A Sensitivity Analysis Comparison of Three Models for the Dynamics of Germinal Centers

A Method to Analyze Models of the Dynamics of Germinal Centers

Germinal centers (GC) are dynamic, short-lived anatomical structures generated within lymphoid follicles during immune responses to protein-containing antigens. There, follicular dendritic cells, antigen-specific B cells, and follicular T helper cells engage with each other in an antigen dependent way, setting into play a mini-evolutionary ecosystem that ultimately lead to antibody affinity maturation, with the resulting GC reaction following a rise-and-fall dynamics. The complexity of the cell-to-cell interaction processes makes very difficult to mechanistically understand the GC dynamics. Different mathematical or computational models have been or can be developed to help clarify the mechanisms driving and regulating the GC dynamics. However, the very important question of which are the dominant model parameters is not frequently studied for most of those models. Here we describe in detail one method to perform such a parameter analysis-known as parameter sensitivity analysis-which can be applied to many models of the GC dynamics.

A Mathematical Model of the Dynamics of Cytokine Expression and Human Immune Cell Activation in Response to the Pathogen Staphylococcus aureus

Cell-based mathematical models have previously been developed to simulate the immune system in response to pathogens. Mathematical modeling papers which study the human immune response to pathogens have predicted concentrations of a variety of cells, including activated and resting macrophages, plasma cells, and antibodies. This study aims to create a comprehensive mathematical model that can predict cytokine levels in response to a gram-positive bacterium, S. aureus by coupling previous models. To accomplish this, the cytokines Tumor Necrosis Factor Alpha (TNF-α), Interleukin 6 (IL-6), Interleukin 8 (IL-8), and Interleukin 10 (IL-10) are included to quantify the relationship between cytokine release from macrophages and the concentration of the pathogen, S. aureus, ex vivo. Partial differential equations (PDEs) are used to model cellular response and ordinary differential equations (ODEs) are used to model cytokine response, and interactions between both components produce a more robust and more complete systems-level understanding of immune activation. In the coupled cellular and cytokine model outlined in this paper, a low concentration of S. aureus is used to stimulate the measured cellular response and cytokine expression. Results show that our cellular activation and cytokine expression model characterizing septic conditions can predict ex vivo mechanisms in response to gram-negative and gram-positive bacteria. Our simulations provide new insights into how the human immune system responds to infections from different pathogens. Novel applications of these insights help in the development of more powerful tools and protocols in infection biology.

Imaging the different timescales of germinal center selection

Germinal centers (GCs) are the site of antibody affinity maturation, a fundamental immunological process that increases the potency of antibodies and thereby their ability to protect against infection. GC biology is highly dynamic in both time and space, making it ideally suited for intravital imaging. Using multiphoton laser scanning microscopy (MPLSM), the field has gained insight into the molecular, cellular, and structural changes and movements that coordinate affinity maturation in real time in their native environment. On the other hand, several limitations of MPLSM have had to be overcome to allow full appreciation of GC events taking place across different timescales. Here, we review the technical advances afforded by intravital imaging and their contributions to our understanding of GC biology.

Germinal Center and Extrafollicular B Cell Responses in Vaccination, Immunity, and Autoimmunity

Activated B cells participate in either extrafollicular (EF) or germinal center (GC) responses. Canonical responses are composed of a short wave of plasmablasts (PBs) arising from EF sites, followed by GC producing somatically mutated memory B cells (MBC) and long-lived plasma cells. However, somatic hypermutation (SHM) and affinity maturation can take place at both sites, and a substantial fraction of MBC are produced prior to GC formation. Infection responses range from GC responses that persist for months to persistent EF responses with dominant suppression of GCs. Here, we review the current understanding of the functional output of EF and GC responses and the molecular switches promoting them. We discuss the signals that regulate the magnitude and duration of these responses, and outline gaps in knowledge and important areas of inquiry. Understanding such molecular switches will be critical for vaccine development, interpretation of vaccine efficacy and the treatment for autoimmune diseases.

Affinity Selection in Germinal Centers: Cautionary Tales and New Opportunities

Our current quantitative knowledge of the kinetics of antibody-mediated immunity is partly based on idealized experiments throughout the last decades. However, new experimental techniques often render contradictory quantitative outcomes that shake previously uncontroversial assumptions. This has been the case in the field of T-cell receptors, where recent techniques for measuring the 2-dimensional rate constants of T-cell receptor–ligand interactions exposed results contradictory to those obtained with techniques measuring 3-dimensional interactions. Recently, we have developed a mathematical framework to rationalize those discrepancies, focusing on the proper fine-grained description of the underlying kinetic steps involved in the immune synapse. In this perspective article, we apply this approach to unveil potential blind spots in the case of B-cell receptors (BCR) and to rethink the interactions between B cells and follicular dendritic cells (FDC) during the germinal center (GC) reaction. Also, we elaborate on the concept of “catch bonds” and on the recent observations that B-cell synapses retract and pull antigen generating a “retracting force”, and propose some testable predictions that can lead to future research.

Modular analysis of the control of flagellar Ca2+-spike trains produced by CatSper and CaV channels in sea urchin sperm

Intracellular calcium ([Ca²⁺]_i) is a basic and ubiquitous cellular signal controlling a wide variety of biological processes. A remarkable example is the steering of sea urchin spermatozoa towards the conspecific egg by a spatially and temporally orchestrated series of [Ca²⁺]_i spikes. Although this process has been an experimental paradigm for reproduction and sperm chemotaxis studies, the composition and regulation of the signalling network underlying the cytosolic calcium fluctuations are hitherto not fully understood. Here, we used a differential equations model of the signalling network to assess which set of channels can explain the characteristic envelope and temporal organisation of the [Ca²⁺]_i-spike trains. The signalling network comprises an initial membrane hyperpolarisation produced by an Upstream module triggered by the egg-released chemoattractant peptide, via receptor activation, cGMP synthesis and decay. Followed by downstream modules leading to intraflagellar pH (pH_i), voltage and [Ca²⁺]_i fluctuations. The Upstream module outputs were fitted to kinetic data on cGMP activity and early membrane potential changes measured in bulk cell populations. Two candidate modules featuring voltage-dependent Ca²⁺-channels link these outputs to the downstream dynamics and can independently explain the typical decaying envelope and the progressive spacing of the spikes. In the first module, [Ca²⁺]_i-spike trains require the concerted action of a classical Ca_V-like channel and a potassium channel, BK (Slo1), whereas the second module relies on pH_i-dependent CatSper dynamics articulated with voltage-dependent neutral sodium-proton exchanger (NHE). We analysed the dynamics of these two modules alone and in mixed scenarios. We show that the [Ca²⁺]_i dynamics observed experimentally after sustained alkalinisation can be reproduced by a model featuring the CatSper and NHE module but not by those including the pH-independent Ca_V and BK module or proportionate mixed scenarios. We conclude in favour of the module containing CatSper and NHE and highlight experimentally testable predictions that would corroborate this conclusion.

Fertilisation in marine invertebrates, such as the sea urchin, occurs during broadcast spawning events in which males and females of co-localised species ejaculate sperm and spawn eggs synchronously. During these events, spermatozoa have to find and fertilise conspecific eggs in the midst of all the other ones, which is a remarkable navigation and mating choice achievement. Sperm cells do this by navigating towards the source of species-specific peptides released by the egg, steered by spatial and temporally orchestrated peaks in intracellular calcium concentration that trigger sudden reorientations. How these calcium spikes are regulated and timed remains elusive. Different calcium channels have been implicated by indirect experimental evidence giving rise to a complex network of putative interacting components. We gained insight into the structure and function of this network by modelling it as a set of candidate modules that could be studied separately. By using this ‘divide and conquer’ approach to the complexity of the network, we could characterise the potential dynamics of each module and confront these dynamics with specific quantitative data. Our results indicate that the channel mediating calcium signals in sea urchin sperm is likely CatSper, a calcium channel necessary for human male fertility.