DOI: 10.5061/dryad.s4mw6m9cb
title: Data from: Linking continuous and discrete models of cell birth and migration

description: Self-organization of individuals within large collectives occurs
 throughout biology, with examples including locust swarming and cell
 formation of embryonic tissues. Mathematical models can help elucidate the
 individual-level mechanisms behind these dynamics, but analytical
 tractability often comes at the cost of biological intuition. Discrete
 models provide straightforward interpretations by tracking each individual
 yet can be computationally expensive. Alternatively, continuous models
 supply a large-scale perspective by representing the "effective"
 dynamics of infinite agents, but their results are often difficult to
 translate into experimentally relevant insights. We address this challenge
 by quantitatively linking spatio-temporal dynamics of discrete and
 continuous models in settings with biologically realistic, time-varying
 cell numbers. Motivated by zebrafish-skin pattern formation, we create a
 continuous framework describing the movement and proliferation of a single
 cell population by upscaling rules from a discrete model. We introduce and
 fit scaling parameters to account for discrepancies between these two
 frameworks in terms of cell numbers, considering movement and birth
 separately. Our resulting continuous models accurately depict ensemble
 average agent-based solutions when migration or proliferation act alone.
 Interestingly, the same parameters are not optimal when both processes act
 simultaneously, highlighting a rich difference in how combining migration
 and proliferation affects discrete and continuous dynamics.
description: Individual-based data were produced by simulating the ABM listed
 in the paper, using Matlab (version 2022a). The PDEs were simulated using
 either Matlab programmes (for the cell birth-only model) or using custom
 codes in Julia (version 1.7.3). Codes used to simulate the models may be
 found at the following Github repository: https://github.com/wdmartinson/Self-Organization-One-Species .
description: # Data from: Linking continuous and discrete models of cell birth and
 migration
 [https://doi.org/10.5061/dryad.s4mw6m9cb](https://doi.org/10.5061/dryad.s4mw6m9cb) This dataset contains the ABM and PDE results for the mathematical models discussed in the article "Linking continuous and discrete models of cell birth and migration" by W. D. Martinson, A. Volkening, M. Schmidtchen, C. Venkataraman, and J. A. Carrillo. ## Description of the data and file structure The data file contains three folders: "Birth" refers to results obtained using mathematical models incorporating only cell birth, "Migration" to those obtained from models incorporating migration alone, and "Combined (Birth and Migration)" to results obtained from models in which both mechanisms occur simultaneously. Each of these folders contains subfolders detailing whether they contain results from the discrete agent-based model (ABM Results) or continuous model (IDE Results or PDE results). Results are grouped by the initial condition used for the ABM/PDE/IDE solutions. Data are MAT files or, equivalently, BSON files (the latter of which can be converted to a MAT file format). For the Combined and Movement-only models, we also include additional folder that contain data on experiments listed in the SI (these include experiments in which we fit all three parameters of the Combined model simultaneously, where we vary the amount of data used to parameterise the ABM, where we determine how small perturbations to ABM parameter values or initial densities alters the optimal parameter values, and whether fitting to data on coarser meshes leads to similar results). Some of these data consist of files labeled as "optimal_alpha_...", which contain a variable ("x") that gives the value of the optimal scaling parameter, "fun", a vector containing the difference between the ensemble average ABM and PDE solutions across all time and space voxels, and "jac", which is a matrix corresponding to an estimated Jacobian and which is used to estimate 95% confidence intervals using the "nlparci" function in Matlab (version 2022a). **Movement-only model**: PDE results are given as a N_T x N_hist x N_hist array describing the number density of cells for N_T time steps on a spatial histogram of resolution N_hist x N_hist. The mesh resolution used to simulate the PDE is equal to that of the spatial histogram unless otherwise noted. For the ABM results, there are two types of MAT files: one gives a N_cells*N_realizations x 2 x N_T array of all melanophore and/or xanthophore cell positions for each recorded time, while the other is a N_hist x N_hist x N_T array containing the ensemble average ABM results for that particular set of initial data. For each time step, the transpose of the matrix must be taken to align with the corresponding PDE results, due to the way in which the cell sorting is carried out in the MATLAB files. **Birth-only model:** ABM results are given as 1 x 7 cells, with each cell corresponding to a particular value of N_bir. (The particular values of N_bir are listed in the "Nvals" vector which can be found upon loading the MAT file.) The ensemble average ABM data are given by the "densityX" or "densityM" variable. For a particular value of N_bir, the "densityM" cell yields a N_hist x N_hist x N_T array similar to the data of the movement only model. The boundaries of the spatial voxels that are constructed for the N_hist x N_hist mesh are are given by the "stepBnd" vector. The "numMelBornc" or "numXanBornc" variable corresponds to the mean number of melanophores or xanthophores, respectively, that are born on average within a single time step. The "numMelc" or "numXanc" variable corresponds to the total number of melanophores or xanthophores, respectively. The "meanNumber1DM" or "meanNumber1DX" variable corresponds to the number of melanophores or xanthophores, respectively, that are found by taking a column average over one of the spatial dimensions. Finally, the "spreadAbs" variable corresponds to the radius of ensemble average ABM solution support along the x = 0 axis; for each value of N_bir this gives a N_real x N_T matrix, where N_real is the number of realisations of the ABM. The solutions for the continuous birth-only model are listed under the "densityMODE" or "densityXODE" variable. These variables are 1 x length(Nvals) cells that corresponds to each value of N_bir used. For each value of N_bir, one can extract a N_hist x N_hist x N_T array corresponding to the ensemble average ABM solution, in a similar way as the above. The "cells_per_day" structure corresponds to the number of cells born per day in the IDE, the "densitydiff" structure to the difference between the IDE and EA ABM solution, the "mass_rho" structure to the integral of the number density in space per unit time (i.e., the total number of cells per unit time), and the "spread" structure to the radius of the IDE solution support as measured along the x = 0 axis. The "r_growth" and "c_max" variables corresponds to the value of gamma and c+ in Eqn. (7) which are used to simulate the IDE model (see the main text for details). Note that IDE solutions are found both for cases in which parameters were estimated by fitting to the difference in total cell numbers (the "..._mass.mat" files) or to the L2 error between solution densities (the "..._bin.mat" files). **Combined model**: Ensemble average ABM results are found in the "..._BirthMoveFixed.mat" files, whereas example ABM simulations are given in the "..._ABMExample.mat" files. Data in the ABM files have similar names and structures as those for the Birth-only model. PDE results for the Combined model are named based on their value of N_bir and whether we have used the value of c+ estimated from the birth-only model data or whether we have instead used a heuristic value of c+. The PDE data is given on either a 240 x 240 x N_T mesh, which is what was used to simulate the original PDE, or on a coarser N_hist x N_hist x N_T histogram that matches the ensemble average ABM data. ## Code/Software Data were generated using MATLAB (version 2022a) and Julia (version 1.7.3). Software for generating these simulations can be found at [https://github.com/wdmartinson/Self-Organization-One-Species](https://github.com/wdmartinson/Self-Organization-One-Species).