Abstract

Ratiometric fluorescent indicators are used for making quantitative measurements of a variety of physiological variables. Their utility is often limited by noise. This is the second in a series of papers describing statistical methods for denoising ratiometric data with the aim of obtaining improved quantitative estimates of variables of interest. Here, we outline a statistical optimization method that is designed for the analysis of ratiometric imaging data in which multiple measurements have been taken of systems responding to the same stimulation protocol. This method takes advantage of correlated information across multiple datasets for objectively detecting and estimating ratiometric signals. We demonstrate our method by showing results of its application on multiple, ratiometric calcium imaging experiments.

© 2008 Optical Society of America

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References

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  1. J. Broder, A. Majumder, E. Porter, G. Srinivasmoorthy, C. Keith, J. Lauderdale, and A. Sornborger, “Estimating weak ratiometric signals in imaging data. I. Dual-channel data,” J. Opt. Soc. Am. A 24, 2921-2931 (2007).
    [CrossRef]
  2. H. Hotelling, “Relations between two sets of variates,” Biometrika 32, 38-45 (1936).
  3. J. D. Carroll, “Generalization of canonical correlation analysis to three or more sets of variables,” Proceedings of 76th Annual Convention of American Psychological Association (APA, 1968), pp. 227-228.
  4. J. Hanks, “Hanks' balanced salt solution and pH control,” Tissue Culture Association Manual 3, 3-5 (1976).
  5. K. Truong, A. Sawano, A. Mizuno, H. Hama, K. Tong, T. Mal, A. Miyawaki, and M. Ikura, “FRET-based in vivoCa2+ imaging by a new calmodulin-GFP fusion molecule,” Nat. Struct. Biol. 8, 1069-1073 (2001).
    [CrossRef] [PubMed]
  6. T. Anderson, An Introduction to Multivariate Statistical Analysis (Wiley, 1984).

2007 (1)

2001 (1)

K. Truong, A. Sawano, A. Mizuno, H. Hama, K. Tong, T. Mal, A. Miyawaki, and M. Ikura, “FRET-based in vivoCa2+ imaging by a new calmodulin-GFP fusion molecule,” Nat. Struct. Biol. 8, 1069-1073 (2001).
[CrossRef] [PubMed]

1976 (1)

J. Hanks, “Hanks' balanced salt solution and pH control,” Tissue Culture Association Manual 3, 3-5 (1976).

1936 (1)

H. Hotelling, “Relations between two sets of variates,” Biometrika 32, 38-45 (1936).

Anderson, T.

T. Anderson, An Introduction to Multivariate Statistical Analysis (Wiley, 1984).

Broder, J.

Carroll, J. D.

J. D. Carroll, “Generalization of canonical correlation analysis to three or more sets of variables,” Proceedings of 76th Annual Convention of American Psychological Association (APA, 1968), pp. 227-228.

Hama, H.

K. Truong, A. Sawano, A. Mizuno, H. Hama, K. Tong, T. Mal, A. Miyawaki, and M. Ikura, “FRET-based in vivoCa2+ imaging by a new calmodulin-GFP fusion molecule,” Nat. Struct. Biol. 8, 1069-1073 (2001).
[CrossRef] [PubMed]

Hanks, J.

J. Hanks, “Hanks' balanced salt solution and pH control,” Tissue Culture Association Manual 3, 3-5 (1976).

Hotelling, H.

H. Hotelling, “Relations between two sets of variates,” Biometrika 32, 38-45 (1936).

Ikura, M.

K. Truong, A. Sawano, A. Mizuno, H. Hama, K. Tong, T. Mal, A. Miyawaki, and M. Ikura, “FRET-based in vivoCa2+ imaging by a new calmodulin-GFP fusion molecule,” Nat. Struct. Biol. 8, 1069-1073 (2001).
[CrossRef] [PubMed]

Keith, C.

Lauderdale, J.

Majumder, A.

Mal, T.

K. Truong, A. Sawano, A. Mizuno, H. Hama, K. Tong, T. Mal, A. Miyawaki, and M. Ikura, “FRET-based in vivoCa2+ imaging by a new calmodulin-GFP fusion molecule,” Nat. Struct. Biol. 8, 1069-1073 (2001).
[CrossRef] [PubMed]

Miyawaki, A.

K. Truong, A. Sawano, A. Mizuno, H. Hama, K. Tong, T. Mal, A. Miyawaki, and M. Ikura, “FRET-based in vivoCa2+ imaging by a new calmodulin-GFP fusion molecule,” Nat. Struct. Biol. 8, 1069-1073 (2001).
[CrossRef] [PubMed]

Mizuno, A.

K. Truong, A. Sawano, A. Mizuno, H. Hama, K. Tong, T. Mal, A. Miyawaki, and M. Ikura, “FRET-based in vivoCa2+ imaging by a new calmodulin-GFP fusion molecule,” Nat. Struct. Biol. 8, 1069-1073 (2001).
[CrossRef] [PubMed]

Porter, E.

Sawano, A.

K. Truong, A. Sawano, A. Mizuno, H. Hama, K. Tong, T. Mal, A. Miyawaki, and M. Ikura, “FRET-based in vivoCa2+ imaging by a new calmodulin-GFP fusion molecule,” Nat. Struct. Biol. 8, 1069-1073 (2001).
[CrossRef] [PubMed]

Sornborger, A.

Srinivasmoorthy, G.

Tong, K.

K. Truong, A. Sawano, A. Mizuno, H. Hama, K. Tong, T. Mal, A. Miyawaki, and M. Ikura, “FRET-based in vivoCa2+ imaging by a new calmodulin-GFP fusion molecule,” Nat. Struct. Biol. 8, 1069-1073 (2001).
[CrossRef] [PubMed]

Truong, K.

K. Truong, A. Sawano, A. Mizuno, H. Hama, K. Tong, T. Mal, A. Miyawaki, and M. Ikura, “FRET-based in vivoCa2+ imaging by a new calmodulin-GFP fusion molecule,” Nat. Struct. Biol. 8, 1069-1073 (2001).
[CrossRef] [PubMed]

Biometrika (1)

H. Hotelling, “Relations between two sets of variates,” Biometrika 32, 38-45 (1936).

J. Opt. Soc. Am. A (1)

Nat. Struct. Biol. (1)

K. Truong, A. Sawano, A. Mizuno, H. Hama, K. Tong, T. Mal, A. Miyawaki, and M. Ikura, “FRET-based in vivoCa2+ imaging by a new calmodulin-GFP fusion molecule,” Nat. Struct. Biol. 8, 1069-1073 (2001).
[CrossRef] [PubMed]

Tissue Culture Association Manual (1)

J. Hanks, “Hanks' balanced salt solution and pH control,” Tissue Culture Association Manual 3, 3-5 (1976).

Other (2)

T. Anderson, An Introduction to Multivariate Statistical Analysis (Wiley, 1984).

J. D. Carroll, “Generalization of canonical correlation analysis to three or more sets of variables,” Proceedings of 76th Annual Convention of American Psychological Association (APA, 1968), pp. 227-228.

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Figures (6)

Fig. 1
Fig. 1

Simulation of a brain region with four spatiotemporally distinct regions of activity. Each region of activity is represented by a bright area (Gaussian profile) with corresponding timecourses at two wavelengths. The intensity at one wavelength increases in response to activity, while at the other wavelength the intensity decreases in response to the same activity. The activity in regions 1, 2, and 4 represents sinusoidal responses at different frequencies. Active region 3 represents an isolated event. These regions of activity are combined to form one simulated imaging dataset. Low-amplitude, normally distributed noise was added to the dataset. This dataset represents simulated activity in a single brain. SOARS/CCA is designed to analyze a number of such datasets at the same time. The SOARS/CCA analysis shown in Fig. 3 was performed on four datasets, all of which were constructed in a similar way.

Fig. 2
Fig. 2

Intermediate results from a standard SOARS analysis on simulated data. The first five eigenimages and their associated timecourses from a SOARS analysis of the simulated data shown in Fig. 1 are depicted. The first four eigenimages are statistically significant and capture most of the ratiometric activity in the data; however, the activity is mixed among the eigenimages and therefore the active regions appear in various superpositions within the eigenimages. The timecourses display similar superpositions of the temporal activity. The fifth eigenimage and timecourse are much less statistically significant and represent noise.

Fig. 3
Fig. 3

Intermediate results from a SOARS/CCA analysis on simulated data. Four simulated ratiometric imaging datasets of the type depicted in Fig. 1 were analyzed using the SOARS/CCA method. Dataset 1 in this figure is the dataset depicted in Fig. 1. In each of these datasets, there was a single temporally correlated region of activity of similar size and timecourse to that in active region 4 in Fig. 1. The other regions of activity had randomly assigned sizes, locations, and timecourses. In the leftmost column, we present the first two eigenvectors Z 1 and Z 2 . In the rows to the right of the eigenvectors, we depict their associated transformation vectors ϕ 11 , 12 , 13 , 14 and ϕ 21 , 22 , 23 , 24 . Below the transformation vectors, we depict the projections π 11 , 12 , 13 , 14 and π 21 , 22 , 23 , 24 .

Fig. 4
Fig. 4

Assigning a statistical threshold for a SOARS/CCA analysis. Panel A The first 20 correlations ρ as a function of CCA eigenvector index for the four simulated datasets. An arrow shows the location of the “knee,” which represents the location of an abrupt change in the correlations. Panels B–E Histograms of the first four eigenvectors Z 1 , 2 , 3 , 4 with normal distributions with the same mean and standard deviation superimposed.

Fig. 5
Fig. 5

Intermediate results from a SOARS/CCA analysis on experimental data. Eight ratiometric imaging datasets were analyzed using the SOARS/CCA method. Datasets 1–3 and 5–8 represent measurements of the calcium response of PC12 cells loaded with fluo-4 and Fura-Red to ten cycles of stimulation ( 2 min on, 2 min off) with 50 mM of K + solution, followed by 5 min with ionomycin + EGTA (low calcium clamp), followed by 5 min of ionomycin + saturating Ca + + . Dataset 4 is a control in which the high K + solution was replaced by an iso-osmotic solution. The switching profile is shown at the bottom of panel Z 3 and in bar form in other timecourse plots. In the leftmost column, we present the first three eigenvectors Z 1 , Z 2 , and Z 3 . In the rows to the right of the eigenvectors, we depict their associated transformation vectors ϕ 11 , 12 , , ϕ 21 , 22 , , and ϕ 31 , 32 , . Below the transformation vectors, we depict the projections π 11 , 12 , , etc. In the bottom row, we plot the projections of the transformation vectors of the third eigenvector in the fluo-4 and Fura-red channels. ρ = R i 2 8 with i 1 , 2 , 3 (see text) are plotted in the rightmost column. Scale bar, 20 μ m .

Fig. 6
Fig. 6

A comparison of SOARS versus SOARS/CCA estimates. Panel A depicts the eigenimage best correlated with our stimulus protocol resulting from a SOARS analysis on dataset 1. Panel B shows the associated timecourses in the fluo-4 and Fura-red channels. Panel C depicts the transformation vector from a SOARS/CCA analysis associated with the eigenvector best correlated with our stimulus protocol ( Z 3 ) . Panel D shows the associated timecourses in the fluo-4 and Fura-red channels. The SOARS/CCA eigenvector has a higher contrast than the SOARS eigenvector and the SOARS/CCA timecourse lacks the initial transient obvious in the fluo-4 channel of the SOARS timecourses and has better resolved oscillations. The stimulus switching paradigm is depicted at the bottom of both timecourse plots. Note that there is a delay between switching stimulants and the calcium response (See [1]). Scale bar, 20 μ m .

Equations (15)

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X 1 ( t ) = α b ( t ) ( 1 f ( t ) ) + η 1 ( t ) ,
X 2 ( t ) = β b ( t ) ( 1 + f ( t ) ) + η 2 ( t ) .
X 1 ( t ) X 2 ( t ) = α β 1 f ( t ) 1 + f ( t )
X i ( t ) = X i ( t ) X ¯ i V ¯ i ,
E { ϵ ( t ) } E { X 1 ( t ) X 2 ( t ) } = 0 ,
r ( Z , ϕ j T ϵ j ) = Z ϵ j T ( ϵ j ϵ j T ) 1 ϵ j Z T Z Z T .
R 2 = Z Q Z T Z Z T ,
Q = j = 1 n ϵ j T ( ϵ j ϵ j T ) 1 ϵ j .
ϕ k j = Z k ϵ j T ( ϵ j ϵ j T ) 1 .
ϵ j = u j s j v j .
X j 1 denoise ( t ) i R k 2 > threshold ( ψ i j , X j 1 ( t ) ) ψ i ,
X j 2 denoise ( t ) i R k 2 > threshold ( ψ i j , X j 2 ( t ) ) ψ i .
R j m estimate ( t ) V ¯ ̂ j 1 , m X j 1 , m denoise ( t ) + X ¯ ̂ j 1 , m V ¯ ̂ j 2 , m X j 2 , m denoise ( t ) + X ¯ ̂ j 2 , m ,
X 1 ( x , y , t ) = B + i = 1 4 G i ( x , y ) T i ( t ) + η 1 ( t ) ,
X 2 ( x , y , t ) = B i = 1 4 G i ( x , y ) T i ( t ) + η 2 ( t ) .

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