# Introduction to magrene

Fabricio Almeida-Silva1,2 and Yves Van de Peer1,2,3,4

1VIB-UGent Center for Plant Systems Biology, Ghent, Belgium
2Department of Plant Biotechnology and Bioinformatics, Ghent University, Ghent, Belgium
3College of Horticulture, Academy for Advanced Interdisciplinary Studies, Nanjing Agricultural University, Nanjing, China
4Center for Microbial Ecology and Genomics, Department of Biochemistry, Genetics and Microbiology, University of Pretoria, Pretoria, South Africa

#### 2023-04-25

# Contents

# 1 Introduction

Network motifs are the building blocks of complex networks, and they can be interpreted as small genetic circuits. Identifying and counting motifs in gene regulatory networks can reveal important aspects of the evolution of transcriptional regulation. In particular, they can be used to explore the impact of gene duplication in the rewiring of regulatory interactions (Mottes et al. 2021). `magrene` aims to identify and analyze motifs in (duplicated) gene regulatory networks to better comprehend the role of gene duplication in network evolution. The figure below shows the networks motifs users can identify with `magrene`.

(#fig:intro_motifs)Network motifs and functions to identify them. Shaded boxes indicate paralogs. Regulators and targets are indicated in purple and green, respectively. Arrows indicate directed regulatory interactions, while dashed lines indicate protein-protein interaction.

# 2 Installation

You can install `magrene` with:

    if (!requireNamespace("BiocManager", quietly = TRUE)) {
          install.packages("BiocManager")
      }

    BiocManager::install("magrene")

Then, you can load the package with:

    library(magrene)
    set.seed(123) # for reproducibility

# 3 Data description

For this vignette, we will use three example data sets:

1. gma_grn: A gene regulatory network inferred with `BioNERO` (Almeida-Silva and Venancio 2022) using soybean RNA-seq data from the Soybean Expression Atlas (Machado et al. 2020).

2. gma_ppi: A protein-protein interaction (PPI) network for soybean obtained from the STRING database (Szklarczyk et al. 2021), filtered to contain only physical interactions with confidence score > 0.4.

3. gma_paralogs: Soybean paralogous gene pairs derived by whole-genome, tandem, proximal, transposed, and dispersed duplications (WGD, TD, PD, TRD, and DD, respectively). This data set was obtained from Almeida-Silva et al. (2020).

Networks are represented as edge lists. Let’s take a look at the data.

    data(gma_grn)
    head(gma_grn)
    #>                  Node1           Node2     Weight
    #> 615572 Glyma.20G042000 Glyma.13G027300 0.09219597
    #> 177435 Glyma.06G225200 Glyma.04G205700 0.09054177
    #> 208602 Glyma.09G278100 Glyma.05G084800 0.08846030
    #> 190115 Glyma.18G039200 Glyma.05G002100 0.08382400
    #> 78998  Glyma.01G075800 Glyma.02G221200 0.08271081
    #> 870271 Glyma.09G231700 Glyma.18G220700 0.08208945

    data(gma_ppi)
    head(gma_ppi)
    #>                node1           node2
    #> 4971 Glyma.19G213200 Glyma.01G004300
    #> 4985 Glyma.14G061100 Glyma.01G004300
    #> 4995 Glyma.03G179000 Glyma.01G004300
    #> 5021 Glyma.03G097900 Glyma.01G004300
    #> 5037 Glyma.07G047700 Glyma.01G004300
    #> 5053 Glyma.19G051900 Glyma.01G004300

    data(gma_paralogs)
    head(gma_paralogs)
    #>        duplicate1      duplicate2 type
    #> 1 Glyma.01G007200 Glyma.01G007300   TD
    #> 2 Glyma.01G007400 Glyma.01G007500   TD
    #> 3 Glyma.01G011500 Glyma.01G011600   TD
    #> 4 Glyma.01G012100 Glyma.01G012200   TD
    #> 5 Glyma.01G015200 Glyma.01G015300   TD
    #> 6 Glyma.01G020600 Glyma.01G020700   TD

# 4 Finding motifs

Motifs can be found using `find_*` motifs, as shown in Figure 1. Each function returns a character vector of motifs, and each motif has its own character representation. Let’s demonstrate how they work. For the sake of demonstration, we will only use WGD-derived paralogs. For GRN motifs, we will only use a smaller subset of the edge list.

    # Include only WGD-derived paralogs
    paralogs <- gma_paralogs[gma_paralogs$type == "WGD", 1:2]

    # Keep only the top 30k edges of the GRN, remove "Weight" variable
    grn <- gma_grn[1:30000, 1:2]

## 4.1 PPI V

PPI V motifs are paralogous proteins that share an interaction partner. To find them, you will use `find_ppi_v()`. The character representation of PPI V motifs is:

\[ \text{paralog1-partner-paralog2} \]

    # Find PPI V motifs
    ppi_v <- find_ppi_v(gma_ppi, paralogs)
    head(ppi_v)
    #> [1] "Glyma.19G213200-Glyma.01G004300-Glyma.03G216600"
    #> [2] "Glyma.07G034300-Glyma.01G004300-Glyma.08G207800"
    #> [3] "Glyma.14G101700-Glyma.01G004300-Glyma.06G061100"
    #> [4] "Glyma.12G028500-Glyma.01G004300-Glyma.11G103700"
    #> [5] "Glyma.19G189600-Glyma.01G004300-Glyma.03G189300"
    #> [6] "Glyma.15G006700-Glyma.01G004300-Glyma.07G029200"

## 4.2 V

V motifs are paralogous regulators that regulate the same target. These motifs can be created when a regulator undergoes a small-scale duplication. To find them, you will use `find_v()`. The character representation of V motifs is:

\[ \text{regulator->target<-regulator} \]

    # Find V motifs
    v <- find_v(grn, paralogs)
    head(v)
    #> [1] "Glyma.01G177200<-Glyma.16G128800->Glyma.02G058700"
    #> [2] "Glyma.01G177200<-Glyma.08G362400->Glyma.02G058700"
    #> [3] "Glyma.01G177200<-Glyma.02G103700->Glyma.02G058700"
    #> [4] "Glyma.01G177200<-Glyma.09G152400->Glyma.02G058700"
    #> [5] "Glyma.01G177200<-Glyma.20G151800->Glyma.02G058700"
    #> [6] "Glyma.01G177200<-Glyma.08G020300->Glyma.02G058700"

## 4.3 Lambda

Lambda motifs are the opposite of V motifs: a single regulator that regulates two target genes that are paralogous. These motifs can be created when an ancestral target gene undergoes a small-scale duplication. To find them you will use `find_lambda()`. The character representation of lambda motifs is:

\[ \text{target1<-regulator->target2} \]

    lambda <- find_lambda(grn, paralogs)
    head(lambda)
    #> [1] "Glyma.01G132800<-Glyma.07G126500->Glyma.03G035400"
    #> [2] "Glyma.01G205300<-Glyma.07G126500->Glyma.05G064600"
    #> [3] "Glyma.01G205300<-Glyma.08G227000->Glyma.05G064600"
    #> [4] "Glyma.01G205300<-Glyma.07G128700->Glyma.05G064600"
    #> [5] "Glyma.01G205300<-Glyma.07G087900->Glyma.05G064600"
    #> [6] "Glyma.01G205300<-Glyma.19G022200->Glyma.05G064600"

## 4.4 Delta

Delta motifs are pretty similar to lambda motifs, but here we take protein-protein interactions between targets into account. Thus, they are represented by a regulator that regulates two targets that interact at the protein level. They can be created by the same evolutionary mechanism of lambda motifs. To find them, you will use `find_delta()`. The character representation of delta motifs is:

\[ \text{target1<-regulator->target2} \]

To find delta motifs, you have two options:

1. Pass PPI edge list + a vector of previously identified lambda motifs (recommended).

2. Pass PPI edge list + GRN edge list + paralogs data frame. In this option, `find_delta()` will find lambda motifs first, then use the lambda vector to find delta motifs. If you have identified lambda motifs beforehand, it is way faster to pass the lambda vector to `find_delta()`, so you don’t have to do double work.

    # Find delta motifs from lambda motifs
    delta <- find_delta(edgelist_ppi = gma_ppi, lambda_vec = lambda)
    head(delta)
    #> [1] "Glyma.10G242600<-Glyma.04G221800->Glyma.20G151500"

## 4.5 Bifan

Bifan motifs are the most complex: they are represented by two paralogous regulators that regulate the same set of two paralogous targets. They can be created when both the ancestral regulator and the ancestral target are duplicated by small-scale duplications, or when the genome undergoes a whole-genome duplication event. To find these motifs, you will use `find_bifan()`. The character representation of bifan motifs is:

\[ \text{regulator1,regulator2->target1,target2} \]

Under the hood, what `find_bifan()` does it to find lambda motifs involving the same targets and check if their regulators are paralogs. Thus, if you have identified lambda motifs beforehand, it is much faster to simply give them to `find_bifan()`, so it doesn’t have to find them again.

    # Find bifans from lambda motifs
    bifan <- find_bifan(paralogs = paralogs, lambda_vec = lambda)
    head(bifan)
    #> [1] "Glyma.01G177200,Glyma.02G058700->Glyma.09G152400,Glyma.20G151800"

# 5 Counting motifs and evaluating significance

As motifs are simple character vectors, one can count their frequencies with the base R `length()` function. For example, let’s count the frequency of each motif in our example data set:

    count_df <- data.frame(
        Motif = c("PPI V", "V", "Lambda", "Delta", "Bifan"),
        Count = c(
            length(ppi_v), length(v), length(lambda), length(delta), length(bifan)
        )
    )

    count_df
    #>    Motif Count
    #> 1  PPI V  1521
    #> 2      V   112
    #> 3 Lambda   163
    #> 4  Delta     1
    #> 5  Bifan     1

However, unless you have another data set to which you can compare your frequencies, counting is not enough. You need to evaluate the significance of your motif frequencies. One way to do that is by comparing your observed frequencies to a null distribution generated by counting motifs in N (e.g., 1000) simulated networks.1 1 1 NOTE: Simulated networks are created by node label permutation (i.e., resampling node labels without replacement). This method allows you to have random networks that preserve the same degree of the original network. Hence, networks are called degree-preserving simulated networks. `magrene` allows you to generate null distributions of motif frequencies for each motif type with the function `generate_nulls()`. As generating the null distributions takes a bit of time, we will demonstrate `generate_nulls()` with 5 permutations only. As a rule of thumb, you would probably want N >= 1000.

    generate_nulls(grn, paralogs, gma_ppi, n = 5)
    #> $lambda
    #> lambda lambda lambda lambda lambda
    #>    106     96     74     90     95
    #>
    #> $delta
    #> delta delta delta delta delta
    #>     4     3     3     3     5
    #>
    #> $V
    #>   V   V   V   V   V
    #> 137 138 123 134 141
    #>
    #> $PPI_V
    #> PPI_V PPI_V PPI_V PPI_V PPI_V
    #>   219   185   197   185   167
    #>
    #> $bifan
    #> bifan bifan bifan bifan bifan
    #>     0     0     0     0     0

As you can see, the output of `generate_nulls()` is a list of numeric vectors with the frequency of each motif type in the simulated networks2 2 2 Note on performance: The function `generate_nulls()` can be parallelized thanks to the Bioconductor package BiocParallel. However, keep in mind that parallelization is not always the best choice, because it may take longer to create multiple copies of your data to split into multiple cores than it takes to find motifs with a single core.. You can use the null distribution to calculate Z-scores for your observed frequencies, which are defined as below:

\[ Z = \frac{ n_{observed} - \bar{n}_{null} }{ \sigma_{null} } \]

To calculate Z-scores, you can use the function `calculate_Z()`. As input, you need to give a list of observed frequencies and a list of nulls. Here, we will load pre-computed null distributions of N = 100.

    # Load null distros
    data(nulls)
    head(nulls)
    #> $lambda
    #> lambda lambda lambda lambda lambda lambda lambda lambda lambda lambda lambda
    #>     76     84     80     88     77     83     81     85     83     79     89
    #> lambda lambda lambda lambda lambda lambda lambda lambda lambda lambda lambda
    #>     77     87     78     88     82     87     70     89     82     78     78
    #> lambda lambda lambda lambda lambda lambda lambda lambda lambda lambda lambda
    #>     93     75     69     92     86     95     78     90     86     95     94
    #> lambda lambda lambda lambda lambda lambda lambda lambda lambda lambda lambda
    #>     80    100     93     92    100     79     71     83     92     74     78
    #> lambda lambda lambda lambda lambda lambda lambda lambda lambda lambda lambda
    #>    103     89     91     84     85     89     78     81     71     89     83
    #> lambda lambda lambda lambda lambda lambda lambda lambda lambda lambda lambda
    #>    103     84    101     93     98     77     95     88     78     98     86
    #> lambda lambda lambda lambda lambda lambda lambda lambda lambda lambda lambda
    #>     70     82     82     78     75     86    100     83     96     78     93
    #> lambda lambda lambda lambda lambda lambda lambda lambda lambda lambda lambda
    #>     94     82     65     85     83     83     74    100     87     78     78
    #> lambda lambda lambda lambda lambda lambda lambda lambda lambda lambda lambda
    #>     94     84     97     82     90    113     81     76     96     88     87
    #> lambda
    #>     91
    #>
    #> $delta
    #> delta delta delta delta delta delta delta delta delta delta delta delta delta
    #>     1     4     2     8     6     5     3     0     1     7     2     1     3
    #> delta delta delta delta delta delta delta delta delta delta delta delta delta
    #>     1     2     1     2     1     2     3     7     1     4     4     3     1
    #> delta delta delta delta delta delta delta delta delta delta delta delta delta
    #>     3     3     3     4     3     2     1     2     2     6     7     4     1
    #> delta delta delta delta delta delta delta delta delta delta delta delta delta
    #>     2     5     1     1     0     1     2     6     6     6     9     2     2
    #> delta delta delta delta delta delta delta delta delta delta delta delta delta
    #>     3     4     3     3     4     4     4     3     2     1     2     8     7
    #> delta delta delta delta delta delta delta delta delta delta delta delta delta
    #>     4     4     4     1     3     2     7     4     2     2     4     5     6
    #> delta delta delta delta delta delta delta delta delta delta delta delta delta
    #>     3     2     3     2     4     3     3     1     2     1     2     2     7
    #> delta delta delta delta delta delta delta delta delta
    #>     0     4     4     3     1     1     5     3     2
    #>
    #> $V
    #>   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V
    #> 133 131 136 130 123 112 126 117 124 130 128 141 111 115 132 137 129 139 151 134
    #>   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V
    #>  99 133 133 123 127 128 141 109 141 140 136 102 114 126 136 120 144 132 127 122
    #>   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V
    #> 142 129 114 139 136 129 153 150 139 148 113 106 132 143 139 129 147 119 140 125
    #>   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V
    #> 126 121 119 134 142 145 117 135 127 133 111 133 136 132 130 106 132 123 140 130
    #>   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V   V
    #> 128 131 127 138 129 132 126 121 120 115 137 123 140 131 110 133 149 129 139 149
    #>
    #> $PPI_V
    #> PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V
    #>   198   178   179   191   173   172   210   192   219   192   178   194   159
    #> PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V
    #>   188   193   186   182   202   166   190   192   178   189   176   189   161
    #> PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V
    #>   178   199   186   172   178   171   196   199   195   207   190   188   163
    #> PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V
    #>   170   162   186   192   184   207   198   201   199   186   201   169   178
    #> PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V
    #>   192   180   175   192   192   178   183   184   179   195   177   185   187
    #> PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V
    #>   173   205   191   185   192   174   175   195   193   188   201   191   182
    #> PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V
    #>   190   209   193   199   183   186   194   181   191   199   187   201   197
    #> PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V PPI_V
    #>   201   182   200   180   169   204   200   198   185
    #>
    #> $bifan
    #> bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan
    #>     0     0     0     0     0     0     0     0     0     0     0     0     0
    #> bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan
    #>     0     0     0     0     0     0     0     0     0     1     0     0     0
    #> bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan
    #>     0     0     0     0     0     0     0     0     0     0     0     0     0
    #> bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan
    #>     0     0     0     0     0     0     0     0     0     0     0     0     0
    #> bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan
    #>     0     0     0     0     0     0     0     0     0     0     0     0     0
    #> bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan
    #>     0     0     0     0     0     0     0     0     0     0     0     0     0
    #> bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan bifan
    #>     0     0     0     0     0     0     0     0     0     0     0     0     0
    #> bifan bifan bifan bifan bifan bifan bifan bifan bifan
    #>     0     0     0     0     0     0     0     0     0

    # Create list of observed frequencies
    observed <- list(
        lambda = length(lambda),
        bifan = length(bifan),
        V = length(v),
        PPI_V = length(ppi_v),
        delta = length(delta)
    )
    calculate_Z(observed, nulls)
    #>     lambda      delta          V      PPI_V      bifan
    #>   8.856408  -1.068636  -1.560671 113.300559   9.900000

Now that you have Z-scores, you can use a cut-off of your choice to define significance.

# 6 Evaluting interaction similarity

Finally, another interesting pattern you may want to analyze is the interaction similarity between paralogous gene pairs. Previous studies have demonstrated that the Sorensen-Dice similarity is a suitable index for interaction similarity (Defoort, Van de Peer, and Carretero-Paulet 2019; Mottes et al. 2021), which is defined as:

\[ S(A,B) = \frac{2 \left| A \cap B \right|}{ \left|A \right| + \left| B \right|} \]

where A and B are the interacting partners of nodes A and B. To calculate the Sorensen-Dice similarity for paralogous gene pairs, you can use the function `sd_similarity()`. Let’s demonstrate it by calculating the similarity between paralogs in the PPI network.

    sim <- sd_similarity(gma_ppi, paralogs)
    head(sim)
    #>            duplicate1      duplicate2 sorensen_dice
    #> 11631 Glyma.01G004300 Glyma.05G223800             1
    #> 11696 Glyma.01G205300 Glyma.05G064600             1
    #> 11706 Glyma.01G207700 Glyma.05G061400             1
    #> 12157 Glyma.01G004300 Glyma.08G030900             1
    #> 12803 Glyma.01G207700 Glyma.11G034700             1
    #> 13298 Glyma.01G163300 Glyma.16G123100             1

    summary(sim$sorensen_dice)
    #>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
    #>  0.2200  1.0000  1.0000  0.9812  1.0000  1.0000

# Session information

This document was created under the following conditions:

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    #> ──────────────────────────────────────────────────────────────────────────────

# References

Almeida-Silva, Fabricio, Kanhu C Moharana, Fabricio B Machado, and Thiago M Venancio. 2020. “Exploring the Complexity of Soybean (Glycine Max) Transcriptional Regulation Using Global Gene Co-Expression Networks.” Planta 252 (6): 1–12.

Almeida-Silva, Fabricio, and Thiago M Venancio. 2022. “BioNERO: An All-in-One R/Bioconductor Package for Comprehensive and Easy Biological Network Reconstruction.” Functional & Integrative Genomics 22 (1): 131–36.

Defoort, Jonas, Yves Van de Peer, and Lorenzo Carretero-Paulet. 2019. “The Evolution of Gene Duplicates in Angiosperms and the Impact of Protein–Protein Interactions and the Mechanism of Duplication.” Genome Biology and Evolution 11 (8): 2292–2305.

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