Author: Ang Dong
Date: 2023-04-18

# idopNetwork_vignette

#### Ang Dong

#### 04/18/2023

    library(idopNetwork)
    backup_options <- options()
    #load pre-computered results
    test_result = idopNetwork:::test_result

# About

idopNetwork is packed as a cartographic tool that performs power curve fitting, classification, variable selection, microbial abundance decomposition, and network visualization based on microbial 16S rRNA gene sequencing metadata.

For complete details on the use and execution of this protocol, please refer to Chen et al and Cao et al.

# 1. Input data

Before running idopNetwork, user need to provide datasets, and they should be cleaned and merged to exactly the same format of the example data.

## 1.1 gut microbe OTUs data

Microbe Operational taxonomic unit dataset must have first column as IDs for microbes.

    data("gut_microbe")
    View(gut_microbe)

name | GQ_UC1 | GQ_UC3 | GQ_UC4 | GQ_UC5
--- | --- | --- | --- | ---
Escherichia_coli | 176 | 1701 | 28506 | 177
Arthrobacter_oxydans | 3 | 215 | 1578 | 1
Ruminococcus_gnavus | 744 | 0 | 4 | 733
Bacteroides_plebeius | 271 | 4401 | 7 | 8394
Bacteroides_pyogenes | 0 | 15 | 0 | 0
Clostridium_tertium | 305 | 1 | 1 | 270
Bacteroides_stercoris | 24 | 0 | 0 | 11
Flavisolibacter_ginsengisoli | 0 | 123 | 831 | 0
Bacteroides_massiliensis | 0 | 6098 | 6 | 0
Prevotella_heparinolytica | 0 | 6 | 0 | 0

## 1.2 mustard microbe OTUs data

    data("mustard_microbe")
    View(mustard_microbe)

| ID | 8_0149.leaf | 8_0157.leaf | 8_0158.leaf | 8_0160.leaf
--- | --- | --- | --- | --- | ---
2 | OTU_1 | 3 | 2 | 59 | 268
3 | OTU_2 | 8 | 13 | 1890 | 44
4 | OTU_3 | 5485 | 3302 | 4522 | 6581
5 | OTU_4 | 42 | 351 | 7 | 4
6 | OTU_5 | 16 | 2002 | 24 | 43
7 | OTU_6 | 1645 | 1170 | 10636 | 31686
8 | OTU_7 | 4050 | 4094 | 13 | 66
9 | OTU_8 | 6 | 282 | 124 | 27
10 | OTU_9 | 102 | 5 | 10 | 18
11 | OTU_10 | 25 | 1393 | 35 | 19

| JAM_E4.leaf | 8_0149.root | 8_0157.root | 8_0158.root | 8_0160.root
--- | --- | --- | --- | --- | ---
2 | 7 | 28861 | 122 | 1 | 9
3 | 4 | 11618 | 19 | 14 | 1894
4 | 4981 | 481 | 755 | 17914 | 15998
5 | 622 | 862 | 1072 | 11 | 103
6 | 10014 | 447 | 758 | 19 | 289
7 | 577 | 1010 | 490 | 25030 | 29443
8 | 2233 | 13 | 5575 | 26 | 64
9 | 19 | 3348 | 36 | 33 | 46
10 | 362 | 6269 | 228 | 2 | 459
11 | 3071 | 4 | 6454 | 6 | 234

# 2. power curve fitting

The first major step in our idopNetwork reconstruction is to fit allometric growth curves to the data using the power function. This is easily done by using the related function power_fit. This function needs cleaned dataset as input and will return fitted OTUs for given dataset. Then the fitted output with original dataset can be transfer into function power_equation_plot for quick visualization

    df = data_cleaning(gut_microbe)

    result1 = power_equation_fit(df)

    power_equation_plot(result1)

# 3. Functional clustering

In this step we implement the power equation into functional clustering to detect different microbial modules. If after clustering there are still too many microbes within a certain module for network reconstruction, we can rerun functional clustering to further classify this module into distinct submodules.

## 3.1 mean curve

we fit mean vector of each cluster center by power equation(assume k=3)

    matplot(t(power_equation(x = 1:30, matrix(c(2,1,3,0.2,0.5,-0.5),nrow = 3, ncol = 2))),
            type = "l",
            xlab = "time",
            ylab = "population")
    legend("topright",
           c("cluster 1", "cluster 2", "cluster 3"),
           lty = c(1,2,3),
           col = c(1,2,3),
           box.lwd = 0)

## 3.2 covariance matrix

we fit covariance matrix of multivariate normal distribution with SAD1, it can be showed as

    get_SAD1_covmatrix(c(2,0.5), n = 5)
    #>      [,1]  [,2]  [,3]  [,4]  [,5]
    #> [1,] 0.25  0.50  1.00  2.00  4.00
    #> [2,] 0.50  1.25  2.50  5.00 10.00
    #> [3,] 1.00  2.50  5.25 10.50 21.00
    #> [4,] 2.00  5.00 10.50 21.25 42.50
    #> [5,] 4.00 10.00 21.00 42.50 85.25

## 3.3 initial parameters

we can check initial parameters (k=4)

    get_par_int(X = log10(df+1), k = 4, times = as.numeric(log10(colSums(df)+1)))
    #> Error in nls(y ~ a * x^b, start = list(a = a, b = b), control = nls.control(maxiter = 1000,  :
    #>   number of iterations exceeded maximum of 1000
    #> Error in nls(y ~ a * x^b, start = list(a = a, b = b), control = nls.control(maxiter = 1000,  :
    #>   number of iterations exceeded maximum of 1000
    #> Error in nls(y ~ a * x^b, start = list(a = a, b = b), control = nls.control(maxiter = 1000,  :
    #>   number of iterations exceeded maximum of 1000
    #> Error in nls(y ~ a * x^b, start = list(a = a, b = b), control = nls.control(maxiter = 1000,  :
    #>   number of iterations exceeded maximum of 1000
    #> Error in nls(y ~ a * x^b, start = list(a = a, b = b), control = nls.control(maxiter = 1000,  :
    #>   number of iterations exceeded maximum of 1000
    #> Error in nls(y ~ a * x^b, start = list(a = a, b = b), control = nls.control(maxiter = 1000,  :
    #>   number of iterations exceeded maximum of 1000
    #> Error in nls(y ~ a * x^b, start = list(a = a, b = b), control = nls.control(maxiter = 1000,  :
    #>   number of iterations exceeded maximum of 1000
    #> Error in nls(y ~ a * x^b, start = list(a = a, b = b), control = nls.control(maxiter = 1000,  :
    #>   number of iterations exceeded maximum of 1000
    #> Error in nls(y ~ a * x^b, start = list(a = a, b = b), control = nls.control(maxiter = 1000,  :
    #>   number of iterations exceeded maximum of 1000
    #> Error in nls(y ~ a * x^b, start = list(a = a, b = b), control = nls.control(maxiter = 1000,  :
    #>   number of iterations exceeded maximum of 1000
    #> $initial_cov_params
    #> [1] 0.50000 1.09618
    #>
    #> $initial_mu_params
    #>                 a           b
    #> [1,] 1.222888e+15 -21.7299072
    #> [2,] 1.275768e+01  -0.8462041
    #> [3,] 1.930472e+02  -3.6046396
    #> [4,] 2.241205e-35  50.4617985
    #>
    #> $initial_probibality
    #>
    #>          1          2          3          4
    #> 0.24590164 0.06557377 0.40983607 0.27868852

    #use kmeans to get initial centers
    tmp = kmeans(log10(df+1),4)$centers
    tmp2 = power_equation_fit(tmp, trans = NULL)
    power_equation_plot(tmp2, label = NULL,n = 4)

## 3.4 functional clustering (power-equation,SAD1)

idopNetowrk already wrapped the mean curve modelling, covariance matrix modelling and likelihood ratio calculation into a function `fun_clu()`.

    options(max.print = 10)
    fun_clu(result1$original_data, k = 3, iter.max = 5)
    #> Error in nls(y ~ a * x^b, start = list(a = a, b = b), control = nls.control(maxiter = 1000,  :
    #>   number of iterations exceeded maximum of 1000
    #> Error in nls(y ~ a * x^b, start = list(a = a, b = b), control = nls.control(maxiter = 1000,  :
    #>   number of iterations exceeded maximum of 1000
    #> initial  value 1225.809010
    #> iter  10 value 1131.213445
    #> final  value 1125.868093
    #> converged
    #>
    #>  iter = 1
    #>  Log-Likelihood =  1125.868
    #> initial  value 1118.632904
    #> iter  10 value 1115.215807
    #> iter  10 value 1115.215807
    #> iter  10 value 1115.215807
    #> final  value 1115.215807
    #> converged
    #>
    #>  iter = 2
    #>  Log-Likelihood =  1115.216
    #> initial  value 1113.154283
    #> final  value 1112.161477
    #> converged
    #>
    #>  iter = 3
    #>  Log-Likelihood =  1112.161
    #> initial  value 1111.441671
    #> final  value 1111.183070
    #> converged
    #>
    #>  iter = 4
    #>  Log-Likelihood =  1111.183
    #> initial  value 1110.923567
    #> final  value 1110.833860
    #> converged
    #>
    #>  iter = 5
    #>  Log-Likelihood =  1110.834
    #> initial  value 1110.730889
    #> final  value 1110.694611
    #> converged
    #>
    #>  iter = 6
    #>  Log-Likelihood =  1110.695
    #> $cluster_number
    #> [1] 3
    #>
    #> $Log_likelihodd
    #> [1] 1110.695
    #>
    #> $AIC
    #> [1] 2241.389
    #>
    #> $BIC
    #> [1] 2260.101
    #>
    #> $cov_par
    #> [1] 0.3550810 0.7415481
    #>
    #> $mu_par
    #>              [,1]       [,2]
    #> [1,] 4.803180e+01  -1.683863
    #> [2,] 1.595905e+13 -19.044478
    #> [3,] 4.142849e-26  36.857826
    #>
    #> $probibality
    #> [1] 0.08309011 0.46813139 0.44877850
    #>
    #> $omega
    #>               [,1]         [,2]         [,3]
    #>  [1,] 3.154770e-10 9.998988e-01 1.011960e-04
    #>  [2,] 1.159313e-10 9.978411e-01 2.158879e-03
    #>  [3,] 9.560057e-18 9.398919e-01 6.010813e-02
    #>  [ reached getOption("max.print") -- omitted 45 rows ]
    #>
    #> $cluster
    #>      MC_UC3 MC_UC4 YZJC_UC3 GQ_UC3 ZC_UC3 MC_UC2 GQ_UC1 GQ_UC4 HJC_UC3 ZC_UC4
    #>      HJC_UC1 HJC_UC4 HC_UC2 ZC_UC1 HC_UC4 JJC_UC2 HJC_UC2 MC_UC1 GQ_UC5 ZC_UC5
    #>      apply(omega, 1, which.max)
    #>  [ reached 'max' / getOption("max.print") -- omitted 48 rows ]
    #>
    #> $cluster2
    #>      MC_UC3 MC_UC4 YZJC_UC3 GQ_UC3 ZC_UC3 MC_UC2 GQ_UC1 GQ_UC4 HJC_UC3 ZC_UC4
    #>      HJC_UC1 HJC_UC4 HC_UC2 ZC_UC1 HC_UC4 JJC_UC2 HJC_UC2 MC_UC1 GQ_UC5 ZC_UC5
    #>      apply(omega, 1, which.max)
    #>  [ reached 'max' / getOption("max.print") -- omitted 48 rows ]
    #>
    #>  [ reached getOption("max.print") -- omitted 2 entries ]

Usually we use multithread to calcuation k = 2-n and then to decide best k , `fun_clu_BIC` uses BIC to select best cluster number by default.

    result2 = fun_clu_parallel(result1$original_data, start = 2, end = 5)

    best.k = which.min(sapply( result2 , "[[" , 'BIC' )) + 1 #skipped k = 1
    best.k
    #> [1] 7

    fun_clu_BIC(result = result2)

    #we can direct give other k value
    fun_clu_plot(result = result2, best.k = best.k)

## 3.5 bi-functional clustering (power-equation,biSAD1)

In this part we select part of mustard data for demonstration purpose.

    data("mustard_microbe")
    df2 = data_cleaning(mustard_microbe, x = 160)

    res_l = power_equation_fit(df2[,1:5]
    res_r = power_equation_fit(df2[,89:95])
    res1 = data_match(result1 = res_l, result2 = res_r)

    res2 = bifun_clu_parallel(data1 = res1$dataset1$original_data,
                              data2 = res1$dataset2$original_data,
                              Time1 = res1$dataset1$Time,
                              Time2 = res1$dataset2$Time,
                              start = 2,
                              end = 10,
                              thread = 9,
                              iter.max = 10)

    res2 = bifun_clu_parallel(data1 = res1$dataset1$original_data,
                              data2 = res1$dataset2$original_data,
                              Time1 = res1$dataset1$Time,
                              Time2 = res1$dataset2$Time,
                              start = 2,
                              end = 10,
                              thread = 9,
                              iter.max = 10)

    fun_clu_BIC(result = res2)

    #we can set best.k directly
    bifun_clu_plot(result = res2, best.k = 3, color1 = "#C060A1", color2 = "#59C1BD")

## 3.6 sub-clustering

Sometimes a module is still too large for network reconstruction, which is determined by Dunbar’s number, we can further cluster it into sub-modules.

    res3 = bifun_clu_convert(res2, best.k = 3)
    large.module = order(sapply(res3$a$Module.all,nrow))[5]

    res_suba = fun_clu_select(result_fit = res1$dataset1, result_funclu = res3$a, i = large.module)
    res_subb = fun_clu_select(result_fit = res1$dataset2, result_funclu = res3$b, i = large.module)
    dfsuba_l = power_equation_fit(res_suba$original_data)
    dfsubb_r = power_equation_fit(res_subb$original_data)
    ressub1 = data_match(result1 = dfsuba_l, result2 = dfsubb_r)
    ressub2 = bifun_clu_parallel(data1 = ressub1$dataset1$original_data,
                                 data2 = ressub1$dataset2$original_data,
                                 Time1 = ressub1$dataset1$Time,
                                 Time2 = ressub1$dataset2$Time,
                                 start = 2,
                                 end = 5,
                                 iter.max = 3)

    ressub2 = test_result$d2_subcluster

    fun_clu_BIC(result = ressub2)

    bifun_clu_plot(result = ressub2, best.k = 2, color1 = "#C060A1", color2 = "#59C1BD", degree = 1)

# 4. LASSO-based variable selection

idopNetwork implements a LASSO-based procedure to choose a small set of the most significant microbes/module that links with a given microbes/modules. `get_interaction()`return a compound list contain the target microbe/module name, the most relevant Modules/microbes names and the coefficients.

## 4.1 For Modules

    result3 = fun_clu_convert(result2,best.k = best.k)
    df.module = result3$original_data
    get_interaction(df.module,1)
    #> $ind.name
    #> [1] "M1"
    #>
    #> $dep.name
    #> [1] "M5"
    #>
    #> $coefficient
    #> [1] 5.951174

## 4.2 For Microbes

    #we can the microbial relationship in Module1
    df.M1 = result3$Module.all$`1`
    get_interaction(df.M1,1)
    #> Warning: Option grouped=FALSE enforced in cv.glmnet, since < 3 observations per
    #> fold

    #> Warning: Option grouped=FALSE enforced in cv.glmnet, since < 3 observations per
    #> fold

    #> Warning: Option grouped=FALSE enforced in cv.glmnet, since < 3 observations per
    #> fold
    #> $ind.name
    #> [1] "Arthrobacter_oxydans"
    #>
    #> $dep.name
    #> [1] "Fusobacterium_necrophorum"
    #>
    #> $coefficient
    #> [1] 0.08263525

# 5 qdODE solving

qdODE system is build based on variable selection results, it has unique ability to decompose the observed module/microbe abundance level into its independent component and dependent component, which can be used for inferring idopNetwork.

## 5.1 solving qdODE between modules

    options(max.print = 10)
    # first we test solving a qdODE
    module.relationship = lapply(1:best.k, function(c)get_interaction(df.module,c))
    ode.test = qdODE_all(result = result3, relationship = module.relationship, 1, maxit = 100)
    #> iter   10 value 0.004336
    #> final  value 0.000002
    #> converged
    # we can view the result
    qdODE_plot_base(ode.test)

    # then we solve all qdODEs
    ode.module = qdODE_parallel(result3)

    qdODE_plot_all(ode.module)

## 5.2 solving qdODE within a module

    result_m1 = fun_clu_select(result_fit = result1, result_funclu = result3, i = 1)
    ode.M1 = qdODE_parallel(result_m1)

    qdODE_plot_all(ode.M1)

# 6 idopNetwork reconstruction

The final step of this guide is to visualization the multilayer network, and our package provide network_plot function to easily draw our idopNetwork. We can simply plug previous qdODE results into network_conversion function, and it will convert qdODE result for network visualization

## 6.1 network between modules

    net_module = lapply(ode.module$ode_result, network_conversion)
    network_plot(net_module, title = "Module Network")

## 6.2 network within a module

    net_m1 = lapply(ode.M1$ode_result, network_conversion)
    network_plot(net_m1, title = "M1 Network")

## 6.3 network comparison

    mustard_module_a = qdODE_parallel(res3$a)
    mustard_module_b = qdODE_parallel(res3$b)

    res_m1a = fun_clu_select(result_fit = res1$dataset1, result_funclu = res3$a, i = 3)
    res_m1b = fun_clu_select(result_fit = res1$dataset2, result_funclu = res3$b, i = 3)
    mustard_M1a = qdODE_parallel(res_m1a)
    mustard_M1b = qdODE_parallel(res_m1b)

    mustard_m_a <- lapply(mustard_module_a$ode_result, network_conversion)
    mustard_m_b <- lapply(mustard_module_b$ode_result, network_conversion)

    #set seed to make same random layout
    layout(matrix(c(1,2),1,2,byrow=TRUE))
    set.seed(1)
    network_plot(mustard_m_a, title = "Module Network a")
    set.seed(1)
    network_plot(mustard_m_b, title = "Module Network b")

###

    result_m1a = fun_clu_select(result_fit = res1$dataset1, result_funclu = res3$a, i = 1)
    result_m1b = fun_clu_select(result_fit = res1$dataset2, result_funclu = res3$b, i = 1)
    ode.m1a = qdODE_parallel(result_m1a, thread = 16)
    ode.m1b = qdODE_parallel(result_m1b, thread = 16)

    net_m1a = lapply(ode.m1a$ode_result, network_conversion)
    net_m1b = lapply(ode.m1b$ode_result, network_conversion)

    #set seed to make same random layout
    layout(matrix(c(1,2),1,2,byrow=TRUE))
    set.seed(1)
    network_plot(net_m1a, title = "Module1 a")
    set.seed(1)
    network_plot(net_m1b, title = "Module1 b")

# Troubleshooting

object “LL.next” not found
This happens when parameters optimization failure, try rerun cluster.

plot failure when using `fun_clu_plot()` or `bifun_clu_plot()`
This often happens when bad initial parameters is given and some cluster is lost, try rerun cluster or use a smaller k.

# Session info

    sessionInfo()
    #> R version 4.2.1 (2022-06-23 ucrt)
    #> Platform: x86_64-w64-mingw32/x64 (64-bit)
    #> Running under: Windows 10 x64 (build 22621)
    #>
    #> 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
    #>
    #> attached base packages:
    #> [1] stats     graphics  grDevices utils     datasets  methods   base
    #>
    #> other attached packages:
    #> [1] idopNetwork_0.1.2
    #>
    #> loaded via a namespace (and not attached):
    #>  [1] deSolve_1.34         shape_1.4.6          tidyselect_1.2.0
    #>  [4] xfun_0.37            bslib_0.4.2          reshape2_1.4.4
    #>  [7] splines_4.2.1        lattice_0.20-45      colorspace_2.1-0
    #> [10] vctrs_0.5.2          generics_0.1.3       htmltools_0.5.4
    #> [13] yaml_2.3.7           utf8_1.2.3           survival_3.3-1
    #> [16] rlang_1.0.6          orthopolynom_1.0-6.1 jquerylib_0.1.4
    #> [19] pillar_1.8.1         withr_2.5.0          glue_1.6.2
    #> [22] foreach_1.5.2        lifecycle_1.0.3      plyr_1.8.8
    #> [25] stringr_1.5.0        munsell_0.5.0        gtable_0.3.1
    #> [28] mvtnorm_1.1-3        codetools_0.2-18     evaluate_0.20
    #> [31] labeling_0.4.2       knitr_1.42           fastmap_1.1.1
    #> [34] parallel_4.2.1       fansi_1.0.4          highr_0.10
    #> [37] Rcpp_1.0.10          polynom_1.4-1        scales_1.2.1
    #> [40] cachem_1.0.7         jsonlite_1.8.4       farver_2.1.1
    #> [43] ggplot2_3.4.1        digest_0.6.31        stringi_1.7.12
    #> [46] dplyr_1.1.0          grid_4.2.1           cli_3.6.0
    #> [49] tools_4.2.1          magrittr_2.0.3       sass_0.4.5
    #> [52] glmnet_4.1-6         patchwork_1.1.2      tibble_3.1.8
    #> [55] pkgconfig_2.0.3      Matrix_1.5-3         rmarkdown_2.20
    #> [58] rstudioapi_0.14      iterators_1.0.14     R6_2.5.1
    #> [61] igraph_1.4.1         compiler_4.2.1