A Mathematical Model of Collective Cell Migration in a Three-Dimensional, Heterogeneous Environment

<div><p>Cell migration is essential in animal development, homeostasis, and disease progression, but many questions remain unanswered about how this process is controlled. While many kinds of individual cell movements have been characterized, less effort has been directed towards understanding how clusters of cells migrate collectively through heterogeneous, cellular environments. To explore this, we have focused on the migration of the border cells during Drosophila egg development. In this case, a cluster of different cell types coalesce and traverse as a group between large cells, called nurse cells, in the center of the egg chamber. We have developed a new model for this collective cell migration based on the forces of adhesion, repulsion, migration and stochastic fluctuation to generate the movement of discrete cells. We implement the model using Identical Math Cells, or IMCs. IMCs can each represent one biological cell of the system, or can be aggregated using increased adhesion forces to model the dynamics of larger biological cells. The domain of interest is filled with IMCs, each assigned specific biophysical properties to mimic a diversity of cell types. Using this system, we have successfully simulated the migration of the border cell cluster through an environment filled with larger cells, which represent nurse cells. Interestingly, our simulations suggest that the forces utilized in this model are sufficient to produce behaviors of the cluster that are observed <i>in vivo</i>, such as rotation. Our framework was developed to capture a heterogeneous cell population, and our implementation strategy allows for diverse, but precise, initial position specification over a three- dimensional domain. Therefore, we believe that this model will be useful for not only examining aspects of <i>Drosophila</i> oogenesis, but also for modeling other two or three-dimensional systems that have multiple cell types and where investigating the forces between cells is of interest.</p></div

Effects of DBP genotype on free 25OHD and 1,25(OH)₂D <i>in vitro</i> and <i>in vivo</i>.

<p>eSS-predicted levels of free 25OHD and 1,25(OH)₂D relative to total serum levels of these metabolites for in vitro tissue culture conditions (5% serum) and in vivo (100% serum) according to DBP genotype (GC allele combinations). X-axis indicates total serum concentrations of 25OHD (nM) or 1,25(OH)₂D (pM) and Y-axis indicates concentration of free 25OHD or 1,25(OH)₂D. Concentration of (A) 1,25(OH)₂D = 5 pM (5% serum), (B) 25OHD = 2.5 nM (5% serum), (C) 1,25(OH)₂D = 100 pM (100% serum) or (D) 25OHD = 50 nM (100% serum) were fixed.</p

Parameters for iSS mathematical model.

<p>i. rate has only been measured for 1,25(OH)₂D <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030773#pone.0030773-Eil1" target="_blank">[50]</a>.</p><p>ii. K_r1 = 500*K_r2.</p><p>iii. K_r2 = 1/K_d where K_d = 1×10⁻¹⁰<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030773#pone.0030773-Pike1" target="_blank">[42]</a>.</p><p>iv. K_cat*Y_T = 0.1 µM/hr <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030773#pone.0030773-Tang1" target="_blank">[46]</a>.</p><p>v. estimate based on <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030773#pone.0030773-Sakaki1" target="_blank">[29]</a>, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030773#pone.0030773-Urushino1" target="_blank">[47]</a>, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030773#pone.0030773-Eto1" target="_blank">[48]</a>, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030773#pone.0030773-Vieth1" target="_blank">[49]</a>.</p><p>vi. 3000 molecules/cell <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030773#pone.0030773-Hewison2" target="_blank">[40]</a> and spherical cell of 10 µm radius <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030773#pone.0030773-Krombach1" target="_blank">[41]</a>.</p><p>vii. K_cc1 = 10*K_cc2.</p><p>viii. fit to in vitro data.</p

Schematic framework of parameters used to produce extracellular steady state (eSS) and intracellular (iSS) mathematical models for vitamin D metabolism and function.

<p>Free 25OHD and 1,25(OH)₂D interacting with extra-cellular vitamin D binding protein (DBP) or albumin indicated in black text and arrows (eSS model). Intra-cellular interactions involving the vitamin D-activating enzyme (CYP27B1), the vitamin D receptor (VDR) and transcriptional induction of the antibacterial protein CAMP via interaction between VDR and the CAMP gene promoter (CAMP-DNA) indicated by grey text and arrows (iSS model).</p

Predicted effects of vitamin D status and DBP genotype (Gc allelic combinations) on <i>in vivo</i> monocyte expression of CAMP under basal or immune activated conditions.

<p>Predicted effects of vitamin D status and DBP genotype (Gc allelic combinations) on <i>in vivo</i> monocyte expression of CAMP under basal or immune activated conditions.</p

Comparison of iSS-predicted effects of 25OHD or 1,25(OH)₂D on monocyte expression of CAMP with observed <i>in vitro</i> responses of monocytes to treatment with these metabolites.

<p>Adherent human monocytes were incubated for 6 hrs in media containing 5% serum with doses of (A) 25OHD (1–300 nM) and (B) 1,25(OH)₂D (0.1–6 nM). The experimental data is indicated by blue dots and error bars (± SD). Black lines indicate data predicted by the iSS mathematical model assuming basal levels of VDR and CYP27B1 (i.e. no activation). For the purpose of this modeling, DBP was represented by the GC1F/1F allelic combination.</p

Simulating the three dimensional model results in collective migration.

<p>A simulation showing six border cells (green), two polar cells (red), the epithelium (transparent green), and the surface of the oocyte (black, right) at three time points during the migration. Fifteen nurse cells are situated inside the egg chamber, but are not plotted so as to maintain clarity of this three dimensional structure. Polar cells are surrounded by border cells, making them hard to distinguish. (A) At 2 minutes, cells are beginning to invade between nurse cells. (B) At 2.4 hours, the cluster is about halfway to its destination. (C) At 5.6 hours, the border cell cluster has reached the edge of the oocyte. See also Supplemental <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0122799#pone.0122799.s003" target="_blank">S3 Movie</a>.</p

The total time and relative time taken for computational clusters of 4, 6 and 8 border cells to complete migration.

<p>The total time and relative time taken for computational clusters of 4, 6 and 8 border cells to complete migration.</p

Simulations with four, six, and eight border cells at the same time point (<i>t</i> = 1.8 hours) during migration.

<p>The cluster with four border cells (A) has moved significantly less distance than the cluster with six (B) or eight (C) border cells.</p

Binding protein and ligand biochemical parameters for eSS mathematical model.

<p>Binding protein and ligand biochemical parameters for eSS mathematical model.</p