TCPNet: Topology-Complete Perceptron Network

• TCPNet is an SE(3)-equivariant topological neural network that leverages a protein combinatorial complex to integrate residues, interactions, secondary structure elements, and protein-level features.
• It employs a four-step hierarchical message passing scheme using the TCP module for scalar-vector processing, ensuring full topological coverage across multiple ranks.
• TCPNet shows competitive performance on protein tasks, excelling in fold classification and antibody developability by effectively modeling secondary structure organization.

Searching arXiv for the primary TCPNet paper and closely related topological neural network work. The Topology-Complete Perceptron Network (TCPNet) is an SE(3)-equivariant topological neural network introduced within the Topotein framework for protein representation learning. In that formulation, TCPNet operates on a Protein Combinatorial Complex (PCC) whose ranks encode residues, residue-interaction edges, secondary structure elements (SSEs), and the whole protein, and it uses the Topology-Complete Perceptron (TCP) module as its basic building block (Wang et al., 4 Sep 2025). The designation “topology-complete” does not appear in that work as a formal universality theorem; rather, it denotes an architectural commitment to exploiting the full topological structure of the PCC across all ranks, instead of restricting computation to a residue graph alone (Wang et al., 4 Sep 2025).

1. Definition, scope, and nomenclature

In the Topotein formulation, TCPNet is the main encoder built on the PCC. It is defined as a network that is SE(3)-equivariant, operates on multi-rank cells—residues, residue-interaction edges, secondary structure elements, and whole proteins—and performs a four-step hierarchical message passing scheme across ranks $0$–$3$ (Wang et al., 4 Sep 2025). Formally, the input is a PCC

$\mathcal{C}=(S,\mathcal{X},\mathrm{rk}),$

with scalar and vector features at each rank $r\in\{0,1,2,3\}$, and the output is a protein-level embedding, optionally accompanied by residue-level embeddings for downstream tasks (Wang et al., 4 Sep 2025).

The motivation stated for TCPNet is specific to protein representation learning. The authors identify three deficiencies in prior model classes: sequence-only protein LLMs do not use 3D structure directly and do not explicitly model SSEs or multi-scale geometric hierarchy; standard residue-level geometric graph neural networks (GGNNs) create information bottlenecks between SSEs and do not provide a unified representation of residues, interactions, SSEs, and proteins; and existing topological deep learning models based on simplicial, cellular, or hypergraph constructions either impose strict boundary constraints or do not encode hierarchy in a protein-specific way (Wang et al., 4 Sep 2025).

Within the broader literature, TCPNet is best understood as a domain-specific instance of a topological neural network rather than a generic synonym for all topology-aware perceptron systems. The general idea of a neural network whose inputs are drawn from a topological space rather than a finite-dimensional Euclidean space was formalized earlier as a Topological Neural Network (TNN), with related Distributional Neural Networks (DNNs) on spaces of measures (Kouritzin et al., 2023). That earlier work is close in spirit, but it addresses universal approximation on Tychonoff spaces and spaces of measures, whereas TCPNet in Topotein is a concrete protein architecture on combinatorial complexes (Kouritzin et al., 2023).

2. Protein Combinatorial Complex as the computational domain

The PCC is a combinatorial complex

$(S,\mathcal{X},\mathrm{rk}),$

where $S$ is a finite vertex set, $$\mathcal{X}\subseteq\mathcal{P}(S)\setminus\{\emptyset\}$$ is a set of cells, and $\mathrm{rk}:\mathcal{X}\to\mathbb{Z}_{\ge 0}$ is an order-preserving rank function such that

$$x\subseteq y \Rightarrow \mathrm{rk}(x)\le\mathrm{rk}(y),$$

with every singleton $\{s\}$ assigned rank $3$0 (Wang et al., 4 Sep 2025). Unlike simplicial complexes, the PCC imposes no requirement that the boundary of a cell be present, which is presented as crucial for biological structures such as irregular or non-contiguous SSEs (Wang et al., 4 Sep 2025).

For proteins, the rank set is $3$1. The specialization is as follows (Wang et al., 4 Sep 2025):

Rank	Cell type	Interpretation
$3$2	residues	individual residues
$3$3	directed edges	residue interactions
$3$4	SSEs	secondary structure elements
$3$5	protein cell	the whole protein

The PCC uses directed pairwise edges as 1-cells rather than undirected hyperedges. Each residue is connected to its 16 nearest neighbors in 3D, and directionality permits asymmetric incidence between residues and edges (Wang et al., 4 Sep 2025). Rank-2 cells are SSEs formed from sequentially consecutive residues of the same DSSP label with minimum length $3$6, and each residue belongs to at most one SSE, so the 2-cells do not overlap (Wang et al., 4 Sep 2025). Rank $3$7 is a single protein cell containing all residues (Wang et al., 4 Sep 2025).

Incidence matrices organize the relations between ranks. For $3$8,

$3$9

and the corresponding adjacency construction is

$\mathcal{C}=(S,\mathcal{X},\mathrm{rk}),$0

These matrices support within-rank and cross-rank neighborhood definitions for message passing (Wang et al., 4 Sep 2025).

A distinctive PCC construction is the outer-edge neighborhood, introduced because non-overlapping SSEs cannot communicate by shared residues. It is defined by

$\mathcal{C}=(S,\mathcal{X},\mathrm{rk}),$1

$\mathcal{C}=(S,\mathcal{X},\mathrm{rk}),$2

These matrices identify edges that originate in an SSE and exit it, or enter it from outside, while excluding internal SSE edges (Wang et al., 4 Sep 2025). In practice, TCPNet uses the intersection

$\mathcal{C}=(S,\mathcal{X},\mathrm{rk}),$3

for scalarization at the SSE level (Wang et al., 4 Sep 2025).

The PCC is also hierarchical at the feature level. Residues carry amino-acid, angular, positional, and geometric vector features; edges carry distance and displacement features; SSEs carry type, size, angular, eigenspectral, and shape-descriptor features; and the protein cell carries global statistics such as size, covariance eigenvalues, shape descriptors, radius of gyration, and contact density (Wang et al., 4 Sep 2025). This hierarchy is explicitly combinatorial and geometric.

3. The Topology-Complete Perceptron module and SE(3)-equivariant processing

The Topology-Complete Perceptron (TCP) is the fundamental layer used by TCPNet. It is described as an SE(3)-equivariant adaptation of the GCP module from GCPNet, generalized to arbitrary PCC ranks (Wang et al., 4 Sep 2025). For a cell of rank $\mathcal{C}=(S,\mathcal{X},\mathrm{rk}),$4, the TCP takes scalar features $\mathcal{C}=(S,\mathcal{X},\mathrm{rk}),$5, vector features $\mathcal{C}=(S,\mathcal{X},\mathrm{rk}),$6, and a rank-specific local frame $\mathcal{C}=(S,\mathcal{X},\mathrm{rk}),$7, or in practice associated edge frames (Wang et al., 4 Sep 2025).

The module performs vector-channel processing and scalarization through three stages (Wang et al., 4 Sep 2025): $\mathcal{C}=(S,\mathcal{X},\mathrm{rk}),$8

$\mathcal{C}=(S,\mathcal{X},\mathrm{rk}),$9

followed by

$r\in\{0,1,2,3\}$0

and output maps

$r\in\{0,1,2,3\}$1

$r\in\{0,1,2,3\}$2

$r\in\{0,1,2,3\}$3

The scalar gate modulates vector channels while preserving equivariance, because vector features remain geometric vectors and only scalar functions are used to scale them (Wang et al., 4 Sep 2025).

A central implementation choice is edge-centric scalarization. For an edge $r\in\{0,1,2,3\}$4, the local frame is

$r\in\{0,1,2,3\}$5

with $r\in\{0,1,2,3\}$6, and scalarization is

$r\in\{0,1,2,3\}$7

For nodes and SSEs, scalarization averages projections onto incident edge frames or outer-edge frames, respectively (Wang et al., 4 Sep 2025). At rank $r\in\{0,1,2,3\}$8, the protein frame is obtained from PCA eigenvectors of residue positions, with sign disambiguation using the farthest residue direction (Wang et al., 4 Sep 2025).

The paper attributes SE(3)-equivariance to three ingredients: vector features are treated as geometric vectors, scalarized quantities are obtained relative to local orthonormal frames and are therefore rotation- and translation-invariant, and vector computations remain linear with scalar-only gating (Wang et al., 4 Sep 2025). A plausible implication is that equivariance is not attached to any single rank in isolation, but is maintained across the entire rank hierarchy through a common scalar–vector processing template.

4. Four-step hierarchical message passing in TCPNet

TCPNet consists of an embedding module, repeated interaction layers, residual normalization, and a readout stage (Wang et al., 4 Sep 2025). The interaction stack has $r\in\{0,1,2,3\}$9 layers in the reported implementation, and each interaction layer executes a four-step hierarchical message passing scheme (Wang et al., 4 Sep 2025).

Step 1: edge-level messaging. For each directed edge $(S,\mathcal{X},\mathrm{rk}),$0, the network computes a message from the source residue, target residue, edge features, and the SSE features of the parent SSEs of the two residues: $(S,\mathcal{X},\mathrm{rk}),$1 where $(S,\mathcal{X},\mathrm{rk}),$2 is the parent-SSE feature if residue $(S,\mathcal{X},\mathrm{rk}),$3 belongs to an SSE and $(S,\mathcal{X},\mathrm{rk}),$4 otherwise (Wang et al., 4 Sep 2025). The edge state itself is updated by a TCP block with residual structure,

$(S,\mathcal{X},\mathrm{rk}),$5

and then transformed into a final message with scalar attention on the scalar channel (Wang et al., 4 Sep 2025).

Step 2: SSE-level aggregation. Each SSE aggregates from four sources: its own features, constituent residues through $(S,\mathcal{X},\mathrm{rk}),$6, internal edges through $(S,\mathcal{X},\mathrm{rk}),$7, and outer-edge messages through $(S,\mathcal{X},\mathrm{rk}),$8: $(S,\mathcal{X},\mathrm{rk}),$9 This is the stage at which inter-SSE contact geometry is explicitly incorporated without collapsing SSEs into coarse supernodes (Wang et al., 4 Sep 2025).

Step 3: residue refinement. Updated SSE information is pushed back to residues, and incoming edge messages are aggregated: $S$0 This stage fuses local residue context with the meso-scale structural context represented at the SSE level (Wang et al., 4 Sep 2025).

Step 4: protein-level update. The protein cell is updated from the lower-rank states,

$S$1

with aggregation over residues and SSEs in practice (Wang et al., 4 Sep 2025).

Residual updates with GVP-style layer normalization are then applied rank-wise: $S$2 while edges are updated through their own TCP-plus-residual pathway (Wang et al., 4 Sep 2025). Readout can be performed either by pooling residue embeddings or by using the rank-$S$3 protein embedding directly (Wang et al., 4 Sep 2025).

At the level of topological deep learning formalism, TCPNet instantiates a generic neighborhood-based update

$S$4

where $S$5 contains incidence, adjacency, and outer-edge neighborhood types on the PCC (Wang et al., 4 Sep 2025). This places TCPNet within the general message-passing paradigm for topological neural networks on combinatorial complexes.

5. Experimental performance and task profile

TCPNet was evaluated on four protein representation learning tasks: inverse folding on CATH 4.4, fold classification on SCOP 1.75, cellular component prediction on Gene Ontology data, and antibody developability on SabDab (Wang et al., 4 Sep 2025). The reported training configuration used Adam with learning rate $S$6, ReduceLROnPlateau with factor $S$7 and patience $S$8, early stopping with patience $S$9, batch size $$\mathcal{X}\subseteq\mathcal{P}(S)\setminus\{\emptyset\}$$0, and NVIDIA A100 80GB hardware (Wang et al., 4 Sep 2025).

Against GCPNet, GVP-GNN, and EGNN, TCPNet showed a task-dependent profile (Wang et al., 4 Sep 2025). On inverse folding, GVP-GNN remained best with perplexity $$\mathcal{X}\subseteq\mathcal{P}(S)\setminus\{\emptyset\}$$1 and accuracy $$\mathcal{X}\subseteq\mathcal{P}(S)\setminus\{\emptyset\}$$2, while TCPNet was second-best with perplexity $$\mathcal{X}\subseteq\mathcal{P}(S)\setminus\{\emptyset\}$$3 and accuracy $$\mathcal{X}\subseteq\mathcal{P}(S)\setminus\{\emptyset\}$$4 (Wang et al., 4 Sep 2025). On fold classification, TCPNet was best on all three SCOP test splits:

• Fold: $$\mathcal{X}\subseteq\mathcal{P}(S)\setminus\{\emptyset\}$$5
• Superfamilies: $$\mathcal{X}\subseteq\mathcal{P}(S)\setminus\{\emptyset\}$$6
• Families: $$\mathcal{X}\subseteq\mathcal{P}(S)\setminus\{\emptyset\}$$7

The paper notes that on the hardest Fold split, TCPNet improves over second-best EGNN from $$\mathcal{X}\subseteq\mathcal{P}(S)\setminus\{\emptyset\}$$8 to $$\mathcal{X}\subseteq\mathcal{P}(S)\setminus\{\emptyset\}$$9, approximately $\mathrm{rk}:\mathcal{X}\to\mathbb{Z}_{\ge 0}$0 absolute (Wang et al., 4 Sep 2025). On cellular component prediction, TCPNet was best in the structure-only setting with $\mathrm{rk}:\mathcal{X}\to\mathbb{Z}_{\ge 0}$1, though slightly behind GCPNet and GVP-GNN when sequence features were added (Wang et al., 4 Sep 2025). On antibody developability, TCPNet was best in the structure-only case with AUPRC $\mathrm{rk}:\mathcal{X}\to\mathbb{Z}_{\ge 0}$2, and second-best with structure plus sequence at $\mathrm{rk}:\mathcal{X}\to\mathbb{Z}_{\ge 0}$3, narrowly behind GCPNet at $\mathrm{rk}:\mathcal{X}\to\mathbb{Z}_{\ge 0}$4 (Wang et al., 4 Sep 2025).

A second comparison examined topological enhancements relative to their GGNN baselines. TCPNet consistently improved over GCPNet across all reported tasks, including inverse folding, fold classification, cellular component prediction, and antibody developability (Wang et al., 4 Sep 2025). By contrast, GVP-TNN improved strongly on fold classification but degraded on inverse folding and some other tasks, while ETNN was significantly worse than EGNN on most tasks (Wang et al., 4 Sep 2025). This directly supports a nontrivial design conclusion stated in the source: topological enhancement is not automatically beneficial and must be architecturally integrated with care (Wang et al., 4 Sep 2025).

The task profile emphasizes fold classification. The reported gains are largest where secondary structure organization is central, especially in the hardest fold-level split on SCOP (Wang et al., 4 Sep 2025). The authors interpret this as evidence that explicit SSE-level modeling and multi-rank communication capture structural patterns that residue-only GNNs represent less effectively (Wang et al., 4 Sep 2025). By contrast, inverse folding remains dominated by local residue environments, where a strong residue-level GGNN can still lead (Wang et al., 4 Sep 2025).

6. Relation to prior theory, misconceptions, and limitations

TCPNet sits at the intersection of several earlier lines of work, but it should not be conflated with them. In the general topological neural network literature, a TNN is a network on a topological input space $\mathrm{rk}:\mathcal{X}\to\mathbb{Z}_{\ge 0}$5 that evaluates finitely many bounded continuous test functions and feeds the resulting finite-dimensional vector into an ordinary neural network. Under separation and strong-separation conditions on the test family, such networks are uniformly dense in the uniformly continuous functions $\mathrm{rk}:\mathcal{X}\to\mathbb{Z}_{\ge 0}$6 associated with a canonical uniformity (Kouritzin et al., 2023). That result is a genuine universal approximation theorem, but it is not the theorem proved for TCPNet in protein learning (Kouritzin et al., 2023).

A second relevant antecedent is the topological view of classification networks as continuous maps that separate labeled subsets by sending them into disjoint embedded discs or, for softmax networks, into Voronoi cells of simplex vertices (Hajij et al., 2020). That framework emphasizes that architectural choices cannot be made independently of the topology of the underlying data and that too-early dimensionality reduction can destroy separability (Hajij et al., 2020). TCPNet is compatible with that broader perspective, but its construction is specific: it uses a combinatorial complex, explicit multi-rank incidence structure, and SE(3)-equivariant scalar–vector processing tailored to proteins (Wang et al., 4 Sep 2025).

A common misconception is to read “topology-complete” in TCPNet as a formal completeness or universality guarantee. The Topotein source explicitly does not present the term in that sense. It states that the term is a design philosophy: the model uses a combinatorial complex that includes all biologically relevant ranks $\mathrm{rk}:\mathcal{X}\to\mathbb{Z}_{\ge 0}$7–$\mathrm{rk}:\mathcal{X}\to\mathbb{Z}_{\ge 0}$8 and neighborhoods enabling communication within and across those ranks, and the TCP module is rank-agnostic in the sense that it can process scalar and vector features at any rank given appropriate frames (Wang et al., 4 Sep 2025). The phrase therefore refers to architectural coverage of the hierarchy, not to a theorem that all target functions or all protein tasks are representable.

The reported limitations are correspondingly practical rather than foundational. The PCC and outer-edge neighborhoods are computationally heavier than simple residue graphs; the construction depends on reliable 3D coordinates and DSSP assignments; improvements over strong GGNN baselines are typically in the $\mathrm{rk}:\mathcal{X}\to\mathbb{Z}_{\ge 0}$9–$$x\subseteq y \Rightarrow \mathrm{rk}(x)\le\mathrm{rk}(y),$$0 range rather than orders of magnitude; and the method assumes a single static structure, without explicit handling of disordered regions or multiple conformations (Wang et al., 4 Sep 2025). The paper also identifies future directions: integration with large sequence or structure models such as ESM3, extension to protein complexes and other biomolecular systems, improved theoretical analysis of expressivity and invariance on combinatorial complexes, and richer hierarchy designs such as adding domains as an intermediate rank (Wang et al., 4 Sep 2025).

Taken together, these works place TCPNet in a precise position. It is neither merely a renamed GGNN nor a generic topological universal approximator. It is a hierarchical, SE(3)-equivariant topological neural network on a protein combinatorial complex, designed to represent residues, contacts, SSEs, and proteins within a single message-passing system, and it is empirically strongest on tasks where secondary-structure arrangement and multi-scale structural organization are decisive (Wang et al., 4 Sep 2025).