Citation: Tamara S. Wilson, Benjamin M. Sleeter, Jason Sherba, Dick Cameron. Land-use impacts on water resources and protected areas: applications of state-and-transition simulation modeling of future scenarios[J]. AIMS Environmental Science, 2015, 2(2): 282-301. doi: 10.3934/environsci.2015.2.282
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The production of high-affinity antibodies capable of broad neutralization, viral inactivation, and protection against viral infections or disease requires activation, expansion, and maturation of B-cells into virus specific long-lived plasma and memory cells [46]. Germinal centers (GC) are the anatomical structures in which B-cells undergo somatic hypermutation, immunoglobulin class switching, and antigen-specific selection [30]. Somatic hypermutations are random and, therefore, the emergence of non-autoreactive, high-affinity B-cell clones requires strong selection through competition for survival signals [48]. The exact nature of these survival signals is poorly understood. T follicular helper (Tfh) cells have been identified as an important factor in driving B-cell hypermutation inside germinal centers [45]. Indeed, recent experiments have identified correlations between the density, function, and infection status of Tfh cells and the development of mature germinal centers [18,31,32,33,44,43,36].
Determining the characteristics of germinal centers such as their formation, size, and composition is important in understanding the protective mechanisms against pathogens that induce chronic infections. During HIV infections, only approximately 15-20% of chronically infected subjects develop antibodies with neutralization breadth [25,40]. These antibodies are highly mutated compared to antibodies induced by most viral infections in vivo [41] or through vaccination [26]. For example, the high-affinity human antibody VRC01, which neutralizes 90% of HIV-1, has 70-90 somatic mutations [47] compared to the natural 5-10 somatic mutations [14]. The mechanisms that allow for the production of protective antibodies in some patients but not others are still under investigation [15,39,7,16,3].
Mathematical models have been used in the past to investigate the mechanisms responsible for B-cell somatic hypermutations inside the germinal centers [10,28,22,21,6,12,19,23,1,36,34]. Early studies hypothesized that re-entry into new GCs of B-cells from previous GCs may explain the affinity maturation process [10,28]. Others showed that affinity maturation requires cyclic transition of B-cells between the two anatomical structures of the germinal center: the dark and light zones [22,21,6,12,19,23,1]. The models that incorporate dark and light zones investigated the role of molecular mechanisms such as competition for Tfh cells [23,36,34], antigen on the surface of follicular dendritic cells [37], binding sites [12], and clonal competition [35] in facilitating movement between the two zones. Lastly, they investigated internal and external stimuli that lead to germinal center termination [20,27,1]. These studies have not considered the mechanisms behind the emergence of large number of B-cell somatic hypermutations inside germinal centers as seen in some HIV patients [47]. Nor did they present hypotheses behind the absence of broadly neutralizing antibodies in the majority of HIV patients. Understanding the mechanistic interactions inside GCs that lead to production of plasma cells capable of producing antibodies with neutralization breadth forms the focus of this paper.
To address this, we develop mathematical models of germinal center formation that investigate the role of B-cell competition, Tfh cells, and antigen in inducing large numbers of B-cell somatic hypermutations, as seen in the few HIV patients that produce broadly neutralizing antibodies. We first develop a deterministic model of Tfh cell-B cell interactions to determine how B-cell selection and competition influences GC formation in acute infections. We fit the model to published germinal center B-cell data to estimate parameters. We then investigate the mechanisms that allow for emergence of highly mutated B-cell clones that are capable of protecting against chronic infections with non-mutating antigen, i.e. substances that do not mutate but stimulate antibody generation. Finally, we investigate how our predictions change when we consider antigenic mutation.
For a non-mutating pathogen, we predict that when only a few rounds of somatic hypermutations are needed for the clearance of a pathogen, as in acute infections, the Tfh cells are not limiting the emergence of high affinity B-cell clones. When large numbers of somatic hypermutations arise, however, a limitation in the number of Tfh cells may prevent B-cell clones of higher affinity from emerging and becoming the dominant B-cell population inside the germinal centers. Moreover, we predict that for a mutating pathogen which drives the somatic hypermutation of B-cells, emergence of B clones of highest affinity may be hindered not only through a limitation in the number of Tfh cells but also by the speed of the viral mutation.
We develop a mathematical model of B-Tfh cell dynamics which considers the interaction between the naive CD4 T-cells (
Primed follicular B-cells
The system describing these interactions is given by:
$ \frac{dN}{dt} = s_N -d_N N - \alpha_N V N, $ | (1a) |
$ \frac{dH}{dt} = \alpha_N V N -d_H H - \gamma H B_0, $ | (1b) |
$ \frac{dG}{dt} = \gamma H B_0 - d_G G - \eta G \sum\limits_{i=1}^n B_i, $ | (1c) |
$ \frac{dB_0}{dt} = -d B_0 - \sigma B_0 H, $ | (1d) |
$ \frac{dB_1}{dt} = \alpha \sigma B_{0} H - \sigma B_1 G - d B_1, $ | (1e) |
$ \frac{dB_i}{dt} = \alpha \sigma B_{i-1} G - \sigma B_i G - d B_i, $ | (1f) |
$ \frac{dB_n}{dt} = \alpha \sigma B_{n-1} G -d B_n - \kappa B_n, $ | (1g) |
$ \frac{dP}{dt} = \kappa B_n, $ | (1h) |
$ \frac{dV}{dt} = - \mu V P, $ | (1i) |
for
$ B_t=\sum\limits^{n}_{i=1} B_i, \label{sum} $ | (2) |
for acute infections and for chronic infections where many rounds of affinity maturation lead to development of broadly neutralizing antibody-producing plasma cells, as seen in a few HIV infections [25,40].
In acute infections, B clones undergo between 5 and 10 steps of affinity maturation [14,29,27]. Without loss of generality, we set
The
In our model, the per capita death rates of all CD4 T-cells are equal,
B-cells in each B clone die at rate
Name | Value | Units | Description | Citation |
cells per ml per day | Naive CD4 T-cell recruitment rate | [38] | ||
per day | Naive CD4 T-cell death rate | [38] | ||
ml per day per cell | Pre-Tfh cell production rate | |||
per day | Pre-Tfh cell death rate | [38] | ||
per day | Tfh cell death rate | [38] | ||
per day | B-cell death rate | [11] | ||
per day | Plasma cells production rate | |||
per cell per day | Pre-Tfh cell differentiation rate | [36] | ||
per cell per day | Antigen removal rate | |||
per cell per day | Tfh competition rate | |||
cells per ml | Initial amount of CD4 T cells | [38] | ||
0 | cells per ml | Initial amount of Pre-Tfh cells | ||
0 | cells per ml | Initial amount of Tfh cells | ||
3 | cells | Initial amount of B-cells | [13,11] | |
0 | cells | Initial amount of B-cell clones | ||
0 | cells | Initial amount of plasma cells | ||
per ml | Initial amount of non-mutating antigen | [9] |
We estimate the remaining parameters
Name | Units | Value | Description | Confidence Intervals |
27.469 | B-cell offspring production rate | [14.015 40.924] | ||
ml per cell per day | Affinity maturation rate | [4.8 |
The dynamics of all variables of system (1) over time for parameters in Tables 1 and 2 are shown in Figure 1. The number of offspring produced by each B-cell clone is
The total number of B-cells in the germinal center,
The pre-Tfh and Tfh populations,
For
We performed a focused analysis of the time-dependent sensitivity of model (1)'s trajectories to parameter variation, known as a semi-relative sensitivity analysis. We start by looking at the sensitivity of variables
In Figure 2 we compared the semi-relative sensitivity curves
We further looked for parameters that have antagonistic effects on the
We next want to understand the size and B-cell clone compositions of germinal centers during prolonged antigenic stimuli. During chronic virus infections with viruses like HIV, the development of broadly neutralizing antibodies, with high mutation levels, can occur after many years of infection. For example, the high-affinity human antibody VRC01 has 70-90 mutations [47]. We will use model (1) and parameters in Tables 1 and 2 as a starting point for understanding how the B-cell and Tfh cell dynamics change when many rounds of somatic hypermutations are allowed. Most importantly, we want to determine the mechanistic interactions that allow for the emergence of a large enough B-clone with the highest level of mutation, which is capable of removing the antigen.
We represent highly mutated antibodies by increasing the level of admissible B-cell somatic hypermutations to
We observe that the Tfh cell population is smaller compared to the acute case during the contraction time, i.e. past
To gain an understanding on the role of competition for Tfh cell help we compute and plot the distribution of B-cell clones for
Experimental data suggests that the key to developing therapies againstchronic HIV infection lies in creating B-cells of the highest allowed level of somatic hypermutation [30]. Such later clones are instrumental for creating plasma and memory cells that produce highly mutated antibodies capable of neutralizing HIV virus. Our model is such that only the B-cells in the last clone become plasma cells that remove the virus, and since few
Not surprisingly, clone
Under the adjusted values,
Our model does not consider the effect of a mutating antigen, nor does it consider the need of both antigenic stimuli and Tfh cell help at each stage of B-cell somatic hypermutation. Previous studies predict that B-cell hypermutation is dependent on not only the ability of B-cells to recruit Tfh cell help, but also on the ability of the B-cells to retrieve and present antigen deposited on follicular dendritic cells [42,4,23,34].
We extend model (1) to account for a mutating virus. In particular, we model a sequential mutation from virus
$ \frac{dV_0}{dt} = - f V_0 -\mu V_0 P, $ | (3a) |
$ \frac{dV_i}{dt} = f V_{i-1}-f V_i -\mu V_i P, $ | (3b) |
$ \frac{dV_{n-1}}{dt} =f V_{n-2}-\mu V_{n-1} P, $ | (3c) |
$ \frac{dN}{dt} = s_N -d_N N - \alpha_N^\phi \sum\limits_{i=0}^{n-1} V_i N, $ | (3d) |
$ \frac{dH}{dt} = \alpha_N^\phi \sum\limits_{i=0}^{n-1} V_i N -d_H H - \gamma H B_0, $ | (3e) |
$ \frac{dG}{dt} = \gamma H B_0 - d_G G - \eta G \sum\limits_{i=1}^n B_i, $ | (3f) |
$ \frac{dB_0}{dt} = -d_0 B_0 - \sigma B_0 H V_0, $ | (3g) |
$ \frac{dB_1}{dt} =\alpha \sigma B_0 H V_0 -\sigma V_1 B_1 G -d B_1, $ | (3h) |
$ \frac{dB_j}{dt} = \alpha \sigma B_{j-1} G V_{j-1} - \sigma B_j V_j G - d B_j, $ | (3i) |
$ \frac{dB_n}{dt} = \alpha \sigma B_{n-1} G V_{n-1} -d B_n - \kappa B_n, $ | (3j) |
$ \frac{dP}{dt} = \kappa B_n, $ | (3k) |
for
We numerically solve model (3), using parameters in Tables 1 and 2,
We look in detail at the slow mutation case. For
$ \frac{dB_j}{dt}=\alpha \sigma B_{j-1} G V_{j-1} - \sigma B_j V_j G +r B_j- d B_j, $ | (4) |
where
Lastly, when we consider that the number of somatic hypermutations needed to produce plasma cells is
We developed a mathematical model of germinal center formation that includes competition between B-cell clones for Tfh cell stimulation. When we model responses to an acute pathogen requiring eight rounds of somatic hypermutations, the model reproduces the dynamics observed during germinal center formation, such as the size of the B-cell population, the time of germinal center termination, and the ratio between pre-Tfh and Tfh populations following antigenic challenge. We fit the model to data, and found that there are enough Tfh cells to allow for B-cell clones of the highest level of somatic hypermutations to emerge.
We then extended our model to allow for as many as
When modeling a mutating antigen that drives the rate of B-cell somatic hypermutations, plasma cell production is dependent on the speed of viral mutation. For eight rounds of somatic hypermutations, fast and intermediate mutating plasma cells capable of removing the virus are always produced. A slow mutating virus, however, requires an additional antigen-independent B-cell expansion that maintains enough B-cells inside germinal centers to induce the next round of somatic hypermutation even when the antigenic stimulus is delayed. As in the non-mutating case Tfh are not limiting the emergence of all B-cell clones. For
Our models assume that the B-cell division rate is exponentially distributed (as in [2]), and disregarded the inherent cell cycle delay shown experimentally and considered in previous modeling studies [24,5,17]. One of the reasons for this assumption is the fact that the
$f(t)=\left\{0,t<τ,1,t≥τ, \right.$
|
(5) |
and
Our work assumes that B-cells must undergo a strict number of mutations before maturing into plasma cells. We found that modeling the breadth of the response, through creating plasma cells of different affinities at each stage of B-cell somatic hypermutations did not change our results. Further work is needed to determine the tradeoff between the need of high mutation numbers and the breadth of the immune response in fighting chronic infections.
In summary, we have developed models of Tfh-B-cell interactions to examine the dynamics of germinal centers in both acute and chronic infections. We found that T follicular helper cells are a limiting factor in the emergence of extremely high rounds of B-cell somatic hypermutations for both non-mutating and mutating virus. Moreover, we found that this limitation can be removed by inducing faster transition between clones and limiting the sizes of individual clones. Lastly, for a mutating virus that drives the somatic hypermutations, additional factors such as antigen-independent B-cell proliferation may be needed for plasma cell production and virus neutralization. These results may provide insight into the germinal center role during chronic infections.
We would like to thank the anonymous reviewers for the valuable comments and suggestions.
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1. | Samantha Erwin, Lauren M. Childs, Stanca M. Ciupe, Mathematical model of broadly reactive plasma cell production, 2020, 10, 2045-2322, 10.1038/s41598-020-60316-8 | |
2. | Zishuo Yan, Hai Qi, Yueheng Lan, The role of geometric features in a germinal center, 2022, 19, 1551-0018, 8304, 10.3934/mbe.2022387 | |
3. | Komlan Atitey, Benedict Anchang, Mathematical Modeling of Proliferative Immune Response Initiated by Interactions Between Classical Antigen-Presenting Cells Under Joint Antagonistic IL-2 and IL-4 Signaling, 2022, 9, 2296-889X, 10.3389/fmolb.2022.777390 |
Name | Value | Units | Description | Citation |
cells per ml per day | Naive CD4 T-cell recruitment rate | [38] | ||
per day | Naive CD4 T-cell death rate | [38] | ||
ml per day per cell | Pre-Tfh cell production rate | |||
per day | Pre-Tfh cell death rate | [38] | ||
per day | Tfh cell death rate | [38] | ||
per day | B-cell death rate | [11] | ||
per day | Plasma cells production rate | |||
per cell per day | Pre-Tfh cell differentiation rate | [36] | ||
per cell per day | Antigen removal rate | |||
per cell per day | Tfh competition rate | |||
cells per ml | Initial amount of CD4 T cells | [38] | ||
0 | cells per ml | Initial amount of Pre-Tfh cells | ||
0 | cells per ml | Initial amount of Tfh cells | ||
3 | cells | Initial amount of B-cells | [13,11] | |
0 | cells | Initial amount of B-cell clones | ||
0 | cells | Initial amount of plasma cells | ||
per ml | Initial amount of non-mutating antigen | [9] |
Name | Units | Value | Description | Confidence Intervals |
27.469 | B-cell offspring production rate | [14.015 40.924] | ||
ml per cell per day | Affinity maturation rate | [4.8 |
Name | Value | Units | Description | Citation |
cells per ml per day | Naive CD4 T-cell recruitment rate | [38] | ||
per day | Naive CD4 T-cell death rate | [38] | ||
ml per day per cell | Pre-Tfh cell production rate | |||
per day | Pre-Tfh cell death rate | [38] | ||
per day | Tfh cell death rate | [38] | ||
per day | B-cell death rate | [11] | ||
per day | Plasma cells production rate | |||
per cell per day | Pre-Tfh cell differentiation rate | [36] | ||
per cell per day | Antigen removal rate | |||
per cell per day | Tfh competition rate | |||
cells per ml | Initial amount of CD4 T cells | [38] | ||
0 | cells per ml | Initial amount of Pre-Tfh cells | ||
0 | cells per ml | Initial amount of Tfh cells | ||
3 | cells | Initial amount of B-cells | [13,11] | |
0 | cells | Initial amount of B-cell clones | ||
0 | cells | Initial amount of plasma cells | ||
per ml | Initial amount of non-mutating antigen | [9] |
Name | Units | Value | Description | Confidence Intervals |
27.469 | B-cell offspring production rate | [14.015 40.924] | ||
ml per cell per day | Affinity maturation rate | [4.8 |