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June  2017, 14(3): 655-671. doi: 10.3934/mbe.2017037

## Germinal center dynamics during acute and chronic infection

 460 McBryde Hall, Virginia Tech, Blacksburg, VA 24061, USA

* Corresponding author: Stanca M. Ciupe

Received  February 16, 2016 Accepted  October 12, 2016 Published  December 2016

The ability of the immune system to clear pathogens is limited during chronic virus infections where potent long-lived plasma and memory B-cells are produced only after germinal center B-cells undergo many rounds of somatic hypermutations. In this paper, we investigate the mechanisms of germinal center B-cell formation by developing mathematical models for the dynamics of B-cell somatic hypermutations. We use the models to determine how B-cell selection and competition for T follicular helper cells and antigen influences the size and composition of germinal centers in acute and chronic infections. We predict that the T follicular helper cells are a limiting resource in driving large numbers of somatic hypermutations and present possible mechanisms that can revert this limitation in the presence of non-mutating and mutating antigen.

Citation: Samantha Erwin, Stanca M. Ciupe. Germinal center dynamics during acute and chronic infection. Mathematical Biosciences & Engineering, 2017, 14 (3) : 655-671. doi: 10.3934/mbe.2017037
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##### References:
Dynamics of model (2) applied to an acute infection

(a) $B_t$ as given by (2) versus data (•); (b) B-cells that underwent different levels of somatic hypermutations and plasma cells, for $\mu=2$.; (c) Pre-Tfh cells $H$ per ml and Tfh cells $G$ per ml; and (d) Antigen per ml; for $\mu=2$ per cell per day (solid lines), $\mu=2\times 10^{-2}$ per cell per day (dashed lines) and $\mu=2\times 10^{-4}$ per cell per day (dotted lines). The dashed-dotted line is the antigen limit of detection of $3\times 10^{-4}$ sRBC per ml. The other parameter values are given in Tables 1 and 2.

Sensitivity Analysis

$B_t$ and $G$ (grey lines) and the semi-relative sensitivity curves, $q \frac{\partial B_t}{\partial q}$and $q \frac{\partial G}{\partial q}$, over time for $q=\alpha$ (solid lines), $q=\gamma$ (dotted lines) and $q=\sigma$ (dashed lines).

Comparison of model (1)'s dynamics when $n$, $\alpha$ and $\sigma$ are varied

(a) $B_t$ as given by (2) versus data (•); and (b) $G$ per ml as given by model (1) for $n=8$ (solid lines); $n=50$, $\alpha=27.5$, $\sigma=1.1\times 10^{-5}$ ml per cell per day (dashed lines); and $n=50$, $\alpha=1.6$, $\sigma=\times 10^{-3}$ ml per cell per day (dotted lines). The other parameters are given in Tables 1 and 2.

B clone distribution in acute and chronic infections

Clonal distribution $B_i/B_t$ for $0\leq i \leq n$, $t=10$, $t=20$, $t=30$ days following infection for $n=8$ (top row) and $n=50$ (second row). Note that for $n=8$ the germinal center contains the B clone with the highest level of somatic hypermutation $B_8$, while for $n=50$ case the germinal center is dominated by middle clones and the $B_{50}$ clone is absent. We then show two mechanisms to achieve the $B_{50}$ clone: (third row) $\eta=0$ and (bottom row) $\alpha=1.6$, $\sigma=10^{-3}$ ml per cell per day, $\eta=10^{-5}$ per cell per day. $B_t$ is given by (2), and the other parameters are given in Tables 1 and 2. In both situations clone $B_{50}$ dominates the germinal center B-cell population $20$ days following infection.

Comparison of model (3)'s dynamics when $f$ is varied

(a) $B_t$ and $P$, (b) $G$ per ml, and (c) $V_t=\sum_{i=0}^{n-1} V_i$ as given by (3) for $n=8$ and $f=0.9$ (solid lines); $f=0.1$ (dashed lines); $f=0.01$ dotted lines. The other parameters are given in Tables 1, 2, $\alpha_N^\phi=3.6 \times 10^{-6}$ and $V_0^\phi=10^3$. Note that $P$ for $f=0.01$ is negligible.

Virus strains and B clone dynamics for slow mutating virus

(a) $V_i$ per ml, (b) $B_i$ as given by (3) for $n=8$ and $f=0.01$. The other parameters are given in Tables 1, 2, $\alpha_N^\phi=3.6 \times 10^{-6}$ and $V_0^\phi=10^3$.

Models (3)-(4)'s dynamics

(a) $B_t$, (b) $G$ per ml, and (c) $V_t=\sum_{i=0}^{n-1} V_i$ as given by (3) and (4) for $n=8$, $f=0.01$ and $r=0$ (solid lines); $r=0.75$ (dashed lines); and $r=0.8$ (dotted lines); for $n=50$, $f=0.1$ and $r=0.75$ (dashed-dotted line). The other parameters are given in Tables 1, 2, $\alpha_N^\phi=3.6 \times 10^{-6}$ and $V_0^\phi=10^3$.

Variables and fixed parameter values.
 Name Value Units Description Citation $s_N$ $10^4$ cells per ml per day Naive CD4 T-cell recruitment rate [38] $d_N$ $0.01$ per day Naive CD4 T-cell death rate [38] $\alpha_N$ $1.8\times 10^{-11}$ ml per day per cell Pre-Tfh cell production rate $d_H$ $0.01$ per day Pre-Tfh cell death rate [38] $d_G$ $0.01$ per day Tfh cell death rate [38] $d$ $0.8$ per day B-cell death rate [11] $\kappa$ $1.2$ per day Plasma cells production rate $\gamma$ $2$ per cell per day Pre-Tfh cell differentiation rate [36] $\mu$ $2$ per cell per day Antigen removal rate $\eta$ $10^{-5}$ per cell per day Tfh competition rate $N(0)$ $10^6$ cells per ml Initial amount of CD4 T cells [38] $H(0)$ 0 cells per ml Initial amount of Pre-Tfh cells $G(0)$ 0 cells per ml Initial amount of Tfh cells $B_0(0)$ 3 cells Initial amount of B-cells [13,11] $B_i(0)$ 0 cells Initial amount of B-cell clones $P(0)$ 0 cells Initial amount of plasma cells $V(0)$ $2\times 10^8$ per ml Initial amount of non-mutating antigen [9]
 Name Value Units Description Citation $s_N$ $10^4$ cells per ml per day Naive CD4 T-cell recruitment rate [38] $d_N$ $0.01$ per day Naive CD4 T-cell death rate [38] $\alpha_N$ $1.8\times 10^{-11}$ ml per day per cell Pre-Tfh cell production rate $d_H$ $0.01$ per day Pre-Tfh cell death rate [38] $d_G$ $0.01$ per day Tfh cell death rate [38] $d$ $0.8$ per day B-cell death rate [11] $\kappa$ $1.2$ per day Plasma cells production rate $\gamma$ $2$ per cell per day Pre-Tfh cell differentiation rate [36] $\mu$ $2$ per cell per day Antigen removal rate $\eta$ $10^{-5}$ per cell per day Tfh competition rate $N(0)$ $10^6$ cells per ml Initial amount of CD4 T cells [38] $H(0)$ 0 cells per ml Initial amount of Pre-Tfh cells $G(0)$ 0 cells per ml Initial amount of Tfh cells $B_0(0)$ 3 cells Initial amount of B-cells [13,11] $B_i(0)$ 0 cells Initial amount of B-cell clones $P(0)$ 0 cells Initial amount of plasma cells $V(0)$ $2\times 10^8$ per ml Initial amount of non-mutating antigen [9]
Parameter estimates and confidence intervals
 Name Units Value Description Confidence Intervals $\alpha$ 27.469 B-cell offspring production rate [14.015 40.924] $\sigma$ ml per cell per day $1.1 \times 10^{-5}$ Affinity maturation rate [4.8 $\times 10^{-6}$ 1.7 $\times 10^{-5}$]
 Name Units Value Description Confidence Intervals $\alpha$ 27.469 B-cell offspring production rate [14.015 40.924] $\sigma$ ml per cell per day $1.1 \times 10^{-5}$ Affinity maturation rate [4.8 $\times 10^{-6}$ 1.7 $\times 10^{-5}$]
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