Abstract

Quantification of protein abundance and subcellular localization dynamics from fluorescence microscopy images is of high contemporary interest in cell and molecular biology. For large-scale studies of cell populations and for time-lapse studies, such quantitative analysis can not be performed effectively without some kind of automated image analysis tool. Here, we present fast algorithms for automatic cell contour recognition in bright field images, optimized to the model organism budding yeast (Saccharomyces cerevisiae). The cell contours can be used to effectively quantify cell morphology parameters as well as protein abundance and subcellular localization from overlaid fluorescence data.

© 2008 Optical Society of America

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2007 (5)

S. Di Talia, J. Skotheim, J. Bean, E. Siggia, and F. Cross, "The effects of molecular noise and size control on variability in the budding yeast cell cycle," Nature 448, 947-951 (2007).
[CrossRef] [PubMed]

S.-C. Chen, T. Zhao, G. J. Gordon, and R. F. Murphy, "Automated image analysis of protein localization in budding yeast," Bioinformatics 23, 66-71 (2007).
[CrossRef]

M. A. de Carvalho, R. de A. Lotufo, and M. Couprie, "Morphological segmentation of yeast by image analysis," Image Vis. Comput. 25, 34-39 (2007).
[CrossRef]

A. Gordon, A. Colman-Lerner, T. Chin, K. Benjamin, R. Yu, and R. Brent, "Single-cell quantification of molecules and rates using open-source microscope-based cytometry," Nat. Methods 4, 175-181 (2007).
[CrossRef] [PubMed]

P. Jorgensen, N. P. Edgington, B. L. Schneider, I. Rupes, M. Tyers, and B. Futcher, "The Size of the Nucleus Increases as Yeast Cells Grow," Mol. Biol. Cell 18, 3523-3532 (2007).
[CrossRef] [PubMed]

2006 (2)

Y. Sheu and B. Stillman, "Cdc7-Dbf4 phosphorylates MCM proteins via a docking site-mediated mechanism to promote S phase progression," Mol. Cell 24, 101-113 (2006).
[CrossRef] [PubMed]

J. Bean, E. Siggia, and F. Cross, "Coherence and timing of cell cycle start examined at single-cell resolution," Molecular Cell 21, 3-14 (2006).
[CrossRef] [PubMed]

2005 (2)

A. Colman-Lerner, A. Gordon, E. Serra, T. Chin, O. Resnekov, D. Endy, C. Pesce, and R. Brent, "Regulated cell-to-cell variation in a cell-fate decision system," Nature 437, 699-706 (2005).
[CrossRef] [PubMed]

T. Saito, J. Sese, Y. Nakatani, F. Sano, M. Yukawa, Y. Ohya, and S. Morishita, "Data mining tools for the Saccharomyces cerevisiae morphological database," Nucleic Acids Res. 33, 753-757 (2005).
[CrossRef]

2004 (1)

S. Forsburg, "Eukaryotic MCM proteins: Beyond replication initiation," Microbiol. Mol. Biol. Rev. 68, 109-131 (2004).
[CrossRef] [PubMed]

2003 (1)

W. Huh, J. Falvo, L. Gerke, A. Carroll, R. Howson, J. Weissman, and E. O�??Shea, "Global analysis of protein localization in budding yeast," Nature 425, 686-691 (2003).
[CrossRef] [PubMed]

2001 (1)

S. Forsburg, "The art and design of genetic screens: Yeast," Nat. Rev. Genet. 2, 659-668 (2001).
[CrossRef] [PubMed]

2000 (1)

V. Nguyen, C. Co, K. Irie, and J. Li, "Clb/Cdc28 kinases promote nuclear export of the replication initiator proteins Mcm2-7," Curr. Biol. 10, 195-205 (2000).
[CrossRef] [PubMed]

1999 (1)

K. Labib, J. Diffley, and S. Kearsey, "G1-phase and B-type cyclins exclude the DNA-replication factor Mcm4 from the nucleus," Nat. Cell Biol. 1, 415-422 (1999).
[CrossRef] [PubMed]

1998 (2)

C. Xu and J. Prince, "Snakes, Shapes, and Gradient Vector Flow," IEEE Trans. Image Proc. 7, 359-369 (1998).
[CrossRef]

H. Rue and O. K. Husby, "Identification of partly destroyed objects using deformable templates," Stat. Comput. 8, 221-228 (1998).
[CrossRef]

1997 (2)

K. V. Mardia, W. Qian, D. Shah, and K. M. de Souza, "Deformable Template Recognition of Multiple Occluded Objects," IEEE Trans. Pattern Anal. and Mach. Intell. 19, 1035-1042 (1997).
[CrossRef]

D. Botstein, S. Chervitz, and M. Cherry, "Yeast as a model organism," Science 277, 1259-1260 (1997).
[CrossRef] [PubMed]

1979 (1)

N. Otsu, "A threshold selection method from gray-level histograms," IEEE Trans. on Systems, Man, and Cybernetics 9, 62-66 (1979).
[CrossRef]

1977 (1)

L. Hartwell and L. Unger, "Unequal division in Saccharomyces cerevisiae and its implications for the control of cell division," J. Cell Biol. 75, 422-435 (1977).
[CrossRef] [PubMed]

Bean, J.

S. Di Talia, J. Skotheim, J. Bean, E. Siggia, and F. Cross, "The effects of molecular noise and size control on variability in the budding yeast cell cycle," Nature 448, 947-951 (2007).
[CrossRef] [PubMed]

J. Bean, E. Siggia, and F. Cross, "Coherence and timing of cell cycle start examined at single-cell resolution," Molecular Cell 21, 3-14 (2006).
[CrossRef] [PubMed]

Benjamin, K.

A. Gordon, A. Colman-Lerner, T. Chin, K. Benjamin, R. Yu, and R. Brent, "Single-cell quantification of molecules and rates using open-source microscope-based cytometry," Nat. Methods 4, 175-181 (2007).
[CrossRef] [PubMed]

Botstein, D.

D. Botstein, S. Chervitz, and M. Cherry, "Yeast as a model organism," Science 277, 1259-1260 (1997).
[CrossRef] [PubMed]

Brent, R.

A. Gordon, A. Colman-Lerner, T. Chin, K. Benjamin, R. Yu, and R. Brent, "Single-cell quantification of molecules and rates using open-source microscope-based cytometry," Nat. Methods 4, 175-181 (2007).
[CrossRef] [PubMed]

A. Colman-Lerner, A. Gordon, E. Serra, T. Chin, O. Resnekov, D. Endy, C. Pesce, and R. Brent, "Regulated cell-to-cell variation in a cell-fate decision system," Nature 437, 699-706 (2005).
[CrossRef] [PubMed]

Carroll, A.

W. Huh, J. Falvo, L. Gerke, A. Carroll, R. Howson, J. Weissman, and E. O�??Shea, "Global analysis of protein localization in budding yeast," Nature 425, 686-691 (2003).
[CrossRef] [PubMed]

Chen, S.-C.

S.-C. Chen, T. Zhao, G. J. Gordon, and R. F. Murphy, "Automated image analysis of protein localization in budding yeast," Bioinformatics 23, 66-71 (2007).
[CrossRef]

Cherry, M.

D. Botstein, S. Chervitz, and M. Cherry, "Yeast as a model organism," Science 277, 1259-1260 (1997).
[CrossRef] [PubMed]

Chervitz, S.

D. Botstein, S. Chervitz, and M. Cherry, "Yeast as a model organism," Science 277, 1259-1260 (1997).
[CrossRef] [PubMed]

Chin, T.

A. Gordon, A. Colman-Lerner, T. Chin, K. Benjamin, R. Yu, and R. Brent, "Single-cell quantification of molecules and rates using open-source microscope-based cytometry," Nat. Methods 4, 175-181 (2007).
[CrossRef] [PubMed]

A. Colman-Lerner, A. Gordon, E. Serra, T. Chin, O. Resnekov, D. Endy, C. Pesce, and R. Brent, "Regulated cell-to-cell variation in a cell-fate decision system," Nature 437, 699-706 (2005).
[CrossRef] [PubMed]

Co, C.

V. Nguyen, C. Co, K. Irie, and J. Li, "Clb/Cdc28 kinases promote nuclear export of the replication initiator proteins Mcm2-7," Curr. Biol. 10, 195-205 (2000).
[CrossRef] [PubMed]

Colman-Lerner, A.

A. Gordon, A. Colman-Lerner, T. Chin, K. Benjamin, R. Yu, and R. Brent, "Single-cell quantification of molecules and rates using open-source microscope-based cytometry," Nat. Methods 4, 175-181 (2007).
[CrossRef] [PubMed]

A. Colman-Lerner, A. Gordon, E. Serra, T. Chin, O. Resnekov, D. Endy, C. Pesce, and R. Brent, "Regulated cell-to-cell variation in a cell-fate decision system," Nature 437, 699-706 (2005).
[CrossRef] [PubMed]

Cross, F.

S. Di Talia, J. Skotheim, J. Bean, E. Siggia, and F. Cross, "The effects of molecular noise and size control on variability in the budding yeast cell cycle," Nature 448, 947-951 (2007).
[CrossRef] [PubMed]

J. Bean, E. Siggia, and F. Cross, "Coherence and timing of cell cycle start examined at single-cell resolution," Molecular Cell 21, 3-14 (2006).
[CrossRef] [PubMed]

de Carvalho, M. A.

M. A. de Carvalho, R. de A. Lotufo, and M. Couprie, "Morphological segmentation of yeast by image analysis," Image Vis. Comput. 25, 34-39 (2007).
[CrossRef]

de Souza, K. M.

K. V. Mardia, W. Qian, D. Shah, and K. M. de Souza, "Deformable Template Recognition of Multiple Occluded Objects," IEEE Trans. Pattern Anal. and Mach. Intell. 19, 1035-1042 (1997).
[CrossRef]

Di Talia, S.

S. Di Talia, J. Skotheim, J. Bean, E. Siggia, and F. Cross, "The effects of molecular noise and size control on variability in the budding yeast cell cycle," Nature 448, 947-951 (2007).
[CrossRef] [PubMed]

Diffley, J.

K. Labib, J. Diffley, and S. Kearsey, "G1-phase and B-type cyclins exclude the DNA-replication factor Mcm4 from the nucleus," Nat. Cell Biol. 1, 415-422 (1999).
[CrossRef] [PubMed]

Edgington, N. P.

P. Jorgensen, N. P. Edgington, B. L. Schneider, I. Rupes, M. Tyers, and B. Futcher, "The Size of the Nucleus Increases as Yeast Cells Grow," Mol. Biol. Cell 18, 3523-3532 (2007).
[CrossRef] [PubMed]

Endy, D.

A. Colman-Lerner, A. Gordon, E. Serra, T. Chin, O. Resnekov, D. Endy, C. Pesce, and R. Brent, "Regulated cell-to-cell variation in a cell-fate decision system," Nature 437, 699-706 (2005).
[CrossRef] [PubMed]

Falvo, J.

W. Huh, J. Falvo, L. Gerke, A. Carroll, R. Howson, J. Weissman, and E. O�??Shea, "Global analysis of protein localization in budding yeast," Nature 425, 686-691 (2003).
[CrossRef] [PubMed]

Forsburg, S.

S. Forsburg, "Eukaryotic MCM proteins: Beyond replication initiation," Microbiol. Mol. Biol. Rev. 68, 109-131 (2004).
[CrossRef] [PubMed]

S. Forsburg, "The art and design of genetic screens: Yeast," Nat. Rev. Genet. 2, 659-668 (2001).
[CrossRef] [PubMed]

Futcher, B.

P. Jorgensen, N. P. Edgington, B. L. Schneider, I. Rupes, M. Tyers, and B. Futcher, "The Size of the Nucleus Increases as Yeast Cells Grow," Mol. Biol. Cell 18, 3523-3532 (2007).
[CrossRef] [PubMed]

Gerke, L.

W. Huh, J. Falvo, L. Gerke, A. Carroll, R. Howson, J. Weissman, and E. O�??Shea, "Global analysis of protein localization in budding yeast," Nature 425, 686-691 (2003).
[CrossRef] [PubMed]

Gordon, A.

A. Gordon, A. Colman-Lerner, T. Chin, K. Benjamin, R. Yu, and R. Brent, "Single-cell quantification of molecules and rates using open-source microscope-based cytometry," Nat. Methods 4, 175-181 (2007).
[CrossRef] [PubMed]

A. Colman-Lerner, A. Gordon, E. Serra, T. Chin, O. Resnekov, D. Endy, C. Pesce, and R. Brent, "Regulated cell-to-cell variation in a cell-fate decision system," Nature 437, 699-706 (2005).
[CrossRef] [PubMed]

Gordon, G. J.

S.-C. Chen, T. Zhao, G. J. Gordon, and R. F. Murphy, "Automated image analysis of protein localization in budding yeast," Bioinformatics 23, 66-71 (2007).
[CrossRef]

Hartwell, L.

L. Hartwell and L. Unger, "Unequal division in Saccharomyces cerevisiae and its implications for the control of cell division," J. Cell Biol. 75, 422-435 (1977).
[CrossRef] [PubMed]

Howson, R.

W. Huh, J. Falvo, L. Gerke, A. Carroll, R. Howson, J. Weissman, and E. O�??Shea, "Global analysis of protein localization in budding yeast," Nature 425, 686-691 (2003).
[CrossRef] [PubMed]

Huh, W.

W. Huh, J. Falvo, L. Gerke, A. Carroll, R. Howson, J. Weissman, and E. O�??Shea, "Global analysis of protein localization in budding yeast," Nature 425, 686-691 (2003).
[CrossRef] [PubMed]

Husby, O. K.

H. Rue and O. K. Husby, "Identification of partly destroyed objects using deformable templates," Stat. Comput. 8, 221-228 (1998).
[CrossRef]

Irie, K.

V. Nguyen, C. Co, K. Irie, and J. Li, "Clb/Cdc28 kinases promote nuclear export of the replication initiator proteins Mcm2-7," Curr. Biol. 10, 195-205 (2000).
[CrossRef] [PubMed]

Jorgensen, P.

P. Jorgensen, N. P. Edgington, B. L. Schneider, I. Rupes, M. Tyers, and B. Futcher, "The Size of the Nucleus Increases as Yeast Cells Grow," Mol. Biol. Cell 18, 3523-3532 (2007).
[CrossRef] [PubMed]

Kearsey, S.

K. Labib, J. Diffley, and S. Kearsey, "G1-phase and B-type cyclins exclude the DNA-replication factor Mcm4 from the nucleus," Nat. Cell Biol. 1, 415-422 (1999).
[CrossRef] [PubMed]

Labib, K.

K. Labib, J. Diffley, and S. Kearsey, "G1-phase and B-type cyclins exclude the DNA-replication factor Mcm4 from the nucleus," Nat. Cell Biol. 1, 415-422 (1999).
[CrossRef] [PubMed]

Li, J.

V. Nguyen, C. Co, K. Irie, and J. Li, "Clb/Cdc28 kinases promote nuclear export of the replication initiator proteins Mcm2-7," Curr. Biol. 10, 195-205 (2000).
[CrossRef] [PubMed]

Mardia, K. V.

K. V. Mardia, W. Qian, D. Shah, and K. M. de Souza, "Deformable Template Recognition of Multiple Occluded Objects," IEEE Trans. Pattern Anal. and Mach. Intell. 19, 1035-1042 (1997).
[CrossRef]

Morishita, S.

T. Saito, J. Sese, Y. Nakatani, F. Sano, M. Yukawa, Y. Ohya, and S. Morishita, "Data mining tools for the Saccharomyces cerevisiae morphological database," Nucleic Acids Res. 33, 753-757 (2005).
[CrossRef]

Murphy, R. F.

S.-C. Chen, T. Zhao, G. J. Gordon, and R. F. Murphy, "Automated image analysis of protein localization in budding yeast," Bioinformatics 23, 66-71 (2007).
[CrossRef]

Nakatani, Y.

T. Saito, J. Sese, Y. Nakatani, F. Sano, M. Yukawa, Y. Ohya, and S. Morishita, "Data mining tools for the Saccharomyces cerevisiae morphological database," Nucleic Acids Res. 33, 753-757 (2005).
[CrossRef]

Nguyen, V.

V. Nguyen, C. Co, K. Irie, and J. Li, "Clb/Cdc28 kinases promote nuclear export of the replication initiator proteins Mcm2-7," Curr. Biol. 10, 195-205 (2000).
[CrossRef] [PubMed]

O???Shea, E.

W. Huh, J. Falvo, L. Gerke, A. Carroll, R. Howson, J. Weissman, and E. O�??Shea, "Global analysis of protein localization in budding yeast," Nature 425, 686-691 (2003).
[CrossRef] [PubMed]

Ohya, Y.

T. Saito, J. Sese, Y. Nakatani, F. Sano, M. Yukawa, Y. Ohya, and S. Morishita, "Data mining tools for the Saccharomyces cerevisiae morphological database," Nucleic Acids Res. 33, 753-757 (2005).
[CrossRef]

Otsu, N.

N. Otsu, "A threshold selection method from gray-level histograms," IEEE Trans. on Systems, Man, and Cybernetics 9, 62-66 (1979).
[CrossRef]

Pesce, C.

A. Colman-Lerner, A. Gordon, E. Serra, T. Chin, O. Resnekov, D. Endy, C. Pesce, and R. Brent, "Regulated cell-to-cell variation in a cell-fate decision system," Nature 437, 699-706 (2005).
[CrossRef] [PubMed]

Prince, J.

C. Xu and J. Prince, "Snakes, Shapes, and Gradient Vector Flow," IEEE Trans. Image Proc. 7, 359-369 (1998).
[CrossRef]

Qian, W.

K. V. Mardia, W. Qian, D. Shah, and K. M. de Souza, "Deformable Template Recognition of Multiple Occluded Objects," IEEE Trans. Pattern Anal. and Mach. Intell. 19, 1035-1042 (1997).
[CrossRef]

Resnekov, O.

A. Colman-Lerner, A. Gordon, E. Serra, T. Chin, O. Resnekov, D. Endy, C. Pesce, and R. Brent, "Regulated cell-to-cell variation in a cell-fate decision system," Nature 437, 699-706 (2005).
[CrossRef] [PubMed]

Rue, H.

H. Rue and O. K. Husby, "Identification of partly destroyed objects using deformable templates," Stat. Comput. 8, 221-228 (1998).
[CrossRef]

Rupes, I.

P. Jorgensen, N. P. Edgington, B. L. Schneider, I. Rupes, M. Tyers, and B. Futcher, "The Size of the Nucleus Increases as Yeast Cells Grow," Mol. Biol. Cell 18, 3523-3532 (2007).
[CrossRef] [PubMed]

Saito, T.

T. Saito, J. Sese, Y. Nakatani, F. Sano, M. Yukawa, Y. Ohya, and S. Morishita, "Data mining tools for the Saccharomyces cerevisiae morphological database," Nucleic Acids Res. 33, 753-757 (2005).
[CrossRef]

Sano, F.

T. Saito, J. Sese, Y. Nakatani, F. Sano, M. Yukawa, Y. Ohya, and S. Morishita, "Data mining tools for the Saccharomyces cerevisiae morphological database," Nucleic Acids Res. 33, 753-757 (2005).
[CrossRef]

Schneider, B. L.

P. Jorgensen, N. P. Edgington, B. L. Schneider, I. Rupes, M. Tyers, and B. Futcher, "The Size of the Nucleus Increases as Yeast Cells Grow," Mol. Biol. Cell 18, 3523-3532 (2007).
[CrossRef] [PubMed]

Serra, E.

A. Colman-Lerner, A. Gordon, E. Serra, T. Chin, O. Resnekov, D. Endy, C. Pesce, and R. Brent, "Regulated cell-to-cell variation in a cell-fate decision system," Nature 437, 699-706 (2005).
[CrossRef] [PubMed]

Sese, J.

T. Saito, J. Sese, Y. Nakatani, F. Sano, M. Yukawa, Y. Ohya, and S. Morishita, "Data mining tools for the Saccharomyces cerevisiae morphological database," Nucleic Acids Res. 33, 753-757 (2005).
[CrossRef]

Shah, D.

K. V. Mardia, W. Qian, D. Shah, and K. M. de Souza, "Deformable Template Recognition of Multiple Occluded Objects," IEEE Trans. Pattern Anal. and Mach. Intell. 19, 1035-1042 (1997).
[CrossRef]

Sheu, Y.

Y. Sheu and B. Stillman, "Cdc7-Dbf4 phosphorylates MCM proteins via a docking site-mediated mechanism to promote S phase progression," Mol. Cell 24, 101-113 (2006).
[CrossRef] [PubMed]

Siggia, E.

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S. Di Talia, J. Skotheim, J. Bean, E. Siggia, and F. Cross, "The effects of molecular noise and size control on variability in the budding yeast cell cycle," Nature 448, 947-951 (2007).
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S.-C. Chen, T. Zhao, G. J. Gordon, and R. F. Murphy, "Automated image analysis of protein localization in budding yeast," Bioinformatics 23, 66-71 (2007).
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Y. Sheu and B. Stillman, "Cdc7-Dbf4 phosphorylates MCM proteins via a docking site-mediated mechanism to promote S phase progression," Mol. Cell 24, 101-113 (2006).
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T. Saito, J. Sese, Y. Nakatani, F. Sano, M. Yukawa, Y. Ohya, and S. Morishita, "Data mining tools for the Saccharomyces cerevisiae morphological database," Nucleic Acids Res. 33, 753-757 (2005).
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Figures (9)

Fig. 1.
Fig. 1.

Illustration of the main steps of the cell recognition method: (a) the original bright field image showing a population of budding yeast cells; (b) the gradient, or edge map, image, where bright pixels represent a large gradient magnitude; (c) the segmented image with identified circles in red. The centers of these circles will serve as candidate cell centers; (d) the result after cell contour extraction.

Fig. 2.
Fig. 2.

Illustration of how differences in illumination level and cell densities in the bright field images, (a) and (c), affect the gradient image histograms, shown in (b) and (d). The red curves represent the corresponding Rayleigh distributions that have been fitted to values in the histogram that lay below the median value, here displayed as blue vertical lines. The horizontal axis has been truncated at 30 for ease of display.

Fig. 3.
Fig. 3.

Final steps of the segmentation; (a) thresholded gradient image, where white pixels are above β; (b) the final segmented image after filling of holes and after applying a morphological opening to the image in (a).

Fig. 4.
Fig. 4.

Illustration of the method for finding candidate cell centers. The outer boundary of the cell cluster and the approximate normals are shown in red and blue, respectively. The accumulation matrix is displayed as a grey scale image in the background. The identified circles are displayed in green.

Fig. 5.
Fig. 5.

(a) Directional derivatives along M=32 rays, each sampled at N=30 radial distances emanating from a candidate cell center; (b) Polar plot of the criterion function f in Eq. (2) at the points along the rays in (a). The angles ϕ are measured in the counter-clockwise direction, starting from the black arrow in (a). The rows in (b) correspond to the radial distances r in (a) and the columns in (b) correspond to the the angles of the rays ϕ. Bright pixels in (b) represent good contour points as measured by the criterion function.

Fig. 6.
Fig. 6.

Illustration of the refined contour extraction algorithm using two different transition rules. On the left are two optimal paths in three consecutive copies of the polar plot. The upper transition rule only allows for straight or diagonal transitions in the polar plot, whereas the transition rule in the lower example is the local convexity condition that penalizes transitions that corresponds to turning right in the image as the candidate center is encircled. The solution to the middle copy of the polar plot is used as resulting contour. The resulting contours to the examples on the left are displayed in the sub-images on the right as green squares.

Fig. 7.
Fig. 7.

Performance of cell contour recognition. Figures (a)–(d) illustrate the contour classes used in the success rate estimation. Of >1000 cells analyzed, 96% were defined correctly, 1% incorrectly, 1% were false hits and 2% were missed. The two missed cells in (d) were lost because they only have a minor part of their border free to fit to a circle, as shown in (e).

Fig. 8.
Fig. 8.

Analysis of Mcm4p nuclear localization dynamics. In (a) bright field and fluorescence images have been overlaid to show the localization over the different phases of the cell cycle. The fluorescence signal is color coded in red, green and blue, representing decreasing intensity levels. From early G1 phase, Mcm4p is found in the nucleus and it is then gradually translocating to the cytoplasm during S phase. In (b) the intensity variation over the cell cycle has been measured for a mother cell and two emerging buds. The plotted intensity is the mean of the two most intense pixel values within the cell contour in three image planes. The emergence of the daughter cell occurs approximately 10–20 minutes before the first daughter cell data point and seems to be contemporary to the start of translocation to the cytoplasm. In (c) and (d) additional algorithms were utilized to determine Mcm4p residence times in the nucleus and the cytoplasm.

Fig. 9.
Fig. 9.

(a) Definition of the angle θ between two consecutive enhanced states. (x 0,y 0) is the candidate cell center. Left-turns are defined as positive angles. Here M=12, so γ = 2 π M = π 6 . (b) Plot of the penalty function g(θ) in Eq. (3).

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

f R ( x ) = x σ 2 exp { x 2 2 σ 2 } ,
f ( r i , ϕ m ) = ( I x ( r i + 1 , ϕ m ) I x ( r i 1 , ϕ m ) ) cos ϕ m + ( I y ( r i + 1 , ϕ m ) I y ( r i 1 , ϕ m ) ) sin ϕ m
g ( θ ) = { θ 4 for θ < 0 0 if 0 θ < θ 0 ( θ θ 0 ) 2 for θ 0 θ

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