PRL 110, 159801 (2013)
PHYSICAL REVIEW LETTERS
Comment on ‘‘Polymer Dynamics, Fluorescence Correlation Spectroscopy, and the Limits of Optical Resolution’’ We and others have been applying fluorescence correlation spectroscopy (FCS) to measure the dynamics of biological polymers [1–5]. Under rather general conditions, a simple derivation [3,6] relates the monomer mean square displacement (MSD) hr2 ðtÞi to the FCS autocorrelation function (ACF) GðtÞ ¼ F ½hr2 ðtÞi, where the function F is given by (skipping numeric coefficients) F ðhÞ ¼ ð1 þ h=w2xy Þ1 ð1 þ h=w2z Þ1
(1)
and wxy and wz characterize the optical transfer function (OTF) of the setup. Equation (1) allows us to extract the monomer MSD from the measured ACF. The part of the motion range we resolve is below the optical resolution limit wxy . In a recent Letter [7], Enderlein purports to prove theoretically that such resolution is not possible. He considers two models—(1) 1D diffusion of a particle confined to 2a interval and (2) a dumbbell of two beads on a spring—and concludes that ‘‘an effective MSD. . .can produce any power-law behavior with little connection to the actual underlying physical process.’’ It is not clear what type of dynamics the author expects from these systems, but our approach, as applied by him, reflects the physical processes correctly. The situation is most egregious for the dumbbell model: The author is unaware that his long calculation involving 12 integrations in fact rederives our general result for this particular case. Indeed, Enderlein’s ACF for the dumbbell is G ¼ F f½1 expðD0 t=Þ þ Dtg, with F given by Eq. (1) and the notation of Ref. [7]. This matches our expression exactly since the MSD of a bead in the dumbbell is hr2 ðtÞi ½1 expðD0 t=Þ þ Dt, as obtained by a simple substitution of 2 for the number of beads in textbook Rouse chain treatments [8,9] or derived directly [9]. Thus, our approach here extracts the real MSD of a bead, not an effective MSD as claimed by Enderlein. Enderlein observes that in a limited range of parameters the uncovered dumbbell dynamics is reminiscent of that of polymers. However, since the FCS approach assesses the dynamics correctly, this observation does not concern the method but data interpretation only. We interpret our data in the context of polymer dynamics theories since DNA is a polymer, not a dumbbell. As expected for polymers, the observed power laws do not change in a wide temporal range and do not depend on DNA length. Likewise, Fig. 2 of Ref. [7] illustrates a qualitatively [10] correct result of the application of our approach to the confined motion model: hr2 i / t , with changing from 1 for pure diffusion at short time scales (t a2 =D) to 0 for t a2 =D when the MSD saturates because of the confinement. Thus, the two considered models support the FCS method rather than undermine it. Enderlein concludes that FCS ‘‘is governed by the same spatial resolution limit as any linear optical imaging 0031-9007=13=110(15)=159801(2)
week ending 12 APRIL 2013
system,’’ implying Abbe’s resolution limit. However, it has long been recognized that the measurement of molecular positions and displacements in microscopy is limited by the number of collected photons rather than by Abbe’s criterion [11,12]. The same holds for FCS: Any small displacement of a molecule causes some change in emission, and with enough photons it can be picked up reliably. FCS excels at accumulating photon statistics. Enderlein’s own Fig. 2 (inset) contradicts his stated conclusion: As a particle diffuses within the ‘‘subresolution’’ 100 nm interval, its ACF decays by 40%. Such and much smaller changes can easily be picked by FCS. Formally, rather simple derivations based on Ref. [13] lead us to a direct relation and hr2 ðtÞi for R between2 GðtÞ 2 hr2 ðtÞi=6 q ~ qÞ: ~ qÞj ~ GðtÞ / dqj ~ Ið ~ e [6]. The any OTF Ið assumption of the Gaussian OTF leads to Eq. (1) here and in Refs. [1,2]. In this case, the finite support of the OTF discussed in Ref. [7] is reflected in the finite width of the Gaussian. However, even if the OTF is not Gaussian ~ > qmax Þ ¼ 0 for strictly finite support], the [e.g., if Iðq integral form above expresses G directly through hr2 i. So, if Gðhr2 iÞ dependence is calibrated by some simple process such as diffusion of small dye molecules, this calibration can be then used as a look-up table to analyze more complicated situations; e.g., a 10% decay in the ACF corresponds to the same MSD for a dye molecule and for a tagged monomer in a chain. Practically, since Gaussian approximation of the OTF describes the diffusion of dye molecules very well, as indeed used by every FCS lab (even if the OTF is not really Gaussian), it can be applied as an empirical calibration for ACF to MSD conversion in other cases. Nothing here implies any fundamental optical limits on the resolution of MSD measurement. This is well supported by simulations [14]. As with any experimental technique, our method has its limitations [14]. However, those addressed by Ref. [7] are not among them. I am grateful to J. Langowski and J. Krieger for discussions. This work was supported by Israel Science Foundation Grant No. 984/09 and German-Israeli Foundation Grant No. 972-146.14/2007.
Oleg Krichevsky* Physics Department and Ilse Kats Center for Nanoscience Ben-Gurion University Beer-Sheva 84105, Israel Received 23 March 2012; revised manuscript received 9 November 2012; published 9 April 2013 DOI: 10.1103/PhysRevLett.110.159801 PACS numbers: 87.64.kv, 87.57.cf
*[email protected] [1] R. Shusterman, S. Alon, T. Gavrinyov, and O. Krichevsky, Phys. Rev. Lett. 92, 048303 (2004).
159801-1
Ó 2013 American Physical Society
PRL 110, 159801 (2013)
PHYSICAL REVIEW LETTERS
[2] R. Shusterman, T. Gavrinyov, and O. Krichevsky, Phys. Rev. Lett. 100, 098102 (2008). [3] A. Bernheim-Groswasser, R. Shusterman, and O. Krichevsky, J. Chem. Phys. 125, 084903 (2006). [4] E. P. Petrov, T. Ohrt, R. G. Winkler, and P. Schwille, Phys. Rev. Lett. 97, 258101 (2006). [5] T. Kalkbrenner, A. Arnold, and S. J. Tans, Biophys. J. 96, 4951 (2009). [6] See Supplemental Material of Ref. [2]. [7] J. Enderlein, Phys. Rev. Lett. 108, 108101 (2012). [8] M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Clarendon, Oxford, 1986). [9] T. C. B. McLeish, Adv. Phys. 51, 1379 (2002). [10] In the confined diffusion case, our analysis cannot be applied quantitatively since one of the conditions for
[11]
[12]
[13] [14]
159801-2
week ending 12 APRIL 2013
our derivation, the normal distribution of displacements, is not satisfied in the long run here. None of our experimental papers applied our approach to such situations. T. Schmidt, G. J. Schu¨tz, W. Baumgartner, H. J. Gruber, and H. Schindler, Proc. Natl. Acad. Sci. U.S.A. 93, 2926 (1996). E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, Science 313, 1642 (2006). B. J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976). T. Wocjan, J. Krieger, O. Krichevsky, and J. Langowski, Phys. Chem. Chem. Phys. 11, 10 671 (2009).
PHYSICAL REVIEW LETTERS
Comment on ‘‘Polymer Dynamics, Fluorescence Correlation Spectroscopy, and the Limits of Optical Resolution’’ We and others have been applying fluorescence correlation spectroscopy (FCS) to measure the dynamics of biological polymers [1–5]. Under rather general conditions, a simple derivation [3,6] relates the monomer mean square displacement (MSD) hr2 ðtÞi to the FCS autocorrelation function (ACF) GðtÞ ¼ F ½hr2 ðtÞi, where the function F is given by (skipping numeric coefficients) F ðhÞ ¼ ð1 þ h=w2xy Þ1 ð1 þ h=w2z Þ1
(1)
and wxy and wz characterize the optical transfer function (OTF) of the setup. Equation (1) allows us to extract the monomer MSD from the measured ACF. The part of the motion range we resolve is below the optical resolution limit wxy . In a recent Letter [7], Enderlein purports to prove theoretically that such resolution is not possible. He considers two models—(1) 1D diffusion of a particle confined to 2a interval and (2) a dumbbell of two beads on a spring—and concludes that ‘‘an effective MSD. . .can produce any power-law behavior with little connection to the actual underlying physical process.’’ It is not clear what type of dynamics the author expects from these systems, but our approach, as applied by him, reflects the physical processes correctly. The situation is most egregious for the dumbbell model: The author is unaware that his long calculation involving 12 integrations in fact rederives our general result for this particular case. Indeed, Enderlein’s ACF for the dumbbell is G ¼ F f½1 expðD0 t=Þ þ Dtg, with F given by Eq. (1) and the notation of Ref. [7]. This matches our expression exactly since the MSD of a bead in the dumbbell is hr2 ðtÞi ½1 expðD0 t=Þ þ Dt, as obtained by a simple substitution of 2 for the number of beads in textbook Rouse chain treatments [8,9] or derived directly [9]. Thus, our approach here extracts the real MSD of a bead, not an effective MSD as claimed by Enderlein. Enderlein observes that in a limited range of parameters the uncovered dumbbell dynamics is reminiscent of that of polymers. However, since the FCS approach assesses the dynamics correctly, this observation does not concern the method but data interpretation only. We interpret our data in the context of polymer dynamics theories since DNA is a polymer, not a dumbbell. As expected for polymers, the observed power laws do not change in a wide temporal range and do not depend on DNA length. Likewise, Fig. 2 of Ref. [7] illustrates a qualitatively [10] correct result of the application of our approach to the confined motion model: hr2 i / t , with changing from 1 for pure diffusion at short time scales (t a2 =D) to 0 for t a2 =D when the MSD saturates because of the confinement. Thus, the two considered models support the FCS method rather than undermine it. Enderlein concludes that FCS ‘‘is governed by the same spatial resolution limit as any linear optical imaging 0031-9007=13=110(15)=159801(2)
week ending 12 APRIL 2013
system,’’ implying Abbe’s resolution limit. However, it has long been recognized that the measurement of molecular positions and displacements in microscopy is limited by the number of collected photons rather than by Abbe’s criterion [11,12]. The same holds for FCS: Any small displacement of a molecule causes some change in emission, and with enough photons it can be picked up reliably. FCS excels at accumulating photon statistics. Enderlein’s own Fig. 2 (inset) contradicts his stated conclusion: As a particle diffuses within the ‘‘subresolution’’ 100 nm interval, its ACF decays by 40%. Such and much smaller changes can easily be picked by FCS. Formally, rather simple derivations based on Ref. [13] lead us to a direct relation and hr2 ðtÞi for R between2 GðtÞ 2 hr2 ðtÞi=6 q ~ qÞ: ~ qÞj ~ GðtÞ / dqj ~ Ið ~ e [6]. The any OTF Ið assumption of the Gaussian OTF leads to Eq. (1) here and in Refs. [1,2]. In this case, the finite support of the OTF discussed in Ref. [7] is reflected in the finite width of the Gaussian. However, even if the OTF is not Gaussian ~ > qmax Þ ¼ 0 for strictly finite support], the [e.g., if Iðq integral form above expresses G directly through hr2 i. So, if Gðhr2 iÞ dependence is calibrated by some simple process such as diffusion of small dye molecules, this calibration can be then used as a look-up table to analyze more complicated situations; e.g., a 10% decay in the ACF corresponds to the same MSD for a dye molecule and for a tagged monomer in a chain. Practically, since Gaussian approximation of the OTF describes the diffusion of dye molecules very well, as indeed used by every FCS lab (even if the OTF is not really Gaussian), it can be applied as an empirical calibration for ACF to MSD conversion in other cases. Nothing here implies any fundamental optical limits on the resolution of MSD measurement. This is well supported by simulations [14]. As with any experimental technique, our method has its limitations [14]. However, those addressed by Ref. [7] are not among them. I am grateful to J. Langowski and J. Krieger for discussions. This work was supported by Israel Science Foundation Grant No. 984/09 and German-Israeli Foundation Grant No. 972-146.14/2007.
Oleg Krichevsky* Physics Department and Ilse Kats Center for Nanoscience Ben-Gurion University Beer-Sheva 84105, Israel Received 23 March 2012; revised manuscript received 9 November 2012; published 9 April 2013 DOI: 10.1103/PhysRevLett.110.159801 PACS numbers: 87.64.kv, 87.57.cf
*[email protected] [1] R. Shusterman, S. Alon, T. Gavrinyov, and O. Krichevsky, Phys. Rev. Lett. 92, 048303 (2004).
159801-1
Ó 2013 American Physical Society
PRL 110, 159801 (2013)
PHYSICAL REVIEW LETTERS
[2] R. Shusterman, T. Gavrinyov, and O. Krichevsky, Phys. Rev. Lett. 100, 098102 (2008). [3] A. Bernheim-Groswasser, R. Shusterman, and O. Krichevsky, J. Chem. Phys. 125, 084903 (2006). [4] E. P. Petrov, T. Ohrt, R. G. Winkler, and P. Schwille, Phys. Rev. Lett. 97, 258101 (2006). [5] T. Kalkbrenner, A. Arnold, and S. J. Tans, Biophys. J. 96, 4951 (2009). [6] See Supplemental Material of Ref. [2]. [7] J. Enderlein, Phys. Rev. Lett. 108, 108101 (2012). [8] M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Clarendon, Oxford, 1986). [9] T. C. B. McLeish, Adv. Phys. 51, 1379 (2002). [10] In the confined diffusion case, our analysis cannot be applied quantitatively since one of the conditions for
[11]
[12]
[13] [14]
159801-2
week ending 12 APRIL 2013
our derivation, the normal distribution of displacements, is not satisfied in the long run here. None of our experimental papers applied our approach to such situations. T. Schmidt, G. J. Schu¨tz, W. Baumgartner, H. J. Gruber, and H. Schindler, Proc. Natl. Acad. Sci. U.S.A. 93, 2926 (1996). E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, Science 313, 1642 (2006). B. J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976). T. Wocjan, J. Krieger, O. Krichevsky, and J. Langowski, Phys. Chem. Chem. Phys. 11, 10 671 (2009).
