The movement of a molecule of interest in a cell nucleus is influenced by diffusion and by interactions, including binding reactions, the molecule is involved in (Mueller et al., 2010; Hemmerich et al., 2011; van Royen et al., 2011; Mazza et al., 2012). It is possible to model this process by using the full reaction-diffusion equations (Carrero et al., 2004; Sprague et al., 2004; Beaudouin et al., 2006). As we strive for an analytical solution of the equations describing the movement of the molecule of interest, we use a simplification of the full reaction-diffusion equations. Usually, one of the following three simplifications is employed: the pure-diffusion scenario, the effective diffusion scenario or the reaction dominant scenario (Sprague et al., 2004). A pure-diffusion dominant scenario (Sprague et al., 2004) is present when most of the fluorescent molecules are free and interactions can be ignored. An effective diffusion scenario (Sprague et al., 2004; Beaudouin et al., 2006; Mueller et al., 2008; van Royen et al., 2009) occurs “when the reaction process is much faster than diffusion” (Sprague et al., 2004). A reaction dominant scenario is present, when diffusion is very fast compared to the timescale of the image acquisition and to the reaction process (Sprague et al., 2004).

The interactions Dnmt1 is involved in are described by on- and off-rates. In Schneider et al. (2013), where a correction value for diffusion was used, it was found that, in S phase, Dnmt1 is involved in interactions with relatively small off-rates, which means in the case of binding reactions, that the molecules of interest have a relatively long residence time (about 10–20 s) at their binding sites. We have no sufficient information about the magnitude of the on-rate. For these reasons and because we aim to have an analytical solution to the ordinary differential equations describing the movement of Dnmt1, we assume a reaction dominant scenario for our data.

In a reaction dominant FRAP scenario, diffusion is very fast in comparison to reaction processes and the time scale of the FRAP measurement (Bulinski et al., 2001; Coscoy et al., 2002; Dundr et al., 2002) and the recovery curve in the bleached part of the cell nucleus can be modeled using a nonlinear regression model (Sprague et al., 2004).

Here, we regard cases with two or three MCs (Schneider et al., 2013). In all considered cell cycle phases, a MC with a very long residence time compared to the time of image acquisition is indicated (Schneider et al., 2013). For this MC, we estimate only one parameter, and it is later also referred to as “immobile fraction.” In cells with diffuse localization and in early S phase, one additional MC has been identified. For the late S phase, two additional MCs with different off-rates were found.

The binding sites to which the molecules of interest bind are assumed to be part of large complexes, which are relatively immobile on the time scale of the FRAP measurement and the molecular movement (Carrero et al., 2004; Sprague et al., 2004). A compartment model with two or three compartments (Figure 2; the immobile fraction is ignored in this representation) is used to describe the change of the concentration of unbleached molecules in the bleached part of the cell nucleus. In a compartment model with two compartments, the molecules can be either free or bound. Exchange between the compartment of the free and the compartment of the bound molecules occurs with rates $${b}_{1}^{on\mathrm{*}}$$ and $${b}_{1}^{off}.$$ In a compartment model with three compartments, the molecules can be either free or bound in one of two discriminable binding states. Exchange between the compartment of the free molecules and the compartments of the bound molecules occurs with rates $${b}_{1}^{on\mathrm{*}}$$ and $${b}_{1}^{off}$$ and $${b}_{2}^{on\mathrm{*}}$$ and $${b}_{2}^{off},$$ respectively. A similar procedure based on the reaction equation of a binding interaction was proposed by Sprague et al. (2004).

Figure 2 (A) Compartment model with two compartments and (B) compartment model with three compartments.

The on- and off-rates of the binding reaction are denoted by $${b}_{k}^{on\mathrm{*}}$$ and $${b}_{k}^{off},$$ *k*=0,…, *K*. As stated in Sprague et al. (2004), $${b}_{k}^{on\mathrm{*}}$$ is actually a pseudo-on-rate. It is the product of the actual on-rate $${b}_{k}^{on}$$ and the concentration of vacant bindings sites belonging to MC *k*. It is constant during the entire recovery process, because we assume that the biological system is in equilibrium before the bleaching and because bleaching does not affect the number of vacant binding sites (Sprague et al., 2004).

Let *f*(*t*)=[*Free*](*t*) denote the concentration of the free molecules and *a*_{k}(*t*)=[*Bound*_{k}](*t*) the concentration of the bound molecules in MC *k* at time *t*. We can describe the change of the concentration of the free and bound molecules based on the compartment model by the two differential equations

$$\frac{d}{dt}f\mathrm{(}t\mathrm{)}={\displaystyle \sum _{k=0}^{K}}\mathrm{(}-{b}_{k}^{on\mathrm{*}}f\mathrm{(}t\mathrm{)}+{b}_{k}^{off}{a}_{k}\mathrm{(}t\mathrm{)}\mathrm{)}+{D}_{f}{\nabla}^{2}f\mathrm{(}t\mathrm{}\mathrm{)}\mathrm{,}\text{\hspace{1em}(1)}$$(1)

with ∇^{2} the Laplacian operator and *D*_{f} the diffusion coefficient for free proteins, and

$$\frac{d}{dt}{a}_{k}\mathrm{(}t\mathrm{)}={b}_{k}^{on\mathrm{*}}f\mathrm{(}t\mathrm{)}-{b}_{k}^{off}{a}_{k}\mathrm{(}t\mathrm{}\mathrm{)}\mathrm{.}\text{\hspace{1em}(2)}$$(2)

The molecules in the cell nucleus are in equilibrium before the bleaching. In a diffusion-uncoupled FRAP scenario, the free molecules are moreover assumed to be in equilibrium again immediately after the bleaching. Therefore *f*(*t*)=*f*_{eq}, a constant, and equation (2) can be written as

$$\frac{d}{dt}{a}_{k}\mathrm{(}t\mathrm{)}={b}_{k}^{on\mathrm{*}}{f}_{eq}-{b}_{k}^{off}{a}_{k}\mathrm{(}t\mathrm{}\mathrm{)}\mathrm{.}\text{\hspace{1em}(3)}$$(3)

Moreover, we do not have to model the change of the concentration of the free molecules, it suffices to model the change of concentration of the bound molecules, which means that equation (1) can be ignored.

With boundary condition *a*_{k}(0)=0, which means that at time *t*=0 (the time of the bleaching) the concentration of unbleached bound molecules in MC *k* in the bleached area equals zero, the solution of equation (3) is

$${a}_{k}\mathrm{(}t\mathrm{)}=\frac{{b}_{k}^{on\mathrm{*}}{f}_{eq}}{{b}_{k}^{off}}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{{b}_{k}^{on\mathrm{*}}{f}_{eq}}{{b}_{k}^{off}}exp\mathrm{(}-{b}_{k}^{off}t\mathrm{)}\mathrm{.}\text{\hspace{1em}(4)}$$(4)

As the system is in equilibrium before bleaching we have $$\frac{d}{dt}f\mathrm{(}t\mathrm{)}=0,$$ $$\frac{d}{dt}{a}_{k}\mathrm{(}t\mathrm{)}=0$$ and constant steady-state intensities *f*_{eq}, *a*_{k,eq}. Together with equation (3) we get

$${a}_{k\mathrm{,}eq}=\frac{{b}_{k}^{on\mathrm{*}}{f}_{eq}}{{b}_{k}^{off}}\mathrm{,}\text{\hspace{1em}(5)}$$(5)

and can therefore write equation (4) as

$${a}_{k}\mathrm{(}t\mathrm{)}={a}_{k\mathrm{,}eq}\mathrm{(}1-exp\mathrm{(}-{b}_{k}^{off}t\mathrm{)}\mathrm{)}\mathrm{.}\text{\hspace{1em}(6)}$$(6)

The observed value during FRAP recovery is the total fluorescence intensity in the bleached area. It can be described by the sum of the bound and the free unbleached molecules plus an error. The sum of the bound and the free unbleached molecules is denoted by *total*(*t*):

$$total\mathrm{(}t\mathrm{)}={f}_{eq}+{\displaystyle \sum _{k=0}^{K}}{a}_{k}\mathrm{(}t\mathrm{}\mathrm{)}\mathrm{.}\text{\hspace{1em}(7)}$$(7)

For our analysis, in each cell nucleus, the fluorescence intensity has been averaged over the bleached part of the cell nucleus. Therefore, in our analysis, *f*_{eq} is the average of the intensity of the free fluorescent molecules in the bleached half, and *a*_{k}(*t*) is the average of the intensity of the bound fluorescent molecules in the bleached part of the nucleus. With equation (6) we can then write

$$total\mathrm{(}t\mathrm{)}={f}_{eq}+{\displaystyle \sum _{k=0}^{K}}{a}_{k\mathrm{,}eq}\mathrm{(}1-exp\mathrm{(}-{b}_{k}^{off}t\mathrm{)}\mathrm{)}\mathrm{.}\text{\hspace{1em}(8)}$$(8)

With $${f}_{eq}+{\displaystyle {\sum}_{k=0}^{K}}{a}_{k\mathrm{,}eq}=1,$$ which holds because the concentration of the unbleached molecules has been normalized to one, we arrive at

$$total\mathrm{(}t\mathrm{)}=1-{\displaystyle \sum _{k=0}^{K}}{a}_{k\mathrm{,}eq}exp\mathrm{(}-{b}_{k}^{off}t\mathrm{}\mathrm{)}\mathrm{,}\text{\hspace{1em}(9)}$$(9)

which is the deterministic approximation of the model with multiple mobility classes in Fuchs (2013).

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