Preprints.org logo
Preprint
Article

This version is not peer-reviewed.

A Spectral and Geometric Perspective on Immune Repertoire Selection

Submitted:

12 August 2026

Posted:

13 August 2026

You are already at the latest version

Abstract
The immune system delivers a highly diverse receptor repertoire while preserving self-tolerance, functional robustness and effective pathogen recognition. Although the mechanisms governing clonal selection are characterized, the mathematical principles potentially underlying repertoire-wide organization are less understood. We propose a geometric, probabilistic and spectral perspective in which immune repertoire selection could be interpreted as constrained sampling from a high-dimensional space of receptor configurations. Within this perspective, we connect developments in modern probability and combinatorics with distinct, testable aspects of repertoire organization: polynomial stability could define a global landscape of admissible repertoires; negative dependence could constrain simultaneous selection of similar or redundant clonotypes; entropy maximization could preserve the broadest receptor distribution compatible with self-tolerance and antigen recognition; log-concavity could organize compatible configurations into broad basins supporting gradual adaptation. Still, Kadison–Singer subset preservation could explain how comparatively small lymphocyte samples retain statistical and spectral properties of larger repertoires, whereas random-forest connectivity could quantify collective integration generated by local receptor relationships. Finally, spectral independence could constrain propagation of clonal perturbations, allowing local responses without uncontrolled repertoire-wide cascades. These mathematical issues could provide measurable descriptors of diversity, redundancy, representativeness, connectivity and perturbation propagation, generating falsifiable predictions accessible to repertoire sequencing and network analysis. Rather than replacing established immunological mechanisms, we introduce a complementary ensemble-level description shifting the mathematical object of interest from the individual clonotype to the organization of the repertoire as a whole.
Keywords: 
clonality
;  
lymphocytes
;  
immunodynamics
;  
bioinformatics
;  
systems biology
Subject: 
Biology and Life Sciences  -   Biophysics

Introduction

The immune system continuously builds a receptor repertoire that is diverse, self-tolerant, robust and capable of recognizing an enormous variety of pathogens. Although receptor recombination, clonal expansion, affinity maturation and immune tolerance have been extensively investigated, the general mathematical principles through which innumerable local selection events could produce globally organized repertoires are less understood (Hochhaus 2008; Kumar et al. 2019; Adams, Grassmann and Sun 2020; Deimel et al. 2025; Kocher et al. 2025; Ibáñez-Molero et al. 2026). Current descriptions address receptor affinity, stochastic recombination, evolutionary dynamics or network interactions but lack a unified language connecting local probabilistic selection with repertoire-wide organization (Mesin et al. 2020; Daëron 2022; Domínguez-Andrés et al. 2023; Ke et al. 2023; Sjöström et al. 2023; Chen, Liu and Cao 2025).
Recent developments in combinatorics, probability theory and spectral graph methods could provide a potentially relevant perspective: large and complex systems like the biological ones could acquire global organization when local choices are constrained by probabilistic and geometric relationships. Several mathematical advances converge toward this perspective. Strongly Rayleigh probability measures established a relationship between stable multivariate polynomials and negative dependence, providing a mathematical basis for probabilistic selection that limits excessive correlations among selected elements (Borcea, Brändén and Liggett 2009). Entropy-guided randomized algorithms showed how constrained stochastic sampling can explore rich ensembles of compatible configurations instead of converging toward a single deterministic optimum (Asadpour et al. 2010; Oveis Gharan, Saberi and Singh 2011). Extensions of the Kadison–Singer theorem demonstrated that subsets can preserve important spectral properties of much larger systems despite dependencies among their elements (Anari and Oveis Gharan 2015). Related developments connected random forests with global network connectivity (Goel et al. 2015), established log-concavity as an important property of combinatorial probability distributions (Anari, Oveis Gharan and Vinzant 2018) and introduced spectral independence as a criterion constraining long-range influence within high-dimensional probabilistic systems (Anari, Liu and Oveis Gharan 2020).
Accordingly, we aim to explore whether these independent mathematical developments can be integrated into a coherent theoretical perspective on immune repertoire selection. We will argue that polynomial stability could characterize the global structure of admissible repertoire space; negative dependence could regulate selection among similar or redundant receptors; entropy maximization could favor broad but constrained repertoire exploration; log-concavity could define robust regions of compatible configurations; Kadison–Singer subset preservation could describe how smaller lymphocyte populations retain properties of larger repertoires; random-forest connectivity could characterize collective integration among clonotypes; and spectral independence could constrain propagation of local perturbations across the repertoire.
We will proceed as follows. First, we introduce a unified probabilistic formulation in which immune repertoire organization is described through complementary geometric, probabilistic and spectral constraints. Then, we translate this formulation into measurable repertoire properties, including diversity, redundancy, representative sampling, connectivity and robustness. Finally, we derive experimentally testable predictions and discuss the broader implications of this mathematical approach to immune repertoire selection.

The Mathematical Perspective on Immune Repertoire Selection

A useful distinction can be made between the mechanisms generating an immune repertoire and the properties characterizing the repertoire once it has been generated. Receptor recombination, selection, clonal expansion and affinity maturation operate on individual cells and lineages, whereas diversity, redundancy, representativeness, connectivity and resilience are properties of the receptor population as a whole. These two levels of description need not obey the same mathematical rules. Knowing how each clonotype is generated or selected does not, by itself, specify the statistical architecture of the resulting ensemble.
Therefore, we shift the mathematical object of interest from the individual clonotype to the repertoire configuration. Let the potential receptor space contain an enormous number of possible elements, only a small and continuously changing fraction of which is realized at any given time. Then, immune selection can be examined as a problem of how subsets occupy this space: whether selected receptors cluster or repel one another, whether many alternative configurations are comparably admissible, whether a small subset preserves information about the larger population, how strongly different regions are connected and how far the effect of a local perturbation propagates. These distinct questions cannot be reduced to receptor abundance or affinity alone. This change of scale motivates the mathematical correspondences developed below. Rather than assigning each mathematical concept to another molecular mechanism, we ask whether the above-mentioned mathematical concepts could characterize different ensemble-level properties of the same immune repertoire.
Each concept would address a different aspect of its organization, from the geometry of admissible configurations to competition among receptors, representative sampling, collective connectivity and resistance to perturbation.
  • Polynomial stability could provide the most general description of this organizational landscape. Stable multivariate polynomials generate probability distributions whose geometry imposes strong constraints on admissible configurations even in very high-dimensional spaces (Borcea, Brändén and Liggett 2009). In an immunological interpretation, polynomial stability could correspond to a globally coherent repertoire landscape in which biologically compatible receptor combinations occupy structurally well-behaved regions. Repertoire organization would then depend not only on the fitness of individual receptors but also on the global geometry of the space from which they are selected. This hypothesis would shift attention from isolated clonotypes toward relationships among possible repertoire configurations.
  • Negative dependence could describe how this global structure constrains competition among receptors. In Strongly Rayleigh distributions, selection of one element decreases the probability of simultaneously selecting correlated alternatives (Borcea, Brändén and Liggett 2009). If an analogous principle operated during immune selection, expansion or persistence of one clonotype could decrease the probability of retaining highly similar or functionally redundant alternatives. Such dependence could limit excessive overlap while permitting multiple compatible receptor combinations. Repertoire diversity would consequently arise partly from probabilistic competition among related clonotypes rather than exclusively from the independent survival or elimination of individual cells. Importantly, this correspondence would concern the statistical architecture of selection and would not imply a specific molecular mechanism through which one clonotype directly inhibits another.
  • Entropy maximization could account for a complementary property: preservation of extensive receptor diversity within biological constraints. Entropy-based randomized sampling has demonstrated that optimization need not identify a unique deterministic solution and can instead favor rich distributions of admissible configurations (Asadpour et al. 2010; Oveis Gharan, Saberi and Singh 2011). Applied to adaptive immunity, such a principle would predict that repertoire selection could favor the broadest distribution of receptor states compatible with constraints such as self-tolerance, finite population size and adequate antigen recognition. Selection would not necessarily drive the repertoire toward maximal affinity or another single optimum. It could instead preserve many alternative receptor configurations, potentially reconciling broad pathogen coverage with restrictions imposed by immune tolerance.
  • Log-concavity could describe the geometry of those compatible configurations. Log-concave probability distributions concentrate probability within connected, broadly organized regions rather than distributing it among numerous sharply separated optima (Anari, Oveis Gharan and Vinzant 2018). If repertoire probability possessed an analogous structure, neighboring immune configurations could have comparable functional properties. Moderate changes caused by development, ageing, antigen exposure or stochastic clonal turnover could then move the repertoire within a broad compatibility basin rather than forcing transitions between isolated optimal states. Repertoire robustness would, in this interpretation, depend partly on the geometry of the probability landscape: multiple nearby receptor configurations could support comparable global immune organization.
  • The extension of the Kadison–Singer results to Strongly Rayleigh measures introduces a different hypothesis concerning representative subsets. Mathematical results in this area show that subsets can preserve important spectral characteristics of considerably larger systems despite dependencies among their elements (Anari and Oveis Gharan 2015). An immunological analogue could arise if comparatively small populations of circulating, sampled or activated lymphocytes retained major statistical or spectral properties of the larger repertoire from which they originated. Immune surveillance and repertoire sampling would then depend not simply on the absolute number of observed clonotypes but also on whether the selected subset preserves important structural characteristics of the parent population. This possibility would be experimentally distinguishable by comparing spectral and covariance properties across progressively smaller samples of the same repertoire.
  • Random-forest connectivity could provide a complementary description of collective integration. Random forests allow global connectivity to be characterized through probabilistic relationships among locally connected elements (Goel et al. 2015). Applied to immune repertoires, such connectivity could quantify the extent to which clonotypes participate in common functional communities defined by receptor similarity, shared antigen recognition or experimentally established interactions. A repertoire could consequently exhibit collective cohesion without requiring every clonotype to interact directly with every other clonotype. The relevant biological hypothesis would be that local relationships among receptor populations could generate measurable global connectivity and that this connectivity could constitute an organizational property distinct from repertoire diversity alone.
  • Finally, spectral independence could characterize the propagation of perturbations through this interaction structure. Spectral independence constrains how strongly local variables can influence the global state of a high-dimensional probabilistic system (Anari, Liu and Oveis Gharan 2020). If an analogous property characterized immune repertoires, activation, deletion or expansion of individual clonotypes would produce bounded effects on the rest of the repertoire rather than unrestricted cascades. Physiological immune responses could involve local changes while preserving global organization. Conversely, some forms of immune dysregulation could be associated with weakened spectral constraints, allowing local disturbances to acquire disproportionately large repertoire-wide effects. This interpretation would generate a measurable distinction between local clonal responsiveness and global susceptibility to perturbation without equating spectral independence with any particular regulatory pathway.
Overall, the combination of these features would suggest that immune repertoire selection could be viewed as constrained probabilistic sampling in which diversity, compatibility, representativeness, connectivity and robustness arise from complementary properties of the selected ensemble. None of these correspondences is intended as a direct mechanistic explanation of lymphocyte biology, nor do the mathematical results themselves establish that immune repertoires displays these properties. Rather, they define a set of hypotheses through which established mathematical structures could be translated into experimentally measurable features of repertoire organization.

A Unified Mathematical Formulation of Immune Repertoire Selection

Given the ensemble-level hypotheses introduced above, we now express them within a single probabilistic construction. Let
R = { 1 , … , N }
denote a finite candidate receptor space and let
Ω ⊆ R ,     Ω = m ,
denote one realized repertoire. The state variable is not an individual clonotype but the subset Ω . A probability measure
μ Ω = P Ω
is assigned over admissible subsets, with
∑ Ω ⊆ R Ω = m μ Ω = 1 .
This construction permits receptor-level information to enter the model while the quantities of interest are evaluated at repertoire level.
To encode dependencies among selections, introduce binary variables
X i Ω = 1 , i ∈ Ω , 0 , i ∉ Ω .
The generating polynomial of the repertoire distribution is
G μ z 1 , … , z N = ∑ Ω ⊆ R μ Ω ∏ i ∈ Ω z i .
A Strongly Rayleigh version of the proposal would require G μ to be real stable (Borcea, Brändén and Liggett 2009). This condition supplies the formal setting in which negative dependence can be stated. For distinct receptors i and j , a basic pairwise consequence is
C o v μ X i , X j = P i , j ∈ Ω − P i ∈ Ω P j ∈ Ω ≤ 0 .
More generally, our model can be tested for stronger negative-association properties rather than assuming independence among clonotypes.
Repertoire breadth is described by the entropy of the distribution over admissible subsets,
H μ = − ∑ Ω μ Ω l o g μ Ω .
This differs from the Shannon diversity of clone abundances within one observed repertoire: H μ measures how broadly probability is distributed across alternative repertoire configurations. Constrained maximum-entropy selection can be formulated as
μ * = a r g m a x μ H μ
subject to biologically specified expectation constraints
E μ f k Ω = c k ,     k = 1 , … , q ,
and normalization of μ . Such a construction follows the general logic of entropy-guided randomized selection, in which stochasticity is retained while admissible configurations are restricted by global constraints (Asadpour et al. 2010; Oveis Gharan, Saberi and Singh 2011). The corresponding exponential-family form is
μ θ Ω = 1 Z θ e x p   ∑ k = 1 q θ k f k Ω ,
where
Z θ = ∑ Ω e x p   ∑ k = 1 q θ k f k Ω .
Geometric concentration can be introduced by representing a repertoire through a feature vector
x Ω ∈ R d
and assigning it a positive density g x . The proposed log-concavity condition is
g   τ x + 1 − τ y ≥ g x τ g y 1 − τ ,     0 ≤ τ ≤ 1 ,
or equivalently
l o g g   τ x + 1 − τ y ≥ τ l o g g x + 1 − τ l o g g y .
This provides a direct mathematical criterion for testing whether high-probability repertoire states form geometrically coherent regions, consistent with log-concavity results for combinatorial probability distributions (Anari, Oveis Gharan and Vinzant 2018).
Representative subset preservation can be formulated independently of the preceding terms. Associate each receptor i with a feature vector
v i ∈ R d .
For the complete candidate set define
C R = ∑ i ∈ R w i   v i v i ⊤ ,
and for a selected repertoire
C Ω = ∑ i ∈ Ω w ~ i   v i v i ⊤ ,
where the weights are normalized within their respective sets. A spectral preservation error is
E s p e c Ω = C Ω − C R 2 .
The subset-preservation hypothesis predicts the existence of comparatively small Ω for which E s p e c stays bounded. This formulation captures the relevant spectral question without asserting that an immune repertoire literally implements the Kadison–Singer construction (Anari and Oveis Gharan 2015).
Connectivity is introduced by defining a weighted receptor graph
G Ω = Ω , W ,
where W i j ≥ 0 encodes an experimentally chosen relationship between receptors i and j . Let
P i j R F
denote the probability that i and j belong to the same tree component under a specified random-forest measure. Repertoire connectivity is then summarized by
C R F Ω = 2 m m − 1 ∑ i , j ∈ Ω i < j P i j R F .
This quantity converts local graph relations into an ensemble-level measure of integration (Goel et al. 2015).
To formalize bounded propagation of perturbations, let
Ψ Ω = ψ i j
be an influence matrix, where ψ i j quantifies the change in the conditional probability of receptor i associated with a change in receptor j . A spectral-independence condition can be expressed through
λ m a x Ψ Ω ≤ κ ,
for a finite bound κ across the relevant conditional repertoire distributions (Anari, Liu and Oveis Gharan 2020). The quantity
Λ Ω = λ m a x Ψ Ω
becomes the principal spectral observable for perturbation propagation.
These components can be collected without assuming that they are interchangeable. Let
Φ Ω = H , D , L , − E s p e c , C R F , − Λ , U
denote a vector of repertoire-level descriptors, where D is an empirically defined diversity or nonredundancy statistic and U represents biological constraints such as antigen recognition and self-reactivity. A scalarized representation can be written as
F α Ω = α H H + α D D Ω + α L L Ω − α E E s p e c Ω + α C C R F Ω − α Λ Λ Ω + α U U Ω ,
with
α k ≥ 0 .
The coefficients are deliberately left unspecified. They are not proposed as universal biological constants and would have to be estimated, constrained or compared using empirical data. Alternatively, no scalarization need be imposed: repertoires can be compared directly in the multidimensional descriptor space Φ , allowing trade-offs among organizational properties to be measured rather than hidden within predetermined weights.
The resulting probabilistic model can be expressed in Gibbs form,
P α , T Ω = e x p   F α Ω / T ∑ Ω ' e x p   F α Ω ' / T ,
where T > 0 controls concentration over repertoire configurations. This equation is not intended as a quantitative law of immune physiology. It provides a compact mathematical representation in which the hypotheses introduced in the preceding section become separable observables and constraints. Our proposal is falsifiable at several levels: the repertoire distribution can be tested for negative dependence and log-concavity; subsamples can be evaluated for spectral preservation; receptor graphs can be examined for random-forest connectivity; influence matrices can be tested for bounded spectral radius; and the joint descriptor vector can be compared across physiological states and against null models based on independent or affinity-only selection.

Experimentally Testable Predictions

Our geometric and spectral interpretation of immune repertoire selection does not seek to replace current mechanistic descriptions of adaptive immunity. Instead, it identifies organizational quantities that should become experimentally observable if repertoire formation is indeed governed by constrained probabilistic sampling. Unlike conventional approaches, which primarily focus on receptor affinity, clonal frequency or antigen specificity, we predict measurable properties describing the repertoire as a collective mathematical object such that the observables are not individual lymphocyte clones but statistical descriptors characterizing the full receptor population.
The first prediction concerns repertoire redundancy. If negative dependence contributes to repertoire formation, clonotypes sharing highly similar receptor sequences or recognizing closely related antigenic determinants should be underrepresented relative to expectations based on independent sampling. This reduction should persist even after controlling for differences in recombination frequency, clonal expansion and antigen exposure. Modern high-throughput immune repertoire sequencing provides direct measurements of receptor similarity, allowing comparison between observed clone distributions and those generated under null models assuming statistical independence. Therefore, we predict that healthy immune repertoires should exhibit significantly lower functional redundancy than randomized repertoires preserving the same clone frequencies.
A second prediction concerns representative immune sampling. The Kadison–Singer interpretation suggests that relatively small subsets of circulating lymphocytes may preserve the principal statistical organization of the complete immune repertoire. Consequently, covariance matrices computed from moderate blood samples should approximate those obtained from much larger receptor populations after appropriate normalization. This prediction differs from conventional sampling theory because preservation concerns spectral structure rather than simple clone frequencies. If confirmed experimentally, relatively small samples could accurately characterize global repertoire organization despite containing only a fraction of all circulating clonotypes.
Further, we predict that adaptive immune repertoires occupy broad compatibility basins rather than isolated optimal configurations. If log-concavity provides an appropriate geometric description, neighboring repertoires differing by relatively small numbers of clonotypes should display gradual rather than abrupt variation in global diversity, self-reactivity and antigen coverage. Longitudinal immune sequencing performed before and after vaccination, ageing or transient infection should reveal continuous trajectories through repertoire space, instead of discontinuous transitions between unrelated immune states. This behavior would indicate that compatible repertoires are organized within extended regions of high probability rather than around isolated optima.
Another prediction concerns entropy and immune diversity. Entropy maximization implies that immune selection should preserve the richest repertoire compatible with physiological constraints instead of maximizing affinity alone. Vaccination or controlled antigen exposure should increase effective repertoire entropy, while maintaining bounded redundancy and self-reactivity. Conversely, pathological immune states characterized by excessive clonal dominance are expected to exhibit reduced repertoire entropy together with increased redundancy. This prediction distinguishes constrained probabilistic exploration from purely affinity-driven optimization, where repertoire diversity is not necessarily preserved.
Also, we predict measurable consequences for network organization. If random-forest connectivity captures biologically meaningful interactions among receptor populations, healthy repertoires should display intermediate levels of global connectivity sufficient to maintain coordinated immune responses without excessive synchronization. Chronic inflammatory diseases or severe immune deficiencies may instead shift repertoire connectivity toward opposite extremes, either producing fragmented receptor communities or excessively interconnected interaction networks. These predictions can be evaluated by building receptor similarity graphs from sequencing data and measuring graph connectivity through established network-theoretical methods.
The most distinctive prediction concerns spectral organization. Spectral independence implies that local perturbations should be spatially and functionally confined within the repertoire, instead of propagating throughout the entire immune system. Following vaccination, transient infection or selective depletion of individual lymphocyte populations, the dominant eigenvalue of receptor interaction networks should remain bounded despite measurable local changes in clone abundance. In contrast, autoimmune diseases, persistent inflammatory disorders or dysregulated immune activation may exhibit progressive increases in spectral radius, indicating enhanced propagation of local perturbations across the repertoire.
Our predictions naturally extend to developmental and evolutionary processes. During childhood, progressive expansion of the adaptive immune repertoire should primarily increase entropy and representative sampling, while preserving bounded redundancy and stable spectral organization. Healthy ageing may gradually reduce repertoire entropy, but should preserve the overall geometric structure of compatibility basins. Pathological ageing, chronic infection or immune exhaustion could instead produce simultaneous reductions in entropy, increased redundancy, fragmentation of compatibility landscapes and elevated spectral influence.
Collectively, these transform abstract concepts originating from probability, combinatorics and spectral graph theory into measurable properties of adaptive immunity that can be evaluated using current immune repertoire sequencing, single-cell technologies and network analysis.

Conclusions

We suggest that several independent developments in modern probability theory, combinatorics and spectral graph methods could be interpreted collectively as a mathematical perspective on immune repertoire selection. Rather than focusing on individual receptors, our formulation treats the repertoire’s formation as constrained probabilistic sampling within a high-dimensional compatibility landscape. Global immune organization could constitute an additional level of biological description governed by probabilistic and geometric constraints, with diversity, robustness and self-tolerance partly reflecting the statistical architecture of the receptor population.
Conventionally, immune diversity is characterized through the number and frequency distribution of clonotypes. Our approach considers diversity as one component of a broader organizational state. Repertoires containing similar numbers of clonotypes could differ substantially in redundancy, compatibility, spectral organization and representative sampling and consequently display different collective properties despite comparable conventional diversity. Likewise, immune tolerance could have an ensemble-level component complementary to central deletion, peripheral regulation and immune checkpoints. Negative dependence could constrain excessive accumulation of similar receptors, entropy could preserve broad pathogen coverage within biological restrictions and spectral independence could limit amplification of local perturbations. Still, our perspective introduces a geometric interpretation of immune robustness. Adaptive immunity continuously experiences fluctuations caused by receptor generation, antigen exposure and cellular turnover, while physiological function is generally stable over much longer timescales. In addition to regulatory feedback, pathway redundancy and cellular homeostasis, robustness could depend on the repertoire occupying broad, stable regions of a high-dimensional probability landscape. Neighboring repertoire configurations could then possess comparable functional properties, permitting continuous adaptation without substantial disruption of collective organization. Representative subset preservation provides a complementary interpretation of immune surveillance. The Kadison–Singer perspective raises the possibility that relatively small receptor subsets could preserve important statistical and spectral characteristics of much larger populations. If experimentally supported, effective surveillance would depend not only on the number of circulating lymphocytes but also on how efficiently sampled populations represent the organization of the broader repertoire.
Our perspective could extend beyond adaptive immunity. Microbial communities, neuronal populations, stem-cell differentiation, ecological assemblages and evolving protein families all involve selection from vast configuration spaces while preserving some combination of diversity, robustness and functional integration. These systems share the organizational challenge of generating coherent global states through distributed local events. Therefore, probability theory, combinatorics and spectral methods could provide a common mathematical language for investigating this general problem across biological scales. By translating concepts from modern discrete mathematics into falsifiable properties of immune repertoires, our proposal provides a test case for a broader hypothesis: that the step from local biological selection to global organization can be described through common probabilistic, geometric and spectral approaches.

Author Contributions

The Author performed: study concept and design, acquisition of data, analysis and interpretation of data, drafting of the manuscript, critical revision of the manuscript for important intellectual content, statistical analysis, obtained funding, administrative, technical and material support, study supervision.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial or not-for-profit sectors.

Availability: of data and materials

All data and materials generated or analyzed during this study are included in the manuscript. The Author had full access to all the data in the study and took responsibility for the integrity of the data and the accuracy of the data analysis.

Declaration: of generative AI and AI-assisted technologies in the writing process

During the preparation of this work, the author used ChatGPT 5.3 to assist with data analysis and manuscript drafting and to improve spelling, grammar and general editing. After using this tool, the author reviewed and edited the content as needed, taking full responsibility for the content of the publication.

References

  1. Adams, N. M.; Grassmann, S.; Sun, J. C. Clonal Expansion of Innate and Adaptive Lymphocytes. Nat. Rev. Immunol. 2020, 20(11), 694–707. [Google Scholar] [CrossRef] [PubMed]
  2. Anari, Nima; Gharan, Shayan Oveis. The Kadison-Singer Problem for Strongly Rayleigh Measures and Applications to Asymmetric TSP. arXiv 2015, arXiv:1412.1143. [Google Scholar]
  3. Anari, Nima; Gharan, Shayan Oveis; Vinzant, Cynthia. Log-Concave Polynomials I: Entropy and a Deterministic Approximation Algorithm for Counting Bases of Matroids. In Proceedings of the IEEE Symposium on Foundations of Computer Science, 2018. [Google Scholar]
  4. Anari, Nima; Liu, Kuikui; Gharan, Shayan Oveis. Spectral Independence in High-Dimensional Expanders and Applications to the Hardcore Model. arXiv 2020, arXiv:2001.00303. [Google Scholar]
  5. Asadpour, Arash; Goemans, Michel X.; Mądry, Aleksander; Gharan, Shayan Oveis; Saberi, Amin. An O(log n/log log n)-Approximation Algorithm for the Asymmetric Traveling Salesman Problem. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, 2010; pp. 379–389. [Google Scholar]
  6. Borcea, Julius; Brändén, Petter; Liggett, Thomas M. Negative Dependence and the Geometry of Polynomials. J. Am. Math. Soc. 2009, 22, 521–567. [Google Scholar] [CrossRef]
  7. Chen, J.; Liu, J.; Cao, X. Functional and Metabolic Heterogeneity of Dendritic Cells in Self-Tolerance and Autoimmunity. Immunol. Rev. 2025, 336(1), e70068. [Google Scholar] [CrossRef] [PubMed]
  8. Daëron, M. The Immune System as a System of Relations. Front. Immunol. 2022, 13, 984678. [Google Scholar] [CrossRef] [PubMed]
  9. Deimel, L. P.; Nishimura, Y.; Silva Santos, G. S.; Baharani, V. A.; Hernandez, B.; Oliveira, T. Y.; MacLean, A. J.; et al. Clonal Expansion and Diversification of Germinal Center and Memory B Cell Responses to Booster Immunization in Primates. Cell Rep. 2025, 44(8), 116142. [Google Scholar] [CrossRef] [PubMed]
  10. Domínguez-Andrés, J.; Dos Santos, J. C.; Bekkering, S.; Mulder, W. J. M.; van der Meer, J. W. M.; Riksen, N. P.; Joosten, L. A. B.; Netea, M. G. Trained Immunity: Adaptation within Innate Immune Mechanisms. Physiol. Rev. 2023, 103(1), 313–46. [Google Scholar] [CrossRef] [PubMed]
  11. Goel, Ashish; Khanna, Sanjeev; Raghvendra, Sharath; Zhang, Hongyang. Connectivity in Random Forests and Credit Networks. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, 2015; pp. 2037–2048. [Google Scholar]
  12. Hochhaus, G. Relative Receptor Affinity Comparisons among Inhaled/Intranasal Corticosteroids: Perspectives on Clinical Relevance. Respir. Res. 2008, 9(1), 75. [Google Scholar] [CrossRef] [PubMed]
  13. Ibáñez-Molero, S.; Veldman, J.; Nieto, J. Simon; Traets, J. J. H.; George, A.; Hoefakker, K.; Karomi, A.; et al. Tumour-Reactive Heterotypic CD8 T Cell Clusters from Clinical Samples. Nature 2026, 649(8096), 467–76. [Google Scholar] [CrossRef] [PubMed]
  14. Ke, Q.; Greenawalt, A. N.; Manukonda, V.; Ji, X.; Tisch, R. M. The Regulation of Self-Tolerance and the Role of Inflammasome Molecules. Front. Immunol. 2023, 14, 1154552. [Google Scholar] [CrossRef] [PubMed]
  15. Kocher, K.; Drost, F.; Tesfaye, A. M.; Moosmann, C.; Schülein, C.; Grotz, M.; D’Ippolito, E.; et al. Vaccination-Induced T Cell Responses Maintain Polyclonality with High Antigen Receptor Avidity. Sci. Immunol. 2025, 10(112), eadu6730. [Google Scholar] [CrossRef] [PubMed]
  16. Kumar, P.; Saini, S.; Khan, S.; Lele, S. Surendra; Prabhakar, B. S. Restoring Self-Tolerance in Autoimmune Diseases by Enhancing Regulatory T-Cells. Cell. Immunol. 2019, 339, 41–49. [Google Scholar] [CrossRef] [PubMed]
  17. Mesin, L.; Schiepers, A.; Ersching, J.; Barbulescu, A.; Cavazzoni, C. B.; Angelini, A.; Okada, T.; Kurosaki, T.; Victora, G. D. Restricted Clonality and Limited Germinal Center Reentry Characterize Memory B Cell Reactivation by Boosting. Cell 2020, 180(1), 92–106.e11. [Google Scholar] [CrossRef] [PubMed]
  18. Gharan, Oveis; Shayan; Saberi, Amin; Singh, Mohit. A Randomized Rounding Approach to the Traveling Salesman Problem. In Proceedings of the IEEE Symposium on Foundations of Computer Science, 2011. [Google Scholar]
  19. Sjöström, D. J.; Grill, B.; Ambrosetti, E.; Veetil, A. A.; Mohlin, C.; Teixeira, A. I.; Oberdofer, G.; Bjelic, S. Affinity Maturated Transferrin Receptor Apical Domain Blocks Machupo Virus Glycoprotein Binding. J. Mol. Biol. 2023, 435(20), 168262. [Google Scholar] [CrossRef] [PubMed]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.