# Vignette 1: Getting Started with BANDLE

Oliver M. Crook1 and Lisa M. Breckels2

1Department of Statistics, University of Oxford, UK
2Cambridge Centre for Proteomics, University of Cambridge, UK

#### 6 November 2024

#### Abstract

This vignette provides an introduction to the BANDLE package (Crook et al. 2021) and follows a short theortical example of how to perform differential localisation analysis of quantitative proteomics data using the BANDLE model. Explanation and general recommendations of the input parameters are provided here. For a more comprehensive workflow which follows a real-life use case, please see the second vignette in this package.

#### Package

bandle 1.10.0

# 1 Introduction

Bayesian ANalysis of Differential Localisation Experiments (BANDLE) is an integrative semi-supervised functional mixture model, developed by (Crook et al. 2021), to obtain the probability of a protein being differentially localised between two conditions.

In this vignette we walk users through how to install and use the R (R Development Core Team 2011) Bioconductor (Gentleman et al. 2004) `bandle` package by simulating a well-defined differential localisation experiment from spatial proteomics data from the `pRolocdata` package (Gatto et al. 2014).

The BANDLE method uses posterior Bayesian computations performed using Markov-chain Monte-Carlo (MCMC) and thus uncertainty estimates are available (Gilks, Richardson, and Spiegelhalter 1995). Throughout this vignette we use the term differentially localised to pertain to proteins which are assigned to different sub-cellular localisations between two conditions. One of the main outputs of BANDLE is the probability that a protein is differentially localised between two conditions.

# 2 Installation

The package can be installed with the `BiocManager` package:

    if (!requireNamespace("BiocManager", quietly=TRUE))
        install.packages("BiocManager")
    BiocManager::install("bandle")

and then loaded,

    library("bandle")

For visualisation we also load the packages,

    library("pheatmap")
    library("viridis")

# 3 The data

In this vignette and Crook et al. (2021), the main data source that we use to study differential protein sub-cellular localisation are data from high-throughput mass spectrometry-based experiments. The data from these types of experiments traditionally yield a matrix of measurements wherein we have, for example, PSMs, peptides or proteins along the rows, and samples/channels/fractions along the columns. The `bandle` package uses the `MSnSet` class as implemented in the Bioconductor MSnbase package and thus requires users to import and store their data as a `MSnSet` instance. For more details on how to create a `MSnSet` see the relevant vignettes in pRoloc. The pRolocdata experiment data package is a good starting place to look for test data. This data package contains tens of quantitative proteomics experiments, stored as `MSnSet`s.

## 3.1 A well-defined theoretical example

To get started with the basics of using `bandle` we begin by generating a simple example dataset which simulates a differential localisation experiment (please see the second vignette in this package for a full real-life biological use case). In this example data, the key elements are replicates, and a perturbation of interest. There is code within the bandle package to simulate an example experiment.

In the code chunk below we begin by loading the pRolocdata package to obtain a spatial proteomics dataset. This will be the basis of our simulation which will use boostrapping to generate new datasets. The dataset we have chosen to load is a dataset from 2009 (`tan2009r1`). This is data from a early LOPIT experiment performed on Drosophila embryos (Tan et al. 2009). The aim of this experiment was to apply LOPIT to an organism with heterogeneous cell types. This experiment used four isotopes across four distinct fractions and thus yielded four measurements (features) per protein profile. We visualise the data by using principal components analysis.

    library("pRolocdata")
    data("tan2009r1")

    ## Let's set the stock colours of the classes to plot to be transparent
    setStockcol(NULL)
    setStockcol(paste0(getStockcol(), "90"))

    ## Plot the data using plot2D from pRoloc
    plot2D(tan2009r1,
           main = "An example spatial proteomics datasets",
           grid = FALSE)
    addLegend(tan2009r1, where = "topleft", cex = 0.7, ncol = 2)

The following code chuck simulates a differential localisation experiment. It will generate `numRep/2` of each a control and treatment condition. We will also simulate relocalisations for `numDyn` proteins.

    set.seed(1)
    tansim <- sim_dynamic(object = tan2009r1,
                          numRep = 6L,
                          numDyn = 100L)

    ## [1] "markers"

The list of the 6 simulated experiments are found in `tansim$lopitrep`. Each one is an `MSnSet` instance (the standard data container for proteomics experimental data). The first 3 are the simulated control experiments (see `tansim$lopitrep[1:3]`), and the following 3 in the list are the treatment condition simulated experiments (see `tansim$lopitrep[4:6]`). We can plot them using the `plot2D` function from `pRoloc`.

    plot_title <- c(paste0("Replicate ", seq(3), " condition", " A"),
                   paste0("Replicate ", seq(3), " condition", " B"))

    par(mfrow = c(2, 3))
    out <- lapply(seq(tansim$lopitrep), function(z)
        plot2D(tansim$lopitrep[[z]], grid = FALSE, main = plot_title[z]))

For understanding, exploring and visualizing individual spatial proteomics experiments, see the vignettes in `pRoloc` and `MSnbase` packages.

    tansim$lopitrep[[1]]

    ## MSnSet (storageMode: lockedEnvironment)
    ## assayData: 888 features, 4 samples
    ##   element names: exprs
    ## protocolData: none
    ## phenoData
    ##   sampleNames: X114 X115 X116 X117
    ##   varLabels: Fractions
    ##   varMetadata: labelDescription
    ## featureData
    ##   featureNames: P20353 P53501 ... P07909 (888 total)
    ##   fvarLabels: FBgn Protein.ID ... knn.scores (18 total)
    ##   fvarMetadata: labelDescription
    ## experimentData: use 'experimentData(object)'
    ##   pubMedIds: 19317464
    ## Annotation:
    ## - - - Processing information - - -
    ## Added markers from  'mrk' marker vector. Thu Jul 16 22:53:44 2015
    ## Performed knn prediction (k=10) Wed Nov  6 17:10:36 2024
    ##  MSnbase version: 1.17.12

# 4 Preparing for `bandle` analysis

The main function of the package is `bandle`, this uses a complex model to analyse the data. Markov-Chain Monte-Carlo (MCMC) is used to sample the posterior distribution of parameters and latent variables. From which statistics of interest can be computed. Here we only run a few iterations for brevity but typically one needs to run thousands of iterations to ensure convergence, as well as multiple parallel chains.

## 4.1 Fitting Gaussian processes

First, we need to fit non-parametric regression functions to the markers profiles, upon which we place our analysis. This uses Gaussian processes. The `fitGPmaternPC` function can be used and fits some default penalised complexity priors (see `?fitGP`), which works well. However, these can be altered, which is demonstrated in the next code chunk

    par(mfrow = c(4, 3))
    gpParams <- lapply(tansim$lopitrep, function(x)
      fitGPmaternPC(x, hyppar = matrix(c(10, 60, 250), nrow = 1)))

We apply the `fitGPmaternPC` function to each datasets by calling `lapply` over the `tansim$lopitrep` list of datasets. The output of `fitGPmaternPC` returns a list of posterior predictive means and standard deviations. As well as MAP hyperparamters for the GP.

Note here we the use the default `hyppar = matrix(c(10, 60, 250), nrow = 1)` as a starting point for fitting the GPs to the marker profile distributions. In the Crook et al. (2021) manuscript we found that these values worked well for smaller spatial proteomics datasets. This was visually assessed by passing these values and visualising the GP fit using the `plotGPmatern` function.

The `plotGPmatern` function can be used to plot the profiles for each class in each replicate condition with the posterior predictive distributions overlayed with the markers protein profiles.

For example, to plot the predictive distributions of the first dataset,

    par(mfrow = c(4, 3))
    plotGPmatern(tansim$lopitrep[[1]], params = gpParams[[1]])

The prior needs to form a `K*3` matrix. `K` corresponds to the number of subcellular classes in the data, and 3 columns for (1) the prior, (2) length-scale amplitude and (3) standard deviation parameters (see `hyppar` in `?fitGPmaternPC`). Increasing these values, increases the shrinkage. For more details see the manuscript by Crook et al. (2021). We strongly recommend users start with the recommended `hyppar` parameters and change and assess them as necessary for their dataset by visually evaluating the fit of the GPs using the `plotGPmatern` function.

    K <- length(getMarkerClasses(tansim$lopitrep[[1]], fcol = "markers"))
    pc_prior <- matrix(NA, ncol = 3, K)
    pc_prior[seq.int(1:K), ] <- matrix(rep(c(10, 60, 250),
                                           each = K), ncol = 3)

Now we have generated these complexity priors we can pass them as an argument to the `fitGPmaternPC` function. For example,

    gpParams <- lapply(tansim$lopitrep,
                       function(x) fitGPmaternPC(x, hyppar = pc_prior))

By looking at the plot of posterior predictives using the `gpParams` we can see the GP fit looks sensible.

## 4.2 Setting the prior on the weights

The next step is to set up the matrix Dirichlet prior on the mixing weights. These weights are defined across datasets so these are slightly different to mixture weights in usual mixture models. The \((i,j)^{th}\) entry is the prior probability that a protein localises to organelle \(i\) in the control and \(j\) in the treatment. This mean that off-diagonal terms have a different interpretation to diagonal terms. Since we expect re-localisation to be rare, off-diagonal terms should be small. The following functions help set up the priors and how to interpret them. The parameter `q` allow us to check the prior probability that more than `q` differential localisations are expected.

    set.seed(1)
    dirPrior = diag(rep(1, K)) + matrix(0.001, nrow = K, ncol = K)
    predDirPrior <- prior_pred_dir(object = tansim$lopitrep[[1]],
                                   dirPrior = dirPrior,
                                   q = 15)

The mean number of re-localisations is small:

    predDirPrior$meannotAlloc

    ## [1] 0.3457542

The prior probability that more than `q` differential localisations are expected is small

    predDirPrior$tailnotAlloc

    ## [1] 4e-04

The full prior predictive can be visualised as histogram. The prior probability that proteins are allocated to different components between datasets concentrates around 0.

    hist(predDirPrior$priornotAlloc, col = getStockcol()[1])

For most use-cases we indeed expect the number of differential localisations to be small. However, there may be specific cases where one may expect the either a smaller or larger number of differential localisations. Users could try testing different values for the `dirPrior` for example, replacing 0.001 with 0.0005 or smaller, for larger datasets to bring the number of expected re-localisations inline with the biological expectation, and visa-versa when we expect the number of proteins to have changed to be higher.

# 5 Running the `bandle` function

We are now ready to run the main `bandle` function. Remember to carefully select the datasets and replicates that define the control and treatment. As a reminder, in this introductory vignette we have used a small dataset and generated theoretical triplicates of each theoretical condition. Please see the second vignette in this package for a more detailed workflow and real biological use-case. In the below code chunk we run `bandle` for only 50 iterations for the convenience of building the vignette, but typically we’d recommend you run the number of iterations (`numIter`) in the 1000s.

Remember: the first 3 datasets are the first 3 elements of `tansim` and the final 3 elements are the “treatment” triplicate datasets.

    control <- tansim$lopitrep[1:3]
    treatment <- tansim$lopitrep[4:6]

    bandleres <- bandle(objectCond1 = control,
                        objectCond2 = treatment,
                        numIter = 20,  # usually 10,000
                        burnin = 5L,   # usually 5,000
                        thin = 1L,     # usually 20
                        gpParams = gpParams,
                        pcPrior = pc_prior,
                        numChains = 1,  # usually >=4
                        dirPrior = dirPrior)

The `bandle` function generates an object of class `bandleParams`. The `show` method indicates the number of parallel chains that were run, this should typically be greater than 4 (here we use 1 just as a demo). Normally, we should also assess convergence but this omitted for the moment so that we can move forward with the analysis. Please see the end of the vignette for convergence plots.

    bandleres

    ## Object of class "bandleParams"
    ## Method: bandle
    ## Number of chains: 1

# 6 Analysing `bandle` output

Before we can begin to extract protein allocation information and a list of proteins which are differentially localised between conditions, we first need to populate the `bandleres` object by calling the `bandleProcess` function.

## 6.1 Populating a `bandleres` object

Currently, the summary slots of the `bandleres` object are empty. The `summaries` function accesses them.

    summaries(bandleres)

    ## [[1]]
    ## An object of class "bandleSummary"
    ## Slot "posteriorEstimates":
    ## <S4 Type Object>
    ## attr(,"elementType")
    ## [1] "ANY"
    ## attr(,"elementMetadata")
    ## `\001NULL\001`
    ## attr(,"metadata")
    ## list()
    ##
    ## Slot "diagnostics":
    ## <0 x 0 matrix>
    ##
    ## Slot "bandle.joint":
    ## <0 x 0 matrix>
    ##
    ##
    ## [[2]]
    ## An object of class "bandleSummary"
    ## Slot "posteriorEstimates":
    ## <S4 Type Object>
    ## attr(,"elementType")
    ## [1] "ANY"
    ## attr(,"elementMetadata")
    ## `\001NULL\001`
    ## attr(,"metadata")
    ## list()
    ##
    ## Slot "diagnostics":
    ## <0 x 0 matrix>
    ##
    ## Slot "bandle.joint":
    ## <0 x 0 matrix>

These can be populated as follows

    bandleres <- bandleProcess(bandleres)

These slots have now been populated

    summary(summaries(bandleres))

    ##      Length Class         Mode
    ## [1,] 1      bandleSummary S4
    ## [2,] 1      bandleSummary S4

### 6.1.1 `bandle` results

We can save the results by calling `summaries`. We see that it is of length 2. 1 for control and 1 for treatment.

    res <- summaries(bandleres)
    length(res)

    ## [1] 2

There are a number of slots,

    str(res[[1]])

    ## Formal class 'bandleSummary' [package "bandle"] with 3 slots
    ##   ..@ posteriorEstimates:Formal class 'DFrame' [package "S4Vectors"] with 6 slots
    ##   .. .. ..@ rownames       : chr [1:677] "P20353" "P53501" "Q7KU78" "P04412" ...
    ##   .. .. ..@ nrows          : int 677
    ##   .. .. ..@ elementType    : chr "ANY"
    ##   .. .. ..@ elementMetadata: NULL
    ##   .. .. ..@ metadata       : list()
    ##   .. .. ..@ listData       :List of 7
    ##   .. .. .. ..$ bandle.allocation               : Named chr [1:677] "PM" "Golgi" "Ribosome 40S" "PM" ...
    ##   .. .. .. .. ..- attr(*, "names")= chr [1:677] "P20353" "P53501" "Q7KU78" "P04412" ...
    ##   .. .. .. ..$ bandle.probability              : Named num [1:677] 1 0.992 1 0.951 0.992 ...
    ##   .. .. .. .. ..- attr(*, "names")= chr [1:677] "P20353" "P53501" "Q7KU78" "P04412" ...
    ##   .. .. .. ..$ bandle.outlier                  : num [1:677] 0 0.0667 0 0 0 ...
    ##   .. .. .. ..$ bandle.probability.lowerquantile: Named num [1:677] 1 0.976 1 0.904 0.975 ...
    ##   .. .. .. .. ..- attr(*, "names")= chr [1:677] "P20353" "P53501" "Q7KU78" "P04412" ...
    ##   .. .. .. ..$ bandle.probability.upperquantile: Named num [1:677] 1 0.998 1 0.976 0.998 ...
    ##   .. .. .. .. ..- attr(*, "names")= chr [1:677] "P20353" "P53501" "Q7KU78" "P04412" ...
    ##   .. .. .. ..$ bandle.mean.shannon             : Named num [1:677] 2.67e-06 1.18e-02 0.00 9.65e-03 0.00 ...
    ##   .. .. .. .. ..- attr(*, "names")= chr [1:677] "P20353" "P53501" "Q7KU78" "P04412" ...
    ##   .. .. .. ..$ bandle.differential.localisation: Named num [1:677] 0 0 0 0 0 0 0 0 0 0 ...
    ##   .. .. .. .. ..- attr(*, "names")= chr [1:677] "P20353" "P53501" "Q7KU78" "P04412" ...
    ##   ..@ diagnostics       : logi [1, 1] NA
    ##   ..@ bandle.joint      : num [1:677, 1:11] 2.53e-95 1.07e-167 1.96e-17 5.88e-124 7.57e-65 ...
    ##   .. ..- attr(*, "dimnames")=List of 2
    ##   .. .. ..$ : chr [1:677] "P20353" "P53501" "Q7KU78" "P04412" ...
    ##   .. .. ..$ : chr [1:11] "Cytoskeleton" "ER" "Golgi" "Lysosome" ...

The main one of interest is the `posteriorEstimates` slot,

    posteriorEstimates(res[[1]])

    ## DataFrame with 677 rows and 7 columns
    ##        bandle.allocation bandle.probability bandle.outlier
    ##              <character>          <numeric>      <numeric>
    ## P20353                PM           0.999997      0.0000000
    ## P53501             Golgi           0.991990      0.0666667
    ## Q7KU78     Ribosome 4...           0.999991      0.0000000
    ## P04412                PM           0.951037      0.0000000
    ## Q7KJ73     Ribosome 6...           0.991917      0.0000000
    ## ...                  ...                ...            ...
    ## Q95TL8                ER           1.000000              0
    ## P25007        Proteasome           0.999994              0
    ## P41374     Cytoskelet...           0.999989              0
    ## Q8SZM1        Peroxisome           0.810216              0
    ## P07909           Nucleus           0.999995              0
    ##        bandle.probability.lowerquantile bandle.probability.upperquantile
    ##                               <numeric>                        <numeric>
    ## P20353                         0.999994                         0.999998
    ## P53501                         0.975504                         0.997563
    ## Q7KU78                         0.999939                         1.000000
    ## P04412                         0.904319                         0.976460
    ## Q7KJ73                         0.974572                         0.998445
    ## ...                                 ...                              ...
    ## Q95TL8                         1.000000                         1.000000
    ## P25007                         0.999987                         0.999998
    ## P41374                         0.999963                         0.999998
    ## Q8SZM1                         0.354690                         0.977439
    ## P07909                         0.999960                         1.000000
    ##        bandle.mean.shannon bandle.differential.localisation
    ##                  <numeric>                        <numeric>
    ## P20353         2.67267e-06                                0
    ## P53501         1.17928e-02                                0
    ## Q7KU78         0.00000e+00                                0
    ## P04412         9.65289e-03                                0
    ## Q7KJ73         0.00000e+00                                0
    ## ...                    ...                              ...
    ## Q95TL8         0.00000e+00                         0.000000
    ## P25007         1.23894e-05                         0.000000
    ## P41374         6.47245e-06                         0.000000
    ## Q8SZM1         8.59218e-02                         0.733333
    ## P07909         0.00000e+00                         0.000000

This output object is a `data.frame` containing the protein allocations and associated localisation probabilities (including the upper and lower quantiles of the allocation probability distribution), the mean Shannon entropy and the `bandle.differential.localisation` probability.

## 6.2 Extracting posteriors and allocation results

We create two new objects `pe1` and `pe2` in the below code chunk which contain the output of the `posteriorEstimates` slot.

    pe1 <- posteriorEstimates(res[[1]])
    pe2 <- posteriorEstimates(res[[2]])

One quantity of interest is the protein allocations, which we can plot as a barplot.

    alloc1 <- pe1$bandle.allocation
    alloc2 <- pe2$bandle.allocation

    par(mfrow = c(1, 2), oma = c(6,2,2,2))
    barplot(table(alloc1), col = getStockcol()[2],
            las = 2, main = "Protein allocation: control")
    barplot(table(alloc2), col = getStockcol()[2],
            las = 2, main = "Protein allocation: treatment")

The barplot tells us for this example that `bandle` has allocated the majority of unlabelled proteins to the ER, followed by the Golgi (irrespective of the posterior probability).

The associated posterior estimates are located in the `bandle.probability` column.

    pe_alloc1 <- pe1$bandle.probability
    pe_alloc2 <- pe1$bandle.probability

## 6.3 Allocation probabilities

The full allocation probabilities are stored in the `tagm.joint` slot. These can be visualised in a heatmap

    bjoint_control <- bandleJoint(summaries(bandleres)[[1]])
    pheatmap(bjoint_control, cluster_cols = FALSE, color = viridis(n = 25))

    bjoint_treatment <- bandleJoint(summaries(bandleres)[[2]])
    pheatmap(bjoint_treatment, cluster_cols = FALSE, color = viridis(n = 25))

## 6.4 Predicting the subcellular location

We can append the results to our original `MSnSet` datasets using the `bandlePredict` function.

    xx <- bandlePredict(control,
                        treatment,
                        params = bandleres,
                        fcol = "markers")
    res_control <- xx[[1]]
    res_treatment <- xx[[2]]

The output is a `list` of `MSnSets`. In this example, we have 3 for the control and 3 for the treatment.

    length(res_control)

    ## [1] 3

    length(res_treatment)

    ## [1] 3

The results are appended to the first `MSnSet` feature data slot for each condition.

    fvarLabels(res_control[[1]])

    ##  [1] "FBgn"                             "Protein.ID"
    ##  [3] "Flybase.Symbol"                   "AccessionNo"
    ##  [5] "EntryName"                        "AccessionNoAll"
    ##  [7] "EntryNameAll"                     "No.peptide.IDs"
    ##  [9] "Mascot.score"                     "No.peptide.quantified"
    ## [11] "PLSDA"                            "pd.2013"
    ## [13] "pd.markers"                       "markers.orig"
    ## [15] "markers"                          "markers.tl"
    ## [17] "knn"                              "knn.scores"
    ## [19] "bandle.allocation"                "bandle.probability"
    ## [21] "bandle.probability.lowerquantile" "bandle.probability.upperquantile"
    ## [23] "bandle.mean.shannon"              "bandle.differential.localisation"
    ## [25] "bandle.outlier"                   "bandle.joint"

To access them use the `fData` function

    fData(res_control[[1]])$bandle.probability
    fData(res_control[[1]])$bandle.allocation

### 6.4.1 Thresholding on protein allocations

It is common practice in supervised machine learning to set a specific threshold on which to define new assignments/allocations, below which classifications are left unassigned/unknown. Indeed, we do not expect the whole subcellular diversity to be represented by the 11 niches defined here, we expect there to be many more, many of which will be multiply localised within the cell. It is important to allow for the possibility of proteins to reside in multiple locations (this information is available in the `bandle.joint` slot - see above for more details on multiple location).

As we are using a Bayesian model the outputs of the classifier are probabilities. This not only allows us to look at the distribution of probabilities over all subcellular classes but also allows us to extract a probability threshold on which we can define new assignments.

The subcellular allocations are located in the `bandle.allocation` column of the `fData` slot and the posteriors are located in the `bandle.probability` slot. We can use the `getPredictions` function from the `pRoloc` package to return a set of predicted localisations according to if they meet a probability threshold.

For example, in the below code chunk we set a 1% FDR for assigning proteins a subcellular nice, below which we leave them unlabelled.

    res_control[[1]] <- getPredictions(res_control[[1]],
                                       fcol = "bandle.allocation",
                                       scol = "bandle.probability",
                                       mcol = "markers",
                                       t = .99)

    ## ans
    ##  Cytoskeleton            ER         Golgi      Lysosome       Nucleus
    ##            13           221           134             8            39
    ##            PM    Peroxisome    Proteasome  Ribosome 40S  Ribosome 60S
    ##           134             9            43            75            50
    ## mitochondrion       unknown
    ##            81            81

    res_treatment[[1]] <- getPredictions(res_treatment[[1]],
                                       fcol = "bandle.allocation",
                                       scol = "bandle.probability",
                                       mcol = "markers",
                                       t = .99)

    ## ans
    ##  Cytoskeleton            ER         Golgi      Lysosome       Nucleus
    ##            18           193           123            13            41
    ##            PM    Peroxisome    Proteasome  Ribosome 40S  Ribosome 60S
    ##           136            14            45            79            49
    ## mitochondrion       unknown
    ##            87            90

We may also wish to take into account the probability of the protein being an outlier and thus use the results in the `bandle.outlier` column of the feature data. We could calculate the product of the posterior and the outlier (as they are both probabilities) to obtain a localisation score that takes into account the outlier model. More details on this are found in the second vignette of this package.

## 6.5 Differential localisation probability

As previously mentioned the term “differentially localised” is used to pertain to proteins which are assigned to different sub-cellular localisations between two conditions. For the majority of users this is the main output they are keen to extract using the BANDLE method.

Following on from the above example, after extracting posterior estimates for each condition using the `summaries` function we can also access the differential localisation probability as it is stored in the `bandle.differential.localisation` column of the `data.frames` of `pe1` and `pe2`, in the above sections.

The differential localisation probability tells us which proteins are most likely to differentially localise. We can for example, examine how many proteins get a differential probability greater than 0.99 to look for the most confident differentially localised candidates.

    diffloc_probs <- pe1$bandle.differential.localisation
    head(diffloc_probs, 50)

    ##     P20353     P53501     Q7KU78     P04412     Q7KJ73     Q9VM65     Q9VCK0
    ## 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
    ##     B7Z0W3     Q9V415     Q00174     Q9V769     Q27593     Q9V780     P19109
    ## 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
    ##     Q95NR4     O77047     Q8SXD0     Q9VBV5     Q9VBU5     Q8IA62     Q9Y105
    ## 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
    ##     Q9W2M4     P26308     Q9VP77     Q5U0Z2     Q960E2     Q9VLJ6     Q9VJ39
    ## 0.00000000 1.00000000 1.00000000 0.00000000 0.00000000 1.00000000 1.00000000
    ##     Q9VTX8     Q9VTZ5     B7Z0C1     Q9VRJ4     M9PCB7     P46150     A1ZBH5
    ## 0.00000000 0.26666667 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
    ##     Q9W3M7     A8DZ29     Q9VN14     Q9VZL3     M9PC99     Q86BP3     Q9W3N9
    ## 0.00000000 0.80000000 0.60000000 0.00000000 0.06666667 0.00000000 0.00000000
    ##     Q7K5M6     P16620     P48375     Q9VMB9     Q9VI55     Q9VU58     Q9V498
    ## 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.53333333 0.00000000
    ##     Q9V496
    ## 1.00000000

    length(which(diffloc_probs[order(diffloc_probs, decreasing = TRUE)] > 0.99))

    ## [1] 90

We find there are 90 proteins above this threshold.

This can also be seen on a rank plot

    plot(diffloc_probs[order(diffloc_probs, decreasing = TRUE)],
         col = getStockcol()[3], pch = 19, ylab = "Probability",
         xlab = "Rank", main = "Differential localisation rank plot")

In-line with our expectations, the rank plot indicates that most proteins are not differentially localised.

### 6.5.1 Estimating uncertainty

#### 6.5.1.1 Applying the `bootstrapdiffLocprob` function

We can examine the top `n` proteins (here we use an example of `top = 100`) and produce bootstrap estimates of the uncertainty (note here the uncertainty is likely to be underestimated as we did not produce many MCMC samples). These can be visualised as ranked boxplots.

    set.seed(1)
    boot_t <- bootstrapdiffLocprob(params = bandleres, top = 100,
                                   Bootsample = 5000, decreasing = TRUE)

    boxplot(t(boot_t), col = getStockcol()[5],
            las = 2, ylab = "Probability", ylim = c(0, 1),
            main = "Differential localisation \nprobability plot (top 100 proteins)")

#### 6.5.1.2 Applying the `binomDiffLoc` function

Instead of applying the `bootstrapdiffLocprob` we could use the `binomDiffLoc` function to obtain credible intervals from the binomial distribution.

    bin_t <- binomialDiffLocProb(params = bandleres, top = 100,
                                 nsample = 5000, decreasing = TRUE)

    boxplot(t(bin_t), col = getStockcol()[5],
            las = 2, ylab = "Probability", ylim = c(0, 1),
            main = "Differential localisation \nprobability plot (top 100 proteins)")

#### 6.5.1.3 Obtaining probability estimates

There are many ways we could obtain probability estimates from either of the above methods. We could, for example, take the mean of each protein estimate, or compute the cumulative error (there is not really a false discovery rate in Bayesian statistics) or we could threshold on the interval to reduce the number of differential localisations if you feel the model has been overconfident.

    # more robust estimate of probabilities
    dprobs <- rowMeans(bin_t)

    # compute cumulative error, there is not really a false discovery rate in
    # Bayesian statistics but you can look at the cumulative error rate
    ce <- cumsum(1  - dprobs)

    # you could threshold on the interval and this will reduce the number of
    # differential localisations
    qt <- apply(bin_t, 1, function(x) quantile(x, .025))

#### 6.5.1.4 The expected false discovery rate

Instead of estimating the false discovery rate we can estimate the expected false discovery rate from the posterior probabilities at a particular threshold. This mean that for fixed threshold, we compute the expected proportion of false discoveries. Here is an example below. We can see that setting a probability threshold of 0.9 leads to an expected false discovery rate of less than \(0.5\%\)

    EFDR(diffloc_probs, threshold = 0.90)

    ## [1] 0.002150538

# 7 Visualising differential localisation

We can visualise the changes in localisation between conditions on an alluvial plot using the `plotTranslocations` function

    plotTranslocations(bandleres)

Or alternatively, on a chord (circos) diagram

    plotTranslocations(bandleres, type = "chord")

### 7.0.1 Important consideration

These visualisations are showing the change in class label between the two conditions (as assigned by `bandle` i.e. the result stored in `bandle.allocation`). The results are taken directly from `bandleres` and thus no thresholding on the class label and posterior to allow for proteins to be left “unassigned” or unknown, is conducted. Furthermore, there is no thresholding on the `bandle.differential.localisation` probability.

It would be better to re-plot these figures with some thresholds on the above quantities, to get a better representation of what is moving. The easiest way to do this is to pass the `MSnSets` output after performing `bandlePredict` and `getPredictions`.

For example, first let’s identify which proteins get a high differential localisation probability,

    ## identify which proteins get a high differential localisation probability
    ind <- which(fData(res_control[[1]])$bandle.differential.localisation > 0.99)

    ## create two new MSnSets with only these proteins
    res_dl_control <- res_control[[1]][ind, ]
    res_dl_treatment <- res_treatment[[1]][ind, ]

Now we can plot only these 90 proteins which are deemed to move/differentially localise. We also specify where the prediction results are located e.g. `fcol = "bandle.allocation.pred"`.

    ## specify colour palette
    mycols <- c(getStockcol()[seq(getMarkerClasses(res_control[[1]]))], "grey")
    names(mycols) <- c(getMarkerClasses(res_control[[1]]), "unknown")

    ## Create a list of the datasets for plotTranslocations
    res <- list(res_dl_control, res_dl_treatment)

    plotTranslocations(res, fcol = "bandle.allocation.pred", col = mycols)

    ## 90 features in common

    ## ------------------------------------------------
    ## If length(fcol) == 1 it is assumed that the
    ## same fcol is to be used for both datasets
    ## setting fcol = c(bandle.allocation.pred,bandle.allocation.pred)
    ## ----------------------------------------------

We can also use the function `plotTable` to display a summary table of the number of proteins that have changed in location between conditions.

    (tbl <- plotTable(res, fcol = "bandle.allocation.pred"))

    ##        Condition1    Condition2 value
    ## 1              ER         Golgi     1
    ## 2              ER       Nucleus     1
    ## 3              ER            PM     3
    ## 4              ER    Peroxisome     7
    ## 5              ER    Proteasome     4
    ## 6              ER  Ribosome 40S     5
    ## 7              ER  Ribosome 60S     1
    ## 9              ER       unknown     5
    ## 10             ER  Cytoskeleton     2
    ## 11             ER      Lysosome     2
    ## 12          Golgi            ER     1
    ## 13          Golgi       Nucleus     3
    ## 14          Golgi            PM     1
    ## 18          Golgi  Ribosome 60S     2
    ## 19          Golgi mitochondrion     1
    ## 25        Nucleus            PM     1
    ## 26        Nucleus    Peroxisome     1
    ## 35             PM         Golgi     1
    ## 38             PM    Proteasome     2
    ## 39             PM  Ribosome 40S     4
    ## 43             PM  Cytoskeleton     1
    ## 44             PM      Lysosome     1
    ## 46     Peroxisome         Golgi     2
    ## 56     Proteasome            ER     1
    ## 59     Proteasome            PM     2
    ## 61     Proteasome  Ribosome 40S     2
    ## 63     Proteasome mitochondrion     2
    ## 64     Proteasome       unknown     2
    ## 68   Ribosome 40S         Golgi     1
    ## 69   Ribosome 40S       Nucleus     1
    ## 70   Ribosome 40S            PM     1
    ## 74   Ribosome 40S mitochondrion     1
    ## 75   Ribosome 40S       unknown     1
    ## 76   Ribosome 40S  Cytoskeleton     1
    ## 77   Ribosome 40S      Lysosome     2
    ## 83   Ribosome 60S    Proteasome     2
    ## 84   Ribosome 60S  Ribosome 40S     2
    ## 85   Ribosome 60S mitochondrion     1
    ## 92  mitochondrion            PM     1
    ## 94  mitochondrion    Proteasome     1
    ## 97  mitochondrion       unknown     1
    ## 100       unknown            ER     1
    ## 101       unknown         Golgi     2
    ## 103       unknown            PM     1
    ## 105       unknown    Proteasome     2
    ## 106       unknown  Ribosome 40S     1
    ## 107       unknown  Ribosome 60S     1
    ## 108       unknown mitochondrion     1
    ## 109       unknown  Cytoskeleton     1

# 8 Assessing convergence

In this section, we demonstrate how to visually assess convergence of the MCMC algorithm. In the chunk below, we use 4 chains so that we can assess the convergence of the method.

    control <- tansim$lopitrep[1:3]
    treatment <- tansim$lopitrep[4:6]

    bandleres <- bandle(objectCond1 = control,
                        objectCond2 = treatment,
                        numIter = 50, # usually 10,000
                        burnin = 10L, # usually 5,000
                        thin = 1L, # usually 20
                        gpParams = gpParams,
                        pcPrior = pc_prior,
                        numChains = 4,
                        dirPrior = dirPrior)

We then use the `plotConvergence` function which plots the ranks of the total number of outliers within in each of the chains. If convergence has been reached these plots should be uniform. As we can see these plots are not uniform and are skewed towards the extremes, so they have not converged. Clearly one of the chains has higher values on average than the other chains producing the skew. Running the algorithm for more iterations (typically 10,000) should produce convergence. This is not done here for brevity of the vignette.

    par(mfrow = c(2, 2))
    out <- plotConvergence(bandleres)

# 9 Description of `bandle` parameters

The `bandle` function has a significant number of parameters to allow flexible and bespoke analysis. Here, we describe these parameters in more detail to allow user to make decisions on the level of flexibility they wish to exploit.

1. `objectCond1`. This is a list of `MSnSets` containing the first condition.

2. `objectCond2`. This is a list of `MSnSets` containing the second condition.

   1. These object should have the same observations and features. These will be checked during bandle analysis.

3. `fcol` indicates the feature column in the `MSnSets` that indicated the markers. Proteins that are not markers should be labels `unknown`. The default is `markers`.

4. `hyperLearn` is the algorithm used to learn the hyperparameters of the Gaussian processes. For speed the default is an optimization algorithm called “LBFGS”, however is users want to perform uncertainty quantification on these parameters we can use Markov-chain Monte Carlo (MCMC) methods. This is implemented using the Metropolis-Hastings algorithm. Though this latter methodology provides more information, it is much more costly. The analysis is expected to take several days rather than hours.

5. `numIter` is the number of MCMC iterations for the algorithm. We typically suggest around 10,000 iterations is plenty for convergence. Though some cases may take longer. If resources are constrained, we suggest 4,000 iterations as acceptable. A minimum number of iterations is around 1,000 though at this level we expect the posterior estimates to suffer considerably. If possible more parallel chains should be run in this case by changing `numChains` to, say, 9. The more chains and iterations the more computationally expensive the algorithm. The time taken for the algorithm scales roughly linearly in the number of iterations

6. `burnin` is the number of samples that should be discarded from the beginning of the chain due to the bias induced by the starting point of the algorithm. We suggest sensible `burnin` values to be roughly \(10-50\%\) of the number of iterations

7. `thin` reduces auto-correlation in the MCMC samples. The default is \(5\), which means every 5th sample is taken. If memory requirements are an issue, we suggest to increase the thinning amount. Though above \(20\), you will see a decrease in performance.

8. `u` and `v` represent the prior hyperparameters of the proportion of outliers. This is modelled using a `Beta(u,v)` with `u = 2` and `v = 10` a default. This suggest that roughly \(\frac{u}{u = V} = 16%\) of proteins are believed to be outliers and that it is quite unlikely that more than \(50%\) of proteins are outliers. Users can examine the quantiles of the `Beta(u,v)` distribution if they wish to place a more bespoke prior. For example, increasing `u` will increase the number of a prior believed outliers.

9. `lambda` is a ridge parameter used for numerical stability and is set to \(0.01\). If you experience the algorithm fails due to numerical issue then you can set this value larger. If you require values above \(1\) it is likely that there are other issues with the analysis. We suggest checking the method is appropriate for your problem and opening issue detailing the problems.

10. `gpParams` results from fitting Gaussian proccess (Gaussian random fields). We refer the users to those functions. The default is `NULL` which will fit GPs internally but we recommend setting these outside the bandle function because it leads to more stable results.

11. `hyperIter` if the hyperparameters of the GP are learnt using MH algorithm then this is the frequency at which these are updated relative to the bandle algorithm. By default this is unused, but if `hyperLearn` is set to `MH` then this proceed at every 20 iterations.

12. `hyperMean` is the mean of the log normal prior used on the hyperparameters. Though by default this is not used unless `PC` is set to false

13. `hyperSd` is the standard deviation of the log normal prior used on the hyperparameters. The default is `c(1,1,1)` for the 3 hyperparameters, increasing these values increases the uncertainty in the prior values of the hyperparameters.

14. `seed` is the random-number seed.

15. `pg` indicates whether or not to use the Polya-Gamma (PG) prior. The default is false and a Dirichlet prior is used instead. If set to true the `pg` is used. In which case a default PG prior is used. This prior attempts to match the default Dirichlet prior that is used when PG prior is set to false. The PG prior is more computationally expensive but can provide prior information on correlations

16. `pgPrior` is by default NULL. We suggest using the `pg_prior` function to help set this parameter and the documentation therein. This function uses an empirical approach to compute a sensible default.

17. `tau` is a parameter used by the Polya-Gamma prior and we refer to BANDLE manuscript for details. By default it is only used if `pg` prior is true, when the default becomes `0.2`. At this value the `pg` prior is similar to the Dirichlet prior but with information on correlations.

18. `dirPrior` is the Dirichlet matrix prior on the correlations. This should be provided as a K by K matrix, where K is the number of subcellular niches. The diagonal component should represent the prior belief that organelles do not re-localise (same compartment), where as the off-diagonal terms represent the prior terms of re-localisation. The `prior_pred_dir` can be used to provide a prior predictive check based on the provided prior. It is recommended that the off-diagonal terms are at least two orders of magnitude smaller than the diagonal terms. An example is given in the vignette.

19. `maternCov` is this true the covariance function is the matern covariance, otherwise a Gaussian covariance is used.

20. `PC` indicates whether a penalised complexity (PC) is used. The default is true and otherwise log normal priors are used.

21. `pcPrior` is a numeric of length 3 indicating the parameters of the PC prior. The prior is placed on the parameters of length-scale, amplitude, and variance in that order. The default values are \(0.5,3,100\), and increasing the value increases the shrinkage towards straight-lines with zero variance.

22. `nu` which defaults to 2 is the smoothness of the matern covariance. By increasing `nu` you encourage smoother solutions. `nu` should be an integer, though for values of `nu` above 3, we have observed numerical instability.

23. `propSd` is the standard deviation of the random-walk update used in the MH algorithm. We do not recommend changing this unless you are familiar with Bayesian analysis. The default is `c(0.3,0.1,0.05)` for the 3 hyperparameters. Changing these will alter the efficiency of the underlying samplers.

24. `numChains` is the number of parrallel chains and defaults to 4. We recommend using as much processing resources as you have and frequently have used 9 in practise.

25. `BPPARAM` is the BiocParallel back-end which defaults to `BiocParallel::bpparam()`. We refer you to the BiocParallel package for details on setting this dependent on your computing system.

# 10 Session information

All software and respective versions used to produce this document are listed below.

    sessionInfo()

    ## R version 4.4.1 (2024-06-14 ucrt)
    ## Platform: x86_64-w64-mingw32/x64
    ## Running under: Windows Server 2022 x64 (build 20348)
    ##
    ## Matrix products: default
    ##
    ##
    ## locale:
    ## [1] LC_COLLATE=C
    ## [2] LC_CTYPE=English_United States.utf8
    ## [3] LC_MONETARY=English_United States.utf8
    ## [4] LC_NUMERIC=C
    ## [5] LC_TIME=English_United States.utf8
    ##
    ## time zone: America/New_York
    ## tzcode source: internal
    ##
    ## attached base packages:
    ## [1] stats4    stats     graphics  grDevices utils     datasets  methods
    ## [8] base
    ##
    ## other attached packages:
    ##  [1] pRolocdata_1.44.0    viridis_0.6.5        viridisLite_0.4.2
    ##  [4] pheatmap_1.0.12      bandle_1.10.0        pRoloc_1.46.0
    ##  [7] BiocParallel_1.40.0  MLInterfaces_1.86.0  cluster_2.1.6
    ## [10] annotate_1.84.0      XML_3.99-0.17        AnnotationDbi_1.68.0
    ## [13] IRanges_2.40.0       MSnbase_2.32.0       ProtGenerics_1.38.0
    ## [16] mzR_2.40.0           Rcpp_1.0.13-1        Biobase_2.66.0
    ## [19] S4Vectors_0.44.0     BiocGenerics_0.52.0  BiocStyle_2.34.0
    ##
    ## loaded via a namespace (and not attached):
    ##   [1] splines_4.4.1               filelock_1.0.3
    ##   [3] tibble_3.2.1                hardhat_1.4.0
    ##   [5] preprocessCore_1.68.0       pROC_1.18.5
    ##   [7] rpart_4.1.23                lifecycle_1.0.4
    ##   [9] httr2_1.0.6                 doParallel_1.0.17
    ##  [11] globals_0.16.3              lattice_0.22-6
    ##  [13] MASS_7.3-61                 MultiAssayExperiment_1.32.0
    ##  [15] dendextend_1.18.1           magrittr_2.0.3
    ##  [17] limma_3.62.1                plotly_4.10.4
    ##  [19] sass_0.4.9                  rmarkdown_2.29
    ##  [21] jquerylib_0.1.4             yaml_2.3.10
    ##  [23] MsCoreUtils_1.18.0          DBI_1.2.3
    ##  [25] RColorBrewer_1.1-3          lubridate_1.9.3
    ##  [27] abind_1.4-8                 zlibbioc_1.52.0
    ##  [29] GenomicRanges_1.58.0        purrr_1.0.2
    ##  [31] mixtools_2.0.0              AnnotationFilter_1.30.0
    ##  [33] nnet_7.3-19                 rappdirs_0.3.3
    ##  [35] ipred_0.9-15                circlize_0.4.16
    ##  [37] lava_1.8.0                  GenomeInfoDbData_1.2.13
    ##  [39] ggrepel_0.9.6               listenv_0.9.1
    ##  [41] gdata_3.0.1                 parallelly_1.38.0
    ##  [43] ncdf4_1.23                  codetools_0.2-20
    ##  [45] DelayedArray_0.32.0         xml2_1.3.6
    ##  [47] shape_1.4.6.1               tidyselect_1.2.1
    ##  [49] farver_2.1.2                UCSC.utils_1.2.0
    ##  [51] matrixStats_1.4.1           BiocFileCache_2.14.0
    ##  [53] jsonlite_1.8.9              caret_6.0-94
    ##  [55] e1071_1.7-16                ggalluvial_0.12.5
    ##  [57] survival_3.7-0              iterators_1.0.14
    ##  [59] foreach_1.5.2               segmented_2.1-3
    ##  [61] tools_4.4.1                 progress_1.2.3
    ##  [63] snow_0.4-4                  lbfgs_1.2.1.2
    ##  [65] glue_1.8.0                  prodlim_2024.06.25
    ##  [67] gridExtra_2.3               SparseArray_1.6.0
    ##  [69] xfun_0.49                   MatrixGenerics_1.18.0
    ##  [71] GenomeInfoDb_1.42.0         dplyr_1.1.4
    ##  [73] withr_3.0.2                 BiocManager_1.30.25
    ##  [75] fastmap_1.2.0               fansi_1.0.6
    ##  [77] digest_0.6.37               timechange_0.3.0
    ##  [79] R6_2.5.1                    colorspace_2.1-1
    ##  [81] gtools_3.9.5                lpSolve_5.6.21
    ##  [83] biomaRt_2.62.0              RSQLite_2.3.7
    ##  [85] utf8_1.2.4                  tidyr_1.3.1
    ##  [87] generics_0.1.3              hexbin_1.28.4
    ##  [89] data.table_1.16.2           recipes_1.1.0
    ##  [91] FNN_1.1.4.1                 class_7.3-22
    ##  [93] prettyunits_1.2.0           PSMatch_1.10.0
    ##  [95] httr_1.4.7                  htmlwidgets_1.6.4
    ##  [97] S4Arrays_1.6.0              ModelMetrics_1.2.2.2
    ##  [99] pkgconfig_2.0.3             gtable_0.3.6
    ## [101] timeDate_4041.110           blob_1.2.4
    ## [103] impute_1.80.0               XVector_0.46.0
    ## [105] htmltools_0.5.8.1           bookdown_0.41
    ## [107] MALDIquant_1.22.3           clue_0.3-65
    ## [109] scales_1.3.0                png_0.1-8
    ## [111] gower_1.0.1                 knitr_1.48
    ## [113] reshape2_1.4.4              coda_0.19-4.1
    ## [115] nlme_3.1-166                curl_6.0.0
    ## [117] GlobalOptions_0.1.2         proxy_0.4-27
    ## [119] cachem_1.1.0                stringr_1.5.1
    ## [121] parallel_4.4.1              mzID_1.44.0
    ## [123] vsn_3.74.0                  pillar_1.9.0
    ## [125] grid_4.4.1                  vctrs_0.6.5
    ## [127] pcaMethods_1.98.0           randomForest_4.7-1.2
    ## [129] dbplyr_2.5.0                xtable_1.8-4
    ## [131] evaluate_1.0.1              magick_2.8.5
    ## [133] tinytex_0.54                mvtnorm_1.3-2
    ## [135] cli_3.6.3                   compiler_4.4.1
    ## [137] rlang_1.1.4                 crayon_1.5.3
    ## [139] future.apply_1.11.3         LaplacesDemon_16.1.6
    ## [141] mclust_6.1.1                QFeatures_1.16.0
    ## [143] affy_1.84.0                 plyr_1.8.9
    ## [145] stringi_1.8.4               munsell_0.5.1
    ## [147] Biostrings_2.74.0           lazyeval_0.2.2
    ## [149] Matrix_1.7-1                hms_1.1.3
    ## [151] bit64_4.5.2                 future_1.34.0
    ## [153] ggplot2_3.5.1               KEGGREST_1.46.0
    ## [155] statmod_1.5.0               highr_0.11
    ## [157] SummarizedExperiment_1.36.0 kernlab_0.9-33
    ## [159] igraph_2.1.1                memoise_2.0.1
    ## [161] affyio_1.76.0               bslib_0.8.0
    ## [163] sampling_2.10               bit_4.5.0

# References

Crook, Oliver M, Colin TR Davies, Laurent Gatto, Paul DW Kirk, and Kathryn S Lilley. 2021. “Inferring Differential Subcellular Localisation in Comparative Spatial Proteomics Using Bandle.” bioRxiv. doi: https://doi.org/10.1101/2021.01.04.425239.

Gatto, Laurent, Lisa M. Breckels, Samuel Wieczorek, Thomas Burger, and Kathryn S. Lilley. 2014. “Mass-Spectrometry Based Spatial Proteomics Data Analysis Using pRoloc and pRolocdata.” Bioinformatics.

Gentleman, Robert C., Vincent J. Carey, Douglas M. Bates, Ben Bolstad, Marcel Dettling, Sandrine Dudoit, Byron Ellis, et al. 2004. “Bioconductor: Open Software Development for Computational Biology and Bioinformatics.” Genome Biol 5 (10): –80. https://doi.org/10.1186/gb-2004-5-10-r80.

Gilks, Walter R, Sylvia Richardson, and David Spiegelhalter. 1995. Markov Chain Monte Carlo in Practice. CRC press.

R Development Core Team. 2011. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. http://www.R-project.org/.

Tan, Denise JL, Heidi Dvinge, Andrew Christoforou, Paul Bertone, Alfonso Martinez Arias, and Kathryn S Lilley. 2009. “Mapping Organelle Proteins and Protein Complexes in Drosophila Melanogaster.” Journal of Proteome Research 8 (6): 2667–78.