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Increase in intensity at observed area could be measured to know the net increase in the protein at that region. To characterize this dynamics, we can apply traditional biochemical kinetics. Example case: Kinetics of VSVG protein accumulation to ER exit site. | Increase in intensity at observed area could be measured to know the net increase in the protein at that region. To characterize this dynamics, we can apply traditional biochemical kinetics. Example case: Kinetics of VSVG protein accumulation to ER exit site. | ||

- | <jsmath>{dI(t)\over dt}=k_{on}[VSVG_{free}]- k_{off}[VSVG_{ERES}]</jsmath> | + | $${dI(t)\over dt}=k_{on}[VSVG_{free}]- k_{off}[VSVG_{ERES}]$$ |

Here, | Here, | ||

- | * <jsm>k_{on}</jsm> is the binding rate of VSVG protein to ER exit site | + | * $k_{on}$ is the binding rate of VSVG protein to ER exit site |

- | * <jsm>[VSVG_{free}]</jsm> is the concentration of unbound VSVG protein | + | * $[VSVG_{free}]$ is the concentration of unbound VSVG protein |

- | * <jsm>k_{off}</jsm> is the dissociation rate of VSVG protein from ER exit site | + | * $k_{off}$ is the dissociation rate of VSVG protein from ER exit site |

- | * <jsm>[VSVG_{ERES}]</jsm> is the density of VSVG protein bound to the ER exit site | + | * $[VSVG_{ERES}]$ is the density of VSVG protein bound to the ER exit site |

- | During the initial phase of binding, when there is almost no VSVG protein bound to ER exit site, we can approximate the initial speed of the density increase at ERES site depends only on binding reaction: <jsm>k_{off}[VSVG_{ERES}]\simeq0</jsm>. \\Then | + | During the initial phase of binding, when there is almost no VSVG protein bound to ER exit site, we can approximate the initial speed of the density increase at ERES site depends only on binding reaction: $k_{off}[VSVG_{ERES}]\simeq0$. \\Then |

- | <jsmath> | + | $$ |

{dI(t)\over{dt}}=k_{on}[VSVG_{free}] | {dI(t)\over{dt}}=k_{on}[VSVG_{free}] | ||

- | </jsmath> | + | $$ |

- | Since there are enough free VSVG, we consider that <jsm>[VSVG_{free}]</jsm> is constant, we are able to simply calculate the slope of initial increase of intensity, measure the free VSVG intensity and then calculate <jsm>k_{on}</jsm>. | + | Since there are enough free VSVG, we consider that $[VSVG_{free}]$ is constant, we are able to simply calculate the slope of initial increase of intensity, measure the free VSVG intensity and then calculate $k_{on}$. |

For details, see [[http://www.ncbi.nlm.nih.gov/pubmed/16794576?dopt=Abstract |Runz et al (2006)]]. | For details, see [[http://www.ncbi.nlm.nih.gov/pubmed/16794576?dopt=Abstract |Runz et al (2006)]]. | ||

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=== pure diffusion === | === pure diffusion === | ||

Theoretical curve of the diffusion mediated fluorescence recovery was proposed by Soumpasis (1984) and has been widely used. \\ | Theoretical curve of the diffusion mediated fluorescence recovery was proposed by Soumpasis (1984) and has been widely used. \\ | ||

- | <jsmath> | + | $$ |

f(t)=e^{- \frac{\tau_D}{2t}}\left(I_{0}(\frac{\tau_D}{2t})+I_{1}(\frac{\tau_D}{2t})\right) | f(t)=e^{- \frac{\tau_D}{2t}}\left(I_{0}(\frac{\tau_D}{2t})+I_{1}(\frac{\tau_D}{2t})\right) | ||

- | </jsmath> | + | $$ |

This theoretical equation assumes: | This theoretical equation assumes: | ||

* 2D | * 2D | ||

* circular (cylindrical) bleaching | * circular (cylindrical) bleaching | ||

- | when above equation could be fitted nicely (evaluated by goodness of fit, such as Pearson's coefficient //r// or gamma-Q value), one could calculate diffusion coeffecient by using the obtained <jsm>\tau_D</jsm> and radius of the circular ROI <jsm>w</jsm>.\\ | + | when above equation could be fitted nicely (evaluated by goodness of fit, such as Pearson's coefficient //r// or gamma-Q value), one could calculate diffusion coeffecient by using the obtained $\tau_D$ and radius of the circular ROI $w$.\\ |

- | <jsmath> | + | $$ |

D=\frac{w^{2}}{\tau_D} | D=\frac{w^{2}}{\tau_D} | ||

- | </jsmath> | + | $$ |

For strip-ROI bleaching, empirical formula used by Ellenberg et al. (1997) could be used, and is also possible to use Gaussian curve fitting that **Christian Tischer** developed. For Christian's method, [[:documents:100426FRAPgaussfit]]. | For strip-ROI bleaching, empirical formula used by Ellenberg et al. (1997) could be used, and is also possible to use Gaussian curve fitting that **Christian Tischer** developed. For Christian's method, [[:documents:100426FRAPgaussfit]]. | ||

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For a simple chemical reaction with singular type of interaction, we could again think of the reaction model that was already explained above, the first-order chemical reaction modeled as a compartment system (see figure right)\\ | For a simple chemical reaction with singular type of interaction, we could again think of the reaction model that was already explained above, the first-order chemical reaction modeled as a compartment system (see figure right)\\ | ||

- | <jsmath> | + | $$ |

\frac {df(t)} {dt} = k_{on}[free] - k_{off}[bound] | \frac {df(t)} {dt} = k_{on}[free] - k_{off}[bound] | ||

- | </jsmath> | + | $$ |

where | where | ||

- | * <jsm>k_{on}</jsm> Binding constant | + | * $k_{on}$ Binding constant |

- | * <jsm>k_{off}</jsm> Dissociation constant | + | * $k_{off}$ Dissociation constant |

- | * <jsm>[free]</jsm> Density of free molecules | + | * $[free]$ Density of free molecules |

- | * <jsm>[bound]</jsm> Density of bound-molecules | + | * $[bound]$ Density of bound-molecules |

We solve the differential equation | We solve the differential equation | ||

- | <jsmath> | + | $$ |

f(t)=A(1-e^{- \tau t}) | f(t)=A(1-e^{- \tau t}) | ||

- | </jsmath> | + | $$ |

where | where | ||

- | * <jsm>\tau = k_{on} + k_{off}</jsm> | + | * $\tau = k_{on} + k_{off}$ |

- | * <jsm>A = \frac {k_{on}}{k_{on} + k_{off}}</jsm> | + | * $A = \frac {k_{on}}{k_{on} + k_{off}}$ |

=== Reaction Dominant Recovery with Immobile Binding Partner=== | === Reaction Dominant Recovery with Immobile Binding Partner=== | ||

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[{{ :documents:reactiondominant05_immobileentity.jpg?150| Modeling fluorescence recovery at immobile binding partner 02}}] | [{{ :documents:reactiondominant05_immobileentity.jpg?150| Modeling fluorescence recovery at immobile binding partner 02}}] | ||

Next we modify above model to consider a situation a bit more frequently we see in cell biology. The protein we are analyzing is either freely diffusing in cytoplasm or bound to an immobile structure inside cell. We FRAP this structure, to know the kinetic constants of the protein interaction with the structure (e.g. microtubule binding protein, structure = microtubule) | Next we modify above model to consider a situation a bit more frequently we see in cell biology. The protein we are analyzing is either freely diffusing in cytoplasm or bound to an immobile structure inside cell. We FRAP this structure, to know the kinetic constants of the protein interaction with the structure (e.g. microtubule binding protein, structure = microtubule) | ||

- | <jsmath> | + | $$ |

\frac {df(t)} {dt} = k_{on}[free][s] - k_{off}[bound] | \frac {df(t)} {dt} = k_{on}[free][s] - k_{off}[bound] | ||

- | </jsmath> | + | $$ |

where | where | ||

- | * <jsm>k_{on}</jsm> Binding constant | + | * $k_{on}$ Binding constant |

- | * <jsm>k_{off}</jsm> Dissociation constant | + | * $k_{off}$ Dissociation constant |

- | * <jsm>[free]</jsm> Density of free molecules | + | * $[free]$ Density of free molecules |

- | * <jsm>[s]</jsm> Density of immobile binding partner | + | * $[s]$ Density of immobile binding partner |

- | * <jsm>[bound]</jsm> Density of bound-molecules | + | * $[bound]$ Density of bound-molecules |

- | Since [s] is immobile and constant during experiment, we define <jsm>k*_{on}</jsm> as | + | Since [s] is immobile and constant during experiment, we define $k*_{on}$ as |

- | <jsmath> | + | $$ |

k*_{on}=k_{on}[s] | k*_{on}=k_{on}[s] | ||

- | </jsmath> | + | $$ |

- | in addition, density of free molecule in cytoplasm is almost constant so we assume <jsm>[free] = F</jsm> and does not change. We then solve | + | in addition, density of free molecule in cytoplasm is almost constant so we assume $[free] = F$ and does not change. We then solve |

- | <jsmath> | + | $$ |

\frac {df(t)} {dt} = k*_{on}F - k_{off}[bound] | \frac {df(t)} {dt} = k*_{on}F - k_{off}[bound] | ||

- | </jsmath> | + | $$ |

We get | We get | ||

- | <jsmath> | + | $$ |

f(t)=1-Ce^{- \tau t} | f(t)=1-Ce^{- \tau t} | ||

- | </jsmath> | + | $$ |

where | where | ||

- | * <jsm>\tau = k_{off}</jsm> | + | * $\tau = k_{off}$ |

- | ... note that the shape of recovery curve now only depends on <jsm>k_{off}</jsm> | + | ... note that the shape of recovery curve now only depends on $k_{off}$ |

==== Diffusion and Reaction combined Recovery ==== | ==== Diffusion and Reaction combined Recovery ==== | ||

[{{ :documents:reactiondiffusion.jpg?150| Diffusion-Reaction Combined Model}}] | [{{ :documents:reactiondiffusion.jpg?150| Diffusion-Reaction Combined Model}}] | ||

- | <jsmath> | + | $$ |

\frac{\partial [free]}{\partial t} = D_{free} \nabla ^2[free]-k_{on}[free][s]+k_{off}[bound] | \frac{\partial [free]}{\partial t} = D_{free} \nabla ^2[free]-k_{on}[free][s]+k_{off}[bound] | ||

- | </jsmath> | + | $$ |

- | <jsmath> | + | $$ |

\frac{\partial [s]}{\partial t} = D_s \nabla ^2[s]-k_{on}[free][s]+k_{off}[bound] | \frac{\partial [s]}{\partial t} = D_s \nabla ^2[s]-k_{on}[free][s]+k_{off}[bound] | ||

- | </jsmath> | + | $$ |

- | <jsmath> | + | $$ |

\frac{\partial [bound]}{\partial t} = D_{bound} \nabla ^2[bound]+k_{on}[free][s]-k_{off}[bound] | \frac{\partial [bound]}{\partial t} = D_{bound} \nabla ^2[bound]+k_{on}[free][s]-k_{off}[bound] | ||

- | </jsmath> | + | $$ |

Since | Since | ||

* [s] is constant and immobile | * [s] is constant and immobile | ||

- | * <jsm>k*_{on} = k_{on}[s]</jsm> | + | * $k*_{on} = k_{on}[s]$ |

- | * <jsm>\frac{\partial [s]}{\partial t}=0 </jsm> | + | * $\frac{\partial [s]}{\partial t}=0 $ |

- | * bound molecules do not diffuse so <jsm>D_{bound}=0</jsm> | + | * bound molecules do not diffuse so $D_{bound}=0$ |

Then we solve only | Then we solve only | ||

- | <jsmath> | + | $$ |

\frac{\partial [free]}{\partial t} = D_{free} \nabla ^2[free]-k*_{on}[free]+k_{off}[bound] | \frac{\partial [free]}{\partial t} = D_{free} \nabla ^2[free]-k*_{on}[free]+k_{off}[bound] | ||

- | </jsmath> | + | $$ |

- | <jsmath> | + | $$ |

\frac{\partial [bound]}{\partial t} = k_{on}[free][s]-k_{off}[bound] | \frac{\partial [bound]}{\partial t} = k_{on}[free][s]-k_{off}[bound] | ||

- | </jsmath> | + | $$ |

We could solve this either analytically (Sprague et al, 2004) or numerically (Beaudouin et al, 2006). In the latter paper, calculation involves spatial context (on-rate was spatially varied; see also "geometry" section below). | We could solve this either analytically (Sprague et al, 2004) or numerically (Beaudouin et al, 2006). In the latter paper, calculation involves spatial context (on-rate was spatially varied; see also "geometry" section below). | ||

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Analytical solution was made in Laplace transformed equation. | Analytical solution was made in Laplace transformed equation. | ||

- | <jsmath> | + | $$ |

\overline{frap(p)} = \frac 1 p - \frac{F_{eq}}{p}\left(1-2K_1(qw)I_1(qw)\right)\times\left(1+\frac{k*_{on}}{p+k_{off}}\right)-\frac C {p+k_{off}} | \overline{frap(p)} = \frac 1 p - \frac{F_{eq}}{p}\left(1-2K_1(qw)I_1(qw)\right)\times\left(1+\frac{k*_{on}}{p+k_{off}}\right)-\frac C {p+k_{off}} | ||

- | </jsmath> | + | $$ |

=== Beaudouin Method === | === Beaudouin Method === | ||

documents/100420frapinternal.txt · Last modified: 2016/05/24 08:53 by kota