1.1 Requirements - Hardware

• A 32 or 64 bit processor (recommended: 64bit)
• 1GHz processor (recommended: multicore for a comprehensive analysis)
• 2GB MB RAM (recommended: 4GB+, depends on the size of experimental data)

1.2 Requirements - Software

The main software requirement is the installation of the R environment (version: >= 3.2), which can be downloaded from R project website and is distributed for all common operating systems. We tested the package in R environment installed on Windows 7, 10; Mac OS X 10.11 - 10.13 and Ubuntu 18.04 with no significant differences in the performance. The use of a dedicated Integrated development environment (IDE), e.g. posit is recommended.

Apart from a base installation of R, SLEMI requires the following R packages:

1. for installation

• devtools

2. for estimation

• e1071
• Hmisc
• nnet
• glmnet
• caret
• doParallel (if parallel computation are needed)

3. for visualisation

• ggplot2
• ggthemes
• gridExtra
• corrplot

4. for data handling

• reshape2
• stringr
• plyr

Each of the above packages can be installed by executing

install.packages("name_of_a_package")

in the R console.

Importantly, during installation availability of the above packages will be verified and missing packages will be automatically installed.

1.3 Installation

The package can be directly installed from GitHub. For installation, open RStudio (or base R) and run following commands in the R console

install.packages("devtools") # run if 'devtools' is not installed
library(devtools)
install_github("sysbiosig/SLEMI")

Are required packages not found, they will be installed automatically.

1.4 Citing and support

The package implements methods published in PLOS Computational Biology, please cite:

Jetka T, Nienałtowski K, Winarski T, Błoński S, Komorowski M (2019) Information-theoretic analysis of multivariate single-cell signaling responses. PLoS Comput Biol 15(7): e1007132. https://doi.org/10.1371/journal.pcbi.1007132

All problems, issues and bugs can be reported here:

https://github.com/sysbiosig/SLEMI/issues

or directly via e-mail: t.jetka a t gmail.com.

2.1 Formulation of the problem

SLEMI package is designed to estimate information-theoretic measures between a discrete-valued input, \(X\), and multivariate, continuous output, \(Y\). In a typical experiment aimed to quantify information flow a given signaling system, input values \(x_1\leq x_2 \ldots... \leq x_m\), ranging from 0 to saturation are considered.

Then, for each input level, \(x_i\),\(n_i\) observations are collected, which are represented as vectors \[y^i_j \sim P(Y|X = x_i)\] Within information theory the degree of information transmission is measured as the mutual information \[MI(X,Y) = \sum_{i=1}^{m} P(x_i)\int_{R^k} P(y|X = x_i)log_2\frac{P(y|X = x_i)}{P(y)}dy\] where \(P(y)\) is the marginal distribution of the output. MI is expressed in bits and \(2^{MI}\) can be interpreted as the number of inputs that the system can resolve on average.

The maximization of mutual information with respect to the input distribution, \(P(X)\), defines the information capacity, \(C^*\). Formally, \[C^* = max_{P(X)} MI(X,Y)\] Information capacity is expressed in bits and \(2^{C^*}\) can be interpreted as the maximal number of inputs that the system can effectively resolve. For details regarding information theory or its application in systems biology please see Methods section and Supplementary Information of the corresponding paper (Jetka et al. 2019).

2.2 Input data

Functions mi_logreg_main(), capacity_logreg_main(), prob_discr_pairwise() require data in the form of the object data.frame with a specific structure of rows and columns. Responses \(y^i_j\) are assumed to be measured for a finite set of stimuli levels \(x_1,x_2,\ldots,x_m\). The responses \(y^i_j\) can be multidimensional. Usually, experimental dataset is represented as a table with rows and columns organized as shown in Figure 2.

Therefore, the input data frame is expected to have the form represented by the above table, which can be formally described by the following conditions

• each row represent a response of a single cell
• first column contains values of the input (X).
• second and subsequent columns contain values of the measured output(s); these columns should be of type numeric; order and number of outputs should be the same for all cells.
• the number of unique values of the input should be finite
• a large number of observations, possibly >100, per input value is required.

An example of the input data.frame, which contains the measurements of the NfkB system presented in the MP is available within the package under the variable data_nfkb. It has the following format

where each row represents measurements of a single-cell, the column named signal specifies the level of stimulation, while response_T is the response of the NfkB system in an individual cell at time point T. The above table can be shown in R by calling

library(SLEMI)
rbind(data_nfkb[1:3,1:4],data_nfkb[10001:10003,1:4],tail(data_nfkb[,1:4],3))

2.3 Calculation of the information capacity

Calculation of the information capacity with default settings is performed by the command

capacity_logreg_main(dataRaw, signal, response, output_path)

where the required arguments are

• dataRaw - data frame with column of type factor containing values of input (X) and columns of type numeric containing values of output (Y), where each row represents a single observation
• signal - a character which indicates the name of the column in dataRaw with values of input (X)
• response - a character vector which indicates names of columns in dataRaw with values of output (Y)
• output_path - a character with the directory, to which output should be saved

The function returns a list with the following elements

• cc - a numeric scalar with channel capacity estimate (in bits)
• p_opt - a numeric vector with the optimal input distribution
• model - a nnet object describing fitted logistic regression model
• data - a data.frame with the raw experimental data (if data_out=TRUE)
• time - processing time of the algorithm
• params - a vector of parameters used in the algorithm
• regression - a confusion matrix of logistic regression predictions

By default, all returned elements are saved in output_path directory in a file output.rds. Along with the output data, results of the computations are visualised as the graphs listed below

• MainPlot.pdf - a simple summary plot with basic distribution visualization and capacity estimate
• capacity.pdf - a diagram presenting the capacity estimates
• data_boxplots.pdf - boxplots of data
• data_MeanViolin.pdf - violin plots of data with input-output relation curve (of means)

2.4 Calculation of the mutual information

The function mi_logreg_main() takes a similar list of arguments and generates analogous plots to the function capacity_logreg_main(). The differences are listed below.

Firstly, user must specify the distribution of input that should be used for calculation of the mutual information. It is done by passing a numeric vector via the argument pinput of mi_logreg_main() function. Secondly, the returned list stores the value of the computed mutual information (in bits) under the element mi.

2.5 Calculation of the probabilities of correct discrimination

Calculation of the probabilities of correct discrimination between pairs of input values is performed by running the following command

prob_discr_pairwise(dataRaw, signal, response, output_path)

where the required arguments are analogous to the arguments of the functions capacity_logreg_main() and mi_logreg_main(). The probabilities of correct discrimination are computed for each pair of unique input values and returned as a list with the following elements

• prob_matr - a symmetric numeric matrix with a probability of discriminating between \(i\)-th and \(j\)-th input values in cell (i,j)
• diagnostics - a list of summaries describing fitted logistic regression models of classification between each pair of input values.

In addition, a plot of corresponding pie charts is created in output_path in the pdf format.

5.1 Minimal example

Below, we present a minimal model that may serve as a quick introduction to computations within the package. Precisely, we consider a system

1. with four different input values \(X\): 0, 0.1, 1 and 10
2. with the conditional output, \(Y|X=x\), give by a one-dimensional log-normal distribution \(\exp\{\mathcal{N}(10\cdot\frac{x}{1+x},1)\}\)
3. and the sample consisting of 1000 observations for each input value.

The example is analogous to the Test scenario 2 of the Supplementary Information of (Jetka et al. 2019) (Section 3.2).

Input data

Firstly, we generate a a synthetic dataset. The data corresponding to the model can be generated, and represented as the data frame tempdata with columns input and output, by running

xs=c(0,0.1,1,10) # concentration of input.
tempdata = data.frame(input = factor(c(t(replicate(1000,xs))),
                      levels=xs),
                      output =  c(matrix(rnorm(4000, mean=10*(xs/(1+xs)),sd=c(1,1,1,1)),
                                            ncol=4,byrow=TRUE) ))

The generated data.frame has the following structure

Calculation of the information capacity

The Information capacity can be calculated using the capacity_logreg_main() function that takes the data frame “tempdata” as dataRaw argument. Column names “input” and “output” are used as arguments signal and response, respectively. The output_path is set as “minimal_example/”. Therefore, the function is run as follows

tempoutput  <- capacity_logreg_main(dataRaw=tempdata, 
                                    signal="input", response="output", 
                                    output_path="minimal_example/")

Results of the computations are returned as a data structure described before. In addition, results are presented in the form of the following graph (by default saved as MainPlot.pdf in minimal_example/ directory). It represents the input-output data and gives the corresponding channel capacity.

Calculation of the mutual information

To compare mutual information of experimental data with its channel capacity, we can run (uniform distribution of input values is assumed, as default)

tempoutput_mi  <- mi_logreg_main(dataRaw=tempdata, 
                                    signal="input", response="output", 
                                    output_path="minimal_exampleMI/",
                                    pinput=rep(1/4,4)) 

and display results

print(paste("Mutual Information:", tempoutput_mi$mi,"; ",
            "Channel Capacity:", tempoutput$cc, sep=" "))

## [1] "Mutual Information: 1.48828226407725 ;  Channel Capacity: 1.54565042478486"

Alternatively, the distribution of the input can be defined with probabilities \((0.4,0.1,0.4,0.1)\)

tempoutput_mi  <- mi_logreg_main(dataRaw=tempdata, 
                                    signal="input", response="output", 
                                    output_path="minimal_exampleMI/",
                                    pinput=rc(0.4,0.1,0.4,0.1)) 

and display results

print(paste("Mutual Information:", tempoutput_mi$mi,"; ",
            "Channel Capacity:", tempoutput$cc, sep=" "))

## [1] "Mutual Information: 1.33866963524711 ;  Channel Capacity: 1.54565042478486"

Calculation of the probabilities of correct discrimination

Probabilities of correct discrimination between input values are calculated as follows

tempoutput_probs  <- prob_discr_pairwise(dataRaw=tempdata, 
                                    signal="input", response="output", 
                                    output_path="minimal_exampleProbs/") 

The above command generates graph shown in Figure 4 in the output directory

Diagnostics

The diagnostic test can be performed as follows

dir.create("example1_testing/")
outputCLR=capacity_logreg_main(dataRaw=data_example1,
                            signal="signal",response="response",
                            output_path="example1_testing/",
                            testing=TRUE, TestingSeed = 1234, testing_cores = 4, 
                            boot_num = 40, boot_prob = 0.8, 
                            traintest_num = 40,partition_trainfrac = 0.6)

It will run diagnostics with 40 re-sampling of the data, where bootstrap is calculated using 80% of the data, while the over-fitting test uses 60% of the original dataset.

Its results are provided in graph presented in Figure 5.

The top diagram shows the value of the capacity estimate (in black) obtained from the complete dataset and the mean value of bootstrap repetitions with indicated +/- standard deviation (in red). Plots that follow show histograms of calculated capacities for different diagnostic regimes. The black dot represents the estimate of the channel capacity based on the complete dataset. In addition, corresponding empirical p-values of both tests (left- and right-sided) are calculated to assess the randomness of obtained results (PV in the plots).

A reliable estimation of the information capacity should yield the following results of the bootstrap and overfitting tests.

1. The bootstrap test should yield distribution of the capacity estimates with small variance. In addition, the capacity estimated based on the complete dataset should not be an outlier (p-value>0.05). Otherwise, it would indicate that the sample size is too low for an accurate estimation of the channel capacity.

2. The over-fitting test should provide similar results. The capacity estimate obtained based on the complete dataset should lie within the distribution of capacities generated in the test. In the opposite case, it could mean that the logistic regression model does not fully grasp the essential aspects of input-output dependencies in the data.

5.2 Further step-by-step introductory examples

Two step-by-step examples that further illustrate the applicability of the SLEMI package are provided in the Section 6 of the ‘Testing procedures’ pdf file that is added to the publication (Jetka et al. 2019) and can be found here.

5.3 Examples in paper

To reproduce results of the NFkB analysis presented in the publication, see Section 7 of the ‘Testing procedures’ pdf file that is added to the publication (Jetka et al. 2019) and can be found here.