We consider the lifetime of a T cell clonotype, the set of T cells with the same T cell receptor, from its thymic origin to its extinction in a multiclonal repertoire. Using published estimates of total cell numbers and thymic production rates, we calculate the mean number of cells per TCR clonotype, and the total number of clonotypes, in mice and humans. When there is little peripheral division, as in a mouse, the number of cells per clonotype is small and governed by the number of cells with identical TCR that exit the thymus. In humans, peripheral division is important and a clonotype may survive for decades, during which it expands to comprise many cells. We therefore devise and analyse a computational model of homeostasis of a multiclonal population. Each T cell in the model competes for self pMHC stimuli, cells of any one clonotype only recognising a small fraction of the many subsets of stimuli. A constant mean total number of cells is maintained by a balance between cell division and death, and a stable number of clonotypes by a balance between thymic production of new clonotypes and extinction of existing ones. The number of distinct clonotypes in a human body may be smaller than the total number of naive T cells by only one order of magnitude.

We seek numerical methods for second‐order stochastic differential equations that reproduce the stationary density accurately for all values of damping. A complete analysis is possible for scalar linear second‐order equations (damped harmonic oscillators with additive noise), where the statistics are Gaussian and can be calculated exactly in the continuous‐time and discrete‐time cases. A matrix equation is given for the stationary variances and correlation for methods using one Gaussian random variable per timestep. The only Runge–Kutta method with a nonsingular tableau matrix that gives the exact steady state density for all values of damping is the implicit midpoint rule. Numerical experiments, comparing the implicit midpoint rule with Heun and leapfrog methods on nonlinear equations with additive or multiplicative noise, produce behavior similar to the linear case.

Homeostasis of lymphocyte numbers is believed to be due to competition between cellular populations for a common niche of restricted size, defined by the combination of interactions and trophic factors required for cell survival. Here we propose a new mechanism: homeostasis of lymphocyte numbers could also be achieved by the ability of lymphocytes to perceive the density of their own populations. Such a mechanism would be reminiscent of the primordial quorum-sensing systems used by bacteria, in which some bacteria sense the accumulation of bacterial metabolites secreted by other elements of the population, allowing them to “count” the number of cells present and adapt their growth accordingly. We propose that homeostasis of CD4⁺ T cell numbers may occur via a quorum-sensing-like mechanism, where IL-2 is produced by activated CD4⁺ T cells and sensed by a population of CD4⁺ Treg cells that expresses the high-affinity IL-2Rα-chain and can regulate the number of activated IL-2-producing CD4⁺ T cells and the total CD4⁺ T cell population. In other words, CD4⁺ T cell populations can restrain their growth by monitoring the number of activated cells, thus preventing uncontrolled lymphocyte proliferation during immune responses. We hypothesize that malfunction of this quorum-sensing mechanism may lead to uncontrolled T cell activation and autoimmunity. Finally, we present a mathematical model that describes the key role of IL-2 and quorum-sensing mechanisms in CD4⁺ T cell homeostasis during an immune response.

T-cell activation in lymph nodes relies on encounters with antigen (Ag)–bearing dendritic cells (DCs) but the number of DCs required to initiate an immune response is unknown. Here we have used a combination of flow cytometry, 2-photon imaging, and computational modeling to quantify the probability of T cell–DC encounters. We calculated that the chance for a T cell residing 24 hours in a murine popliteal lymph nodes to interact with a DC was 8%, 58%, and 99% in the presence of 10, 100, and 1000 Ag-bearing DCs, respectively. Our results reveal the existence of a threshold in DC numbers below which T-cell responses fail to be elicited for probabilistic reasons. In mice and probably humans, we estimate that a minimum of 85 DCs are required to initiate a T-cell response when starting from precursor frequency of 10−6. Our results have implications for the rational design of DC-based vaccines.

We investigate the nucleation, annihilation, and dynamics of kinks in a classical $(1+1)-\mathrm{dimensional}$ ${\phi }^4$ field theory at finite temperature. From large scale Langevin simulations, we establish that the nucleation rate is proportional to the square of the equilibrium density of kinks. We identify two annihilation time scales: one due to kink-antikink pair recombination after nucleation, the other from nonrecombinant annihilation. We introduce a mesoscopic model of diffusing kinks based on “paired” and “survivor” kinks and antikinks. Analytical predictions for the dynamical time scales, as well as the corresponding length scales, are in good agreement with the simulations.