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From Sparse AFM Observations to Probabilistic Macroscale Mechanics: A Generative-Physics Framework for Plant Cell Walls
Samundra Karki, Angona Biswas, Nabila Masud, Md Hasibul Hasan Hasib, Anwesha Sarkar, Baskar Ganapathysubramanian
TL;DR
Sparse AFM scans capture only part of the variability in developing cotton cell walls, and matrix stiffness is uncertain. The paper expands 8 DPA scans into candidate microstructures, maps them to elastic fields, and propagates homogenized property pairs into a larger-scale simulation. The effective-property distributions shift with matrix-to-fibril stiffness ratio, their joint relationship changes sign across cases, and several estimated densities appear multimodal.
Problem
Sparse AFM scans capture only part of variable wall microstructure, and matrix stiffness is not directly measurable at this scale.
Source: §1 · pp. 1–2Method
LoRA-based image generation, image-to-property mapping, ensemble homogenization, and stochastic upscaling connect sparse 8 DPA AFM scans to larger-scale mechanics.
Source: §5 · p. 14Results
Effective-property distributions shift with matrix-to-fibril stiffness ratio, the Eeff–νeff relationship changes sign across cases, and several estimated densities appear multimodal.
Source: §4 · p. 13Takeaways & Limitations
A single nominal constitutive pair cannot encode the distributional and joint-property variation that the workflow carries into larger-scale mechanics.
Source: §4 · pp. 12–13The three-image comparison is not established as an independent held-out test, and 197 training images derive from only six raw scans.
Source: §4 · p. 13
Abstract
from arXiv · showhide
Nanoscale imaging of developing plant cell walls is expensive, and a limited number of atomic force microscopy (AFM) scans cannot capture the full structural variability of the wall. We present a generative-physics workflow that links sparse AFM observations of cotton (Gossypium hirsutum) fiber cell walls at 8 days post-anthesis (DPA) to distributions of effective elastic properties and a larger-scale mechanical response. We adapt a Stable Diffusion model with Low-Rank Adaptation (LoRA) to expand the experimental scans into an ensemble of AFM-like microstructures and assess the synthetic structures using microfibril crossover count and crossover angle. Each microstructure is mapped to spatially varying Young's modulus and Poisson's ratio fields and analyzed using strain-controlled finite element homogenization. Repeating this process across the image ensemble and a prescribed sweep of matrix-to-fibril stiffness ratios produces distributions of effective Young's modulus and Poisson's ratio. A single constitutive pair cannot capture this variation. The resulting distributions depend strongly on the matrix-to-fibril stiffness ratio and, in several cases, exhibit apparent multimodality. We further propagate the paired effective properties into 20 stochastic realizations of a tensile simulation of a larger specimen, yielding a distribution of macroscale stress response. The framework provides a probabilistic link between sparse nanoscale observations and continuum-scale mechanics while retaining variability across scales. The present results establish the computational workflow, while calibration of the intensity-to-property mapping against nanomechanical measurements remains necessary for quantitative prediction at the fiber scale.
1. Introduction
Sparse AFM scans sample only part of an intrinsically variable cell wall, while uncertain matrix properties complicate prediction of its larger-scale mechanical response. The workflow expands the scans into candidate microstructures and propagates structural and property variability through homogenization to a larger-scale simulation.
1. Introduction
- The workflow links sparse AFM observations to an ensemble of effective responses rather than a single deterministic material description.Its contributions span LoRA-based microstructure generation, image-to-property mapping and homogenization, and stochastic upscaling.Source: §1 · pp. 2–3, Figure 1 · p. 3
2.1. Plant Material, AFM Imaging, and Nanomechanical Measurements
- The study uses six raw AFM scans across three biological replicates of 8 DPA cotton fibers, with crossover count, crossover angle, and microfibril modulus measured from images or force curves.The scans cover 1.05 µm × 1.05 µm at 576 × 576 pixels.Source: §2.1 · pp. 3–4
2.2. Matrix-to-Fibril Stiffness Ratio as an Uncertain Model Input
- The measured fibril modulus (55.1 MPa) anchors four prescribed matrix-to-fibril stiffness ratios (0.18, 0.91, 1.45, 1.81), while matrix stiffness remains uncertain.The matrix is treated as nearly incompressible with νm = 0.49.Source: §2.2 · p. 4
2.3. Generative Augmentation of the AFM Dataset
- The authors use text- and image-conditioned latent diffusion with LoRA after SinGAN produced insufficient morphological diversity.Six scans were patch- and flip-augmented to 197 training images; inference generates image-conditioned variants with different random seeds.Source: §2.3 · pp. 4–5, Figure 2 · p. 5
2.4. Structural Comparison of Real and Synthetic Images
- Synthetic microfibril crossover counts and angles are compared with real images using each image’s unweighted mean across its variants, with 23 variants nested within three conditioning images.The supplied measurements do not establish whether validation images are scans or cropped patches, so the comparison does not report area-normalized density.Source: §2.4 · pp. 5–6, Figure 3 · p. 8
2.5. Image-to-Property Mapping
- Normalized grayscale intensity is continuously interpolated between matrix- and fibril-side endpoints to assign local elastic properties, but intensity is an uncalibrated proxy rather than a direct composition or modulus measurement.The study uses continuous rather than threshold-based mapping because measured modulus varies locally without a clear binary phase separation.Source: §2.5 · p. 6
2.6. Computational Homogenization and Simulation Ensemble
- Each mapped image is treated as a plane-strain heterogeneous RVE and homogenized under three independent strain states, with approximate isotropy screened using a 1% tolerance.The completed database contains 798 runs across four matrix-stiffness cases, and paired effective properties are summarized using Gaussian kernel density estimation.Source: §2.6 · pp. 6–7
2.7. Stochastic Upscaling to a Larger-Scale Simulation
- The larger-scale simulation samples homogenized Young’s modulus and Poisson’s ratio jointly at Gauss points, preserving their one-point joint distribution but imposing no spatial correlation.The demonstration uses the Em = 50 MPa case and 20 independent realizations with identical geometry, mesh, and loading but redrawn material assignments.Source: §2.7 · p. 7
3. Results
The results assess synthetic AFM structures, map them to spatially varying elastic fields, and derive effective-property distributions across stiffness ratios. They then propagate those paired properties to a macroscale tensile simulation, retaining variability across scales.
3.1. Crossover Count and Crossover Angle in Real and Synthetic Images
- Synthetic images had a 5.86° higher mean crossover angle (95% CI 0.95°–10.77°, p=0.036), while the crossover-count comparison did not establish equivalence.Both exploratory, unadjusted comparisons use only three image-level pairs.Source: §3.1 · p. 8
3.2. Mapped Property Fields and Homogenized Response
- Mapped intensity-proxy fields produce microscale displacement fluctuations under uniform strain; mesh refinement reduces Eeff deviation from 1.57% to 0.076%, with νeff varying less than 0.15%.All 798 completed runs satisfy reciprocity, but the isotropy screen establishes small directional normal-stiffness mismatch and negligible normal-shear coupling, not full isotropy.Source: §3.2 · pp. 8–9
3.3. Effective Property Distributions Depend Strongly on the Matrix-to-Fibril Stiffness Ratio
- Mean Eeff rises across Em/Ef = 0.18, 0.91, 1.45, 1.81: 22.40, 51.61, 72.65, 86.55 MPa; coefficients of variation are 4.95%, 0.24%, 0.73%, 1.10%.The distributions shift upward with matrix-to-fibril stiffness ratio, while their relative spread varies nonmonotonically.Source: §3.3 · p. 9, Figure 5 · p. 10
- The Eeff–νeff correlation is negative at Em/Ef = 0.18 and 0.91, but positive at 1.45 and 1.81, consistent with the phase-stiffness contrast changing across ratio 1.The proposed explanation is that higher fibril content decreases νeff, while its effect on Eeff reverses when the stiffer phase changes from fibrils to matrix.Source: §3.3 · p. 9, Figure 6 · p. 10
- Several Eeff kernel-density estimates have multiple peaks, including two for Em/Ef = 1.45, but parent-image grouping, finite sampling, and bandwidth can contribute to apparent modes.Visible modes do not establish distinct physical microstructure classes; the authors use the full empirical distribution in upscaling.Source: §3.3 · pp. 10–11, Figure 7 · p. 11
3.4. Extremal Microstructures
- At Em/Ef = 1.45, the minimum-, near-mean-, and maximum-Eeff RVEs show visible differences in the organization of high- and low-stiffness regions.Visual differences alone do not establish the cause of the stiffness ranking.Source: Figure 8 · p. 11, §3.4 · p. 11
3.5. Propagation to a Macroscale Tensile Specimen
- Across 20 realizations, maximum element-averaged von Mises stress averaged 0.561 MPa (SD 0.003; CV 0.52%), while maximum |σxx| had a 3.3% CV.The peak von Mises stress was about 9% above nominal axial stress; independent Gauss-point sampling attenuated the spread, making it a lower bound conditional on no spatial correlation.Source: §3.5 · p. 12
4. Discussion
The workflow links sparse 8 DPA cotton-fiber AFM observations to distributions of effective elastic properties and carries those paired properties into a macroscale mechanics calculation. The results vary with the prescribed matrix-to-fibril stiffness ratio, while limitations in validation, property mapping, and mechanical idealizations constrain quantitative interpretation.
4. Discussion
- The effective-property distributions shift with matrix-to-fibril stiffness ratio, the Eeff–νeff relationship changes sign across cases, and several estimated densities appear multimodal.A single nominal constitutive pair does not encode this distributional and joint-structure information.Source: §4 · pp. 12–13
- The synthetic-image evidence remains bounded by training and validation data derived from 197 patches from six raw scans, with validation independence unestablished.The crossover-count comparison does not establish agreement, and the crossover-angle difference is exploratory at an unadjusted 0.05 level.Source: §4 · p. 13
- Normalized AFM grayscale intensity is an uncalibrated proxy for elastic properties, so mapped fields should not be read as quantitatively resolved constituent mechanics.Co-registered modulus data and independent composition measurements could help calibrate the mapping.Source: §4 · p. 13
- Reducing effective behavior to (Eeff, νeff) assumes isotropy despite oriented fibrils, while the surface images do not establish the imposed plane-strain constraint.The isotropy screen does not test the shear identity, and the full effective tensors were not stored.Source: §4 · p. 13
- Apparent modulus modes need bandwidth-sensitivity analysis and links to structural descriptors, while the model also uses 2D microstructures and linear elasticity for a 3D wall.The discussion also identifies possible nonlinear, viscoelastic, poroelastic, and moisture-dependent behavior as outside these idealizations.Source: §4 · p. 14
- In 20 macroscale realizations, only Gauss-point material assignments are redrawn; independent sampling preserves the one-point joint distribution but omits measured spatial correlation length.The narrow stress-response spread therefore reflects independent material sampling, and 20 runs limit tail resolution.Source: Figure 10 · p. 13, §4 · p. 14
5. Conclusions
The paper presents a generative-physics workflow linking sparse AFM observations of 8 DPA cotton fiber walls to effective-property ensembles and a larger-scale mechanics calculation. The results establish a computational template, while quantitative prediction requires calibration and stronger structural validation.
5. Conclusions
- The workflow links sparse AFM observations of 8 DPA cotton fiber walls to an ensemble of effective elastic properties and a larger-scale mechanics calculation.It combines LoRA-adapted image generation, image-to-property mapping, finite element homogenization, empirical probability estimation, and stochastic upscaling.Source: §5 · p. 14
- Effective Young’s modulus and Poisson’s ratio vary across the microstructure ensemble and depend strongly on the prescribed matrix-to-fibril stiffness ratio.The tensile example samples paired properties in 20 realizations, producing a macroscale peak-stress distribution whose narrow spread reflects spatially independent material sampling.Source: §5 · p. 14
- Quantitative predictive use requires experimental calibration of the intensity-to-property mapping and larger structural validation of the synthetic images.The workflow is presented as a computational template for uncertainty-aware multiscale characterization from limited microscopy data.Source: §5 · p. 14
Supplementary Material
The supplementary material collects model details, AFM measurements, homogenization and macroscale formulations, and additional results. Its single-image GAN baseline generates coarse-to-fine images but showed insufficient morphological diversity for this application.
Supplementary Material
- Supplementary Section S1 collects generative-model equations, hyperparameters, the algorithm, crossover measurements, and paired statistical analysis.S2 covers AFM line profiles; S3 and S4 give homogenization and macroscale assembly, while S5 reports additional results and mesh convergence.Source: Supplementary Material · p. 15
S1.1. Single-Image GAN Baseline
- SinGAN learns a coarse-to-fine hierarchy of conditional distributions from a single image.Each conditional generates a finer-scale image from the preceding coarser scale.Source: S1.1. Single-Image GAN Baseline · p. 18
- SinGAN outputs showed insufficient morphological diversity for this application, with the model constrained by patch-level statistics from one training image.Source: S1.1. Single-Image GAN Baseline · p. 18
S1.2. Latent Diffusion
Latent diffusion moves image generation into a compressed latent representation, where a denoising network learns to predict added noise. Stable Diffusion conditions denoising on text embeddings, and LoRA adapts selected attention projections while freezing the base model.
S1.2. Latent Diffusion
- A variational autoencoder maps an input image to a latent representation, where the forward process adds noise and the denoising network learns the reverse process by predicting it.Source: S1.2. Latent Diffusion · p. 18
- Stable Diffusion combines a VAE, latent-space denoising U-Net, and CLIP text encoder, with text embeddings conditioning the U-Net through cross-attention.Cross-attention uses latent image features as queries and text embeddings as keys and values.Source: S1.2. Latent Diffusion · p. 18
S1.3. Low-Rank Adaptation
- LoRA trains low-rank matrices A and B while freezing the base Stable Diffusion parameters, applying updates to the U-Net cross-attention projections.The rank sets adaptation capacity and α regulates adaptation strength.Source: S1.3. Low-Rank Adaptation · p. 18
S1.4. Data Preparation, Hyperparameters, and Algorithm
Six exported AFM images were patchified, augmented, and filtered into 197 training images; a LoRA-adapted Stable Diffusion model then generated image-conditioned variants using varied random seeds.
S1.4. Data Preparation, Hyperparameters, and Algorithm
- Six 576 × 576 AFM images were divided into 192 × 192 patches, augmented by flips, and filtered to 197 training images containing microfibrils.The preparation began with three biological replicates.Source: S1.4. Data Preparation,… · p. 18, Generation prompt and inference settings · p. 20
Generation prompt and inference settings
- The positive prompt specified straight, thin fibrils aligned at approximately 78° and moderate crossover density, while the negative prompt excluded curved or tangled structures.The positive prompt also requested nanoscale AFM texture and high contrast.Source: Generation prompt and inference settings · p. 19
- Image-to-image inference used denoising strength 0.6, guidance scale 6, and 50 steps, with different random seeds for variants.The code draws integer seeds from 0 to 999999; the example loop is not a record of the complete generation batches.Source: Generation prompt and inference settings · pp. 19–20
- LoRA training froze the base U-Net, VAE, and text encoder; the algorithm specifies rank 4 and scale α = 4 for the adapter.The supplied algorithm retains a reported 12,000-iteration budget, but says the optimizer-update convention requires confirmation.Source: Generation prompt and inference settings · p. 20
S1.5. Crossover Validation Measurements
The supplement compares synthetic microfibril crossover descriptors with three conditioning AFM images and motivates continuous property mapping with line-profile measurements.
Paired image-level analysis
- Real-image crossover means were 48.72 ± 2.97° and 27.67 ± 2.08 counts, versus synthetic per-image means of 54.58 ± 1.40° and 24.63 ± 3.14 counts.The comparison comprises three real images and 23 variants: eight each for Images 1 and 2, and seven for Image 3; counts are per image, not per unit area.Source: Table S2 · p. 20, S1.5. Crossover Validation Measurements · p. 20, Table S3 · p. 21
- Paired tests use three image-level pairs and two degrees of freedom; they are exploratory, unadjusted, and cannot establish equivalence from a nonsignificant count test.The supplied data do not permit assessment of normality or biological independence across the three conditioning images.Source: Paired image-level analysis · p. 21, Table S4 · p. 21
- Synthetic-variant variability is distinct from variability across the three synthetic per-image means: within-image angle SDs were 5.51°, 6.85°, and 3.06° and count SDs were 2.45, 2.49, and 2.83.The reported synthetic per-image means weight variants equally within each conditioning image.Source: Paired image-level analysis · p. 21
S2. AFM Line-Profile Measurements Motivating Continuous Mapping
- The 40 plotted modulus measurements show spatial heterogeneity without an obvious binary separation into two phases, motivating the continuous mapping used in the reported results.The threshold-based alternative instead assigns representative properties to segmented regions; continuous fields do not themselves validate intensity-to-modulus calibration.Source: S2. AFM Line-Profile Measurements… · p. 21, Figure S1 · p. 22, Figure S2 · p. 23
S3. Homogenization Formulation
The homogenization formulation solves heterogeneous linear elasticity on an image-derived computational cell, applies prescribed macroscopic strains, and derives effective stiffness from volume-averaged stress.
S3.1. Governing Equations
- The computational cell uses displacement field u on domain Ω to formulate a boundary-value problem in heterogeneous linear elasticity.Source: S3.1. Governing Equations · p. 24
- At each Gauss point, the spatial stiffness tensor is assembled from image-derived Young’s modulus and Poisson’s ratio fields under a plane-strain assumption.The computational cell is called an RVE, but its statistical representativeness is not established.Source: S3.1. Governing Equations · p. 24
S3.2. Voigt Notation and Plane-Strain Stiffness
- The two-dimensional strain and stress vectors use standard Voigt notation with engineering shear.Source: S3.2. Voigt Notation and Plane-Strain… · p. 24
- The local constitutive relation is σV(x) = C(x) εV(x), with a 3 × 3 stiffness matrix for an isotropic material under plane strain.Source: S3.2. Voigt Notation and Plane-Strain… · p. 24
S3.3. Macroscopic Load Cases and Boundary Conditions
- Three linearly independent macroscopic strain states—unit longitudinal, transverse, and engineering shear—identify the columns of effective stiffness.Source: S3.3. Macroscopic Load Cases and… · p. 24
- Each load case is imposed through affine displacement boundary conditions on the cell boundary, using the corresponding macroscopic strain tensor.Source: S3.3. Macroscopic Load Cases and… · p. 24
- Homogenization uses a unit RVE with normalized coordinates and unit-magnitude load cases; this does not establish the image window’s physical size or representativeness.Source: S3.3. Macroscopic Load Cases and… · p. 24
S3.4. Volume-Averaged Stress and Effective Stiffness
The effective constitutive response is obtained by relating prescribed macroscopic strain to the volume-averaged stress from finite-element solutions.
S3.4. Volume-Averaged Stress and Effective Stiffness
- After each sub-problem is solved, the macroscopic stress is computed as the volume average of stress over the computational cell.Source: S3.4. Volume-Averaged Stress and… · p. 25
- In the finite-element calculation, the stress average sums contributions over elements and Gauss points, weighted by the mapping Jacobian and quadrature weights.Source: S3.4. Volume-Averaged Stress and… · p. 25
- The homogenized law ⟨σV⟩ = Ceff ε̄ gives effective stiffness from macroscopic stress and strain; the three unit load cases form the identity matrix, so their averaged stresses give the columns of Ceff.Source: S3.4. Volume-Averaged Stress and… · p. 25
- For a load case with amplitude other than unity, its effective-stiffness column is divided by that amplitude.Source: S3.4. Volume-Averaged Stress and… · p. 25
S3.5. Isotropy Check and Extraction of Effective Constants
The section extracts scalar effective properties after screening homogenized stiffness tensors for approximate isotropy. Most runs pass, but the screen does not establish full in-plane isotropy.
Isotropy screen applied to the ensemble
- The screen requires reciprocity, equal axis-normal stiffness, and low normal-shear coupling; AN measures directional mismatch only and must be below 1%.Source: Isotropy screen applied to the ensemble · p. 25
- Across 798 completed runs, 24 (3.0%) were labeled anisotropic; all 24 occurred at Em/Ef = 0.18 and failed the normal-stiffness criterion.Source: Isotropy screen applied to the ensemble · p. 25, Table S5 · p. 26
- Mean directional mismatch was lowest at ratios 0.91 and 1.45, rising with phase contrast to its maximum at 0.18; pooled percentages describe runs, not independent microstructures.Source: Isotropy screen applied to the ensemble · p. 26
- Passing the screen does not establish full in-plane isotropy: it omits the isotropic shear relation, and missing full tensors prevent recovery of shear residuals or full-tensor indices.Source: Isotropy screen applied to the ensemble · p. 26
S3.5. Isotropy Check and Extraction of Effective Constants
- Under isotropic reduction, plane-strain stiffness relations define scalar Eeff and νeff approximations whose accuracy depends on the isotropy criteria.Source: Isotropy screen applied to the ensemble · p. 26
S4. Macroscale Assembly with Sampled Properties
The macroscale model samples paired effective properties from the homogenization ensemble at integration points, then uses the resulting local constitutive response in the linear-elasticity formulation.
S4. Macroscale Assembly with Sampled Properties
- At each Gauss point, the method samples an ensemble index uniformly and assigns its paired effective E and ν, thereby sampling the property density directly.Source: S4. Macroscale Assembly with Sampled… · p. 26
- The assigned property pair defines a local constitutive tensor that enters the weak form of the macroscale linear-elasticity problem.Source: S4. Macroscale Assembly with Sampled… · pp. 26–27
- Internal virtual work is approximated by quadrature contributions weighted by the Gauss-point weight and element-mapping Jacobian determinant.Source: S4. Macroscale Assembly with Sampled… · p. 27
S5. Additional Results
The additional results show mesh stability of volume-averaged constants over tested refinement levels, local displacement fluctuations under nominally uniform loading, and apparent peaks in one effective-modulus density.
S5.3. Mesh Convergence of the Homogenized Constants
- 1.57% at L = 5 fell to 0.076% at L = 8 for Eeff relative to L = 9; νeff changed less than 0.15% over L = 5–9 and differed by 0.016% at L = 8.The L = 9 mesh is a finite-resolution reference; the results support stability of volume-averaged constants, not convergence of local stress fields.Source: Table S6 · p. 27, S5.3. Mesh Convergence of the… · p. 27, Figure S5 · p. 29
S5.1. Displacement Fields from the Homogenization Load Cases
- Under unit longitudinal, transverse, and shear loading, heterogeneous stiffness produces local displacement fluctuations that enter the volume-averaged stresses used to construct Ceff.Source: S5.1. Displacement Fields from the… · p. 27, Figure S3 · p. 28
S5.2. Effective-Modulus Density for a Single Matrix Stiffness
- At Em/Ef = 1.45 (Em = 80 MPa), the Eeff/Ef marginal has two visible peaks, but their dependence on KDE bandwidth has not been tested.Source: S5.2. Effective-Modulus Density for a… · p. 27, Figure S4 · p. 28