Scaling concepts in ‘omics: Nuclear lamin-B scales with tumor growth and often predicts poor prognosis, unlike fibrosis

Significance

Nonlinear scaling analyses pervade polymer physics and chemistry, yielding characteristic exponents that conceivably apply to expressed genes and their assemblies. Stoichiometric scaling and fractal scaling indeed emerge as gene-gene power laws in human cancer data for key structural polymers in nuclei or extracellular matrix. Only one nuclear envelope factor, lamin-B1, scales with tumor proliferation genes, predicting poor survival for multiple cancers, including liver cancer. These genes include one transcription factor that directly regulates lamin-B1 and broadly categorizes prognosis. In contrast, collagen-1 scales with fibrosis genes, increases in tumors relative to adjacent tissue, and predicts “wound that heals”–type survival only for liver cancer—albeit intertumor comparisons show no correlation without adjacent referencing. Scaling thus identifies and characterizes fundamental gene-gene interrelationships.

Abstract

Physicochemical principles such as stoichiometry and fractal assembly can give rise to characteristic scaling between components that potentially include coexpressed transcripts. For key structural factors within the nucleus and extracellular matrix, we discover specific gene-gene scaling exponents across many of the 32 tumor types in The Cancer Genome Atlas, and we demonstrate utility in predicting patient survival as well as scaling-informed machine learning (SIML). All tumors with adjacent tissue data show cancer-elevated proliferation genes, with some genes scaling with the nuclear filament LMNB1, including the transcription factor FOXM1 that we show directly regulates LMNB1. SIML shows that such regulated cancers cluster together with longer overall survival than dysregulated cancers, but high LMNB1 and FOXM1 in half of regulated cancers surprisingly predict poor survival, including for liver cancer. COL1A1 is also studied because it too increases in tumors, and a pan-cancer set of fibrosis genes shows substoichiometric scaling with COL1A1 but predicts patient outcome only for liver cancer—unexpectedly being prosurvival. Single-cell RNA-seq data show nontrivial scaling consistent with power laws from bulk RNA and protein analyses, and SIML segregates synthetic from contractile cancer fibroblasts. Our scaling approach thus yields fundamentals-based power laws relatable to survival, gene function, and experiments.

Dysmorphic nuclei prevail in tumors and sometimes reflect changes in nuclear lamins that influence nuclear shape and stiffness (1–4). Tumors also generally display abnormal architecture and often differ palpably from normal tissues. Breast tumor stiffness, for example, can be felt in self-examinations and can affect tumor growth in three-dimensional models (5, 6) and fibrotic rigidity is a major risk factor for liver cancer (7). Such physical changes are often attributed to fibrillar collagen accumulation in the extracellular matrix (ECM), especially “stoichiometric” assembly of COL1A1 and COL1A2 proteins (8, 9). With normal tissues, plots of the increased levels of collagen-I from soft brain to rigid bone reveal a characteristic power law increase in levels of Lamin-A,C (LMNA gene), whereas Lamin-B1 (LMNB1) shows little variation normally but is elevated in some cancers, including liver cancer (10, 11). Here, in an analysis across all 32 tumor types and 9,112 patients in The Cancer Genome Atlas (TCGA) as well as a subset with both tumor and normal adjacent tissue data, we seek and indeed discover sets of genes that scale with the above nuclear and ECM factors, and we introduce scaling into machine learning as gene sets are used to predict survival.

Scaling approaches are reasonable to pursue because lamin and collagen polymers assemble into fractal structures with functional properties including stiffness that exhibit characteristic power laws versus concentration (9, 12) (e.g., Fig. 1A). Another structure-function example is provided by hemoglobin binding of oxygen that fits a characteristic power law of ∼3 at half-saturation (the “Hill coefficient”), prompting Pauling to propose a now well-known molecular mechanism (13). We seek similarly reproducible and potentially “universal” scaling exponents rather than correlations measured merely by “goodness of fit.” Scaling pervades physics and includes fiber bending stiffness that scales with fiber thickness (∼h³), and scaling in biology includes Klieber’s power law (14, 15) for metabolic rate ∼ (body mass)^0.75. Scaling in chemistry can reflect stoichiometry (Fig. 1B) and might apply to complex kinetics such as cell cycle (Fig. 1C).

Fig. 1.

Physicochemical scaling concepts in gene-gene expression. (A) Factors A and B distribute differently and scale per fractal physics of a volume and its surrounding surface or according to branched, decorated networks. Mechanical properties that scale with such factors can also yield new scaling relations. (B) Stoichiometric scaling is expected for coregulated factors that form complexes or share a promoter. (C) Genes that follow power laws in time (defined by phase of cell cycle) can also scale with each other. (D) UMAP clustering (i) of tumors relates to organ systems rather than patient survival, based on mRNA expression data for 20,530 genes in 9,112 patients across 32 cohorts of cancer types (SI Appendix, Table S1). However, (ii) SIML reduces the genes to just those that scale with LMNB1 in 17 cancer types, and these are then color coded as “regulated,” which also includes 2 more cancers for which transcription factor FOXM1 scales with LMNB1 expression. (iii) The “regulated” tumors show significantly longer median patient survival.

The characteristic scaling that we discover between some genes in TCGA is supported by our further experiments on tumor proteomics, diverse cancer cell line studies, and single-cell RNA-seq. We show, for example, with two major fibrillar collagen genes that COL5A1 scales substoichiometrically with COL1A1^α across all 32 cancers in TCGA with α = 0.84 (± 0.02, SEM) proving similar to bulk proteomics (α = 0.87) and to single-cell RNA-seq (α = 0.88). Universality across RNA, protein, bulk, and single-cell measurements seems surprising, but we also present a means of scaling-informed machine learning (SIML), which uses the small gene sets that follow characteristic scaling to better understand big ‘omics data. Patient survival predictions based on collagen-I gene sets suggest that, if cancer is a “wound that does not heal” per Virchow, then patients with liver cancer who heal better survive longer.

Results

Survival of Cancer Cohorts Segregate with SIML.

TCGA’s 32 cohorts of cancer types (SI Appendix, Table S1) provides primary tumor mRNA data in a large matrix (9,112 patients × 20,530 genes) well suited to dimensionality reduction and visualization by uniform manifold approximation and projection (UMAP) (16). The machine-learned clusters of patient cohorts distribute based on organ systems (Fig. 1 D, i); all tumors of the digestive system, for example, appear as nearest neighbors. In this unsupervised approach, survival is not an input and no pattern of overall survival of patients is evident; colon cancer has very limited median survival, for example, compared to the nearby cohort of stomach cancer. Moreover, tumor-adjacent “uninvolved” tissue for the 16 cancers with matched data (SI Appendix, Table S1) shows this adjacent tissue clustering close to the corresponding primary tumor tissues (SI Appendix, Fig. S1A). Tissue specialization thus underlies the patterns rather than any clear hallmark of cancer.

SIML was performed as a UMAP analysis of a minimal set of genes that we show below scale with LMNB1 and that proves to be cell cycle centric and includes one cell cycle transcription factor, FOXM1. Proliferation is a hallmark of cancer more so than differentiation, but such scaling will prove evident only in 19 tumor cohorts (of the 32) and might be denoted as “regulated” (Fig. 1 D, ii and SI Appendix, Fig. S1A and Table S1). Compared to the remaining 13 nonscaling tumor types (“dysregulated”), the scaling or regulated cancers show significantly longer median survival (Fig. 1 D, iii). SIML also better distinguishes adjacent-uninvolved tissue for the 16 cancers with such data (SI Appendix, Fig. S1B). This initial analysis motivates a stepwise elaboration of our scaling approach, with a further goal of assessing the relevance of gene-gene scaling to survival of a patient with a specific cancer type.

Expression Scaling across Tumors and Adjacent-Uninvolved Tissues.

The observed coclustering of system-related organs (e.g., digestive system) (Fig. 1 D, i and SI Appendix, Fig. S1A) implies a strong influence of normal tissue gene expression, and so we start with scaling analyses of all 16 cancer cohorts in TCGA (of the 32) that provide expression data for both adjacent-uninvolved patient tissue and tumor (SI Appendix, Table S1). Given that filaments of lamin-B1 assemble quickly around chromatin after mitosis (Fig. 1A), the physics-motivated power law,

$$\left[\mathit{mRNA} \mathit{expression} of \mathit{gene}\right]\sim {\left[\mathit{LMNB}1\right]}^{{\mathit{\beta }}_{\mathit{gene}}},$$ [1]

was first applied to a marker of proliferation, MKI67 (Fig. 2A). Representative tumors show lung adenocarcinoma and breast (not liver) exhibit a reasonably continuous trend from adjacent normal to tumor over a ∼2⁶ range of apparent expression (SI Appendix, Table S2). Given the rule-of-thumb that each decade of data yields one decimal place in a power law, all three tumor types show reasonable scaling with exponents from ∼0.8 to ∼1.3 (R² = 0.6 to 0.8). For normal liver, the lack of scaling and very low MKI67 suggest little to no proliferation.

Fig. 2.

TCGA transcriptomes reveal pan-cancer increases in cell cycle, LMNB1, and collagen-1. (A) RNA reads (RSEM) for proliferation marker MKI67 versus nuclear lamina factor, LMNB1, reveals scaling in all three tumor types (R² > 0.62) but not adjacent lung adenocarcinoma or liver tissue. For clarity, lung adenocarcinoma and breast data are shifted up 5 and 10 units, respectively. (B) RNA reads (RSEM) for the two subunits of the collagen-I heterotrimer scale together across all three cancers and corresponding adjacent tissue (R² > 0.87). For each patient, tumor and adjacent tissue RNA are reported, and tumor shows more collagen-I on average. For clarity, lung adenocarcinoma and breast data are shifted up 3 and 6 units, respectively. (C) Schematic polymer systems of main interest. (D) Standard heatmap of log₂ (fold-change of RNA in tumor relative to adjacent tissue) averaged over all patients for the 16 tumors having >4 patients with adjacent tissue data. Yellow-blue heatmap of log₂ (fold-change of protein in tumor relative to adjacent tissue) as obtained in label-free quantification units from MS for n = 3 liver cancer patients. (E) RNA changes for various matrix mechanosensitive factors including Lamin-A across the 16 cancers; such factors do not vary with either collagen-I or lamin-B. (F) Ratios of (tumor/adjacent) for liver protein and RNA show 85% of data in the first and third quadrants.

Fitting COL1A2 versus COL1A1 (Fig. 2B and SI Appendix, Table S3) yields α_COL1A2 = 0.83 ± 0.11 (SD) per

$$\left[\mathit{mRNA} \mathit{expression} of \mathit{gene}\right] \sim {\left[\mathit{COL}1A1\right]}^{{\mathit{\alpha }}_{\mathit{gene}}}.$$ [2]

The scaling is substoichiometric (α < 1), but R² > 0.84 suggests coregulation over a very large range of apparent expression (i.e., ∼2¹⁰ fold, which is much broader than LMNB1). COL4A2 versus COL4A1 show scaling exponents only ∼12% lower than stoichiometric (R² = 0.84 to 0.95) (SI Appendix, Fig. S2A), and unlike Col1s, these two main genes of basement membrane share a promoter (Fig. 1B) and do not scale well with the fibrous Col1s (SI Appendix, Fig. S2B), consistent with these ECMs being distinct.

Heatmaps of High Cell Cycle, LMNB, and Collagen-I Differ from Varied Cytoskeleton-LMNA.

Conventional heatmaps (tumor/adjacent) (Fig. 2 C–F) always show higher COL1A1 in tumors as well as higher B-type lamins (LMNB1 and LMNB2) and higher cell cycle plus DNA repair genes for progression through cell cycle checkpoints (17). Protein changes for liver (tumor/adjacent) conducted by quantitative mass spectrometry (MS) proteomics analyses (n = 3 patients) show very good concordance with TCGA transcriptome trends. Structural proteins are generally abundant and well suited for accurate quantitation. Overall, >85% of proteins and RNA concurred as up-regulated or down-regulated (Fig. 2F), and LMNB1 and LMNA were up in both analyses as were all three MS-detected, cell cycle–related DNA repair factors.

Unlike LMNB1, LMNA is lower than adjacent in about half the tumors and shows roughly similar variability to some of the cytoskeleton and adhesion structure genes. To relate such observations to a limited literature, we focus on liver and its differences from lung and breast cancers (R² = 0.4; SI Appendix, Fig. S2C). Liver shows relatively higher basement membrane ECM (e.g., collagen-4), adhesions-cytoskeleton, and also LMNA—unlike lung and breast cancer. Low Lamin-A/C in lung (18) and breast (19) cancers potentially facilitates invasion and growth based on knockdown studies in vitro and in vivo with a lung cancer line (1). In various two-dimensional cultures, LMNA promotes adhesive spreading and actomyosin assembly (9) downstream of the serum response factor (SRF) pathway (8, 20), and high LMNA in a few cancers including liver cancer (Fig. 2E) favor coclustering with SRF coactivators MKL1, MKL2, and/or their target nonmuscle myosin-IIA (MYH9) (8). Paxillin is very similar as are basement membrane collagen-4s (unlike collagen-1), but so is YAP1 in the Hippo pathway for growth (21), differing from SRF and its target adhesion factor vinculin (22). Overall, the results are consistent with the known matrix mechanosensing pathway of LMNA-contractility-adhesion downstream of basement membrane density (or stiffness) (Fig. 2E, Sketch), but importantly this pathway segregates from LMNBs.

Proliferative Genes Scale with LMNB1 and Predict Poor Survival.

In analyzing all 32 cancer cohorts, scaling with LMNB1 across patients with a given tumor type is evident for some cancers (such as liver) for well-known cell cycle genes TOP2A, FOXM1, and MKI67 (Fig. 3 A and B and SI Appendix, Fig. S2A, n = 371 patients). The results are consistent with the smaller dataset (Fig. 2A, n = 50). TOP2A is among the top 24 most up-regulated genes (tumor versus adjacent normal) together with 15 other genes (mostly proliferation) that also scale well with LMNB1 (SI Appendix, Table S4). The negative power law for the hepatocyte-expressed, complement-related factor MASP2 (Fig. 3C) is consistent with a dedifferentiation given that such mRNAs are maximized when cell cycle is protracted (23). Such genes differ in lung cancer (SI Appendix, Fig. S2B), consistent with a distinct lineage. Such observations again reinforce proliferation as more of a cancer hallmark than lineage-specialization (Fig. 1D).

Fig. 3.

Scaling of LMNB1 in liver tumors (n = 371 patients) and all 32 tumors. (A–C) Power laws versus LMNB1 RNA for cell cycle genes TOP2A and FOXM1 and opposite trend for differentiation gene MASP2. The R² and scaling exponent (${\mathit{\beta }}_{\mathit{gene}}$) are indicated. (D) Genes that scale with LMNB1 RNA. For all 17,958 genes, ${\mathit{\beta }}_{\mathit{gene}}$ versus R² gives a sideways-volcano plot; 168 genes scale strongly ${\mathit{\beta }}_{\mathit{gene}}$> 0.5 and well R² > 0.5, and most relate to mitosis. (E) Gene copy numbers of LMNB1 in liver cancer patients is almost linear in RNA expression, although patients with a single copy are bimodally distributed and only the lower mode fits the trend. (F) Schematic Venn diagram of overlapping gene sets that scale with LMNB1 in 32 cancers. (G) LMNB1 strong scaling genes show overlap across tumors, decreasing as the number of tumors being considered increases (SI Appendix, Gene Overlap). (Inset) A maximum of 25 genes scale with LMNB1 across 17 tumors. The 19 of 32 tumors where transcription factor FOXM1 scales with LMNB1, denoted as “regulated” tumors include the 17 tumors plus testicular and large B-cell lymphoma. (H) For patients with LMNB1 RNA levels that exceed the respective median levels for all patients, the median time for survival is significantly shorter (P < 0.05) by 2 to 3 y in Kaplan Meier (KM) plots. (I) For all genes, similar KM analyses are summarized by the hazard ratio of the two cohorts plotted against the P value, yielding 3,464 genes that show significant differences, including 161 (of 168) genes that scale with LMNB1 and show poor survival when expression exceeds the median. (J) High expression of FOXM1, TOP2A, and LMNB1 predicts poor survival in 9, 8, or 7 of the 17 tumor types, with the 7 cancers being: liver cancer, pancreatic cancer, adrenocortical carcinoma, lower-grade glioma, sarcoma, kidney clear cell, and kidney papillary cell. Among the other 15 cancers, only melanoma for LMNB1, mesothelioma for FOXM1, and both cancers for TOP2A show significantly poor survival in patients with higher expression.

Scaling exponents ${\mathit{\beta }}_{\mathit{gene}}$ and R² of the fits for all ∼20,000 expressed genes (per Eqs. 1 and 2) yield a “sideways-volcano plot” (Fig. 3D), and the genes that scale best (i.e., ${\mathit{\beta }}_{\mathit{gene}}$ > 0.5, R² > 0.5) are proliferation related (83% of 168 genes for liver cancer). To assess whether LMNB1 transcript levels relate to the number of copies of the LMNB1 gene, genomic data for each liver cancer patient were analyzed (24), noting that copy number variations are consistent with malignancy and can predict patient survival (25–28). Transcript levels binned on gene copy number fit a power law of ∼1 (Fig. 3E)—although patients with one copy of the LMNB1 gene are bimodal distributed and only the low expressors fit the trend. Consistent with gene dosage DNA → mRNA → protein, our proteomics for liver cancer detected LMNB1 plus three proliferation factors that are all up-regulated in tumors (Fig. 2 D and E). Experiments in cancer cells will test directly whether LMNB1 DNA levels indeed affect proliferation.

Scaling with respect to MKI67 yields a similar number of genes that are >95% identical to the LMNB1 gene set (SI Appendix, Fig. S2C), underscoring invertibility of the scaling, whereas fits on linear scales (i.e., Pearson) yield <30% of these genes. When repeated for all 32 cancers, 25 show the mitotic factor KIF20A is the gene that scales most often with LMNB1, and a total of 866 unique genes also scale across 27 of the 32 (with 5 showing no genes that scale), with only 242 genes appearing in more than one tumor (Fig. 3F). The number of genes that are shared across multiple cancers clearly decreases with the number of cancers considered, but the theoretical solution to the maximum number of common set of genes is an open problem in mathematics known as the maximum k-subset intersection problem, which lacks even approximate solutions (29). To compute, for example, the maximum number of strongly scaling genes that overlap across 17 of the 32 cancers, the brute force method must consider all ³²C₁₇ = 565,722,720 combinations.

As a heuristic approach for computation, we represented the genes in a matrix populated with indicator variables (SI Appendix, Supplementary Theory: Gene Overlap). For LMNB1 scaling genes, it so happens that tumors that show maximum overlap of genes for a particular number of tumors (n) being considered are also present in the group that shows maximum gene overlap in n + 1 tumors with the addition of another tumor. We listed the cancer which, when added to a growing group of cancers, gives the maximum number of genes that overlap (Fig. 3G). Based on this approach, for example, the maximum overlap of genes obtained from all combinations of choosing 17 tumors is 25 genes. Notable among the 25 genes that scale with LMNB1 is FOXM1 (Fig. 3E), which is the only transcription factor and a regulator of cell cycle (30, 31). FOXM1 scales strongly with LMNB1 in the 17 cancers plus 2 more (thymoma and a lymphoma), and all were denoted as the “regulated” cancers among TCGA’s 32 cancers (per Fig. 1 D, ii). TOP2A is also a notable mitotic gene regulated directly by FOXM1 (32).

The biological significance of these scaling results is best assessed by patient survival. For liver cancer (371 patients), median survival is ∼3 to 4 y for high LMNB1 patients but almost twice as long for low LMNB1 (Fig. 3H). This partitioning is independent of etiology including alcohol or hepatitis (SI Appendix, Fig. S3D). Across all genes, the median survival for high/low expressors shows a significant hazard ratio for 3,464 genes, and among the genes that predict poor survival when expression is high (2,111), >95% of the genes scale with LMNB1 (Fig. 3I). Staging of primary tumors in terms of size and invasion also shows LMNB1 increasing (SI Appendix, Fig. S3 E and F), whereas LMNA and COL1A1 show no trends. A sideways-volcano plot for LMNA shows only one gene that scales strongly (SI Appendix, Fig. S2F), which underscores the distinctive significance of LMNB1. Across the 32 cancers, LMNA shows strong scaling genes in very few cancers, with only 75 genes appearing in >1 cancer, dropping down to a maximum overlap of one gene in groups of four tumors (SI Appendix, Fig. S3G); LMNA does not predict patient survival (SI Appendix, Fig. S3H). More generally, among the 17 cancers that are regulated with 25 overlapping genes or more scaling with LMNB1 (Fig. 3G), patients with high expression of LMNB1, FOXM1, and TOP2A are predicted to have poor survival in 7 to 9 cancers (Fig. 3J). Only thymoma shows prosurvival with high LMNB1 and high TOP2A, which might reflect a role for Lamin-B1 in thymus development of T cells (33, 34). In contrast, only 1 to 2 cancers among the remaining 15 show high LMNB1, FOXM1, and TOP2A and can have any significance in predicting (poor) survival. This highlights the regulated versus dysregulated distinction even for well-accepted cell cycle markers FOXM1 and TOP2A.

Lamin-B2 is farnesylated like lamin-B1 (unlike mature lamin-A), and tight association of both LMNBs with lipid at the nuclear envelope implies scaling with chromatin mass (Fig. 1A). Studies of proliferating embryonic cardiomyocytes also suggest LMNB2 regulates microtubules (35). In our scaling analyses, LMNB2 is more like LMNA with a maximum overlap of one gene in groups of eight tumors, but genes that scale with LMNB2 nonetheless predict poor survival as per LMNB1 (SI Appendix, Figs. S3I and S4). Moreover, FOXM1 also scales with LMNB2 (SI Appendix, Fig. S4), which again motivates experiments on FOXM1 regulation of LMNB1.

FOXM1 Directly Regulates LMNB1 Expression, But Perturbing LMNB1 Perturbs the Cancer Cell Cycle.

Chromatin-immunoprecipitation with anti-FOXM1 followed by sequencing (ChIP-seq) shows FOXM1 binds the promotor regions of LMNB1 as well as cell cycle genes TOP2A, KIF20A, and KIF11 in two cell lines (Materials and Methods). No signals were detected for either LMNA or the constitutive heat shock gene HSP90AA (Fig. 4A). To test LMNB1 gene regulation by FOXM1, we designed a promoter-reporter plasmid and transfected it into A549 lung adenocarcinoma cells and U2OS osteosarcoma cells for detection and perturbation of green fluorescent protein (GFP) reporter signal (Fig. 4 B and C). Siomycin reduces FOXM1 expression and promotes its degradation (36, 37), and FOXM1-i inhibits DNA binding at the consensus sequence (38). Cells treated with Siomycin (for >2 h) and FOXM1-i (at ∼6 h) show less GFP expression, consistent with reduced LMNB1 promotor binding (Fig. 4 D and E). FOXM1 inhibition decreases mitotic cell numbers (by ∼6 h) as expected (39). FOXM1 inhibition suppresses Lamin-B1 protein and TOP2A (Fig. 4 F and G), whereas mitotic cell accumulation induced with Nocodazole (18 h) increases Lamin-B1 (Fig. 4F), as also occurs with FOXM1 (39). The results are consistent with simple cell cycle trends (Fig. 1C).

Fig. 4.

FOXM1 transcription factor regulates Lamin-B1 expression. (A) ChIP-seq analyses show FOXM1 binds to the promotor regions of cell cycle regulated genes including TOP2A, KIF11, KIF20A, and two sites in LMNB1. No binding is evident in LMNA and HSP90AA. (B–E) GFP promoter-reporter for assessing regulation of LMNB1, and its transfection into A549 and U2OS cancer cell lines shows decreased GFP in immunoblots of cells treated with FOXM1 inhibitors Siomycin for 18 h and FOXM1-i for 5 to 7 h (representative of n = 3). Additional analysis of GFP in images of U2OS cells normalized to DNA, and again shows reduced GFP expression after 7 h of Siomycin, which also reduces mitotic cell counts as expected (average ± SEM for >50 cells measured). (F and G) Lamin-B1 protein is also decreased by FOXM1 inhibition (Siomycin, 18 h) as is TOP2A (5 h), but enriching for mitotic cells with Nocodazole increases Lamin-B1. (H–K) Mitotic and G2-phase A549 cells show high intensities of stained DNA, anti-FOXM1, and gene-edited Lamin-B1, in contrast to G1-phase cells with low to negligible anti-FOXM1 intensity. FOXM1 inhibition (6 h) reduces anti-FOXM1 intensity as expected, but FOXM1 increases in cell cycle as DNA intensity doubles. FOXM1 and LMNB1 scale linearly at protein levels through the cell cycle.

Anti-FOXM1 immunofluorescence and LMNB1 gene-edited with red fluorescent protein (RFP) both show nuclear localization in interphase and chromatin association in mitosis (Fig. 4H), and FOXM1 inhibition decreases anti-FOXM1 intensity as expected. Intensity analyses further show FOXM1 increases through cell cycle (Fig. 4J), with many G0/G1 cells showing no FOXM1, consistent with its degradation at mitotic exit (40, 41). Importantly, Lamin-B1 increasing linearly with FOXM1 (Fig. 4K) is in excellent quantitative agreement with the scaling analyses of 17 regulated cancers in TCGA (Fig. 3G).

Lamin-B1 dynamics differ from FOXM1: LMNB1 disassembles and is inherited by daughter cells (42), remaining nonzero throughout cell cycle as confirmed by imaging of live or fixed cells (Figs. 4K and 5 A and B). Note that endogenous promoters for the gene-edited RFP-LMNB1 and GFP-histone-H2B (43) avoid overexpression artifacts. Power law increases in RFP-LMNB1 and GFP-histone-H2B per nucleus allow one to eliminate the time dependence in gene-gene scaling with cell cycle (per Fig. 1C and SI Appendix, Supplemental Theory). This is important given that tumors show differences in mean doubling time and is also consistent with scaling between cancer patients (Fig. 3 A–D).

Fig. 5.

LMNB1 scales with DNA in vitro and LMNB1 levels modulate cell cycle as a proto-oncogene. (A and B) Live cell imaging of gene-edited lung cancer line expressing GFP-H2B and RFP-LMNB1 shows parallel increases in intensities normalized to telophase in mitosis with a mean cell cycle of 21 ± 0.35 h. (C) Intensity of RFP-LMNB1 is linear versus Hoechst-stained DNA in fixed A549s, even with lamin-A knockdown. Intensities are normalized to cells in the nonreplicated state, “2N.” (D) Anti-LMNB also shows LMNB increases linearly with DNA in osteosarcoma derived U2OS cells. (E–G) EdU incorporation (1 h) in replicating cells in combination with Hoechst-stained DNA identifies cell cycle stage (G1/S/G2). Knockdown with shLMNB1 in U2Os cells reveals a smaller percentage of cells proceeding to S and G2 compared to overexpressing GFP-LMNB1 cells. Error bars indicate SEM values across image fields.

RFP-LMNB1 and anti-LMNB1 intensities not only scale with total DNA staining intensity per nucleus, including high ploidy cells relevant to cancer (Fig. 5 C and D), but LMNA knockdown has no effect. This is consistent with the independence and absence of LMNA from the LMNB1-scaling set (Fig. 3D) and with distinct roles for lamin genes in mechanosensing (LMNA) (Fig. 2F) versus proliferation (LMNBs). To assess effects of LMNB1 on cancer cell proliferation, we transfected GFP-LMNB1 plasmid or shLMNB1 plasmid into U2OS cells and pulsed incorporation of the nucleotide analog EdU (for 1 h) together with DNA staining to quantify cell cycle stage (43). High EdU signal indicates ongoing DNA synthesis (S-phase), and low EdU signal indicates G1 or G2 phases, depending respectively on low or high DNA intensity (i.e., “2N” or “4N”) (Fig. 5E). Knockdown cells show more cells in G1 compared to overexpressing cells (identified by GFP signal at the single-cell level), whereas the latter were more in S and G2 phases (Fig. 5F), implying low LMNB1 suppresses cancer cell cycle (Fig. 5G). These cancer cell line results agree with fibroblast results (44) and imply LMNB1 is a possible therapeutic target to suppress cancer.

Pan-cancer Exponents for Genes Scaling with LMNB1, ACTA2, COL4A1, and COL1A1.

The “universality” of power law scaling exponents across cancers is illustrated by results for distinct sets of 17 cancers (from ref. 31) for LMNB1 (Fig. 3E and SI Appendix, Table S1) and a few other genes, noting also that 17 is a majority of the 32 total tumor types in TCGA. LMNB1 scaling genes yielded 25 genes that are all cell cycle related, including FOXM1, with exponents across tumors that range from 0.83 to 1.23 (Fig. 6A). Higher values (>1) are consistent with a lamina (∼Area) that surrounds replicating DNA (∼Volume) (Fig. 1A) or perhaps a bias from early or late cell cycle genes (Fig. 1C). In comparison, all four genes that scale with the cytoskeleton gene ACTA2 across 17 cancers have superstoichiometric exponents (>1), and three are also cytoskeleton while one is a membrane calcium regulator (Fig. 6B and SI Appendix, Table S1).

Fig. 6.

Pan-cancer power law exponents. (A–D) Circles indicate characteristic exponents for genes in TCGA that show indicated scaling with slope > 0.5 and R² > 0.5 across 17 primary tumor cancer types (SI Appendix, Gene Overlap). Blue squares indicate exponents obtained from proteomics across normal tissues and model tumors. Exponents from single-cell RNA-seq: red squares, hepatocellular carcinoma (HCC); green squares, lung cancer–derived A549 line; and pink square, average exponent of HCC and A549. For LMNB1, 25 genes show maximum overlap in only one grouping of 17 tumors; scaling in scRNA-seq rounds down to the first significant digit due to low reads of LMNB1 (pink). For ACTA2, two combinations of six genes show maximum overlap in groups of 17 tumors and four genes that appear in both combinations. For COL4A1, 21 combinations of seven genes show maximum overlap in groups of 17 tumors; four genes that appear in >18 combinations. For COL1A1, 11 combinations of nine genes show maximum overlap in groups of 17 tumors; nine genes that appear in >6 combinations. (E) Machine-learned, dimensionally reduced projection (UMAP) of single-cell mRNA sequencing of liver cancer (HCC) labeled with standard marker identified lineage. Each point is a cell, with purple indicating detected expression levels of COL1A1, COL1A2, ACTA2, FAP, and CNN1. (F) Power law fits of raw reads from single-cell mRNA sequencing: (i) LMNB1 vs. MKI67 for HCC or A549s. For HCC, ∼45% of cells with nonzero reads of LMNB1 or MKI67 are T cells, and <5% are malignant cells. (ii) COL4A1 vs. COL4A2 from HCC patients or A549 cells. Pie charts for HCC: <5% of malignant cells have nonzero reads for COL4A1 or COL4A2, versus ∼60% of endothelial cells, ∼35% are fibroblasts. (iii) ACTA2 vs. CNN1 from HCC scale together as myofibroblastic genes and show similar cell-type specific expression profiles.

Four genes also scale with COL4A1 for a different set of 17 cancers and again show a range of power laws for these basement membrane ECM genes (Fig. 6C and SI Appendix, Table S1), including a pan-cancer exponent for COL4A2 of 0.92 ± 0.02 within 10% of expected results for a shared promoter (Fig. 1B). Ten fibrillar ECM or membrane genes scale with COL1A1 (Fig. 6D), including COL1A2 as noted previously (Fig. 2B). Membrane factors include one integrin and also fibroblast activating protein (FAP) which is a membrane-bound gelatinase targeted in the clinic (45). As with COL4A1, all COL1A1 exponents are substoichiometric (i.e., <1), consistent with these major ECM components serving as scaffolds for the other factors to decorate like leaves on a tree (Fig. 1A). Equally important, scaling of COL5A1 ∼ COL1A1°^0.84–0.92 from TCGA for 17 cancers (and across all 32 cancers in TCGA α = 0.84 ± 0.02, SEM) agrees with MS-derived scaling proteomic results (9) based on COL1A1 ∼ E^1.5 and COL5A1 ∼ E^1.3 giving COL5A1 ∼ COL1A1°^0.87. Exponents for other genes agree on average within 20%, and agreement of tumor and normal transcript and protein is evidence of “universality” in scaling.

Single-Cell RNA-seq Clarified with SIML.

Bulk sequencing of tumors mashes together a diversity of cell types at different stages of cell cycle, and thus motivates scaling analyses of single-cell RNA-seq data (scRNA-seq). Liver cancer biopsies from chemotherapy-treated patients show many types of stromal and immune cells plus some malignant hepatocellular carcinoma cells (HCC) in recent scRNA-seq (46). UMAP clusters include cancer-associated fibroblasts (CAFs) with detectable reads for COL1A1, COL1A2, FAP, ACTA2, and CNN1 (Fig. 6E and SI Appendix, Fig. S5). COL1A1 and all of its pan-cancer scaling genes (Fig. 6D and SI Appendix, Fig. S5) are high in a subpopulation of CAFs distinct from those expressing ACTA2 and its bulk-scaling genes (Fig. 6B and SI Appendix, Fig. S5). Fibroblasts with high FAP are reportedly distinct from fibroblasts with high ACTA2 (smooth muscle actin protein, αSMA) in the stroma from cancer patients (47–49), tumor xenografts (50), and in vitro experiments (51), but what genes best distinguish the two CAF phenotypes across cancers has been uncertain. Power law scaling discriminates “fibrotic” synthetic CAFs (COL1A1, FAP…) from “fibrotic” contractile CAFs (ACTA2, CNN1…) and shows SIML as a general method to better define and resolve lineages.

LMNB1 scRNA-seq reads are dominated by the easily extracted (nonadherent) T cells (Fig. 6 F, i and SI Appendix, Fig. S6 A and B) even though cancer cells such as A549s clearly express LMNB1 (Fig. 6 F, i and SI Appendix, Fig. S6C). Importantly, LMNB1 and FOXM1 reads are coincident in cell populations (SI Appendix, Fig. S6 B and C) indicative of LMNB1 expression regulation (Fig. 4). Scaling exponents for LMNB1-scaling genes that were averaged for liver cancer and A549 scRNA-seq also correspond well to those from TCGA’s pan-cancer exponents (Fig. 6D), even though only the first decimal place can be considered reproducible for scaling in low scRNA-seq (given reads typically span less than a decade). Collagens are far more abundant, and CAFs and endothelial cells express COL4A1, A2 isoforms that scale with exponent = 0.83 versus 0.86 for A549s (Fig. 6 F, ii and iv) and 0.92 for bulk from TCGA (Fig. 6C). ACTA2 scaling gene CNN1 shows weaker scaling in scRNA-seq (Fig. 6 F, iii) by ∼20% than the pan-cancer exponents in bulk TCGA, which is about typical for deviations. Nonetheless, raw reads in scRNA-seq scaling analyses seem consistent with “universality.”

ECM Fibrosis Genes Scale with COL1A1 across Cancers and Predict Liver Cancer Survival.

Lastly, because fibrotic collagen-1 and cell cycle lamin-B1 are both high in all 16 cancers for which tumor and adjacent TCGA data are available (Fig. 2D), scaling with COL1A1 was scrutinized, starting with primary liver cancer (n = 371). COL1A1 expression varies ∼16,000-fold (versus ∼32-fold for LMNB1 in Fig. 7A) and does not relate to genome copy number changes in COL1A1 (SI Appendix, Fig. S7A), consistent with expression by nonmalignant stromal cells (found also in adjacent and normal tissue). Even though fibrous COL1A1 and most associated ECM are abnormally high within a given patient across all cancers (Fig. 2D), these genes do not scale with LMNB1 (Fig. 7A: α_LMNB1 ∼0 and R² < 0.1), whereas COL1A2 and an ECM protease scale substoichiometrically: α_COL1A2 = 0.86, α_MMP2 = 0.79 (Fig. 7 B and C). Indeed, the majority of the 162 scaling genes are ECM (α_gene > 0.5, R² > 0.5) (Fig. 7D), and even COL4s fit well but with weak scaling (α_gene < 0.5, R² > 0.5) that suggests (bulk/surface) scaling (Fig. 1A). Only COL5A1 and COL3A1 show R² > 0.9 (Fig. 7D), and these associate with collagen-1 fibers (52), scale with COL1A1 in proteomics (9), and scale “universally” across all 32 cancers (Fig. 7D and SI Appendix, Gene Overlap).

Fig. 7.

LMNB1 does not scale with COL1A1 (fibrous ECM) and COL1A1 and myofibroblastic ACTA2 are prosurvival in primary liver tumors (n = 371 patients). (A–C) Plots versus COL1A1 for LMNB1 (which shows no scaling) and for ECM genes COL1A2 and MMP2. The R² and scaling exponent ($a_{\mathit{gene}}$) are indicated. (D) Genes that scale with COL1A1. For all 17,958 genes, $a_{\mathit{gene}}$ versus R² gives a sideways-volcano plot; 162 genes scale strongly $a_{\mathit{gene}}$ > 0.5 and well R² > 0.5, and more than half relate to ECM. (E) For COL1A1 and COL1A2, (tumor/adjacent) RNA levels that exceed the median of all patients (n = 50) show longer survival by 2 to 3 y. (F) For all 17,958 genes, KM analyses are summarized by the fold-change in median survival plotted against the P value, showing that many of the genes that scale with COL1A1 or with ACTA2 predict prolonged survival when the expression ratio (tumor/adjacent) exceeds the median. (Inset) Liver cancer is the only one of the 16 cancers with matched adjacent tissue data that predicts survival in patients. (G) Liver tumor stiffness measured before therapy and 6 wk later (54) was converted to calculated changes in collagen-1 (Col1 ∼ E^1.5) and plotted versus patient survival (n = 7 patients), revealing strong power laws.

To assess survival of cancer patients in relation to “fibrotic” genes, we focused first on the 16 (tumor/adjacent) cancers. This is because, unlike LMNB1, which is expressed by malignant cells (and predicted poor survival in seven cancers including liver cancer; Fig. 3 G and I), COL1A1 and COL1A2 are expressed primarily by fibroblasts found in both tumor (i.e., CAFs) and adjacent tissues (Fig. 6E). Most scaling genes associated with CAFs (i.e., ACTA2 and COL1A1 gene sets per Fig. 6 B and D) predict prolonged survival in liver cancer (∼2 y versus ∼4 y in Fig. 7 E and F). Equally surprising is that no other tumor types show significance with these genes. Alternatively, survival can be assessed by comparing expression in liver tumors from many different patients (with no comparison to adjacent tissue), and this analysis shows prolonged survival for high levels of most (80%) of the contractile genes including ACTA2, whereas no significance was found for survival with high levels of most (89%) of the synthetic genes including COL1A1 (SI Appendix, Fig. S7 B–D). The results underscore contractile versus synthetic made clear by SIML (Fig. 6E).

Discussion

Specific values of power law exponents are informative. For example, high levels of the ACTA2 contractile gene set are prosurvival in liver cancer for (Tumor/Adjacent) data and for larger patient cohorts of tumor-only data except for the contractile gene with the lowest power (TAGLN in SI Appendix, Fig. S7D). All contractile genes scale strongly with ACTA2 (exponents >1). In comparison, the COL1A1 synthetic gene set is prosurvival only in (Tumor/Adjacent) data and is not significant in nearly all of the tumor-only data (Fig. 7 E and F and SI Appendix, Fig. S7D); furthermore, unlike ACTA2 exponents, all (except POSTN) of the synthetic genes scale substoichiometrically with COL1A1 (exponents <1) (Fig. 6D). A reasonable analogy is that collagen-1 forms the woody bulk of a tree that the other factors assemble onto like bark or leaves on the tree. Regardless, the contractile gene set is thus more sensitive to changes in ACTA2.

Our recent experiments (53) show ACTA2 protein increases with a switch-like cooperativity exponent of ∼3 versus a mechano-repressive transcription factor that exits the nucleus in mesenchymal cells spreading on stiff collagenous substrates. COL1A1 shows no change when we overexpressed the same transcription factor in cells on rigid plastic. Specific power law exponents help to indicate strong (ACTA2) or weak (COL1A1) regulation—extending to patient survival data. It should also be noted that treatment by radiation or pharmaceutical for each patient in TCGA with liver, lung, or breast cancer shows no impact on these key scaling results (SI Appendix, Fig. S8).

Prosurvival results for high expression of fibrotic genes in liver cancer patients is surprising but consistent with a recent study of patients having liver stiffness measured by magnetic resonance elastography before immunotherapy and 6 wk afterward (54). Remarkably, patient survival associated with increased local liver stiffness (rather than initial or final stiffness). Such results for stiffness (E) are readily converted to collagen-I changes via COL1A1 ∼ E^1.5 based on MS of tumors and normal tissues (9). Power law fits for survival metrics reveal strong scaling over a ∼2²-fold calculated range of collagen-I that increases or decreases after treatment (Fig. 7G). Virchow’s notion that cancer is a “wound that does not heal” might thus imply cancer patients who heal better survive longer.

Compared to fibrosis scaling genes, the more numerous genes in the LMNB1 scaling gene set are cell cycle related, exhibit a broader range of scaling exponents (>1 and <1), and include exactly one transcription factor FOXM1. Cancers that show scaling of FOXM1 with LMNB1 can be considered “regulated” because we show FOXM1 regulates LMNB1 (Fig. 4). Better median survival is also evident for such cancer cohorts versus (nonscaling) dysregulated (Fig. 1D, iii). However, scaling applies across low and high expression and within a cancer cohort, high expression of FOXM1 and LMNB1 predict poor survival in 7 to 9 cancer types (Fig. 3J). Liver cancer was again appropriate to illustrate survival because Lamin-B1 is not only high in malignant hepatocytes but is even a circulating biomarker (10, 11). Cell culture studies show LMNB1 depletion inhibits DNA replication (55, 56) and LMNB1 knockout undermines development earlier than LMNA knockout (57)—all of which is consistent with cell cycle scaling with LMNB1 but not LMNA (Fig. 3 D and G and SI Appendix, Fig. S3 F–H). Such a difference is also evident in UMAP plots of scRNA-seq data (SI Appendix, Fig. S6), which confirm actively proliferating cells colocalize high LMNB1 with MKI67, FOXM1, and TOP2A, whereas LMNA is more broadly expressed across cell types and more similar to COL4s (SI Appendix, Fig. S6 B and C). Thus, even though all solid tumors show up-regulation of fibrous ECM genes, especially COL1A1 (Fig. 2D and SI Appendix, Table S1), such changes and their impact are independent of lamin-B1. The reason for the FOXM1 driven increase of LMNB1 in cell cycle is likely to protect the increased amount of DNA in the expanding nucleus (Fig. 5 A–D) since low levels of LMNB1 at sites of high curvature favor nuclear rupture, DNA damage, and repression of cell cycle (58).

Scaling Is Interchangeable, But Weak Scaling Is Lost in Noisy Data.

Lastly, conventional heatmap presentations can suggest associations such as between COL1A1 and LMNB1 (Fig. 2 C–F) where none exist. However, a key strength of quantitative power law relationships is the ease of predicting new power law relations:

$$\text{If}\hspace{0.17em}A\sim B^{\text{a}}\hspace{0.17em}\text{and}\hspace{0.17em}\text{if}\hspace{0.17em}C\sim B^{\text{c}}\text{,}\hspace{0.17em}\text{then}\hspace{0.17em}A\sim C^{\text{a/c}}.$$

For example, defining genes A = COL5A1, B = COL1A1, and C = COL1A2 with measured power laws of a = 0.86 and c = 0.93 (Fig. 6A) gives a/c = 1.08. Measurements agree: COL5A1 ∼ COL1A2^1.06 (R² = 0.96). Poorer fits yield poorer predictions of course, but weak scaling genes are also generally problematic.

Genes that scale weakly with LMNB1 and COL1A1 in sideways-volcano plots (i.e., α < 0.5 and R² > 0.5 in Figs. 3D and 7D) are rare. This is surprising given evidence of genes that scale weakly with COL1A1 in small datasets (59), but a likely reason is illustrated by the noise in COL1A2 scaling (Fig. 7B, Inset: RMSE = 0.44). If we apply this noise to generic power law functions f that are weaker (or stronger) power laws versus COL1A1 than the actual data, then power law fits of these equally noisy f yield R² values that decrease strongly from high R² with the best-fit exponents(SI Appendix, Fig. S9). Noisy data thus explains the lack of weak scaling genes in TCGA data; noise in the liver cancer data, for example, could reflect batch effects from 19 patient cohorts analyzed over ∼3 y in several sequencing centers. Nonetheless, the strong scaling genes and especially the “universal” power laws supported by proteomics-rheology and initial scRNA-seq (Fig. 6) should motivate physicochemical theories for the interacting pathways that underlie proliferation and ECM production as well as patient survival.

Materials and Methods

Detailed descriptions are provided in SI Appendix. Briefly, primary tumor and adjacent tissue data such as mRNA-seq in TCGA was downloaded from the University of California Santa Cruz Xena website (60). All codes developed are labeled in the public repository, GitHub. Data from MS (liquid chromatography–tandem MS) of three liver tumors and adjacent tissues (deidentified following our institutional review board–approved protocol) was analyzed using label-free quantification in MaxQuant (version 1.5.3.8, Max Planck Institute of Biochemistry). Our custom-designed human Lamin-B1 promoter-reporter plasmid is available from Genecopoeia.

Data Availability

All study data are included or referenced in the article and/or SI Appendix.