Plan

Refs: easyFRAP, and the manual appendix which has the actual equations and good stuff.

Using their terminology here’s the plan:

1. Perform single normalization.

• Divide by the average pre-bleach intensity. Note that the manual refers to double normalization to correct for differences in fluorescence in the background of each image, which we could take into account later. I don’t think that is important right now though…

2. Perform full scale normalization.

• With the singly normalized data subtract the intensity of the first post-bleach frame from the dataset, then divide by 1 - pre-bleach normalized intensity to scale the data from 0 to about 1 (if recovery is full).

3. Before fitting the dataset, set the bleach to time zero.

• effectively we will set the time of the first post bleach frame to t=0.

4. Fit with a single exponential model using nls (see math below)
5. Use estimated coefficients to infer \(\tau_{1/2}\) then \(D_{ap}\) etc.

Single exponential model:

\[ I_{fit} = I_0 - \alpha e^{- \beta t}\] for this model:

\[\tau_{1/2} = \frac{ln(2)}{\beta}\]

Results

Ok let’s do this.

First, let’s read in the csv of the frap data. All of these scans were acquired in the same way, so they should be pretty comparable. In excel I already normalized the intensity values to the average baseline intensity (before bleach), so we already have the singly normalized data.

Here’s what the csv looks like.

df_single_norm <- read_csv("05_30_19_PYO_FRAP_data.csv") %>% 
    rename(single_norm_int = scaled)


df_single_norm

## # A tibble: 3,500 x 12
##    img_num  time intensity avg_baseline single_norm_int filename condition
##      <dbl> <dbl>     <dbl>        <dbl>           <dbl> <chr>    <chr>    
##  1       1 0         0.032         127.        0.000251 1percen… 1percent…
##  2       2 0.208   128.            127.        1.00     1percen… 1percent…
##  3       3 0.399   128.            127.        1.01     1percen… 1percent…
##  4       4 0.591   126.            127.        0.993    1percen… 1percent…
##  5       5 0.782   127.            127.        0.994    1percen… 1percent…
##  6       6 0.974   127.            127.        1.000    1percen… 1percent…
##  7       7 1.17    127.            127.        1.000    1percen… 1percent…
##  8       8 1.39    127.            127.        0.997    1percen… 1percent…
##  9       9 1.58    127.            127.        0.998    1percen… 1percent…
## 10      10 1.77    128.            127.        1.01     1percen… 1percent…
## # … with 3,490 more rows, and 5 more variables: PYO <chr>,
## #   objective <chr>, resolution <chr>, bleach <chr>, id <dbl>

And here’s a plot of that singly normalized dataset.

# ignore the two frames that have ~ 0 intensity (frame 1 and
# frame post bleach)

ggplot(df_single_norm %>% filter(intensity > 0.1), aes(x = time, 
    y = single_norm_int, color = condition, group = id)) + geom_line(alpha = 0.5) + 
    facet_wrap(~condition, scales = "free")

Now let’s perform the full scale normalization.

df_full_norm <- df_single_norm %>% filter(intensity > 1) %>% 
    group_by(condition, id) %>% mutate(min_single_norm_int = min(single_norm_int)) %>% 
    mutate(full_norm_int = (single_norm_int - min_single_norm_int)/(1 - 
        min_single_norm_int))

And plotting that looks like this:

ggplot(df_full_norm, aes(x = time, y = full_norm_int, color = condition, 
    group = id)) + geom_line(alpha = 0.5) + facet_wrap(~condition, 
    scales = "free")

Now let’s normalize the time to the first post_bleach frame time. Note that the bleaches all occur between 2-6 seconds for all of the conditions, so we can find that frame by searching for the minimum timed frame that exists greater than 5 seconds.

df_full_fit <- df_full_norm %>% filter(time > 5) %>% group_by(condition, 
    id) %>% mutate(min_time = min(time)) %>% mutate(norm_time = time - 
    min_time)

ggplot(df_full_fit, aes(x = norm_time, y = full_norm_int, color = condition, 
    group = id)) + geom_line(alpha = 0.5) + facet_wrap(~condition, 
    scales = "free")

Now we are finally read to fit! First, here’s a plot of what the fits will look like.

ggplot(df_full_fit, aes(x = norm_time, y = full_norm_int)) + 
    geom_point(shape = 21) + geom_smooth(method = "nls", formula = y ~ 
    (I0 - a * exp(-B * x)), method.args = list(start = c(I0 = 0.85, 
    a = 0.5, B = 0.563)), se = F) + facet_wrap(condition ~ id)

Not too bad…seems pretty reasonable. Now let’s just perform the nls command and extract the coefficients in to a dataframe etc.

sing_exp_coef <- df_full_fit %>% group_by(condition, id) %>% 
    do(tidy(nls(formula = full_norm_int ~ I0 - a * exp(-B * norm_time), 
        data = ., start = list(I0 = 0.85, a = 0.5, B = 0.563)), 
        conf.int = T))

ggplot(sing_exp_coef, aes(x = condition, y = estimate, color = condition)) + 
    geom_pointrange(aes(ymin = conf.low, ymax = conf.high), position = position_jitter(height = 0)) + 
    facet_wrap(~term, scales = "free") + scale_color_viridis_d()

There are a few things to notice here. First, notice that the 95% confidence intervals are definitely the tightest for the 4% agarose condition. This makes sense, since we acquired the nicest FRAP curves with that conditions, and you can see that the nls fits those data quite well. Perhaps because the FRAPs were generally noisy, you can see that our parameter estimates definitely overlap between the different conditions for all the parameters (except maybe \(I_0\)). So already, I know that there’s know statistically significant difference between the conditions for \(\beta\), which will determine the \(\tau_{1/2}\).

But let’s go ahead and calculate \(\tau_{1/2}\) from \(\beta\) to see if the values make sense.

tau_est <- sing_exp_coef %>% filter(term == "B") %>% mutate(tau_mean = log(2)/estimate) %>% 
    mutate(tau_low = log(2)/conf.low) %>% mutate(tau_high = log(2)/conf.high)


ggplot(tau_est, aes(x = condition, y = tau_mean, color = condition)) + 
    geom_pointrange(aes(ymin = tau_low, ymax = tau_high), position = position_jitter(height = 0)) + 
    scale_color_viridis_d()

Overall, this estimate for \(\tau_{1/2}\) makes sense. It’s about what we could see by eye for these noisy curves…I think I estimated 5 seconds for my original FRAP acquisition.

For the DNA sample the first run (id = 1) seems to be an outlier. Let’s look at it compared to the next acquisition.

ggplot(df_full_fit %>% filter(condition == "1pctAgaroseDNA" & 
    id == 1), aes(x = norm_time, y = full_norm_int)) + geom_point(shape = 21) + 
    geom_smooth(method = "nls", formula = y ~ (I0 - a * exp(-B * 
        x)), method.args = list(start = c(I0 = 0.85, a = 0.5, 
        B = 0.563)), se = F)

ggplot(df_full_fit %>% filter(condition == "1pctAgaroseDNA" & 
    id == 2), aes(x = norm_time, y = full_norm_int)) + geom_point(shape = 21) + 
    geom_smooth(method = "nls", formula = y ~ (I0 - a * exp(-B * 
        x)), method.args = list(start = c(I0 = 0.85, a = 0.5, 
        B = 0.563)), se = F)

You can see that theres a weird grouping of points that do not decay following the bleach and the nls call basically ignores them giving a very shallow sloping curve. Because the recovery rises above 1 we know there was also significant TCEP equilibration happening, so I think we can safely throw this one away.

Looking at some of the other FRAP curves up close you can see the fits are ok, not great. They often don’t take into account the full decay including the first few points post bleach. There may be a difference with a different analysis scheme, but I’m happy with this for now.

ggplot(df_full_fit %>% filter(condition == "1percentAgarose" & 
    id == 15), aes(x = norm_time, y = full_norm_int)) + geom_point(shape = 21) + 
    geom_smooth(method = "nls", formula = y ~ (I0 - a * exp(-B * 
        x)), method.args = list(start = c(I0 = 0.85, a = 0.5, 
        B = 0.563)), se = F)

ggplot(df_full_fit %>% filter(condition == "4pctAgarose" & id == 
    1), aes(x = norm_time, y = full_norm_int)) + geom_point(shape = 21) + 
    geom_smooth(method = "nls", formula = y ~ (I0 - a * exp(-B * 
        x)), method.args = list(start = c(I0 = 0.85, a = 0.5, 
        B = 0.563)), se = F)

Anyway, let’s end by calculating \(D_{ap}\) from our \(\tau_{1/2}\). Based on the expression from this paper we can calculate \(D_{ap}\) with the following expression:

\[D = 0.224 \frac{r^2}{\tau_{1/2}}\] where \(r\) is the nominal bleach radius from the ROI we bleached. I believe the ROIs were all approximately 50um or 0.005cm. Note that there are definitely biases in the assumptions for this model, but I’m going to use it for its simplicity now. (for example it assumes instantaneous bleach)

r = 0.005

d_est <- tau_est %>% mutate(d_mean = (0.224 * r^2)/tau_mean) %>% 
    mutate(d_low = (0.224 * r^2)/tau_low) %>% mutate(d_high = (0.224 * 
    r^2)/tau_high)


ggplot(d_est, aes(x = condition, y = d_mean, color = condition)) + 
    geom_pointrange(aes(ymin = d_low, ymax = d_high), position = position_jitter(height = 0)) + 
    scale_color_viridis_d()

These values are a little lower than expected, just barely in the \(10^{-6} cm^2 / sec\) range. Perhaps this error is due to the poor assumptions mentioned in the paper. Unfortunately it was not possible to correct using the effective bleach radius as they describe…

“Failure to take this into account can lead to underestimation of D when the Soumpasis or Axelrod equations are used to analyze confocal FRAP data, especially for fast diffusing soluble proteins”

Analysis

Using this analysis scheme, I think we have a reasonable first pass quantifying the FRAP data. Unfortunately we don’t see evidence that the diffusion of the reduced PYO is different (slower) in high percentage agarose or agarose with DNA. That said, this is very good to know before starting the in vitro echem. Moving forward I have the highest confidence in the 4% agarose FRAP estimates, so that may be useful down the road. Now we need to think about what other perturbations we can perform in the agarose to perturb the diffusion of reduced PYO…

FRAP acquisition

I think I have now optimized the acquisition settings as much as possible. We are acquiring as fast as possible, and we are sensitively measuring PYO fluorescence. The part of this FRAP that is still not great is the bleach. I can think of three reasons that may be:

1. For some unknown reason PYO is difficult to bleach.
2. PYOred doesn’t absorb well at 405 and would be better bleached at a lower wavelength.
3. We are bleaching PYO effectively, but because bleaching often causes photooxidation, PYO may be almost instantly re-reduced by the abundant TCEP.

Obviously these explanations lend themselves to different solutions, so in thinking of what to try next I would say we could: * Try FRAP with the adjustable 2-photon laser on the LSM-880, which can be adjusted down into the 300’s nm range. * Use electrochemically reduced PYO. Either generated from the survival chambers or in situ with the ITO coverslips I ordered.

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