A model of immune regulation as a consequence of randomized lymphocyte division and death times

Abstract

The magnitude of an adaptive immune response is controlled by the interplay of lymphocyte quiescence, proliferation, and apoptosis. How lymphocytes integrate receptor-mediated signals influencing these cell fates is a fundamental question for understanding this complex system. We examined how lymphocytes interleave times to divide and die to develop a mathematical model of lymphocyte growth regulation. This model provides a powerful method for fitting and analyzing fluorescent division tracking data and reveals how summing receptor-mediated kinetic changes can modify the immune response progressively from rapid tolerance induction to strong immunity. An important consequence of our results is that intrinsic variability in otherwise identical cells, usually dismissed as noise, may have evolved to be an essential feature of immune regulation.

Fig. 1.

Probabilistic regulation of time to die. Small resting B cells from spleens or lymph nodes were placed in culture under different conditions, and survival was measured by propidium iodide uptake. Cell numbers were counted by reference to beads as described (7). Fitting was performed by using a Matlab fmincon function. (A) Survival curves were fitted by using an exponential decay function [solid line, k = 0.35, 95% confidence interval (C.I.) (+0.011, −0.008)] or a lognormal survival function without T = 0 and T = 1 [dashed line, μ = 48.55, 95% C.I. (+2.3, 2.2), σ = 24.90, 95% C.I. (+4.6, −4.5)]. (B) Survival of B cells isolated from lymph nodes by using the quick preparation method. Data fitted using lognormal survival function [dashed line, μ = 42.59, 95% C.I. (+4.6,−4.5), σ = 29.08, 95% C.I. (+3.8, −3.0)]. The probability distribution function of the fitted lognormal is represented in Inset. (C) Survival curves of B cells isolated from spleen and cultured either alone [circles, μ = 45.40, 95% C.I. (+1.8, −1.7), σ = 36.94, 95% C.I. (+7.3, −5.9)] or with saturating IL-4 [squares, μ = 62.33, 95% C.I.(+2.2, −2.0), σ = 30.02, 95% C.I. (+5.1, −4.2)]. (D) The values for μ and σ of lognormal survival function fit to viability data for three experiments titrating IL-4. Error bars for A–C represent SEM for triplicate samples. Error bars in D represent 95% C.I.s assigned by using a Monte Carlo simulation.

Fig. 2.

Independent times to die and divide. Resting B cells were placed in culture with 10 μg/ml, 3.3 μg/ml, or 0 μg/ml α-CD40 and 500 units/ml IL-4. (A) B cells were cultured in the presence of colcemid, and a time course of 1 h [³H]thymidine pulses was conducted. Cells were harvested and scintillation counted. Lowering α-CD40 concentration delayed μ [60.53, 95% C.I. (+1.1, −1.3) vs. 73.56, 95% C.I. (+3.1, −2.6)] and increased σ [19.01, 95% C.I. (+1.8, −2.5) vs. 24.89, 95% C.I. (+4.9, −4.0)]. (B) B cell number was measured by flow cytometry by using the protocol described in ref. . After 48 h, cell numbers increased in a dose-dependent manner. Before 48 h, cell numbers remained the same regardless of stimulation level. (C) The independent operation of times to divide and times to die for B cells stimulated with 10 μg/ml α-CD40 and 500 units/ml IL-4 is represented here by a cyton plot. The times to divide and times to die are represented as positive and negative values, respectively. Assigning probability distributions to the variations in times to divide and die allows the number of cells dividing and dying in each time interval to be quantitated. The net effect of the two independent timed events is shown in the shaded area.

Fig. 3.

The generalized cyton model. (A) Cell number over time is plotted for five different cyton configurations. Progressing from cyton plot 1 to 5, the median times to divide (φ_i>0) and die (ψ_i>0) in subsequent divisions are increased and decreased, respectively, by <20% (represented by the shaded area in cyton plots). These subtle shifts cause a large net change in the response from expansion (trace 1) to contraction (trace 5). Cyton parameter values: trace 1, φ_i>0 9, ψ_i>0 11; trace 2, φ_i>0 9, ψ_i>0 10.5; trace 3, φ_i>0 10, ψ_i>0 10.5; trace 4, φ_i>0 11, ψ_i>0 10.5; trace 5, φ_i>0 11, ψ_i>0 9. s is kept constant at 0.2 for all cyton plots. (B) B cells were labeled with CFSE and stimulated with 10 μg/ml α-CD40 and 500 units/ml IL-4. Cells were harvested at different times, and the total cell number as well as number of cells in each division was calculated. The optimal cyton solution is shown by dashed lines. Mean and SEM values of three replicates are shown. (C) Cyton solution of data shown in B. By using the data available, the cyton solution for time to death in subsequent divisions cannot be constrained.

Fig. 4.

Modeling division and death. CFSE-labeled B cells were stimulated with 30 μg/ml α-CD40 and 500 units/ml IL-4, and cell numbers were followed by flow cytometry. Identical cells were washed after 40 h of stimulation and recultured either with or without stimulus. (A and B) Cell numbers per division for various harvest times during continual stimulation (A) or after stimulus removal (B) are shown. Increasing culture period before stimulus removal led to an increase in the mean division number before division stops (C, green = 30 h, blue = 40 h, red = 50 h). (D–F) Total cell numbers for 30 h (D, green), 40 h (E, blue), or 50 h (F, red) before stimulus removal. Control cultures with continual stimulation are shown in G (black). (H) The resulting division destiny distributions for each data set (colors representing stimulation time). An optimal cyton solution was determined for each data set by using the modified GCytS in which the division destiny was implemented. The value for φ_i>0 (med = 6.27 h and s = 0.05 h) was determined from continual stimulation data and fixed for all data sets. The resulting fits to experimental data are shown by black lines in D–F. (I) The optimal values for parameter ψ_i>0 obtained for each data set (colors representing stimulation duration). The gray line for ψ_i>0 shown in I is the average value of med and s for 30-, 40-, and 50-h stimulus removal data (med = 30.1 h and s = 0.35 h). This average value for ψ_i>0 was used in an alternative cyton fit (gray lines in D–F) and was also used to fit continual stimulation data G. These model solutions illustrate that the slight variations obtained in ψ_i>0 do not appreciably affect the fit to total cell number data and that the same parameter value can apply to cells in continual stimulus. Data points represent mean and SEM of three replicates.

Fig. 5.

Regulating adaptive immunity. This figure exemplifies the proliferation curves that result from varying parameters known to be affected by signal strength and by external signal regulation such as cytokines. (A–C) Parameters altered are the means of times to division and death (A and B) and the average number of divisions undergone (C). Parameter values varied at equally spaced intervals from: (A) φ₀ med = 30 h (darkest) to 60 h (lightest), s constant at 0.3 h and ψ_i>0 from 100 h to 50 h, s constant at 0.5 h; (B) φ_i>0 from 10 h to 20 h, s constant at 0.2 h and ψ_i>0 from 35 h to 16 h, s constant at 0.75. pF₀ was set to 1.0 in all cases. Average division number (C) was defined by varying the mean of the division destiny distribution from Dμ = 12 (darkest) to Dμ = 5 (lightest), while keeping Dσ constant at 2. Possible outcomes are shown in D.