DOI: 10.5061/dryad.qbzkh18w9
title: Universality in the mechanical behavior of vertex models for biological tissues

description: This dataset contains numerical simulation results. The dataset includes
 tabulated mechanical observables, such as the static shear modulus,
 storage modulus G', and loss modulus G'', computed for
 distinct vertex models under varying geometric and mechanical parameters.
 These include the standard vertex model, active foam model, spring-edge
 model, and dynamical vertex models with internal degrees of freedom. The
 data is organized to directly reproduce figures from the main text. The
 Jupyter notebook demonstrates how to load, process, and visualize this
 data. This repository is intended to support both the direct reproduction
 of published figures and enable further comparative studies of mechanical
 responses in vertex-based models of biological tissues. Detailed
 simulation protocols and modeling assumptions are provided in the
 associated manuscript.
description: # Universality in the mechanical behavior of vertex models for biological
 tissues **Authors:** O. K. Damavandi```*```, S. Arzash```*```, E.
 Lawson-Keister, M. L. Manning ``*``These authors contributed equally to
 this work. **Corresponding publication:** PRX Life (2025) **DOI:**
 [https://doi.org/10.1103/9ktk-6rqc](https://doi.org/10.1103/9ktk-6rqc) ---
 This dataset supports the figures and analysis in the paper *Universality
 in the Mechanical Behavior of Vertex Models for Biological Tissues*. It
 contains simulation outputs for various vertex models used to probe the
 zero-strain-rate shear modulus, frequency-dependent viscoelastic moduli,
 and model universality. Each CSV file corresponds to one or more figure
 panels in the main text. A Jupyter notebook (`plot_data.ipynb`) is
 included for reproducing the figures. --- ## Description of Dataset and
 Files ### 1. **data_fig1a.csv** **Purpose:** Contains viscoelastic moduli
 for the standard vertex model in the **solid regime** (target shape
 factors (p_0 = 3.4) and (p_0 = 3.6)). **Columns:** * `Omega_list`:
 Frequency of applied oscillatory strain ((\omega)). * `G_prime_p034_mean`:
 Mean storage modulus (G') for (p_0 = 3.4). * `G_prime_p034_std`:
 Standard deviation of (G') for (p_0 = 3.4). *
 `G_double_prime_p034_mean`: Mean loss modulus (G'') for (p_0 =
 3.4). * `G_double_prime_p034_std`: Standard deviation of (G'')
 for (p_0 = 3.4). * `G_prime_p036_mean`: Mean storage modulus (G') for
 (p_0 = 3.6). * `G_prime_p036_std`: Standard deviation of (G') for
 (p_0 = 3.6). * `G_double_prime_p036_mean`: Mean loss modulus
 (G'') for (p_0 = 3.6). * `G_double_prime_p036_std`: Standard
 deviation of (G'') for (p_0 = 3.6). --- ### 2.
 **data_fig1b.csv** **Purpose:** Contains viscoelastic moduli for the
 standard vertex model in the **fluid regime** ((p_0 = 4.0) and (p_0 =
 4.2)). **Columns:** * `Omega_list`: Frequency of applied oscillatory
 strain ((\omega)). * `G_prime_p04_mean`: Mean storage modulus (G')
 for (p_0 = 4.0). * `G_prime_p04_std`: Standard deviation of (G') for
 (p_0 = 4.0). * `G_double_prime_p04_mean`: Mean loss modulus
 (G'') for (p_0 = 4.0). * `G_double_prime_p04_std`: Standard
 deviation of (G'') for (p_0 = 4.0). * `G_prime_p042_mean`: Mean
 storage modulus (G') for (p_0 = 4.2). * `G_prime_p042_std`: Standard
 deviation of (G') for (p_0 = 4.2). * `G_double_prime_p042_mean`: Mean
 loss modulus (G'') for (p_0 = 4.2). * `G_double_prime_p042_std`:
 Standard deviation of (G'') for (p_0 = 4.2). --- ### 3.
 **data_fig2d.csv** **Purpose:** Contains frequency-dependent viscoelastic
 moduli for the **confluent active foam model**, for two different values
 of interfacial tension (\Lambda). **Columns:** * `Omega_list`: Frequency
 of applied oscillatory strain ((\omega)). * `G_prime_tension_01_mean`:
 Mean (G') for (\Lambda = 0.1). * `G_prime_tension_01_std`: Standard
 deviation of (G') for (\Lambda = 0.1). *
 `G_double_prime_tension_01_mean`: Mean (G'') for (\Lambda =
 0.1). * `G_double_prime_tension_01_std`: Standard deviation of
 (G'') for (\Lambda = 0.1). * `G_prime_tension_001_mean`: Mean
 (G') for (\Lambda = 0.01). * `G_prime_tension_001_std`: Standard
 deviation of (G') for (\Lambda = 0.01). *
 `G_double_prime_tension_001_mean`: Mean (G'') for (\Lambda =
 0.01). * `G_double_prime_tension_001_std`: Standard deviation of
 (G'') for (\Lambda = 0.01). --- ### 4. **data_fig2d_inset.csv**
 **Purpose:** Provides zero-frequency shear modulus (G) as a function of
 interfacial tension (\Lambda) for the active foam model. **Columns:** *
 `Tension_list`: Values of line tension (\Lambda). * `G_mean`: Mean
 zero-frequency shear modulus (G). * `G_std`: Standard deviation of (G)
 across samples. --- ### 5. **data_fig3c.csv** **Purpose:** Viscoelastic
 moduli for the spring-edge vertex model in the **solid-like regime** (
 rest lengths (l_0 = 0.4) and (l_0 = 0.6)). **Columns:** * `Omega_list`:
 Frequency of applied oscillatory strain ((\omega)). *
 `G_prime_l0_04_mean`: Mean storage modulus (G') for (l_0 = 0.4). *
 `G_prime_l0_04_std`: Standard deviation of (G') for (l_0 = 0.4). *
 `G_double_prime_l0_04_mean`: Mean loss modulus (G'') for (l_0 =
 0.4). * `G_double_prime_l0_04_std`: Standard deviation of (G'')
 for (l_0 = 0.4). * `G_prime_l0_06_mean`: Mean storage modulus (G')
 for (l_0 = 0.6). * `G_prime_l0_06_std`: Standard deviation of (G')
 for (l_0 = 0.6). * `G_double_prime_l0_06_mean`: Mean loss modulus
 (G'') for (l_0 = 0.6). * `G_double_prime_l0_06_std`: Standard
 deviation of (G'') for (l_0 = 0.6). --- ### 6.
 **data_fig3d.csv** **Purpose:** Viscoelastic moduli for the spring-edge
 vertex model in the **fluid-like regime** ( rest lengths (l_0 = 0.75) and
 (l_0 = 0.8)). **Columns:** * `Omega_list`: Frequency of applied
 oscillatory strain ((\omega)). * `G_prime_l0_075_mean`: Mean storage
 modulus (G') for (l_0 = 0.75). * `G_prime_l0_075_std`: Standard
 deviation of (G') for (l_0 = 0.75). * `G_double_prime_l0_075_mean`:
 Mean loss modulus (G'') for (l_0 = 0.75). *
 `G_double_prime_l0_075_std`: Standard deviation of (G'') for
 (l_0 = 0.75). * `G_prime_l0_08_mean`: Mean storage modulus (G') for
 (l_0 = 0.8). * `G_prime_l0_08_std`: Standard deviation of (G') for
 (l_0 = 0.8). * `G_double_prime_l0_08_mean`: Mean loss modulus
 (G'') for (l_0 = 0.8). * `G_double_prime_l0_08_std`: Standard
 deviation of (G'') for (l_0 = 0.8). --- ### 7.
 **data_fig4a.csv** **Purpose:** Frequency-dependent viscoelastic moduli
 for the active spring model, for two different waiting times `tw` in the
 **solid-like regime**. **Columns:** * `Omega_list`: Frequency of applied
 oscillatory strain ((\omega)). * `G_prime_tw_num_0_mean`,
 `G_prime_tw_num_0_std`: Mean and standard deviation of storage modulus
 (G') at short (t_w) highlighted in the inset. *
 `G_double_prime_tw_num_0_mean`, `G_double_prime_tw_num_0_std`: Mean and
 standard deviation of loss modulus (G'') at short (t_w)
 highlighted in the inset. * `G_prime_tw_num_1_mean`,
 `G_prime_tw_num_1_std`: Mean and standard deviation of storage modulus
 (G') at a slightly larger (t_w) highlighted in the inset. *
 `G_double_prime_tw_num_1_mean`, `G_double_prime_tw_num_1_std`: Mean and
 standard deviation of loss modulus (G'') at a slightly larger
 (t_w) highlighted in the inset. --- ### 8. **data_fig4a_inset.csv**
 **Purpose:** Time evolution of moduli at fixed small frequency to show
 fluidization trend in active spring model. **Columns:** * `Tw_list`:
 Waiting times. * `G_prime_omega_num_0_mean`: Mean storage modulus at low
 frequency. * `G_double_prime_omega_num_0_mean`: Mean loss modulus. --- ###
 9. **data_fig4b.csv** **Purpose:** Frequency-dependent viscoelastic moduli
 for the active spring model, for two different waiting times `tw` in the
 **fluid-like regime**. **Columns:** * `Omega_list`: Frequency of applied
 oscillatory strain ((\omega)). * `G_prime_tw_num_9_mean`,
 `G_prime_tw_num_9_std`: Mean and standard deviation of storage modulus
 (G') at long (t_w) highlighted in the inset. *
 `G_double_prime_tw_num_9_mean`, `G_double_prime_tw_num_9_std`: Mean and
 standard deviation of loss modulus (G'') at long (t_w)
 highlighted in the inset. * `G_prime_tw_num_10_mean`,
 `G_prime_tw_num_10_std`: Mean and standard deviation of storage modulus
 (G') at a slightly larger (t_w) highlighted in the inset. *
 `G_double_prime_tw_num_10_mean`, `G_double_prime_tw_num_10_std`: Mean and
 standard deviation of loss modulus (G'') at a slightly larger
 (t_w) highlighted in the inset. --- ### 10. **data_fig5e.csv**
 **Purpose:** Frequency-dependent viscoelastic moduli for the active
 tension network at long waiting time `tw`. **Columns:** * `Omega_list`:
 Frequency of applied oscillatory strain ((\omega)). * `G_prime_mean`,
 `G_prime_std`: Mean and standard deviation of storage modulus (G') at
 long (t_w) highlighted in the inset. * `G_double_prime_mean`,
 `G_double_prime_std`: Mean and standard deviation of loss modulus
 (G'') at long (t_w) highlighted in the inset. --- ### 11.
 **data_fig5e_inset.csv** **Purpose:** Time evolution of moduli at fixed
 small frequency in the active tension network. **Columns:** * `Tw_list`:
 Waiting times. * `G_prime_omega_num_0_mean`: Mean storage modulus at low
 frequency. * `G_double_prime_omega_num_0_mean`: Mean loss modulus at low
 frequency. --- ## Plotting Software The Python notebook `plot_data.ipynb`
 can be used to load the CSV files and reproduce all figures listed above.
 Key libraries: `pandas`, `numpy`, `matplotlib`. --- ## Reference
 **Corresponding publication:** Damavandi*, O. K., Arzash*, S.,
 Lawson-Keister, E., Manning, M.L. *Universality in the Mechanical Behavior
 of Vertex Models for Biological Tissues*, PRX Life (2025). DOI:
 [https://doi.org/10.1103/9ktk-6rqc](https://doi.org/10.1103/9ktk-6rqc)
 **Dataset citation:** Damavandi*, O. K., Arzash*, S., Lawson-Keister, E.,
 Manning, M.L. (2025). *Universality in the Mechanical Behavior of Vertex
 Models for Biological Tissues*. Dryad Digital Repository.
 [https://doi.org/10.5061/dryad.qbzkh18w9](https://doi.org/10.5061/dryad.qbzkh18w9)