A Bayesian mixture modelling approach for spatial proteomics
[['Spatial proteomics is an interdisciplinary field studying the localisation of proteins on a large-scale .', 'Where a protein is localised in a cell is a fundamental question , since a protein must be localised to its required sub-cellular compartment to interact with its binding partners ( for example , proteins , nucleic acids , metabolic substrates ) and carry out its function [1] .', 'Furthermore , mis-localisations of proteins are also critical to our understanding of biology , as aberrant protein localisation have been implicated in many pathologies [2–6] , including cancer [7–10] and obesity [11] .', 'Sub-cellular localisations of proteins can be studied by high-throughput mass spectrometry ( MS ) [12] .', 'MS-based spatial proteomics experiments enable us to confidently determine the sub-cellular localisation of thousands of proteins within in a cell [13] , given the availability of rigorous data analysis and interpretation [12] .', 'In a typical MS-based spatial proteomics experiment , cells first undergo lysis in a fashion which maintains the integrity of their organelles .', 'The cell content is then separated using a variety of methods , such as density separation [13 , 14] , differential centrifugation [15] , free-flow electrophoresis [16] , or affinity purification [17] .', 'In LOPIT [14 , 18 , 19] and hyperLOPIT [13 , 20] , cell lysis is proceeded by separation of the content along a density gradient .', 'Organelles and macro-molecular complexes are thus characterised by density-specific profiles along the gradient [21] .', 'Discrete fractions along the continuous density gradient are then collected , and quantitative protein profiles that match the organelle profiles along the gradient , are measured using high accuracy mass spectrometry [20] .', 'The data are first visualised using principal component analysis ( PCA ) and known sub-cellular compartments are annotated [22] .', 'Supervised machine learning algorithms are then typically employed to create classifiers that associate un-annotated proteins to specific organelles [23] , as well as semi-supervised methods that detect novel sub-cellular clusters using both labelled and un-labelled features [24] .', 'More recently , a state-of-the-art transfer learning ( TL ) algorithm has been shown to improve the quantity and reliability of sub-cellular protein assignments [25] .', 'Applications of such methods have led to organelle-specific localisation information of proteins in plants [14] , Drosophila [26] , chicken [27] , human cell lines [24] , mouse pluripotent embryonic stem cells [13] and cancer cell lines [28] .', 'Classification methods which have previously been used include partial least squares discriminate analysis [14] , K nearest neighbours [29] , random forests [30] , naive Bayes [31] , neural networks [32] and the support vector machine amongst others ( see [23] for an overview ) .', 'Although these methods have proved successful within the field they have limitations .', 'Typically , such classifiers output an assignment of proteins to discrete pre-annotated sub-cellular locations .', 'However , it is important to note that half the proteome cannot be robustly assigned to a single sub-cellular location , which may be a manifestation of proteins in so far uncharaterised organelles or proteins that are distributed amongst multiple locations .', 'These factors lead to uncertainty in the assignment of proteins to sub-cellular localisations , and thus quantifying this uncertainty is of vital importance [33] .', 'To overcome the task of uncertainty quantification , this article presents a probabilistic generative model for MS-based spatial proteomics data .', 'Our model posits that each annotated sub-cellular niche can be modelled by a multivariate Gaussian distribution .', 'Thus , the full complement of annotated proteins is captured by a mixture of multivariate Gaussian distributions .', 'With the prior knowledge that many proteins are not captured by known sub-cellular niches , we augment our model with an outlier component .', 'Outliers are often dispersed and thus this additional component is described by a heavy-tailed distribution: the multivariate Student’s t-distribution , leading us to a T-Augmented Gaussian Mixture model ( TAGM ) .', 'Given our model and proteins with known location , we can probabilistically infer the sub-cellular localisation of thousands of proteins .', 'We can perform inference in our model by finding maximum a posteriori ( MAP ) estimates of the parameters .', 'This approach returns the probability of each protein belonging to each annotated sub-cellular niche .', 'These posterior localisation probabilities can then be the basis for classification .', 'In a more sophisticated , fully Bayesian approach to uncertainty quantification , we can additionally infer the entire posterior distribution of localisation probabilities .', 'This allows the uncertainty in the parameters in our model to be reflected in the posterior localisation probabilities .', 'We perform this inference using Markov-chain Monte-Carlo methods; in particular , we provide an efficient collapsed Gibbs sampler to perform inference .', 'We perform a comprehensive comparison to state-of-the-art classifiers to demonstrate that our method is reliable across 19 different spatial proteomics datasets and find that all classifiers we considered perform competitively .', 'To demonstrate the additional biological advantages our method can provide , we apply our method to a hyperLOPIT dataset on mouse pluripotent embryonic stem cells [13] .', 'We consider several examples of proteins that were unable to be assigned using traditional machine-learning classifiers and show that , by considering the full posterior distribution of localisation probabilities , we can draw meaningful biological results and make powerful conclusions .', 'We then turn our hand to a more global perspective , visualising uncertainty quantification for over 5 , 000 proteins , simultaneously .', 'This approach reveals global patterns of protein organisation and their distribution across sub-cellular compartments .', 'We make extensive use of the R programming language [34] and existing MS and proteomics packages [35 , 36] .', 'We are highly committed to creating open software tools for high quality processing , visualisation , and analysis of spatial proteomics data .', 'We build upon an already extensive set of open software tools [36] as part of the Bioconductor project [37 , 38] and our methods are made available as part of this project .'], ['We model mouse pluripotent embryonic stem cell ( E14TG2a ) data [13] , which contains quantitation data for 5032 proteins .', 'This high-resolution map was produced using the hyperLOPIT workflow [20] , which uses a sophisticated sub-cellular fractionation scheme .', 'This fractionation scheme is made possible by the use of Tandem Mass Tag ( TMT ) 10-plex and high accuracy TMT quantification was facilitated by using synchronous precursor selection MS3 ( SPS-MS3 ) [39] , which reduces well documented issues with ratio distortion in isobaric multiplexed quantitative proteomics [40] .', 'The data resolves 14 sub-cellular niches with an additional chromatin preparation resolving the nuclear chromatin and non-chromatin components .', 'Two biological replicates of the data are concatenated , each with 10 fractions along the density gradient .', 'We defined gold standard organelle markers as those with unambiguous single annotation [23] .', 'A protein marker list for the mouse pluripotent embryonic stem cells was manually curated using information from the UniProt database , the Gene Ontology and the literature , as was performed in [13] .', 'The following section applies our statistical methodology to these data and we explore the results .', 'In this section , we assess the generalisation performance of our methods on several datasets , by computing performance metrics associated with each classifier as detailed in section Classifier assessment .', 'We compare the SVM and KNN classifiers alongside the MAP and MCMC approaches detailed in the methods section .', 'We compute the F1 score and quadratic loss over 100 rounds of stratified 5-fold cross-validation .', 'The hyperparameter for the KNN algorithm , the number of nearest neighbours , is optimised via an additional internal 5-fold cross-validation and the hyperparameters for the SVM , sigma and cost , are also optimised via internal 5-fold cross validation [45] .', 'We test our methods on the following datasets Drosophila [26] , chicken [27] , mouse pluripotent embryonic stem cells from [13] and [25] , the human bone osteosarcoma epithelial ( U2-OS ) cell line [28] , the HeLa cell line of [15] , the 3 HeLa cell lines from [46] and 10 primary fibroblast datasets from [47] .', 'These datasets represent a great variety of spatial proteomics experiments across many different workflows .', 'The two hyperLOPIT datasets on mouse pluripotent embryonic stem cells and the U2-OS cell line use TMT 10-plex labelling and contain the greatest number of proteins .', 'Earlier LOPIT experiments on the Drosophila and chicken use iTRAQ 4-plex labelling , whilst another LOPIT mouse pluripotent embryonic stem cell dataset uses iTRAQ 8-plex .', 'The datasets of [15] and [46] employ a different methodology completely—separating cellular content using differential centrifugation ( as opposed to along a density-gradient ) .', 'Furthermore , the methods use SILAC rather than iTRAQ or TMT for labelling .', 'The experiments of [46] were designed to explore the functional role of AP-5 by coupling CRISPR-CAS9 knockouts with spatial proteomics methods .', 'We analysed all three datasets from [46] , which includes a wild type HeLa cell line as a control , as well as two CRISPR-CAS9 knockouts: AP5Z1-KO1 and AP5Z1-KO2 respectively .', 'In addition , we analyse the spatio-temporal proteomics experiments of [47] , which uses TMT-based MS quantification .', 'This experiment explored infecting primary fibroblasts with Human cytomegalovirus ( HMCV ) and the goal of these experiments was to explore the dynamic perturbation of host proteins during infection , as well as the sub-cellular localisation of viral proteins throught the HCMV life-cycle .', 'They produced spatial maps at different time points: 24 , 48 , 72 , 96 , 120 hours post infection ( hpi ) , as well as mock maps at these same time points to serve as a control—this results in 10 different spatial proteomics maps .', 'In each case , a dataset specific marker list was used , which is curated specifically for the each cell line .', 'We removed “high-curvature ER” annotations from the HeLa dataset [15] , as well as the “ER Tubular” , “Nuclear pore complex” and “Peroxisome” annotations from the HeLa CRISPR-CAS9 knockout experiments [46] as there are too few proteins to correctly perform cross-validation .', 'Table 1 summarises these datasets , including information about number of quantified proteins , the workflow used and the number of fractions .', 'Fig 5 compares the Macro-F1 scores across the datasets for all classifiers and demonstrates that no single classifier consistently outperforms any other across all datasets , with results being highly consistent across all methods , as well as across datasets .', 'We perform a pairwise unpaired t-test with multiple testing correction applied using the Benjamini-Höchberg procedure [48] to detect differences between classifier performance .', 'In the Drosophila dataset only the KNN algorithm outpeforms the SVM at significance level of 0 . 01 , whilst no other significant differences exist between the classifiers .', 'In the chicken DT40 dataset only the MCMC method outperforms the KNN classifier at significance level of 0 . 01 , no other significant conclusion can be drawn .', 'In the mouse dataset the MAP based method outperforms the MCMC method at significance level of 0 . 01 , no other significant conclusions can be drawn .', 'In the HeLa dataset all classifiers are significantly different at a 0 . 01 level .', 'These differences may exist because the dataset does not fit well with our modelling assumptions; in particular , this dataset set has been curated to have a class called “Large Protein Complex” , which likely describes several sub-cellular structures .', 'These might include nuclear compartments and ribosomes , as well as any cytosolic complex and large protein complex which pellets during the centrifugation conditions used to capture this mixed sub-cellular fraction .', 'Moreover , the cytosolic and nuclear fraction were processed separately leading to possible imbalance with comparisions with other datasets .', 'Thus , the large protein complex component might be better described as itself a mixture model or more detailed curation of these data may be required .', 'We do not consider further modelling of this dataset in this manuscript .', 'For the U2-OS all classifiers are significantly different at a significance level of 0 . 01 except for the SVM classifier and the MCMC method , with the MAP method performing the best .', 'Fig 5 shows that for this dataset all classifiers are performing extremely well .', 'In the three Hirst datasets the MAP method significantly outperforms all other methods ( p < 0 . 01 ) , whilst in the wild type HeLa and in the CRISPR-CAS9 KO1 there is no significant difference between the KNN and MCMC method .', 'In the CRISPR-CAS9 KO2 the MCMC method outperforms the SVM and KNN methods ( p < 0 . 01 ) .', 'In the interest of brevity , the remaining results for the t-tests can be found in tables in supporting information , section S5 .', 'The Macro-F1 scores do not take into account that whilst the TAGM model may misclassify , it may do so with low confidence .', 'We therefore additionally compute the quadratic loss , which allows us to make use of the probabilitic information provided by the classifiers .', 'The lower the quadratic loss the closer the probabilitic predicition is to the true value .', 'We plot the distributions of quadratic losses for each classifier in Fig', '6 . We observe highly consistent performance across all classifiers across all datasets .', 'Again , we perform a pairwise unpaired t-test with multiple testing correction .', 'We find that in 16 out of 19 datasets ( all of those except HeLa Wild type , HeLa KO1 and HeLa KO2 ) the MCMC method achieves the lowest quadratic loss at a signifiance level < 0 . 0001 over the SVM and KNN classifiers .', 'In 6 out of these 16 datasets there is no significant difference between the MCMC and the MAP methods .', 'In the three Hirst datasets in which the MCMC did not acheive the lowest quadratic loss , the SVM outperformed .', 'However , in two of these datasets ( HeLa Wild type and KO1 ) the MAP method and SVM classifier were not significantly different .', 'In the Hirst KO2 dataset there were no signicant differences between the MAP and MCMC methods .', 'In the vast majority of cases , we observe that if the TAGM model , using the MCMC methodology , makes an incorrect classification it does so with lower confidence than the SVM classifier , the KNN classifier and the MAP based classifier , whilst if it is correct in its assertion it does so with greater confidence .', 'Additionally , a fully Bayesian methodology provides us with not only point estimates of classification probabilities but uncertainty quantification in these allocations , and we show in the following section that this provides deeper insights into protein localisation .', 'Computing distributions of F1 scores and quadratic losses , which can only be done on the marker proteins , can help us understand whether a classifier might have greater generalised performance accuracy .', 'However , we are interested in whether there is a large disagreement between classifiers when prediction is performed on proteins for which we have no withheld localisation information .', 'This informs us about a systematic bias for a particular classifier or whether a classifier ensemble could increase performance .', 'To maintain a common set of proteins we set thresholds for each classifier in turn and compare to the other classifier without thresholding .', 'Firstly , we set a global threshold of 0 . 95 for the TAGM-MCMC and then for these proteins plot a contingency table against the classification results from the SVM .', 'Secondly , we set a 5% FDR for the SVM and then for these proteins plot a contingency table against the classification results from the TAGM-MCMC .', 'We visualise the contingency tables as heat plots in Fig', '7 . In general , we see an extremely high level of coherence between the TAGM and the SVM , with almost all proteins predicted to concordant sub-cellular compartments .', 'Fig 7 shows there is some disagreement between assigning proteins to the lysosome and plasma membrane , to the cytosol and proteasome , and between the large and small ribosomal subunits .', 'However , we have not used the uncertainty in the probabilitic assignments to produce the contingency tables above .', 'In the next sections , we explore examples of proteins with uncertainty in their posterior localisation probabilities .', 'Selecting biologically relevant thresholds is important for any classifier and exploring uncertainty is of vital importance when drawing biological conclusions .', 'Protein sub-cellular localisation can be uncertain for a number of reasons .', 'Technical variations and unknown biological novelty , such as yet uncharacterised functional compartments , can be some of the reasons why a protein might have an unknown or uncertain localisation .', 'Furthermore many proteins are known to reside in multiple locations with possibly different functional duties in each location [49] .', 'With these considerations in mind , it is pertinent to quantify the uncertainty in our allocation of proteins to organelles .', 'This section explores several situations where proteins display uncertain localisation and considers the biological factors that influence uncertainty .', 'We later explore and visualise whole proteome uncertainty quantification .', 'Exportin 5 ( Q924C1 ) forms part of the micro-RNA export machinery of the nucleus , transporting miRNA from the nucleus to the cytoplasm for further processing .', 'It then translocates back through the nuclear pore complex to return to the nucleus .', 'Exportin 5 can then continue to mediate further transport between nucleus and cytoplasm .', 'The SVM was unable to assign a localisation of Exportin 5 , with its assignment falling below a 5% FDR to wrongly assign this protein to the proteasome .', 'This incorrect assertion by the SVM was confounded by the similarity between the cytosol and proteasome profiles .', 'Fig 8 demonstrates , according to the TAGM-MCMC model , that Exportin 5 most likely localises to the cytosol but there is some uncertainty with this assignment .', 'This uncertainty is reflected in possible assignment of Exportin 5 to the nucleus non-chromatin and this uncertainty is a manifestation of the the fact that the function of this protein is to shuttle between the cytosol and nucleus .', 'The Phenylalanine–tRNA ligase beta subunit protein ( Q9WUA2 ) has an uncertain localisation between the 40S ribosome and the nucleus non-chromatin demonstrated in Fig 9 .', 'This protein was left unclassified by the SVM because its score fell below a 5% FDR threshold to assign it to the 40S ribosome .', 'Considering that this protein is involved in the acylation of transfer RNA ( tRNA ) with the amino acid phenylalanine to form tRNA–Phe to be used in translation of proteins , it is therefore unsurprising that this protein’s steady state location is ribosomal .', 'Whilst the SVM is unable to make an assignment , TAGM-MCMC is able to suggest an assignment and quantify our uncertainty .', 'Relatively little is known about the Dedicator of cytokinesis ( DOCK ) protein 6 ( Q8VDR9 ) , a guanine nucleotide exchange factor for CDC42 and RAC1 small GTPases .', 'The SVM could not assign localisation to the ER/Golgi , since its score fell below a 5% FDR .', 'Furthermore , the TAGM-MCMC model assigned this DOCK 6 to the outlier component with posterior probability > 0 . 95 .', 'Fig 10 shows possible localisation to several components along the secretory pathway .', 'As an activator for CDC42 and RAC1 we may expect to see them with similar localisation .', 'CDC42 , a plasma membrane associated protein , regulates cell cycle and division and is found with many localisations .', 'Furthermore RAC1 , a small GTPase , also regulates many cellular processes and is found in many locations .', 'Thus the steady-state distribution of DOCK6 is unlikely to be in a single location , since its interaction partners are found in many locations .', 'This justifies including an outlier component in our model , else we may erroneously assign such proteins to a single location .', 'The advantage of the TAGM-MCMC model is its ability to provide proteome wide uncertainy quantification .', 'Regions where organelle assignments overlap are areas were uncertainty is expected to be the greatest , as well as areas with no dominant component .', 'We take an information theoretic approach to summarising uncertainty in protein localisation by computing the Shannon entropy [50] for each Monte-Carlo sample t = 1 , … , T of the posterior localisation probabilities of each protein', '{ H ( t ) = - ∑ k = 1 K p i k ( t ) log ( p i k ( t ) ) } t = 1 T , ( 1 )', 'where p i k ( t ) denotes the posterior localisation probabilty of protein i to component k at iteration t .', 'We then summarise this as a Monte-Carlo averaged Shannon entropy .', 'The greater the Shannon entropy the more uncertainty associated with the assignment of this protein .', 'The lower the Shannon entropy the lower the uncertainty associated with the assignment of this protein .', 'In Fig 11 panel ( a ) , we visualise the Shannon entropy of each protein in a PCA plot , by scaling the pointers in accordance to this metric .', 'We also note that while localisation probability ( of a protein to its most probable location ) and the Shannon entropy are correlated , Fig 11 panel ( c ) , it is not perfect .', 'Thus it is important to use both the localisation probabilities and the uncertainty in these assignments to make conclusions .', 'Fig 11 demonstrates that the regions of highest uncertainty are those in regions where organelles assignments overlap .', 'The conclusions from this plot are manifold .', 'Firstly , many proteins are assigned unambiguously to sub-cellular localisations; that is , not only are some proteins assigned to organelles with high probability but also with low uncertainty .', 'Secondly , there are well defined regions with high uncertainty , for example proteins in the secretory pathway or proteins on the boundary between cytosol and proteasome .', 'Finally , some organelles , such as the mitochondria , are extremely well resolved .', 'This observed uncertainty in the secretory pathway and cytosol could be attributed to the dynamic nature of these parts of the cell with numerous examples of proteins that traffic in and out of these sub-cellular compartments as part of their biological role .', 'Moreover , the organelles of the secretory pathway share similar and overlapping physical properties making their separation from one another using biochemical fractionation more challenging .', 'Furthermore , there is a region located in the centre of the plot where proteins simultaneously have low probability of belonging to any organelle and high uncertainty in their localisation probability .', 'This suggests that these proteins are poorly described by any single location .', 'These proteins could belong to multiple locations or belong to undescribed sub-cellular compartments .', 'The information displayed in these plots and the conclusion therein would be extremely challenging to obtain without the use of Bayesian methodology .'], ['We have demonstrated that a Bayesian framework , based on Gaussian mixture models , for spatial proteomics can provide whole sub-cellular proteome uncertainty quantification on the assignment of proteins to organelles and such information is invaluable .', 'Performing MAP inference using our generative model provides fast and straightforward approach , which is vital for quality control and early data exploration .', 'Full posterior inference using MCMC provides not only point estimates of the posterior probability that a protein belongs to a particular sub-cellular niche , but uncertainty in this assignment .', 'Then , this uncertainty can be summarised in several ways , including , but not limited to , equi-tailed credible intervals of the Monte-Carlo samples of posterior localisation probabilities .', 'Posterior distributions for indivdual proteins can then be rigorously interrogated to shed light on their biological mechanisms; such as , transport , signalling and interactions .', 'As well as the local uncertainty seen by exploring individual proteins , we further explored using a Monte-Carlo averaged Shannon entropy to visualise global uncertainty .', 'Regions of high uncertainty , as measured using this Shannon entropy , reflect highly dynamics regions of the sub-cellular environment .', 'Hence , biologists can now explore uncertainty at different levels and then are able to make quantifiable conclusions and insights about their data .', 'Furthermore , our Bayesian model is interpretable and our inferences are fully conditional on our data , allowing them to be easily modified with changing experimental design .', 'In addition , we produced competitive classifier performance to the state-of-the-art classifiers .', 'We considered two traditional machine-learning methods: the SVM and KNN classifiers; as well as two classifiers based on our model: a MAP classifier and classification based on MCMC .', 'We compared all methods on 19 different spatial proteomics datasets , across four different organisms .', 'When considering the macro-F1 score as a performance metric , no single classifier outperformed another across all datasets .', 'However , using MCMC based inference our method significantly outperforms the SVM and KNN classifiers with respect to the quadratic loss in 16 out of 19 datasets .', 'This allows us to have greater confidence in our conclusions when they are drawn from our Bayesian inferences .', 'Furthermore , using MCMC provides a wealth of additional information , and so becomes the method of choice for analysing spatial proteomics data .', 'Analysis of a hyperLOPIT experiment applied to mouse pluripotent embryonic stem cells demonstrated that the additional layer of information that our model provides is biologically relevant and provides further avenues for additional exploration .', 'Moreover , applying our method to a biologically significant dataset now provides the scientific community with localisation information on up to 4000 proteins for the mouse pluripotent stem cell proteome .', 'Fig 12 demonstrates that from an initial input of roughly 1000 marker proteins with a priori known location and 4000 unknown proteins with unknown location , SVM and TAGM-MCMC can provide rigorous localisation information on roughly 2000 proteins .', 'However , our methodology , by also considering uncertainty , allows us to obtain information on another 1000 proteins .', 'Thus , we have augmented this dataset by providing uncertainty quantification on the localisation of proteins to their sub-cellular niches , which had been previously unavailable .', 'We note that our method is general enough to be applied to many MS-based spatial proteomics protocols including: LOPIT , hyperLOPIT , protein correlation profiling ( PCP ) [51] , differential centrifugation approaches and spatio-temporal proteomics methods .', 'In our flexible software implementation , all hyperparameters for the priors can be changed if users have precise priors they wish to specify .', 'We have also provided a new set of visualisation methods to accompany our model , which allow us to easily interrogate our data .', 'High quality visualisation tools are essential for rigorous quality control and sound biological conclusions .', 'Our methods have been developed in the R statistical programming language and we continue to contribute to the Bioconductor project [37 , 38] with inclusion of our methods within the pRoloc package ( > = 1 . 21 . 1 ) [36] .', 'The underlying source code used to generate this document is available at https://github . com/lgatto/2018-TAGM-paper .', 'Currently , our model does not integrate localisation information from different data sources , nor does it explicitly model proteins with multiple localisation .', 'However , one ( of many ) biological explanations for the uncertainty that we model in the allocation probabilities is provided by multiple localisation .', 'Thus a protein for which it is uncertain to which two sub-cellular niches it is resident within it is perhaps resident of both niches .', 'In further work , we plan to explicitly model such cases to deconvolute different sources of uncertainty .', 'In addition , extensions to semi-supervised non-parametric methods are under consideration to detect novel sub-cellular niches .', 'These are the subjects of further work .'], ['We observe N protein profiles each of length L , corresponding to the number of quantified fractions along the gradient density , including combining replicates .', 'For i = 1 , … , N , we denote the profile of the i-th protein by x i = [x1i , … , xLi] .', 'We suppose that there are K known sub-cellular compartments to which each protein could localise ( e . g . cytoplasm , endoplasmic reticulum , mitochondria , … ) .', 'Henceforth , we refer to these K sub-cellular compartments as components , and introduce component labels zi , so that zi = k if the i-th protein localises to the k-th component .', 'We denote by XL the set of proteins whose component labels are known , and by XU the set of unlabelled proteins .', 'If protein i is in XU , we desire the probability that zi = k for each k = 1 , … , K . That is , for each unlabelled protein , we want the probability of belonging to each component ( given a model and the observed data ) .', 'We initially model the distribution of profiles associated with proteins that localise to the k-th component as multivariate normal with mean vector μk and covariance matrix Σk , so that:', 'x i | z i = k ∼ N ( μ k , Σ k ) .', '( 2 ) For any i , we define the prior probability of the i-th protein localising to the k-th component to be p ( zi = k ) = πk .', 'Letting θ = { μ k , Σ k } k = 1 K denote the set of all component mean and covariance parameters , and π = { π k } k = 1 K denote the set of all mixture weights , it follows ( from the law of total probability ) that:', 'p ( x i | θ , π ) = ∑ k = 1 K π k f ( x i | μ k , Σ k ) , ( 3 )', 'where f ( x|μ , Σ ) denotes the density of the multivariate normal with mean vector μ and covariance matrix Σ evaluated at x .', 'Eq ( 3 ) defines a generative probabilistic model known as a mixture model .', 'Such models are useful for describing populations that are composed of a number of distinct homogeneous subpopulations .', 'In our case , we model the full complement of measured proteins as being composed of K subpopulations , each corresponding to a different organelle or sub-cellular compartment .', 'The literature of mixture model applications to biology is rich and some recent example include applications to retroviral integration sites [52] , genome-wide associations studies [53] , single-cell transcriptomics [54] and affinity purification MS proteomics [55] .', 'Though some proteins are well described as belonging to a single component , many proteins multi-localise or might belong to uncharacterised organelles .', 'In order to allow the model to better account for these “outliers” that cannot be straightforwardly allocated to any single known component , we extend it by introducing an additional “outlier component” .', 'To do this , we augment our model by introducing a further indicator latent variable ϕ .', 'Each protein xi is now described by an additional variable ϕi , with ϕi = 1 indicating that protein xi belongs to a organelle derived component and ϕi = 0 indicating that protein xi is not well described by these known components .', 'This outlier component is modelled as a multivariate T distribution with degrees of freedom κ , mean vector M , and scale matrix V . Thus Eq ( 2 ) becomes', 'x i | z i = k , ϕ i ∼ N ( μ k , Σ
['Introduction', 'Results', 'Discussion', 'Model']
['Analysis of the spatial sub-cellular distribution of proteins is of vital importance to fully understand context specific protein function .', 'Some proteins can be found with a single location within a cell , but up to half of proteins may reside in multiple locations , can dynamically re-localise , or reside within an unknown functional compartment .', 'These considerations lead to uncertainty in associating a protein to a single location .', 'Currently , mass spectrometry ( MS ) based spatial proteomics relies on supervised machine learning algorithms to assign proteins to sub-cellular locations based on common gradient profiles .', 'However , such methods fail to quantify uncertainty associated with sub-cellular class assignment .', 'Here we reformulate the framework on which we perform statistical analysis .', 'We propose a Bayesian generative classifier based on Gaussian mixture models to assign proteins probabilistically to sub-cellular niches , thus proteins have a probability distribution over sub-cellular locations , with Bayesian computation performed using the expectation-maximisation ( EM ) algorithm , as well as Markov-chain Monte-Carlo ( MCMC ) .', 'Our methodology allows proteome-wide uncertainty quantification , thus adding a further layer to the analysis of spatial proteomics .', 'Our framework is flexible , allowing many different systems to be analysed and reveals new modelling opportunities for spatial proteomics .', 'We find our methods perform competitively with current state-of-the art machine learning methods , whilst simultaneously providing more information .', 'We highlight several examples where classification based on the support vector machine is unable to make any conclusions , while uncertainty quantification using our approach provides biologically intriguing results .', 'To our knowledge this is the first Bayesian model of MS-based spatial proteomics data .']
['Sub-cellular localisation of proteins provides insights into sub-cellular biological processes .', 'For a protein to carry out its intended function it must be localised to the correct sub-cellular environment , whether that be organelles , vesicles or any sub-cellular niche .', 'Correct sub-cellular localisation ensures the biochemical conditions for the protein to carry out its molecular function are met , as well as being near its intended interaction partners .', 'Therefore , mis-localisation of proteins alters cell biochemistry and can disrupt , for example , signalling pathways or inhibit the trafficking of material around the cell .', 'The sub-cellular distribution of proteins is complicated by proteins that can reside in multiple micro-environments , or those that move dynamically within the cell .', 'Methods that predict protein sub-cellular localisation often fail to quantify the uncertainty that arises from the complex and dynamic nature of the sub-cellular environment .', 'Here we present a Bayesian methodology to analyse protein sub-cellular localisation .', 'We explicitly model our data and use Bayesian inference to quantify uncertainty in our predictions .', 'We find our method is competitive with state-of-the-art machine learning methods and additionally provides uncertainty quantification .', 'We show that , with this additional information , we can make deeper insights into the fundamental biochemistry of the cell .']
['ellipses', 'geometry', 'membrane proteins', 'probability distribution', 'mathematics', 'stem cells', 'cellular structures and organelles', 'research and analysis methods', 'cell potency', 'animal cells', 'embryonic stem cells', 'biological databases', 'proteomics', 'pluripotency', 'cell membranes', 'probability theory', 'biochemistry', 'proteomic databases', 'cell biology', 'database and informatics methods', 'biology and life sciences', 'cellular types', 'physical sciences', 'cytosol']