Elsevier

Journal of Theoretical Biology

Volume 454, 7 October 2018, Pages 215-230
Journal of Theoretical Biology

Physiological factors leading to a successful vaccination: A computational approach

https://doi.org/10.1016/j.jtbi.2018.06.008Get rights and content

Highlights

  • A hybrid mathematical model of T cell priming by dendritic cells in a lymph node is developed.

  • We identify the relative importance of a number of physiological factors in determining the success of T cell activation and immune response time, post vaccination.

  • Models like this should be used as an adjunctive tool to medical experiments for optimisation of the efficiency of a given vaccination protocol.

Abstract

The immune system mounts a response to an infection by activating T cells. T cell activation occurs when dendritic cells, which have already interacted with the pathogen, scan a T cell that is cognate for (responsive to) the pathogen. This often occurs inside lymph nodes. The time it takes for this scanning event to occur, indeed the probability that it will occur at all, depends on many factors, including the rate that T cells and dendritic cells enter and leave the lymph node as well as the geometry of the lymph node and of course other cellular and molecular parameters. In this paper, we develop a hybrid stochastic-deterministic mathematical model at the tissue scale of the lymph node and simulate dendritic cells and cognate T cells to investigate the most important physiological factors leading to a successful and timely immune response after a vaccination. We use an agent-based model to describe the small population of cognate naive T cells and a partial differential equation description for the concentration of mature dendritic cells. We estimate the model parameters based on the known literature and measurements previously taken in our lab. We perform a parameter sensitivity analysis to quantify the sensitivity of the model results to the parameters. The results show that increasing T cell inflow through high endothelial venules, restricting cellular egress via the efferent lymph and increasing the total dendritic cell count by improving vaccinations are the among the most important physiological factors leading to an improved immune response. We also find that increasing the physical size of lymph nodes improves the overall likelihood that an immune response will take place but has a fairly weak effect on the response rate. The nature of dendritic cell trafficking through the LN (either passive or active transport) seems to have little effect on the overall immune response except if a change in overall egress time is observed.

Introduction

The immune response is the body’s biological mechanism to defend against infections and comprises of both innate and adaptive immunity. While innate immunity, the first line of defence, is often sufficient to control an infection, developing essential long-term immunity requires an adaptive response. The adaptive immune response is multifaceted, however, central to the process is the activation of immune cells known as T cells.

Inactivated (or naive) T cells patrol through blood and lymphatic systems and are activated by the detection of antigen, a marker for a specific infection. T cells express receptors (TCRs) which are only specific to an associated (or cognate) antigen. T cell activation occurs when TCRs engage with a processed form of their cognate antigen known as pMHC (antigenic peptide bound to major histocompatibility complex) (Kindt et al., 2007). Efficient activation of T cells requires macrophages and dendritic cells (DCs) which express large numbers of MHC proteins. Activation causes rapid T cell proliferation and the daughter cells subsequently mount an antigen-specific response (Kindt et al., 2007).

Each T cell expresses a specific TCR and each organism needs many T cells and a wide variety of TCRs to help identify a wide range of pathogenic threats. The organism therefore has a TCR repertoire. Talking about T cells, TCRs, cognate T cells/TCRs can get confusing without a well defined language to describe them. We will use the terms defined by Laydon, Bangham and Asquith (Asquith et al., 2015). The total number of distinct TCRs which are created by an organism we will refer to as the TCR ‘diversity’ and we will use the term ‘species’ to describe the collection of T cells which share a common TCR (also known as clonotypes). We shall use the symbol NTCR to describe the TCR diversity and the symbol Nt to describe the copy number of clonotypes of a given species. The copy number of clonotypes of the various species of T cell in an organism is heavily skewed but also specific to the individual. We treat Nt therefore as the effective ‘average’ copy number of clonotypes for each species. We can relate Nt and NTCR therefore with the total number of T cells in the whole organism Ntotal=NTCRNt. As an example, for the αβ T cell repertoire in mice the diversity is estimated at NTCR2×106 (Casrouge et al., 2000) whilst humans have genes which encode for a larger number of TCRs NTCR2×107 (Naylor et al., 2005). Estimates for the total number of naive T cells in humans vary depending on the body mass and age of the individual, but are on the order of Ntotal ∼ 1011 (Bains et al., 2009) and therefore a given T cell species has approximately Nt ∼ 10, 000 clonotypes. Finding an antigen-specific naive T cell (a T cell of a specific species) amongst the total pool of T cells is like finding a needle in a haystack.

T cell activation is supported by dendritic cells (DCs). After processing antigen at the site of an infection, DCs undergo a maturation phase during which they grow long dendrites which can extend up to twice their body length (Miller et al., 2004b). Fragments of the processed antigen are displayed on MHC molecules found on the outer membrane of the DCs and the purpose of the dendrites is to interact with as many neighbouring T cells as possible, optimising the likelihood that a T cell specific to the antigen will be primed (von Andrian, Mempel, 2003, Randolph, Angeli, Swartz, 2005). For this reason, DCs are often called antigen presenting cells (APCs).

Lymph nodes (LNs) are secondary lymphoid tissues. Their main function is to facilitate the otherwise unlikely event that an antigen-specific T cell (a T cell belonging to a species cognate for the exposed antigen) will encounter its cognate antigen on a mature DC and be activated. Until the mid 2000s, theoretical studies of LNs considered them to be spatially homogeneous in which naive T cells could encounter mature DCs in a closed, concentrated environment at a rate determined only by copy number (and not spatial distribution). However, LNs are complex tissues with a spatially inhomogeneous internal structure that supports the interaction between DCs and T cells, although how this is achieved is poorly understood. A schematic cross-section of the anatomy of a LN is presented in Fig. 1. Our schematic is based on detailed diagrams presented by Willard-Mack (2006).

At the site of an infection (or vaccination), immature DCs ingest antigen, mature and are actively transported through lymphatic vessels to LNs. The lymph (the fluid in the lymphatic vessels), which carries the mature DCs, drains into a nearby LN and enters the capsule (‘top’) of the LN via one of the afferent lymphatic (AL) vessels (Fig. 1). Lymph that enters the LN flows into the subcapsular sinus and preferentially around the lobules, through transverse sinuses, towards the efferent lymphatic (EL) vessel where it leaves the LN. The DCs which are carried into the LN via the AL deposit on the interface between the superficial cortex and the subcapsular sinus. Scanning electron micrographs of rat LNs have revealed that the floor of the subcapsular sinus possesses pores leading to the paracortex (Ohtani and Ohtani, 2008). DCs are transported rather directly through the superficial cortex, which consists largely of follicles filled with B cells (which are not the focus of this manuscript), into the paracortex and the deep cortical unit (DCU) which contains densely packed naive T cells which are constantly being trafficked through the LNs of the organism. Two-photon experiments have provided evidence that the DCs, at a cellular level, gradually move through the cortex, in a random fashion (Grigorova, Panteleev, Cyster, 2010, Miller, Wei, Parker, Cahalan, 2002). This microscopic movement is consistent with Brownian motion and diffusion constants can be calculated from experimental data (Miller et al., 2004a). However, spread throughout the DCU are sinuses carrying lymph which flows from the AL to the medulla and eventually into the EL. DCs can find these sinuses and actively move with a bias towards the EL. Therefore, it is expected that there may be some net active transport of cells from the AL to the medulla. The network of sinuses may also contribute to an increase in tissue scale dispersion of cells as well as active transport from AL to EL, but it is unclear to what extent this occurs (Murphy, Travers, Walport, 2007, Sainte-Marie, Peng, Belisle, 1982). There is no evidence that mature DCs exit the LN; they are not present in efferent lymph or circulating blood and typically reside only within the LN after entering. Mature DCs in the murine LN are known to reside for an average of about 60 h (Haig, Hopkins, Miller, 1999, Neeland, 2015). It is likely DCs are triggered to apoptose (die) within the medulla and are then scavenged by resident macrophages (Tomura et al., 2014). DC fate within the LN is rather poorly understood. However, assuming DCs are removed on exiting the DCU and entering the medulla, the time spent in the LN provides us with a more reliable source of approximate transport parameters on the tissue scale than the microscopic scale transport parameters determined from 2-photon experiments.

Naive T cells originate from haematopoietic stem cells in the bone marrow and develop in the thymus before being distributed around the body. T cells typically enter a LN through the blood supply. The blood vessels branch into small vessels known as high endothelial venules (HEVs) that are distributed throughout the lobules in the LN. T cells in the HEVs adhere to the endothelial walls leading to transendothelial migration into the LN and particularly into the DCU (Sage and Carman, 2009). Interestingly, it has recently been shown that mature DCs can activate HEVs so that they become more permeable to T cells entering the DCU (Moussion and Girard, 2011). In the DCU tissue, T cells have also been shown to move randomly using two photon experiments but substantially more rapidly than their DC counterparts and show no evidence of chemotactic attraction to DCs (Miller et al., 2004a). Recently, it has been shown that T cells in the DCU may enter very small lymphatic vessels known as cortical sinuses which permeate through the DCU and are often found close to HEVs (Cyster, Schwab, 2012, Förster, Braun, Worbs, 2012). These vessels drain into the medullary sinuses where T cells are carried into the EL with the bulk flow of the lymph to recirculate through the body. The full nature of how cortical sinuses transport T cells in the DCU is not well understood. Whilst it may be assumed that T cells flow out of the LN once they enter a cortical sinus, it has been suggested that T cells may also roll along the outer surface of these small vessels before detaching later and even enter the cortical sinus for a short time before re-entering the DCU, ultimately contributing to some component of active transport of T cells in the DCU at the tissue scale in the direction of lymph flow (Cyster, Schwab, 2012, Grigorova, Panteleev, Cyster, 2010, Grigorova, Schwab, Phan, Pham, Okada, Cyster, 2009). As with DCs, the transport of T cells at the tissue scale is best characterised by known macroscopic trafficking rates and dwell times (rates of inflow and outflow). There is no evidence to suggest that cognate T cells have an affinity for a particular lymph node. We will therefore consider the number of T cells inside a given DCU is some unbiased random sample of the T cells in the whole system. We will define ntotal=ϵNtotal as the total (large) number of T cells in the DCU where ϵ is the fraction of T cells present in the given DCU compared to the whole lymphatic system. We also define nt=ϵNt as the expected number of T cells of a single species in the DCU at any moment in time, however as this number is usually very small we will intrinsically observe substantial fluctuations of these copy numbers.

In the DCU, slowly moving DCs and relatively motile antigen-specific naive T cells can interact when a TCR from a T cell engages with the antigen presented by a DC. Despite being surrounded by naive T cells, there may only be a few T cells specific for the antigen being presented by the DC in the entire LN and the necessary encounter is subsequently unlikely. We shall use the symbol r to denote the (small) frequency/proportion of T cells which are cognate for a given antigen. Thus, NAg=rNTCR (the ‘cognate TCR diversity’) is the approximate number of distinct T cell species which are cognate for a given antigen (and for which there needs to be at least one encounter each with the cognate antigen for optimal immune response). Thus, after an infection or vaccination (which usually involves giving a patient a reduced strength pathogen to elicit an immune response) the LN typically experiences a large influx of mature DCs depending on the size/dose and type of infection/vaccination. How many mature DCs enter the LN varies greatly between infections and vaccination regimes. As DCs move slowly through the densely packed T cells they are said to ‘scan the repertoire’ of TCRs as they interact with individual T cells, that is, they scan as much of the full TCR diversity as they can to ensure activation of all cognate T cells. Naive T cells not specific to the antigen presented by a DC take approximately 3 minutes to be scanned and a single DC can scan between 500 and 5000 T cells per hour (Bousso, 2008). Combined with the fact that the DCU is highly concentrated with T cells, the purpose of the DCU in this context of the immune system is clear; the DCU substantially increases the chance that a cognate T cell will encounter a mature DC (Bousso, Robey, 2003, Miller, Hejazi, Wei, Cahalan, Parker, 2004a). Due to the efficiency of DC scanning capabilities, the rate of interaction between T cells and DCs is thought to be predominantly transport-limited. In order to ensure safety yet maintain immunogenicity of vaccine formulations, immunologists are often faced with the question, what is the minimal amount of antigen required in a vaccine formulation to ensure all cognate T cell species are activated. The answer to such a question depends on many factors, including transport rates of both DCs and T cells, the size of the DCU, as well as the extent to which HEVs supply T cells and lymphatic sinuses remove cells from the DCU. In this paper, we will explore the effect of these factors on the immune response.

A number of mathematical models have quantified DC-T cell interactions, using three main modelling approaches. The first and simplest modelling approach treats the LN as a spatially homogeneous compartment and describes the rates of change of DC and T cell concentrations using ordinary differential equations (Bajaria, Webb, Cloyd, Kirschner, 2002, Lee, Mandl, Germain, Yates, 2012, Marino, Pawar, Fuller, Reinhart, Flynn, Kirschner, 2004). The advantages of these models is they are simple to construct and can typically be easily solved numerically. However they do not capture the spatial-heterogeneity within a LN or the stochastic nature of cognate T cell scanning events.

The second modelling approach describes cells (or agents) individually and are known as agent-based models (ABM). Cells move and interact according to probabilistic rules based on the current state and position of the cell. ABMs have had many successes modelling host-pathogen systems (for a review, see Bauer et al., 2009). The main advantage of ABMs is that spatial and environmental detail within the LN can be included. Riggs et al. used an ABM to investigate the effect of HEV-to-cortical sinus average separation on scanning times and turnover times for T cells (Riggs et al., 2008) while Linderman et al. used an ABM to investigate DC scanning of T cells during acute infection on a cellular scale and inflow/outflow of cells was calibrated from human studies (Linderman et al., 2010). These sorts of models are also extensively utilized when investigating the interactions between DCs and T cells when the agents search for each other in space (Bogle, Dunbar, 2010a, Bogle, Dunbar, 2010b, Celli, Day, Müller, Molina-Paris, Lythe, Bousso, 2012, Gong, Mattila, Miller, Flynn, Linderman, Kirschner, 2013). Furthermore, because of the intrinsic probabilistic nature of these models, stochastic effects are naturally incorporated into the ABM framework.

A more detailed modelling approach are cellular Potts models (CPMs), which typically involve discretising the domain and individual cells occupy a number of lattice points. This means that cells have defined sizes/morphologies and rules governing motility can be made on a physical (rather than phenomenological) basis. Whilst CPMs tend to give very detailed results, the computational demand of these models are prohibitively large to model the tissue scale and instead have been used to model the interaction of subpopulations of T cells and DCs at a microscopic level (Beltman, Marée, De Boer, 2007a, Beltman, Marée, Lynch, Miller, de Boer, 2007b). CPMs have been an important tool to shed light on the dynamics of cellular motion of immune cells (Meyer-Hermann and Maini, 2005). A review of previous mathematical models of LN dynamics during infection is by Mirsky et al. (2011).

Later ABM and CPM models focused on individual cells and were performed on scales comparable to HEV to cortical sinus separations, in contrast to earlier ODE models that treated LNs as spatially homogeneous.Furthermore, while cellular scale properties in the LN are critical to the overall function of the LN, to date there have been few attempts at a spatial model of LN function on a tissue scale.

An interesting mystery that seems to not be addressed anywhere in the biological or mathematical literature is that despite the variance in the size of lymphatic organs between animals, adaptive immunity (which operates with the same fundamental cellular mechanisms) tends to occur on the same time frame (Miao, Hollenbaugh, Zand, Holden-Wiltse, Mosmann, Perelson, Wu, Topham, 2010, Neeland, Elhay, Nathanielsz, Meeusen, de Veer, 2014). How can a mouse and a sheep elicit an immune response on the same temporal scale despite clearly different spatial scales if T cell activation is transport-limited? Volume changes in secondary organs like the LNs can also occur within the same individual if it becomes inflamed during an immune response. Inflamed LNs commonly undergo a temporary process known as shut down which is driven by prostaglandin production within the LN and it significantly reduces the egress of cells via the EL (Hopkins, McConnell, Pearson, 1981, Neeland, de Veer, Scheerlinck, 2017). Given the hypothesis that phagocytosis of DCs is transport-governed, geometry changes during events such as shut down may have significant implications for DC-T cell interactions. In this paper, we construct and analyse a coarse-grained hybrid ABM model of a DCU in a LN after vaccination to see how the scanning times change due to the effect of tissue-scale morphological changes as well as cellular transport and copy number parameters in the DCU. In order to construct the model, microscopic features have been averaged and replaced by macroscopic properties. Furthermore, the model does not consider all ntotal naive T cells but rather only those that are cognate for the antigen presented by the DCs. Since this is usually a small number, we model each of these T cells individually and stochastically within the DCU. Due to the large number of DCs that flood a LN after inflammation or vaccination, we model the DCs using a partial differential equation (PDE) solved on an irregular lattice over the DCU.

The manuscript is structured as follows. In the next section, the methods are detailed. This is followed by simulation results and parameter sensitivity tests, and finally, we discuss the physiological factors which play the most significant role in the immune response. It should be clear at this point that capitalised N (such as Ntotal and Nt) are used for expected T cell copy numbers in the whole lymphatic system of the organism, calligraphic N (such as NTCR and NAg) are used to describe a number of T cell species and lower case n (such as ntotal and nt) as equivalent to their capitalised counterpart but specifically for the DCU rather than the whole system.

Section snippets

Parameters

As definitive parameters for our model are difficult to obtain, we will focus specifically on modelling sheep lymph nodes. We have obtained cellular trafficking data for sheep lymph nodes in our lab and can be relatively certain about the magnitude of cellular copy numbers. We model only a single DCU. We do this for two reasons. First, as previously mentioned DCs are escorted rather efficiently to the paracortex and this is where the majority of TC interactions occur. It is not immediately

Results and discussion

In this section we subdivide the results and discuss the sensitivity of immune response to changes in various physiological parameters. For each set of parameter sensitivity tests, each parameter is varied by various factors (given on the x-axis of each figure). The main graph shows the distribution of immune response times Tˇ using violin plots. For reference, it is important to remember that DCs are not dumped into the simulation all at once. We distribute the inflow of total DCs N˜ over time

Conclusion

In this manuscript we present a tissue scale spatially resolved mathematical model of the LN (focusing on the DCU specifically) during a vaccination. We use a novel hybrid mathematical model which treats the large number of DCs which arrive post-vaccination/post-infection as a continuum whilst naive cognate T cells of any given species is treated in a stochastic manner (one cell at a time). We base all of our parameters for this model on values from the literature or fit to data we have

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