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- | ====== Notes: FRAP internal course ====== | + | ====== Lecture Notes: FRAP internal course ====== |

- | April 20, 2010 @EMBL | + | April 20, 2010 @EMBL\\ |

Kota Miura\\ | Kota Miura\\ | ||

+ | |||

will be also further added by Sebastian Huet and Christian Tischer\\ | will be also further added by Sebastian Huet and Christian Tischer\\ | ||

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===== Introduction ===== | ===== Introduction ===== | ||

- | in vivo protein kinetics could be analyzed in two ways: measuring particular movement or averaged movement. By tracking labeled single protein molecules, it is possible to estimate their diffusion and transport behavior. Single molecule studies of membrane proteins enabled us to analyze how they are organized with their dynamics, through corrals of membrane territory. Motor protein moving along cytoskeletal tracks were analyzed in detail how they convert chemical energy into physical force was only possible by probing their singular movement and steps. While single particle tracking requires high-temporal and spatial resolution setup for analysis, analysis of averaged movement measured through fluorescence intensity dynamics, could be achieved with larger spatial and temporal resolution (typically in micrometer scale). | + | in vivo protein kinetics could be analyzed in two ways: measuring particular movement or averaged movement. By tracking labeled single protein molecules, we could estimate their diffusion and transport behavior. Such single molecule studies of membrane proteins, for example, enabled us to analyze how they are organized with their dynamics, such as boundary for movement constrained by membrane corrals. Motor protein moving along cytoskeletal tracks were analyzed in detail to know how they convert chemical energy into physical force. This was only possible by probing their singular movement and steps. While single particle tracking requires high-temporal and spatial resolution setup for analysis, analysis of averaged movement, measured by temporal changes in fluorescence intensity, could be achieved with larger spatial and temporal resolution (typically in micrometer scale). |

- | Here, we focus on one of such averaged movement analysis technique, Fluorescence Recovery After Photobleaching (FRAP). We first start with a simpler case of monitoring averaged movement that does not need to bleach. | + | Here, we focus on one of such averaged movement analysis technique: Fluorescence Recovery After Photobleaching (FRAP). We first start the explanation with a simpler case of monitoring averaged movement that does not need to bleach. |

===== Fluorescence intensity and Protein Dynamics ===== | ===== Fluorescence intensity and Protein Dynamics ===== | ||

- | [{{ :150|Measurement of VSV-G protein exit dynamics}}] | + | [{{ :200|Measurement of VSV-G protein exit dynamics}}] |

- | [{{ :150| First-order Chemical Reaction}}] | + | [{{ :200| First-order Chemical Reaction}}] |

- | Increase in intensity at observed area could be measured to know the net increase in the protein at that region. To calculate biochemical kinetics, 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. |

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

Here, | Here, | ||

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

- | * <jsm>k_{off}</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}]</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 and assume <jsm>k_{off}[VSVG_{ERES}]\simeq0</. 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 |

- | < | + | $$ |

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

- | </ | + | $$ |

- | Since there are enough free VSVG, we consider that <jsm>[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 <jsm>k_{on}</. | + | 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:// | For details, see [[http:// | ||

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===== FRAP Simple Measures ===== | ===== FRAP Simple Measures ===== | ||

[{{ : | [{{ : | ||

- | Unlike the example explained above, dynamics of protein are not observable in many cases. Even though proteins are exchanging in system, the flux of protein constituting the system is not evident if the flux is steady and constant (e.g. liver). In such cases, we need to some how experimentally measure the system. One way is FRAP. In FRAP, we bleach some population of fluorescence-labeled protein and evaluate the mobility of the protein. Typically, we use confocal microscopy and bleach fluorescence of small area of the system by short pulse of strong laser beam and measure changes in the fluorescence intensity over time at that bleached spot over time. Detail on these measurement protocol has been presented in Stefan and Yury's talks (link?). Here, we focus on how to analyze the curve we obtained by such measurements. | + | Unlike the example shown above, dynamics of protein are not observable to eyes (through microscope) in many cases. Even though proteins are exchanging in system, the flux of protein constituting the system is not evident if the in/flux of protein is steady and constant (e.g. liver). In such cases, we need to some how experimentally treat the system. One way is FRAP.\\ |

+ | In FRAP, we bleach some population of fluorescence-labeled protein and evaluate the mobility of the protein. Typically we use confocal microscopy and bleach fluorescence of small area of the system by short pulse of strong laser beam and measure following changes in the fluorescence intensity at that bleached spot over time. Detail on these measurement protocol has been presented in Stefan and Yury's talks (link?). Here, we focus on how to analyze the curve we obtained through such measurements. | ||

- | From measured temporal changes in the intensity at the bleached Region of Interest (this curve indeed is the Fluorescence Recovery After Photobleaching, | + | From measured temporal changes in the intensity at the bleached Region of Interest (this curve indeed is the Fluorescence Recovery After Photobleaching,we can measure two parameters which represents speed of recovery, and fraction of molecules that is moving around in the system. |

- | Half Max and Mobile-Immobile fraction\\ | + | * Half Max |

- | Fitting to Exponential curve | + | * Mobile-Immobile fraction |

+ | |||

+ | Fitting the curve to exponential equation eases us to calculate these parameters. Half-Max value (time) is rather qualitative value, but is a simple and straightforward index for comparing different systems. | ||

===== FRAP Measurements based on Modelling ===== | ===== FRAP Measurements based on Modelling ===== | ||

- | FRAP curve reflects the mobility of proteins. In dilute solution of single protein species, mobility of protein could probably be considered as pure-diffusion. But in many cases, this is not the case. The mobility is affected by the system. | + | FRAP curve reflects the mobility of proteins. In dilute solution of with single protein solute, mobility of protein could probably be considered as pure-diffusion. But in many cases, this is does not hold. The mobility is often affected by the system. |

* reaction with other proteins | * reaction with other proteins | ||

* geometry of the system, that constrains the mobility | * geometry of the system, that constrains the mobility | ||

* active transport process | * active transport process | ||

- | By modeling how the mobility is (generate some hypothesis how the protein mobility is affected in the system), we can set up equation/s to that should describe the FRAP curve. This is done by fitting the theoretical curves to the experimental curves. By evaluating the goodness of fit, we can discuss which models would be the most likely hypothesis. | + | By modeling how the mobility is (generate some hypothesis how the protein mobility is affected in the system), we can set up equation/s to hypothesize what is the bases of FRAP curve. To test the hypothesis, we fit the experimental curves with theoretical curve. By evaluating the goodness of fit, we can discuss which models would be the most likely hypothesis. If the fit is good, then we could know the value of biochemical parameters which governs the recovery curve. |

- | Currently we have more-or-less standardized protocol to analyze FRAP curve. Starting with simple model of diffusion, we test the fit of different curves and proceed to more complex models. See next section for the protocol. | + | Currently we have more-or-less standardized protocol to analyze FRAP curves. Starting with simple model of diffusion, we test the fit of different curves and proceed to more complex models. See next section for the protocol. |

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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. \\ | ||

- | < | + | $$ |

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) | ||

- | </ | + | $$ |

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'<jsm>\tau_D</and radius of the circular ROI <jsm>w</.\\ | + | when above equation could be fitted nicely (evaluated by goodness of fit, such as Pearson'$\tau_D$ and radius of the circular ROI $w$.\\ |

- | < | + | $$ |

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

- | </ | + | $$ |

- | 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'maybe we should wait for him to construct another chapter or page for this... | + | 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'[[:. |

=== effective diffusion === | === effective diffusion === | ||

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(almost diffusion) | (almost diffusion) | ||

- | === special cases: anomalous diffusion === | + | === anomalous diffusion === |

==== Reaction Dominant Recovery ==== | ==== Reaction Dominant Recovery ==== | ||

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[{{ : | [{{ : | ||

- | If molecule under study is binding/this interaction. There are two cases on how these two events, diffusion and reaction, are combined in the curve. We could think of two cases. | + | If molecule under study is binding/these interactions. There are two cases on how these two events, diffusion and reaction, are combined in the curve. We could think of two cases. |

- | - **Reaction-dominant recovery** Cases when reaction binding rate (meaning "be become brighter due to diffusion, and then there will be a slower recovery of intensity due to none-bleached fluorescence exchange with the bleached fluorescence. In such cases, we could consider that the recovery curve is dominated by reaction since duration of diffusion-recovery phase and reaction-recovery phase is much shorter for the diffusion-recovery phase, these two phases could be assumed to be separated(call this **reaction-dominant** or **diffusion uncoupled**; | + | - **Reaction-dominant recovery** Cases when reaction binding rate (meaning "first become brighter due to diffusion, and then there will be a slower recovery of intensity due to none-bleached fluorescence exchange with the bleached fluorescence. In such cases, we could consider that the recovery curve is dominated by reaction since duration of diffusion-recovery phase is much shorter compared to reaction-recovery phase. In such case, these two phases are considered to be separable (call this **reaction-dominant** or **diffusion uncoupled**;. We even might not ". |

- | - **Reaction-Diffusion Recovery** Other cases would be when durations of diffusion-recovery phase and reaction-recovery phase are comparable. Then recovery curve consists of a combination of fluorescence that came in with diffusion, and also by the binding of fluorescence molecule at the FRAP bleached field (**" | + | - **Reaction-Diffusion Recovery** Other cases would be when durations of diffusion-recovery phase and reaction-recovery phase are with comparable duration. Then recovery curve consists of a combination of fluorescence that came in with diffusion, and also by the binding of fluorescence molecule at the FRAP bleached field (**" |

- | For a simple chemical reaction with singular type of interaction,could be modeled as compartment system (see figure right)\\ | + | For a simple chemical reaction with singular type of interaction,a compartment system (see figure right)\\ |

- | < | + | $$ |

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

- | </ | + | $$ |

where | where | ||

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

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

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

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

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

- | < | + | $$ |

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

- | </ | + | $$ |

where | where | ||

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

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

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

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[{{ : | [{{ : | ||

[{{ : | [{{ : | ||

- | 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 diffusion in cytoplasm or bound to an immobile structure inside cell. We FRAP at this structure, to know the kinetic constants of the protein against 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) |

- | < | + | $$ |

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

- | </ | + | $$ |

where | where | ||

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

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

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

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

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

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

- | < | + | $$ |

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

- | </ | + | $$ |

- | in addition, density of free molecule in cytoplasm is almost constant so we assume <jsm>[free] = F</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 |

- | < | + | $$ |

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

- | </ | + | $$ |

We get | We get | ||

- | < | + | $$ |

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

- | </ | + | $$ |

where | where | ||

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

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

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

[{{ : | [{{ : | ||

- | < | + | $$ |

\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] | ||

- | </ | + | $$ |

- | < | + | $$ |

\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] | ||

- | </ | + | $$ |

- | < | + | $$ |

\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] | ||

- | </ | + | $$ |

Since | Since | ||

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

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

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

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

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

- | < | + | $$ |

\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] | ||

- | </ | + | $$ |

- | < | + | $$ |

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

- | </ | + | $$ |

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 " | 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 " | ||

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

- | < | + | $$ |

\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}} | ||

- | </ | + | $$ |

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

==== Diffusion and Transport combined Recovery ==== | ==== Diffusion and Transport combined Recovery ==== | ||

- | We did not talk about this issue in the course, but there is another factor that could interfere with recovery curve in vivo: active transport. There is some trial on including issue by Hallen and Endow (2009). | + | We did not talk about this issue in the course, but there is another factor that could interfere with recovery curve in vivo: active transport. There is some trial on including this factor by Hallen and Endow (2009). |

==== Diffusion and Reaction, along with Spatial Context, Geometry ==== | ==== Diffusion and Reaction, along with Spatial Context, Geometry ==== | ||

[{{ : | [{{ : | ||

- | Since molecular behavior inside cell is constrained largely by structure and geometry of intracellular architecture,in interpreting the measurement you did. | + | Since molecular behavior inside cell is constrained largely by structure and geometry of intracellular architecture,when interpreting the FRAP measurement. |

- | Physical parameters such as Diffusion coefficient measured through FRAP is affected largely by geometrical constraint. Even if the geometry is rather simple, there are many obstacles in intracellular space which will cause longer time for molecules to reach from one point to the other. In such cases (which probably is frequently the case), estimated diffusion coefficient would be smaller than that of the actual. Presence of such obstacle should be somehow taken into account. For this reason, estimation of Diffusion coefficient could be more precise if one analyzes the structural geometry of where the molecule is constrained and calculate the actual diffusion coefficient. Such protocol would be especially important if Diffusion and Reaction coupled recovery curve since wrong estimates on Diffusion coefficient would end up in wrong reaction parameters as well. | + | Physical parameters such as Diffusion coefficient measured by FRAP is affected largely by geometrical constraint. Even if the geometry is rather simple, there are many obstacles in intracellular space which will cause longer time for molecules to reach from one point to the other. In such cases (which probably is frequently the case), estimated diffusion coefficient would calculated to be smaller than that of the "true diffusion coefficient". Presence of such obstacles should be somehow taken into account. For this reason, estimation of Diffusion coefficient could be more precise if one analyzes the structural geometry of where the molecule is constrained. Such protocol would be especially important if Diffusion and Reaction coupled recovery curve since wrong estimates on Diffusion coefficient would end up in wrong reaction parameters as well. |

- | Joel Beaudouin who did PhD study at Ellenberg lab in EMBL actually encountered such question in his project on nuclear protein study and solved the problem by using initial image frames of the FRAP experiment and let the molecule to diffuse by simulation, compared with the experimental FRAP image sequence to fit the data to diffusion-reaction model (see above) using ODE solver. Excellent idea. For more details, see Beaudouin et al. (2006). We will later add some protocol to this lecture notes on how to fit FRAP image sequence data using ODE solver (with help of Sebastian Huet). Joel's method was actually made into application called " | + | Joel Beaudouin who did PhD study in the Ellenberg lab in the EMBL actually encountered such question in his project on nuclear protein study and solved the problem by using initial image frames of the FRAP experiment and let the molecule to diffuse by simulation, then fit the simulation with the experimental FRAP image sequence. Diffusion-reaction model was used and simulation was done (see above) using ODE solver to scan through the parameter space. Excellent idea. For more details, see Beaudouin et al. (2006). We will later add some protocol to this lecture notes on how to fit FRAP image sequence data using ODE solver (with help of Sebastian Huet). Joel's method was actually made into application called " |

Other papers we could refer to are Sbalzarini et. al. (2005, 2006). In these papers, authors did two things in parallel: 3D reconstruction of ER membrane structure and FRAP of certain molecule moving around along ER membrane. Using the 3D structure they reconstructed, | Other papers we could refer to are Sbalzarini et. al. (2005, 2006). In these papers, authors did two things in parallel: 3D reconstruction of ER membrane structure and FRAP of certain molecule moving around along ER membrane. Using the 3D structure they reconstructed, | ||

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===== List of Tools for FRAP Analysis ===== | ===== List of Tools for FRAP Analysis ===== | ||

- | |||

==== Basic ==== | ==== Basic ==== | ||

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* [[http:// | * [[http:// | ||

* Import data output from Zeiss, Leica, Olympus measurements and do FRAP fitting. | * Import data output from Zeiss, Leica, Olympus measurements and do FRAP fitting. | ||

+ | * [[http:// | ||

+ | * Similar to above, but stand alone and also incorporated diffusion-reaction model. | ||

* [[http:// | * [[http:// | ||

* Does measurement and fitting. Sprague et al. (2004) Reaction-Diffusion Full model is implemented. | * Does measurement and fitting. Sprague et al. (2004) Reaction-Diffusion Full model is implemented. | ||

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Requires your own coding, customization\\ | Requires your own coding, customization\\ | ||

- | === Analytical Approach === | + | ==== Analytical Approach ==== |

Sprague et. al. (explained above) is an example case of analytically solving the model for the fitting. | Sprague et. al. (explained above) is an example case of analytically solving the model for the fitting. | ||

+ | |||

==== ODE Simulation ==== | ==== ODE Simulation ==== | ||

[{{ : | [{{ : |

documents/100420frapinternal.1272009911.txt.gz · Last modified: 2016/05/24 12:46 (external edit)