Are Assumptions about the Model Type Necessary in Reaction-Diffusion Modeling? A FRAP Application

Abstract

At present, fluorescence recovery after photobleaching (FRAP) data are interpreted using various types of reaction-diffusion (RD) models: the model type is usually fixed first, and corresponding model parameters are inferred subsequently. In this article, we describe what we believe to be a novel approach for RD modeling without using any assumptions of model type or parameters. To the best of our knowledge, this is the first attempt to address both model-type and parameter uncertainties in inverting FRAP data. We start from the most general RD model, which accounts for a flexible number of molecular fractions, all mobile, with different diffusion coefficients. The maximal number of possible binding partners is identified and optimal parameter sets for these models are determined in a global search of the parameter-space using the Simulated Annealing strategy. The numerical performance of the described techniques was assessed using artificial and experimental FRAP data. Our general RD model outperformed the standard RD models used previously in modeling FRAP measurements and showed that intracellular molecular mobility can only be described adequately by allowing for multiple RD processes. Therefore, it is important to search not only for the optimal parameter set but also for the optimal model type.

Introduction

In recent years, interest in using noninvasive methods to observe and analyze intracellular molecular mobility has increased dramatically (1–7). One technique widely used for this purpose is fluorescence recovery after photobleaching (FRAP) (8–16).

To study the behavior of a fluorescent molecule in a live cell by FRAP, a specific region (i.e., bleaching spot) in the cell is defined and exposed to an intense excitation pulse sufficient to irreversibly inactivate fluorescence emission (Fig. 1). The recovery of fluorescence reflects the movement of new fluorescent molecules into the photobleached region. This increase in fluorescence over time is described by a so-called recovery curve, which can be used to extract information on mobility and binding of the monitored fluorescent molecules.

Figure 1.

Schematic representation of FRAP. (A–D) A FRAP experiment is based on the bleaching of fluorescent molecules (gray area) in a predefined region of interest (ROI, black circle) and the subsequent recovery of fluorescence intensity in this region in a predefined region of interest.

Several reaction-diffusion models have been suggested for the analysis of such recovery curves. However, diffusion coefficients and reaction parameters can be deduced only if an analytical or numerical solution for these models can be determined. Until now, it was possible to calculate an analytical solution only by simplifying the model using assumptions about particular parameter values (8,9,17–26). These assumptions are reflected in various model types, such as the reaction-dominant model, where the diffusion coefficients of the acting molecules are fixed, or the diffusion-dominant model, where all reaction processes are neglected. As implied by the model names, particular mechanisms are supposed for the underlying biological process. To eliminate the necessity for these restrictions, we herein introduce the solution of the general reaction-diffusion model, which includes an unconfined number of reacting and diffusing compounds.

A generalization is attended by an increasing number of model parameters, which raises the concern of overfitting. To address this potential problem, we adapted a method introduced by de Prony (27–29) to include a preprocessing step in which the maximal number of parameters that can be fitted reliably is deduced.

After this preprocessing, the parameter values of diffusion and reaction still have to be determined. Hitherto, local search algorithms like the Levenberg-Marquardt algorithm were used for inversion. However, a major drawback of such a local approach is the sensitivity to the initial parameter settings (17,30,31). Therefore, a global search strategy that is robust against initial settings is preferable. As an alternative to the restricted local search algorithms, we used Simulated Annealing (SA) as one of the possible global approaches to infer unbiased and more reliable parameter sets (32).

In summary, our new approach consists of a multiple-reaction-diffusion model with predetermination of the possible number of parameters, which then are inverted by the global search algorithm SA. To demonstrate the strength of our new approach, we applied our generalized reaction-diffusion model to artificial and real FRAP data sets and compared the results with those from previously used models. To our knowledge, this is the first attempt to use an analytical approach to address both model type and parameter uncertainties in inverting FRAP data.

Materials and Methods

Analysis of FRAP data: A theoretical approach

In this section, we first introduce the system of reaction-diffusion equations used to describe motion as well as chemical conversion, specify the general initial and boundary conditions to describe FRAP experiments, and review known solutions of special cases (17).

General model

The molecules may undergo S different binding reactions with other substances. Each reaction is described by

$$F+V_i\underset{k_{{\text{off}}_i}}{\overset{k_{{\text{on}}_i}}{\rightleftharpoons }}{B}_i,i=1\dots S.$$ (1)

The unbound (free) fraction of molecules, F, binds with V_i vacant binding sites to form the bound fraction of molecules, B_i. k_oni and k_offi are the corresponding association and dissociation rates in mol s^–1 and s^–1, respectively.

The most general set of reaction-diffusion equations that describe one unbound fraction and S bound fractions is

$$\frac{\partial c_F}{\partial t}=D_F{\nabla }^2{c}_F-\sum _{i=1}^{\text{S}}\left(k_{{\text{on}}_i}{c}_F{c}_{V_i}-k_{{\text{off}}_i}{c}_{B_i}\right)$$ (2a)

$$\frac{\partial c_{V_i}}{\partial t}=D_{V_i}{\nabla }^2{c}_{V_i}-\sum _{i=1}^{\text{S}}\left(k_{{\text{on}}_{\text{i}}}{c}_F{c}_{V_i}-k_{{\text{off}}_i}{c}_{B_i}\right)$$ (2b)

$$\frac{\partial c_{B_i}}{\partial t}=D_{B_i}{\nabla }^2{c}_{B_i}+k_{{\text{on}}_i}{c}_F{c}_{V_i}-k_{{\text{off}}_i}{c}_{B_i},i=\mathrm{1}\dots \mathit{S},$$ (2c)

where ∇² is the Laplacian operator, c represents the concentration of a given molecule type at a certain time and place, and D is the diffusion coefficient. Index i denotes the reaction as it is described in Eq. 1.

Equation 2 can be further simplified by assuming that the fluorescent molecules are in equilibrium before photobleaching. The bleaching procedure changes the number of fluorescent molecules (c_F, c_Bi), whereas the fraction of the nonfluorescent binding sites, $c_{V_{\text{i}}}$, is constant over time. Therefore, we can eliminate Eq. 2b. The variable V_i in the remaining equations is constant over time. Thus, we define a new variable called the pseudo association rate, $k_{{\text{on}}_{\text{i}}}^{\ast }$, as

$$k_{{\text{on}}_i}^{\ast }=k_{{\text{on}}_i}{c}_{V_i}.$$ (3)

Thus, Eq. 2 reduces to

$$\frac{\partial c_F}{\partial t}=D_F{\nabla }^2{c}_F-\sum _{i=1}^{\text{S}}\left(k_{{\text{on}}_i}^{\ast }{c}_F-k_{{\text{off}}_i}{c}_{B_i}\right)$$ (4a)

$$\frac{\partial c_{B_i}}{\partial t}=D_{B_i}{\nabla }^2{c}_{B_i}+k_{{\text{on}}_i}^{\ast }{c}_F-k_{{\text{off}}_i}{c}_{B_i},i=1\dots S.$$ (4b)

As mentioned above, the system (Eq. 2) is in steady state before bleaching.

$$\frac{\partial c_F}{\partial t}=\frac{\partial c_{B_i}}{\partial t}=0\text{if}t=0.$$ (5)

Since the total mass has to be conserved, it follows that

$$c_F+\sum _{i=1}^S{c}_{B_i}=1.$$ (6)

Hence, the equilibrium concentrations of free fraction, F_eq, and bound fraction, B_eqi, are

$$\frac{\partial c_{B_i}}{\partial t}=0\Rightarrow B_{{\text{eq}}_i}=\frac{k_{{\text{on}}_i}^{\ast }}{k_{{\text{off}}_i}}{F}_{\text{eq}}$$ (7a)

$$c_{\text{F}}+\sum _{i=1}^S{c}_{B_i}=1\Rightarrow F_{\text{eq}}=\frac{1}{1+\sum _{j=1}^S\frac{k_{{\text{on}}_j}^{\ast }}{k_{{\text{off}}_j}}}.$$ (7b)

To characterize the initial and boundary conditions, we divide the space into two regions: 1), the bleaching spot area, a; and 2), the region outside the bleaching spot, $\overline{a}$. The initial conditions of concentrations inside the bleaching spot, cⁱⁿ, are zero, whereas initial conditions of concentrations outside this region, c^out, are at equilibrium:

$$c_F^{\mathrm{in}}=0\mathrm{if}t=0$$ (8a)

$$c_F^{\mathrm{out}}=F_{\mathrm{eq}}\mathrm{if}t=0$$ (8b)

$$c_{B_i}^{\mathrm{in}}=0\mathrm{if}t=0$$ (8c)

$$c_{B_i}^{\mathrm{out}}=B_{{\mathrm{eq}}_i}\mathrm{if}t=0.$$ (8d)

A more general assumption is a constant initial value, θ, inside the bleaching spot whereby Eq. 8 converts to

$$c_F^{\mathrm{in}}=\theta \times F_{\mathrm{eq}}\mathrm{if}t=0$$ (9a)

$$c_F^{\mathrm{out}}=F_{\mathrm{eq}}\mathrm{if}t=0$$ (9b)

$$c_{B_i}^{\mathrm{in}}=\theta \times B_{{\mathrm{eq}}_i}\mathrm{if}t=0$$ (9c)

$$c_{B_i}^{\mathrm{out}}=B_{{\mathrm{eq}}_i}\mathrm{if}t=0.$$ (9d)

For simplicity, we assume that the bleaching spot is a radial area with radius R, which yields the boundary conditions

$$\frac{\partial c_F^{\text{in}}}{\partial r}=0\text{if}r=0$$ (10a)

$$\frac{\partial c_{B_i}^{\text{in}}}{\partial r}=0i=1\dots S\text{if}r=0$$ (10b)

$$\frac{\partial c_F^{\text{out}}}{\partial r}=0\text{if}r\to \infty$$ (10c)

$$\frac{\partial c_{B_i}^{\text{out}}}{\partial r}=0i=1\dots S\text{if}r\to \infty$$ (10d)

$$c_F^{\text{in}}=c_F^{\text{out}}\text{if}r=R$$ (10e)

$$c_{B_i}^{\text{in}}=c_{B_i}^{\text{out}}i=1\dots S\text{if}r=R$$ (10f)

$$\frac{\partial c_F^{\text{in}}}{\partial r}=\frac{\partial c_F^{\text{out}}}{\partial r}\text{if}r=R$$ (10g)

$$\frac{\partial c_{B_i}^{\text{in}}}{\partial r}=\frac{\partial c_{B_i}^{\text{out}}}{\partial r}i=1\dots S\text{if}r=R.$$ (10h)

Unlike the reactions (Eq. 1) where all bound states B_i depend directly on the free fraction F, a more realistic scenario is to model reactions as a chain

$$B_{i-1}+V_{i-1}\underset{k_{{\text{off}}_i}}{\overset{k_{{\text{on}}_i}}{\rightleftharpoons }}{B}_i,i=1\dots S,$$ (11)

where B₀ represents the free molecular fraction, F.

This leads to modified differential equations and equilibrium concentrations, respectively.

$$B_{{\text{eq}}_0}=F_{\text{eq}}=\frac{1}{1+\sum _{i=1}^{\text{S}}\left(\prod _{j=1}^i\frac{k_{{\text{on}}_j^{\ast }}}{k_{{\text{off}}_j}}\right)}$$ (12a)

$$B_{{\text{eq}}_i}=\left(\prod _{j=1}^i\frac{k_{{\text{on}}_j}^{\ast }}{k_{{\text{off}}_j}}\right)F_{\text{eq}},i=1\dots S.$$ (12b)

The initial and boundary conditions (Eqs. 8–10) are equal for both reaction models.

Review of solutions for special cases

Different special cases for Eq. 4 with Eq. 8 and Eq. 10 have already been solved and published (17). Since we need these results for comparison, we briefly present the analytical results in the following sections.

Reaction Dominant Model

The first simplified scenario assumes that diffusion is very fast compared to reaction. Thus, diffusion can be neglected, and models that employ this mechanism are called reaction-dominant. For the analysis of FRAP experiments, the solution of such a reaction-dominant model was introduced by Sprague et al. (17). These authors assumed a zero initial condition (Eq. 8) and treated models with one- and two-binding-state solutions. We generalize their approach to get the reaction-dominant solution with S binding states.

The S reaction equations,

$$F+V_i\underset{k_{{\text{off}}_i}}{\overset{k_{{\text{on}}_i}^{\ast }}{\rightleftharpoons }}{B}_i,i=1\dots S,$$ (13)

yield the relationship

$$\text{frap}\left(t\right)=1-\sum _{i=1}^S{B}_{{\text{eq}}_i}{e}^{-k_{{\text{off}}_i}t},$$ (14)

where frap(t) represents the recovery curve of the FRAP experiment and the equilibrium concentration of the bound molecular fraction, B_eqi, is given by Eq. 7.

Mueller et al. (25) solved this model under the condition of a constant initial value, θ (Eq. 9). The general solution with S binding states is given by

$$\text{frap}\left(t\right)=1-(1-\theta )\times \sum _{i=1}^{\text{S}}{B}_{{\text{eq}}_i}{e}^{-k_{{\text{off}}_i}t}.$$ (15)

Reaction Diffusion Model with Single Diffusion

The second scenario describes the case in which the free molecules, F, are moving with a diffusion coefficient, D_F, and the reaction products, B_i are immobile (D_Bi = 0).

$$\underset{D_{\text{F}}}{\underbrace{F}}+V_i\underset{k_{{\text{off}}_i}}{\overset{k_{{\text{on}}_i}^{\ast }}{\rightleftharpoons }}{B}_i,i=1\dots S.$$ (16)

The Laplace transformed solution $\widehat{\text{frap}}\left(s\right)$ with an assumed zero initial condition (Eq. 8) is given in Sprague et al. (17). The authors derive a solution for one- and two-binding-state models, which we extend to S binding states:

$$\widehat{\text{frap}}\left(s\right)=\frac{1}{s}-\frac{F_{\text{eq}}}{s}\left(1-2{\mathbf{K}}_{\mathbf{1}}\left(qR\right){\mathbf{I}}_{\mathbf{1}}\left(qR\right)\right)\times \left(1+\sum _{i=1}^{\text{S}}\frac{k_{{\text{on}}_i}}{s+k_{{\text{off}}_i}}\right)-\sum _{i=1}^{\text{S}}\frac{B_{{\text{eq}}_i}}{s+k_{{\text{off}}_i}},$$ (17)

with

$$q^2=\frac{s}{D_{\text{F}}}\left(1+\sum _{i=1}^{\text{S}}\frac{k_{{\text{on}}_i}}{s+k_{{\text{off}}_i}}\right),$$

where R represents the radius of the circular bleaching spot and I₁ and K₁ are modified Bessel functions of the first and second kind. The equilibrium concentrations of the free molecular fraction, F_eq, and the bound molecular fractions, B_eqi, are given by Eq. 7.

The corresponding solution with a constant initial condition was calculated using the strategy described by Sprague et al. (17) and is given by

$$\begin{array}{c}\widehat{\text{frap}}\left(s\right)=\frac{1}{s}-\sum _{i=1}^{\text{S}}\frac{(1-\theta )B_{{\text{eq}}_i}}{s+k_{{\text{off}}_i}}\\ -\frac{(1-\theta )F_{\text{eq}}}{s}\left(1-2K_1\left(qR\right)I_1\left(qR\right)\right)\times \left(1+\sum _{i=1}^{\text{S}}\frac{k_{{\text{on}}_i}}{s+k_{{\text{off}}_i}}\right),\end{array}$$ (18)

with

$$q^2=\frac{s}{D_{\text{F}}}\left(1+\sum _{i=1}^{\text{S}}\frac{k_{{\text{on}}_i}}{s+k_{{\text{off}}_i}}\right).$$

The averaged fluorescence intensity within the bleaching spot (Eqs. 17 and 18) is still in the Laplace transformed form. The Laplace transform is defined as

$$\widehat{\text{frap}}\left(s\right)={\int }_0^{\infty }{e}^{-st}\times \text{frap}\left(t\right)dt.$$ (19)

Equations 17 and 18 have to be inverted to real time t. An analytical back transform is not possible in closed form; therefore, the Stehfest algorithm was used for numerical inversion (33,34).

Derivation of the general solution: Reaction Diffusion Model with Multiple Diffusion

After solving some simplified cases we now describe the derivation of the general coupled reaction-diffusion equations (Eq. 4 with Eqs. 8 and 10). We call this model full reaction-diffusion with multiple diffusion. This model describes reactive coupled molecular fractions that could all be mobile. We assume that both the free fraction, F, and the S bound fractions B₁…B_S are moving diffusively. The motion is described by diffusion coefficients for the free and bound molecular fractions, D_F and $D_{B_1}\dots D_{B_S}$, respectively. The reactions are characterized by the association rates $k_{{\text{on}}_i}^{\ast }$ and the dissociation rates $k_{{\text{off}}_{\mathit{i}}}$ (i = 1, …, S). We assume a chain of reactions so that the free molecular fraction (F and B₀, respectively) converts only to B₁. Molecules of B₁ are able to dissociate to B₀ or associate to B₂.

$$\underset{D_{B_{\text{i}-1}}}{\underbrace{B_{i-1}}}+V_i\underset{k_{{\text{off}}_i}}{\overset{k_{{\text{on}}_i}^{\ast }}{\rightleftharpoons }}\underset{D_{B_i}}{\underbrace{B_i}},i=1\dots S.$$ (20)

First, we elucidate one-binding-state models and derive their solution. The detailed derivation is given in the Supporting Material. Second, we show that our solution is also applicable to the more general case of S binding states.

Semi-analytical recovery function (1-binding-state)

The general set of coupled reaction-diffusion equations to describe a one-binding-state model is

$$\frac{\partial c_{B_0}}{\partial t}=D_{B_0}{\nabla }^2{c}_{B_0}-k_{\text{on}}^{\ast }{c}_{B_0}+k_{\text{off}}{c}_{B_1}$$ (21a)

$$\frac{\partial c_{B_1}}{\partial t}=D_{B_1}{\nabla }^2{c}_{B_1}+k_{\text{on}}^{\ast }{c}_{B_0}-k_{\text{off}}{c}_{B_1},$$ (21b)

where B₀ represents the free molecular fraction, F.

The Laplace transformed average fluorescence intensity inside the bleaching spot is described by (for derivation, see Supporting Material)

$$\begin{array}{c}\widehat{\text{frap}}\left(s^{\ast }\right)=\frac{\theta }{s}+\frac{2{\mathbf{I}}_1\left(u\right){\mathbf{K}}_1\left(u\right)\left(K-s^{\ast }+p-1\right)\left(s^{\ast }-q\right)\left(1-\theta \right)}{\left(K-1\right)s^{\ast }\left(p-q\right)}\\ -\frac{2{\mathbf{I}}_1\left(v\right){\mathbf{K}}_1\left(v\right)\left(K-s^{\ast }+q-1\right)\left(s^{\ast }-p\right)\left(1-\theta \right)}{\left(K-1\right)s^{\ast }\left(p-q\right)}\end{array}$$ (22)

with

$$u=R\sqrt{\frac{k_{\text{on}}^{\ast }\times p}{D_F}}\text{and}v=R\sqrt{\frac{k_{\text{on}}^{\ast }\times q}{D_F}},$$ (23)

where R represents the radius of the circular bleaching spot, θ denotes the initial fluorescence value inside the bleaching spot, and I₁ and K₁ are modified Bessel functions of the first and second kind. The original time variable t^∗ = k_on × t changed to the Laplace variable s^∗ due to the transformation. For the description of variables p, q, D, and K, see Supporting Material. Since the solution is Laplace-transformed, we have to transform it back to frap(t^∗) using the Stehfest algorithm (33,34).

Generalized semi-analytical recovery function (S-binding-states)

In the next step, the solution has to be generalized to S binding states. The matrix notation of the Laplace transformed general differential equation set (Eq. 4) is given by

$$\left({\nabla }^2{\widehat{c}}_B\right)=\underset{:=A}{\underbrace{\left(\begin{array}{ccccc}\frac{s+k_{{\text{on}}_1}}{D_{B_0}}& -\frac{k_{{\text{off}}_1}}{D_{B_0}}& 0& \dots & 0\\ -\frac{k_{{\text{on}}_1}}{D_{B_1}}& \frac{s+k_{{\text{off}}_1}+k_{{\text{on}}_2}}{D_{B_1}}& -\frac{k_{{\text{off}}_2}}{D_{B_1}}& \dots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \dots & \frac{s+k_{{\text{off}}_{\text{S}}}}{D_{B_{\text{S}}}}\end{array}\right)}}\left({\widehat{c}}_B\right),$$ (24)

where $\left({\nabla }^2{\widehat{c}}_B\right)$ and $\left({\widehat{c}}_B\right)$ are vectors.

First, the eigenvalues of A have to be derived. Let ${\lambda }_{B_0}$, ${\lambda }_{B_1}$, …, ${\lambda }_{B_S}$ be the S + 1 eigenvalues of A and EV = {EV(${\lambda }_{B_0}$), EV(${\lambda }_{B_1}$), …, EV(${\lambda }_{B_S}$)} the matrix of their eigenvectors. The Laplace transformed solution of Eq. 4 inside the bleaching-spot area with constant initial values θ (Eq. 9) is given by

$$\left({\widehat{c}}_B^{\text{in}}\right)=\frac{\theta }{s}\left(B_{\text{eq}}\right)-EV\left(\begin{array}{c}{\alpha }_0\times {\mathbf{I}}_{\mathbf{0}}\left(r\sqrt{{\lambda }_{B_0}}\right)+{\beta }_0\times {\mathbf{K}}_{\mathbf{0}}\left(r\sqrt{{\lambda }_{B_0}}\right)\\ {\alpha }_1\times {\mathbf{I}}_{\mathbf{0}}\left(r\sqrt{{\lambda }_{B_1}}\right)+{\beta }_1\times {\mathbf{K}}_{\mathbf{0}}\left(r\sqrt{{\lambda }_{B_1}}\right)\\ ⋮\\ {\alpha }_{\text{S}}\times {\mathbf{I}}_{\mathbf{0}}\left(r\sqrt{{\lambda }_{B_S}}\right)+{\beta }_S\times {\mathbf{K}}_{\mathbf{0}}\left(r\sqrt{{\lambda }_{B_S}}\right)\end{array}\right),$$ (25)

where $\left({\widehat{c}}_B^{\text{in}}\right)$ and (B_eq) are vectors and the upper index, ${\widehat{c}}^{\text{in}}$, represents the concentrations inside the bleaching spot.

Solving the set of equations with the constant initial values outside the bleaching spot (Eq. 9) yields

$$\left({\widehat{c}}_B^{\text{out}}\right)=\frac{1}{s}\left(B_{\text{eq}}\right)-EV\left(\begin{array}{c}{\gamma }_0\times {\mathbf{I}}_{\mathbf{0}}\left(r\sqrt{{\lambda }_{B_0}}\right)+{\delta }_0\times {\mathbf{K}}_{\mathbf{0}}\left(r\sqrt{{\lambda }_{B_0}}\right)\\ {\gamma }_1\times {\mathbf{I}}_{\mathbf{0}}\left(r\sqrt{{\lambda }_{B_1}}\right)+{\delta }_1\times {\mathbf{K}}_{\mathbf{0}}\left(r\sqrt{{\lambda }_{B_1}}\right)\\ ⋮\\ {\gamma }_S\times {\mathbf{I}}_{\mathbf{0}}\left(r\sqrt{{\lambda }_{B_S}}\right)+{\delta }_S\times {\mathbf{K}}_{\mathbf{0}}\left(r\sqrt{{\lambda }_{B_S}}\right)\end{array}\right),$$ (26)

where $\left({\widehat{c}}_B^{\text{out}}\right)$ and (B_eq) are vectors and the upper index, ${\widehat{c}}^{\text{out}}$, represents the concentrations outside the bleaching spot.

After determination of the coefficients α_k, β_k, γ_k, and δ_k by the given boundary conditions (Eq. 10), we have to calculate the average fluorescence of all molecular fractions inside the bleaching spot

$$\widehat{\text{frap}}\left(s\right)=\frac{1}{\pi R^2}\times \underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{R}{\int }}r\times \sum _{i=0}^S{\widehat{c}}_{{\text{B}}_i}^{\text{in}}drd\varphi .$$ (27)

We used Eq. 22 to analyze recovery curves with one binding state and Eq. 27 to analyze recovery curves with two binding states.

FRAP Inversion Algorithms

Our optimization problem is to find diffusion coefficients and reaction rates within our solutions that fit a given recovery curve best.

Preprocessing Algorithm: Adapted Prony's method

To infer the maximal number of hidden binding sites, we used a preprocessing step by adapting Prony's algorithm (27,29,35). This preprocessing step allowed us to limit the number of models to be fitted and to estimate the effects of overfitting.

In the next paragraph, we describe the general strategy of Prony's method and how we made use of this strategy to deduce the maximal number of S. Prony's technique is based on exponential decay curves, in our case the loss of bleached molecules inside the bleaching spot (loss(t) = 1 – frap(t)). The measured decay curve is approximated by

$$\begin{array}{c}\text{loss}\left(\text{t}\right)=f\left(t\right)\approx C_1{e}^{a_{1}t}+C_2{e}^{a_{2}t}+\dots +C_S{e}^{a_{S}t}\\ =C_1{\mu }_1^t+C_2{\mu }_2^t+\dots +C_S{\mu }_S^t\end{array},$$ (28)

which can be rewritten as a set of equations,

$$C_1+C_2+\dots +C_S=f_0$$

$$C_1{\mu }_1+C_2{\mu }_2+\dots +C_S{\mu }_S=f_1$$ (29)

⋮

$$C_1{\mu }_1^{N-1}+C_2{\mu }_2^{N-1}+\dots +C_S{\mu }_S^{N-1}=f_{N-1},$$

if a set of N equally spaced measurements is given. Since ${\mu }_i=e^{a_i}$ and C_i are unknown, at least 2S equations are needed. The difficulty of solving the problem is caused by the nonlinearity in μ_i. Therefore, the Eq. 29 set of equations is linearized (28):

$$f_S+f_{S-1}{\alpha }_1+f_{S-2}{\alpha }_2+\dots +f_0{\alpha }_S=0$$

$$f_{S+1}+f_S{\alpha }_1+f_{S-1}{\alpha }_2+\dots +f_1{\alpha }_S=0$$ (30)

⋮

$$f_{N-1}+f_{N-2}{\alpha }_1+f_{N-3}{\alpha }_2+\dots +f_{N-S-1}{\alpha }_S=0,$$

where the α_i are determined and the μ_i are found as the roots of

$${\mu }^S+{\alpha }_1{\mu }^{S-1}+{\alpha }_2{\mu }^{S-2}+\dots +{\alpha }_{S-1}\mu +{\alpha }_S=0$$ (31)

Equation 29 then becomes a set of linear equations for C_i where the coefficients are known. C_i can then be inferred by applying a least-squares method.

Prony's original suggestion was the least-squares algorithm

$$A={\left[X^{T}X\right]}^{-1}{X}^{T}Y,$$ (32)

where

$$\begin{array}{c}X=\left(\begin{array}{ccc}{f}_{S-1}& f_{S-2}& f_0\\ f_S& f_{S-1}& f_1\\ ⋮& ⋮& ⋮\\ f_{N-2}& f_{N-3}& f_{N-S-1}\end{array}\right)\\ Y^T=\left(\begin{array}{cccc}{f}_S& f_{S+1}& \cdots & f_{N-1}\end{array}\right)\\ A^T=\left(\begin{array}{cccc}-{\alpha }_0& -{\alpha }_1& \cdots & -{\alpha }_S\end{array}\right),\end{array}$$

for the identification of α_i in Eq. 30. Since simulations showed that this strategy results in nonoptimal values of A, we used the optimized correlation method by Sun et al. (35), introducing the following auxiliary matrix, Z:

$$Z=\left(\begin{array}{ccc}{f}_{S-1+k}& f_{S-2+k}& f_k\\ f_{S+k}& f_{S-1+k}& f_{1+k}\\ ⋮& ⋮& ⋮\\ f_{N-2+k}& f_{N-3+k}& f_{N-S-1+k}\end{array}\right),$$

which is then applied to calculate the unknowns as follows

$$A={\left[Z^{T}X\right]}^{-1}{Z}^{T}Y.$$ (33)

The α_i determined by this method can then be used to calculate the C_i as described earlier.

Inversion Algorithm: Simulated Annealing

For inversion purposes, an objective function describing the difference between the measured and the calculated data set has to be minimized. We use the mean absolute error (MAE) as our objective function:

$$\text{MAE}=\sum _{i=1}^N|x_i-x_i^e|,$$ (34)

in which N is the number of measured time points, x_i represents the measured recovery value at the ith time point, and x_i^e is the estimated recovery value at the ith time point.

Unlike the Gauss-Newton and Levenberg-Marquardt algorithms commonly applied to fit data sets (17,30,31), we choose the SA technique to solve this problem (32). SA is a heuristic optimization technique based on the metropolis algorithm (36). The advantages of this algorithm are that

• 1.it can deal efficiently with cost or objective functions characterized by quite arbitrary degrees of nonlinearities, discontinuities, and stochasticity;
• 2.it can process quite arbitrary boundary conditions and constraints imposed on these cost functions;
• 3.it can be implemented quite easily in comparison with other nonlinear optimization algorithms;
• 4.it is independent of the initial parameter settings; and finally
• 5.it converges to the optimum solution, so that finding a near-optimum solution is statistically guaranteed.

SA has the great benefit that results are not constrained by initial parameter settings, since it is known that initial values far from the true values cause fitting failure by using the Gauss-Newton or the Levenberg-Marquardt algorithm (17,37).

We take the same initial parameter settings and parameter ranges to generate a feasible solution for every fitting (Table 1).

Table 1.

Inital values and range of diffusion coefficients and reaction rates

	Initial value	Minimum	Maximum
DF	1.0	1.0·10−2	100.0
$D_{B_i}$	1.0	1.0·10−2	100.0
$k_{{\text{on}}_i}$	0.5	1.0·10−8	1.0
$k_{{\text{off}}_i}$	0.5	1.0·10−8	1.0

The values shown were used for parameter search by SA. The search space is defined by the given range.

Experiments

To test the reliability of our preprocessing and inversion algorithm we created artificial data sets from our (semi)analytical solutions. The aim was to identify the correct model type as well as to estimate the model parameters. Subsequently, we applied our fitting algorithm to real FRAP data to prove that diffusion coefficients and reaction rates can be robustly identified from real (i.e., noisy) data sets.

Artificial FRAP data

Preprocessing Algorithm: Setup and Results

The parameter values were chosen to correspond to the experimental setup and the achieved values. For testing the preprocessing step of the adapted Prony's method, we used artificial datasets with parameter values corresponding to the experimental setup and the achieved values (Table 2). To account for realistic experimental data, we added a Gaussian-distributed noise value (n ∼ N (0.00, 0.03)). In summary, we tested six different model functions: three model types each with and without added noise. Since noise is a random process, we used 100 different sample data sets for the model functions. The model types chosen were reaction-dominant (M₁), full reaction-diffusion with single diffusion (M₂), and full reaction-diffusion with multiple diffusion (M₃), all with S = 2.

Table 2.

Artificial data set values for testing the preprocessing step of the adapted Prony's method

Radius of bleaching spot	R = 2 (LU)
Simulated time steps	N = 250 (TU)
Reaction rates	$k_{{\text{on}}_1}$ = 0.9, $k_{{\text{on}}_2}$ = 0.9
	$k_{{\text{off}}_1}$ = 0.9, $k_{{\text{off}}_2}$ = 0.3
Diffusion coefficients	DF = 10.0, $D_{B_1}=5.0$
Unit of diffusion coefficients	$\left[L{U}^2/TU\right]$
Unit of association and dissociation rate	$\left[1/TU\right]$

Prony's original method was designed to determine the number of molecular fractions that are coupled by a reaction-dominant scheme. For the reaction-dominant model type with and without noise, the number of binding sites, S, was estimated correctly (M₁; see Table 3). Considering only one diffusive fraction does not influence the correct prediction. However, increasing the number of diffusive components leads to an overestimation of S. As expected, the noise level imposed on the artificial data sets influenced the predictive power. However, in every case, the method predicted either the correct number or one additional S. The results for all model types with and without noise are shown in Table 3.

Table 3.

Number of binding sites estimated by Prony's method

		M1	M2	M3
No noise	1S	0%	0%	0%
	2S	100%	100%	0%
	3S	0%	0%	100%
	>3S	0%	0%	0%
With noise	1S	2%	14%	0%
	2S	87%	84%	56%
	3S	11%	2%	44%
	>3S	0%	0%	0%

A data set without noise and 100 noisy data sets (n ∼ N (0.00, 0.03)) with different underlying model types (M1, reaction-dominant (2S); M2, full reaction-diffusion with single diffusion (2S); and M3, full reaction-diffusion with multiple diffusion (2S)) were analyzed by Prony's method. Entries in bold highlight the most likely number of binding sites S.

Inversion Algorithm: Setup and Results

To test our inversion routine, we tested the model functions using a SA algorithm applied to artificial datasets with settings specified in Table 4. To allow for a more realistic description of measurements, different noise values were added. A low noise value, n_L ∼ N (0.00, 0.01), and a high noise value, n_H ∼ N (0.00, 0.03), were taken, leading to nine artificial datasets in total (Fig. 2). They were abbreviated with M_k^σ, where k indicates the model type: reaction-dominant model (M₁ (Eq. 14)), full reaction-diffusion model with single diffusion (M₂ (Eq. 17)), and full reaction-diffusion model with multiple diffusion (M₃ (Eq. 22)) all with S = 1. σ represents the standard deviation of the added Gaussian-distributed noise signal. Every data set was fitted against the solutions of the three model types mentioned above.

Table 4.

Artificial data set used with SA algorithm to test inversion routine

Radius of bleaching spot	R = 15 [LU]
Simulated time steps	N = 100 [TU]
Reaction rates	kon = 0.3, koff = 0.05
Diffusion coefficients	DF = 10.0, $D_{B_1}=1.0$
Unit of diffusion coefficients	$\left[L{U}^2/TU\right]$
Unit of association- and dissociation rate	$\left[1/TU\right]$

Figure 2.

Artificial data sets. Calculation of the model function values (M₁, M₂, M₃) with different noise levels (σ). (A) σ = 0.00, no noise. (B) σ = 0.01, low noise level. (C) σ = 0.03, high noise level.

The error-function value was determined as the sum of absolute differences between fitted functions and data sets (Eq. 34). As expected, the error-function values for all model types increased along with the noise level. Even with a high noise level the model type was determined reliably (Table 5). In some cases, differentiation between model types was based on marginal differences of the best error-function values. To prove that even these differences were significant, we investigated the distribution of the error-function values of 500 simulated-annealing runs. Plotting the absolute frequencies of error-function values clearly distinguished the investigated model types. The correct model function always shows an accumulation of the lowest error-function values (Fig. 3). These results demonstrate the fact that even small differences in least error-function values are significant.

Table 5.

Error-function values of analyzed artificial data sets

	Fitted model type
	Model 1	Model 2	Model 3
M10.00	0.00000	1.92217	2.07131
M10.01	0.78581	2.09649	2.24620
M10.03	2.11161	3.03402	3.16262
M20.00	1.50488	0.00885	0.06056
M20.01	1.70500	0.78243	0.78933
M20.03	2.87615	2.23449	2.35603
M30.00	2.17175	0.03437	0.00092
M30.01	2.26306	0.77783	0.77610
M30.03	2.97956	2.48055	2.45850

Artificial data sets with different noise levels (σ₁ = 0.00, σ₂ = 0.01, and σ₃ = 0.03) were fitted by three model functions. Least error-function values are printed in bold.

Figure 3.

Error-value histogram of analyzed artificial data sets. (A–I) Artificial data sets were generated by three different model types to which different noise levels were added (0.00–0.03). The distributions of error-function values of 500 SA runs are shown for the three model types (Model 1, black bars; Model 2, gray bars; and Model 3, white bars).

We next tested the ability of the model to correctly predict parameters. As an example, we compared the prediction of the reaction rates k_on and k_off, since these are the only parameters present in all three model types. Similar to the model-type prediction, the noise level of the artificial data sets influenced the prediction power of the parameter values. Fig. 4 shows the mean and variance of these parameters determined by the best 100 (out of 500) SA runs with least error-function values. In summary, not only the correct underlying model type but also the correct diffusion coefficients and reaction rates were determined. For further analysis of the artificial data set and information about computational expenses of SA, see the Supporting Material.

Figure 4.

Robustness of estimated reaction parameters. Box plots are shown for the 100 best of the 500 SA runs (mean ± SD). Black dots represent the parameters yielding the least error-function values. The parameter value used for artificial data set creation is represented by the dotted line.

Real FRAP data

Setup

The murine hepatoma cells Tao BpRc1, deficient in endogenous aryl hydrocarbon receptor (AhR), were stably transfected with a green-fluorescent-protein (GFP)-labeled AhR construct (38) under tetracycline control. Cells were grown in phenol-free DMEM containing 10% fetal bovine serum and 2 mM L-alanyl-L-glutamine and cultured in 35 mm Ibidi μ-dishes. To induce expression of GFP-AhR, cells were taken off tetracycline 24 h before exposure. To induce AhR translocation, cells were exposed to 50 nM Benzo(a)pyrene (BaP) for 15 min.

The FRAP experiments were performed on a Zeiss LSM 510 META confocal microscope (Carl Zeiss, Jena, Germany) with a 100×/1.4 NA oil-immersion objective. Bleaching was performed with a circular spot (radius 1.12305 μm) using the 488- and 514-nm lines from an argon laser operating at 74% laser power. A single iteration was used for the bleach pulse. Five prebleach images were taken and the fluorescence recovery was monitored in 83.2-ms intervals. During all FRAP experiments, cells were kept at 37°C using a heated stage plate (Carl Zeiss). In total, 50 FRAP experiments were performed in the nucleus. The raw image data were used to extract the fluorescence-recovery curves. Afterward, each recovery curve was double-normalized using the prebleach images as well as two reference areas, as described by Phair et al. (8) (see also Supporting Material). The average recovery shown in Fig. 5 was calculated by taking the mean of all 50 individual recovery curves.

Figure 5.

Graphical comparison of experimental FRAP data with model functions. Dots represent the average GFP-AhR recovery from 50 independent FRAP experiments (nucleus, 15min, 50 nM BaP) and lines represent model functions. It can be seen clearly that the reaction-diffusion model with multiple diffusion (2S) with the least error-function value fits the experimental data best.

To estimate the constant initial value, we extracted the bleaching profile out of the raw data and fitted a Gaussian function. We used this function to determine the initial value of θ as described by Hinow et al. (39) and readjusted the bleaching-spot radius according to the procedure of Mueller et al. (25).

For fitting, we used the model functions with the estimated constant initial value, θ, and the adapted bleaching-spot radius (Eqs. 15, 18, and 22).

As a reference measurement, we performed FRAP experiments in Tao BpRc1 cells expressing GFP only.

Results

To infer the correct underlying model type for the movement of GFP-tagged AhR inside the nucleus, we compared all available model types. Data were described best by a full reaction-diffusion model with multiple diffusion and two binding sites (Table 6 and Fig. 5). As already described for the artificial data, the distribution of the error-function values confirmed the model type determined by the least error-function value (data not shown). Deduced from the model type, three molecular fractions were predicted for the GFP-AhR data set: one fraction is moving diffusively with a diffusion coefficient of D_B0 = 5.1 μm² s⁻¹; this fraction then converts to a second fraction with a diffusion coefficient of D_B1 = 3.3 μm² s⁻¹, and this slower fraction finally becomes immobile (D_B2 = 0.1 μm² s⁻¹; due to the model restrictions, a diffusion coefficient equal to zero is not defined). Calculated from the fitted reaction rates, the percentage of the fractions are $B_{{\text{eq}}_0}$ = 93%, $B_{{\text{eq}}_1}$ = 1%, and $B_{{\text{eq}}_2}$ = 6% for the immobile fraction (Eq. 7).

Table 6.

Fitted parameters of models with least error function values

	DF	DB1	Feq	$B_{{\text{eq}}_1}$
		DB2		$B_{{\text{eq}}_2}$
		DB3		$B_{{\text{eq}}_3}$
Full reaction-diffusion with multiple diffusion (1S)	4.930	0.101	0.882	0.118
		—		—
Full reaction-diffusion with single diffusion (2S)	4.997	0.000	0.951	0.046
		0.000		0.004
Full reaction-diffusion with multiple diffusion (2S)	5.067	3.340	0.934	0.007
		0.091		0.060
		—		—

Analysis of nuclear FRAP data on GFP-AhR yielded three models with comparable error-function values. The least error-function value was obtained by the reaction-diffusion model with multiple diffusion (2S). (F_eq = B_eq0 and F = B₀).

Although the models became increasingly complex, all deduced diffusion and reaction parameters were consistent among model types (Table 6). Verifying the determined model type, Prony's method yielded a maximal number of binding partners, S = 3. For further analysis of the real data set, see the Supporting Material.

Discussion

FRAP is a powerful technique to investigate the dynamic behavior of proteins in living cells. Mathematical modeling of FRAP data allows determination of dissociation and association rates, distribution of mobile and immobile fractions, and corresponding diffusion coefficients. A number of simplified models that describe motion of reactively coupled fluorescent molecules observed by FRAP have been described (17–21). However, all of these existing approaches include a priori assumptions to allow for the determination of an analytical solution. To circumvent this bias, we established a generalized reaction-diffusion model that comprises a flexible number of reacting and diffusing fractions. More specifically, we impose constraints on neither model type nor parameters.

Generalizing the model type obviously yields a higher number of parameters. To address this issue, we introduce a preliminary approximation of the number of acting molecular fractions by applying the adapted Prony's method (27,29,35). This method was tested using artificial as well as real data sets and proved to identify the correct number of binding sites, either S or S + 1. Since the Prony's method is based on a reaction-dominant scheme, the introduction of diffusive components results in an overestimation of binding sites. Therefore, the Prony's method always deduces the upper limit of molecular fractions, i.e., parameters that can be fitted reliably. Therefore, we can rule out that the decreasing error-function values of our more complex models are due to overfitting effects. In addition, for future studies, the deduced number S could be used to limit the models to be fitted.

To infer optimal parameter sets, local search algorithms like Levenberg-Marquardt are commonly used (17,30,31). However, Sprague and colleagues (17,37) showed that the Levenberg-Marquardt algorithm is very sensitive to the choice of initial parameters. To allow for a global search independent of these initial settings, we employed the SA strategy (32). The performance test of the SA algorithm using artificial data sets demonstrated that SA indeed is able to predict the correct model type reliably. Although the variance of estimated parameter values (e.g., diffusion coefficients) increased with the complexity of the model, the prediction of the parameters was still satisfactory with respect to their variance.

We demonstrated the consistency of our new approach by fitting existing (17) and new models to nuclear FRAP measurement in a murine hepatoma cell line stably transfected with a GFP-labeled AhR construct. AhR is a soluble cytoplasmic transcription factor. After ligand binding, the receptor-ligand complex (AhR/L) translocates into the nucleus where it associates with its cofactor, AhR nuclear translocator (ARNT). Association of ARNT with AhR/L is necessary for binding to so-called xenobiotic-response elements (XREs) to regulate transcription.

For nuclear FRAP data on GFP-AhR, our model outperformed previous models (11) and suggested the existence of three different molecular fractions and diffusion coefficients. In correspondence with common knowledge of AhR signaling (40–43), these molecular fractions should represent AhR/L, AhR/L/ARNT, and AhR/L/ARNT/XRE.

The relation D ∝ m^–1/3 (17), where D is the diffusion coefficient and m is the mass of a molecule, allows us to deduce the mass for a particular diffusion coefficient or vice versa, as long as a reference measurement is available. An empty vector expressing the fluorophore tagging the protein of interest is usually used for this purpose (11,17,39). The fitted GFP diffusion coefficient of our in vitro system correlates very well with published values to date (17,44–46) and helped us to identify the following molecular fractions: a diffusion coefficient of GFP-Ahr/L (124 kDa) equal to 4.8 μm² s⁻¹, and a diffusion coefficient of GFP-AhR/L/ARNT (211 kDa) equal to 4.0 μm² s⁻¹. Since all fitted diffusion models are restricted to D > 0, we consider the slowest fraction, with a diffusion coefficient close to zero, as GFP-AhR/L/ARNT/XRE. These estimated values are in close agreement to the fitted values (D₁ = 5.1 μm² s⁻¹ and D₂ = 3.3 μm² s⁻¹) inside the nucleus.

In addition, our estimated parameters are similar to those reported for the glucocorticoid receptor, which has a mass comparable to that of AhR (25).

Our new approach shows that intracellular molecular mobility can only be described adequately by allowing for multiple reaction-diffusion processes, as shown by our application to GFP-AhR FRAP data. Our general reaction-diffusion model performed significantly better than the standard types of reaction-diffusion model used previously. Coming back to the question we posed at the beginning, “Are assumptions of the model type necessary in reaction-diffusion modeling?”, we give the provocative answer, it may be not only not necessary but also too restrictive to describe the processes sufficiently, since we may use an assumption that is not justified. Therefore, we argue that optimizing parameter sets for predetermined model types is too restrictive to describe biological processes sufficiently.