Mechanistic Insights into Compositional Change

• Mechanistic understanding of compositional change is a framework that quantifies how systems evolve using rigorous mathematical and computational models.
• It integrates methodologies such as graph transformations, process algebra, and causal models to decompose micro-to-macro interactions and predict phase behavior.
• The field utilizes quantitative metrics like Shapley decomposition and autoregressive models to precisely map and control compositional fluctuations in diverse systems.

Mechanistic understanding of compositional change encompasses the precise, often quantitative, accounting of how the constituents of a complex chemical, physical, material, or biological system evolve, interact, or rearrange—either under natural dynamics or in response to interventions. In leading research, this mechanistic perspective is operationalized through mathematical modeling frameworks such as graph transformation (chemistry), random matrix/free probability theory (complex mixtures), machine learning and explainability techniques (materials design), causal category theory (model abstraction), persistent topological descriptors (nucleation), process algebra (systems biology), and spatiotemporal autoregression (regional economics). This article surveys contemporary approaches for mechanistically describing, tracking, or reasoning about compositional change across these domains, emphasizing the technical and formal tools that allow the decomposition, visualization, and quantification of micro-to-macro compositional effects, phase selection, mechanistic interdependencies, and information flow.

1. Formalisms for Mechanistic Representation and Rule Composition

Graph-based and category-theoretic frameworks have become foundational for formalizing compositional change at the mechanistic level.

Graph Transformation in Chemistry:

In the double-pushout (DPO) approach, reaction mechanisms are encoded as sequences of elementary graph transformation rules $p = (L \xleftarrow{l} K \xrightarrow{r} R)$, where $L$ is the molecular educt graph, $R$ the product graph, and $K$ is the interface graph specifying atoms and bonds that are preserved. Rule composition proceeds via categorical constructions—pullbacks and pushouts—to yield composite rules that rigorously coarse-grain multi-step mechanisms. The “overlay graph” (OG) is introduced as a superposed educt–product pattern, color-coding all atoms and bonds according to their persistent, created, deleted, or transiently changed status, generalizing Fujita’s Imaginary Transition Structure to multi-step pathways. This formalism captures all transient edits, not merely net changes, providing a detailed, automated map from elementary steps to overall mechanism (Andersen et al., 2022).

Process Algebraic Modeling in Systems Biology:

BlenX uses a typed process calculus to modularize biochemical circuits. Elementary reactions are specified using templates and typed binders, ensuring compositional assembly of reaction modules and systematic “unpacking” of phenomenological laws (e.g., Michaelis–Menten, Hill kinetics) into minimal sets of elementary steps compatible with stochastic simulation. Type systems enforce binder compatibility, and compositionality is preserved under module joining, supporting tractable inference in large biochemical networks (Zámborszky et al., 2010).

Category Theory and Causal Model Abstraction:

A formal category of interventional causal models allows compositional abstraction morphisms, mapping fine-grained variable spaces and interventions onto coarse-grained representations. The abstraction error is quantified by Jensen–Shannon divergence between the high-level and lifted predicted distributions under all interventions; the composition of abstractions has subadditive error, enabling mechanistic modularity and certification of abstraction pipelines (Rischel et al., 2021).

2. Quantitative Metrics and Decomposition of Compositional Effects

Mechanistic frameworks often rely on rigorous decomposition of changes into microscopically interpretable terms, statistical metrics, or explainable ML attributions.

Compositional Property Models via Shapley Decomposition in Glasses:

Composition–property predictions are built using XGBoost regressors on mol % vectors spanning hundreds of oxide/halide components. Each predicted property is then exactly decomposed into aligned Shapley additive explanations: $f(x) = E[f(X)] + \sum_{i=1}^n \phi_i(x),$ with $\phi_i(x)$ the marginal contribution of component $i$, efficiently computed via TreeSHAP. Interaction terms ($\phi_{i,j}$) further quantify non-additive effects, revealing mechanistic interdependencies such as the boron anomaly, mixed-modifier effects, and Loewenstein’s rule (Ravinder et al., 2021).

Price Equation for Decomposing Aggregate Change:

In the analysis of urban demographic or economic systems, changes in average trait $Z$ at any aggregation can be decomposed: $$\Delta Z = \sum p_i \Delta z_i + \mathrm{Cov}(p_i, z_i)$$ where the first term is the endogenous within-unit effect, and the covariance term quantifies selection—i.e., compositional reshuffling due to sorting of subunits (e.g., migration to higher-income tracts). This exact and recursive decomposition can be extended across all nested spatial levels, capturing the propagation and dilution of compositional change from micro to macro scales (Kemp et al., 8 Nov 2025).

Mechanistic Metrics in Alloy and Defect Chemistry:

In multicomponent alloys, the local composition vector $\mathbf{C}_{\rm local}^i$ is constructed for atom $i$, with distance $$\lambda_i = \|\mathbf{C}_{\rm global} - \mathbf{C}_{\rm local}^i\|$$ quantifying deviation from the global mean. Distributions of defect energies or short-range order parameters are analyzed as functions of $\lambda$, revealing how property distributions—especially for localized phenomena—are mechanistically controlled by local composition fluctuations within a global background (McCarthy et al., 2023).

3. Mechanistic Drivers of Compositional Fluctuations, Instabilities, and Phase Selection

A central theme is the explication of physical mechanisms—entropic, enthalpic, boundary-induced, or dynamical—that drive transitions, amplify fluctuations, or select among alternative compositional states.

Amplified Instabilities in Multi-Component Mixtures:

Random matrix and free probability theory yield the exact spinodal of multi-component mixtures with arbitrary component densities. The minimal eigenvalue of the Hessian, comprising entropic, diagonal (composition) and random Wigner interaction blocks, determines onset of condensation, random demixing, or composition-driven demixing. Composition imbalance is amplified: even a slight enrichment of a single species (high $y_\alpha$) can localize the instability almost entirely onto that species, exceeding naive entropy scaling. The analytic spinodal is given by a transcendental equation

$\psi(-\rho\,\tilde s^2 z/T) = \tilde s^2 z^2$

with explicit expressions for the instability vector and amplification factor depending on the regime (Thewes et al., 2022).

Compositional Selection and Polymorph Interpenetration in Nucleation:

In model colloidal crystallization, compositional change in growing nuclei is tracked on a two-dimensional free energy surface $\Delta G(N, \chi)$ (N = nucleus size, $\chi$ = BCC fraction), with minima determined by bulk chemical potentials and interfacial terms. Near triple points, nearly flat barriers allow large compositional wandering, leading to mixed/interpenetrating polymorphs rather than core–shell nucleation. Persistent homology–based structural descriptors reveal pre-nucleation polymorph bias in the metastable fluid, showing that the fluid's compositional fluctuations prefigure the compositional outcome of nucleation (Kumari et al., 17 Jun 2025).

Compositional Uniformity and Superspace Descriptions of Nanopatterns:

Nanoscale domain patterning (stripes, snub-square, chessboard) in binary lattices arises not from phase separation but from a constrained uniformity drive—the repulsion among minority motifs under lattice occupation constraints. Modulation vectors from diffraction specify a step-function occupational domain in superspace; this fully encodes the long-range compositional order, allowing prediction and design of the nano-pattern as a deterministic function of overall composition (Gonzalez et al., 2012).

4. Mechanistic Interpretability and Compositional Generalization in Machine Learning Models

Understanding compositional change in high-dimensional representations—particularly in neural networks—necessitates both new interpretability methods and circuit-level causal analysis.

Mechanistic Analysis via Spectral Tracking (CAST):

Layerwise composition in transformer models is mapped by estimating the realized linear transformation matrices $T_i$ between hidden states. Spectral properties—effective rank, spectral decay, entropy—reveal a compression–expansion cycle in decoder architectures: initial feature extraction (high ER), bottlenecked abstraction (low ER, high SDR), and subsequent specialization (re-expansion). Kernel CKA analysis aligns these transitions with stratified phases of information processing, providing a mechanistic blueprint for how composition is sequentially built, compressed, and recomposed (Fu et al., 16 Oct 2025).

Explicit Attention Circuits for Compositional Induction:

In compact transformers tasked with compositional generalization, a minimal circuit of attention heads implements interpretable lookup-and-scan routines. Each attention head is causally validated: from question-index tagging to argument retrieval, and relative–absolute position mapping, every compositional transformation coincides with the flow of information through specific subcircuits. Precise activation swaps at the top of the circuit deterministically control model outputs, confirming the sufficiency and necessity of each mechanistic component (Tang et al., 19 Feb 2025).

Failure Mechanisms for Compositional Binding in VLMs:

In VLMs such as CLIP, mechanistic failure arises from superposition—MLP neurons simultaneously coding for multiple visual features—hindering object binding and clean compositional embeddings. Gradient- and entropy-based analysis of neuron activations links superposition directly to reduced cluster separability and increased misclassification. Suggested interventions include loss terms penalizing co-affinity and architecture modifications for explicit disentanglement, though such fixes remain to be validated (Aravindan et al., 20 Aug 2025).

5. Hierarchical, Multiscale, and Dynamic Compositional Change

Compositional change is rarely flat or static: it unfolds across scales, hierarchies, and through time.

Spatiotemporal Multivariate Autoregressive Models:

Compositional time series—e.g., regional sector shares—are modeled in isometric log-ratio transformed space. The core dynamic is

$$y_{i,t} = A\, y_{i,t-1} + B\, \sum_j w_{ij} y_{j,t-1} + \epsilon_{i,t}$$

with $A$ encoding intra-unit temporal feedback (own and cross-part), and $B$ encoding spatial spillover among neighbor regions or units. Compositional constraints are enforced via back-transformation, and stability and identifiability are analytically characterized. Applications to property transactions and sectoral compositions confirm the model’s ability to partition local inertia and neighborhood effects—a dynamic, mechanistic decomposition across both composition and space (Eckardt et al., 18 Jul 2025).

Recursive Multiscale Price Equation:

In socio-economic systems with nested spatial units, the change in aggregate properties is explicitly decomposed at each scale into transmission (endogenous, trait-changing) and selection (composition-sorting) effects, with recursive nesting from block groups up to counties and entire cities. Scaling laws and spatial concentration of selection become apparent, providing a statistical mechanistic atlas of how micro-level sorting manifests (or cancels) at the macro scale (Kemp et al., 8 Nov 2025).

6. Mechanistic Synthesis, Design Principles, and Outlook

The technical frameworks highlighted above enable direct mechanistic reasoning and principled design of systems for specified compositional outcomes.

• Overlay graphs and mechanistic fingerprints, as in EC hydrolase mechanisms, allow automated classification and interpretable mapping of enzymatic “catalytic entanglement.”
• Shapley-decomposition–based models directly recommend recipe-level interventions for glass design, quantifying both component and interaction effects for targeted property optimization.
• In alloys and materials, explicit mapping from global-to-local composition space guides both sampling strategies for ML-IAPs and defect/property prediction, as well as experimental comparisons using atom-probe maps and short-range order metrics.
• The categorical abstraction error formalism provides modular error bounds for stepwise or hierarchical model reduction and abstraction, restoring mechanistic transparency even in complex multi-layered or multi-scale systems.

A recurring principle is that mechanistic understanding of compositional change—whether in chemistry, materials, biology, social systems, or neural representations—is achievable only via rigorous, often hierarchical and modular, formalisms that preserve fidelity, traceability, and quantitative interpretability at every stage. These mechanistic paradigms underpin both the analysis of naturally occurring complexity and the rational design of novel functional systems.