Quantification of biological interactions with particle image cross-correlation spectroscopy (PICCS)

Abstract

A multitude of biological processes that involve multiple interaction partners are observed by two-color microscopy. Here we describe an analysis method for the robust quantification of correlation between signals in different color channels: particle image cross-correlation spectroscopy (PICCS). The method, which exploits the superior positional accuracy obtained in single-object and single-molecule microscopy, can extract the correlation fraction and length scale. We applied PICCS to correlation measurements in living tissues. The morphogen Decapentaplegic (Dpp) was imaged in wing imaginal disks of fruit fly larvae and we quantified what fraction of early endosomes contained Dpp.

Figure 1

PICCS algorithm. For all YFP signals (solid circles) the number of CFP signals (open circles) is counted which fall into a circle of radius l from a YFP signal. The total number is subsequently divided by the number of YFP signals. By increasing l from 0 to l_max the correlation function C_cum(l) is constructed. (Dashed line) Area in which the YFP signals are used for analysis. This area is separated from the edges of the image by l_max (in this figure l_max = 2 μm is taken). The signal positions were simulated with the following parameters: density of YFP signals c_YFP = 1 μm⁻², correlation fraction α = 0.5 (results in a density of CFP signals of c_CFP = 0.5 μm⁻²), and correlation length σ = 150 nm.

Figure 2

The cumulative correlation function. (Open circles) C_cum(l) calculated from simulated data. (Solid circles) C_cum(l) after subtraction of the linear contribution (dotted line). To determine the slope π · c_CFP of the linear contribution, a straight line is fitted to C_cum(l) between l_min and l_max. The offset of this straight line is equal to the correlation fraction α. The value σ is equal to the distance l where the function C_cum(l) − π · c_CFPl² has the value α (1 − √e).

Figure 3

Dependence of the relative errors on the number of images M. (a) The relative errors of α (circles), σ (triangles), and c_CFP (squares) all scaled approximately like 1/√M (solid line). M = 10, α = 0.5, c_YFP = 0.5 μm⁻², and σ = 0.15 μm. (b) Dependence of the relative errors on the interaction fraction α. The legend is the same as in panel a, where c_YFP = 1 μm⁻² (solid symbols), c_YFP = 10 μm⁻² (open symbols), M = 10, and σ = 0.15 μm in both cases. The errors of all determined parameters approximately scaled like 1/√α (solid line), independent of the density c_YFP. (c) Dependence of the relative errors on the density c_YFP. The legend is the same as in panel a. The relative error of α (circles) and σ (triangles) were fitted with the model A · (c_YFP/μm⁻²)^−0.5 + B · (c_YFP/μm⁻²)^0.25 (solid and dashed line, respectively). For α A = 0.04, B = 0.12, which resulted in a minimum at 0.6 μm⁻² and for σ A = 0.07, B = 0.14, which gave a minimum at 0.5 μm⁻². The relative error of c_CFP (squares) scaled approximately like $c_{\text{YFP}}^{-2/3}$. (Shaded line) Linear fit in the logarithmic plot given by y = −0.66(c_CFP/μm⁻²) − 2.9). M = 10, α = 0.5, and σ = 0.15 μm. (d) Dependence of the relative errors on the density c_CFP. The legend is the same as in panel a. The relative error of α and σ scaled approximately like √c_CFP (solid line), the relative error of c_CFP scaled like 1/√c_CFP (dashed line). M = 10, c_YFP = 1 μm−2, α = 0.5, and σ = 0.15 μm. (e) Dependence of the relative errors on σ. The legend is the same as in panel a. The relative error of α and c_CFP did not change significantly with σ. The relative error of σ scaled approximately like 1/√σ (solid line is a linear fit in the logarithmic plot given by y = −0.52(c_YFP/μm⁻²)−2.8). M = 50, c_YFP = 1 μm⁻², and α = 0.5.

Figure 4

(a) Dependence of the relative error of α on the center of the fit interval l_center = (l_max − l_min)/2. The legend is the same as in Fig. 3a, where l_center = 0.925 μm (solid symbols), l_center = 1.175 μm (shaded symbols), and l_center = 1.375 μm (open symbols). The relative error of α was fitted with the model A · (c_YFP/μm⁻²)^−0.5 + B · (c_YFP/μm⁻²)^0.25. A = 0.04, B = 0.05 (solid line), A = 0.04, B = 0.09 (shaded line), and A = 0.04, B = 0.13 (dashed line). That resulted in minima at 1.9 μm⁻², 0.9 μm⁻², and 0.5 μm⁻², respectively. M = 10, α = 0.5, and σ = 0.05μm. (b) Dependence of the relative error of σ on the center of the fit interval l_center = (l_max − l_min)/2. The legend is the same as in Fig. 3a, where l_center = 0.925 μm (solid symbols), l_center = 1.175 μm (shaded symbols), and l_center = 1.375 μm (open symbols). The relative error of σ is fitted with the model A · (c_YFP/μm⁻²)^−0.5 + B · (c_YFP/μm⁻²)^0.25. A = 0.07, B = 0.15 (solid line), A = 0.08, B = 0.25 (shaded line), and A = 0.09, B = 0.34 (dashed line). That resulted in minima at 0.9 μm⁻², 0.6 μm⁻², and 0.4 μm⁻², respectively. M = 10, α = 0.5, and σ = 0.05 μm. (c) Dependence of the relative error of c_CFP on the center of the fit interval l_center = (l_max − l_min)/2. The legend is the same as in Fig. 3a, where l_center = 0.925 μm (solid symbols), l_center = 1.175 μm (shaded symbols), and l_center = 1.375 μm (open symbols). The relative error of σ was fitted with the straight line (in the logarithmic plot). The slope is −0.77 (solid line), −0.67 (shaded line), and −0.62 (dashed line). M = 10, α = 0.5, and σ = 0.05 μm. (d) Dependence of the relative errors on the length of the fit interval (l_max − l_min). The legend is the same as in Fig. 3a. M = 50, α = 0.5, σ = 0.15 μm, and c_YFP = 1 μm⁻². (e) Dependence of the relative errors on the step size Δl (Fig. 2). The legend is the same as in Fig. 3a. M = 50, α = 0.5, σ = 0.15 μm, and c = 1 μm⁻².

Figure 5

Control experiment with fluorescent beads. (a) Image from red excitation channel showing only the tetraspeck beads. (Scale bar, 2 μm.) (b) Image from the green excitation channel showing both tetraspeck and yellow-green beads. (c) Cumulative correlation function for a correlation fraction of α = 0.56. (d) Correlation fractions determined in experiments with five different ratios of single color and dual color fluorescent beads. (Dashed line) y = x. Errors were determined from simulations; see Error Scaling Simulation.

Figure 6

Correlation fraction, signal density, and correlation length determined from experimental data. A wing imaginal disk was imaged for 300 s using an alternating excitation method (described in Materials and Methods). Each image stack consists of five image planes (10 × 10 μm) separated by 0.7 μm in axial direction. Background of low spatial frequency was eliminated by applying a high-pass filter. (a) Raw image stack from the Dpp-YFP channel (scale bar = 2 μm). (b) Raw image stack from the Rab5-CFP channel. (c) Correlation function C_cum(l) obtained by PICCS. Fitting of the linear part yielded a Dpp-YFP density of c = 0.12 ± 0.02 endosomes · μm⁻² (solid line) and a correlation fraction of α_e,Dpp = 0.46 ± 0.04 (offset of the fitted line). (d) P_cum(l) which resulted from subtraction of the linear contribution from C_cum(l) and division by α_e,Dpp. The correlation lengths σ₁, σ₂ and the fraction σ were determined by fitting Eq. 5 which gave σ₁ = 71 ± 17 nm, σ₂ = 161 ± 34 nm, and σ = 0.44 ± 0.17, respectively. All errors were determined from simulations; see Error Scaling Simulation.

Figure 7

Number of detected endosomes per image stack for (a) early endosomes and (b) Dpp-containing endosomes. The number of endosomes (signals) in both channels stayed approximately constant. The average number of endosomes and the standard deviation are indicated for both cases.