Agent-Based Model of Heterogeneous T-Cell Activation in Vitro

Abstract

We combine modelling and in vitro measurement of T-cell properties at the level of individual cells. Rather than seek to model kinetics using one differential equation per cell population, our computational model describes the dynamics of a cohort of cells in terms of a series of discrete events, one after the other, occurring to one cell at a time. Two types of heterogeneity are explicitly simulated: cell surface marker expression that is time-dependent and varies from cell to cell and spatial heterogeneity that arises from local influences such as the proximity of IL-2-producing cells. Neither storing 50,000 individual cell instances and their attributes nor following the kinetics over times of order 48 h and producing simulated flow cytometry plots, is too taxing for a modern desktop computer. In vitro, the kinetics of the activation of cohorts of CD4⁺ T cells from BALB/c mice was studied, under stimulus with soluble or plate-bound anti-CD3 or a combination of PMA and ionomycin.

A Python Code

The T-cell class is a subclass of the cell class, and the CD4 class is a subclass of the T-cell class. A class definition can be thought of as a template. Each time it is used, to create an in silico cell, the initial values of the attributes are set in the __init__ method. A CellPopulation is a collection of cells.

• The method placecellsinwell assigns each unactivated cell a radius that is a random variable, uniformly distributed between r ₀ − 0.5 μm and r ₀ − 0.5 μm, where r ₀ = 4 μm. Activated cells have radius r ₀ + u r ₁, where u is uniformly distributed between 0 and 1 and r ₁ = 3 μm. To place N cells without overlap, each cell in turn is assigned a tentative position. If it results in no overlap with any cell already assigned position and radius, the tentative position is accepted. If not, a new tentative position is generated, where x and y coordinates are drawn independently from Gaussian distributions with mean 0 and standard deviation $\sigma = 0.45r_0\sqrt {N}\,\upmu$m.

• A timestep of duration dt is carried out in the method step. Each cell, independently, may die with probability μdt. Each may also receive a signal, with probability proportional to the current value of its IL2 attribute. There are two types of signal: cytokine and TCR. The parameter γ is the fraction of signalling that is TCR signalling.

  If a non-cycling cell receives a cytokine signal, its attributes are updated according to the following rules:

  $$\displaystyle \begin{aligned} \mathtt{cell.cd25} \to \sqrt{U \times \mathtt{cell.cd25} \times CD25max}\\ \mathtt{cell.cd62l} \to \sqrt{U \times \mathtt{cell.cd62l} \times CD62Lmin},\end{aligned} $$

  where the U are drawn, independently from the uniform distribution on (0, 1). Note that the random variable $\sqrt {U}$ has range (0, 1), mean 2∕3 and standard deviation 1∕3.

  If a non-activated cell receives a TCR signal, it becomes an activated cell. If an activated cell receives a TCR signal, it enters the cell cycle. The elapsed time from entering cell cycle to division is 24 h for the first division and 12 h for subsequent divisions.

• The CellPopulation method localil2 calculates a nondimensionalised measure of local IL-2 concentration in each cell’s neighbourhood, which depends on the status of all cells within radius ril2, stored in the cell attribute vic. The IL-2 attribute of a cell at time t is given by

  $$\displaystyle \begin{aligned} \mathtt{cell.IL2} = \frac{n_3}{n_6+1},\end{aligned} $$

  where n ₃ is the number of IL-2 producing cells in cell.vic, and n ₆ is the number of CD25⁺ cells in cell.vic.

• The method neighbourhood compiles a list of all cells whose centre is within radius r _IL2 = 7r ₀ of a given cell, stored in the attribute vic.

• The method getstats returns the following cell counts:

  n ₁ :

  all live cells

  n ₂ :

  activated cells

  n ₃ :

  IL-2 producing cells

  n ₄ :

  cycling cells

  n ₅ :

  cells in generation greater than 0.

  n ₆ :

  CD25⁺ cells