Abstract

Biological stimulation of living cells is sometimes associated with morphological changes. A practical method is developed to monitor cell stimulation by means of their conformational changes through interpretation of the pattern of light scattered from a cell population. For this purpose a mathematical model is suggested that predicts the power spectrum from a population of elliptic objects with a given eccentricity. A computer simulation of that model is presented together with supporting experimental results of the simulation. The predicted and the measured spectra are in good agreement. This technique was applied to elongated cells that become circular on exposure to a human hormone, indicating the potential applicability of the method in biology and medicine. The method and the apparatus presented in this study could be applied to bioassays of cell systems that respond to a variety of stimulants and to trace quantitatively the structural changes that occur during biological processes.

© 1999 Optical Society of America

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References

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  1. I. A. Schafer, A. M. Jamieson, M. Petrelli, B. J. Price, G. C. Salzman, “Multiangle light scattering flow photometry of cultured human fibroblasts. Comparison of normal cells with a mutant cell line containing cytoplasmic inclusions,” J. Histochem. Cytochem. 27, 359–365 (1979).
    [CrossRef] [PubMed]
  2. G. C. Salzman, P. F. Mullaney, B. J. Price, “Light-scattering approaches to cell characterization,” in Flow Cytometry and Sorting, M. R. Melamed, P. F. Mullaney, M. R. Mendelsohn, eds. (Wiley, New York, 1979), pp. 108–124.
  3. G. C. Salzman, J. M. Crowell, C. A. Goad, K. M. Hansen, R. D. Hiebert, P. M. LaBauve, J. C. Martin, M. Ingram, P. F. Mullaney, “A flow-system multiangle light scattering instrument for cell characterization,” Clin. Chem. 21, 1297–1304 (1975).
    [PubMed]
  4. G. Seger, M. Achatz, W. Heinze, F. Sinsel, “Quantitative extraction of morphological cell parameters from the diffraction pattern,” J. Histochem. Cytochem. 25, 707–718 (1977).
    [CrossRef] [PubMed]
  5. B. Turke, G. Seger, M. Achatz, V. S. Werner, “Fourier optical approach to the extraction of morphological parameters from the diffraction pattern of biological cells,” Appl. Opt. 17, 2754–2761 (1978).
    [CrossRef] [PubMed]
  6. D. E. Burger, J. H. Jett, P. F. Mullaney, “Extraction of morphological features from biological models and cells by Fourier analysis of static light scatter measurements,” Cytometry 2, 327–336 (1982).
    [CrossRef] [PubMed]
  7. A. Brunsting, P. F. Mullaney, “Light scattering from coated spheres: models for biological cells,” Appl. Opt. 11, 675–680 (1972).
    [CrossRef] [PubMed]
  8. S. Asano, M. Sato, “Light scattering by randomly oriented spheroidal particles,” Appl. Opt. 19, 972–974 (1980).
    [CrossRef]
  9. R. A. Dobbins, C. M. Megaridis, “Absorption and scattering of light by polydisperse aggregates,” Appl. Opt. 30, 4747–4754 (1991).
    [CrossRef] [PubMed]
  10. S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
    [CrossRef] [PubMed]
  11. J. B. Riley, Y. C. Agrawal, “Sampling and inversion of data in diffraction particle sizing,” Appl. Opt. 30, 4800–4817 (1991).
    [CrossRef] [PubMed]
  12. S. Evans, “Comparison of the diffraction theory of image formation with the three-dimensional, first Born scattering approximation in lens systems,” Opt. Commun. 2, 317–320 (1970).
    [CrossRef]
  13. S. D. Coston, N. George, “Particle sizing by inversion of the optical transform pattern,” Appl. Opt. 30, 4785–4794 (1991).
    [CrossRef] [PubMed]
  14. I. S. Gradshteyn, I. M. Ryzhik, ed., Tables of Integrals, Series and Products, 5th ed., (Academic, New York, 1965), p. 954.
  15. I. Keren-Tal, A. Dantes, R. Sprengel, A. Amsterdam, “Establishment of steroidiogenic granulosa cell lines expressing follicle stimulating hormone receptors,” Mol. Cell Endocrinol. 95, R1–R10 (1993).
    [CrossRef] [PubMed]
  16. J. B. Riaey, Y. C. Agrawal, “Sampling and inversion of data in diffraction particle sizing,” Appl. Opt. 30, 4800–4816 (1991).
    [CrossRef]
  17. A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1986), p. 79.
  18. A. Amsterdam, D. Aharoni, “Plasticity of cell organization during differentiation of normal and oncogene transformed granulosa cells,” Microsc. Res. Tech. 27, 108–124 (1994).
    [CrossRef] [PubMed]
  19. A. Brunsting, P. F. Mullaney, “Differential light scattering from spherical mammalian cells,” Biophys. J. 14, 439–453.
  20. M. Zohar, Y. Salomon, “Melanocortins stimulate proliferation and induce morphological changes in cultured rat astrocytes by distinct transducing mechanisms,” Brain Res. 576, 49–58 (1992).
    [CrossRef] [PubMed]

1994

A. Amsterdam, D. Aharoni, “Plasticity of cell organization during differentiation of normal and oncogene transformed granulosa cells,” Microsc. Res. Tech. 27, 108–124 (1994).
[CrossRef] [PubMed]

1993

I. Keren-Tal, A. Dantes, R. Sprengel, A. Amsterdam, “Establishment of steroidiogenic granulosa cell lines expressing follicle stimulating hormone receptors,” Mol. Cell Endocrinol. 95, R1–R10 (1993).
[CrossRef] [PubMed]

1992

M. Zohar, Y. Salomon, “Melanocortins stimulate proliferation and induce morphological changes in cultured rat astrocytes by distinct transducing mechanisms,” Brain Res. 576, 49–58 (1992).
[CrossRef] [PubMed]

1991

1982

D. E. Burger, J. H. Jett, P. F. Mullaney, “Extraction of morphological features from biological models and cells by Fourier analysis of static light scatter measurements,” Cytometry 2, 327–336 (1982).
[CrossRef] [PubMed]

1980

S. Asano, M. Sato, “Light scattering by randomly oriented spheroidal particles,” Appl. Opt. 19, 972–974 (1980).
[CrossRef]

1979

I. A. Schafer, A. M. Jamieson, M. Petrelli, B. J. Price, G. C. Salzman, “Multiangle light scattering flow photometry of cultured human fibroblasts. Comparison of normal cells with a mutant cell line containing cytoplasmic inclusions,” J. Histochem. Cytochem. 27, 359–365 (1979).
[CrossRef] [PubMed]

1978

1977

G. Seger, M. Achatz, W. Heinze, F. Sinsel, “Quantitative extraction of morphological cell parameters from the diffraction pattern,” J. Histochem. Cytochem. 25, 707–718 (1977).
[CrossRef] [PubMed]

1975

G. C. Salzman, J. M. Crowell, C. A. Goad, K. M. Hansen, R. D. Hiebert, P. M. LaBauve, J. C. Martin, M. Ingram, P. F. Mullaney, “A flow-system multiangle light scattering instrument for cell characterization,” Clin. Chem. 21, 1297–1304 (1975).
[PubMed]

S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
[CrossRef] [PubMed]

1972

1970

S. Evans, “Comparison of the diffraction theory of image formation with the three-dimensional, first Born scattering approximation in lens systems,” Opt. Commun. 2, 317–320 (1970).
[CrossRef]

Achatz, M.

B. Turke, G. Seger, M. Achatz, V. S. Werner, “Fourier optical approach to the extraction of morphological parameters from the diffraction pattern of biological cells,” Appl. Opt. 17, 2754–2761 (1978).
[CrossRef] [PubMed]

G. Seger, M. Achatz, W. Heinze, F. Sinsel, “Quantitative extraction of morphological cell parameters from the diffraction pattern,” J. Histochem. Cytochem. 25, 707–718 (1977).
[CrossRef] [PubMed]

Agrawal, Y. C.

Aharoni, D.

A. Amsterdam, D. Aharoni, “Plasticity of cell organization during differentiation of normal and oncogene transformed granulosa cells,” Microsc. Res. Tech. 27, 108–124 (1994).
[CrossRef] [PubMed]

Amsterdam, A.

A. Amsterdam, D. Aharoni, “Plasticity of cell organization during differentiation of normal and oncogene transformed granulosa cells,” Microsc. Res. Tech. 27, 108–124 (1994).
[CrossRef] [PubMed]

I. Keren-Tal, A. Dantes, R. Sprengel, A. Amsterdam, “Establishment of steroidiogenic granulosa cell lines expressing follicle stimulating hormone receptors,” Mol. Cell Endocrinol. 95, R1–R10 (1993).
[CrossRef] [PubMed]

Asano, S.

S. Asano, M. Sato, “Light scattering by randomly oriented spheroidal particles,” Appl. Opt. 19, 972–974 (1980).
[CrossRef]

S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
[CrossRef] [PubMed]

Brunsting, A.

A. Brunsting, P. F. Mullaney, “Light scattering from coated spheres: models for biological cells,” Appl. Opt. 11, 675–680 (1972).
[CrossRef] [PubMed]

A. Brunsting, P. F. Mullaney, “Differential light scattering from spherical mammalian cells,” Biophys. J. 14, 439–453.

Burger, D. E.

D. E. Burger, J. H. Jett, P. F. Mullaney, “Extraction of morphological features from biological models and cells by Fourier analysis of static light scatter measurements,” Cytometry 2, 327–336 (1982).
[CrossRef] [PubMed]

Coston, S. D.

Crowell, J. M.

G. C. Salzman, J. M. Crowell, C. A. Goad, K. M. Hansen, R. D. Hiebert, P. M. LaBauve, J. C. Martin, M. Ingram, P. F. Mullaney, “A flow-system multiangle light scattering instrument for cell characterization,” Clin. Chem. 21, 1297–1304 (1975).
[PubMed]

Dantes, A.

I. Keren-Tal, A. Dantes, R. Sprengel, A. Amsterdam, “Establishment of steroidiogenic granulosa cell lines expressing follicle stimulating hormone receptors,” Mol. Cell Endocrinol. 95, R1–R10 (1993).
[CrossRef] [PubMed]

Dobbins, R. A.

Evans, S.

S. Evans, “Comparison of the diffraction theory of image formation with the three-dimensional, first Born scattering approximation in lens systems,” Opt. Commun. 2, 317–320 (1970).
[CrossRef]

George, N.

Goad, C. A.

G. C. Salzman, J. M. Crowell, C. A. Goad, K. M. Hansen, R. D. Hiebert, P. M. LaBauve, J. C. Martin, M. Ingram, P. F. Mullaney, “A flow-system multiangle light scattering instrument for cell characterization,” Clin. Chem. 21, 1297–1304 (1975).
[PubMed]

Hansen, K. M.

G. C. Salzman, J. M. Crowell, C. A. Goad, K. M. Hansen, R. D. Hiebert, P. M. LaBauve, J. C. Martin, M. Ingram, P. F. Mullaney, “A flow-system multiangle light scattering instrument for cell characterization,” Clin. Chem. 21, 1297–1304 (1975).
[PubMed]

Heinze, W.

G. Seger, M. Achatz, W. Heinze, F. Sinsel, “Quantitative extraction of morphological cell parameters from the diffraction pattern,” J. Histochem. Cytochem. 25, 707–718 (1977).
[CrossRef] [PubMed]

Hiebert, R. D.

G. C. Salzman, J. M. Crowell, C. A. Goad, K. M. Hansen, R. D. Hiebert, P. M. LaBauve, J. C. Martin, M. Ingram, P. F. Mullaney, “A flow-system multiangle light scattering instrument for cell characterization,” Clin. Chem. 21, 1297–1304 (1975).
[PubMed]

Ingram, M.

G. C. Salzman, J. M. Crowell, C. A. Goad, K. M. Hansen, R. D. Hiebert, P. M. LaBauve, J. C. Martin, M. Ingram, P. F. Mullaney, “A flow-system multiangle light scattering instrument for cell characterization,” Clin. Chem. 21, 1297–1304 (1975).
[PubMed]

Jamieson, A. M.

I. A. Schafer, A. M. Jamieson, M. Petrelli, B. J. Price, G. C. Salzman, “Multiangle light scattering flow photometry of cultured human fibroblasts. Comparison of normal cells with a mutant cell line containing cytoplasmic inclusions,” J. Histochem. Cytochem. 27, 359–365 (1979).
[CrossRef] [PubMed]

Jett, J. H.

D. E. Burger, J. H. Jett, P. F. Mullaney, “Extraction of morphological features from biological models and cells by Fourier analysis of static light scatter measurements,” Cytometry 2, 327–336 (1982).
[CrossRef] [PubMed]

Keren-Tal, I.

I. Keren-Tal, A. Dantes, R. Sprengel, A. Amsterdam, “Establishment of steroidiogenic granulosa cell lines expressing follicle stimulating hormone receptors,” Mol. Cell Endocrinol. 95, R1–R10 (1993).
[CrossRef] [PubMed]

LaBauve, P. M.

G. C. Salzman, J. M. Crowell, C. A. Goad, K. M. Hansen, R. D. Hiebert, P. M. LaBauve, J. C. Martin, M. Ingram, P. F. Mullaney, “A flow-system multiangle light scattering instrument for cell characterization,” Clin. Chem. 21, 1297–1304 (1975).
[PubMed]

Martin, J. C.

G. C. Salzman, J. M. Crowell, C. A. Goad, K. M. Hansen, R. D. Hiebert, P. M. LaBauve, J. C. Martin, M. Ingram, P. F. Mullaney, “A flow-system multiangle light scattering instrument for cell characterization,” Clin. Chem. 21, 1297–1304 (1975).
[PubMed]

Megaridis, C. M.

Mullaney, P. F.

D. E. Burger, J. H. Jett, P. F. Mullaney, “Extraction of morphological features from biological models and cells by Fourier analysis of static light scatter measurements,” Cytometry 2, 327–336 (1982).
[CrossRef] [PubMed]

G. C. Salzman, J. M. Crowell, C. A. Goad, K. M. Hansen, R. D. Hiebert, P. M. LaBauve, J. C. Martin, M. Ingram, P. F. Mullaney, “A flow-system multiangle light scattering instrument for cell characterization,” Clin. Chem. 21, 1297–1304 (1975).
[PubMed]

A. Brunsting, P. F. Mullaney, “Light scattering from coated spheres: models for biological cells,” Appl. Opt. 11, 675–680 (1972).
[CrossRef] [PubMed]

A. Brunsting, P. F. Mullaney, “Differential light scattering from spherical mammalian cells,” Biophys. J. 14, 439–453.

G. C. Salzman, P. F. Mullaney, B. J. Price, “Light-scattering approaches to cell characterization,” in Flow Cytometry and Sorting, M. R. Melamed, P. F. Mullaney, M. R. Mendelsohn, eds. (Wiley, New York, 1979), pp. 108–124.

Papoulis, A.

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1986), p. 79.

Petrelli, M.

I. A. Schafer, A. M. Jamieson, M. Petrelli, B. J. Price, G. C. Salzman, “Multiangle light scattering flow photometry of cultured human fibroblasts. Comparison of normal cells with a mutant cell line containing cytoplasmic inclusions,” J. Histochem. Cytochem. 27, 359–365 (1979).
[CrossRef] [PubMed]

Price, B. J.

I. A. Schafer, A. M. Jamieson, M. Petrelli, B. J. Price, G. C. Salzman, “Multiangle light scattering flow photometry of cultured human fibroblasts. Comparison of normal cells with a mutant cell line containing cytoplasmic inclusions,” J. Histochem. Cytochem. 27, 359–365 (1979).
[CrossRef] [PubMed]

G. C. Salzman, P. F. Mullaney, B. J. Price, “Light-scattering approaches to cell characterization,” in Flow Cytometry and Sorting, M. R. Melamed, P. F. Mullaney, M. R. Mendelsohn, eds. (Wiley, New York, 1979), pp. 108–124.

Riaey, J. B.

Riley, J. B.

Salomon, Y.

M. Zohar, Y. Salomon, “Melanocortins stimulate proliferation and induce morphological changes in cultured rat astrocytes by distinct transducing mechanisms,” Brain Res. 576, 49–58 (1992).
[CrossRef] [PubMed]

Salzman, G. C.

I. A. Schafer, A. M. Jamieson, M. Petrelli, B. J. Price, G. C. Salzman, “Multiangle light scattering flow photometry of cultured human fibroblasts. Comparison of normal cells with a mutant cell line containing cytoplasmic inclusions,” J. Histochem. Cytochem. 27, 359–365 (1979).
[CrossRef] [PubMed]

G. C. Salzman, J. M. Crowell, C. A. Goad, K. M. Hansen, R. D. Hiebert, P. M. LaBauve, J. C. Martin, M. Ingram, P. F. Mullaney, “A flow-system multiangle light scattering instrument for cell characterization,” Clin. Chem. 21, 1297–1304 (1975).
[PubMed]

G. C. Salzman, P. F. Mullaney, B. J. Price, “Light-scattering approaches to cell characterization,” in Flow Cytometry and Sorting, M. R. Melamed, P. F. Mullaney, M. R. Mendelsohn, eds. (Wiley, New York, 1979), pp. 108–124.

Sato, M.

S. Asano, M. Sato, “Light scattering by randomly oriented spheroidal particles,” Appl. Opt. 19, 972–974 (1980).
[CrossRef]

Schafer, I. A.

I. A. Schafer, A. M. Jamieson, M. Petrelli, B. J. Price, G. C. Salzman, “Multiangle light scattering flow photometry of cultured human fibroblasts. Comparison of normal cells with a mutant cell line containing cytoplasmic inclusions,” J. Histochem. Cytochem. 27, 359–365 (1979).
[CrossRef] [PubMed]

Seger, G.

B. Turke, G. Seger, M. Achatz, V. S. Werner, “Fourier optical approach to the extraction of morphological parameters from the diffraction pattern of biological cells,” Appl. Opt. 17, 2754–2761 (1978).
[CrossRef] [PubMed]

G. Seger, M. Achatz, W. Heinze, F. Sinsel, “Quantitative extraction of morphological cell parameters from the diffraction pattern,” J. Histochem. Cytochem. 25, 707–718 (1977).
[CrossRef] [PubMed]

Sinsel, F.

G. Seger, M. Achatz, W. Heinze, F. Sinsel, “Quantitative extraction of morphological cell parameters from the diffraction pattern,” J. Histochem. Cytochem. 25, 707–718 (1977).
[CrossRef] [PubMed]

Sprengel, R.

I. Keren-Tal, A. Dantes, R. Sprengel, A. Amsterdam, “Establishment of steroidiogenic granulosa cell lines expressing follicle stimulating hormone receptors,” Mol. Cell Endocrinol. 95, R1–R10 (1993).
[CrossRef] [PubMed]

Turke, B.

Werner, V. S.

Yamamoto, G.

Zohar, M.

M. Zohar, Y. Salomon, “Melanocortins stimulate proliferation and induce morphological changes in cultured rat astrocytes by distinct transducing mechanisms,” Brain Res. 576, 49–58 (1992).
[CrossRef] [PubMed]

Appl. Opt.

Biophys. J.

A. Brunsting, P. F. Mullaney, “Differential light scattering from spherical mammalian cells,” Biophys. J. 14, 439–453.

Brain Res.

M. Zohar, Y. Salomon, “Melanocortins stimulate proliferation and induce morphological changes in cultured rat astrocytes by distinct transducing mechanisms,” Brain Res. 576, 49–58 (1992).
[CrossRef] [PubMed]

Clin. Chem.

G. C. Salzman, J. M. Crowell, C. A. Goad, K. M. Hansen, R. D. Hiebert, P. M. LaBauve, J. C. Martin, M. Ingram, P. F. Mullaney, “A flow-system multiangle light scattering instrument for cell characterization,” Clin. Chem. 21, 1297–1304 (1975).
[PubMed]

Cytometry

D. E. Burger, J. H. Jett, P. F. Mullaney, “Extraction of morphological features from biological models and cells by Fourier analysis of static light scatter measurements,” Cytometry 2, 327–336 (1982).
[CrossRef] [PubMed]

J. Histochem. Cytochem.

G. Seger, M. Achatz, W. Heinze, F. Sinsel, “Quantitative extraction of morphological cell parameters from the diffraction pattern,” J. Histochem. Cytochem. 25, 707–718 (1977).
[CrossRef] [PubMed]

I. A. Schafer, A. M. Jamieson, M. Petrelli, B. J. Price, G. C. Salzman, “Multiangle light scattering flow photometry of cultured human fibroblasts. Comparison of normal cells with a mutant cell line containing cytoplasmic inclusions,” J. Histochem. Cytochem. 27, 359–365 (1979).
[CrossRef] [PubMed]

Microsc. Res. Tech.

A. Amsterdam, D. Aharoni, “Plasticity of cell organization during differentiation of normal and oncogene transformed granulosa cells,” Microsc. Res. Tech. 27, 108–124 (1994).
[CrossRef] [PubMed]

Mol. Cell Endocrinol.

I. Keren-Tal, A. Dantes, R. Sprengel, A. Amsterdam, “Establishment of steroidiogenic granulosa cell lines expressing follicle stimulating hormone receptors,” Mol. Cell Endocrinol. 95, R1–R10 (1993).
[CrossRef] [PubMed]

Opt. Commun.

S. Evans, “Comparison of the diffraction theory of image formation with the three-dimensional, first Born scattering approximation in lens systems,” Opt. Commun. 2, 317–320 (1970).
[CrossRef]

Other

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1986), p. 79.

G. C. Salzman, P. F. Mullaney, B. J. Price, “Light-scattering approaches to cell characterization,” in Flow Cytometry and Sorting, M. R. Melamed, P. F. Mullaney, M. R. Mendelsohn, eds. (Wiley, New York, 1979), pp. 108–124.

I. S. Gradshteyn, I. M. Ryzhik, ed., Tables of Integrals, Series and Products, 5th ed., (Academic, New York, 1965), p. 954.

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Figures (7)

Fig. 1
Fig. 1

Scattering and diffraction planes. The scattering object is defined in the xy plane (Z = 0). A scattering pattern is created at the diffraction plane xy′ (Z = Z S ) with each representative point p also defined by the scattering azimuthal angles; Z S r max. (The scatterer dimension is exaggerated for illustrative purposes; sin θ ≅ θ, sin ϕ ∼ ϕ.)

Fig. 2
Fig. 2

Diffraction pattern created at the plane Z = Z S by an elliptic scatterer located at the object plane Z = 0. The density of the diffraction fringes is highest in the direction parallel to the major axis of the scattering ellipse.

Fig. 3
Fig. 3

Measuring system arrangement. A He–Ne laser beam, λ = 632 µm, illuminates a sample cell population attached to a microscope slide. The diffraction pattern is scanned along an arclike trajectory (S.T) by a movable optical fiber (O.F), which transmits the light to a photomultiplier tube (P.M.T). The successive signals are analyzed by the computer.

Fig. 4
Fig. 4

Scattering pattern of 1,500 optical scatterers obtained by computer simulation (a/ b = 3). The screen angular dimensions are approximately 0.8 rad. Black areas, full constructive interference; white regions, full destructive interference; gray areas, intermediate levels of interference.

Fig. 5
Fig. 5

Power spectra of populations of elliptic scatterers as obtained by computer simulation (solid curves) and theory (dashed curves). The abscissa is the weighted power spectrum S(ω) = II(ω)ω2, and the ordinate is the spatial frequency (ω) axis. Results obtained for ∊ = a/ b = 6/5, 2, 3, 4 are presented in a, b, c, d, respectively. For a numerical comparison, see Table 1. Note that there is no dashed curve for a because it is almost spherical (homogeneous); the result is a δ function, which is just a point on the graph.

Fig. 6
Fig. 6

Morphological changes in granulosa follicle-stimulating hormone rounding-17 cells following follicle sensitive hormone stimulation. Phase-contrast images of cells stimulated for 24 h with, A, 0.24 nM follicle-stimulating hormone or, B, 2.4 nm of the same hormone.

Fig. 7
Fig. 7

Effect of follicle-stimulating hormone concentrations on weighted power spectra S(ω) = II(ω)ω2. The theoretical (dashed) curve expected for the cells according to their appearance in the light microscope, and S(ω), which we calculated by subjecting the real scattering light intensity patterns to a Fourier transform and smoothing it by using a spline procedure (solid curve) of two cell samples, incubated for 24 h with, a, 0.24 nm or, b, 2.4 nM of this hormone, and subsequently fixed on a microscope slide on which they were cultivated. Note that there is no dashed curve for b, for the same reason as given in the caption for Fig. 5a.

Tables (1)

Tables Icon

Table 1 Comparison between Theoretical and Simulated PHR

Equations (40)

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

T/λΔn<1,
λ/D  1,
Iθ, ϕ=β2-- Tx, yexpiKx sin θ+y sin ϕdxdy2,
Iθ, ϕ=12 βT¯2S expiKxθ+yϕdxdy2,
Iθ, ϕ=2βT¯πR22J1KRθ2+ϕ21/2KRθ2+ϕ21/22=2βVcell2J1[KRθ2+ϕ21/2KRθ2+ϕ21/22,
E=m,
I=1/2m22,
J1W2πW1/2cosW-3π4,
J12W=2πWcos2W-3π4=1πW1-sin 2W.
Iθ, ϕ=2βVcell2π1-sin2KRθ2+ϕ21/2KRθ2+ϕ21/23,
Iω=IαKα3=- IαKα3 expiωαdα,
Iω=- R1-sin2KRαexpiωαdα=2δω+δω-2KR+δω+2KRR,
IIω  NIω.
IIω=0r=Rmax- r1-sin2Krαρr×expiωαdαdr.
IIω=0r=Rmax2δω+δω-2Kr+δω+2Krρrrdr.
IIω=2δω0r=Rmax rρrdr+ρω/2Kω/2K.
Iθ, ϕ=12 βT¯ell2x1=-ax2=+ay1=-b1-x/a21/2y2=+b1-x/a21/2expiKxθ+yϕdxdy2,
Iθ, ϕ=12 βT¯22Kϕx1=-ax2=+asinKbϕ1-xa21/2×expiKxθdx2.
Iθ, ϕ=2βT¯πab2J1Kaθ2+bϕ21/2Kaθ2+bϕ21/22.
θ=θ cos Ω+ϕ sin Ω, ϕ=-θ sin Ω+ϕ cos Ω.
Iθ, Ω=2βVcell2J1Kθa2 cos2 Ω+b2 sin2 Ω1/2Kθa2 cos2 Ω+b2 sin2 Ω1/22.
Iθ=2βVcel2π1-sin2Kθb2-a2sin2 Ω+a21/2Kθb2-a2sin2 Ω+a21/23.
IIω=IθKθ3=-0π IΩ, θpΩKθ3 expiωθdΩdθ,
IIω=12π-Ω=0πdΩdθ×1-sin2Kθb2-a2sin2 Ω+a21/2expiωθb2-a2sin2 Ω+a23/2.
γ=b2-a2sin2 Ω+a21/2,
dΩ=γdγγ2-b21/2a2-γ21/2.
IIω=2-abdθdγ/γ21-sin2Kγθexpiωθγ2-b21/2a2-γ21/2.
IIω=×abdγ/γ22δω+δω-2Kγ+δω+2Kγγ2-b21/2a2-γ21/2.
IIω=2Cδω+2K/ω22Kω2-2Kb21/22Ka2-ω21/2,
C=lim0b+a- γ2-b21/2a2-γ21/2dγγ2.
Ratio=limΔω0Sω2Ka-ΔωSω2Kb+Δω=limΔω02Kb+Δω2-2Kb21/22Ka-Δω2-2Kb21/2×2Ka2-2Kb-Δω21/22Ka2-2Ka-Δω21/2=2Ka2-2Kb21/22Ka2-2Kb21/24KbΔω1/24KaΔω1/2=ba1/2.
Wθ=1|θ|<θmax0|θ|>θmax.
IIMθ=IIθWθ,
IIMθ=IIθWθ=1/2πIIθ * Wθ=1/2πIIω * sin cωθ,
IP=Iθmin+pΔθP,
Ep, q-x0/2x0/2-y0/2y0/2 TAPx, yexpiKxθ×expiKyθdxdy2limN,Mn=0N-1m=0M-1 TAPn, m×expi2π/Nnpexpi2π/Mmq,
i2π/Nnp=i2π/λXθ.
i2π/Nnp=i2π/λL0/Nnθ.
P=L0/λθmax
Ratiolim sΔω0Sω2Ka-ΔωSω2Kb+Δω=ba1/2.

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