Mathematical Biosciences and Engineering, 2020, 17(1): 28-55. doi: 10.3934/mbe.2020002 Research article Special Issues

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Dynamics of T cell receptor distributions following acute thymic atrophy and resumption

Stephanie M. Lewkiewicz, Yao-Li Chuang, Tom Chou

1 Department of Mathematics, UCLA, Los Angeles, CA, 90095-1555, USA
2 Department of Mathematics, CalState Northridge, Northridge, CA 91330, USA
3 Department of Biomathematics, UCLA, Los Angeles, CA, 90095-1766, USA

Received: , Accepted: , Published:

Special Issues: Mathematical Methods in the Biosciences

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Naive human T cells are produced and developed in the thymus, which atrophies abruptly and severely in response to physical or psychological stress. To understand how an instance of stress affects the size and “diversity” of the peripheral naive T cell pool, we derive a mean-field autonomous ODE model of T cell replenishment that allows us to track the clone abundance distribution (the mean number of different TCRs each represented by a specific number of cells). We identify equilibrium solutions that arise at different rates of T cell production, and derive analytic approximations to the dominant eigenvalues and eigenvectors of the mathematical model linearized about these equilibria. From the forms of the eigenvalues and eigenvectors, we estimate rates at which counts of clones of different sizes converge to and depart from equilibrium values—that is, how the number of clones of different sizes ”adjusts” to the changing rate of T cell production. Under most physiological realizations of our model, the dominant eigenvalue (representing the slowest dynamics of the clone abundance distribution) scales as a power law in the thymic output for low output levels, but saturates at higher T cell production rates. Our analysis provides a framework for quantitatively understanding how the clone abundance distribution evolves under small changes in the overall T cell production rate.

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