Abstract

Three-dimensional dynamic deformation of a red blood cell in a dual-trap optical tweezers is computed with the elastic membrane theory and is compared with the experimental results. When a soft particle is trapped by a laser beam, the particle is deformed depending on the radiation stress distribution whereas the stress distribution on the particle in turn depends on the deformation of its morphological shape. We compute the stress re-distribution on the deformed cell and its subsequent deformations recursively until a final equilibrium state solution is achieved. The experiment is done with the red blood cells in suspension swollen to spherical shape. The cell membrane elasticity coefficient is obtained by fitting the theoretical prediction with the experimental data. This approach allows us to evaluate up to 20% deformation of cell’s shape

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References

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  1. A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330(6150), 769–771 (1987).
    [Crossref] [PubMed]
  2. S. M. Block, “Making light work with optical tweezers,” Nature 360(6403), 493–495 (1992).
    [Crossref] [PubMed]
  3. J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
    [Crossref] [PubMed]
  4. M. Dao, C. T. Lim, and S. Suresh, “Mechanics of the human red blood cell deformed by optical tweezers,” J. Mech. Phys. Solids 51(11-12), 2259–2280 (2003).
    [Crossref]
  5. S. Hénon, G. Lenormand, A. Richert, and F. Gallet, “A new determination of the shear modulus of the human erythrocyte membrane using optical tweezers,” Biophys. J. 76(2), 1145–1151 (1999).
    [Crossref] [PubMed]
  6. P. J. H. Bronkhorst, G. J. Streekstra, J. Grimbergen, E. J. Nijhof, J. J. Sixma, and G. J. Brakenhoff, “A new method to study shape recovery of red blood cells using multiple optical trapping,” Biophys. J. 69(5), 1666–1673 (1995).
    [Crossref] [PubMed]
  7. W. G. Lee, H. Bang, J. Park, S. Chung, K. Cho, C. Chung, D.-C. Han, and J. K. Chang, “Combined microchannel-type erythrocyte deformability test with optical tweezers,” Proc. of SPIE.6088, 608813–1-12, (2006).
  8. G. B. Liao, P. B. Bareil, Y. Sheng, and A. Chiou, “One-dimensional jumping optical tweezers for optical stretching of bi-concave human red blood cells,” Opt. Express 16(3), 1996–2004 (2008).
    [Crossref] [PubMed]
  9. P. B. Bareil, Y. Sheng, and A. Chiou, “Local scattering stress distribution on surface of a spherical cell in optical stretcher,” Opt. Express 14(25), 12503 (2006).
    [Crossref] [PubMed]
  10. J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
    [Crossref] [PubMed]
  11. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9(8), S196–S203 (2007).
    [Crossref]
  12. T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1005–1017 (2003).
    [Crossref]
  13. M. Mansuripur, “Electromagnetic stress tensor in ponderable media,” Opt. Express 16(8), 5193–5198 (2008).
    [Crossref] [PubMed]
  14. F. Xu, J. A. Lock, G. Gouesbet, and C. Tropea, “Optical stress on the surface of a particle: homogeneous sphere,” Phys. Rev. A 79(5), 053808 (2009).
    [Crossref]
  15. Y. C. Fung, “Theoretical considerations of the elasticity of red cells and small blood vessels,” Fed. Proc. 25(6), 1761–1772 (1966).
    [PubMed]
  16. P. B. Bareil, Y. Sheng, Y. Q. Chen, and A. Chiou, “Calculation of spherical red blood cell deformation in a dual-beam optical stretcher,” Opt. Express 15(24), 16029–16034 (2007).
    [Crossref] [PubMed]
  17. E. Ventsel, and T. Krauthammer, Thin plate and shells (Marcel Dekket, New York, 2001).

2009 (1)

F. Xu, J. A. Lock, G. Gouesbet, and C. Tropea, “Optical stress on the surface of a particle: homogeneous sphere,” Phys. Rev. A 79(5), 053808 (2009).
[Crossref]

2008 (2)

2007 (2)

P. B. Bareil, Y. Sheng, Y. Q. Chen, and A. Chiou, “Calculation of spherical red blood cell deformation in a dual-beam optical stretcher,” Opt. Express 15(24), 16029–16034 (2007).
[Crossref] [PubMed]

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9(8), S196–S203 (2007).
[Crossref]

2006 (1)

2003 (2)

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1005–1017 (2003).
[Crossref]

M. Dao, C. T. Lim, and S. Suresh, “Mechanics of the human red blood cell deformed by optical tweezers,” J. Mech. Phys. Solids 51(11-12), 2259–2280 (2003).
[Crossref]

2001 (2)

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[Crossref] [PubMed]

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[Crossref] [PubMed]

1999 (1)

S. Hénon, G. Lenormand, A. Richert, and F. Gallet, “A new determination of the shear modulus of the human erythrocyte membrane using optical tweezers,” Biophys. J. 76(2), 1145–1151 (1999).
[Crossref] [PubMed]

1995 (1)

P. J. H. Bronkhorst, G. J. Streekstra, J. Grimbergen, E. J. Nijhof, J. J. Sixma, and G. J. Brakenhoff, “A new method to study shape recovery of red blood cells using multiple optical trapping,” Biophys. J. 69(5), 1666–1673 (1995).
[Crossref] [PubMed]

1992 (1)

S. M. Block, “Making light work with optical tweezers,” Nature 360(6403), 493–495 (1992).
[Crossref] [PubMed]

1987 (1)

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330(6150), 769–771 (1987).
[Crossref] [PubMed]

1966 (1)

Y. C. Fung, “Theoretical considerations of the elasticity of red cells and small blood vessels,” Fed. Proc. 25(6), 1761–1772 (1966).
[PubMed]

Ananthakrishnan, R.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[Crossref] [PubMed]

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[Crossref] [PubMed]

Ashkin, A.

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330(6150), 769–771 (1987).
[Crossref] [PubMed]

B. Bareil, P.

Bareil, P. B.

Block, S. M.

S. M. Block, “Making light work with optical tweezers,” Nature 360(6403), 493–495 (1992).
[Crossref] [PubMed]

Brakenhoff, G. J.

P. J. H. Bronkhorst, G. J. Streekstra, J. Grimbergen, E. J. Nijhof, J. J. Sixma, and G. J. Brakenhoff, “A new method to study shape recovery of red blood cells using multiple optical trapping,” Biophys. J. 69(5), 1666–1673 (1995).
[Crossref] [PubMed]

Branczyk, A. M.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9(8), S196–S203 (2007).
[Crossref]

Bronkhorst, P. J. H.

P. J. H. Bronkhorst, G. J. Streekstra, J. Grimbergen, E. J. Nijhof, J. J. Sixma, and G. J. Brakenhoff, “A new method to study shape recovery of red blood cells using multiple optical trapping,” Biophys. J. 69(5), 1666–1673 (1995).
[Crossref] [PubMed]

Chen, Y. Q.

Chiou, A.

Cunningham, C. C.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[Crossref] [PubMed]

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[Crossref] [PubMed]

Dao, M.

M. Dao, C. T. Lim, and S. Suresh, “Mechanics of the human red blood cell deformed by optical tweezers,” J. Mech. Phys. Solids 51(11-12), 2259–2280 (2003).
[Crossref]

Dziedzic, J. M.

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330(6150), 769–771 (1987).
[Crossref] [PubMed]

Fung, Y. C.

Y. C. Fung, “Theoretical considerations of the elasticity of red cells and small blood vessels,” Fed. Proc. 25(6), 1761–1772 (1966).
[PubMed]

Gallet, F.

S. Hénon, G. Lenormand, A. Richert, and F. Gallet, “A new determination of the shear modulus of the human erythrocyte membrane using optical tweezers,” Biophys. J. 76(2), 1145–1151 (1999).
[Crossref] [PubMed]

Gouesbet, G.

F. Xu, J. A. Lock, G. Gouesbet, and C. Tropea, “Optical stress on the surface of a particle: homogeneous sphere,” Phys. Rev. A 79(5), 053808 (2009).
[Crossref]

Grimbergen, J.

P. J. H. Bronkhorst, G. J. Streekstra, J. Grimbergen, E. J. Nijhof, J. J. Sixma, and G. J. Brakenhoff, “A new method to study shape recovery of red blood cells using multiple optical trapping,” Biophys. J. 69(5), 1666–1673 (1995).
[Crossref] [PubMed]

Guck, J.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[Crossref] [PubMed]

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[Crossref] [PubMed]

Heckenberg, N. R.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9(8), S196–S203 (2007).
[Crossref]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1005–1017 (2003).
[Crossref]

Hénon, S.

S. Hénon, G. Lenormand, A. Richert, and F. Gallet, “A new determination of the shear modulus of the human erythrocyte membrane using optical tweezers,” Biophys. J. 76(2), 1145–1151 (1999).
[Crossref] [PubMed]

Käs, J.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[Crossref] [PubMed]

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[Crossref] [PubMed]

Knöner, G.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9(8), S196–S203 (2007).
[Crossref]

Lenormand, G.

S. Hénon, G. Lenormand, A. Richert, and F. Gallet, “A new determination of the shear modulus of the human erythrocyte membrane using optical tweezers,” Biophys. J. 76(2), 1145–1151 (1999).
[Crossref] [PubMed]

Liao, G. B.

Lim, C. T.

M. Dao, C. T. Lim, and S. Suresh, “Mechanics of the human red blood cell deformed by optical tweezers,” J. Mech. Phys. Solids 51(11-12), 2259–2280 (2003).
[Crossref]

Lock, J. A.

F. Xu, J. A. Lock, G. Gouesbet, and C. Tropea, “Optical stress on the surface of a particle: homogeneous sphere,” Phys. Rev. A 79(5), 053808 (2009).
[Crossref]

Loke, V. L. Y.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9(8), S196–S203 (2007).
[Crossref]

Mahmood, H.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[Crossref] [PubMed]

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[Crossref] [PubMed]

Mansuripur, M.

Moon, T. J.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[Crossref] [PubMed]

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[Crossref] [PubMed]

Nieminen, T. A.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9(8), S196–S203 (2007).
[Crossref]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1005–1017 (2003).
[Crossref]

Nijhof, E. J.

P. J. H. Bronkhorst, G. J. Streekstra, J. Grimbergen, E. J. Nijhof, J. J. Sixma, and G. J. Brakenhoff, “A new method to study shape recovery of red blood cells using multiple optical trapping,” Biophys. J. 69(5), 1666–1673 (1995).
[Crossref] [PubMed]

Richert, A.

S. Hénon, G. Lenormand, A. Richert, and F. Gallet, “A new determination of the shear modulus of the human erythrocyte membrane using optical tweezers,” Biophys. J. 76(2), 1145–1151 (1999).
[Crossref] [PubMed]

Rubinsztein-Dunlop, H.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9(8), S196–S203 (2007).
[Crossref]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1005–1017 (2003).
[Crossref]

Sheng, Y.

Sixma, J. J.

P. J. H. Bronkhorst, G. J. Streekstra, J. Grimbergen, E. J. Nijhof, J. J. Sixma, and G. J. Brakenhoff, “A new method to study shape recovery of red blood cells using multiple optical trapping,” Biophys. J. 69(5), 1666–1673 (1995).
[Crossref] [PubMed]

Stilgoe, A. B.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9(8), S196–S203 (2007).
[Crossref]

Streekstra, G. J.

P. J. H. Bronkhorst, G. J. Streekstra, J. Grimbergen, E. J. Nijhof, J. J. Sixma, and G. J. Brakenhoff, “A new method to study shape recovery of red blood cells using multiple optical trapping,” Biophys. J. 69(5), 1666–1673 (1995).
[Crossref] [PubMed]

Suresh, S.

M. Dao, C. T. Lim, and S. Suresh, “Mechanics of the human red blood cell deformed by optical tweezers,” J. Mech. Phys. Solids 51(11-12), 2259–2280 (2003).
[Crossref]

Tropea, C.

F. Xu, J. A. Lock, G. Gouesbet, and C. Tropea, “Optical stress on the surface of a particle: homogeneous sphere,” Phys. Rev. A 79(5), 053808 (2009).
[Crossref]

Xu, F.

F. Xu, J. A. Lock, G. Gouesbet, and C. Tropea, “Optical stress on the surface of a particle: homogeneous sphere,” Phys. Rev. A 79(5), 053808 (2009).
[Crossref]

Yamane, T.

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330(6150), 769–771 (1987).
[Crossref] [PubMed]

Biophys. J. (4)

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[Crossref] [PubMed]

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81(2), 767–784 (2001).
[Crossref] [PubMed]

S. Hénon, G. Lenormand, A. Richert, and F. Gallet, “A new determination of the shear modulus of the human erythrocyte membrane using optical tweezers,” Biophys. J. 76(2), 1145–1151 (1999).
[Crossref] [PubMed]

P. J. H. Bronkhorst, G. J. Streekstra, J. Grimbergen, E. J. Nijhof, J. J. Sixma, and G. J. Brakenhoff, “A new method to study shape recovery of red blood cells using multiple optical trapping,” Biophys. J. 69(5), 1666–1673 (1995).
[Crossref] [PubMed]

Fed. Proc. (1)

Y. C. Fung, “Theoretical considerations of the elasticity of red cells and small blood vessels,” Fed. Proc. 25(6), 1761–1772 (1966).
[PubMed]

J. Mech. Phys. Solids (1)

M. Dao, C. T. Lim, and S. Suresh, “Mechanics of the human red blood cell deformed by optical tweezers,” J. Mech. Phys. Solids 51(11-12), 2259–2280 (2003).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9(8), S196–S203 (2007).
[Crossref]

J. Quant. Spectrosc. Radiat. Transf. (1)

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1005–1017 (2003).
[Crossref]

Nature (2)

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330(6150), 769–771 (1987).
[Crossref] [PubMed]

S. M. Block, “Making light work with optical tweezers,” Nature 360(6403), 493–495 (1992).
[Crossref] [PubMed]

Opt. Express (4)

Phys. Rev. A (1)

F. Xu, J. A. Lock, G. Gouesbet, and C. Tropea, “Optical stress on the surface of a particle: homogeneous sphere,” Phys. Rev. A 79(5), 053808 (2009).
[Crossref]

Other (2)

E. Ventsel, and T. Krauthammer, Thin plate and shells (Marcel Dekket, New York, 2001).

W. G. Lee, H. Bang, J. Park, S. Chung, K. Cho, C. Chung, D.-C. Han, and J. K. Chang, “Combined microchannel-type erythrocyte deformability test with optical tweezers,” Proc. of SPIE.6088, 608813–1-12, (2006).

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Figures (6)

Fig. 6
Fig. 6

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Fig. 1
Fig. 1

3D stress distribution (N/m2) on spherical surface of a cell in suspension in the dual-beam optical tweezers with D as separation distance of two trapping beams, computed with ray-tracing (a)–(d), with T-matrix (e) and with FDTD (f). The trapping beams of NA = 1.25, power P = 89mW are along the + z-axis. The cell radius ρ = 3.86 μm, the refractive index in buffer n1 = 1.335 and inside the cell n2 = 1.378

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Fig. 5
Fig. 5

Images of a trapped and stretched RBC: (a) without drug treatment; (b) with 1mM N-ethylmaleimide (NEM) treatment for 30 minutes as a function of the dual beam separation D = 0, 1.27, 2.54, 3.80, 5.07 and 6.34 μm (from left to right); Theoretical results fit to experimental data for the length of the major axis as a function of D: (c) without drug treatment, Eh = 15.99 (μN/m) ρ = 3.554 μm; (d) with drug treatment, Eh = 24.39 (μN/m) ρ = 3.697 μm. Error bars show the root-mean-square standard deviation of the measurements over 30 samples under the same experimental condition.

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Fig. 2
Fig. 2

Top-left: stress distribution on a spherical cell with radius ρ = 3.86 μm and beam separation D = 5.07 μm. In clock-wise order: stress redistribution as the cell is deformed gradually, computed in iterations of COMSOLTM modules.

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Fig. 3
Fig. 3

3D deformation and its 2D projection along the z-axis to the x-y plane of a spherical RBC in the dual-trap optical tweezers computed in the dynamic regime and. D: separation of the two beams along ± x directions; Initial spherical cell radius ρ = 3.658 μm, refractive index n2 = 1.378 and that in buffer n1 = 1.335, Eh = 23.77 μN/m. Power of 89 mW for each beam. The colors encode the radial displacements of the membrane, which are positive (outward from the cell) or negative (inward to the cell)

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Fig. 4
Fig. 4

A schematic diagram of the experimental setup.

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Equations (24)

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σ→=1cEiAΔtn1(a→i−(nTa→t+Ra→r))≡n1cPAQ→
σfront=[(n1(1+R)−n2T]PcA=−0.042PcA
σrear=T[n2(1+R)−n1T]PcA=−0.047PcA
∂Nθ∂θ+sinφ∂S∂φ+2Scosφ=0
∂Nφ∂φsinφ+(Nφ−Nθ)cosφ+∂S∂θ=0
Nθ+Nφ+ρσr=0
∂υ∂φ−υcotφ−1sinφ∂u∂θ=R(Nφ−vNθEh−Nθ−vNφEh)
1sinφ∂υ∂θ+∂u∂φ−ucotφ=RSGh
w=∂υ∂φ−REh(Nφ−vNθ)
Qx=n1(cos(δ+ϕ)−nT×cos(β+ϕ)+R×cos(ϕ−δ))Qy=n1(sin(δ+ϕ)−nTsin(β+ϕ)+Rsin(ϕ−δ))y/y2+z2Qz=n1(−sin(δ+ϕ)+nTsin(β+ϕ)−Rsin(ϕ−δ))z/y2+z2Q=Qx2+Qy2+Qz2
r⊥=n1cos(δ)−n2cos(β)n1cos(δ)+n2cos(β)  and  r∥=n2cos(δ)−n1cos(β)n2cos(δ)+n1cos(β)
∂S∂φ+2Scotφ−1sinφ∂Nφ∂θ−ρ1sinφ∂σρ∂θ=0
∂Nφ∂φ+2Nφcotφ+1sinφ∂S∂θ+ρcotφσρ=0
σr=∑σrmexp(−jmθ),S=∑Smexp(jmθ),Nφ=∑Nφmexp(−jmθ),N=∑Nmexp(−jmθ)
∂Nm∂φ+(2cotφ+msinφ)Nm+ρσρm(cotφ+m)=0
Nm=ρsin−2φcotm(φ/2)(∫(cosφ+m)sinφtanm(φ/2)σρmdφ+Am)
Lm=ρsin−2φcot−m(φ/2)(∫(cosφ−m)sinφtan−m(φ/2)σρmdφ+Bm)
Nφm=−(ρ/2)(ψm++ψm−)
Sm=−(ρ/2)(ψm+−ψm−)
ψm±=ρsin−2φcot±m(φ/2)(∫(cosφ±m)sinφtan±m(φ/2)σρmdφ+AmBm)
υm=R2Eh(ηm++ηm−)
um=R2Eh(ηm++ηm−)
ηm±=sinφtan±m(φ/2)(∫[(1+2υ+α)Nφm±EGSm+(1+υ)Rσρm]cot±m(φ/2)sin−1φdφ+CmDm)
wm=dυmdφ−REh[(1+v)Nφm+vρσρm]

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