Sizing-up finite fluorescent particles with nanometer-scale precision by convolution and correlation image analysis

Abstract

Determining the positions, shapes and sizes of finite living particles such as bacteria, mitochondria or vesicles is of interest in many biological processes. In fluorescence microscopy, algorithms that can simultaneously localize such particles as a function of time and determine the parameters of their shapes and sizes at the nanometer scale are not yet available. Here we develop two such algorithms based on convolution and correlation image analysis that take into account the position, orientation, shape and size of the object being tracked, and we compare the precision of the two algorithms using computer simulations. We show that the precision of both algorithms strongly depends on the object’s size. In cases where the diameter of the object is larger than about four to five times the beam waist radius, the convolution algorithm gives a better precision than the correlation algorithm (it leads to more precise parameters), while for smaller object diameters, the correlation algorithm gives superior precision. We apply the convolution algorithm to sequences of confocal laser scanning micrographs of immobile Escherichia coli bacteria, and show that the centroid, the front end, the rear end, the left border and the right border of a bacterium can be determined with a signal-to-noise-dependent precision down to ~5 nm.

Appendices

Appendix A

Derivation of Eq. 10 leads to the three intensity contributions of the first cap, the shaft, and the second cap of the model bacterium: $I^{{{\text{ell1}}}}_{{{\text{obj}}}} (\tilde{x},\tilde{y})$ , $I^{{{\text{cyl}}}}_{{{\text{obj}}}} (\tilde{x},\tilde{y})$ and $I^{{{\text{ell2}}}}_{{{\text{obj}}}} (\tilde{x},\tilde{y})$ , respectively,

$$ \begin{aligned} I^{{{\text{ell1}}}}_{{{\text{obj}}}} (\tilde{x},\tilde{y}) = & q{\int\limits_{ - d_{1} }^0 {{\text{d}}{x}'} }{\int\limits_0^1 {{\text{d}}\eta } }{\int\limits_0^{2\pi } {{\text{d}}\theta } }\,\eta {\left( {\frac{{d_{y} }} {2}} \right)}^{2} {\left( {1 - \frac{{{x}'^{2} }} {{d^{2}_{1} }}} \right)} \\ & \quad \times \exp {\left[ {{ - 2(\tilde{x} - \tilde{x}_{0} + {d_{x} } \mathord{\left/ {\vphantom {{d_{x} } 2}} \right. \kern-\nulldelimiterspace} 2 - {x}')^{2} } \mathord{\left/ {\vphantom {{ - 2(\tilde{x} - \tilde{x}_{0} + {d_{x} } \mathord{\left/ {\vphantom {{d_{x} } 2}} \right. \kern-\nulldelimiterspace} 2 - {x}')^{2} } {r^{2}_{{xy}} }}} \right. \kern-\nulldelimiterspace} {r^{2}_{{xy}} }} \right]} \\ & \quad \times \exp {\left[ {{ - 2{\left\{ {\tilde{y} - \tilde{y}_{0} - {\left( {{d_{y} } \mathord{\left/ {\vphantom {{d_{y} } 2}} \right. \kern-\nulldelimiterspace} 2} \right)}{\left( {{1 - {x}'^{2} } \mathord{\left/ {\vphantom {{1 - {x}'^{2} } {d^{2}_{1} }}} \right. \kern-\nulldelimiterspace} {d^{2}_{1} }} \right)}^{{1/2}} \,\eta \cos \theta } \right\}}^{2} } \mathord{\left/ {\vphantom {{ - 2{\left\{ {\tilde{y} - \tilde{y}_{0} - {\left( {{d_{y} } \mathord{\left/ {\vphantom {{d_{y} } 2}} \right. \kern-\nulldelimiterspace} 2} \right)}{\left( {{1 - {x}'^{2} } \mathord{\left/ {\vphantom {{1 - {x}'^{2} } {d^{2}_{1} }}} \right. \kern-\nulldelimiterspace} {d^{2}_{1} }} \right)}^{{1/2}} \,\eta \cos \theta } \right\}}^{2} } {r^{2}_{{xy}} }}} \right. \kern-\nulldelimiterspace} {r^{2}_{{xy}} }} \right]} \\ & \quad \times \exp {\left[ { - 2{\left( {{d_{y} } \mathord{\left/ {\vphantom {{d_{y} } 2}} \right. \kern-\nulldelimiterspace} 2} \right)}^{2} {\left( {{1 - {x}'^{2} } \mathord{\left/ {\vphantom {{1 - {x}'^{2} } {d^{2}_{1} }}} \right. \kern-\nulldelimiterspace} {d^{2}_{1} }} \right)}\,\eta ^{2} {\sin ^{2} \theta } \mathord{\left/ {\vphantom {{\sin ^{2} \theta } {r^{2}_{z} }}} \right. \kern-\nulldelimiterspace} {r^{2}_{z} }} \right]}, \\ \end{aligned} $$

(34)

$$ \begin{aligned} I^{{{\text{cyl}}}}_{{{\text{obj}}}} (\tilde{x},\tilde{y}) = & q{\int\limits_{ - d_{x} /2}^{d_{x} /2} {{\text{d}}{x}'} }{\int\limits_0^{d_{y} /2} {{\text{d}}r} }{\int\limits_0^{2\pi } {{\text{d}}\theta \cdot r} } \\ & \quad \times \exp {\left[ { - 2{(\tilde{x} - \tilde{x}_{0} - {x}')^{2} } \mathord{\left/ {\vphantom {{(\tilde{x} - \tilde{x}_{0} - {x}')^{2} } {r^{2}_{{xy}} }}} \right. \kern-\nulldelimiterspace} {r^{2}_{{xy}} }} \right]} \\ & \quad \times \exp {\left[ { - 2{{\left( {\tilde{y} - \tilde{y}_{0} - r\cos \theta } \right)}^{2} } \mathord{\left/ {\vphantom {{{\left( {\tilde{y} - \tilde{y}_{0} - r\cos \theta } \right)}^{2} } {r^{2}_{{xy}} }}} \right. \kern-\nulldelimiterspace} {r^{2}_{{xy}} }} \right]} \\ & \quad \times \exp {\left[ { - 2r^{2} {\sin ^{2} \theta } \mathord{\left/ {\vphantom {{\sin ^{2} \theta } {r^{2}_{z} }}} \right. \kern-\nulldelimiterspace} {r^{2}_{z} }} \right]}, \\ \end{aligned} $$

(35)

and

$$ \begin{aligned} I^{{{\text{ell2}}}}_{{{\text{obj}}}} (\tilde{x},\tilde{y}) = & q{\int\limits_0^{d_{2} } {{\text{d}}{x}'} }{\int\limits_0^1 {{\text{d}}\eta } }{\int\limits_0^{2\pi } {{\text{d}}\theta } }\,\eta {\left( {\frac{{d_{y} }} {2}} \right)}^{2} {\left( {1 - \frac{{{x}'^{2} }} {{d^{2}_{2} }}} \right)} \\ & \quad \times \exp {\left[ {{ - 2(\tilde{x} - \tilde{x}_{0} - {d_{x} } \mathord{\left/ {\vphantom {{d_{x} } 2}} \right. \kern-\nulldelimiterspace} 2 - {x}')^{2} } \mathord{\left/ {\vphantom {{ - 2(\tilde{x} - \tilde{x}_{0} - {d_{x} } \mathord{\left/ {\vphantom {{d_{x} } 2}} \right. \kern-\nulldelimiterspace} 2 - {x}')^{2} } {r^{2}_{{xy}} }}} \right. \kern-\nulldelimiterspace} {r^{2}_{{xy}} }} \right]} \\ & \quad \times \exp {\left[ {{ - 2{\left\{ {\tilde{y} - \tilde{y}_{0} - {\left( {{d_{y} } \mathord{\left/ {\vphantom {{d_{y} } 2}} \right. \kern-\nulldelimiterspace} 2} \right)}{\left( {1 - {{x}'^{2} } \mathord{\left/ {\vphantom {{{x}'^{2} } {d^{2}_{2} }}} \right. \kern-\nulldelimiterspace} {d^{2}_{2} }} \right)}^{{1/2}} \,\eta \cos \theta } \right\}}^{2} } \mathord{\left/ {\vphantom {{ - 2{\left\{ {\tilde{y} - \tilde{y}_{0} - {\left( {{d_{y} } \mathord{\left/ {\vphantom {{d_{y} } 2}} \right. \kern-\nulldelimiterspace} 2} \right)}{\left( {1 - {{x}'^{2} } \mathord{\left/ {\vphantom {{{x}'^{2} } {d^{2}_{2} }}} \right. \kern-\nulldelimiterspace} {d^{2}_{2} }} \right)}^{{1/2}} \,\eta \cos \theta } \right\}}^{2} } {r^{2}_{{xy}} }}} \right. \kern-\nulldelimiterspace} {r^{2}_{{xy}} }} \right]} \\ & \quad \times \exp {\left[ { - 2{\left( {{d_{y} } \mathord{\left/ {\vphantom {{d_{y} } 2}} \right. \kern-\nulldelimiterspace} 2} \right)}^{2} {\left( {1 - {{x}'^{2} } \mathord{\left/ {\vphantom {{{x}'^{2} } {d^{2}_{2} }}} \right. \kern-\nulldelimiterspace} {d^{2}_{2} }} \right)}\,\eta ^{2} {\sin ^{2} \theta } \mathord{\left/ {\vphantom {{\sin ^{2} \theta } {r^{2}_{z} }}} \right. \kern-\nulldelimiterspace} {r^{2}_{z} }} \right]}, \\ \end{aligned} $$

(36)

where

$$ q = {\left\langle C \right\rangle }gQI_{{\text{a}}} . $$

(37)

Appendix B

Carrying out the numerical optimization procedure for the hemiellipsoidal caps leads to the following parameters for the slices:

$$ \begin{aligned} x_{{\text{A}}} = & 0.305579 \times d_{1} \\ x_{{\text{B}}} = & 0.585352 \times d_{1} \\ x_{{\text{D}}} = & 0.585352 \times d_{2} \\ x_{{\text{E}}} = & 0.305579 \times d_{2} \\ y_{{\text{A}}} = & y_{{\text{E}}} = 0.593244 \times d_{y} \\ y_{{\text{B}}} = & y_{{\text{D}}} = 0.843075 \times d_{y} , \\ \end{aligned} $$

(38)

whereas the width

$$ y_{{\text{C}}} = {{\sqrt \pi }d_{y} } \mathord{\left/ {\vphantom {{{\sqrt \pi }d_{y} } 2}} \right. \kern-\nulldelimiterspace} 2 $$

(39)

of the middle slice of length x _C =d _x (see Fig. 3b) follows directly from the requirement that the cylinder and slice volume be identical, in other words π d ² _y d _x /4=y ² _C d _x .

Appendix C

The five intensity distributions describing the the slice model (Eq. 11), are given by

$$ \begin{aligned} \mathcal{I}_{{\text{A}}} (\tilde{x},\tilde{y}) = p & \times {\text{erf}}{\left( {\frac{{z_{{\text{A}}} }} {{{\sqrt 2 }r_{z} }}} \right)}\left\{ {{\text{erf}}{\left( {\frac{{{\sqrt 2 }}} {{r_{{xy}} }}{\left[ {\tilde{x} - \tilde{x}_{0} + \frac{{x_{{\text{C}}} }} {2} + x_{{\text{B}}} + x_{{\text{A}}} } \right]}} \right)}} \right. \\ & \quad \left. { - {\text{erf}}{\left( {\frac{{{\sqrt 2 }}} {{r_{{xy}} }}{\left[ {\tilde{x} - \tilde{x}_{0} + \frac{{x_{{\text{C}}} }} {2} + x_{{\text{B}}} } \right]}} \right)}} \right\} \\ & \times {\left\{ {{\text{erf}}{\left( {\frac{{{\sqrt 2 }}} {{r_{{xy}} }}{\left[ {\tilde{y} - \tilde{y}_{0} + \frac{{y_{{\text{A}}} }} {2}} \right]}} \right)} - {\text{erf}}{\left( {\frac{{{\sqrt 2 }}} {{r_{{xy}} }}{\left[ {\tilde{y} - \tilde{y}_{0} - \frac{{y_{{\text{A}}} }} {2}} \right]}} \right)}} \right\}}, \\ \end{aligned} $$

(40)

$$ \begin{aligned} \mathcal{I}_{{\text{B}}} (\tilde{x},\tilde{y}) = p & \times {\text{erf}}{\left( {\frac{{z_{{\text{B}}} }} {{{\sqrt 2 }r_{z} }}} \right)}\left\{ {{\text{erf}}{\left( {\frac{{{\sqrt 2 }}} {{r_{{xy}} }}{\left[ {\tilde{x} - \tilde{x}_{0} + \frac{{x_{{\text{C}}} }} {2} + x_{{\text{B}}} } \right]}} \right)}} \right. \\ & \quad \left. { - {\text{erf}}{\left( {\frac{{{\sqrt 2 }}} {{r_{{xy}} }}{\left[ {\tilde{x} - \tilde{x}_{0} + \frac{{x_{{\text{C}}} }} {2}} \right]}} \right)}} \right\} \\ & \times {\left\{ {{\text{erf}}{\left( {\frac{{{\sqrt 2 }}} {{r_{{xy}} }}{\left[ {\tilde{y} - \tilde{y}_{0} + \frac{{y_{{\text{B}}} }} {2}} \right]}} \right)} - {\text{erf}}{\left( {\frac{{{\sqrt 2 }}} {{r_{{xy}} }}{\left[ {\tilde{y} - \tilde{y}_{0} - \frac{{y_{{\text{B}}} }} {2}} \right]}} \right)}} \right\}}, \\ \end{aligned} $$

(41)

$$ \begin{aligned} \mathcal{I}_{{\text{C}}} (\tilde{x},\tilde{y}) = p & \times {\text{erf}}{\left( {\frac{{z_{{\text{C}}} }} {{{\sqrt 2 }r_{z} }}} \right)}\left\{ {{\text{erf}}{\left( {\frac{{{\sqrt 2 }}} {{r_{{xy}} }}{\left[ {\tilde{x} - \tilde{x}_{0} + \frac{{x_{{\text{C}}} }} {2}} \right]}} \right)}} \right. \\ & \quad \left. { - {\text{erf}}{\left( {\frac{{{\sqrt 2 }}} {{r_{{xy}} }}{\left[ {\tilde{x} - \tilde{x}_{0} - \frac{{x_{{\text{C}}} }} {2}} \right]}} \right)}} \right\} \\ & \times {\left\{ {{\text{erf}}{\left( {\frac{{{\sqrt 2 }}} {{r_{{xy}} }}{\left[ {\tilde{y} - \tilde{y}_{0} + \frac{{y_{{\text{C}}} }} {2}} \right]}} \right)} - {\text{erf}}{\left( {\frac{{{\sqrt 2 }}} {{r_{{xy}} }}{\left[ {\tilde{y} - \tilde{y}_{0} - \frac{{y_{{\text{C}}} }} {2}} \right]}} \right)}} \right\}}, \\ \end{aligned} $$

(42)

$$ \begin{aligned} \mathcal{I}_{{\text{D}}} (\tilde{x},\tilde{y}) = p & \times {\text{erf}}{\left( {\frac{{z_{{\text{D}}} }} {{{\sqrt 2 }r_{z} }}} \right)}\left\{ {{\text{erf}}{\left( {\frac{{{\sqrt 2 }}} {{r_{{xy}} }}{\left[ {\tilde{x} - \tilde{x}_{0} - \frac{{x_{{\text{C}}} }} {2}} \right]}} \right)}} \right. \\ & \quad \left. { - {\text{erf}}{\left( {\frac{{{\sqrt 2 }}} {{r_{{xy}} }}{\left[ {\tilde{x} - \tilde{x}_{0} - \frac{{x_{{\text{C}}} }} {2} - x_{{\text{D}}} } \right]}} \right)}} \right\} \\ & \times {\left\{ {{\text{erf}}{\left( {\frac{{{\sqrt 2 }}} {{r_{{xy}} }}{\left[ {\tilde{y} - \tilde{y}_{0} + \frac{{y_{{\text{D}}} }} {2}} \right]}} \right)} - {\text{erf}}{\left( {\frac{{{\sqrt 2 }}} {{r_{{xy}} }}{\left[ {\tilde{y} - \tilde{y}_{0} - \frac{{y_{{\text{D}}} }} {2}} \right]}} \right)}} \right\}} \\ \end{aligned} $$

(43)

and

$$ \begin{aligned} \mathcal{I}_{{\text{E}}} (\tilde{x},\tilde{y}) = p & \times {\text{erf}}{\left( {\frac{{z_{E} }} {{{\sqrt 2 }r_{z} }}} \right)}\left\{ {{\text{erf}}{\left( {\frac{{{\sqrt 2 }}} {{r_{{xy}} }}{\left[ {\tilde{x} - \tilde{x}_{0} - \frac{{x_{{\text{C}}} }} {2} - x_{{\text{D}}} } \right]}} \right)}} \right. \\ & \quad \left. { - {\text{erf}}{\left( {\frac{{{\sqrt 2 }}} {{r_{{xy}} }}{\left[ {\tilde{x} - \tilde{x}_{0} - \frac{{x_{{\text{C}}} }} {2} - x_{{\text{D}}} - x_{{\text{E}}} } \right]}} \right)}} \right\} \\ & \times {\left\{ {{\text{erf}}{\left( {\frac{{{\sqrt 2 }}} {{r_{{xy}} }}{\left[ {\tilde{y} - \tilde{y}_{0} + \frac{{y_{{\text{E}}} }} {2}} \right]}} \right)} - {\text{erf}}{\left( {\frac{{{\sqrt 2 }}} {{r_{{xy}} }}{\left[ {\tilde{y} - \tilde{y}_{0} - \frac{{y_{{\text{E}}} }} {2}} \right]}} \right)}} \right\}}, \\ \end{aligned} $$

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where

$$ p = {\left\langle C \right\rangle }gQI_{{\text{a}}} \frac{1} {4}{\left( {\frac{\pi } {2}} \right)}^{{3/2}} r^{2}_{{xy}} r_{z} . $$

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Appendix D

The slice model can only be preferred if the error introduced by approximating the sausage model by slices is acceptable. The question to be answered here is therefore: how well does the slice model approximate the sausage model?

Let us adopt the sausage model assuming morphologically-realistic E.coli parameters, and then calculate the resulting fluorescence intensity I_obj(x,y) according to Eq. 10 (Fig. 12a). A cross-section at y=32 through its maximum is shown by the continuous curves shown in (b) and (c) of this figure. Then we fit the intensity distribution $\mathcal{I}_{{{\text{obj}}}} (x,y)$ (Eq. 11 with Eqs.38 and 39) of the corresponding slice model to the calculated intensity distribution of the sausage model. A section through this function at y=32 gives the dashed curve shown in Fig. 12b. Though the maximum amplitude of the sausage object is underestimated by the slice model fit, the front and rear edge of the intensity distribution are almost perfectly fitted (Fig. 12b, dashed versus continuous curve) due to the good approximation of the bacterium’s caps by the stack of slices (Fig. 3c). As the front and rear edge of the intensity distribution are well described by the intensity distribution of the slice model, the resulting length parameter of d=36.018 pixels is in good agreement with the length of d=36 pixels of the underlying sausage model. For a given pixel size of 88 nm, this corresponds to a deviation of 1.6 nm, much smaller than the experimental precision that can be achieved under our conditions (see below).

Fig. 12a–c

Intensity distribution of the sausage model and its approximation by the intensity distributions of a slice and a box model, respectively.a Theoretical distribution I_obj(x,y) (Eq. 10, $x = \tilde{x}$ and $y = \tilde{y}$) of the sausage model, calculated with the parameters: d _x =25 pixels, d₁=d₂=5.5 pixels, d _y =11 pixels, φ=0 and p₀=(49,32). E. coli length: d=36 pixels. b x-section through the intensity distribution of the sausage model shown in (a) (solid line) and through the fitted intensity distribution $\mathcal{I}_{{{\text{obj}}}} (x,y)$ (Eq. 11 with Eqs. 6, 7, 38 and 39) of the slice model (dashed line) both at y=32. Result of the fit: d _x =24.7 pixels, d₁=5.66 pixels, d₂=5.658 pixels, d _y =10.492, x₀=49 pixels, y₀=32 pixels and φ=0 (χ²=5456.97). These data result in an object length of d=36.018 pixels.c Fitting the intensity distribution of a simple box model (Eq. 11 with d₁=d₂=0) gave (dashed line): d _x =32.886 pixels, d _y =10.3 pixels and φ=0 (χ²=15250.22)

The negligible error of the slice model approximation suggests that a simple box model (a slice model with just one slice) might be sufficient to approximate the sausage model. However, assuming a slice model with d₁=d₂=0 (just one box without the hemiellipsoidal caps attached to it, Fig. 3), a clear deviation between the distributions is seen at the edges (Fig. 12c). A box model is thus too coarse for assessing the bacterium’s edges. This is also reflected in the larger deviation between the resulting width of d _y =10.492 of the box model and the exact width of d _y =11 pixels of the sausage model. For the same pixel size, this would correspond to a bias of ~51 nm. The reason for this discrepancy is obviously the mismatch between the rectangular cross-sections of the slice model and the circular cross-sections of the sausage model. If the slice model is used instead for determining the relative displacements of the left and right borders of the sausage model, the bias becomes much smaller: increasing the width d _y of the sausage model from 11 pixels to 11.1 pixels (all other parameters kept unchanged, see legend of Fig. 12), and analyzing the resulting intensity distribution I_obj(x,y) with the distribution $\mathcal{I}_{{{\text{obj}}}} (x,y)$ of the slice model, results in a width of d _y =10.59 pixels. The detected increase of the width of the object of Δd _y =0.098 pixels is in excellent agreement with the exact change of 0.1 pixels. For a pixel size of 88 nm, the deviation would therefore be a few Angstroms. Another convenient feature of the slice model is that small length changes of a non-symmetric sausage model (d₁≠ d₂) can be detected using a symmetric slice model (d₁=d₂, fixed). This feature allows us to reduce the number of fit parameters. For example, increasing the length of the left hemiellipsoidal cap of the sausage model from d₁=5.5 pixels to d₁=6 pixels (Δd₁=Δd=0.5) while the other parameters stay constant, and analyzing the corresponding intensity distribution with the distribution of a symmetric slice model leads to a detected length change of Δd=0.503. For 88 nm pixel size, the bias is therefore again a few Angstroms.

Taken together, these results show that the slice model gives reliable results when used to determine the absolute length of sausage-like objects such as E. coli bacteria, or when used to detect relative length and width changes.