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.
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Acknowledgements
This collaboration began thanks to the Royal Society’s Frontiers of Science programme. We acknowledge the help of all members of the DpN laboratory and appreciate the efforts of the Central Animal Facility and Divisional Flow cytometry facility in IISc. In particular the assistance of Dr. Santosh Poddar and Mr. Vasista Adiga with imaging and flow cytometry studies is gratefully acknowledged. This study was possible due to grant support from the DBT-IISc programme and infrastructural support from grants from UGC-SAP and DST-FIST. We acknowledge support from European Union FP7-PEOPLE-2012-IRSES 317893 Mathematics for Health and Disease.
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A Python Code
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.
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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.
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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.
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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.
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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.
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The method getstats returns the following cell counts:
- n 1 :
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all live cells
- n 2 :
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activated cells
- n 3 :
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IL-2 producing cells
- n 4 :
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cycling cells
- n 5 :
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cells in generation greater than 0.
- n 6 :
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CD25+ cells
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Majumdar, S., Molina-París, C., Nandi, D., Lythe, G. (2021). Agent-Based Model of Heterogeneous T-Cell Activation in Vitro. In: Molina-París, C., Lythe, G. (eds) Mathematical, Computational and Experimental T Cell Immunology. Springer, Cham. https://doi.org/10.1007/978-3-030-57204-4_14
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