Repulsive expansion dynamics in colony growth and gene expression

Abstract

Spatial expansion of a population of cells can arise from growth of microorganisms, plant cells, and mammalian cells. It underlies normal or dysfunctional tissue development, and it can be exploited as the foundation for programming spatial patterns. This expansion is often driven by continuous growth and division of cells within a colony, which in turn pushes the peripheral cells outward. This process generates a repulsion velocity field at each location within the colony. Here we show that this process can be approximated as coarse-grained repulsive-expansion kinetics. This framework enables accurate and efficient simulation of growth and gene expression dynamics in radially symmetric colonies with homogenous z-directional distribution. It is robust even if cells are not spherical and vary in size. The simplicity of the resulting mathematical framework also greatly facilitates generation of mechanistic insights.

Fig 1. The repulsive expansion model captures colony growth dynamics.

A. Illustration of the repulsive expansion dynamics. Each orange dot represents a cell at that given location. As cells in the colony grow and divide, cells in the interior will push the outer cells, generating a velocity field u(x,t). Given a location, $\widehat{r}$ is the normal vector along radial direction. Therefore $u\bullet \widehat{r}$ represents the radial expansion velocity at the location. Since the model is assumed to be radial symmetric, the most outer layer of cells determines the size of the colony. At time t, the colony radius is expressed as R(t). The different color lines indicate different cell trajectories. Colors are chosen simply for distinguishable visualization effect. B. Colony growth over time. The black line indicates the radius of the colony, the colored lines represent the trajectories of pre-selected 10 locations within the colony. C. Sample trajectories of individual cells in the ABM simulation. Each trajectory shows the movement of a single cell. Here, cells are approximately spherical; time is measured in generations (~30 minutes). D. Average trajectories of cells in the ABM simulation over time. Cells were assigned to 5 different groups according to their radial positions when the total population of the colony reached 1000 cells; subsequently, all the 1000 cells were tracked over time in terms of their radial positions. Each of the colored lines shows the average trajectory of the cells in each group.

Fig 2. The repulsive expansion model under different initial model settings.

A. Average trajectories of cells in the ABM simulation with different length-to-width ratios. Similar to Fig 1D, cells were assigned to 5 different groups according to their radial positions when the total population reached 1000 cells and, subsequently, tracked over time in terms of their radial positions. The left, middle and right panels correspond to the length-to-width ratios of 1.5:1, 2:1 and 2.5:1 respectively. The trajectories show similar behaviors despite the variation of aspect ratio. B. Simulated colony growth using the repulsive expansion model, assuming different initial colony sizes. The top panel shows four cell density distribution with the same shape, the same initial selected positions, but different initial colony size. The bottom panel shows the colony radius expansion over time with different initial colony size (same color code). C. Simulated colony growth using the repulsive expansion model, assuming different initial cell-density distributions. The top panel shows four cell density distributions with the same initial colony size and the same initial selected positions. The bottom panel shows the colony radius expansion over time with different cell density distribution shape (same color code). Each subfigure in the panel has the same x-axis and y-axis. For visual clarity, the axis label and title were shown only in the left most subfigure.

Fig 3. Simulated spatial-temporal dynamics of the pattern formation in engineered bacteria with a high metabolic burden and a sharp gene expression capacity profile.

A. Left: circuit logic. Middle: the model was simulated under the condition of high metabolic burden. The cell growth rate is a decaying function of the production of T7 RNAP and T7 lysozyme. Right: Gene expression capacity. The x-axis represents the distance from the colony edge. B. Top left to bottom right: Simulated spatial-temporal dynamics of colony radius, AHL, T7 RNAP and T7 lysozyme for varying distance (x-axis) over time (y-axis), respectively. The parameters used in the simulation are listed in S4 Table. C. Simulated pattern formation with different environmental factors. The top panels from left to right shows the AHL dynamic under the base case (left, same with (B)), with initial AHL concentration is 0.3 (middle), and with domain size which is twice as large as the base case (right). The bottom panels are the T7 lysozyme distribution at the time when nutrient is exhausted under each condition, respectively. The x axis represents the distance from the colony edge. D. Simulated double rings. Left: the AHL dynamic with initial AHL concentration 0.8. Right top panel: the lysozyme distribution at time point 1, which is labeled in the left panel. Right bottom panel: the lysozyme distribution at time point 2.

Fig 4. Simulated spatial-temporal dynamics of the pattern formation process with a high metabolic burden and a flat gene expression capacity profile.

A. Simulation assumption with a low metabolic burden and a moderate gene expression capacity. The notations are the same with Fig 3. B. Simulated spatial-temporal dynamics of T7 RNAP and T7 lysozyme for varying distance (x-axis) over time (y-axis). The parameters used in the simulation are listed in S4 Table, except K_σ = 0.1; n_σ = 2; n_φ = 2; K_φ = 0.1. C. Simulated pattern formation with different environmental factors. The top panels from left to right shows the AHL dynamic under the base case (left), with initial AHL concentration 0.3 (middle), and with domain size which is twice as large as base case (right). The bottom panels are the T7 lysozyme distribution under each condition, respectively. The x-axis represents the distance from the colony edge. D. Simulated scale invariance in pattern formation. Dependence of the ring width (red circles) and the colony radius (green circles) on the domain radius from 1 to 4. The lines represent the linear regression of the colony radius and the ring width with respect to the domain radius in the white region. E. Phase diagram of T7 lysozyme (y axis) to T7 RNAP (x axis). Based on the phase plane diagram, given an initial condition (T7RNAP = 0.1, T7 lysozyme = 0), T7 RNAP increases first then decreases, while T7 lysozyme keeps increasing. F. Simulated spatial-temporal dynamics of T7 RNAP and T7 lysozyme for varying distance (x-axis) over time (y-axis). The parameters used in the simulation are listed in the S4 Table, except K_σ = 0.1; n_σ = 2. G. Ring pattern formation in the butterfly wings. Since eyespot signals to from inner and outer rings are released at different time-points. The outer rings form first, then the inner rings form within the outer rings at a later time point.