Algorithm & Biological Foundations

Nick Borcherding

Washington University in St. Louis, School of Medicine, St. Louis, MO, USA
ncborch@gmail.com

2026-06-26

Overview

This article describes the mathematical formulation and biological motivation for every component of bHIVE. It is intended as a reference companion to the tutorial vignettes and is organized to mirror the algorithm’s execution flow: initialization, affinity computation, clonal selection, mutation, regulation, and assignment.

Throughout, we use the following notation:

Symbol	Meaning
$\mathbf{X} \in \mathbb{R}^{n \times d}$	Data matrix ($n$ observations, $d$ features)
$\mathbf{A} \in \mathbb{R}^{m \times d}$	Antibody (prototype) matrix ($m$ antibodies)
$\mathbf{x}_i$	The $i$-th data point (row of $\mathbf{X}$)
$\mathbf{a}_j$	The $j$-th antibody (row of $\mathbf{A}$)
$f(\mathbf{x}, \mathbf{a})$	Affinity between a data point and an antibody
$d(\mathbf{x}, \mathbf{a})$	Distance between a data point and an antibody

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Shape Space and Affinity

Biological context

In immunology, shape space (Perelson & Oster 1979) is the abstract space in which an antibody’s binding site (paratope) and an antigen’s epitope are represented as points. Affinity is a function of distance in this space, with closer shapes bind more tightly. The shape space framework provides the geometric foundation for all AIS algorithms: data points are antigens, prototypes are antibodies, and learning is the process of moving antibodies through shape space to maximize affinity to the data.

Affinity kernels

bHIVE computes an $n \times m$ affinity matrix $\mathbf{F}$ where $F_{ij} = f(\mathbf{x}_i, \mathbf{a}_j)$. Five kernels are available:

Gaussian (RBF)

$$f_{\text{gauss}}(\mathbf{x}, \mathbf{a}) = \exp\!\bigl(-\alpha \|\mathbf{x} - \mathbf{a}\|^2\bigr)$$

The default kernel. The bandwidth parameter $\alpha$ controls how quickly affinity decays with distance. Larger $\alpha$ produces sharper, more localized affinity peaks.

Laplace

$$f_{\text{lap}}(\mathbf{x}, \mathbf{a}) = \exp\!\bigl(-\alpha \|\mathbf{x} - \mathbf{a}\|\bigr)$$

Similar to Gaussian but decays with L1-like (linear) distance rather than squared distance. Produces heavier tails, making it more robust to outliers.

Polynomial

$$f_{\text{poly}}(\mathbf{x}, \mathbf{a}) = (\mathbf{x} \cdot \mathbf{a} + c)^p$$

A dot-product kernel where $c$ is a bias term and $p$ is the polynomial degree. Captures non-Euclidean similarity structure. Useful when the relevant signal is in the angle and magnitude of feature vectors rather than their Euclidean distance.

Cosine

$$f_{\text{cos}}(\mathbf{x}, \mathbf{a}) = \frac{\mathbf{x} \cdot \mathbf{a}}{\|\mathbf{x}\| \|\mathbf{a}\| + \epsilon}$$

Direction-based similarity, invariant to vector magnitude. The small $\epsilon = 10^{-12}$ prevents division by zero. Appropriate when feature magnitudes are uninformative (e.g., normalized expression profiles).

Hamming

$$f_{\text{ham}}(\mathbf{x}, \mathbf{a}) = 1 - \frac{1}{d}\sum_{k=1}^d \mathbf{1}[x_k \neq a_k]$$

Proportion of matching features. Used for categorical or binary data.

Distance functions

For clustering assignment and network suppression, bHIVE uses distance rather than affinity. Six metrics are available:

Distance	Formula	Notes
Euclidean	$\sqrt{\sum_k (x_k - a_k)^2}$	Default; L2 norm
Manhattan	$\sum_k |x_k - a_k|$	L1 norm; robust to outliers
Minkowski	$\bigl(\sum_k |x_k - a_k|^p\bigr)^{1/p}$	Generalized; $p=2$ delegates to Euclidean
Cosine	$1 - f_{\text{cos}}(\mathbf{x}, \mathbf{a})$	Angular distance
Mahalanobis	$\sqrt{(\mathbf{x}-\mathbf{a})^\top \Sigma^{-1} (\mathbf{x}-\mathbf{a})}$	Accounts for feature covariance
Hamming	$\frac{1}{d}\sum_k \mathbf{1}[x_k \neq a_k]$	Categorical features

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Initialization

Standard methods

bHIVE provides four initialization strategies for the starting antibody population:

Method	Algorithm	When to use
"sample"	Random rows from $\mathbf{X}$	Default; fast, data-representative
"random"	$\mathbf{a} \sim \mathcal{N}(\bar{\mathbf{x}}, \text{diag}(\sigma^2_k))$	When you want diversity beyond data support
"random_uniform"	$a_k \sim U(\min_i x_{ik}, \max_i x_{ik})$	Uniform coverage of feature ranges
"kmeans++"	D2-weighted sampling	Better coverage; $O(nmd)$ cost

The kmeans++ initialization (Arthur & Vassilvitskii 2007) selects the first antibody uniformly at random, then each subsequent antibody with probability proportional to its squared distance to the nearest existing antibody: $P(\mathbf{x}_i) \propto \min_{j < \text{current}} \|\mathbf{x}_i - \mathbf{a}_j\|^2$.

V(D)J Gene Library (VDJLibrary)

Biological context

Real antibody diversity is generated by V(D)J recombination: the variable region of an immunoglobulin gene is assembled by randomly combining one V (variable), one D (diversity), and one J (joining) gene segment from a germline library. This combinatorial mechanism generates approximately $10^{11}$ distinct antibodies from only $$300 gene segments.

Algorithm

VDJLibrary translates this to feature space:

1. Segment the feature space into three groups (V, D, J) by splitting the $d$ dimensions. Three methods are available:

   • PCA: The first $\lfloor d/3 \rfloor$ principal components form V, the next form D, the remainder form J
   • Cluster: k-means clustering within each dimension group creates alleles
   • Random partition: Dimensions randomly assigned to V, D, J

2. Create alleles for each segment by clustering the data projected onto that segment’s dimensions (using k-means with nV, nD, nJ centers respectively)

3. Generate antibodies by combinatorial sampling: select one V allele, one D allele, and one J allele, then concatenate to form a complete $d$-dimensional antibody vector

This produces structured coverage of the data manifold with $n_V \times n_D \times n_J$ potential combinations, analogous to the combinatorial diversity of the real immune system.

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The Core Loop: Clonal Selection

Biological context

Clonal selection theory (Burnet 1959) states that when an antigen enters the body, the B cells whose receptors best match that antigen are selected to proliferate (clonal expansion) and undergo mutation (somatic hypermutation). The result is an iterative refinement process: the population of antibodies evolves toward higher affinity for the antigen.

This is the mechanism that de Castro & Von Zuben (2001) formalized in the AI-Net algorithm and that bHIVE implements with C++ acceleration.

Algorithm

For each iteration $t = 1, \ldots, T$:

Step 1: Affinity computation. Compute the $n \times m$ affinity matrix $\mathbf{F}$ using the chosen kernel. This is the most expensive step and is dispatched to BLAS via RcppArmadillo.

Step 2: Clonal expansion and mutation. For each data point $\mathbf{x}_i$:

1. Select the $k$ antibodies with highest affinity: $\mathcal{K}_i = \text{top-}k_j\, f(\mathbf{x}_i, \mathbf{a}_j)$

2. For each selected antibody $\mathbf{a}_j \in \mathcal{K}_i$, generate $n_c$ clones where: $$n_c = \text{round}\!\left(\beta \cdot f(\mathbf{x}_i, \mathbf{a}_j)\right), \quad n_c \leq \texttt{maxClones}$$

3. Mutate each clone: $$\mathbf{a}' = \mathbf{a}_j + \boldsymbol{\epsilon}$$ where $\boldsymbol{\epsilon}$ is drawn from the active SHM strategy (see below). If the mutant has higher affinity than the parent, it replaces the parent.

Step 3: Network regulation. Suppress redundant antibodies (see Idiotypic Network section) or apply simple $\epsilon$-threshold suppression: remove $\mathbf{a}_j$ if $d(\mathbf{a}_i, \mathbf{a}_j) < \epsilon$ for any $i < j$.

Step 4: Task-specific assignment.

• Clustering: each data point assigned to the nearest antibody by distance
• Classification: each antibody labeled by majority vote of its assigned data points; predictions via affinity-weighted nearest antibody

Step 5: Convergence check. Early stopping if the antibody population size has not changed by more than stopTolerance for noImprovementLimit consecutive iterations.

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Somatic Hypermutation (SHM)

Biological context

Somatic hypermutation is the process by which activated B cells introduce point mutations into their antibody genes at a rate approximately $10^6$ times higher than the background mutation rate. The enzyme activation-induced cytidine deaminase (AID) preferentially targets WRCY/RGYW DNA motifs (hotspots), creating a non-uniform mutation landscape. Mutations that improve antigen binding are selected for; those that reduce it lead to cell death. Over multiple rounds of mutation and selection (affinity maturation), the antibody population converges on high-affinity binders.

Five strategies

1. Uniform

$$\mathbf{a}' = \mathbf{a} + \mathcal{N}\!\left(0,\; (1 - f) \cdot \gamma^{t-1}\right)$$

where $f$ is the affinity of the parent antibody, $\gamma$ is mutationDecay, and $t$ is the iteration. The mutation rate decreases with both affinity (better antibodies mutate less) and time (the population stabilizes).

When to use: Default baseline. Simple and robust.

2. AIRS (Affinity-Proportional)

$$\text{rate} = c \cdot \exp(-f / T)$$

where $c$ is a scaling constant (c_rate) and $T$ is a temperature parameter. From Watkins & Timmis (2004), this achieves approximately 50% better data reduction than uniform mutation by concentrating mutation on low-affinity antibodies.

When to use: When you want the mutation rate to be strictly controlled by affinity, with a tunable temperature.

3. Hotspot (Feature-Weighted)

$$\epsilon_k = \text{base\_rate} \cdot |g_k| \cdot \mathcal{N}(0, 1)$$

where $g_k = x_k - a_k$ is the per-feature “gradient” toward the matched data point. Features with larger discrepancies mutate more, analogous to AID targeting specific DNA motifs.

When to use: When some features are more informative than others and you want the algorithm to discover this automatically.

4. Energy-Budget

$$\|\boldsymbol{\epsilon}\|^2 \leq E_0 \cdot (1 - f)^2$$

The total mutation magnitude is bounded by an energy budget that shrinks as affinity increases. Inspired by Kleinstein’s observation that the energetic cost of SHM scales quadratically with the number of mutations ($E_{\text{SHM}} \sim N_{\text{mut}}^2$). Individual feature perturbations are sampled uniformly, then rescaled to satisfy the budget constraint.

When to use: When you want hard bounds on how much any single mutation event can perturb an antibody.

5. Adaptive (Adam-Like)

Implementation of a convergence hypothesis. For each feature $k$ of each antibody:

$$g_k = x_k - a_k \quad \text{(gradient toward matched antigen)}$$

$$m_k \leftarrow \beta_1 \, m_k + (1 - \beta_1)\, g_k \quad \text{(first moment)}$$

$$v_k \leftarrow \beta_2 \, v_k + (1 - \beta_2)\, g_k^2 \quad \text{(second moment)}$$

$$\epsilon_k = \text{base\_rate} \cdot \frac{m_k}{\sqrt{v_k} + \epsilon_{\text{adam}}}$$

This is identical to the Adam optimizer (Kingma & Ba 2015) applied per-feature. The biological interpretation: $m_k$ represents the cell’s “memory” of which direction to mutate (accumulated selection pressure), while $v_k$ tracks the variance of past selection signals (stability of the gradient). Features with consistent directional pressure get larger mutations; features with noisy signals are damped.

When to use: Complex landscapes where different features have different scales and signal-to-noise ratios. Best general-purpose choice.

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Idiotypic Network Regulation

Biological context

Jerne’s idiotypic network theory (1974) proposes that antibodies don’t just recognize antigens, but also recognize each other. The variable region of one antibody (its idiotype) can serve as an epitope for another antibody (an anti-idiotype). This creates a regulatory network with emergent properties: memory, tolerance, and self-organized repertoire structure.

Varela & Coutinho (1991) formalized this as a “second generation” immune network with a bell-shaped activation function: below a lower threshold $\theta_{\text{low}}$, insufficient stimulation leads to cell death (neglect); between thresholds, moderate stimulation leads to activation; above $\theta_{\text{high}}$, excessive stimulation leads to suppression.

This is biologically more realistic than the simple $\epsilon$-threshold suppression used in classical AIS, and it produces qualitatively different repertoire dynamics.

Algorithm

Given the antibody matrix $\mathbf{A}$ with $m$ rows:

Step 1: Compute the $m \times m$ pairwise affinity matrix $\mathbf{S}$ where $S_{ij} = f(\mathbf{a}_i, \mathbf{a}_j)$.

Step 2: For each pair $(i, j)$, compute the activation signal:

$$h_{ij} = \begin{cases} -1 & \text{if } S_{ij} < \theta_{\text{low}} \quad \text{(death signal)} \\ +1 & \text{if } \theta_{\text{low}} \leq S_{ij} \leq \theta_{\text{high}} \quad \text{(activation)} \\ -1 & \text{if } S_{ij} > \theta_{\text{high}} \quad \text{(suppression)} \end{cases}$$

Step 3: Euler-integrate the population dynamics ODE for $T$ steps with step size $\Delta t$:

$$\frac{dA_i}{dt} = s - \delta \cdot A_i + A_i \sum_{j: h_{ij} > 0} S_{ij} - A_i \sum_{j: h_{ij} < 0} S_{ij}$$

where: - $s$ = source_rate (basal production of new cells) - $\delta$ = decay_rate (natural death) - The activation and suppression terms are weighted by the actual affinity values

Step 4: Remove antibodies whose population level $A_i$ falls below survival_threshold.

Parameter guidance

Parameter	Default	Effect of increasing
theta_low	0.01	More antibodies die from neglect
theta_high	0.5	Narrower activation window; more suppression
source_rate	0.5	More new cells enter; larger final repertoire
decay_rate	0.1	Faster natural death; smaller repertoire
dt	0.1	Larger integration steps (less stable if too large)
timeSteps	20	Longer simulation; more time for dynamics to settle
survival_threshold	0.5	Stricter survival; fewer antibodies remain

---

Germinal Center Selection

Biological context

The germinal center (GC) is a specialized microstructure within lymph nodes where B cells undergo iterative rounds of mutation (in the dark zone) and selection (in the light zone). In the light zone, B cells compete for help from T follicular helper (Tfh) cells. A Tfh cell evaluates the quality of the B cell’s antigen presentation and provides survival signals only to the best competitors. B cells that fail to receive Tfh help undergo apoptosis.

This is a quality-control bottleneck: it ensures that only improved antibodies survive each round.

Algorithm

The GerminalCenter module implements this as resource competition:

1. Compute a quality score $q_j$ for each antibody $\mathbf{a}_j$. The scoring is task-aware:

   • Classification: $q_j$ = proportion of assigned data points whose label matches the antibody’s majority label
   • Clustering: $q_j$ = mean affinity to assigned data points (cohesion)

2. Apply selection pressure $p \in [0, 1]$ to compute a survival threshold: $$\tau = q_{\min} + p \cdot (q_{\max} - q_{\min})$$

3. For each of nTfh helper cells, the Tfh selects the antibody with highest quality among those not yet helped. Antibodies below the threshold $\tau$ that do not receive Tfh help are removed.

4. Repeat for rounds cycles.

Parameter guidance

Parameter	Default	Effect
nTfh	10	Determines maximum survivors per round
selectionPressure	0.5	0 = everything survives; 1 = only the best
rounds	1	Multiple rounds compound the selection effect

---

Microenvironment

Biological context

B cell fate is strongly influenced by the tissue microenvironment: chemokine gradients direct migration, cytokines influence differentiation, and local cell density affects competition for resources. In the germinal center, the dark zone (proliferation/mutation) and light zone (selection) create spatially distinct functional regions.

Algorithm

The Microenvironment module classifies each antibody’s local context by computing kernel density estimates from the data:

$$\rho_j = \frac{1}{n} \sum_{i=1}^n f(\mathbf{x}_i, \mathbf{a}_j)$$

Based on the percentile of $\rho_j$ within the antibody population:

Zone	Condition	Mutation modifier	Biological analog
Stable	$\rho_j > P_{\text{high}}$	$\times 0.3$ (reduce)	Light zone; protect good solutions
Explore	$\rho_j < P_{\text{low}}$	$\times 2.0$ (increase)	Dark zone; search sparse regions
Boundary	otherwise	$\times 1.0$ (no change)	Transitional; potential class switch

where $P_{\text{high}}$ and $P_{\text{low}}$ are the high_density_threshold and low_density_threshold percentiles, respectively.

Chemokine-like gradients

The module also computes a directional gradient for each antibody, pointing toward the local center of mass of nearby data:

$$\mathbf{g}_j = \frac{\sum_i f(\mathbf{x}_i, \mathbf{a}_j) \cdot \mathbf{x}_i}{\sum_i f(\mathbf{x}_i, \mathbf{a}_j)} - \mathbf{a}_j$$

This gradient can bias mutation direction toward higher-density regions, analogous to chemokine-directed B cell migration.

---

Two-Signal Activation Gate

Biological context

In real immunity, B cell activation requires two signals:

• Signal 1: Specific antigen recognition by the B cell receptor (affinity-dependent)
• Signal 2: Costimulatory signals from helper T cells, danger signals (DAMPs/PAMPs), or cytokines

This two-signal model (Bretscher & Cohn 1970, formalized for AIS by Freitas 2006) prevents autoimmune activation: a B cell that binds self-antigen without costimulation is anergized (silenced) rather than activated. It serves as a biologically-principled regularization mechanism.

Algorithm

An antibody-antigen interaction $(i, j)$ is activated only if both signals exceed their thresholds:

$$\text{activated}_{ij} = \mathbf{1}\!\left[f(\mathbf{x}_i, \mathbf{a}_j) > \tau_1\right] \;\wedge\; \mathbf{1}\!\left[\text{Signal2}_j > \tau_2\right]$$

Three options for Signal 2:

Type	Computation	Use case
"density"	Local data density around $\mathbf{a}_j$	Default; prevents activation in sparse regions
"danger"	User-provided per-data-point danger scores	When you have external quality annotations
"entropy"	Local label entropy among $\mathbf{a}_j$’s neighbors	Classification; high entropy = uncertain region

---

Class Switching

Biological context

During an immune response, B cells can switch the constant region of their antibody (isotype) while retaining the same antigen-binding variable region. The functional consequence is a change in effector properties:

• IgM: Pentameric; broad, low-affinity binding. First responder.
• IgG: Monomeric; high-affinity, highly specific. Dominant in secondary responses.
• IgA: Dimeric; secreted at mucosal surfaces. Boundary patrol.

Algorithm

In bHIVE, class switching changes the effective kernel width of the affinity function, modifying the matching breadth without changing the antibody’s position in feature space:

Isotype	$\alpha$	Matching behavior
IgM	0.1 (broad)	Large receptive field; captures general patterns
IgG	5.0 (narrow)	Small receptive field; fine-grained discrimination
IgA	1.0 (medium)	Intermediate; boundary patrol

Switching rules are driven by the Microenvironment zone assignments:

• Stable zone $\to$ IgG: data-dense regions benefit from specific matching
• Explore zone $\to$ IgM: sparse regions need broad coverage to find signal
• Boundary zone $\to$ IgA: intermediate regions need balanced matching

---

Memory Pool

Biological context

After an immune response resolves, a subset of activated B cells differentiate into long-lived memory B cells that persist for years. Upon re-encounter with the same antigen, memory cells mount a faster and stronger secondary response.

Algorithm

The MemoryPool module maintains an archive of high-performing antibodies:

Archive: After each iteration, antibodies whose mean affinity to their assigned data exceeds archive_threshold are copied into the memory pool, up to max_memory total cells. If the pool is full, the lowest- affinity memories are evicted.

Recall: When new data arrive (or when the data distribution shifts), memory antibodies whose affinity to the new data exceeds recall_threshold are reintroduced into the active repertoire. This provides a warm start for the next round of adaptation.

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Convergent Selection

Biological context

In adaptive immunology, certain TCR/BCR sequences appear in multiple unrelated individuals responding to the same pathogen. These public clonotypes are driven by convergent selection: the structure of the antigen imposes such strong selective pressure that independent immune systems arrive at the same solution.

Algorithm

The ConvergentSelector module finds public antibodies across multiple independent bHIVE runs:

1. Run $R$ independent bHIVE analyses (with different random seeds) on the same data
2. For each antibody from run 1, check how many other runs contain an antibody within distance tolerance
3. Antibodies that appear in $\geq$min_appearances runs are declared public and retained as consensus prototypes

This implements a biologically-motivated ensemble: rather than averaging predictions, it identifies the prototypes that multiple independent optimization trajectories converge on.

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Parameter Guidance

The tables below provide practical starting points and tuning advice for the most important parameters. These are rules of thumb – the optimal settings depend on your data.

Core Parameters

Parameter	Default	Start here	Tune if…
nAntibodies	20	$\sqrt{n}$ to $2\sqrt{n}$	Underfitting (too few) or slow (too many)
beta	5	3–10	Low affinity at convergence (increase)
epsilon	0.01	0.01–0.1	Too many or too few final antibodies
maxIter	50	20–100	Not converged (increase) or slow (decrease)
k	3	2–5	Underfitting (increase) or overfitting (decrease)
affinityFunc	"gaussian"	"gaussian"	Non-Euclidean data (try cosine, polynomial)
alpha	1	$1/(2\sigma^2_{\text{data}})$	All affinities $\approx 0$ (decrease) or $\approx 1$ (increase)

SHM Parameters

Parameter	Default	When to change
method	"uniform"	Try "adaptive" for complex landscapes
decay	1.0	Set $<1$ if population oscillates and won’t converge
mutationMin	0.01	Increase if algorithm stalls at local optima
temperature	0.5 (airs)	Lower = less mutation on high-affinity antibodies
base_rate	0.1 (adaptive)	Scale with feature magnitudes

Module Interactions

Combination	Effect	Recommendation
SHM adaptive + Microenvironment	Zone-dependent adaptive mutation rates	Strong synergy; recommended for complex data
Idiotypic + GerminalCenter	Network regulation + quality selection	Complementary; idiotypic controls diversity, GC controls quality
VDJLibrary + ActivationGate	Structured init + regularized activation	Good for high-dimensional data
ClassSwitcher + Microenvironment	Zone-driven kernel width changes	Requires Microenvironment to define zones

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References

• Arthur, D. & Vassilvitskii, S. (2007). k-means++: The advantages of careful seeding. SODA.
• Bretscher, P. & Cohn, M. (1970). A theory of self-nonself discrimination. Science, 169, 1042–1049.
• Burnet, F.M. (1959). The Clonal Selection Theory of Acquired Immunity. Cambridge University Press.
• de Castro, L.N. & Von Zuben, F.J. (2001). aiNet: Artificial immune network for data analysis. In Data Mining: A Heuristic Approach, 231–260.
• de Castro, L.N. & Timmis, J. (2003). Artificial immune systems as a novel soft computing paradigm. Soft Computing, 7, 526–544.
• Freitas, A.A. (2006). Immunoinformatics. Springer.
• Jerne, N.K. (1974). Towards a network theory of the immune system. Annales d’Immunologie (Institut Pasteur), 125C, 373–389.
• Kingma, D.P. & Ba, J. (2015). Adam: A method for stochastic optimization. ICLR.
• Perelson, A.S. & Oster, G.F. (1979). Theoretical studies of clonal selection: Minimal antibody repertoire size and reliability of self-non-self discrimination. Journal of Theoretical Biology, 81, 645–670.
• Varela, F.J. & Coutinho, A. (1991). Second generation immune networks. Immunology Today, 12, 159–166.
• Watkins, A. & Timmis, J. (2004). Artificial immune recognition system (AIRS): An immune-inspired supervised learning algorithm. Genetic Programming and Evolvable Machines, 5, 291–317.

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