Bayesian inference of multi-point macromolecular architecture mixtures at nanometre resolution

Gaussian spot fitting methods have significantly extended the spatial range where fluorescent microscopy can be used, with recent techniques approaching nanometre (nm) resolutions. However, small inter-fluorophore distances are systematically over-estimated for typical molecular scales. This bias can be corrected computationally, but current algorithms are limited to correcting distances between pairs of fluorophores. Here we present a flexible Bayesian computational approach that infers the distances and angles between multiple fluorophores and has several advantages over these previous methods. Specifically it improves confidence intervals for small lengths, estimates measurement errors of each fluorophore individually and infers the correlations between polygon lengths. The latter is essential for determining the full multi-fluorophore 3D architecture. We further developed the algorithm to infer the mixture composition of a heterogeneous population of multiple polygon states. We use our algorithm to analyse the 3D architecture of the human kinetochore, a macro-molecular complex that is essential for high fidelity chromosome segregation during cell division. Using triple fluorophore image data we unravel the mixture of kinetochore states during human mitosis, inferring the conformation of microtubule attached and unattached kinetochores and their proportions across mitosis. We demonstrate that the attachment conformation correlates with intersister tension and sister alignment to the metaphase plate.

Uninformative priors 794 We use uninformative (improper) priors, namely: dπ 0 {θ j } j , {ϑ n } n := π X {X j } j ·   j dX j · χ σj;xy≥0 dσ j;xy · χ σj;z≥0 dσ j;z   · n dT n · dπ R (R n ) , (A) where χ is the characteristic function, π X {X j } j is the prior distribution on the 795 template positions and dπ R [R n ] is the probability measure on the rotation matrices 796 R n ∈ R 3×3 as follows. 797 Uninformative priors are typically chosen to be flat (uniform); however this is not 798 re-parametrisation independent, and there are many choices of polygon properties where 799 a uniform prior can be imposed. Because our interest is in the polygon side lengths, a 800 flat prior on each length l ij := |X j − X i | would appear appropriate, giving the 801 (improper) prior ∝ i̸ =j dl ij . However, the marginal prior for any length would in 802 fact not be flat for J > 2, because of the constraints on the lengths to form a J-polygon. 803 Priors with approximately flat marginal densities in the lengths can be constructed for 804 the triangle case (J = 3) using the density (see Flat prior on marginals of 805 lengths in S1 Text for a derivation): 806 π X {X j } j ∝ 1 l 12 · l 13 · l 23 · min ({l 12 , l 13 , l 23 }) .
Three priors on the template positions are compared in Fig A in S1 Text, namely 807 j dX j , ( j dXj ) l12·l13·l23 and π X {X j } j · j dX j . l12·l13·l23 · j dX j , π X {X j } j · j dX j . The support is bounded by the triangle inequality, e.g. the front face is the bounding simplex l 23 = l 12 + l 13 . A cut-off for each of the lengths was used, forcing them to be within [0,1]. The symmetry of the prior with respect to re-labelling fluorophores j was used when sampling.
The measure dπ R on the rotation matrix R n is defined, such that for an arbitrary 809 vector on the R 3 -unit sphere, v ∈ S 3 , the image R n · v · dπ R [R n ] is uniform on the 810 surface of S 3 (i.e. the Lebesgue measure on that surface, sin(ψ)dψdφ in spherical 811 polars).

812
Flat prior on marginals of lengths 813 Here we show, that the distribution given in Eq (B) in S1 Text corresponds to a flat 814 prior on the marginals of the lengths in the limit of unconstrained triangle sizes. 815 Triangles are specified by the three 3D positional vectors X j , j ∈ {1, 2, 3}, giving the 816 unbounded parameter space R 9 . Consider the stripe subspaces, corresponding to length l ij := |X j − X i | constrained to a small ϵ-neighbourhood of a fixed value l 0 > 0, and all lengths being less than a > 0. This stripe region is infinite, because of the translational degrees of freedom, but becomes finite once the translations are fixed. We want to show that for the prior from Eq (B) in S1 Text the probability to be in such a domain (and constraining X 1 to a bounded domain) is independent of the value l 0 in the limit of narrow stripes and unbounded triangles: where we used A := A ⊂ R 3 0 < A dX < ∞ to confine node X 1 to a domain of 818 finite volume. 819 We first show, that gives a flat prior on the lengths space (within the boundaries of the triangle inequality), 821 i.e.: withÃ the set of all lengths for which {X j } j ∈ A and the set of all rotation symmetric positions with X 1 ∈ A 1 ⊂ A. It is well-known that -for 824 two positions X 1 , X 2 ∈ R 3 -we can switch from Cartesian to spherical coordinates via: 825 where dΩ = sin (ψ) · dφdψ is the differential of the solid angle (for azimuth and inclination angles φ, ψ). On the other hand, we know from the transition from Cartesian to cylinder coordinates (with the X 1 -X 2 -edge as the axis of the cylinder), that dX 3 = h 3 · dh 3 dadθ, where h 3 ≥ 0 is the distance of X 3 from the cylinder axis, a ∈ R is the coordinate along the cylinder axis and θ ∈ [0, 2π] is the rotation angle of X 3 around the cylinder axis (relative to some fixed plane containing that axis). Expressing l 13 , l 23 through the cylinder coordinates a, h 3 as well as l 12 we have: yielding: Combining this with our results for dX 1 dX 2 and dX 3 we get: dX 1 dX 2 dX 3 = l 2 12 · dX 1 dl 12 dΩ · (h 3 · dh 3 dadθ) = = (l 12 · l 13 · l 23 ) · dl 12 dl 13 dl 23 · (dX 1 dΩdθ) , where the term in the last bracket just gives a constant, thus proving Eq (F) in S1 Text. 827 Without loss of generality, we show Eq (D) in S1 Text for ij = 23, i.e. the marginal over l 23 . Note, that the additional factor min (l 12 , l 13 , l 23 ) −1 in Eq (B) in S1 Text versus d f latlengths merely comes from the lengths space only being occupied within the boundaries of the triangle inequality (other points are not contained inÃ in Eq (F) in S1 Text). To see how the extra factor fixes this, we assume an upper cut-off for all lengths a ≫ l 0 > 0 and divide the marginal integral into different parts depending on which length is shortest: χ max(l12,l13,l23)<a · χ l23∈[l0−ϵ;l0+ϵ] l 12 · l 13 · l 23 · min (l 12 , l 13 , l 23 ) where the terms in the third row correspond to l 0 ≤ l 12 ≤ l 13 , l 12 ≤ l 0 ≤ l 13 and 828 l 12 ≤ l 13 ≤ l 0 in this order (and then similarly with 2 and 3 swapped). For a → ∞ (i.e. 829 no cut-off of lengths) the l 0 -dependent terms o (a) become negligible, thus proofing our 830 claim from Eq (D) in S1 Text.

831
Note firstly, this is only an asymptotic argument and for finite cut-offs there are  Here we give the details of the Markov Chain Monte Carlo approach taken to sample 841 from the posterior, firstly for the single-state model in Eq (8) and secondly for the 842 multi-state model. A Matlab version (Matlab 2019b) of our code is available. Some of 843 the results presented in the paper were obtained using a different implementation of the 844 MCMC algorithm in Julia (particularly: updating more parameters sequentially rather 845 than jointly), but gives identical posterior distributions.

846
Samplers of single-state model 847 We sequentially update the various parameters, {X j } j , {τ j } j , {T n } n , {R n } n , using the 848 following samplers: Random walk samplers Random walk samplers are used for the template positions 850 and perspective parameters, implemented sequentially for each of the parameters X j ,

851
T n and R n . We use block updates, blocking together the three coordinates in each of 852 the vectors X j , and block together the six parameters in T n and R n . For the template 853 positions X j and translations T n the proposals are uniform balls around the current 854 position. For the rotations R n the proposal is implemented as follows: 855 choose a reference vector r ∈ S 3 uniformly from the unit-sphere 856 draw a sphere locationr displaced from r by choosing it uniformly within a 857 spherical cap around r (the radius of the cap is given by the step-size (adjusted 858 during burnin)).

859
determine a rotation perturbationR n as the "geodesic" rotation that maps r onto 860 r, i.e. the rotation with invariant axis r ×r around the angle arccos (r · r) 861 the proposed rotation is the combinedR n · R n , where R n is the current rotation. 862 During burnin the step-sizes of all random walk samplers are adjusted with a target 863 rejection rate of ρ µ := 77.5% (motivated by [1]). The step-size adjustment procedure 864 (given a current step-size of s > 0, current adjustment factor of a > 1, current deviation 865 δρ ∈ R from target acceptance rate ρ µ and a tolerance of ρ σ := 7.5%): We initialise with a = 2 and the above five-step procedure was repeated every 874 ⌈ (#burnin)⌉ Markov chain iterations through burnin, and initial step sizes are chosen 875 heuristically in the range of 10 −2 and 10 (in nm for lengths, measurement errors).

876
Gibbs sampler for the translations T n The translations T n are also updated 877 using a Gibbs sampler. Here we sample from the target: where: Single-state experimental examples (see Table 3 for results):  Table 4 for results): In all examples the first 40% of the iterations is burn-in, and the remaining 60% the posterior samples.
Gibbs sampler for the measurement errors τ j A Gibbs sampler is used for the  of the measurement errors of the three 882 fluorescent spots. Note that the two components xy, z as well as the three fluorophores 883 j are independent from each other, so sequential and joint updates coincide. We have a 884 Γ-distribution for each of the conditional distributions: where d ∈ {xy, z} and number of dimensions (#d) = 2, 1, respectively.

886
Initialisation and convergence monitoring Unless otherwise stated, the variables of all chains are initialised randomly as follows (independent for each j, n): These are over-dispersed compared to the anticipated width of the posterior distribution 887 (confirmed after the run). The total number of iterations per Markov chain for each 888 example is reported in Table A in S1 Text; sub-sampling (equally spaced) was used to 889 give a final sample size of no more than 10000 samples (3000 in Example 4.1).

890
A multi-chain convergence diagnostic was used ( (see [2,Ch 11.6]). We used a threshold of 1.1; if converged, the five chains were then 894 pooled to reconstruct the posterior.

895
The computation time on an ordinary desktop computer is about 1 day for datasets 896 with N = 200 samples (our Matlab implementation may be further improved for speed). 897 Samplers of multi-state model 898 All parameters already present in the single-state model from subsection Polygon 899 parametrisation and inference can be inferred using the same updates, if confining the 900 measured data points to the subset in the currently updated state ζ n = ζ. The new 901 variables {ζ n } n and p (ζ) ζ are updated sequentially with random walk and Gibbs 902 samplers, respectively.

903
Random walk sampler for state-affiliations {ζ n } n The hidden state-affiliations 904 {ζ n } n are sampled with a random walk proposal, equiprobable on all states except the 905 current one (assigned zero probability). To achieve a higher acceptance rate for a new 906 state-affiliation proposal, the translation T n for measurement n are altered such that 907 the centre of mass of the triangle (with equal weights for all nodes j) of the proposed 908 true positions X n j j remain unchanged.

909
Gibbs sampler for state proportions P = p (ζ) ζ The state proportions P are 910 sampled via a Gibbs sampler from a Dirichlet distribution: where #ζ ′ := |{ n ∈ {1, . . . , N }| ζ n = ζ ′ }| are the number of measurements in the 912 respective states. The Dirichlet probability density for the state proportions p (ζ) ζ is 913 with the state proportions all constrained within [0; 1] and summing to one.

914
Initialisation and convergence monitoring The parameters were randomly initialised (independent for each j, n; for Eq (AF) in S1 Text priors): where Cat p (ζ) ζ is the multinoulli distribution, state ζ being drawn with 915 probability p (ζ) . The values l (1) ij;0 and τ (1)  In the simulated multi-state examples, Table 2, we utilised more confining priors on one 925 or more of the states to improve convergence. Specifically, the triangle side lengths are 926 independently constrained to a cuboid-shaped domain in the lengths-space and the 927 precisions are Gamma distributed. For instance, to impose this prior on state ζ = 1, the 928 following factors are included in the prior Eq (7), where the parameters l (1) ij;0 ≥ 0 and σ (1) j;d;0 > 0 are specified by prior knowledge (analysis 930 of previous data or structural data), and the size 2L is taken as 24nm. The second and 931 third parameter of the Γ distribution are the shape and scale parameters. The values for 932 L and the shape parameter are a particular choice for our examples, to be informative 933 enough to help mixing during burnin (see supplementary section Markov Chain Monte 934 Carlo samplers for parameter inference in S1 Text) with negligible effect on the 935 posterior distribution.

937
The total number of inferred parameters can be calculated as follows: where we calculated χ Z>1 degrees of freedom for each state affiliation ζ n (indicator 938 function χ).
and the rotations are sampled from converged chains of a Markov Chain Monte Carlo 955 algorithm using the same random walk update as described in subsection Samplers of 956 single-state model in S1 Text. The random variables modelling the measurement error, 957 γ n j j,n , are sampled independently from their respective Gaussians: Four-fluorophore simulated example 959 We present a simulated example for J = 4 fluorophores and N = 400 measurements.

960
Original and inferred values for lengths and measurement errors are given in 961 Table B in S1 Text, demonstrating good agreement.

962
Having more than three markers requires an extension to the prior on the template 963 positions given in supplementary section Flat prior on marginals of lengths in S1 Text. 964 We suggest to generalise that procedure (without proof; see Fig B in S1 Text for a 965 numerical evaluation) by the recursive Algorithm A in S1 Text. As before, this prior is 966 meant to be asymptotically (for unbounded domains) flat on the length-marginals, but 967 is not uniquely defined by this property. Table B. Single-state simulated example for four fluorophores (N = 400).

(lengths and errors in nm)
|X2 Rows are the simulated input values and the posterior means ± standard deviations of the four-fluorophore version of our algorithm, respectively. . A cut-off for each of the lengths was used, forcing them to be within [0, 1] -note that this boundary effect becomes more pronounced for higher dimensions. The symmetry of the prior with respect to re-labelling fluorophores j was used when sampling.
Algorithm A: Recursive algorithm for prior on template positions, Implementation of pair-wise correction methods 970 We compared our method to existing length inference techniques between two 971 fluorophores, namely by Churchman et al from [3] and BEDCA, [4]. Here we describe 972 our implementation of these methods:  (6) in [3]. 976 For error estimates we calculated the Hessian of the likelihood and used its negative for 977 the inverse covariance matrix of the Gaussian approximation.

978
Mean and standard deviation of the full posterior: we used the same likelihood 979 function and assumed flat priors on the lengths |X j − X i | and measurement errors σ ij . 980 The marginal distributions were computed using numerical integration on a rectangular 981 grid.

982
BEDCA This is the algorithm used in [4] and [5]; however we used different priors 983 analogous to the ones used in the triangle method (see Eq (A) in S1 Text). Specifically, 984 measurement errors have flat priors on the positive real axis (dσ j;d ) and lengths have 985 almost flat priors on the positive real axis (∝ χ lij ≥0 · N 60; 10 10 · dl ij for length 986 l ij = |X j − X i |). Note, the latter gives (approximately) flat priors on the lengths (i.e. 987 the length marginals for triangles) but if the three lengths are used to form a triangle 988 under an independence assumption (rejecting those combinations that violate the 989 triangle inequality), the prior would lack the min ({l 12 , l 13 , l 23 }) term from 990 Eq (B) in S1 Text (see subsection Flat prior on marginals of lengths in S1 Text for 991 details).

992
Small lengths are increasingly difficult to infer 993 Here we give an argument, why for a given measurement error, shorter true lengths 994 exhibit a much larger inference error than longer lengths. This phenomenon has been 995 reported in [6] for the length correction based on [3] before.

996
There are two stochastically independent processes contributing to the distribution of the vectors between two fluorophores,X n j −X n i . First, the isotropic distribution of the displacement vectors between the true positions in space (due to the rotations R n ) and second, the measurement error (a Gaussian measurement error assumption is not required). Computing the second moment of the distance we obtain: where the expectation and covariance are taken with respect to the fluorophore specific 997 random variables (parametrised with n), i.e. measurement errors γ n j , rotations R n and 998 translations T n ; the template positions X j and measurement errors σ j;d are fixed. In term is the covariance of the Lebesgue measure on a sphere of radius |X j − X i |.

1003
Expression (AK) in S1 Text is the 3D analogue to Eq (4) in [7]. To quantify how 1004 strongly |X j − X i | depends on the measurement error, we take the derivative of 1005 (AK) in S1 Text: when the true length is much smaller than the measurement error. Thus, small changes 1007 in the inferred measurement error lead to very large changes in the inferred true length, 1008 with increasing effect for small lengths. See length of 15nm and input (true) measurement error of √ 15 2 + 25 2 nm ≈ 29nm. Plotted is the likelihood function, Eq (6) from [3]. Probability density is colour coded as key. The red line is Eq (AK) in S1 Text with the second moment (left hand side) estimated from the data.
for length l ij inferred by methods 1, 2. If both methods had identical posteriors, we 1012 have p = 50%.

1013
Experimental methods: prometaphase-metaphase dataset 1014 For the prometaphase-metaphase data of Example 4.1 we used immortalised (hTERT) 1015 diploid human retinal pigment epithelial (RPE1) cells in which the Ndc80 C-terminus 1016 was labelled at the endogenous locus with eGFP (MC191). These RPE1 Ndc80-EGFP 1017 (MC191) were cultured and imaged for CenpC-Ndc80C-Ndc80N as [5]. In brief, the 1018 MC191 cells were cultured in a humidified incubator at 37 degrees, 5% CO2, in New Castle, NY). The images were acquired using 488nm, 561nm, 647nm, and 405nm 1034 wavelength lasers. The sampling was set to z-spacing of 0.2µm and 61 z-slices. Spinning 1035 disk images were exported in OME.TIFF format (The Open Microscopy Environment, 1036 UK) from Volocity 6.0 and deconvolved using Huygens 4.1. The images were then 1037 analysed in KiDv1.0.1 for spot detection, approximate chromatic shift correction, 1038 kinetochore sister-sister pairing, and spot quality control as detailed in [4]. Chromatic 1039 shift correction samples were prepared and used as detailed in [4]. For any (proper) probability distribution on the triangle side lengths, l ij ≥ 0, i, j ∈ {1, 2, 3}, the length means necessarily satisfy the triangle inequality (provided they are finite). To prove this, we use Jensen's inequality ( [8,Thm 3.1.3]) in combination with convexity of the max function to write: prometaphase the same procedure was followed upon swapping the state labels 1 ↔ 2): 1053 Apart from the state proportions p (ζ) ζ , the single-state model is a nested sub-model 1054 of the two-state model (i.e. it is absolutely continuous) with all state affiliations {ζ n } n 1055 equal to the same state ζ. We can therefore compute the Bayes factor (see e.g. [9]) of 1056 these two models based on our MCMC samples of the two-state model alone. Let ϕ ts 1057 and ϕ ss denote the model parameters of the two-or single-state models, respectively, 1058 except for the state affiliations {ζ n } n . Their corresponding parameter spaces are 1059 denoted Φ ts , Φ ss ). The Bayes factor is then given by: where the last identity comes from the fact that when we confine the multi-state model 1061 to the case where all measurements are in one state, {ζ n } n ∈ {1} N , the likelihood times 1062 the prior is the same as its single-state counterpart, apart from the extra factor from 1063 the Dirichlet distribution, χ p (2) ∈[0,1] · 1 − p (2) N · dp (2) (note we have the same priors 1064 on lengths and measurement errors in both models). The right-hand ratio in 1065 Eq (AO) in S1 Text can be estimated from our MCMC of the two-state model by 1066 counting how often the chain is in the pure state 1. We abbreviate this ratio by ω. Our 1067 discussion so far has assumed a flat prior on the state proportion for the two-state 1068 model, χ p (2) ∈[0,1] · dp (2) . For analysis of a heterogeneous population with a minor 1069 sub-population, a flat prior on the smaller interval [0, α] for α ∈ [0, 1] appears more 1070 realistic. In this case the Bayes factor in Eq (AO) in S1 Text changes to: · ω α .

1072
From the Bayes factor we immediately get the probability of the two-state model 1073 (assuming a-priori equiprobable models): This is plotted as a function of α in Fig E in S1 Text (for Example 4.1). The metaphase 1075 value p two-states;meta in the main text is computed for α = 20%, while for the 1076 prometaphase values p two-states;late prometa , p two-states;early prometa we used α = 100% (i.e. 1077 for metaphase we expected the attached state to be dominant, while for prometaphase 1078 we were indifferent). conformation towards smaller sister-sister distances, l ss (correlation −0.319% with 1095 p = 2.0 × 10 −9 ), and larger swivels, ϑ (correlation +0.300% with p = 1.9 × 10 −8 ). See 1096 Table K in S1 Text for results for each mitotic phase.

1097
We did not pool this with the data presented in the main text, due to the Marginal lengths (in nm; pooled over metaphase and prometaphase) in panels a)-d), the majority state in metaphase is depicted in red, the minority state in green. See supplement Uninformative priors in S1 Text for the joint prior on the lengths. Marginals of the internal angles of the two states are shown in panels e), f). The marginal state proportions of the metaphase-minority state is depicted for each mitotic phase in panel g). The prior on each state proportion is flat in [0,1]. The mean state affiliations ζn n for each mitotic phase is shown in panel h), exhibiting clearly separated preferences for kinetochores in metaphase and early prometaphase, while late prometaphase contains some kinetochores that are likely attached, some likely unattached and some undecided. The shown mean state affiliations are with respect to the natural unattached state, i.e. kinetochores with a value close to one would be most likely in jthe natural unattached state. The bottom row compares the inferred mean state affiliations of each kinetochore with the tension parameters: Panel i) shows the mean sister-sister distance l ss , panel j) the mean swivel ϑ, where the mitotic phase is colour-coded and the straight lines show the best fit linear model for each phase. For significance tests, see Table K in S1 Text.  (attached, naturally unattached and jack-knifed), combining results of Examples 3.2,3.3,4.1,C.4,and C.5. All lengths are given in nm and have flat priors. See supplement Uninformative priors in S1 Text for the joint priors. Abbreviations are CC for CenpC, NC for Ndc80C and NN for Ndc80N. NAS denotes the (natural) attached state 1 (dominant in metaphase, DMSO), NUS the natural unattached state 2 (dominant in early prometaphase, DMSO) and JKS the jack-knifed state (in nocodazole). The single-state results are shown in blue (Example 3.2,3.3), red is the two-state phase-based analysis (Example 4.1), yellow the second experiment of the two-state phase-based analysis (Example C.4) and purple the two-state cell-based analysis (Example C.5). Example C.2, Jack-knifed conformation metaphase. Nocodazole. N = 118, C = 5: (lengths and errors in nm) CC-NC CC-NN NC-NN triangle, uninformative prior 55.1 ± 3.5 60.9 ± 3.6 13.1 ± 6.7 Example C.3, Prometaphase-metaphase mitotic-phase subsets. DMSO. N meta = 570, C meta = 17, N late prometa = 134, C late prometa = 4, N early prometa = 302, C early prometa = 10: Here the p (ζ) ζ denote the a-posteriori estimated state proportions, see subsection Cell-based subset analysis inference of mitotic phase variables in S1 Text. The priors for p (ζ) ζ are not flat as is the case for the mitotic-phase analysis. Posterior distributions for these lengths and states are compared in Fig G in S1 Text. could be changed a-posteriori by using an importance weight. Fig H. A-posteriori estimate of state proportions and affiliations for cell-based experimental prometaphase-metaphase dataset, Example C.5. Panel a) shows the a-posteriori estimated state proportions for each mitotic phase,p (2) (see subsection Cell-based subset analysis inference of mitotic phase variables in S1 Text for definition). The state proportions are pulled away from the boundaries by the prior. Panel b) shows the mean state affiliations pooled within each of the three mitotic phases. The shown mean state affiliations are with respect to the natural unattached state, i.e. kinetochores with a value close to one would be most likely in natural unattached state. Panel c) shows the numerically estimated prior on the a-posteriori estimates of the state proportionsp (2) for our cell-numbers and -sizes in each mitotic phase (same for all states). For the cell-based analysis the prior on the state proportions is independent between cells and flat in [0, 1] for each cell. As we have multiple cells in the same mitotic phase, this means that the prior on the a-posteriori estimates of the state proportionsp (2) in a mitotic phase are not uniform anymore, but more concentrated around 50%.  no substantial evidence according to [9]). Bottom row shows again one cell in metaphase (d), late prometaphase (e) and early prometaphase (f), where the examples were chosen to illustrate cells in each mitotic phase that have a mean state proportion closest to 0 (or 1 in early prometaphase). In this case almost no kinetochores are undecided. State labels are assigned based on prevalence in the datasets, i.e. p (1) ≥ p (2) ≥ p (3) . It can be seen that the additional state is not well-informed and spreads over several hundred nanometres. Note that due to our prior to have at least three measurements in each state, the lengths stay localised to some extent. See supplement Uninformative priors in S1 Text for the joint priors on the lengths.

Additional supplementary tables
1114 Due to an indistinguishability for the pair-wise methods, these parameters can only be inferred, if at least three fluorophores J ≥ 3 are used.  Table 3 for results): The number of kinetochores and number of cells are given after image processing and quality control.