Simultaneous Measurement of Anisotropic Solute Diffusivity and Binding Reaction Rates in Biological Tissues by FRAP

Abstract

Several solutes (e.g., growth factors, cationic solutes, etc.) can reversibly bind to the extracellular matrix (ECM) of biological tissues. Binding interactions have significant implications on transport of such solutes through the ECM. In order to fully delineate transport phenomena in biological tissues, knowledge of binding kinetics is crucial. In this study, a new method for the simultaneous determination of solute anisotropic diffusivity and binding reaction rates was presented. The new technique was solely based on Fourier analysis of fluorescence recovery after photobleaching (FRAP) images. Computer-simulated FRAP tests were used to assess the sensitivity and the robustness of the method to experimental parameters, such as anisotropic solute diffusivity and rates of binding reaction. The new method was applied to the determination of diffusivity and binding rates of 5-dodecanoylaminofluorescein (DAF) in bovine coccygeal annulus fibrosus (AF). Our findings indicate that DAF reversibly binds to the ECM of AF. In addition, it was found that DAF diffusion in AF is anisotropic. The results were in agreement with those reported in previous studies. This study provides a new tool for the simultaneous determination of solute anisotropic diffusion tensor and rates of binding reaction that can be used to investigate diffusive–reactive transport in biological tissues and tissue engineered constructs.

Appendix

Dimensionless Analysis of the Field Equations for Diffusive–reactive Transport

Now, for the sake of simplicity, the ideal case of isotropic diffusive–reactive fluorescence recovery is considered. The extension of this analysis to the more general case of anisotropic diffusion can be easily derived.

In the case of isotropic diffusive–reactive transport, recovery of fluorescence intensity within the images is controlled by four parameters: the diffusion coefficient D, the pseudo-rate of binding association $k_{a}^{*}$, the rate of binding dissociation k _d , and the initial diameter of the bleach spot (d). Let Da and R be two dimensionless numbers defined as42

$$ Da = {\frac{{d^{2} k_{a}^{*} }}{D}}, $$

(4a)

$$ R = {\frac{{k_{a}^{*} }}{{k_{d} }}}. $$

(4b)

The number Da represents the ratio of the diffusion time to the characteristic time of binding association, and, therefore, has the physical meaning of the Damkhöler number.36 The number R represents the ratio of the characteristic time of binding dissociation to that of binding association. Let d be the characteristic length and τ the characteristic time of the diffusive–reactive process, defined as

$$ \tau = {\frac{{d^{2} }}{D}}\left( {1 + R} \right). $$

(A1)

Equations (3a–b) can be rewritten in the following dimensionless form:

$$ {\frac{{\partial \bar{c}^{f} }}{{\partial \hat{t}}}} = \left( {1 + R} \right) \cdot \hat{\nabla }^{2} \bar{c}^{f} - {\frac{{\partial \bar{c}^{b} }}{{\partial \hat{t}}}}, $$

(A2a)

$$ {\frac{{\partial \bar{c}^{b} }}{{\partial \hat{t}}}} = (1 + R)Da \cdot \bar{c}^{f} - \left( {1 + \frac{1}{R}} \right)Da \cdot \bar{c}^{b} , $$

(A2b)

where $\bar{c}^{f}$ and $\bar{c}^{b}$ are the concentrations of free and bound solutes normalized by the total pre-bleach concentration of fluorescent solute; $\hat{t}$(defined as $\hat{t}$ = t/τ) and $\hat{\nabla }^{2}$ (defined as $\hat{\nabla }^{2}$ = $\nabla^{2}$·d ²) are the dimensionless time and the dimensionless Laplacian operator, respectively. Eqs. (A2a–b) indicate that the two dimensionless numbers, Da and R, govern the diffusive–reactive fluorescence recovery.

Solution of the Diffusion–Reaction Equations in the Fourier Space

The system of Eqs. (3a–b) is transformed in the 2D Fourier space defined by the dimensionless frequencies (u, v):

$$ {\frac{{dC_{f} (u,v,t)}}{dt}} = - 4\pi^{2} {\frac{{(u^{2} + v^{2} )}}{{L^{2} }}}D(\xi )C_{f} (u,v,t) - k_{a}^{*} C_{f} (u,v,t) + k_{d} C_{b} (u,v,t) $$

(A3a)

$$ {\frac{{dC_{b} (u,v,t)}}{dt}} = k_{a}^{*} C_{f} (u,v,t) - k_{d} C_{b} (u,v,t) $$

(A3b)

where C _f and C _b are the 2D Fourier transforms of the concentrations of free and bound solutes, normalized with respect to the pre-bleach total concentration of solute (free + bound); L is the dimension of the video-FRAP image, corresponding to eight times the value of the initial diameter of the bleach spot (d) used in the experiments. The function D(ξ), defined as47

$$ D(\xi ) = D_{xx} \cos^{2} \xi + 2D_{xy} \cos \xi \sin \xi + D_{yy} \sin^{2} \xi , $$

(A4)

with

$$ \xi = \tan^{ - 1} \frac{v}{u}, $$

(A5)

includes the components of the 2D diffusion tensor D (i.e., D _xx , D _xy , and D _yy ). Assuming that, before bleaching (t = 0), the system is at equilibrium (both dC _f /dt and dC _b /dt equal zero), from Eqs. (A3b) we have

$$ k_{a}^{*} C_{f} (u,v,0) = k_{d} C_{b} (u,v,0). $$

(A6)

Since C _f (u, v, 0) + C _b (u, v, 0) = 1, it follows that

$$ C_{f} (u,v,0) = {\frac{{k_{d} }}{{k_{a}^{*} + k_{d} }}}, $$

(A7a)

$$ C_{b} (u,v,0) = {\frac{{k_{a}^{*} }}{{k_{a}^{*} + k_{d} }}}. $$

(A7b)

Equations (A7a–b) represent the initial conditions for Eqs. (3a–b). Note that, in the Fourier space, the intensity of the fluorescence emission is proportional to the total concentration of the fluorescent solute (C = C _f  + C _b ) according to the following relationship2:

$$ {\frac{I(u,v,t)}{I(u,v,0)}} = {\frac{C(u,v,t)}{C(u,v,0)}}. $$

(A8)

Combining the solution of Eqs. (3a–b) with (A8), it follows that

$$ {\frac{I(u,v,t)}{I(u,v,0)}} = Ae^{{\lambda_{1} t}} + Be^{{\lambda_{2} t}} , $$

(A9)

where

$$ A = {\frac{{(\lambda_{2} - a - c)(\lambda_{1} - a + b)}}{{(b + c)(\lambda_{2} - \lambda_{1} )}}} $$

(A10a)

$$ B = {\frac{{(a + c - \lambda_{1} )(\lambda_{2} - a + b)}}{{(b + c)(\lambda_{2} - \lambda_{1} )}}} $$

(A10b)

$$ \lambda_{1} = (a + d - \sqrt {a^{2} + d^{2} - 2ad + 4bc} )\Big/2 $$

(A10c)

$$ \lambda_{2} = (a + d + \sqrt {a^{2} + d^{2} - 2ad + 4bc} )\Big/2 $$

(A10d)

$$ a = - 4\pi^{2} {\frac{{(u^{2} + v^{2} )}}{{L^{2} }}}D(\xi ) - k_{a}^{*} $$

(A10e)

$$ b = k_{d} $$

(A10f)

$$ c = k_{a}^{*} $$

(A10g)

$$ d = - k_{d} $$

(A10h)

Curve-fitting the 2D Fourier transform of the fluorescence emission of a video-FRAP image series with (A9) yields the binding rates $k_{a}^{*}$ and k _d , together with D(ξ).

Determination of the Principal Components of the 2D Anisotropic Diffusion Tensor

Let (x, y) stand for the fixed coordinate system of the microscope, and (x′, y′) be the principal directions of D (i.e., material coordinate system), see Fig. 7a. The components of D in the fixed coordinate system (i.e., D _xx , D _xy , and D _yy ) are calculated by curve-fitting the experimental values of D(ξ), estimated at points (u, v) describing an arc of circumference (u ² + v ² = constant) in the Fourier space spanning from 0 to π, with Eq. (A4) (Figs. 7b–7c). The components D _xx , D _xy , and D _yy are related to the principal components of D ($D^{\prime}_{xx}$ and $D^{\prime}_{yy}$) by the following relationship:

Figure 7

(a) The orientation of the material coordinate system (x′, y′) with respect to the fixed coordinate system (x, y) is θ. (b) D(ξ) is defined over an arc of circumference (u ² + v ² = constant) in the Fourier space spanning from 0 to π. (c) Representative curve-fitting of the values of D(ξ) (open circles) with Eq. (A4) (solid line) to yield D _xx , D _xy , and D _yy . Note that D(ξ) was determined by curve-fitting of computer-simulated FRAP data (D _xx  = 5 × 10⁻⁷ cm² s⁻¹, D _xy  = 1.667 × 10⁻⁷ cm² s⁻¹, D _yy  = 5 × 10⁻⁷ cm² s⁻¹, Da = 500, and R = 1) with Eq. (6)

$$ \left[ {\begin{array}{*{20}c} {D_{xx} } & {D_{xy} } \\ {D_{xy} } & {D_{yy} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\cos \theta } & { - \sin \theta } \\ {\sin \theta } & {\cos \theta } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {D^{\prime}_{xx} } & 0 \\ 0 & {D^{\prime}_{yy} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\cos \theta } & {\sin \theta } \\ { - \sin \theta } & {\cos \theta } \\ \end{array} } \right], $$

(A11)

where θ is the orientation of the principal directions of D with respect to (x,y) (Fig. 7a). It follows that

$$ D^{\prime}_{xx} = {\frac{{D_{xx} + D_{yy} + \sqrt {4D_{xy}^{2} + (D_{xx} - D_{yy} ){}^{2}} }}{2}}, $$

(A12a)

$$ D^{\prime}_{yy} = {\frac{{D_{xx} + D_{yy} - \sqrt {4D_{xy}^{2} + (D_{xx} - D_{yy} ){}^{2}} }}{2}}, $$

(A12b)

$$ \tan(2\theta ) = {\frac{{2D_{xy} }}{{D_{xx} - D_{yy} }}}. $$

(A12c)