## Photon statistics in single molecule orientational imaging

Optics Express, Vol. 15, Issue 21, pp. 13597-13606 (2007)

http://dx.doi.org/10.1364/OE.15.013597

Acrobat PDF (242 KB)

### Abstract

Optical techniques in single molecule imaging rely heavily on photon counting for data acquisition. Extraction of information from the recorded readings is often done by means of statistical signal processing, however this requires a full knowledge of the photoelectron statistics. In addition to counting statistics we include a specific form of random signal variations namely reorientational dynamics, or wobble to derive the general probability density function of the number of detected photons. The relative importance of the two factors is dependent upon the total number of photons in the system and results are given in all regimes.

© 2007 Optical Society of America

## 1. Introduction

1. K. D. Weston and L. S. Goldner “Orientation imaging and reorientation dynamics of single dye molecules,” J. Phys. Chem. B **105**3453-3462 (2001) [CrossRef]

2. D.M. Warshaw, E. Hayes, D. Gaffney, A.M. Lauzon, J.R. Wu, G. Kennedy, K. Trybus, S. Lowey, and C. Berger, “Myosin conformational states determined by single fluorophore polarization,” Proc. Natl. Acad. Sci. U.S.A. **95**8034-8039 (1998) [CrossRef] [PubMed]

3. H. P. Lu, L. Y. Xun, and X. S. Xie, “Single molecule enzymatic dynamics,” Science **282**1877–1882 (1998) [CrossRef] [PubMed]

4. R. E. Dale and S. C. Hopkins “Model-Independent analysis of the orientation of fluorescent probes with restricted mobility in muscle fibers, Biophys. J. **76**1606–1618 (1999) [CrossRef] [PubMed]

5. T. M. Jovin, M. Bartholdi, W. L. C. Vaz, and R. H. Austin “Rotational diffusion of biological macromolecules by time-resolved delayed luminescence (phosphorescence, fluorescence) anisotropy,” Ann. N.Y. Acad. Sci. **366**176–196 (1981) [CrossRef] [PubMed]

6. T. Ha, T. Enderle, D. F. Ogletree, D. S. Chemla, P. R. Selvin, and S. Weiss, “Probing the interaction between two single molecules: Fluorescence resonance energy transfer between a single donor and a single acceptor” Proc. Natl. Acad. Sci. U.S.A. **93**6264–6268 (1996). [CrossRef] [PubMed]

7. T. Ha, J. Glass, T. Enderle, D. S. Chemla, and S. Weiss “Hindered rotational diffusion and rotational jumps of single molecules,” Phys. Rev. Lett. **80**2093–2096 (1998). [CrossRef]

## 2. Signal-to-noise considerations

8. Th. Basché, W. P. Ambrose, and W. E. Moerner, “Optical spectra and kinetics of single impurity molecules in a polymer: spectral diffusion and persistent spectral hole burning,” J. Opt. Soc. Am. B. **9**829–836 (1992). [CrossRef]

*D*is an instrument dependent collection factor typically ranging from 1-8%,

*q*is the fluorescence quantum yield, σ is the peak absorption crosssection,

*P*is the laser power,

*t*

_{0}is the integration time,

*A*is the beam area,

*E*is the energy of a photon in the beam,

_{p}*C*is the background count per watt of excitation power (typically around 2×10

_{b}^{8}photons/Ws in confocal experiments) and

*N*is the dark count of the detector. Figure 1 shows the behaviour of the SNR over a range of experimental conditions from which it can be seen that a value no better than around 15dB is to be expected. Consequently noise properties of the detection process play an important role in determining the statistical behaviour of the detected signal.

_{d}## 3. Probability density function of the number of detected photons

*t*

_{0}by

*N*(

*t*

_{0}), the output reading is of the form

*I*=

_{out}*GN*, where G is some gain factor. The arrival of photons at the detector is a Poisson random process [11] and we can hence write the probability mass function of

*N*

*p*(

_{N}*n*) denotes the probability that

*N*(

*t*

_{0}) =

*n*. We note here that we use the convention whereby an upper case letter denotes a random process and/or variable, whilst the lower case equivalent denotes a particular outcome.

*I*(

*t*

_{0}) is the average rate of arrival of photons (intensity) or equivalently the time average of ℐ(

*t*) where ℐ(

*t*) is the instantaneous rate of arrival of photons at the detector at time

*t*i.e.

12. T. Ha, T. Enderle, D. S. Chemla, P. R. Selvin, and S. Weiss “Single molecule dynamics studied by polarization modulation,” Phys. Rev. Lett. **77**3979–3982 (1996). [CrossRef] [PubMed]

*t*) is the transverse orientation of the dipole at time

*t*,

*β*is the transverse angle of the plane of polarisation of incident light and

*A*is a constant.

*I*(

*t*

_{0}) is a random variable and the probabilities as given by Eq. (2) differ for each possible value. As such we recast Eq. (2) by conditioning the probabilities on a particular outcome

*i*(

*t*

_{0}) i.e.

*p*is now a conditional probability and we have dropped the functional dependence on

_{N}*t*

_{0}for clarity.

*fI*(

*i*) (see Section 4), we can use the identity [11]

*N*and

*I*i.e. the probability that

*N*=

*n*and

*I*=

*i*. Integrating over the joint PDF gives the PDF of the number of detected photons

7. T. Ha, J. Glass, T. Enderle, D. S. Chemla, and S. Weiss “Hindered rotational diffusion and rotational jumps of single molecules,” Phys. Rev. Lett. **80**2093–2096 (1998). [CrossRef]

*f*(

_{I}*i*) which is discussed in the following section.

## 4. Probability density function of time averaged intensity

### 4.1. Discrete reorientational jumps

*f*(

_{I}*i*). We first consider the case when changes in the orientation of a dipole occur discretely. This could for example be associated with the desorption and readsorption of fluorophores from and onto a glass surface [12

12. T. Ha, T. Enderle, D. S. Chemla, P. R. Selvin, and S. Weiss “Single molecule dynamics studied by polarization modulation,” Phys. Rev. Lett. **77**3979–3982 (1996). [CrossRef] [PubMed]

*θ*to the

*x*-axis in the

*x*-

*y*plane as illustrated in Fig. 2. The dipole then remains fixed at this angle for a time τ before moving to a new state. It is this transverse angle that the signal ℐ in many experimental techniques is dependent on (c.f. Eq. 4). Techniques based on structured illumination and total internal reflection [13

13. B. Sick, B. Hecht, and L. Novotny “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. **85**4482–4485 (2000). [CrossRef] [PubMed]

14. R. M. Dickson, D. J. Norris, and W. E. Moerner, “Simultaneous imaging of individual molecules aligned both parallel and perpendicular to the optic axis,” Phys. Rev. Lett. **81**5322–5325 (1998) [CrossRef]

*M*different orientational states are occupied during a single measurement the time averaged intensity is given by:

*θ*

_{j}and

*τ*

_{j}are the parameters corresponding to the

*j*

^{th}occupied angular state. Without loss of generality the dipole is assumed to be initially orientated parallel to the

*x*-axis. It should be further noted that changes in the dipole angle are assumed to occur instantaneously.

*M*be a Poisson random variable or equivalently that the length of time a dipole remains in each state is distributed according to an exponential law i.e.

*f*

_{τ}(τ) denotes the PDF of τ and

*v*is the average rate at which dipole jump events occur.

*I*. The Laplace transform

*X*

^{*}(

*s*) of a random variable

*X*is defined as

*E*[…] denotes the expected value.

*Z*=

_{j}*A*cos

^{2}(Θ

_{j}-β)τ

_{j}. The PDF of the average intensity

*I*is then given by:

*f*(

_{I}*i*|

*m*) = ℒ

^{-1}(

*I*

^{*}

_{M=m}(

*s*)) and the weighted summation over the possible values of

*M*=

*m*is required since the number of reorientations during a measurement is random.

*Z*

^{*}

_{j}(

*s*). From Eqs. (4) and (10) we can write

*f*

_{Θ,τ}(

*θ,τ*) is the joint probability distribution of Θ and τ. Since dipole angle and state occupancy time are independent this is given by the product of the marginal probability distributions

*f*

_{Θ}(θ) and

*f*

_{τ}(τ). Using Eq. (9) we can then write

*f*(

_{I}*i*) would be complicated however in the limits of small and large

*v*we can find simpler results. These limits correspond to only a few, and to many events per measurement respectively. As the rate at which events occur decreases the contribution from later terms in Eq. (12) becomes negligible. In the limit of

*v*≪ 1 only the first term produces a significant contribution and we can consider the dipole as fixed during a single measurement and hence

*v*. Since each value of τ

_{j}is independent each

*Z*term is also independent. There are then two cases to consider; that when each subsequent value of

_{j}*θ*is independent and that when they are not. In the former case we can invoke the Central Limit Theorem which states the PDF of a sum of independent, identically distributed random variables tends to a Gaussian distribution as the number of terms increases. As such the PDF of the average intensity in the limit of large

*v*is given by

*θ*is centered on its previous outcome,

_{j}*θ*

_{j-1}. For a particular realisation of

*θ*, that is to say one possible outcome of the sequence of dipole orientations, we can write

### 4.2. Continuous angular variation

*t*satisfies the differential equation:

*f*(

*t*= 0) =

*δ*(

*θ*-

*θ*

_{0}), where

*δ*represents the Dirac delta function. This diffusion equation holds when subsequent orientations are dependent on the previous orientation. A solution to Eq. (20

20. J. Yguerabide, H. F. Epstein, and L. Stryer “Segmental flexibility in an antibody molecule,” J. Mol. Biol. **51**573–590 (1970). [CrossRef] [PubMed]

*θ*may vary as is set by the diffusion coefficient

*α*.

*fℐ*(

*ℐ*) which we then need only integrate over the length of a measurement to give our desired result. Thus

*θ*

_{k}are the solutions to the equation

*i*=

*A*cos

^{2}(θ - β) and the 1/

*t*

_{0}factor is to ensure correct normalisation of the PDF. The integral can be evaluated using the substitution

*x*

^{2}=

*t*

^{-1}and integration by parts which yields:

*f*

_{Θ}(

*θ,t*) can not depend on time (assuming the physical cause of the wobble does not vary in time) and as such Eq. (22) reduces to

*f*(

_{I}*i*) =

*f*(

_{ℐ}*ℐ*).

^{4}realisations for continuous variation and a diffusion coefficient of

*α*= 5. Various theoretical fits, as based on Eq. (23), are also drawn from which it can be seen that for

*α*= 0 (no dipole wobble) the PDF is identical to that of a Poisson distribution as would be expected. Good agreement can also be seen between the simulated and theoretical results.

*N*taking any value below

*n*as plotted in the inset of Fig. 3b) as a function of

*n*. Confidence levels including or neglecting dipole wobble can then be calculated. Assuming the values

_{0}= 0,

*A*= 10

^{5}photons/s,

*t*

_{0}= 10

^{-3}s and α = 5 we calculated that when neglecting dipole wobble an experimental measurement can determine the orientation of a dipole within a range of 1.78° with 90% confidence. Inclusion of dipole wobble causes this to increase to 2.43°. Such a discrepency further highlights the need to include dipole wobble in statistical processing and error analysis.

## 5. Discussion

*Z*term is reinforced with each additional term in the average.

*n*i.e. slow variation only a few terms significantly contribute to the average performed by the detector. In this case the peaked nature of both the exponentially distributed state occupancy times and the Poisson PDF for the number of events per measurement dominate. For larger

*v*the Poisson PDF becomes smoother and the position of the peak moves to larger

*m*. Low

*m*terms of Eq. (12

12. T. Ha, T. Enderle, D. S. Chemla, P. R. Selvin, and S. Weiss “Single molecule dynamics studied by polarization modulation,” Phys. Rev. Lett. **77**3979–3982 (1996). [CrossRef] [PubMed]

19. P. Wahl, K. Tawada, and J.C. Auchet “Study of tropomyosin labelled with a fluorescent probe by pulse fluorimetry in polarized light. Interaction of that protein with troponin and actin,” Eur. J. Biochem. **88**421–424 (1978) [CrossRef] [PubMed]

5. T. M. Jovin, M. Bartholdi, W. L. C. Vaz, and R. H. Austin “Rotational diffusion of biological macromolecules by time-resolved delayed luminescence (phosphorescence, fluorescence) anisotropy,” Ann. N.Y. Acad. Sci. **366**176–196 (1981) [CrossRef] [PubMed]

7. T. Ha, J. Glass, T. Enderle, D. S. Chemla, and S. Weiss “Hindered rotational diffusion and rotational jumps of single molecules,” Phys. Rev. Lett. **80**2093–2096 (1998). [CrossRef]

20. J. Yguerabide, H. F. Epstein, and L. Stryer “Segmental flexibility in an antibody molecule,” J. Mol. Biol. **51**573–590 (1970). [CrossRef] [PubMed]

21. W. E. Moerner and D. P. Fromm, “Methods of single molecule fluorescence spectroscopy and microscopy,” Rev. Sci. Instrum. **74**3597–3619 (2003) [CrossRef]

22. I. Munro, I. Pecht, and L. Stryer “Subnanosecond motions of Tryptophan residues in proteins,” Proc. Natl. Acad. Sci. USA **76**56–60 (1979) [CrossRef] [PubMed]

*A*), when signal variations from photon counting and dipole wobble are considered separately. Quadratic behaviour can be seen for the case of dipole wobble only, whilst for photon counting the linear behaviour expected from a pure Poisson random variable is evident. The relative importance of the two factors can be seen. At very low light intensities, where it is likely to be impractical to conduct experiments, photon counting dominates. For the intermediate regime both influences are comparable until eventually at higher intensities the molecular wobble dominates.

## Appendix - Three dimensional dipole wobble

*θ*and

*ϕ*as shown in Fig. 2 i.e.

13. B. Sick, B. Hecht, and L. Novotny “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. **85**4482–4485 (2000). [CrossRef] [PubMed]

22. I. Munro, I. Pecht, and L. Stryer “Subnanosecond motions of Tryptophan residues in proteins,” Proc. Natl. Acad. Sci. USA **76**56–60 (1979) [CrossRef] [PubMed]

*a,z*) = ∫

^{∞}

_{z}

*x*

^{a-1}

*e*

^{-x}

*dx*is the incomplete Gamma function and

*θ*and

_{k}*ϕ*are the solutions to the equation

_{k}*i*= ℐ(

*θ*,

*ϕ*).

## References and links

1. | K. D. Weston and L. S. Goldner “Orientation imaging and reorientation dynamics of single dye molecules,” J. Phys. Chem. B |

2. | D.M. Warshaw, E. Hayes, D. Gaffney, A.M. Lauzon, J.R. Wu, G. Kennedy, K. Trybus, S. Lowey, and C. Berger, “Myosin conformational states determined by single fluorophore polarization,” Proc. Natl. Acad. Sci. U.S.A. |

3. | H. P. Lu, L. Y. Xun, and X. S. Xie, “Single molecule enzymatic dynamics,” Science |

4. | R. E. Dale and S. C. Hopkins “Model-Independent analysis of the orientation of fluorescent probes with restricted mobility in muscle fibers, Biophys. J. |

5. | T. M. Jovin, M. Bartholdi, W. L. C. Vaz, and R. H. Austin “Rotational diffusion of biological macromolecules by time-resolved delayed luminescence (phosphorescence, fluorescence) anisotropy,” Ann. N.Y. Acad. Sci. |

6. | T. Ha, T. Enderle, D. F. Ogletree, D. S. Chemla, P. R. Selvin, and S. Weiss, “Probing the interaction between two single molecules: Fluorescence resonance energy transfer between a single donor and a single acceptor” Proc. Natl. Acad. Sci. U.S.A. |

7. | T. Ha, J. Glass, T. Enderle, D. S. Chemla, and S. Weiss “Hindered rotational diffusion and rotational jumps of single molecules,” Phys. Rev. Lett. |

8. | Th. Basché, W. P. Ambrose, and W. E. Moerner, “Optical spectra and kinetics of single impurity molecules in a polymer: spectral diffusion and persistent spectral hole burning,” J. Opt. Soc. Am. B. |

9. | G. H. Patterson, S. N. Knobel, W. D. Sharif, S. R. Kain, and D. W. Piston, “Use of the Green Flurorescent Protein and its mutants in quantitative fluorescence microscopy,” Biophys. J. |

10. | D. J. Pikas, S. M. Kirkpatrick, E. Tewksbury, L. L. Brott, R. R. Naik, M. O. Stone, and W. M. Dennis, “Nonlinear Saturation and Lasing Characteristics of Green Fluorescent Protein,” J. Phys. Chem. B |

11. | A. Leon-Garcia, |

12. | T. Ha, T. Enderle, D. S. Chemla, P. R. Selvin, and S. Weiss “Single molecule dynamics studied by polarization modulation,” Phys. Rev. Lett. |

13. | B. Sick, B. Hecht, and L. Novotny “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. |

14. | R. M. Dickson, D. J. Norris, and W. E. Moerner, “Simultaneous imaging of individual molecules aligned both parallel and perpendicular to the optic axis,” Phys. Rev. Lett. |

15. | K. Itô, |

16. | I. S. Gradshteyn and I. M. Ryzhik, |

17. | W. Feller, |

18. | P. Debye, |

19. | P. Wahl, K. Tawada, and J.C. Auchet “Study of tropomyosin labelled with a fluorescent probe by pulse fluorimetry in polarized light. Interaction of that protein with troponin and actin,” Eur. J. Biochem. |

20. | J. Yguerabide, H. F. Epstein, and L. Stryer “Segmental flexibility in an antibody molecule,” J. Mol. Biol. |

21. | W. E. Moerner and D. P. Fromm, “Methods of single molecule fluorescence spectroscopy and microscopy,” Rev. Sci. Instrum. |

22. | I. Munro, I. Pecht, and L. Stryer “Subnanosecond motions of Tryptophan residues in proteins,” Proc. Natl. Acad. Sci. USA |

**OCIS Codes**

(000.5490) General : Probability theory, stochastic processes, and statistics

(110.4280) Imaging systems : Noise in imaging systems

(180.2520) Microscopy : Fluorescence microscopy

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: May 17, 2007

Revised Manuscript: September 28, 2007

Manuscript Accepted: September 28, 2007

Published: October 2, 2007

**Virtual Issues**

Vol. 2, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Matthew R. Foreman, Sherif S. Sherif, and Peter Török, "Photon statistics in single molecule
orientational imaging," Opt. Express **15**, 13597-13606 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-21-13597

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### References

- K. D. Weston and L. S. Goldner "Orientation imaging and reorientation dynamics of single dye molecules," J. Phys. Chem. B 105, 3453-3462 (2001). [CrossRef]
- D. M. Warshaw, E. Hayes, D. Gaffney, A. M. Lauzon, J. R. Wu, G. Kennedy, K. Trybus, S. Lowey, and C. Berger, "Myosin conformational states determined by single fluorophore polarization," Proc. Natl. Acad. Sci. U.S.A. 95, 8034-8039 (1998). [CrossRef] [PubMed]
- H. P. Lu, L. Y. Xun, and X. S. Xie, "Single molecule enzymatic dynamics," Science 282, 1877-1882 (1998). [CrossRef] [PubMed]
- R. E. Dale and S. C. Hopkins "Model-Independent analysis of the orientation of fluorescent probes with restricted mobility in muscle fibers," Biophys. J. 76, 1606-1618 (1999). [CrossRef] [PubMed]
- T. M. Jovin, M. Bartholdi, W. L. C. Vaz, and R. H. Austin, "Rotational diffusion of biological macromolecules by time-resolved delayed luminescence (phosphorescence, fluorescence) anisotropy," Ann. N. Y. Acad. Sci. 366, 176-196 (1981). [CrossRef] [PubMed]
- T. Ha, T. Enderle, D. F. Ogletree, D. S. Chemla, P. R. Selvin and S. Weiss, "Probing the interaction between two single molecules: Fluorescence resonance energy transfer between a single donor and a single acceptor," Proc. Natl. Acad. Sci. U.S.A. 93, 6264-6268 (1996). [CrossRef] [PubMed]
- T. Ha, J. Glass, T. Enderle, D. S. Chemla, and S. Weiss, "Hindered rotational diffusion and rotational jumps of single molecules," Phys. Rev. Lett. 80, 2093-2096 (1998). [CrossRef]
- Th. Basché, W. P. Ambrose, and W. E. Moerner, "Optical spectra and kinetics of single impurity molecules in a polymer: spectral diffusion and persistent spectral hole burning," J. Opt. Soc. Am. B. 9, 829-836 (1992). [CrossRef]
- G. H. Patterson, S. N. Knobel, W. D. Sharif, S. R. Kain, and D. W. Piston, "Use of the green flurorescent protein and its mutants in quantitative fluorescence microscopy," Biophys. J. 73, 2782-2790 (1997). [CrossRef] [PubMed]
- D. J. Pikas, S. M. Kirkpatrick, E. Tewksbury, L. L. Brott, R. R. Naik, M. O. Stone, and W.M. Dennis, "Nonlinear saturation and lasing characteristics of green fluorescent protein," J. Phys. Chem. B 106, 4831-4837 (2002). [CrossRef]
- A. Leon-Garcia, Probability and Random Processes for Electrical Engineering (Addison-Wesley Publishing Company Inc., 1994).
- T. Ha, T. Enderle, D. S. Chemla, P. R. Selvin, and S. Weiss, "Single molecule dynamics studied by polarization modulation," Phys. Rev. Lett. 77, 3979-3982 (1996). [CrossRef] [PubMed]
- B. Sick, B. Hecht, and L. Novotny "Orientational imaging of single molecules by annular illumination," Phys. Rev. Lett. 85, 4482-4485 (2000). [CrossRef] [PubMed]
- R. M. Dickson, D. J. Norris, and W. E. Moerner, "Simultaneous imaging of individual molecules aligned both parallel and perpendicular to the optic axis," Phys. Rev. Lett. 81, 5322-5325 (1998). [CrossRef]
- K. Itô, Introduction to Probability Theory (Cambridge University Press, 1984).
- I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press, London, 1980).
- W. Feller, Probability Theory and its Applications (John Wiley and Sons Inc., New York, 1950).
- P. Debye, Polar Molecules (Dover Publications, New York, 1945).
- P. Wahl, K. Tawada, and J. C. Auchet, "Study of tropomyosin labelled with a fluorescent probe by pulse fluorimetry in polarized light. Interaction of that protein with troponin and actin," Eur. J. Biochem. 88, 421-424 (1978). [CrossRef] [PubMed]
- J. Yguerabide, H. F. Epstein, and L. Stryer, "Segmental flexibility in an antibody molecule," J. Mol. Biol. 51, 573-590 (1970). [CrossRef] [PubMed]
- W. E. Moerner and D. P. Fromm, "Methods of single molecule fluorescence spectroscopy and microscopy," Rev. Sci. Instrum. 74, 3597-3619 (2003). [CrossRef]
- I. Munro, I. Pecht, and L. Stryer "Subnanosecond motions of Tryptophan residues in proteins," Proc. Natl. Acad. Sci. USA 76, 56-60 (1979). [CrossRef] [PubMed]

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