, in particular for the Biacore and many cell types, the mean free path can be expressed in terms of the effective surface concentration of free receptors as  = . Then , the ratio of the height d of the layer to the mean free path of the ligand in the layer is

(5b)

Consideration of the mean free path clarifies the effect of the model assumption that free receptors are distributed uniformly. In experiments, as ligands enter the layer and bind to receptors, the concentration of free receptors is depleted preferentially near the entrance to the layer. By effectively redistributing free receptors in the series of models we consider, we provide additional opportunities for binding near the entry boundary, shorten the mean free path, and consequently exaggerate the effect of the layer.

Methods for obtaining effective rate coefficients from steady-state profiles

The PDE models for the spherical cell and the Biacore flow cell presented in the Appendix determine the steady-state concentration of free ligand as a function of position, in the layer (C_i) and outside the layer (C_o). There are two equivalent ways to use the ligand concentration functions to calculate an effective association rate coefficient. Both methods equate expressions for the total rate of binding (number of receptors bound per cell per unit time), in the effective rate ODE model and in the full PDE model. In the effective rate model, receptors that are, in fact, distributed in a layer , of volume V, where the concentration of free receptors is R^3D, are taken to be confined to a surface with area A. The effective surface concentration of free receptors is R, where RA = R^3DV. In terms of the effective association rate coefficient k and bulk concentration of ligand, C_T, the rate of binding to the surface is
In the full model, the total rate of binding within the layer can be expressed similarly, as
so that

(6)

The expression for k can also be obtained from the total flux of ligand at the boundary of the binding layer, which can be expressed either in terms of C_i or C_o at the boundary. For example, in the model for binding to a sphere of radius a where the height of the layer is d, we can express C_i and C_o in terms of a radial variable r and find k from

(7)

with V = ((a + d)³  a³) and A = 4(a + d)². In the Biacore model, we express ligand concentrations in terms of a height y, with y = d denoting the interface between the dextran layer and the rest of the flow cell. There, V/A = d and

(8)

Once k is obtained, the effective dissociation rate coefficient, k, is defined by k = Kk, where K denotes the equilibrium association constant, i.e., K = k_a/k_d. Then the effective rate model (Eq. 1) has the correct behavior at long times, as the system approaches equilibrium.

The results we obtain with these steady-state calculations agree, in the special case where a comparison is available, with an effective rate model derived by Edwards (2001, Eq. 66) from a time-dependent system of PDEs (see the discussion of Eq. A40 in the Appendix).

Effective rate coefficients for the case of a spherical cell

To evaluate the effective rate coefficients (Eq. 2) for the spherical cell, we use the diffusion-limited forward rate constant for binding to a sphere of radius a + d, i.e., k₊ = 4D(a + d) (Smoluchowski, 1917), and the corresponding surface area, A = 4(a + d)². Then,

(9a)

In the typical case where d/a 1,

(9b)

In summary, an approximate model that can be used to analyze binding and dissociation data from spherical cells is (Eq. 1)

(10)

where R = R_T  B.

Effective rate coefficients for the Biacore

For the Biacore flow cell, if the transport-limited forward rate constant, k₊, is expressed as a function of the distance x from the inlet of the flow cell, then (Lok et al., 1983)

(11a)

If the transport-limited forward rate constant is averaged over the full length of the flow cell, then

(11b)

Eqs. 11a, 11b, or the intermediate form obtained by averaging over a portion of the flow cell are substituted into Eq. 2 to give effective rate coefficients.

In the Appendix, we consider two limits in which modified expressions for effective rate coefficients (Eqs. A36, A37, A40, A41) give marginally more accurate approximations than Eq. 2, relative to exact expressions obtained from flux calculations. However, the approximations in Eq. 2 differ by no more than 2% from the exact expressions (see Eqs. A32 and A39) for all sets of parameters and experimental conditions.

Under what conditions can the layer be neglected?

The effective rate coefficients (Eq. 2) reduce to the form obtained by treating the layer as a 2D surface when T()  1, or equivalently, when   1. Because  (Eq. 5b) is largest at the start of a binding experiment, when all receptors are free (i.e., R = R_T), a sufficient condition for modeling the layer as a 2D surface is

(12)

That is, we can neglect the layer if the mean free path of the ligand in the layer, when most receptors are free, is long relative to the height of the layer.

There is an additional situation where the thickness factor T is not important in predicting binding or in fitting kinetic data, even though T < 1. In the transport limit, i.e., when T()k_aRA/k₊ is large (1), the dependence of k on T is negligible, because k  k₊/(RA).

To determine conditions under which the correction for the binding layer changes effective rate coefficients by more than a specified amount, we considered the ratio  of effective rate coefficients with and without T() (i.e., using T() = 1 for the case where the layer is ignored),

(13)

Eq. 13 shows that  can be written as a function of two variables. We will find it useful to consider  as a function of  and a quantity  = D_iA/(dk₊) that depends on geometric and transport-related parameters but does not depend on k_a and R. Figure 3 shows level curves of the ratio of effective rate coefficients, . The contour where  = 0.9 shows that, for effective rate coefficients with and without the correction for the layer to differ by more than 10%, we must have both

(14a)

(14b)

The second restriction is very stringent. In the case of a spherical cell with d/a  2 × 10³, as estimated for typical lymphocytes, we have k₊/A  D_o/a (Eq. 9b) and therefore we would need to have

(15)

i.e., D_i would have to be at least 250-fold lower than D_o for the extracellular matrix to make a difference of more than 10% in the effective rate coefficients.

View larger version (1K): [in this window] [in a new window]   FIGURE 3   Contour curves for the ratio  (Eq. 13) of the effective association rate coefficients calculated with and without accounting for the binding layer. The effective rate coefficients are given by Eq. 2. The thickness factor T() corrects for the binding layer. When the binding layer is replaced by a 2D surface, T() = 1. The ratio is considered as a function of  =  and  = DiA/(dk+). The contours show pairs of ,  values for which the ratio of effective rate coefficients is 0.95, 0.9, 0.8, 0.7, or 0.6, corresponding to maximum percentage differences from 5% to 40%.

For the Biacore, in the case where k₊ = k₊ (Eq. 11b), Eq. 14b becomes

(16)

If the layer does not retard diffusion significantly (i.e., if D_i  D_o) and does not reduce significantly the volume fraction of the layer available for reaction (i.e., if   1), then, from Eq. 16, a necessary condition for the layer to make a difference of 10% or more in the effective rate coefficients is that the diffusion coefficient for the ligand outside the layer, D_o, satisfy

(17)

In the Biacore 2000 (Biacore AB), h = 0.005 cm and l = 0.24 cm. For the most commonly used sensor chip, CM5, d = 10⁵ cm (Karlsson et al., 1994). Based on these parameters, and using a flow rate v_c = 10 cm/s, common in Biacore experiments, we find that we would need D_o < 1.4 × 10¹⁰ cm²/s for Eq. 17 to hold. But typical diffusion coefficients for the biological ligands of interest in Biacore experiments are in the range 10⁷  10⁶ cm²/s. Therefore, transport in the layer must be slowed significantly (D_i D_o), or the available volume in the layer reduced significantly (  1), for the binding layer to matter in data analysis.

How much lower than D_o does D_i have to be for there to be a 10% difference between effective rate coefficients in models that treat the dextran layer explicitly and those that assume that binding occurs on a surface? From Eq. 16, using the same parameters as above, we find a necessary condition to be

(18a)

(18b)

In the first case, transport has to be at least 20 times slower in the layer than outside. In the second case, which would correspond to very large asymmetric ligands (see Tanford, 1961, Table 21-1, p. 358), transport has to be slowed 10-fold for the layer to matter.

Recently, the height of the flow chamber (h) has been reduced to speed transport to the sensor surface (Rich and Myszka, 2000). In the Biacore 3000 (Biacore AB), h = 2 × 10³ cm. This expands only slightly the range of parameter values for which the layer is expected to produce observable effects on the binding kinetics.

Proposal for quantifying ligand transport in a dextran layer

Under what experimental conditions is transport in the dextran layer reduced to the extent where we have concluded that the layer must be taken into account to estimate reaction rate constants accurately (Eqs. 14a and 14b)? Existing theory does not provide a reliable way to estimate the quantity D_i that characterizes transport in the dextran layer (Yarmush et al., 1996). Here we propose a way to use experiments like those of Karlsson and Fält (1997), comparing ligand-receptor binding kinetics on sensor chips with and without a dextran layer, in conjunction with the effective rate coefficients we have derived, to estimate D_i in prototypic cases. The results can be used to test models for transport of macromolecules in a polymeric matrix and to provide reasonable estimates of D_i for Biacore experiments similar to the prototypes.

First, we consider the implications of our theory regarding the experimental system investigated by Karlsson and Fält (1997), where the layer did not make a measurable difference. This means that at least one of the conditions for the layer to matter (Eqs. 14a and 14b) does not hold for this system. For the parameters characterizing these experiments, Eq. 14a turns out to be the more stringent condition. The ligand is a 24-kDa antigen (p24) and the immobilized receptor an anti-p24 antibody. The reaction is relatively slow, with estimated binding rate constant k_a = 2.2 × 10⁵ M¹s¹. We estimate that the density of active, accessible receptor binding sites, Rd in Eq. 14a, is equal to the maximal density of bound ligands, approximately 3.85 × 10⁵ M¹ (calculated from the molecular weight of the ligand and the maximal Biacore signal of approximately 100 RU in these experiments). Then the condition given by Eq. 14a is D_i < 3.4 × 10⁹ cm²/s. That is, D_i would have to be 300 times lower than the diffusion coefficient expected for the ligand outside the layer, D_o = 10⁶ cm²/s, for the layer to alter effective rate coefficients by 10% or more. The fact that the layer does not matter in the experiments of Karlsson and Fält tells us that transport in the layer is not slowed 300-fold relative to transport outside the layer. For ligand/receptor pairs that react with a faster forward rate constant, we would not require such an extreme reduction of transport within the layer to see an effect of the layer. If k_a = 10⁷ M¹s¹, Eq. 14b (or equivalently Eq. 18a) is a more stringent condition than Eq. 14a. In this case, we should see a difference in binding to sensor surfaces with and without a dextran layer if the layer reduces ligand transport 20-fold. Despite this, Parsons and Stockley (1997) also found that the presence of the dextran layer had no effect on the binding of a small protein (24 kDa) to DNA on sensor chip surfaces with no dextran layer, a dextran layer of height 30 nm, or a layer of height 100 nm, even though k_a = 3 × 10⁶ M¹s¹ and k_a = 8 × 10⁶ M¹s¹ for the two DNAs studied.

Although the exception rather than the rule, we anticipate that examples will be found where ligand-receptor binding exhibits different kinetics for sensor chips with and without dextran layers. In particular, in experiments with large ligands or where high receptor densities are desirable to increase the Biacore signal, transport of the ligand in the layer may be affected to the extent where we expect the layer to make a difference. In such a case, an estimate of D_i is needed for the accurate estimation of intrinsic binding and dissociation rate constants (see Eqs. 2 and 5b). We propose to obtain estimates for prototype systems where an adequate signal can be measured using a nonlayered chip, so that data can be obtained both in the presence and absence of a dextran layer. From experiments using the nonlayered chip, one would determine the binding and dissociation rate constants, k_a and k_d, and the transport-limited forward rate constant, k₊, by performing a global fit of a series of kinetic curves determined for different ligand concentrations (Myszka, 1997; Myszka et al., 1997; Myszka and Morton, 1998). With these parameters known, the thickness factor T would be determined from the binding kinetics in the presence of a layer. From the value of T(), one obtains the value of  for the layer, and, from , the product D_i (see Eqs. 3 and 5b). Estimates of D_i from the prototype systems can be applied to systems with similar ligand size, ligand geometry, receptor size, and receptor density, but where, as is the common situation, the extent of ligand/receptor binding at a receptor density achievable on a nonlayered chip gives too low a signal for reliable estimation of binding parameters.

A possible complication to this approach arises because the Biacore weights the mass of a bound ligand by a factor that decays exponentially with the distance, y, the ligand is from the sensor surface. Because the decay length of the evanescent wave is 160 nm (Karlsson and Fält, 1997), a bound ligand 100 nm (d for a CM5 chip) from the sensor surface would give a signal only 0.65 that of a ligand bound at the surface. If binding is uniform in the y direction (T()  1) this has no effect on the interpretation of data. However, when the binding is nonuniform and ligands bind first to sites near the top of the layer and then, after these sites are filled, to sites deeper into the layer, the amount bound at the beginning of a binding experiment will be underestimated compared with the amount bound later in the experiment. As a result, under these conditions, the binding kinetics as measured by Biacore may not reflect the true binding kinetics. To get around this potential problem, a chip with a shorter dextran layer could be used to determine D_i. For example, in the experiments of Parsons and Stockley (1997) a chip with d = 30 nm was used. A signal from a bound ligand at the top of this layer is 0.88 that of a ligand bound at the bottom.

Potential effects of slow transport of ligand in a polymeric layer

Figure 4 compares predictions of the time course of binding in a Biacore experiment, based on Eq. 1, under different assumptions regarding ligand transport. The reaction parameters are the same for all simulations. For these parameters, half of the receptors are bound at equilibrium. To compare any two predicted time courses quantitatively, we express the largest absolute difference between fractions of receptors bound as a percentage of 0.5, the maximum fraction of receptors bound in all cases. Curve a is obtained by using the intrinsic rate constants k_a and k_d instead of effective rate coefficients in Eq. 1, thereby ignoring transport completely, inside and outside the layer. Curves b, c, and d, generated using effective rate coefficients given by Eqs. 2 and 11b, predict binding when the ligand's diffusion coefficient outside the layer is D_o = 10⁶ cm²/s. The flow rate and geometric parameters, given in the figure caption, are the same for cases b, c, and d. Curve b is obtained by setting T() = 1 in Eq. 2, thereby ignoring the layer and assuming that receptors are on a 2D surface. Essentially the same curve (with values differing by at most 0.3%) is obtained using the correction for the layer but assuming D_i, which determines transport in the layer, is the same as D_o. For curve c, D_i = 5 × 10⁸ cm²/s, 20-fold lower than D_o. The maximum difference from the prediction when the binding layer is ignored (or where D_i = D_o) is 5% in this case. Curve d is generated using D_i = 10⁸ cm²/s, a factor of 100 lower than D_o. In this case, the maximum difference in predicted binding from the case where the binding layer is ignored is 16%.

View larger version (0K): [in this window] [in a new window]   FIGURE 4   Predictions of Biacore binding, based on an effective rate model (Eq. 1), under different assumptions regarding the effective rate coefficients k and k. Curve a is generated using intrinsic binding and dissociation rate constants in Eq. 1 and therefore ignoring all transport effects. Curves b, c, and d use effective association rate coefficients given by Eq. 2, with transport-limited forward rate constant given by Eq. 11b. For all three curves, the diffusion coefficient for the ligand outside the layer is taken to be Do = 106 cm2/s, but different assumptions are used regarding transport in the dextran layer. For curve b, the dextran layer is ignored, i.e., binding occurs on a 2D surface (Eq. 2 with T() = 1). If we use the correction for the layer but assume that the dextran layer does not slow diffusion or reduce the effective volume in which the ligand diffuses, i.e., if Di = Do, then the predicted binding curve is essentially identical to curve b. In this case, the layer can be ignored. Curves c and d show the predicted binding if (c) Di = 5 × 108 cm2/s or (d) Di = 108 cm2/s. The other parameters used in the simulations are in the typical range for experiments done on a Biacore 2000. The dimensions are standard for a flow cell with a CM5 chip (length l = 0.24 cm, height h = 0.005 cm, height of layer d = 105 cm). We took the maximum flow rate vc = 10 cm/s, the intrinsic rate constants ka = 0.01 nM1s1 and kd = 0.01 s1, and concentrations of ligand and receptor CT = 1 nM, RT = 1 nM cm.

The effective rate coefficients, with and without the correction for the layer, differ by 10% when D_i = 5 × 10⁸ cm²/s = D_o/20 (case c) and 25% when D_i = 10⁸ cm²/s = D_o/100 (case d). These differences are greater than the corresponding differences in predicted binding (5% in case c and 16% in case d). The thickness factor T() shows an even greater effect of ignoring the correction for the dextran layer in cases c and d. The value is T() = 0.63 for case c and T() = 0.32 for case d. Neglecting the layer would result in significant underestimation of binding and dissociation rate constants in these casesan estimate of 0.63k_a in case c or 0.32k_a in case d, instead of the true k_a.

	DISCUSSION

TOP ABSTRACT INTRODUCTION RESULTS DISCUSSION APPENDIX REFERENCES

For a 3D binding layer above a surface to matter in the estimation of reaction rate constants, we have shown that the average distance the ligand travels in the layer before binding occurs (i.e., the mean free path) must be comparable to, or shorter than, the height of the layer. Association and dissociation rate constants determined by methods that ignore the binding layer underestimate the true rates by a "thickness factor" T that depends on the ratio of the height of the layer to the mean free path of the ligand. The layer looks "thin" to the ligand if the mean free path is long relative to the height of the layer. In this case, T  1. The layer is "thick" if the ligand traverses only a small fraction of the layer before binding to a receptor (T 1). In this case, the layer may influence binding kinetics, and the interpretation of binding data, significantly. However, even under conditions where T 1, if transport of the ligand outside the layer is so slow relative to reaction that the system is in the transport limit, the effective association and dissociation rate coefficients become independent of the thickness factor T, and of the intrinsic reaction rate constants.

Therefore, for a reaction layer above a surface to influence binding kinetics observably, two conditions must be met. Transport of the ligand outside the layer must be fast enough relative to binding that the reaction is not transport limited. Transport in the layer must be slow enough relative to binding that gradients arise in the concentrations of free and bound ligand.

One of the conditions imposes a relation between diffusion coefficients for the ligand inside and outside the layer (see Eq. 14b). We have shown that, when ligands bind to sites in the glycocalyx of a spherical cell, diffusion of the ligand must be two orders of magnitude slower inside the glycocalyx than outside for the layer to affect the binding kinetics. For the Biacore biosensor, transport must also be substantially slower (at least an order of magnitude) inside the dextran layer than outside, for the layer to matter.

Experiments comparing ligand-receptor binding on sensor chips with and without dextran layers have failed to detect differences in the kinetics of binding (Karlsson and Fält, 1997; Parsons and Stockley, 1997). The experiments suggest that, under typical Biacore conditions, diffusion of the ligand in the layer is not so slow that the layer influences kinetic data observably. However Schuck (1996) has argued that there are plausible experimental conditions under which transport in the dextran layer is slow enough to affect Biacore data and data analysis. We have proposed a method for estimating the effective diffusion coefficient of a ligand in a dextran layer. Estimates from representative experimental systems can be used to calculate the thickness factor T in related systems, under conditions where the dextran layer is expected to matter. In such cases, the thickness factor is needed to estimate kinetic parameters accurately. Estimates of effective diffusion coefficients for ligands in a dextran layer can also be used to test models that predict macromolecular transport in a polymeric matrix.