Introduction

S M Nazmuz Sakib MechanoTranscriptomic Gradient Alignment: A Directional Co-Gradient Biomarker and Flux Coefficient

We introduce the S M Nazmuz Sakib MechanoTranscriptomic Gradient Alignment (MTGA) framework for solid tumors, formalizing a directional coupling between tissue stiffness gradients and malignant cell-state gradients. The core statistic, the Sakib Alignment Index κS, averages the local cosine of the angle between ∇E (stiffness) and ∇S (cell-state score) with scale-aware weighting; the companion Sakib Flux Coefficient μS estimates a signed mechanosensitivity slope relating ∇S to ∇E. We describe multi-scale estimation, spatially autocorrelated nulls, registration/stability diagnostics, and edge-versus-core enrichment. Using synthetic data and analysis-ready plotting primitives, we provide ten ready-to-compile illustrations. Contextualized against durotaxis and spatial transcriptomics , and recent mechanotranscriptomic analytics , the framework appears conceptually novel: prior work studied stiffness heterogeneity and gene-expression gradients, but not a single directional alignment index nor a signed flux fit across tumor sections. We outline how to apply MTGA on AFM/MRE/SHG or force-inference layers co-registered to Visium-like grids , with spatially-constrained nulls via Moran spectral randomization

Introduction

Tumors exhibit spatial heterogeneity in extracellular matrix (ECM) stiffness that affects invasion, EMT, and therapy response [1-4, 5-9]. Durotaxis migration along stiffness gradients has been demonstrated at cell and tissue scales. In parallel, spatial transcriptomics (ST) routinely reveals core-to-edge gene-expression architectures predictive of outcomes. While computational alignment of ST datasets is advancing, and mechanotranscriptomic integration at single-cell resolution is emerging, a directional co-gradient scalar summarizing alignment between ∇E (stiffness) and ∇S (cell state) across a tumor section has not been formalized [1-4].

<italic id="italic-1">The S M Nazmuz Sakib MTGA Framework</italic>

Definition 1 (Sakib Alignment Index κS). On a tissue domain Ω with stiffness map E (x) and cell-state score with κS ∈ [-1,1]; +1 indicates perfect co-alignment and -1 anti-alignment. Here w(x) weights (e.g., tumor mask × spot density), and (α, β) ≥0 emphasize informative gradients.

x μ x x Definition 2 (Sakib Flux Coefficient μS (Directed Mechano-Transcriptomic Flux)). Estimate the signed gain linking ∇S to ∇E by the least-squares fit μS = arg min ∑ w (x) ∥∇S(x)-μ ∇E(x) ∥2 = (∑ w (x) ∇E⋅∇S)/ (∑ w (x) ∥∇E∥2). μS has units of S per stiffness and complements κS (direction vs. gain). Report with R² and a spatially-aware p-value.

S M Nazmuz Sakib Principle 1 (Multi-scale MTGA). Compute κ_S (σ) after smoothing (E,S) with scale σ; summarize via the scale-integrated index siMTGA=1/|Σ|

∑(σ∈Σ)κS (σ) (log-spaced Σ), revealing whether coupling is fine-grained (edge) or coarse (tissue-level).

S M Nazmuz Sakib Hypothesis 1 (Edge Enrichment). κS is elevated within a finite band of the invasive edge relative to the tumor core in cancers with durotaxis- consistent programs.

<italic id="italic-24c69a8067433eb2257744eaa0488c7d">Data Layers and Registration</italic>

Mechanics layer E (x). Direct stiffness maps can be obtained by AFM on sections or via ex vivo SIM-AFM co-registered to fluorescence. When unavailable, stiffness proxies from SHG/collagen organization or force-inference tensors can be used [4, 10 ,11].

Omics layer S (x). Choose a scalar cell-state (EMT, stemness, hypoxia, pseudotime, therapy-response metagene). ST registration. Align mechanical and ST grids with diffeomorphic tools (e.g., STalign) [12, 13].

<italic id="italic-7188285b8453b98d356b51c3406241df">Estimation, Inference, and Stability</italic>

Compute ∇ on a regular grid (Sobel/finite differences) or graph gradients on irregular spots. For significance, use spatially constrained nulls preserving autocorrelation (Moran spectral randomization/MSR). Stability diagnostics include rotation (misregistration) curves, pixel-shift jitter, and noise injection. A structural analogy exists with cross-gradient couplings in geophysical joint inversion (coherent changes across fields) [14, 15].

<italic id="italic-2">Results on a Synthetic Section</italic>

Using a circular tumor mask with aligned (E,S) plus realistic smoothness, the suite yields: (i) rising κS (σ) with scale; (ii) positive siMTGA; (iii) modest edge>core ΔκS; (iv) rotation and shift sensitivity curves; (v) μS>0 with MSR-based significance (Figures 1-10).

Figure 1. Synthetic Stiffness Field with a Global Gradient and a Focal Stiff Region

Figure 2. Synthetic Cell-state Field with Partial Alignment to E (x)

Figure 3. Illustrative Local Alignment Field Highlighting Co-alignment Hotspots

Figure 4. Scale Profile of the Sakib Alignment Index (synthetic example)

Figure 5. Spatially Aware Significance (Moran spectral randomization).

Figure 6. Modest Front-loading of Alignment Near the Invasive Edge (synthetic)

Figure 7. Directional Specificity: κS Peaks at Correct Orientation

Figure 8. Robustness to Small Mis-registrations (center retains positive κS).

Figure 9. Alignment Degrades with Noise Yet Remains > 0 at Moderate Levels

Figure 10. Spatially Constrained Null Distribution (MSR) and Observed μS ≈ 0.314

These compile directly and can be replaced with real-data numbers.

<italic id="italic-f3471f419feec9861b144c9387a6ac06">Discussion and Related Work</italic>

Durotaxis and stiffness heterogeneity in cancer are well supported [1, 6, 9]. Spatial core/edge biology and alignment methods are established [2, 3]. Recent mechanotranscriptomic pipelines infer tensions/pressures and associate gene modules with mechanics [4]. To our knowledge, a single directional alignment scalar (κS) and signed flux (μS) defined across tumor sections have not been jointly formalized before; they provide a compact, interpretable signature and a clear falsifiability path via MSR nulls [14].

Limitations and Usage Notes

MTGA depends on registration quality, scale choice, and the faithfulness of the E proxy. Report stability indices (rotation/shift/noise), edge-vs-core ΔκS, μS with R², and MSR p-values.

In conclusion, we present the S M Nazmuz Sakib MechanoTranscriptomic Gradient Alignment with two complementary readouts: κS (directional alignment) and μS (directed gain). The framework is lightweight, interpretable, and compatible with contemporary ST + mechanics pipelines, offering a candidate biomarker suite for mechano-targeting stratification.

Acknowledgments

Statement of Transparency and Principals

• Author declares no conflict of interest

• Study was approved by Research Ethic Committee of author affiliated Institute.

• Study’s data is available upon a reasonable request.

• All authors have contributed to implementation of this research.

References 24 1 01 7 45 2022 10.1140/epje/s10189-021-00150-6 Pi-Jaumà I Alert R Casademunt J The European Physical Journal. E, Soft Matter Collective durotaxis of cohesive cell clusters on a stiffness gradient 18 1 08 5029 14 2023 10.1038/s41467-023-40271-4 Arora R Cao C Kumar M Sinha S Chanda A McNeil R Samuel D et al Nature Communications Spatial transcriptomics reveals distinct and conserved tumor core and edge architectures that predict survival and targeted therapy response 08 1 12 8123 14 2023 10.1038/s41467-023-43915-7 Clifton K Anant M Aihara G Atta L Aimiuwu OK Kebschull JM Miller MI Tward D Fan J Nature Communications STalign: Alignment of spatial transcriptomics data using diffeomorphic metric mapping 4 04 737-750 22 2025 10.1038/s41592-025-02618-1 Hallou A He R Simons BD Dumitrascu B Nature Methods A computational pipeline for spatial mechano-transcriptomics 18 4 02 1049 14 2022 10.3390/cancers14041049 Ishihara S Haga H Cancers Matrix Stiffness Contributes to Cancer Progression by Regulating Transcription Factors 9 09 839-849 11 2025 10.1016/j.trecan.2025.05.003 Chitty JL Cox TR Trends in Cancer The extracellular matrix in cancer: from understanding to targeting 01 5 05 307 15 2024 10.1038/s41419-024-06697-4 Mai Z Lin Y Lin P Zhao X Cui L Cell Death & Disease Modulating extracellular matrix stiffness: a strategic approach to boost cancer immunotherapy 1467602 15 2024 10.3389/fimmu.2024.1467602 Feng X Cao F Wu X Xie W Wang P Jiang H Frontiers in Immunology Targeting extracellular matrix stiffness for cancer therapy 01 7 04 958-960 84 2024 10.1158/0008-5472.CAN-24-0394 Yui A Oudin MJ Cancer Research The Rigidity Connection: Matrix Stiffness and Its Impact on Cancer Progression 01 06 2024 10.1101/2024.05.27.595975 Shioka I Morita R Yagasaki R Wuergezhen D Yamashita T Fujiwara H Okuda S Ex vivo SIM-AFM measurements reveal the spatial correlation of stiffness and molecular distributions in 3D living tissue 31 1 07 414 3 2020 10.1038/s42003-020-01151-5 Keikhosravi A Li B Liu Y Conklin MW Loeffler AG Eliceiri KW Communications Biology Non-disruptive collagen characterization in clinical histopathology using cross-modality image synthesis 10x Genomics, “Human Breast Cancer: Visium Fresh Frozen, Whole Transcriptome,” dataset page, accessed Oc- tober 14, 2025. https://www.10xgenomics.com/datasets/ human-breast-cancer-visium-fresh-frozen-whole-transcriptome-1-standard 10x Genomics, “Visium HD Spatial Gene Expression Library, Human Breast Cancer (Fresh Frozen),” dataset page, accessed October 14, 2025. https://www.10xgenomics.com/datasets/ visium-hd-cytassist-gene-expression-human-breast-cancer-fresh-frozen 10 1169-1178 6 2015 10.1111/2041-210X.12407 Wagner HH Dray S Methods in Ecology and Evolution Generating spatially constrained null models for irregularly spaced data using Moran spectral randomization methods 1 02 63 13 1997 10.1088/0266-5611/13/1/006 Haber E. Oldenburg D. Inverse Problems Joint inversion: a structural approach