Statistical mechanical concepts in immunology

RACER-m leverages structural features for sparse T cell specificity prediction

Reliable prediction of T cell specificity against antigenic signatures is a formidable task, complicated by the immense diversity of T cell receptor and antigen sequence space and the resulting limited availability of training sets for inferential models. Recent modeling efforts have demonstrated the advantage of incorporating structural information to overcome the need for extensive training sequence data, yet disentangling the heterogeneous TCR-antigen interface to accurately predict MHC-allele-restricted TCR-peptide interactions has remained challenging. Here, we present RACER-m, a coarse-grained structural model leveraging key biophysical information from the diversity of publicly available TCR-antigen crystal structures. Explicit inclusion of structural content substantially reduces the required number of training examples and maintains reliable predictions of TCR-recognition specificity and sensitivity across diverse biological contexts. Our model capably identifies biophysically meaningful point-mutant peptides that affect binding affinity, distinguishing its ability in predicting TCR specificity of point-mutants from alternative sequence-based methods. Its application is broadly applicable to studies involving both closely related and structurally diverse TCR-peptide pairs.

NoisET: Noise Learning and Expansion Detection of T-Cell Receptors

High-throughput sequencing of T- and B-cell receptors makes it possible to track immune repertoires across time, in different tissues, in acute and chronic diseases and in healthy individuals. However, quantitative comparison between repertoires is confounded by variability in the read count of each receptor clonotype due to sampling, library preparation, and expression noise. We review methods for accounting for both biological and experimental noise and present an easy-to-use python package NoisET that implements and generalizes a previously developed Bayesian method. It can be used to learn experimental noise models for repertoire sequencing from replicates, and to detect responding clones following a stimulus. We test the package on different repertoire sequencing technologies and data sets. We review how such approaches have been used to identify responding clonotypes in vaccination and disease data. Availability: NoisET is freely available to use with source code at github.com/statbiophys/NoisET.

Contact map dependence of a T-cell receptor binding repertoire

The T-cell arm of the adaptive immune system provides the host protection against unknown pathogens by discriminating between host and foreign material. This discriminatory capability is achieved by the creation of a repertoire of cells each carrying a T-cell receptor (TCR) specific to non-self-antigens displayed as peptides bound to the major histocompatibility complex (pMHC). The understanding of the dynamics of the adaptive immune system at a repertoire level is complex, due to both the nuanced interaction of a TCR-pMHC pair and to the number of different possible TCR-pMHC pairings, making computationally exact solutions currently unfeasible. To gain some insight into this problem, we study an affinity-based model for TCR-pMHC binding in which a crystal structure is used to generate a distance-based contact map that weights the pairwise amino acid interactions. We find that the TCR-pMHC binding energy distribution strongly depends both on the number of contacts and the repeat structure allowed by the topology of the contact map of choice; this in turn influences T-cell recognition probability during negative selection, with higher variances leading to higher survival probabilities. In addition, we quantify the degree to which neoantigens with mutations in sites with higher contacts are recognized at a higher rate.

The role of geometric features in a germinal center

The germinal center (GC) is a self-organizing structure produced in the lymphoid follicle during the T-dependent immune response and is an important component of the humoral immune system. However, the impact of the special structure of GC on antibody production is not clear. According to the latest biological experiments, we establish a spatiotemporal stochastic model to simulate the whole self-organization process of the GC including the appearance of two specific zones: the dark zone (DZ) and the light zone (LZ), the development of which serves to maintain an effective competition among different cells and promote affinity maturation. A phase transition is discovered in this process, which determines the critical GC volume for a successful growth in both the stochastic and the deterministic model. Further increase of the volume does not make much improvement on the performance. It is found that the critical volume is determined by the distance between the activated B cell receptor (BCR) and the target epitope of the antigen in the shape space. The observation is confirmed in both 2D and 3D simulations and explains partly the variability of the observed GC size.

DNA self-organization controls valence in programmable colloid design

Just like atoms combine into molecules, colloids can self-organize into predetermined structures according to a set of design principles. Controlling valence-the number of interparticle bonds-is a prerequisite for the assembly of complex architectures. The assembly can be directed via solid "patchy" particles with prescribed geometries to make, for example, a colloidal diamond. We demonstrate here that the nanoscale ordering of individual molecular linkers can combine to program the structure of microscale assemblies. Specifically, we experimentally show that covering initially isotropic microdroplets with N mobile DNA linkers results in spontaneous and reversible self-organization of the DNA into Z(N) binding patches, selecting a predictable valence. We understand this valence thermodynamically, deriving a free energy functional for droplet-droplet adhesion that accurately predicts the equilibrium size of and molecular organization within patches, as well as the observed valence transitions with N Thus, microscopic self-organization can be programmed by choosing the molecular properties and concentration of binders. These results are widely applicable to the assembly of any particle with mobile linkers, such as functionalized liposomes or protein interactions in cell-cell adhesion.

A structural-based machine learning method to classify binding affinities between TCR and peptide-MHC complexes

The activation of T cells is triggered by the interactions of T cell receptors (TCRs) with their epitopes, which are peptides presented by major histocompatibility complex (MHC) on the surfaces of antigen presenting cells (APC). While each TCR can only recognize a specific subset from a large repertoire of peptide-MHC (pMHC) complexes, it is very often that peptides in this subset share little sequence similarity. This is known as the specificity and cross-reactivity of T cells, respectively. The binding affinities between different types of TCRs and pMHC are the major driving force to shape this specificity and cross-reactivity in T cell recognition. The binding affinities, furthermore, are determined by the sequence and structural properties at the interfaces between TCRs and pMHC. Fortunately, a wealth of data on binding and structures of TCR-pMHC interactions becomes publicly accessible in online resources, which offers us the opportunity to develop a random forest classifier for predicting the binding affinities between TCR and pMHC based on the structure of their complexes. Specifically, the structure and sequence of a given complex were projected onto a high-dimensional feature space as the input of the classifier, which was then trained by a large-scale benchmark dataset. Based on the cross-validation results, we found that our machine learning model can predict if the binding affinity of a given TCR-pMHC complex is stronger or weaker than a predefined threshold with an overall accuracy approximately around 75 %. The significance of our prediction was estimated by statistical analysis. Moreover, more than 60 % of binding affinities in the ATLAS database can be successfully classified into groups within the range of 2 kcal/mol. Additionally, we show that TCR-pMHC complexes with strong binding affinity prefer hydrophobic interactions between amino acids with large aromatic rings instead of electrostatic interactions. Our results therefore provide insights to design engineered TCRs which enhance the specificity for their targeted epitopes. Taken together, this method can serve as a useful addition to a suite of existing approaches which study binding between TCR and pMHC.

Modelling the interplay between the CD4[Formula: see text]/CD8[Formula: see text] T-cell ratio and the expression of MHC-I in tumours

Describing the anti-tumour immune response as a series of cellular kinetic reactions from known immunological mechanisms, we create a mathematical model that shows the CD4[Formula: see text]/CD8[Formula: see text] T-cell ratio, T-cell infiltration and the expression of MHC-I to be interacting factors in tumour elimination. Methods from dynamical systems theory and non-equilibrium statistical mechanics are used to model the T-cell dependent anti-tumour immune response. Our model predicts a critical level of MHC-I expression which determines whether or not the tumour escapes the immune response. This critical level of MHC-I depends on the helper/cytotoxic T-cell ratio. However, our model also suggests that the immune system is robust against small changes in this ratio. We also find that T-cell infiltration and the specificity of the intra-tumour TCR repertoire will affect the critical MHC-I expression. Our work suggests that the functional form of the time evolution of MHC-I expression may explain the qualitative behaviour of tumour growth seen in patients.

Fluctuation theory of immune response: A statistical mechanical approach to understand pathogen induced T-cell population dynamics

In this period of intense interest in human immunity, we attempt here to quantify the immune response against pathogen invasion through T-cell population dynamics. Borrowing concepts from equilibrium statistical mechanics, we introduce a new description of the immune response function (IMRF) in terms of fluctuations in the population number of relevant biological cells (effector and regulatory T-cells). We use a coarse-grained chemical reaction network model (CG-CRNM) to calculate the number fluctuations and show that the response function derived as such can, indeed, capture the crossover observed in a T-cell driven immune response. We employ the network model to learn the effect of vitamin-D as an immunomodulator. We solve our CG-CRNM using a stochastic Gillespie algorithm. Depending on the effector T-cell concentration, we can classify immune regulation regimes into three categories: weak, strong, and moderate. The IMRF is found to behave differently in these three regimes. A damped cross-regulatory behavior found in the dynamics of effector and regulatory T-cell concentration in the diseased states correlates well with the same found in a cohort of patients with specific malignancies and autoimmune diseases. Importantly, the crossover from the weakly regulated steady state to the other (the strongly regulated) is accompanied by a divergence-like growth in the fluctuation of both the effector and the regulatory T-cell concentration, characteristic of a dynamic phase transition. We believe such steady-state IMRF analyses could help not only to phase-separate different immune stages but also aid in the valuable connection between autoimmunity, optimal vitamin-D, and consequences of immunosuppressive stress and malignancy.

A Multiscale and Comparative Model for Receptor Binding of 2019 Novel Coronavirus and the Implication of its Life Cycle in Host Cells

The respiratory syndrome caused by a new type of coronavirus has been emerging from China and caused more than one million death globally since December 2019. This new virus, called severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) uses the same receptor called Angiotensin-converting enzyme 2 (ACE2) to attack humans as the coronavirus that caused the severe acute respiratory syndrome (SARS) seventeen years ago. Both viruses recognize ACE2 through the spike proteins (S-protein) on their surfaces. It was found that the S-protein from the SARS coronavirus (SARS-CoV) bind stronger to ACE2 than SARS-CoV-2. However, function of a bio-system is often under kinetic, rather than thermodynamic, control. To address this issue, we constructed a structural model for complex formed between ACE2 and the S-protein from SARS-CoV-2, so that the rate of their association can be estimated and compared with the binding of S-protein from SARS-CoV by a multiscale simulation method. Our simulation results suggest that the association of new virus to the receptor is slower than SARS, which is consistent with the experimental data obtained very recently. We further integrated this difference of association rate between virus and receptor into a mathematical model which describes the life cycle of virus in host cells and its interplay with the innate immune system. Interestingly, we found that the slower association between virus and receptor can result in longer incubation period, while still maintaining a relatively higher level of viral concentration in human body. Our computational study therefore provides, from the molecular level, one possible explanation that this new pandemic by far spread much faster than SARS.

Multi-Lineage Evolution in Viral Populations Driven by Host Immune Systems

Viruses evolve in the background of host immune systems that exert selective pressure and drive viral evolutionary trajectories. This interaction leads to different evolutionary patterns in antigenic space. Examples observed in nature include the effectively one-dimensional escape characteristic of influenza A and the prolonged coexistence of lineages in influenza B. Here, we use an evolutionary model for viruses in the presence of immune host systems with finite memory to obtain a phase diagram of evolutionary patterns in a two-dimensional antigenic space. We find that, for small effective mutation rates and mutation jump ranges, a single lineage is the only stable solution. Large effective mutation rates combined with large mutational jumps in antigenic space lead to multiple stably co-existing lineages over prolonged evolutionary periods. These results combined with observations from data constrain the parameter regimes for the adaptation of viruses, including influenza.

Effects of thymic selection on T cell recognition of foreign and tumor antigenic peptides

The advent of cancer immunotherapy has generated renewed hope for the treatment of many malignancies by introducing a number of novel strategies that exploit various properties of the immune system. These therapies are based on the idea that cytotoxic T lymphocytes (CTLs) directly recognize and respond to tumor-associated neoantigens (TANs) in much the same way as they would to foreign peptides presented on cell surfaces. To date, however, nearly all attempts to optimize immunotherapeutic strategies have been empirical. Here, we develop a model of T cell selection based on the assumption of random interaction strengths between a self-peptide and the various T cell receptors. The model enables the analytical study of the effects of selection on the CTL recognition of TANs and completely foreign peptides and can estimate the number of CTLs that can detect donor-matched transplants. We show that negative selection thresholds chosen to reflect experimentally observed thymic survival rates result in near-optimal production of T cells that are capable of surviving selection and recognizing foreign antigen. These analytical results are confirmed by simulation.

A Perspective on the Role of Computational Models in Immunology

This is an exciting time for immunology because the future promises to be replete with exciting new discoveries that can be translated to improve health and treat disease in novel ways. Immunologists are attempting to answer increasingly complex questions concerning phenomena that range from the genetic, molecular, and cellular scales to that of organs, whole animals or humans, and populations of humans and pathogens. An important goal is to understand how the many different components involved interact with each other within and across these scales for immune responses to emerge, and how aberrant regulation of these processes causes disease. To aid this quest, large amounts of data can be collected using high-throughput instrumentation. The nonlinear, cooperative, and stochastic character of the interactions between components of the immune system as well as the overwhelming amounts of data can make it difficult to intuit patterns in the data or a mechanistic understanding of the phenomena being studied. Computational models are increasingly important in confronting and overcoming these challenges. I first describe an iterative paradigm of research that integrates laboratory experiments, clinical data, computational inference, and mechanistic computational models. I then illustrate this paradigm with a few examples from the recent literature that make vivid the power of bringing together diverse types of computational models with experimental and clinical studies to fruitfully interrogate the immune system.

Can the Immune System Perform a t-Test?

The self-nonself discrimination hypothesis remains a landmark concept in immunology. It proposes that tolerance breaks down in the presence of nonself antigens. In strike contrast, in statistics, occurrence of nonself elements in a sample (i.e., outliers) is not obligatory to violate the null hypothesis. Very often, what is crucial is the combination of (self) elements in a sample. The two views on how to detect a change seem challengingly different and it could seem difficult to conceive how immunological cellular interactions could trigger responses with a precision comparable to some statistical tests. Here it is shown that frustrated cellular interactions reconcile the two views within a plausible immunological setting. It is proposed that the adaptive immune system can be promptly activated either when nonself ligands are detected or self-ligands occur in abnormal combinations. In particular we show that cellular populations behaving in this way could perform location statistical tests, with performances comparable to t or KS tests, or even more general data mining tests such as support vector machines or random forests. In more general terms, this work claims that plausible immunological models should provide accurate detection mechanisms for host protection and, furthermore, that investigation on mechanisms leading to improved detection in “in silico” models can help unveil how the real immune system works.

Anergy in self-directed B lymphocytes: A statistical mechanics perspective

Self-directed lymphocytes may evade clonal deletion at ontogenesis but still remain harmless due to a mechanism called clonal anergy. For B-lymphocytes, two major explanations for anergy developed over the last decades: according to Varela theory, anergy stems from a proper orchestration of the whole B-repertoire, such that self-reactive clones, due to intensive feed-back from other clones, display strong inertia when mounting a response. Conversely, according to the model of cognate response, self-reacting cells are not stimulated by helper lymphocytes and the absence of such signaling yields anergy. Through statistical mechanics we show that helpers do not prompt activation of a sub-group of B-cells: remarkably, the latter are just those broadly interacting in the idiotypic network. Hence Varela theory can finally be reabsorbed into the prevailing framework of the cognate response model. Further, we show how the B-repertoire architecture may emerge, where highly connected clones are self-directed as a natural consequence of ontogenetic learning.

Computing with competition in biochemical networks

Cells rely on limited resources such as enzymes or transcription factors to process signals and make decisions. However, independent cellular pathways often compete for a common molecular resource. Competition is difficult to analyze because of its nonlinear global nature, and its role remains unclear. Here we show how decision pathways such as transcription networks may exploit competition to process information. Competition for one resource leads to the recognition of convex sets of patterns, whereas competition for several resources (overlapping or cascaded regulons) allows even more general pattern recognition. Competition also generates surprising couplings, such as correlating species that share no resource but a common competitor. The mechanism we propose relies on three primitives that are ubiquitous in cells: multiinput motifs, competition for a resource, and positive feedback loops.

A stochastic T cell response criterion

The adaptive immune system relies on different cell types to provide fast and coordinated responses, characterized by recognition of pathogenic challenge, extensive cellular proliferation and differentiation, as well as death. T cells are a subset of the adaptive immune cellular pool that recognize immunogenic peptides expressed on the surface of antigen-presenting cells by means of specialized receptors on their membrane. T cell receptor binding to ligand determines T cell responses at different times and locations during the life of a T cell. Current experimental evidence provides support to the following: (i) sufficiently long receptor–ligand engagements are required to initiate the T cell signalling cascade that results in productive signal transduction and (ii) counting devices are at work in T cells to allow signal accumulation, decoding and translation into biological responses. In the light of these results, we explore, with mathematical models, the timescales associated with T cell responses. We consider two different criteria: a stochastic one (the mean time it takes to have had N receptor–ligand complexes bound for at least a dwell time, τ, each) and one based on equilibrium (the time to reach a threshold number N of receptor–ligand complexes). We have applied mathematical models to previous experiments in the context of thymic negative selection and to recent two-dimensional experiments. Our results indicate that the stochastic criterion provides support to the thymic affinity threshold hypothesis, whereas the equilibrium one does not, and agrees with the ligand hierarchy experimentally established for thymic negative selection.

A multifaceted approach to modeling the immune response in tuberculosis

Tuberculosis (TB) is a deadly infectious disease caused by Mycobacterium tuberculosis (Mtb). No available vaccine is reliable and, although treatment exists, approximately 2 million people still die each year. The hallmark of TB infection is the granuloma, a self-organizing structure of immune cells forming in the lung and lymph nodes in response to bacterial invasion. Protective immune mechanisms play a role in granuloma formation and maintenance; these act over different time/length scales (e.g., molecular, cellular, and tissue scales). The significance of specific immune factors in determining disease outcome is still poorly understood, despite incredible efforts to establish several animal systems to track infection progression and granuloma formation. Mathematical and computational modeling approaches have recently been applied to address open questions regarding host-pathogen interaction dynamics, including the immune response to Mtb infection and TB granuloma formation. This provides a unique opportunity to identify factors that are crucial to a successful outcome of infection in humans. These modeling tools not only offer an additional avenue for exploring immune dynamics at multiple biological scales but also complement and extend knowledge gained via experimental tools. We review recent modeling efforts in capturing the immune response to Mtb, emphasizing the importance of a multiorgan and multiscale approach that has tuneable resolution. Together with experimentation, systems biology has begun to unravel key factors driving granuloma formation and protective immune response in TB. WIREs Syst Biol Med 2011 3 479-489 DOI: 10.1002/wsbm.131