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Phosphorylated AMPK (pAMPK) can activate three major transcription factors (TFs): PGC1α, CREB, and FOXO [1]. Expression of the genes downstream to these three TFs was used to quantify the activity of AMPK. HIF-1 is a transcription factor, and its downstream genes are chosen to evaluate its activity. The downstream genes chosen for AMPK and HIF-1 are listed below. The detailed procedure to obtain these genes can be found in [2]. AMPK downstream gene list (33 genes): ACADL, ACADM, ACOX1, ACSL1, ACSL5, ANGPTL4, APOC3, APOE, CAT, CPT1A, CPT2, CYP27A1, CYP4A11, CYP7A1, EHHADH, FOXA2, G6PC, GADD45A, GADD45G, HNF4A, ONECUT2, PCK1, PCK2, PDK4, TOB1, CCND2, DNMT1, G6PC3, MMP9, PRMT1, RUVBL1, ATF4, BAX. HIF-1 downstream gene list (23 genes): ALDOA, BHLHE40, CA9, CCNB1, DDIT4, EGLN3, EPRS, ETS1, IVNS1ABP, KDM3A, MECOM, MXD1, PGK1, SERPINE1, SSRP1, STC2, TFRC, TGFB3, TMEFF1, TMEM45A, VEGFA, ALDH4A1, BNIP3.


Discrete wavelet transforms (DWT)
ITSPCA transforms the input data into the wavelet domain before denoising. This transformation step is carried out by the discrete wavelet transform (DWT). The wavelet transform decomposes signals over dilated (scaled) and translated wavelets. Compared to the Fourier transform, which decomposes signal into sine waves, the wavelet transform has a better temporal resolution. A mathematical description of DWT is [3]: , ,   [4].
In practice, the fast DWT algorithm obtains coefficients level by level by using filters and therefore reducing the computational complexity. This procedure passes the signal x[n] through a series of filters. A treelike diagram called filter banks can help in understanding the method (see example below). First, the signal is simultaneously decomposed using a low-pass filter g and a high-pass filter h, giving the approximation coefficients and detail coefficients of level 1. Since half of the frequencies of the signal have been removed, the filter outputs are subsequently subsampled by 2. The low-pass filter output is then passed through a new low-pass filter and a new high-pass filter but now the cut-off frequency is halved. This process goes on until it reaches the desired level of detail. Due to the nature of this decomposition, the input signal must have a dimension of 2 m , where m is a positive integer.
Here we provide an example of signal x[n] with a size of 64 under a 3-level decomposition. Frequency range of the signal is 0 to f. The filter banks diagram is shown in Supplementary Figure 1B In our case, the symmlet transform was applied to the original data before a sparse PCA method was applied. The levels and number of vanishing moments are specified in Supplementary Table 2. For an example of a symmlet transform, see Ma section 5.1 [5]. For more information about the symmlet basis, see Mallat [4].

Modularity of the metabolic gene network as a prognostic biomarker for HCC patients with no distant metastasis
To show that modularity of the metabolic gene network is predictive of patients' prognosis independent of metastasis status, we analyzed the association of modularity for 'M0' HCC patients having no spread of tumor to other parts of the body with metabolism phenotypes, stage I-IV, and tumor recurrence. The sample size of 'M0' HCC patients is 266. We analyzed the data of these 266 HCC patients using the same methods as described in the 'Materials and Methods' section.
We observed a modular gene expression pattern of the metabolic genes for these 266 HCC patients, Supplementary Figure 5A. The community structure for these 'M0' HCC patients, Supplementary Figure 5B, identified by the Newman algorithm is the same as that for all the HCC patients, Figure 1C. The association of modularity with metabolism phenotypes, Supplementary Figure 6, tumor stages I or II-IV, Supplementary Figure 7, and recurrence, Supplementary  Figure 8, is consistent with the results for all HCC patients analyzed in the main text. In summary, modularity is a strong predictor, independent of metastatic status, for HCC patients. Adjustable constant in the iterative thresholding steps, β, which is directly related to level sparsity in cleaned eigenvectors.