Calculating Number of Iterations Required to Reach Steady-State

Selcen Ari

2021-10-26

1. Introduction

In the ceRNAnetsim package, regulations of miRNA:target pairs are observed via direct or indirect interactions of elements in network. In this approach, change in expression level of single gene or miRNA can affect the whole network via “ripple effect”. So, when the change is applied the system, it affects to primary neighborhood firstly, and then propagates to further neighborhoods.

In the simple interaction network like minsamp, the ripple effect could be observed when expression level of Gene4 changes and subsequently effecting other genes. In the non-complex networks like minsamp, the steady-state condition can be provided easily, after network disturbed.

In this vignette, first, we demonstrate a suggestion to determine simulation iteration of existing dataset for gaining steady state after perturbing the network. Additionally, new approach which is useful for defining significant of nodes in terms of perturbation of network is elucidated.

library(ceRNAnetsim)
library(png)

2. Installation

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

3. Comparison of gaining steady-state durations of middle and minimal datasets

data("minsamp")

minsamp %>%
  priming_graph(competing_count = Competing_expression, 
                miRNA_count = miRNA_expression) %>%
  update_how("Gene4",2) %>%
  simulate_vis(title = "Minsamp: Common element as trigger", cycle = 15)
#> # A tbl_graph: 8 nodes and 7 edges
#> #
#> # A rooted tree
#> #
#> # Node Data: 8 × 7 (active)
#>   name  type      node_id initial_count count_pre count_current changes_variable
#>   <chr> <chr>       <int>         <dbl>     <dbl>         <dbl> <chr>           
#> 1 Gene1 Competing       1         10000    10061.        10061. Competing       
#> 2 Gene2 Competing       2         10000    10061.        10061. Competing       
#> 3 Gene3 Competing       3          5000     5030.         5030. Competing       
#> 4 Gene4 Competing       4         10000    19528.        19528. Competing       
#> 5 Gene5 Competing       5          5000     5107.         5107. Competing       
#> 6 Gene6 Competing       6         10000    10214.        10214. Competing       
#> # … with 2 more rows
#> #
#> # Edge Data: 7 × 20
#>    from    to Competing_name miRNA_name Competing_expre… miRNA_expression dummy
#>   <int> <int> <chr>          <chr>                 <dbl>            <dbl> <dbl>
#> 1     1     7 Gene1          Mir1                  10000             1000     1
#> 2     2     7 Gene2          Mir1                  10000             1000     1
#> 3     3     7 Gene3          Mir1                   5000             1000     1
#> # … with 4 more rows, and 13 more variables: afff_factor <dbl>,
#> #   degg_factor <dbl>, comp_count_list <list>, comp_count_pre <dbl>,
#> #   comp_count_current <dbl>, mirna_count_list <list>, mirna_count_pre <dbl>,
#> #   mirna_count_current <dbl>, mirna_count_per_dep <dbl>, effect_current <dbl>,
#> #   effect_pre <dbl>, effect_list <list>, mirna_count_per_comp <dbl>


minsamp %>%
  priming_graph(competing_count = Competing_expression, 
                miRNA_count = miRNA_expression) %>%
  update_how("Gene4",2) %>%
  simulate(cycle = 5)
#> # A tbl_graph: 8 nodes and 7 edges
#> #
#> # A rooted tree
#> #
#> # Node Data: 8 × 7 (active)
#>   name  type      node_id initial_count count_pre count_current changes_variable
#>   <chr> <chr>       <int>         <dbl>     <dbl>         <dbl> <chr>           
#> 1 Gene1 Competing       1         10000    10061.        10061. Up              
#> 2 Gene2 Competing       2         10000    10061.        10061. Up              
#> 3 Gene3 Competing       3          5000     5030.         5030. Up              
#> 4 Gene4 Competing       4         10000    19528.        19528. Down            
#> 5 Gene5 Competing       5          5000     5107.         5107. Up              
#> 6 Gene6 Competing       6         10000    10214.        10214. Up              
#> # … with 2 more rows
#> #
#> # Edge Data: 7 × 20
#>    from    to Competing_name miRNA_name Competing_expre… miRNA_expression dummy
#>   <int> <int> <chr>          <chr>                 <dbl>            <dbl> <dbl>
#> 1     1     7 Gene1          Mir1                  10000             1000     1
#> 2     2     7 Gene2          Mir1                  10000             1000     1
#> 3     3     7 Gene3          Mir1                   5000             1000     1
#> # … with 4 more rows, and 13 more variables: afff_factor <dbl>,
#> #   degg_factor <dbl>, comp_count_list <list>, comp_count_pre <dbl>,
#> #   comp_count_current <dbl>, mirna_count_list <list>, mirna_count_pre <dbl>,
#> #   mirna_count_current <dbl>, mirna_count_per_dep <dbl>, effect_current <dbl>,
#> #   effect_pre <dbl>, effect_list <list>, mirna_count_per_comp <dbl>
Minsamp common target perturbation

Minsamp common target perturbation

For example, in minsamp dataset, the steady-state is occurred at iteration-14 (as seen above: after iteration-13, it is observed that there are only orange (miRNAs) and green (competing genes) nodes in network. In this case, genes have new regulated (steady) expression values while expression values of microRNAs are same in comparison with initial case.).

However, when network is larger and interactions are more complex, the number of iterations required to reach steady-state may increase. While at cycle 14 minsamp dataset has reached steady-state, the midsamp (middle sized sample) dataset has not reached steady-state after 15 cycles. In the example below, in midsamp data, Gene17 is upregulated 2 fold as a trigger and simulation is run for 15 cycles.

data("midsamp")

midsamp
#>     Genes miRNAs Gene_expression miRNA_expression seeds targeting_region Energy
#> 1   Gene1   Mir1           10000             1000  0.43             0.30    -20
#> 2   Gene2   Mir1           10000             1000  0.43             0.01    -15
#> 3   Gene3   Mir1            5000             1000  0.32             0.40    -14
#> 4   Gene4   Mir1           10000             1000  0.23             0.50    -10
#> 5   Gene4   Mir2           10000             2000  0.35             0.90    -12
#> 6   Gene5   Mir2            5000             2000  0.05             0.40    -11
#> 7   Gene6   Mir2           10000             2000  0.01             0.80    -25
#> 8   Gene4   Mir3           10000             3000  0.43             0.40     -6
#> 9   Gene6   Mir3           10000             3000  0.35             0.90    -15
#> 10  Gene7   Mir3            7000             3000  0.23             0.30    -20
#> 11  Gene8   Mir3            3000             3000  0.01             0.20    -30
#> 12  Gene6   Mir4           10000             5000  0.05             0.40    -12
#> 13  Gene9   Mir4            6000             5000  0.32             0.80    -18
#> 14 Gene10   Mir4            2000             5000  0.43             0.20    -23
#> 15 Gene11   Mir4            8000             5000  0.35             0.90    -25
#> 16 Gene12   Mir4            1500             5000  0.43             0.40    -30
#> 17 Gene13   Mir4             500             5000  0.23             0.90    -17
#> 18 Gene14   Mir4            7000             5000  0.43             0.80    -15
#> 19 Gene14   Mir3            7000             3000  0.43             0.90    -25
#> 20 Gene15   Mir3            3000             3000  0.35             0.20    -12
#> 21 Gene16   Mir3            2000             3000  0.01             0.80    -18
#> 22 Gene17   Mir3            6000             3000  0.23             0.40    -22
#> 23 Gene17   Mir2            6000             2000  0.35             0.90     -7
#> 24 Gene18   Mir2            1000             2000  0.01             0.01    -30
#> 25 Gene19   Mir2            6500             2000  0.43             0.90    -32
#> 26 Gene20   Mir2            4800             2000  0.35             0.80    -18
midsamp %>%
  priming_graph(Gene_expression, miRNA_expression) %>%
  update_how("Gene17",2) %>%
  simulate_vis(title = "Midsamp: Gene with higher degree as trigger", 15)
#> # A tbl_graph: 24 nodes and 26 edges
#> #
#> # A directed acyclic simple graph with 1 component
#> #
#> # Node Data: 24 × 7 (active)
#>   name  type      node_id initial_count count_pre count_current changes_variable
#>   <chr> <chr>       <int>         <dbl>     <dbl>         <dbl> <chr>           
#> 1 Gene1 Competing       1         10000    10001.        10001. Up              
#> 2 Gene2 Competing       2         10000    10001.        10001. Up              
#> 3 Gene3 Competing       3          5000     5000.         5000. Up              
#> 4 Gene4 Competing       4         10000    10110.        10110. Down            
#> 5 Gene5 Competing       5          5000     5026.         5026. Down            
#> 6 Gene6 Competing       6         10000    10105.        10105. Up              
#> # … with 18 more rows
#> #
#> # Edge Data: 26 × 20
#>    from    to Competing_name miRNA_name Gene_expression miRNA_expression dummy
#>   <int> <int> <chr>          <chr>                <dbl>            <dbl> <dbl>
#> 1     1    21 Gene1          Mir1                 10000             1000     1
#> 2     2    21 Gene2          Mir1                 10000             1000     1
#> 3     3    21 Gene3          Mir1                  5000             1000     1
#> # … with 23 more rows, and 13 more variables: afff_factor <dbl>,
#> #   degg_factor <dbl>, comp_count_list <list>, comp_count_pre <dbl>,
#> #   comp_count_current <dbl>, mirna_count_list <list>, mirna_count_pre <dbl>,
#> #   mirna_count_current <dbl>, mirna_count_per_dep <dbl>, effect_current <dbl>,
#> #   effect_pre <dbl>, effect_list <list>, mirna_count_per_comp <dbl>
Midsamp Gene17 perturbation followed with 15 iterations

Midsamp Gene17 perturbation followed with 15 iterations

3.1. Suggestion for simulation iteration

Guessing or performing trial and error for large networks is not practical, thus we developed a function which calculates optimal iteration in a network after trigger and simulation steps. find_iteration() function analyses the simulated graph and suggests the iteration at which maximum number of nodes are affected. An important argument is limit which sets the threshold below which is considered “no change”, in other words, any node should have level of change greater than the threshold to be considered “changed”. Please be aware that small threshold values will cause excessively long calculation time especially in large networks.

In the example below, Gene2 is upregulated 2-fold and then iteration number at which maximum number of nodes affected will be calculated. The search for iteration number will go up to 50. Also, since we are searching for maximal propagation, limit is set to zero.

NOTE: You can edit the dataset manually. You can change Gene2 expression value as 20000 and save that as a new dataset (midsamp_new_counts).

You can use the dataset that includes new expression values of miRNAs and genes.

find_iteration() function will return a single number: the iteration number at which maximum propagation is achieved. If plot=TRUE argument is used then the function will return a plot which calculates percent of perturbed nodes for each iteration number. The latter can be used for picking appropriate number of cycles for simulate() function.

3.2. Find appropriate iteration number with find_iteration and then simulate accordingly

As shown in plot above, if “Gene17” is upregulated 2-fold, the network will need around 22 iterations to reach the steady-state. Since we have an idea about appropriate iteration number, let’s use simulate() function and iterate for 25 cycles using same trigger (Gene17 2-fold):

Midsamp Gene17 perturbation with 25 iteration

Midsamp Gene17 perturbation with 25 iteration

Note: If you ignore decimal change in gene expression, threshold argument can be used. With this method, system reaches steady-state early.

Midsamp Gene17 perturbation with 6 iteration with threshold

Midsamp Gene17 perturbation with 6 iteration with threshold

The workflow that is aforementioned in this vignette should be considered as suggestion. Because the cycle is a critical argument that is used with simulate() function and affects results of analysis. In light of this vignette and functions, the approach can be developed according to dataset.

4. What is perturbation efficiency?

The perturbation efficiency means that the disturbance and propagation efficiency of an element in the network. In a given network not all nodes have same or similar perturbation efficiency. Changes in some nodes might propagate to whole network and for some nodes the effect might be limited to small subgraph of the network. Not only topology but also miRNA:target interaction dynamics determine perturbation efficiency.

Thus, we developed functions which can calculate perturbation efficiency of a given node or all nodes. calc_perturbation() function calculates perturbation efficiency for given trigger (e.g. Gene17 2-fold). find_node_perturbation() function screens the whole network and calculate perturbation efficiency of all nodes.

4.1. How does the calc_perturbation() work?

This function works for a given node from network. It calculates and returns two values:

In the example below, “Gene17” is up-regulated 3-fold in midsamp dataset where Energy and seeds columns are used for calculating affinity effect and targeting_region columns is used for calculating degradation effect. The network will be iterated over 30 times and number of disturbed nodes (as taking into account nodes that have changed more than the value of the threshold (0.1 percentage in terms of the change)) will be counted.

If you are interested in testing various fold change values of a given node, then we can use map (actually parallelized version future_map) to run function for set of input values.

First, let’s keep the primed version of graph in an object

Now, let’s calculate perturbation efficiency caused by 2-fold to 10-fold increase in Gene17

If you’re interested in screening nodes instead of fold changes then you don’t have to write a complicated map command, there’s already a function available for that purpose.

4.2. A Short-cut: Finding perturbation efficiencies for whole nodes of network

The find_node_perturbation() function calculates the perturbation efficiency and perturbed node count of each node in network.

In the example below, each node is increased 2-fold and tested for perturbation efficiency for 4 cycles with threshold of 0.1

On the other hand, some of nodes in network might not be affected from perturbation because of low expression or weak interaction factors. In this case, fast argument can be used. Argument fast calculate affected expression percent of the targets and perturbation calculation is not ran for these elements in network, if that percentage value is smaller than given fast value.

The results of the find_node_perturbation() will list effectiveness or importance of nodes in the network. This function can help selecting nodes which will have effective perturbation in network.

5. Session Info

sessionInfo()
#> R version 4.1.1 (2021-08-10)
#> Platform: x86_64-pc-linux-gnu (64-bit)
#> Running under: Ubuntu 20.04.3 LTS
#> 
#> Matrix products: default
#> BLAS:   /home/biocbuild/bbs-3.14-bioc/R/lib/libRblas.so
#> LAPACK: /home/biocbuild/bbs-3.14-bioc/R/lib/libRlapack.so
#> 
#> locale:
#>  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
#>  [3] LC_TIME=en_GB              LC_COLLATE=C              
#>  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
#>  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
#>  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
#> [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       
#> 
#> attached base packages:
#> [1] stats     graphics  grDevices utils     datasets  methods   base     
#> 
#> other attached packages:
#> [1] purrr_0.3.4       png_0.1-7         ceRNAnetsim_1.6.0 tidygraph_1.2.0  
#> [5] dplyr_1.0.7      
#> 
#> loaded via a namespace (and not attached):
#>  [1] tidyselect_1.1.1   xfun_0.27          bslib_0.3.1        graphlayouts_0.7.1
#>  [5] listenv_0.8.0      colorspace_2.0-2   vctrs_0.3.8        generics_0.1.1    
#>  [9] viridisLite_0.4.0  htmltools_0.5.2    yaml_2.2.1         utf8_1.2.2        
#> [13] rlang_0.4.12       jquerylib_0.1.4    pillar_1.6.4       glue_1.4.2        
#> [17] DBI_1.1.1          tweenr_1.0.2       lifecycle_1.0.1    stringr_1.4.0     
#> [21] munsell_0.5.0      gtable_0.3.0       future_1.22.1      codetools_0.2-18  
#> [25] evaluate_0.14      labeling_0.4.2     knitr_1.36         fastmap_1.1.0     
#> [29] parallel_4.1.1     fansi_0.5.0        highr_0.9          furrr_0.2.3       
#> [33] Rcpp_1.0.7         scales_1.1.1       jsonlite_1.7.2     farver_2.1.0      
#> [37] parallelly_1.28.1  gridExtra_2.3      ggforce_0.3.3      ggplot2_3.3.5     
#> [41] digest_0.6.28      stringi_1.7.5      ggrepel_0.9.1      polyclip_1.10-0   
#> [45] grid_4.1.1         cli_3.0.1          tools_4.1.1        magrittr_2.0.1    
#> [49] sass_0.4.0         tibble_3.1.5       ggraph_2.0.5       crayon_1.4.1      
#> [53] tidyr_1.1.4        pkgconfig_2.0.3    ellipsis_0.3.2     MASS_7.3-54       
#> [57] viridis_0.6.2      assertthat_0.2.1   rmarkdown_2.11     R6_2.5.1          
#> [61] globals_0.14.0     igraph_1.2.7       compiler_4.1.1