# Benchmarking AlphaFold3’s protein-protein complex accuracy and machine
learning prediction reliability for binding free energy changes upon mutation

JunJie Wee1, and Guo-Wei Wei1,2,3,∗
1Department of Mathematics, Michigan State University, East Lansing, MI 48824,
USA
2Department of Biochemistry and Molecular Biology,
Michigan State University, East Lansing, MI 48824, USA
3Department of Electrical and Computer Engineering,
Michigan State University, East Lansing, MI 48824, USA
∗ Address correspondences to Guo-Wei Wei. E-mail<EMAIL_ADDRESS>
###### Abstract

AlphaFold 3 (AF3), the latest version of protein structure prediction
software, goes beyond its predecessors by predicting protein-protein
complexes. It could revolutionize drug discovery and protein engineering,
marking a major step towards comprehensive, automated protein structure
prediction. However, independent validation of AF3’s predictions is necessary.
Evaluated using the SKEMPI 2.0 database which involves 317 protein-protein
complexes and 8338 mutations, AF3 complex structures give rise to a very good
Pearson correlation coefficient of 0.86 for predicting protein-protein binding
free energy changes upon mutation, slightly less than the 0.88 achieved
earlier with the Protein Data Bank (PDB) structures. Nonetheless, AF3 complex
structures led to a 8.6% increase in the prediction RMSE compared to original
PDB complex structures. Additionally, some of AF3’s complex structures have
large errors, which were not captured in its ipTM performance metric. Finally,
it is found that AF3’s complex structures are not reliable for intrinsically
flexible regions or domains.

Keywords: AlphaFold3, protein-protein interactions,topological deep learning,
Persistent Laplacian, Mutation

###### Contents

  1. 1 Introduction
  2. 2 Results
    1. 2.1 Validation Performance
    2. 2.2 Structural Alignment Performance
  3. 3 Discussion
  4. 4 Conclusion
  5. 5 Methods
    1. 5.1 Feature generation for MT-TopLap

##  1 Introduction

AlphaFold3 (AF3) is the latest iteration of the groundbreaking protein
structure prediction software developed by Google DeepMind and Isomorphic
Labs[1]. Building on the success of its predecessors, AF3 brings significant
advancements to the field of computational biology. One of the key features of
AF3 is its ability to predict protein-molecule complexes, including those
involving DNA and RNA. This represents a major leap from AF2, which primarily
focused on predicting the three-dimensional (3D) structures of proteins. The
remarkable success of AF3 has opened up new possibilities for extending its
purpose beyond fundamental structural prediction tasks to include protein-
molecule complexes.

The significance of accurately predicting 3D protein structures using
AlphaFold cannot be overemphasized. Fundamentally, 3D protein structures are
instrumental in revealing protein functions, comprehending various biological
interactions, and designing drugs. Numerous prevalent diseases, including
Alzheimer’s and Parkinson’s, are linked to aberrant protein structures.
Consequently, the automated prediction of protein folds from sequences has
emerged as a crucial challenge in biology and is often referred to as the holy
grail of molecular biophysics. In 2019, AlphaFold’s [2] leading performance in
predicting 25 structures out of 43 test proteins sparked enthusiasm among
researchers about the potential future of AI-based protein structure
prediction. For instance, DeepFragLib by Wang et al. [3] signifies a novel
progression in ab initio protein structure prediction. Thereafter, the launch
of AlphaFold ignited a transformative shift in the way we model protein
structures and their interactions [2]. AlphaFold also opened up a vast array
of possibilities in protein folding, protein engineering, and design [4, 5, 6,
7, 8, 9, 10].

Data-driven machine learning models have demonstrated great power by utilizing
3D protein-protein complexes. Protein-protein interactions (PPIs) also play a
significant role in nearly all cellular and biological activities. In the
study of PPIs, mutation-induced effects play a paramount role in evolutionary
biology, cancer biology, immunology, directed evolution, and protein
engineering. Data-driven machine learning models have targeted the study of
mutation-induced effects on protein stability and PPI binding affinities.
Computational approaches have primarily served as a viable alternative to
experimental mutagenesis methods. Previously, scientific communities have
naturally extended the capabilities of AlphaFold by expanding the protein
structural database[11]. With the accessibility of AF3 through the AlphaFold
Server, AF3 can potentially become a valuable tool for advancing deep learning
models towards applications of PPIs.

Nonetheless, it was found that the prediction accuracy of protein engineering,
which iteratively optimizes protein fitness by screening the gigantic
mutational space, drops when AlphaFold2 structures were used [5]. Therefore,
it is imperative to independently validate AF3’s accuracy and reliability for
PPI analysis and mutation-induced protein-protein binding free energy (BFE)
change predictions.

One of the very successful approaches for mutation-induced protein-protein BFE
change predictions is the topology-based network tree (TopNetTree) [12], which
is based on topological deep learning (TDL), introduced in 2017 [13]. TDL is
an emerging paradigm in machine learning, based on topological data analysis
(TDA) [14, 15]. TDA, a branch of mathematics, focuses on understanding the
shape and structure of data and is exceptionally successful in improving
standard approaches by contributing novel topological characterization. TDL
captures protein structures, simplifies the structural complexity of
biomolecules, and embeds physical interactions into topological invariants.

TDA has had tremendous success through TopNetTree [12], PerSpect-EL [16], HCML
[17], and TopNetmAb [18] by incorporating topological fingerprints from
persistent homology (PH) and persistent Laplacian to predict PPI BFE changes
upon mutation and subsequently, BFE changes due to mutations in the SARS-CoV-2
receptor binding domain (RBD) - Angiotensin-converting enzyme 2 (ACE2)
complexes. In particular, persistent Laplacian provides both the topological
information in PH and the homotopic shape of evolution [19]. Using persistent
Laplacian, Omicron BA.4 and BA.5 were predicted to be new dominant variants
two months before the World Health Organization (WHO) made the announcement
[18]. Persistent Laplacian is also incorporated in the recent integration
between TDL and pre-trained ESM transformer features for predicting mutation-
induced protein solubility changes, which establishes the state-of-the-art
method for protein-protein BFE change predictions [20].

In this work, we evaluate AF3’s complex structure accuracy and mutation-
induced BFE change prediction reliability using the largest PPI database,
SKEMPI 2.0, which involves 317 protein-protein complexes and 8,330 mutation-
induced BFE changes [21, 12]. First, we consider AF3’s PPI structure accuracy.
Additionally, we examine AF3’s reliability in predicting protein-protein BFE
changes upon mutation using the most advanced TDL method [20]. Our results
demonstrate that AF3-predicted complexes achieved a relatively good Pearson
correlation coefficient of 0.86, which is slightly less than the 0.88 reported
earlier [20]. However, we found that AF3 results in 8.6% increase of root-
mean-square error (RMSE) compared to the original complexes in the Protein
Data Bank (PDB) for BFE change prediction. Through our analysis, we also
discovered that some structurally misaligned AF3 protein-protein complexes are
not captured by AF3’s ipTM performance metric. Finally, AF3 complex
predictions may not be reliable for highly flexible protein domains.

##  2 Results

In this section, we assess the accuracy and reliability of AF3 predictions of
PPI complexes. Here, we use the accessible AlphaFold Server of AF3 to predict
the protein-protein complexes from SKEMPI 2.0 database [21]. This database
contains the S8338 dataset for mutation-induced BFE changes, which is one of
the most comprehensive datasets collected on how mutations can alter the
binding affinity in PPIs, with over 8,338 entries of single mutations.
However, SKEMPI 2.0 database involves certain PPI complexes, namely 3NVN and
4U6H, with restricted viral pathogenic sequences in AlphaFold Server (accessed
11th May 2024). These complexes only make up 8 samples in the S8338 dataset.
Therefore, we incorporate the remaining 8330 single mutation-based samples
generated from 317 PPI complexes. The 317 predicted AF3 complexes are utilized
to predict the 8330 mutation-induced binding free energy changes. More details
on the SKEMPI 2.0 database can be found in the Supplementary Information.

###  2.1 Validation Performance

In the validation test, we conduct a 10-fold cross-validation on MT-TopLapAF3
by predicting the mutation-induced BFE changes using the features extracted
from the remaining 317 AF3 structures. Existing topology-based models like
TopLapNetGBT and TopNetTree, which have proven to be successful, have
undergone training and validation using the S8338 dataset [18, 12].

Figure 1: A: The cartoon representation of ribonuclease inhibitor-angiogenin
complex (PDB ID: 1A4Y). The ribonuclease inhibitor is shown in blue and the
angiogenin is in green. The mutation spots of 1A4Y in the S8338 dataset are
indicated in red. B: The structural alignment of 1A4Y with its AF3 predicted
complex. C: The boxplot for RMSD, ipTM and pTM distributions of 317 predicted
AF3 protein-protein complexes. RMSDs refer to the overall RMSD calculated by
structurally aligning an AF3 complex with its original PDB complex. D: The
breakdown of AF3 protein-protein complexes based on their ipTM and pTM scoring
criteria.

Table 1 shows the 10-fold cross validation results of MT-TopLapAF3 models for
the BFE change prediction for mutations on the S8338 dataset. The performance
of MT-TopLapAF3 model is evaluated based on the Pearson correlation
coefficient ($R_{p}$) and RMSE. Their definitions are included in the
Supplementary Information. MT-TopLapAF3 achieved an $R_{p}$ of 0.86 and RMSE
of 1.025 kcal/mol. In comparison to non-topology-based models such as mCSM-
PPI2 [22], MT-TopLapAF3 displayed an 18% improvement in its RMSE.

Method | $R_{p}$ | Method | $R_{p}$
---|---|---|---
MT-TopLap | 0.88 | mCSM-PPI2[22] | 0.82
TopLapNetGBT[18] | 0.87 | LapNet[18] | 0.81
TopLapNet[18] | 0.87 | LapGBT[18] | 0.80
TopNetGBT[18] | 0.87 |  |
TopNet[18] | 0.86 |  |
MT-TopLapAF3 | 0.86∗ |  |
TopLapGBT[18] | 0.85 |  |
TopGBT[18] | 0.85 |  |
LapNetGBT[18] | 0.83 |  |
Table 1: Comparison of the Pearson correlation coefficients ($R_{p}$) of MT-
TopLap, MT-TopLapAF3 and existing state-of-the-art methods for 10-fold cross-
validation of the S8338 dataset. Results of existing state-of-the-art methods
are obtained from [18, 22]. *Note that MT-TopLapAF3 is validated with 8330
samples as PDB ID: 3NVN and 4U6H cannot be predicted by using AlphaFold server
(accessed 11th May 2024) as they are detected as “restricted sequences from a
small number of viral pathogens.” Figure 2: A: Top 40 protein-protein
complexes with poorest RMSD alignment scores. B: The ipTM scores for protein-
protein complexes in A. C: The mean absolute prediction error for mutation-
induced binding free energy changes of the protein-protein complexes in A. D:
Top 40 protein-protein complexes with ipTM scores below 0.6.

###  2.2 Structural Alignment Performance

Using with the original PDB structures for the S8338 dataset, the original MT-
TopLap achieved an $R_{p}$ of 0.88 and RMSE of 0.937 kcal/mol. This implies an
increase in 8.6% of RMSE due to the AF3 predicted complex structures. To
investigate further, we analyze the AF3 complexes by ranking them based on
their structural alignment RMSD, ipTM, and pTM scores. RMSD scores are
obtained by superimposing AF3 complexes with original PDB complexes. The ipTM
score, which was first introduced in AlphaFold-Multimer, is a metric that
evaluates the accuracy of the predicted interface in a protein-protein complex
[23]. For example, an ipTM score above 0.8 indicates that the PPI complex
generated is highly confident prediction. On the other hand, an ipTM score
below 0.6 suggests that the predicted structure is likely to be incorrect
while any ipTM score between 0.6 and 0.8 includes prediction that are correct
or wrong [23]. Another AF3 scoring is pTM which is a comprehensive measure of
the accuracy of prediction of the entire structure of the complex. It
represents the predicted TM score for a superposition between the predicted
structure and the assumed true structure. However, a good pTM score above 0.5
may be due to a correct prediction of the larger protein dominated in the
protein-protein complex although the other partner protein has a poor
prediction.

Figure 1 shows the statistics of 317 AF3 structures from S8338 dataset. The
average RMSD, average ipTM and average pTM calculated are $1.61$Å, 0.803 and
0.847 respectively. In Figure 1A, complexes that are considered outliers have
a poor RMSD greater than 4Å. Despite this, 71.6% of the complexes have an high
ipTM score of at least 0.8, and 98.7% of the complexes have a pTM score of at
least 0.5. Essentially, this shows that most of the AF3 predicted complexes
have high ipTM but also have low RMSD. The four complexes (i.e. 1DVF, 3LB6,
3LZF and 5TAR) with pTM below 0.5 also have ipTM scores below 0.6.

##  3 Discussion

To analyze which AF3 PPI complex is not well predicted, we first rank the 317
AF3 complexes with their RMSD and ipTM scores. Figure 2A and B shows the top
40 PPI complexes according to its RMSD (from highest to lowest) and ipTM (from
lowest to highest). Note that lower RMSD and higher ipTM indicates better
performance. Clearly, a poor ipTM score does not necessarily translate into
poor RMSD performance. Comparing Figures 2A and B, only six complexes, i.e.,
5XCO, 1DVF, 3EG5, 1KIR, 3BN9, 1VFB, have both low ipTM and low RMSD scores.
Similarly, the same six complexes appear in both Figures 2A and D. Therefore,
there is little correlation between ipTM score and prediction RMSD. In
general, RMSD also influences the final absolute prediction errors in Figure
2C. Future improvements to AF3 should incorporate RMSD as part of its
prediction performance.

Figure 3: A: The B-factor representation of PDB ID: 4YH7. Atoms colored in red
have high B-factors while atoms colored in blue have lower B-factors. B:
Illustration of structural alignment for PDB ID: 4YH7 (in blue) and its AF3
predicted complex (in orange). C: The residue-based B-factor values plotted
against all the residue RMSD calculated from all 317 AF3 predicted complexes.
A $\log$-scale is taken on the $y$-axis.

Another potential limitation of AF3 is illustrated in Figure 3A and B. Figure
3A shows the PPI complex of PDB ID: 4YH7 colored by B-factor values. High
B-factors or high flexible protein regions are colored in red while low
B-factors or more rigid protein regions are colored in blue. Figure 3B shows
the structural alignment of PDB ID: 4YH7 (in blue) with its AF3 predicted
complex (in orange). It can be observed that the regions in Figure 3A with
high B-factors correspond to the region in Figure 3B where the residues are
misaligned between AF3 and PDDB structures. Here, the RMSD per residue is
calculated based on the backbone atoms of the protein. To investigate this
further, we observe a similar pattern when all residue-based B-factors are
plotted against residue-based RMSD for all 317 PPI complexes in Figure 3C. The
colors in Figure 3C represent the 20 canonical amino acid types. In general,
Figure 3C shows a huge amount of residues with high B-factors and high RMSD.
Based on these analyses, it is concluded that AF3 complex prediction is not
reliable for intrinsically flexible regions or domains of PPI complexes.

##  4 Conclusion

We examine the use of AlphaFold3 (AF3) predicted complex structures for the
prediction of protein-protein binding free energy (BFE) changes upon mutation
for the S8338 dataset in the SKEMPI 2.0 database. We noted that AF3 complex
structures lead to a relatively good predicted Pearson correlation coefficient
of 0.86 but have generated 8.6% more in RMSE compared to using PDB complex
structures. Essentially, AF3 is a promising technology for predicting protein-
protein complexes, even capable of contributing to mutation-induced BFE change
predictions.

In total, 317 AF3 complexes inferred by the SKEMPI 2.0 database were
structurally aligned with original ones in the PDB to investigate AF3
predictions that have poor RMSD in structural alignment. Through our analysis,
we discovered that current AF3 performance metrics such as ipTM does not
correlate with RMSD alignment scores. On the other hand, preliminary findings
show that high RMSD are strongly correlated to high B-factors, indicating AF3
predictions are not reliable for highly flexible protein regions or domains.

##  5 Methods

###  5.1 Feature generation for MT-TopLap

Among all features used in MT-TopLap prediction, persistent Laplacians (PL)
plays the most significant role in validating the reliability of AF3
structures. With accurate 3D SARS-CoV-2 receptor binding domain (RBD) -
Angiotensin-converting enzyme 2 (ACE2) complexes, PLs correctly predicted the
dominance of the Omicron BA.4 and BA.5 two months preceding the public
announcement by World Health Organization (WHO) in June 2022 [18].

In this section, we provide the important main mathematical framework
instrumental for understanding element-specific PL descriptors applied in our
feature generation. Element-specific topological approaches were introduced in
earlier work[13, 24]. In the simplicial complex and PL framework [19], we
emphasize their importance in identifying both harmonic and non-harmonic
spectral characteristics that are crucial for understanding PPIs.

To generate the PL features for our MT-TopLap models, we categorize the atoms
in PPIs into several subsets. These subsets are

  * •

Atoms in mutation sites $\mathcal{A}_{m}$,

  * •

Atoms in mutation neighborhood, i.e. Atoms within a distance $r$ from the
mutation site $\mathcal{A}_{mn}(r)$,

  * •

Protein 1’s atoms within distance $r$ of the binding site
$\mathcal{A}_{\text{P}_{1}}(r)$,

  * •

Protein 2’s atoms within distance $r$ of the binding site
$\mathcal{A}_{\text{P}_{2}}(r)$.

Additionally, we also group atoms into different element specific categories
such as $\\{\text{C, N, O}\\}$, denoted as $\mathcal{A}_{\text{ele}}$. These
element-specific groups are crucial in identifying the various types of
interactions in a PPI model, as per biophysical principles. For instance, the
subsets $\mathcal{A}_{\text{C}}\cap\mathcal{A}_{\text{P}_{1}}(r)$ and
$\mathcal{A}_{\text{C}}\cap\mathcal{A}_{\text{P}_{2}}(r)$ capture hydrophobic
C-C PPIs while $\mathcal{A}_{\text{N}}\cap\mathcal{A}_{\text{P}_{1}}(r)$ and
$\mathcal{A}_{\text{O}}\cap\mathcal{A}_{\text{P}_{2}}(r)$ capture hydrophillic
N-O PPIs.

In addition to groups specific to elements, we also utilize distance functions
like ${\rm D}_{\text{mod}}$, which exclude interactions between atoms from the
same subset. For interactions between atoms $A_{i}$ and $A_{j}$ in sets
$\text{P}_{1}$ and $\text{P}_{2}$, ${\rm D}_{\text{mod}}$ is defined as
follows:

${\rm D}_{\text{mod}}(A_{i},A_{j})=\begin{cases}\infty,&\text{ if
}A_{i},A_{j}\in\text{P}_{1}\text{ or }\text{ if }A_{i},A_{j}\in\text{P}_{2}\\\
{\rm DE}(A_{i},A_{j}),&\text{otherwise}.\end{cases}$ (1)

Here, ${\rm DE}(\cdot,\cdot)$ refers to the Euclidean distance between two
atoms.

By utilizing the groups specific to elements/sites, atoms are arranged into
point clouds, which are then used to construct simplicial complexes.
Specifically, a set of $k+1$ atoms from an element/site-specific subset forms
$k+1$ independent points, which can be represented as a set
$S=\\{v_{0},v_{1},v_{2},\cdots,v_{k}\\}$. The convex hull of $k+1$ affinely
independent points essentially forms a $k$-simplex. In simple terms, a point
is a 0-simplex, an edge is a 1-simplex, a triangle is a 2-simplex, a
tetrahedron is a 3-simplex, and so on for higher dimensions, forming a
$k$-simplex. A simplicial complex is created by combining these finite
simplices [25, 26, 27, 28]. There are numerous methods to construct a
simplicial complex. For generating our persistent Laplacian-based features, we
employed the Vietoris Rips complex for dimension 0 and the Alpha complex for
dimensions 1 and 2. A Vietoris-Rips (VR) complex is an abstract simplicial
complex that forms simplices by connecting any subset of points with a
diameter not exceeding a certain threshold. On the other hand, an Alpha
complex is a group of subcomplexes derived from a Delaunay triangulation,
subject to a radius constraint that does not exceed a certain threshold. The
Delaunay triangulation is a geometric structure that divides the convex hull
of a set of points in a plane into triangles.

For a simplicial complex $K$, a $k$-th chain $c_{k}$ is the formal sum of
$k$-simplicies in $K$, i.e. $c_{k}=\sum_{i}\alpha_{i}\sigma_{i}^{k}$. The
$k$-th boundary operator $\partial_{k}:C_{k}\rightarrow C_{k-1}$ defined on a
$k$-th chain $c_{k}$ is
$\partial_{k}c_{k}=\sum_{i=0}^{k}\alpha_{i}\partial_{k}\sigma_{i}^{k}$. Here,
the condition that the boundary of a boundary is empty must be satisfied.
Defining the adjoint operator of $\partial_{k}$, i.e.
$\partial_{k}^{*}:C_{k-1}\to C_{k}$, yields the inner product relation
$\langle\partial_{k}(f),g\rangle=\langle f,\partial_{k}^{*}(g)\rangle,$ for
every $f\in C_{k}$, $g\in C_{k-1}$. From here, the $k$-topological Laplacian,
a linear operator $\Delta_{k}:C_{k}(K)\rightarrow C_{k}(K)$, computed as
$\partial_{k+1}\partial_{k+1}^{*}+\partial_{k}^{*}\partial_{k}$.

In matrix representations, we denote $\mathbf{B}_{k}$ as an $m\times n$ matrix
of the boundary operators under the standard bases
$\\{\sigma_{i}^{k}\\}^{n}_{i=1}$ and $\\{\sigma_{j}^{k-1}\\}^{m}_{j=1}$ of
$C_{k}$ and $C_{k-1}$. In a similar way, the transpose boundary matrix
$\mathbf{B}_{k}^{\top}$ is used to denote the matrix representation of
$\partial_{k}^{*}$ with respect to the same ordered bases of the boundary
operator $\partial_{k}$. This naturally leads to the $k$-combinatorial
Laplacian matrix, which is an $n\times n$ matrix $\mathbf{L}_{k}$ computed as
$\mathbf{B}_{k+1}\mathbf{B}_{k+1}^{\top}+\mathbf{B}_{k}^{\top}\mathbf{B}_{k}$.
For the special case $k=0$,
$\mathbf{L}_{0}=\mathbf{B}_{1}\mathbf{B}_{1}^{\top}$ as $\partial_{0}$ is
understood as a zero map.

In our model, the key topological features are the eigenvalues of
combinatorial Laplacian matrices. These eigenvalues are not dependent on the
orientation choice [29]. Moreover, the total number of zero eigenvalues, or
their multiplicity, in $\mathbf{L}_{k}$ equates to the $k$th Betti number,
$\beta_{k}$, as per the combinatorial Hodge theorem [30]. These Betti numbers
are topological invariants that represent the $k$-dimensional holes in a
$k$-simplicial complex. For instance, $\beta_{0}$, $\beta_{1}$, and
$\beta_{2}$ denote the number of independent components, loops, and cavities,
respectively. Overall, the zero and non-zero eigenvalues embody the harmonic
and non-harmonic spectra of combinatorial Laplacian matrices. The non-harmonic
spectra offer additional homotopic shape information that Betti numbers lack.

A single simplicial complex is not enough to capture all the topological
information from a single protein-protein complex. By combining combinatorial
Laplacian and multiscale filtration, we monitor the variations of harmonic and
non-harmonic spectra by adjusting a filtration parameter such as
radii/diameter for VR complex [19]. For an oriented simplicial complex $K$, a
filtration creates a nested sequence of simplicial complexes
$(K_{t})^{m}_{t=0}$ of $K$,

$\varnothing=K_{0}\subseteq K_{1}\subseteq\cdots\subseteq K_{m}=K.$

As the value of the filtration parameter increases, PL generates a sequence of
simplicial complexes. Based on this nested sequence of simplicial complexes,
we can produce a sequence of combinatorial Laplacian matrices
$\mathbf{L}_{k}^{0},\mathbf{L}_{k}^{1},\mathbf{L}_{k}^{2},\mathbf{L}_{k}^{3}\cdots,\mathbf{L}_{k}^{n}$
where $\mathbf{L}_{k}^{t}=\mathbf{L}_{k}(K_{t})$. By changing the filtration
parameter and performing diagonalization on the $k$-combinatorial Laplacian
matrix, the characteristics of topology and spectrum can be examined from each
$\mathbf{L}_{k}(K_{t})$ ($0\leq t\leq n$). The eigenvalues of
$\mathbf{L}_{k}(K_{t})$ can be sorted in ascending order
$\text{Spectra}(\mathbf{L}_{k}^{t})=\\{(\lambda_{1})_{k}^{t},(\lambda_{2})_{k}^{t},\cdots,(\lambda_{n})_{k}^{t}\\}$
where $\mathbf{L}_{k}^{t}$ is an $n\times n$ matrix. The $p$-persistent
$k$-combinatorial Laplacian can also be extended based on the boundary
operator. Ultimately, these PL descriptors help MT-TopLap track the changes of
harmonic and non-harmonic spectra in 3D protein-protein complexes, capturing
both the intrinsic topological changes and homotopic shape evolution
throughout its filtration process. Figure S2 depicts the harmonic and non-
harmonic spectra generated from a point cloud using Alpha complex and ${\rm
D_{\text{mod}}}$-based filtration. Further details of PL’s mathematical
framework is provided in Supplementary Information.

## Data Availability

The AF3 complexes can be generated using the AlphaFold Server and are readily
available in
https://github.com/ExpectozJJ/MT-TopLap/alphafold/. The original PDB files
used in this study can be downloaded from the official Protein Databank:
https://www.rcsb.org/. The SKEMPI 2.0 database is also readily available from
https://life.bsc.es/pid/skempi2.

## Code Availability

All source codes and models are publicly available at
https://github.com/ExpectozJJ/MT-TopLap/. A detailed set of instructions is
available in the Supporting Information.

## Supporting Information

Supporting Information is available for supplementary tables, figures, and
methods.

## Acknowledgments

This work was supported in part by NIH grants R01GM126189, R01AI164266, and
R01AI146210, NSF grants DMS-2052983, DMS-1761320, and IIS-1900473, NASA grant
80NSSC21M0023, MSU Research Foundation, Bristol-Myers Squibb 65109, and
Pfizer.

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