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Physics of Life Processes, Leiden Institute of Physics, Leiden University, Leiden, The Netherlands
Correspondence: Address reprint requests to T. Schmidt, E-mail: schmidt{at}physics.leidenuniv.nl.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONTHEORYMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX A: BEYOND THE...APPENDIX B: CORRECTION FOR...ACKNOWLEDGEMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX A: BEYOND THE...APPENDIX B: CORRECTION FOR...ACKNOWLEDGEMENTSREFERENCES |
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–6). Driven by recent discussions on the refinement of the classical fluid-mosaic model of the plasma membrane organization (7), both tools were applied to elucidate the contribution of lipid organization and protein interactions to the spatial organization of signaling molecules both in vitro and in vivo. Several structures have been suggested to influence the dynamics of membrane proteins; among these are clathrin-coated pits, caveolae, lipid rafts, and the cytoskeleton. Lipid rafts, especially, have been heavily discussed as possible organizational platforms for molecules involved in cell signaling (8). Their existence and the actual order of lipids in the plasma membrane is, however, still debated (9–12). Recent studies have revealed that protein-protein interactions may play an important role in the spatial organization of signaling proteins (13,14).
Single-particle tracking is ideally suited to study the dynamics of membrane molecules, as this method is able to locate optical probes with a high positional accuracy down to a few nanometers. While gold nanoparticles and fluorescent quantum dots, being relatively large, allow for extremely long observation times (1,3,15,16), labeling of proteins with fluorophores such as, e.g., eGFP or Cy5, is more suitable for biological applications. Those fluorophores, however, suffer from photobleaching. Therefore, tracking of individual molecules results in comparatively short trajectories (typically 10 steps), which makes the retrieval of individual trajectory dynamical information exceedingly difficult. However, given that the biological system is quite stable, the number of observations obtained under the same conditions can be large, to enable determination of dynamic properties of membranes in great detail (17).
For the implementation of SPT, some a priori knowledge about the expected molecular behavior is needed since algorithms have to cope with the probabilistic nature of the tracking problem (3,18). This is especially a drawback for data taken at higher concentrations, where molecular trajectories can be accidentally mixed. Image correlation microscopy (ICM) (5) and fluorescence correlation spectroscopy (2,3) do not need any such prior information. However, both are regular imaging techniques limited in resolution by diffraction and thus by a spatial resolution of 200–300 nm.
To overcome the drawbacks of both SPT and ICM we have developed a robust analysis method that combines both techniques. The method is self-contained on any ensemble of diffusion steps and therefore does not need individual traces to be assigned like in SPT. Consequently, it can deal with arbitrarily high molecule densities and diffusion constants as long as individual molecules can be identified. The starting point of this method is a correlation function, analogous to spatiotemporal image correlation spectroscopy (STICS) (19,20). A qualitative criterion for the general applicability is given. Further, theoretical boundaries for the achievable accuracy are discussed. Finally, the validity of the method is demonstrated by application to data created by Monte Carlo simulations and analysis of experimental data (17). The latter proves the existence of functional domains smaller than 200 nm in the plasma membrane of 3T3-A14 fibroblast cells.
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THEORY |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX A: BEYOND THE...APPENDIX B: CORRECTION FOR...ACKNOWLEDGEMENTSREFERENCES |
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Algorithm
An image I obtained from SPT experiments is described as a sum of delta peaks representing the positions ri of the molecules:
 | (1) |
). For any pair of images, Ia and Ib, which are separated in time by a time lag of 
t, a spatiotemporal correlation function is defined
 | (2) |
...
t denotes the ensemble average over all pairs of images separated by a time-lag t, and A is the area of the field of view of the microscope. The two images are shifted by d with respect to each other and subsequently correlated, i.e., the spatial integral of their product is calculated. If d coincides with a movement during the time-lag t, the correlation will be high. The precise connection to the diffusion dynamics is given below. Note that C(d, t) is basically the correlation function used in STICS (19,20), where the denominator is given by the average number of molecules in image Ia only. This normalization was chosen since it leads directly to the cumulative probability distribution of diffusion steps; see Eq. 5.
In an isotropic medium, the cumulative correlation function only depends on a distance l and time-lag t. By definition of d(
, 
) = ( cos, sin ) with polar coordinates and ,
 | (3) |
. The expression mb(r, l) is the number of molecules in image Ib that lie in a circle with radius l around r.
The algorithm to obtain Ccum(l, t) from experimental data, derived directly from Eq. 3 and the definition of mb(r, l), is illustrated in Fig. 1: for each molecule position rai in image Ia, the number of molecules in image Ib are counted whose distance to rai is smaller or equal to l. Subsequently the contributions from all molecules in image Ia are summed and averaged over all image pairs. The division by the average number of molecules in image Ia finally results in Ccum(l, t).
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t) contains both temporal (i.e., diffusion of molecules) and spatial (i.e., random spatial proximity of molecules) correlations, which will be separated below. The spatial correlations are illustrated by the overlap of the circles in Fig. 1. Given that the molecules are identical, their movement is mutually uncorrelated, and the medium is homogeneous, Ccum(l, t) is simplified to
 | (4) |
is the arbitrary position of a molecule in image Ia. Note that the summation in Eq. 3 cancels out with the denominator ma under the given assumptions. It should be mentioned that a global flow of the molecules is admissible. The same holds for interactions between molecules if they can be sufficiently described by a mean-field approximation. The part of Eq. 4 that is caused by accidental spatial proximity of different molecules is equal to the mean number of molecules in a circle with radius l around a certain fixed but arbitrary molecule. Given that the molecules are distributed uniformly and independently with a density c, the probability to find µ molecules in this circle is given by a Poisson distribution with mean and variance of: 
, where c can be estimated as c = (mb – 1)/A. The latter assumption is justified, given that the ensemble average usually comprises many images of many different cells. Note that the precise definition of c is the density of the neighbors of a certain molecule. For higher densities this equals the total density, since then (mb – 1)/A 
mb/A.
The part of Eq. 4 that contains the diffusion dynamics of the molecules is equal to the cumulative probability Pcum(l, t) to find a diffusion step with a size smaller than l if the time lag is t. For normal diffusion with diffusion coefficient D in two dimensions,
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t):
 | (5) |
The quantity calculated from experimental data by the algorithm described above (Eq. 3) is an estimator for this theoretically expected value. We now define a typical length-scale lcum by
 | (6) |
l2 from Ccum this length scale can be determined and the diffusion constant is calculated as
 | (7) |
Figure of merit and achievable accuracy
Determination of Pcum(l, t) from Eq. 5 is only practical if the variance of the second term cl2 is sufficiently small. Since the average of M statistically uncorrelated pairs of images is taken, the variance is 1/M times the value given above for the single Poisson process. Note that successive pairs of images are statistically uncorrelated since diffusion is a Markov process, whereas successive images are necessarily correlated. To get a qualitative criterion for the number of image pairs to be taken for a significant result, the standard deviation of the spatial correlations at lcum (given by Eq. 7) is compared to the value of Pcum(l, t) at lcum:
 | (8) |
We define a figure of merit 
as twice this standard deviation
 | (9) |
Thus the result will be significant if << 1. Note that molecules may be arbitrarily dense (provided that the overlapping images still allow them to be identified as individual molecules) or diffuse arbitrarily fast if only the number M of image-pairs is sufficiently large.
If the whole correction term is small, 
, i.e.,
 | (10) |
 | (11) |
To get an error estimate for the diffusion constant D the probability density Pcum(l, t) is shifted vertically by ±/2. From the calculation of the typical length scale lcum of the shifted curves, boundaries for the values of D are retrieved,
 | (12) |
. 
designates the mean D.
While this error originates from the method, there is an intrinsic spread of the values obtained for lcum that is due to the stochastic nature of diffusion. If M pairs of images with m molecules on the average are acquired, the number of diffusion steps to be analyzed is N = M m. The probability to find N/2 steps with a step-size smaller than lcum is given by
 | (13) |
. This probability density for lcum is depicted in Fig. 2 for various values of N. For an increasing number of diffusion steps, N, the function becomes symmetric about the value given by Eq. 7 and the width decreases. Hence the more images analyzed the less the spread in lcum. Expansion of the exponentials in Eq. 13 around the maximum and estimation of the relative width for N >> 1 yields 
where 
designates the mean lcum and equals the value given by Eq. 7. Note that lcum is defined analogous to the standard deviation as half the width of Eq. 13. Error propagation gives 
. To determine D with a relative error of ±0.1, N 300 diffusion steps are needed. Since the accuracy scales as 
for N >> 1, a relative error of ±0.01 requires N 30,000 steps. Note that this error estimation is only valid if the diffusion coefficient is determined from the typical length scale lcum of Pcum(l, t). For the scatter inherent to other analysis methods, see the article by Saxton (21).
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 | (14) |
Diffusion modes
Given that the criterion below Eq. 9 is fulfilled, the method developed up to this point is exact for the case of a single, normally diffusing species. For other (anomalous) cases (multiple fractions, intermittent, confined, or anisotropic diffusion, diffusion with trapping or, more generally, diffusion in a potential landscape), the diffusion coefficient determined as described above is only an estimation of the mean diffusion coefficient.
However, since the cumulative probability of step-sizes is intrinsic to the correlation function Eq. 5, analysis of data with more complicated diffusion models is straightforward. E.g., for a two-fraction case, which is important for the data analyzed below, molecules in image Ia are split in a fraction of size 
with diffusion coefficient D1 and one of size 1 – with diffusion coefficient D2. This results in
 | (15) |
. Hence, the probability distribution Pcum(l, t) can faithfully be used to analyze more complex inhomogeneous diffusion behavior. |
MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX A: BEYOND THE...APPENDIX B: CORRECTION FOR...ACKNOWLEDGEMENTSREFERENCES |
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t. The value Asim was taken large enough for the distribution of the molecules to be still approximately uniform in A after each time step t. The average number of molecules in A was fixed at five. Image Ib was obtained by letting each molecule in Ia perform a random step in x and y directions. The step-size in both spatial directions was determined by a Gaussian with variance 2Dt, i.e., all simulated molecules obeyed normal diffusion. Subsequently, all molecules that did not fall into the physical field of view were discarded. Furthermore, it was ensured that diffusion steps up to lmax were adequately represented as detailed in Appendix A. The algorithm derived above was subsequently executed for the values l = 
l, 2 l,..., lmax.
The value lcum was found from Pcum(l, t) by linear interpolation of the distribution at 0.5. The results were normalized to 
such that, according to Eq. 7, a value of 1 corresponds to the most probable lcum. The whole simulation was repeated 1000 times and the results were divided into bins of width 0.05. The number of data points in each bin was subsequently divided by 1000, which resulted in relative frequencies for lcum. For comparison of the simulation with theoretical predictions, the probability density derived in Eq. 13 was integrated over intervals of length 0.05, i.e., the bin size.
Since only a finite number of values for l can be considered, a binning error that depends on l is introduced. Consequently, the distribution of the lcum values will always deviate from Eq. 13. In Fig. 3, results for 
, and 
are compared with lmax = 3. Since we choose a very small density and diffusion coefficient (c = 2.5 x 10–4/µm2, Dt = 0.02 µm2), the deviation from the theoretical distribution Eq. 13 is caused by the binning error alone. Obviously the deviation decreases with decreasing l. The simulations therefore use 
. For smaller or bigger diffusion coefficients or time lags, lmax is scaled accordingly.
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). In short, constitutive active human H-Ras (V12) and constitutive inactive human H-Ras (N17) were coded into pcDNA3.1-eYFP (Qiagen, Hilden, Germany). Cells from a mouse fibroblast cell line stably expressing the human insulin receptor (3T3-A14) (22) were transfected with 1.0 µg DNA and 3 µl FuGENE 6 (Roche Molecular Biochemicals, Indianapolis, IN) per glass slide. 3T3-A14 cells adhered to glass slides were mounted onto the microscope and kept in PBS at 37°C. For the observation of the mobility of individual eYFP-H-Ras molecules, the focus of the microscope was set to the dorsal surface membrane of individual cells (depth of focus 1 µm). The density of fluorescent proteins on the plasma membrane of selected transfected cells was <1 µm–2 to permit imaging and tracking of individual fluorophores. Molecule positions were determined with an accuracy of 35 nm. Fluorescence images were taken consecutively with up to 1000 images per sequence. Typical trajectories were up to nine steps in length, mainly limited by the blinking and photobleaching of the fluorophore (23). Data sets were acquired with different time-lags t between consecutive images. The value t varied from 5 to 60 ms. |
RESULTS |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX A: BEYOND THE...APPENDIX B: CORRECTION FOR...ACKNOWLEDGEMENTSREFERENCES |
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t. The simulated concentrations correspond to a range of 0.1–10 molcules/µm2 for typical experimental values (D 1 µm2/s, t 20 ms).
The results for M = 100 and M = 1000 are presented in Fig. 4. For fixed M, the distribution of lcum values broadens with rising molecule concentration. It should be noted that the distribution of lcum always peaked around the true value. When the correction term for correlations due to random spatial proximity of molecules was omitted (i.e., the second term in Eq. 5), the peak lcum values shifted to a lower value. Likewise the dependence of the method on the diffusion constant D and the number of image pairs M was studied for a fixed molecule density. For typical experimental values (c 1/µm2, t 20 ms), the diffusion constants correspond to a range from 0.1 µm2/s to 10 µm2/s. Results are shown in Fig. 4. The distribution broadens with D, similar to the results for growing molecule density. As predicted by Eq. 12, the distributions become narrower for growing M, which supports the claim that a higher number of image-pairs will compensate for a high molecular density or diffusion constant. The applicability of the method is, therefore, only limited by the number of images that can be acquired for identical conditions. The influence of bleaching and blinking on the distribution of lcum is shown in Fig. 5. Molecules were assumed to turn dark with a probability pdark per time-lag t. The distribution broadens if this probability is increased but stays peaked around the true value. The broadening is fully accounted for by the reduction of the statistical sample size N = Mm. E.g., for pdark = 0.9, only 10% of molecules survive, leaving only 50 visible diffusion steps instead of 500 for pdark = 0. We do not consider explicitly here that molecules can return into the fluorescent state (blinking), since the only effect is an increase in the apparent molecule density c, which was analyzed above.
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), it was found that both the constitutive inactive (N17) as well the constitutive active (V12) variant of the protein displayed an inhomogeneous two-fraction diffusion behavior. In that earlier report the positions of proteins in an image sequence were used to calculate trajectories from which further information on the mobility was extracted. Here the same position data is analyzed with the new algorithm without any a priori knowledge about molecular mobility.
The molecule density c was estimated from the experimental data. The slope of the linear part of Ccum(l) when plotted versus l2 (Fig. 6) directly equals c · . Note that c is by definition of this procedure exactly the density of neighboring molecules introduced above. Subtraction of the correction term cl2 successfully yielded Pcum(l, t) for longer time lags (solid data points in Fig. 6). Artifacts due to diffraction observed for shorter time lags were removed by an empirical, self-consistent algorithm, as detailed in Appendix B.
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t) was subsequently constructed for each time-lag t between 4 and 60 ms. Data were fit according to the two-diffusing-fraction model (Eq. 15) to yield the fraction and respective mean-square displacements 
and 
for both mutants.
Fig. 7 compares the results obtained by the new unbiased method (solid symbols, solid lines) with those obtained by conventional tracking methods (open symbols, dashed lines) in which an initial diffusion constant of D = 1 µm2/s had been assumed. Both data sets excellently match each other within experimental accuracy; see Table 1. For the inactive mutant (N17), 86% of the molecules fell into the highly mobile fraction characterized by a diffusion constant of D1 = 0.94 µm2/s. The slow fraction was characterized by a diffusion constant of D2 = 0.10 µm2/s. Both fractions followed free diffusion as seen by the linear dependence of the mean-square displacements (
) with t. In contrast, the slow diffusing fraction of the active mutant (V12) displayed a confined diffusion behavior (24) characterized by a confinement size of L = 179 nm. In addition, the diffusion constant of the fast, free diffusion fraction of the V12-mutant was reduced to D1 = 0.73 µm2/s and the fraction size decreased to 63% in comparison to the inactive mutant N17.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX A: BEYOND THE...APPENDIX B: CORRECTION FOR...ACKNOWLEDGEMENTSREFERENCES |
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The results of change in mobility on the activation state of H-Ras by the new unbiased method further supports ideas of functional domains in the plasma membrane of mammalian cells. The results agree well with the results of the FRET (25), FRAP (26), EM (27), and single-molecule tracking experiments (17) in all of which functional domains have been observed. Likely localization of active H-Ras to these functional domains is not a static process, but is dynamic as suggested for trapping into cholesterol-independent domains (27) and into more general transient signaling complexes (25), which might be actin-dependent.
In summary, a robust method was presented that is superior to both ICM and SPT, surpassing the first in resolution and largely simplifying the analysis methods required for the second. Another intriguing application is the study of dynamical properties of interacting proteins in model membranes. Because the newly developed method allows the protein concentration to be varied over a wider range, a comparison to theoretical results obtained by a virial expansion is rendered possible.
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APPENDIX A: BEYOND THE IDEAL SITUATION |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX A: BEYOND THE...APPENDIX B: CORRECTION FOR...ACKNOWLEDGEMENTSREFERENCES |
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Finite positional accuracy
The limited positional accuracy makes a fixed molecule appear to move and a free molecule to diffuse faster. Since the real diffusive motion and the apparent motion due to the limited positional accuracy are uncorrelated, the fluctuations simply add so that
 | (16) |
is the standard deviation of a Gaussian distribution that describes the positional error in one dimension. Either the positional accuracy has to be determined independently or the time-lag t must be varied so that the real diffusion coefficient can be obtained from the slope of Eq. 16. Note that this problem does not interfere with the method presented here; e.g., in the case of normal diffusion of one or two molecular species, the functional form of the cumulative probability distribution Pcum remains unchanged. For other diffusion modes, the correct form of Pcum, which might be altered due the finite positional accuracy, has to be employed. An extensive discussion can be found in Martin et al. (28).
Finite exposure/frame integration time
The fact that the fluorescence signal collection and integration time is finite can lead to erroneous results, in particular for confined diffusion (29,30). However, it was shown in Destainville and Salome (30) that the true values for the diffusion coefficient and the size of the confinement area can be retrieved from the data anyway. For the analysis performed above we assume that the influence of confinement or a finite exposure time on the cumulative probability distribution Pcum(l, t) is negligible compared to the experimental error. This is quantified by the criterion given in Destainville and Salome (30): if L is the linear size of the confinement, D is the diffusion coefficient, and T is the exposure/integration time, then 
should be fulfilled. This is indeed the case for the experiments presented above with L 0.18 µm, D = 0.1 µm2/s, and T = 3 ms. So, it is sensible to expect a distribution representing normal diffusion. It should, however, be stressed that our method works in principle for arbitrary forms of Pcum(l, t).
Bleaching and blinking
Because of blinking and bleaching, single-particle trajectories of biologically relevant fluorophores inside cells are usually short (10 steps). Given that poff is the probability per time-lag t that a molecule turns dark or is not found by the peak-fitting algorithm (see also Appendix B), only a fraction (1 – poff) of all diffusion steps is observed. Under the assumption that bleaching is independent of the size of a diffusion step, Pcum is reduced by a factor (1 – poff). One consequence is that the figure of merit (Eq. 9) must be generalized to
 | (17) |
Accordingly Eq. 10 changes to
 | (18) |
The second consequence is that the experimental correlation function Ccum has to be normalized to 1, after subtraction of the correction term cl2, to yield Pcum (see also Appendix B). Correspondingly, the theoretical distribution function has to be divided by Pcum(lmax, t) where lmax is the maximal l included in the analysis.
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APPENDIX B: CORRECTION FOR POSITIONAL CORRELATIONS DUE TO DIFFRACTION |
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0.4 µm) this effect is only observable for small step-sizes, i.e., for slowly diffusing molecules or small time lags. To circumvent this problem, we adapted our algorithm in the following way: in the simple estimation, the number of "wrong" connections that the algorithm makes is described by the quadratic correction term cl2; now the amount of molecules that are found within a certain radius depends on the size of the diffusive step. If the molecule turns dark during the time lag there is no correlation. Therefore Eq. 5 is generalized to
 | (19) |
Pcum(r, t)/r gives the probability for a step of length r. The value pdark—the probability per time-lag that a molecule turns dark—is estimated once and kept fixed for all data sets. For the data analyzed above, pdark = 0.3 was used. The value poff is the probability that a molecule either turns dark or is not found by the molecule-fitting routine, e.g., since it came too close to another molecule. The value 1 – poff can be estimated by the height of Ccum after subtraction of the correction term. The value s(r, l) is determined empirically from the experimental data by application of the algorithm defined in the beginning where, however, images Ia and Ib are identical. Furthermore, the center of the circle, with radius l, in which the molecules are counted, is translated by a vector of length r in arbitrary direction. The average over 20 equally spaced directions results in the array of curves depicted in Fig. 8. Subsequent to the calculation of s(r, l) the correction is determined numerically by the following self-consistent algorithm:Steps 2–4 are repeated until the fit parameters change less than a predefined threshold. Note that this approach to correct for the effective correlation of the peak positions only works because the effect is the same for all molecules. If positional correlations that are different for different molecules become important, the approach is no longer functional.
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX A: BEYOND THE...APPENDIX B: CORRECTION FOR...ACKNOWLEDGEMENTSREFERENCES |
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This work was supported by funds from the Stichting voor Fundamenteel Onderzoek der Materie (FOM) program on Material Properties of Biological Assemblies (grant No. 05MPBA07).
Submitted on July 6, 2006; accepted for publication October 11, 2006.
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REFERENCES |
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for N = 2, 4, 8,..., 1024. The curve for N = 1024 corresponds to the sharpest distribution. For N = 512, expansion around the maximum was used to estimate the width of the distribution (dashed curve). Arrows indicate 
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; square, 
; and circle, 
, and compared to the distribution as given by 
