††thanks: Correspondence should be addressed to: Gian Marco Visani:
<EMAIL_ADDRESS>and Armita Nourmohammad<EMAIL_ADDRESS>Correspondence should be addressed to: Gian Marco Visani:
<EMAIL_ADDRESS>and Armita Nourmohammad<EMAIL_ADDRESS>
# HERMES: Holographic Equivariant neuRal network model for Mutational Effect
and Stability prediction

Gian Marco Visani  Department of Computer Science and Engineering, University
of Washington, Seattle, USA  Michael N. Pun  Department of Physics, University
of Washington, 3910 15th Avenue Northeast, Seattle, WA 98195, USA  William
Galvin  Department of Computer Science and Engineering, University of
Washington, Seattle, USA  Eric Daniel  Department of Computer Science and
Engineering, University of Washington, Seattle, USA  Kevin Borisiak
Department of Physics, University of Washington, 3910 15th Avenue Northeast,
Seattle, WA 98195, USA  Utheri Wagura  Department of Physics, University of
Washington, 3910 15th Avenue Northeast, Seattle, WA 98195, USA  Department of
Physics, Massachusetts Institute of Technology, 182 Memorial Dr, Cambridge, MA
02139  Armita Nourmohammad  Department of Computer Science and Engineering,
University of Washington, Seattle, USA  Department of Physics, University of
Washington, 3910 15th Avenue Northeast, Seattle, WA 98195, USA  Department of
Applied Mathematics, University of Washington, Seattle, USA  Fred Hutchinson
cancer Research Center, 1100 Fairview ave N, Seattle, WA 98109, USA

###### Abstract

Predicting the stability and fitness effects of amino acid mutations in
proteins is a cornerstone of biological discovery and engineering. Various
experimental techniques have been developed to measure mutational effects,
providing us with extensive datasets across a diverse range of proteins. By
training on these data, traditional computational modeling and more recent
machine learning approaches have advanced significantly in predicting
mutational effects. Here, we introduce HERMES, a 3D rotationally equivariant
structure-based neural network model for mutational effect and stability
prediction. Pre-trained to predict amino acid propensity from its surrounding
3D structure, HERMES can be fine-tuned for mutational effects using our open-
source code. We present a suite of HERMES models, pre-trained with different
strategies, and fine-tuned to predict the stability effect of mutations.
Benchmarking against other models shows that HERMES often outperforms or
matches their performance in predicting mutational effect on stability,
binding, and fitness. HERMES offers versatile tools for evaluating mutational
effects and can be fine-tuned for specific predictive objectives.

##  I Introduction

Figure 1: Schematic of HERMES’ architecture and training procedures. (A)
Architecture of the updated Holographic Convolutional Neural Network (HCNN),
which is the backbone of Hermes. The use of an efficient Tensor Product (TP)
operation and foregoing of invariant skip connections makes this architecture
2.75x faster and with better performance on the pre-training task than the
architecture in [1]; we refer the reader to [2] for details of the
architecture. (B) Pre-training procedure. We train HCNN to predict the
identity of the central neighborhood’s amino-acid, whose atoms have been
masked. (C) Fine-tuning procedure for regressing over mutation effects. As
shown in [1] and in this work, after pre-training the quantity
$\log(p_{wt}/p_{mt})$ correlates with experimental $\Delta\Delta
G_{wt\rightarrow mt}$ values of stability - and more broadly with mutation
scores where a lower score indicates higher mutant fitness. Thus, we simply
fine-tune HCNN to make the quantity $\log(p_{wt}/p_{mt})$ regress over
mutation scores. The result is a model whose implementation exactly matches
the pre-trained model’s, making it trivial to substitute them for each other
in downstream pipelines.

Understanding the effects of amino acid mutations on a protein’s function is a
hallmark of biological discovery and engineering. Identifying disease-causing
mutations [3, 4], enhancing enzymes’ catalytic activity [5, 6], forecasting
viral escape [7, 8, 9], and engineering high-affinity antibodies [10], are
just some of the areas of study that rely on accurate modeling of mutational
effects.

Effects on protein stability are likely the most studied, as sufficient
stability is usually a prerequisite of the protein’s successful carrying of
its function [11]. Understanding the impact of mutations on the protein’s
binding affinity to its partner is also crucial, as most functions are
mediated by binding events. The effects on both stability and binding can be
accurately measured experimentally, for example via thermal or chemical
denaturation assays [12], by surface plasmon resonance [13]. However, these
processes are laborious and time-intensive and do not scale up to high numbers
of mutations. In recent years, Deep Mutational Scanning (DMS) emerged as a
technique to simultaneously measure the effects of as many as 1 million
mutations of a single protein, using various assays that usually directly
depend on thermal stability and binding affinity, but often model more complex
cellular activity [14, 15, 16].

Computational modeling of mutational effects remain an attractive alternative,
due to its still much lower cost and required time. Methods based on molecular
dynamics simulations are accurate but have limited benefits in terms of time
and accessibility [17], whereas energy-function-based methods such as FoldX
[18] and Rosetta [19] are well-established and still widely used.

Recently, machine learning models have challenged the dominance of energy-
function-based models, most notably in protein folding [20], but also in
stability and binding mutation effect prediction [4, 21]. In particular,
machine learning models offer great improvements in speed. Several supervised
models have been developed, though they often suffer from overfitting to the
all-too-often limited and biased training data [22, 23, 24, 25]. Self-
supervised learning has emerged as a more robust alternative: instead of
relying on experimental data, these models are usually trained to predict
masked amino-acid labels conditioned on structural and/or sequence
information, thus learning a probability distribution over amino-acids at a
particular site conditioned on its context. The key insight has been that
these learned distributions can be used to approximate mutational effects in
zero-shot [26, 27, 28, 4, 21, 1, 29]. However, zero-shot predictive power
heavily depends upon the training data distribution and context used, and
performance is often worse than that of energy-function-based models [27, 21].

Among the recently-developed self-supervised models is Holographic
Convolutional Neural Network (HCNN) [1]. HCNN is a 3D rotation equivariant
neural network [30] trained to predict amino-acids propensities at individual
sites, using as input the atomic composition of the structural neighborhood
within 10 Åof a focal residue. It was shown that HCNN appears to learn an
effective physical potential of local atomic environments in proteins, and
that it shows high-accuracy zero-shot predictions stability and binding effect
of mutations in a few proteins.
Concurrent to HCNN was the development of RaSP [4], which proposed to
construct representations of masked atomic environments from the output
embeddings of a 3DCNN trained - like HCNN - to predict amino-acid
propensities, and to use these representations as one of the inputs to a small
model trained to predict stability effects ($\Delta\Delta G$) computed with
Rosetta. RaSP was shown to be able to to predict stability effects at the same
level of accuracy of Rosetta, but orders of magnitude faster, enabling a
large-scale analysis of missense-variants.

Building upon HCNN and inspired by RaSP, we propose HERMES, a 3D rotation
equivariant neural network with a more efficient architecture than HCNN, pre-
trained on amino-acid propensity, and easy to fine-tune on custom experimental
or computationally-derived mutational effects using our open-source code. We
develop a suite of models pre-trained with different strategies, and fine-tune
them for predicting the stability effects of mutations computed with Rosetta.
We thoroughly benchmark HERMES models against other zero-shot and fine-tuned
machine learning models in predicting mutational effects on stability, binding
affinity, and various DMS assays. Our contributions are summarized as follows:

  1. 1.

We updated the architecture of HCNN to be $\sim$2.75x faster and with better
amino acid classification performance.

  2. 2.

We developed a fast and effective procedure to fine-tune pre-trained models on
mutational effects. Our procedure leaves the interface of the pre-trained
model unvaried, making it possible to seamlessly swap models in downstream
pipelines.

  3. 3.

We show a thorough benchmarking on stability, binding affinity, and DMS
assays, and provide our evaluation data and code.

  4. 4.

We provide easy-to-use code for extracting amino acid likelihoods and
embeddings of site-centered atomic environments, which can be used from our
github repository 111https://github.com/StatPhysBio/protein_holography-web or,
more simply, from a google colab notebook
222https://colab.research.google.com/drive/1JQxkXeGZJYYcPNglN3rYUYiOuUOkwJPL.

  5. 5.

We provide code and a procedure for fine-tuning our pre-trained models on
arbitrary mutation effect data, available in our github repository.

##  II Methods

Figure 2: Prediction performance of RaSP and HERMES models on 10 Rosetta
$\Delta\Delta G$ test proteins from the RaSP codebase [4]. Zero-shot models on
the left, models fine-tuned on Rosetta $\Delta\Delta G$ on the right. This set
serves as the de-facto test set of models fine-tuned with Rosetta
$\Delta\Delta G$. Pearson correlation is computed between model predictions
and $\Delta\Delta G$ computed with Rosetta. Notably, zero-shot HCNN models
benefit from both noise and from the use of PyRosetta for preprocessing.
Instead, fine-tuned models perform similarly to each other.

###  II.1 Development of HERMES

HERMES models are trained in two steps (Figure 1). First, following [1], we
train an improved version of Holographic Convolutional Neural Network (HCNN)
models to predict the amino acid type of a focal residue from its surrounding
structural neighborhood. Specifically, we remove (mask) all atoms associated
with the focal residue and predict the identity of its amino acid from the the
other atoms within the structural neighborhood of radius 10 Å, centered at the
focal residue’s C$\alpha$ (Figure 1B). Second, inspired by RaSP [4], we
develop a procedure to fine-tune HERMES on mutation effects, specifically
focusing on stability effects measured as $\Delta\Delta G$.

HERMES architecture. We build upon improvements to the HCNN architecture
presented in [2], leading to a model that is $\sim$2.75x faster, more
accurate, with comparable number of parameters ($\sim$3.5M) as the
architecture in [1] (Figure 1A). In short, atomic neighborhoods - i.e.,
featurized point clouds - are first projected onto the orthonormal Zernike
Fourier Basis, centered at the (masked) central residue’s C$\alpha$. We term
the resulting Fourier encoding of the data an holographic encoding, as it
presents a superposition of 3D spherical holograms [1]. Then, the resulting
holograms are fed to a stack of SO(3)-Equivariant layers, which convert the
holograms to an SO(3)-equivariant embedding - i.e. a representation that is
invariant to 3D rotations about the center of the initial holographic
projection. Finally, the embedding is fed to an MLP to generate the desired
predictions. We refer the reader to [2] for details of the architecture, and
for a mathematical introduction to building SO(3)-equivariant models in
Fourier space. We implement HERMES using e3nn [31].

Pre-training data. Following [1], we pre-train HERMES on neighborhoods from
protein chains in ProteinNet’s CASP12 set with 30% similarity cutoff, and
featurize atomic neighborhoods using atom type - including computationally-
added hydrogens - partial charge, and Solvent Accessible Surface Area (SASA).

Preprocessing of protein structures. Two classes of models using different
types of pre-processing for the training data are presented in HERMES: pre-
processing with (i) PyRosetta [32, 1], and (ii) BioPython [33] and other open
source tools, adapting code from [4]. We refer to these models as HERMES PR
and HERMES BP, respectively. To generate our open source pre-processed
training data, we use the following procedure: we use OpenMM [34] to fix the
PDB files, add missing residues and substitute non-canonical residues for
their canonical counterparts; we use the reduce program [35] to add hydrogens;
we take partial charges from the AMBER99sb force field [36]; we use BioPython
to compute SASA [33]. Both preprocessings procedures keep atoms belonging to
non-protein residues and ions, unlike RaSP [4]. Notably, our PyRosetta
preprocessings does not replace non-canonical residues. HERMES BP facilitates
non-academic applications, due to restricted licensing of PyRosetta.

Increasing robustness with ensembling and noise. Each HCNN is an ensemble of
10 individually-trained model instances. Furthermore, we trained versions of
HCNN after adding Gaussian noise to the 3D coordinates, with standard
deviation 0.5 Å, and different random seeds for each of the 10 model
instances.

Predicting mutation effects on protein fitness with HERMES. HERMES, like HCNN,
can be seen as a generative model of amino-acid labels for a residue,
conditioned on the atomic environment surrounding the residue (what we call
the residue’s neighborhood). Prior work [26, 27, 1] has shown that conditional
generative models of amino-acid labels can be successfully used to infer
mutation effects on protein fitness in a zero-shot fashion. Specifically, the
log-likelihood ratio between the probability of observing the mutant amino-
acid at a give residue $i$ ($aa_{i}={\text{mt}}$) conditioned on its
surrounding neighborhood $X_{i}$, which we denote by
$p(aa_{i}={\text{mt}}|X_{i}^{({\text{mt}})})$, and that of the wildtype amino-
acid $p(aa_{i}={\text{wt}}|X_{i}^{({\text{mt}})})$ can well approximate the
fitness difference between the mutant and wildtype protein:

$F_{{\text{mt}}}-F_{{\text{wt}}}\propto\log\frac{p(aa_{i}={\text{mt}}|X_{i}^{({\text{mt}})})}{p(aa_{i}={\text{wt}}|X_{i}^{({\text{wt}})})}$
(1)

where the superscript on $X_{i}$ indicates the structure from which the atomic
neighborhood is extracted from, and the distinction made between the two
neighborhoods in Eq. 1 emphasizes the possible reorganization and relaxation
of the surrounding atoms following the substitution of the focal residue. When
using HERMES to infer mutation effects in practice, the only structure
available is usually that of the wildtype, providing an approximate estimate
for fitness difference (denoted by $\hat{\cdot}$),

$\hat{F}_{{\text{mt}}}-F_{{\text{wt}}}\propto\log\frac{p(aa_{i}={\text{mt}}|X_{i}^{({\text{wt}})})}{p(aa_{i}={\text{wt}}|X_{i}^{({\text{wt}})})}$
(2)

We should emphasize that the presence of the mutant amino-acid and subsequent
energy minimization can lead to a substantially distinct neighborhood compared
to that of the wild type. However, access to experimentally-determined
structures for a large number of mutants is usually limited. Computational
programs like Rosetta [32] can be used to “relax” the wildtype protein after
applying the mutation in-silico, but the procedure can be inaccurate and is
impractical at large scales. We test the use of mutant structures - both
experimentally and in-silico determined- on a few datasets.
While Eq. 1 is specific to a single residue $i$, all methods presented and
benchmarked in this work handle predictions of multiple mutations by simply
adding the individual mutations’ log-likelihood ratios, thereby assuming
independence between the mutations. Although this model can capture some
correlations between the nearby (in structure) residues due to their shared
neighborhoods, it does not fully account for all the epistatic interactions
throughout a protein, and further developments are needed to accurately model
epistasis.
Using the appropriate conditioning ($X_{i}$ in Eq. 1) is likely the most
important factor in modeling a given type of fitness. For example, if
attempting to predict binding affinity between a protein and its binding
partner, it is logical that including the atoms of the binding partner in the
neighborhood of the focal residue would be a prerequisite for successful
modeling.

Fine-tuning on stability effect of mutations. Inspired by RaSP [4], we develop
a procedure to fine-tune HERMES on mutation effects. RaSP was developed by
training an additional regression model for the stability effect of mutations,
taking as inputs embeddings from a pre-trained 3DCNN and the identities of
wildtype and mutant amino-acids. Unlike RaSP, we directly fine-tune the model
itself to make the the predicted log-likelihood ratio in Equation 2 regress
over mutation effects on stability. Importantly, our fine-tuning procedure
preserves the interface to the pre-trained model, thus enabling the two models
to be seamlessly swapped in downstream pipelines.
To greatly speed-up convergence, as a first step of fine-tuning we rescale the
weight matrix and bias vector of the network’s output layer so the mean and
variance of the output logits become the same as that of the training scores.
This step requires one initial pass through the training data to get the mean
and variance, but it makes the model outputs immediately be in the same
distribution as the scores, thus avoiding epochs of fine-tuning just devoted
to rescaling the model outputs. In this work, we fine-tune the models on the
same stability $\Delta\Delta G$ values used by RaSP, which are computed with
Rosetta for 35 training proteins (also we follow the same Fermi function
(sigmoid) transform, as detailed in [4]). Using the re-scaling initialization
outlined above, a batch size of 128 and learning rate of $5e^{-4}$, we find
that the fine-tuning converges on this data within 1-2 epochs only. We provide
easy-to-use code to fine-tune our pre-trained models on arbitrary mutation
effect data. Importantly, as $\Delta\Delta G$ is an energy, with lower values
indicating more stable structures, we implement the fine-tuning with the
physics convention that lower target values indicate higher stability and
fitness, i.e., the negative of Eq. 2 is fit to the data. In practice, to use
the fine-tuning code, just make sure that lower means higher fitness, which
can be done by simply flipping the sign of all the target values.

We pre-train 4 baseline HERMES models, with combinations of different protein
pre-processings - PyRosetta (HERMES PR) vs. BioPython (HERMES BP) - and
varying levels of Gaussian noise injected to the training data - 0.00 vs. 0.50
Å. We then fine-tune each model on Rosetta computed $\Delta\Delta G$ values,
resulting in eight total HERMES models (baseline and fine-tuned).

###  II.2 Alternative methods

We compare both the baseline and the fine-tuned HERMES models with the
following models that are commonly used to predict the effect of mutations.

RaSP [4]. This is the method most similar to ours, developed concurrently to
the first version of HCNN [1]. Similar to HERMES, RaSP is trained in two
steps: first, a neural network - specifically a 3DCNN - is pre-trained to
predict masked amino-acids from their local atomic environment (i.e.
”neighborhood”). Then, a small fully-connected neural network with a single
output is trained to regress over mutation effects, using as input
neighborhoods’ embeddings from the 3DCNN, the one-hot encodings of wildtype
and mutant amino-acids, and the wildtype and mutant amino-acids’ frequencies
in the pre-training data. RaSP is fine-tuned on the stability effect of
mutations $\Delta\Delta G$, computationally determined with Rosetta [32],
which we also use to fine-tune HERMES. We do not reproduce results of RaSP in
this work, and instead show the values reported in the paper.

ProteinMPNN [28]. ProteinMPNN is a tool for protein inverse-folding. The tool
is most commonly used to sample amino-acid sequences conditioned on a
protein’s backbone structure, and optionally a partial sequence. As
ProteinMPNN also outputs probability distributions of amino-acids for the
sites that are to be designed, it can also be used to infer mutation effects
by computing the log-likelihood ratio presented in Eq. 1. Like for HERMES, we
consider ProteinMPNN models trained with two noise levels: 0.02 Å(virtually no
noise) and 0.20 Å. We provide scripts to infer mutation effects built upon a
public fork of the ProteinMPNN repository.

ESM-1v [27]. This is the Protein Language Model (PLM) of the ESM family
trained specifically for improved zero-shot predictions of mutation effects.
As the training objective is predicting amino-acids that have been masked from
the sequence, mutation effects are also predicted using the log-likelihood
ratio (Eq. 1). To our knowledge, this is the strongest representative of PLMs
for inferring mutation effects. We show a mix of previously-reported scores,
and scores computed using their codebase. For our in-house ESM-1V predictions,
wildtype sequences were obtained from the corresponding PDB file and verified
against the European Bioinformatics Institute’s PDBe database via their REST
API [37]. Mutation effect predictions were computed with ESM’s built-in
wildtype marginal method; we attempted using the masked marginal method but
ran into several errors, so we stuck to wildtype marginal as it was more
reliable, and also had very similar performances in the few instances in which
both methods worked.

DeepSequence [26]. This is a state-of-the-art model for inferring mutation
effects from sequence alone. It uses a variational auto-encoder of full
protein sequences to and infers mutation effects via Eq. 1. We only show
previously-reported scores.

##  III Results

###  III.1 Model Validation

Validation on pre-training task. We compute Cross-Entropy (CE) loss and
classification accuracy on the pre-training task of predicting the (wildtype)
amino-acid from its masked neighborhood, on 40 protein crystal structures from
the CASP12 ProteinNet Test Set (Table 1). Accuracy is computed using the
highest-probability amino-acid as prediction, and discarding all other
information. As expected, models trained with noisy structures exhibit
slightly worse CE and accuracy. Interestingly, fine-tuned models exhibit much
worse CE loss, but accompanied by a only slight drop in accuracy. This
indicates that fine-tuned HERMES models output probability distributions with
much higher entropy, while largerly preserving the original relative ranking
of amino-acid preferences. This could, however, just be an artifact of the
scale of predictions for fine-tuning, which, for the fine-tuning data we use,
is in the $[0,1]$ range. Indeed, scaling down loagit values naturally
increases the entropy of the resulting probability distribution after applying
the softmax function.

| CE Loss | Accuracy
---|---|---
HERMES BP 0.00 | 0.69 | 0.76
HERMES BP 0.50 | 1.01 | 0.66
HERMES PR 0.00 | 0.78 | 0.73
HERMES PR 0.50 | 1.06 | 0.64
HERMES BP 0.00 + FT | 2.68 | 0.71
HERMES BP 0.50 + FT | 2.70 | 0.59
HERMES PR 0.00 + FT | 2.69 | 0.69
HERMES PR 0.50 + FT | 2.70 | 0.58
Table 1: Performance of HERMES models on wildtype amino-acid classification on
40 CASP12 test proteins. As expected, models trained with noise have worse
Cross Entropy (CE) Loss and Accuracy. Interestingly, models fine-tuned on
Rosetta Stability $\Delta\Delta G$ values retain most of their accuracy (the
highest-probability amino-acid is usually still the wildtype) but have
significantly worse CE Loss, indicating a significant increase in the entropy
of predictions.

Validation on fine-tuning task. Like RaSP, we test the performance of HERMES
models on inferring the relative Rosetta-computed $\Delta\Delta G$ values on
10 test proteins (Figure 2). HERMES models that have not been fine-tuned
exhibit a zero-shot performance around 0.4/0.5 Pearson r. Models trained with
noise are significantly better and models that use PyRosetta pre-processing
are better than those using BioPython. Fine-tuned HERMES models perform around
0.8 Pearson r, close but slightly below the level of RaSP, and with no
significant difference across noise levels and preprocessing procedure.

Figure 3: Prediction performance on experimental $\Delta\Delta G$ for eight
test proteins. Zero-shot models on the left and center, models fine-tuned on
Rosetta $\Delta\Delta G$ on the right. Each dot represents the value for a
protein, and the horizontal bar is the mean. ESM-1v predictions were made
using the wildtype marginals method. Similarly to Figure 2, zero-shot models -
both HERMES and ProteinMPNN - benefit from training on noisy structures,
whereas fine-tuned models all perform comparably. See Table S1 for detailed
results.

###  III.2 Prediction of stability effects $\Delta\Delta G$

Analysis of single amino acid substitutions. We evaluate the ability of HERMES
to infer experimental $\Delta\Delta G$ values for stability effect of
mutations, first by considering values for eight proteins - curated by [4] \-
that present many single amino-acid substitutions. Specifically, we consider
four proteins from ProTherm [38], the B1 domain of Protein G [39], and three
proteins for which VAMP-seq values are available [40]. Variant Abundance by
Massively Parallel sequencing (VAMP-seq) is a type of deep mutational scanning
that probes mutation effects on cellular protein abundance, and was found to
correlate with in-vitro measurements of stability of mutations $\Delta\Delta
G$, using different experimental methods [40]. Figure 3 shows the Pearson
correlations between the stability effect of mutations $\Delta\Delta G$ and
predictions using ESM-1v, ProteinMPNN, RaSP, and HERMES models. Zero-shot
models perform significantly better when trained with noise, and HERMES models
using PyRosetta preprocessing perform better than BioPython. ESM-1v, the only
sequence-based model we consider here, performs the worst on average, but it
is the best one for one protein from the VAMP-seq set (Table S1). The best
zero-shot model is HERMES with 0.50 Å noise and PyRosetta preprocessing, with
average Pearson r of 0.56. Models fine-tuned on Rosetta $\Delta\Delta G$
values, instead, all perform with a Pearson r larger than 0.60. RaSP and both
of the HERMES models without noise have an average Pearson correlation of
0.62, whereas both HERMES models with 0.50 Å noise have a slightly better
correlation of 0.64; see Table S1 for detailed results.

Using experimentally resolved mutant structures for predictions. As noted in
Eq. 1, our predictions should ideally account for the change in the
surrounding protein structure neighborhoods due to substitutions in the focal
residue. Here, we test the hypothesis as whether access to he true mutant
structure could enhance the accuracy of our predictions relative to the
approximate approach of using only the wildtype structure (Eq. 2). To do so we
will focus on the well-studies protein T4-Lysozyme, for which there exists
many measurements for stability effect of mutations $\Delta\Delta G$ together
with their corresponding protein structures [41, 42, 43, 44, 45, 46, 47, 48,
49, 50, 51, 52, 53, 54, 55, 56, 57, 58]. Table 2 shows prediction based on
ProteinMPNN and HERMES models for these T4-Lysozyme mutants.Interestingly,
HERMES zeros-hot models do not benefit from the use of the mutant structures,
while ProteinMPNN zeros-hot and HERMES fine-tuned models do.

| WT only | WT and MT
---|---|---
ProteinMPNN 0.02 | 0.65 | 0.80
ProteinMPNN 0.20 | 0.69 | 0.77
HERMES BP 0.00 | 0.73 | 0.69
HERMES BP 0.50 | 0.79 | 0.75
HERMES PR 0.00 | 0.71 | 0.70
HERMES PR 0.50 | 0.82 | 0.75
HERMES BP 0.00 + FT | 0.79 | 0.84
HERMES BP 0.50 + FT | 0.83 | 0.87
HERMES PR 0.00 + FT | 0.80 | 0.84
HERMES PR 0.50 + FT | 0.84 | 0.85

Table 2: Pearson correlation with T4 Lysozyme stability $\Delta\Delta G$
values with and without the use of mutant crystal structures. Interestingly,
HERMES fine-tuned models benefit from the use of mutant structures, whereas
HERMES zero-shot models do not. ProteinMPNN zero-shot instead benefits from
mutant structures. Note that these values are computed using 33 mutations.
Figure 4: Anti-symmetry-respecting score for various models (higher is
better). We compute the $R^{2}$ between the predicted forward mutations and
the predicted negative of reverse mutations for the Ssym dataset, which
contains 352 variants from 19 structures (forward) and a structure for each of
the reverse variants. We speculate that the higher performance of ProteinMPNN
is due to it focusing less - and thus overfitting less - on specific atomic
details of the environment (ProteinMPNN does not model side-chains, HERMES
does).

Prediction on additional datasets, and anti-symmetry wrt. the direction of
mutations. We further consider two additional datasets collected by [4]. The
S669 experimental dataset, which includes 669 mutations across 94 protein
structures, and the Ssym+ dataset, which includes 352 mutants across 19
protein structures [59]. In addition, Ssym includes experimentally-determined
structures of all of the 352 mutants. These structures can be used to test
whether our model correctly predicts the anti-symmetry condition, whereby the
$\Delta\Delta G$ of a given mutation must be equal to that of its reverse
mutation with opposite sign. If for example the direct mutation is
$G\rightarrow A$, then the reverse is $A\rightarrow G$.

$\Delta\Delta G_{wt\rightarrow mt}=-\Delta\Delta G_{mt\rightarrow wt}$ (3)

For S669 and Ssym-direct, we report two orthogonal computations of Pearson
correlation: “Per Structure” correlations are computed by averaging
correlations across proteins with at least 10 mutations; “Overall”
correlations are instead computed across all mutations among all proteins. We
consider “Per Structure” correlation as it is more closely related to
practical use when designing highly-stable proteins by mutating from a
starting protein. HERMES PRwith 0.50 Å noise and fine-tuning is consistently
the best-performing model (Table S2). Interestingly, on S669 the Per Structure
correlations are consistently higher than Overall correlations, and the
opposite is true for Ssym-direct.

Figure 5: Performance of models at recovering Binding $\Delta\Delta G$ from
the SKEMPI 2.0 dataset. “Per Structure” correlations are computed by averaging
correlations across proteins with at least 10 mutations; ”Overall”
correlations are instead computed across all mutations among all proteins.
Performance of ESM-1v was taken from [21]. Error bars show the standard error.
We do not show error bars for ESM-1v as we show previously-reported scores.

We evaluate the respecting of the anti-symmetry condition in two ways. First,
we consider the overall Pearson correlation on Ssym-direct vs. Ssym-reverse: a
model that respects the anti-symmetry condition should perform equally well on
both. We find that both ProteinMPNN and HERMES models exhibit a significantly
smaller gap between the two correlations than RaSP, especially after fine-
tuning ({0.64 ; 0.46} vs. {0.58 ; 0.18} on {direct ; reverse} for HERMES PR
0.50 + FT vs. RaSP + FT, Table S2). Second, we compute the $R^{2}$ score
between a model’s predictions on the direct dataset, and the negative of its
predictions on the reverse dataset (Figure 4). We consistently find that
models trained with noise have better anti-symmetry score, that fine-tuning
improves the anti-symmetry score of HERMES models, and that ProteinMPNN models
have the best anti-symmetry scores, even though its predictions are zero-shot
and of lower overall accuracy. We speculate that this is because ProteinMPNN
only considers the backbone atoms and amino-acid labels, unlike HERMES, which
includes all atoms in the structure. Consequently, ProteinMPNN has a reduced
risk of overfitting on spurious details of the atomic environments.
Interestingly, we note that the scores on the reverse dataset are consistently
under-shooting (Figure S1).

Figure 6: Performance of models at recovering Binding $\Delta\Delta G$ from
the ATLAS dataset. “Per Structure” correlations are computed by averaging
correlations across proteins with at least 10 mutations; “Overall”
correlations are instead computed across all mutations among all proteins. We
show both predictions made using the wildtype structure only (Equation 2) and
with using mutant structures (Equation 1). Error bars show the standard error.

###  III.3 Predicting mutational effect on binding, a.k.a. $\Delta\Delta
G^{\text{binding}}$

The $\Delta\Delta G$ value associated with a binding event is slightly
different from $\Delta\Delta G$ of stability, which is associated with a
protein’s folding event. Specifically, for the case of stability, the free
energy of folding $\Delta G^{\text{stability}}$ for both the wildtype and the
mutant is computed from:

$\Delta G^{\text{stability}}=G^{\text{folded}}-G^{\text{unfolded}}$ (4)

resulting in the stability effect of a mutation, $\Delta\Delta
G_{\text{stability}}=\Delta G^{\text{stability}}_{\text{mt}}-\Delta
G^{\text{stability}}_{\text{wt}}$. On the other hand, the free energy
associate with binding follows,

$\Delta G^{\text{binding}}=G^{\text{bound}}-G^{\text{unbound}}$ (5)

resulting in the binding effect if a mutation $\Delta\Delta
G^{\text{binding}}=\Delta G^{\text{binding}}_{\text{mt}}-\Delta
G^{\text{binding}}_{\text{wt}}$. Crucially, the unbound proteins are usually
considered to be folded. In fact, $\Delta G^{\text{binding}}$ is usually
experimentally determined starting from the measurements of the dissociation
constant $K_{d}$:

$\Delta G^{\text{binding}}=RT\log K_{d}$ (6)

where $T$ is usually taken to be room temperature, as most experiments are
performed at that temperature, and $R=8.314\,\text{J K}^{-1}\text{mol}^{-1}$
is the gas constant. Here, we present our predictions for the binding effect
of mutations for a number of datasets.

SKEMPI v2.0 dataset [60]. This dataset, to our knowledge, is the most
comprehensive dataset comprising the effect of mutations on the binding
affinity of protein-protein interactions, with the associated crystal
structures of the wildtype’s bound conformation. After filtering duplicate
experiments, the dataset includes: 5,713 $\Delta\Delta G^{\text{binding}}$
values across 331 structures, of which 4,106 are single-point mutations across
308 structures. Further filtering for mutations that belong to structures with
at least 10 mutations in the dataset, 116 structures remain with 5,025 total
mutations; By restricting to only single-point mutations, we arrive at 93
structures and 3,485 mutations. We consider both “Per Structure” and “Overall”
correlations, and will focus on single-point mutations. For multi-point
mutations, we use an additive model and neglect epistasis.
The best-performing model we consider is HERMES PR0.00 with fine-tuning, with
Per Structure Pearson correlation of 0.32 $\pm$ 0.07 and Overall correlation
of 0.34 (Figure 5 and Table S3). Training with noise does not seem to
consistently provide a boost in performance. Fine-tuning on Rosetta stability
$\Delta\Delta G$ values, instead, does. Notably, the predictive power of all
models - even zero-shot models - is significantly lower here than it is for
protein stability, indicating that the pre-training and fine-tuning strategies
proposed here are more specific to modeling of protein stability than binding;
perhaps a fine-tuning procedure specific to protein-protein interactions would
provide benefit for this task.
We do not expect ESM-1v to have strong predictive power as its training on
single-chain sequences does not easily transfer to reasoning over multi-chain
interactions between protein. Structure based models using local atomic
neighborhoods are agnostic to their provenance, as a neighborhood at the
interface of two proteins (two portions of a single chain) are in principle
indistinguishable from neighborhoods at the interface between two distinct
chains (i.e., domains within a protein). Indeed, ESM-1v cannot predict the
“per structure” binding effect of mutations (Per Structure correlation of
0.01), but it surprisingly has some predictive power, though less than all
other models, across proteins (Overall correlation of 0.19). Finally, all
models are better at predicting $\Delta\Delta G$ of single-point mutations,
and seem to lack accuracy in the epistatic effects of multiple mutations.

Figure 7: Performance of models at recovering values of DMS experimental
assays from [26]. Each point is a study (single protein), and horizontal bars
are mean values. Fine-tuning HERMES models on Rosetta stability $\Delta\Delta
G$ values improves performance, but it does not enable them to reach the
levels of state-of-the-art sequence-based models DeepSequence and ESM-1v.

ATLAS dataset of TCR-pMHC interactions [61]. Similar to SKEMPI, this dataset
gather the effect of mutations on the binding affinity of protein-protein
interactions, with the associated crystal structures of the wildtype’s bound
conformation. There are two main differences between ATLAS with SKEMPI: 1)
ATLAS focuses on TCR-pMHC complexes, thoroughly annotating on which region of
the complex the mutation is taking place, and 2) it provides mutant structures
as well, where most are computationally modeled. After filtering duplicate
experiments, the dataset includes: 372 $\Delta\Delta G^{\text{binding}}$
values across 39 structures, of which 261 are single-point mutations across 36
structures. Further filtering for mutations belonging to structures that have
at least 10 mutations in the dataset, results in 8 remaining structures with
266 total mutations; 7 structures and 180 mutations if considering only
single-point mutations. Therefore, this dataset is consistently smaller than
SKEMPI. Pearson correlations for predictions of binding affinities in ATLAS
are shown in Figure 6, Table S4 and Table S5. Not many consistent patterns can
be identified, probably due to the small sample sizes. HERMES models seem to
benefit from fine-tuning, and proteinMPNN does surprisingly well,
outperforming even fine-tuned HERMES models in some settings.

###  III.4 Prediction of Deep Mutational Scanning assays

We evaluate model performance on 25 out of the 41 Deep Mutational Scanning
(DMS) studies collected by [26] and considered by [27]. To simplify the
analysis, we consider only the 37 studies containing single-point mutations
only. For these, only the proteins’ sequences were available to us a priori.
Starting from the sequences, we augmented the dataset with both experimental
structures that we identified in the RCSB website 333https://www.rcsb.org/ and
AlphaFold2 structures, either from the AlphaFold database
444https://alphafold.ebi.ac.uk/, or folded using the AlphaFold2 [20] google
colab with default parameters. Keeping only studies with at least one high-
quality structure, we were left with 25 studies, many of which with only the
AlphaFold-generated structure. Some proteins have multiple experimental
structures, as in each structure they are bound to different a different and
it was not obvious from the study of origin which ligand was more appropriate.
We provide structures and detailed notes for each study on our github
repository.

In Figure 7 we show absolute Pearson correlations between model predictions
and expreiments for the 25 studies, selected as described above. We use
absolute Pearson correlation for simplicity, as assays may have either
positive or negative sign associated with higher fitness. Patterns are similar
to those we found for the stability effect of mutations $\Delta\Delta G$:
training with noise improves zero-shot models, and so does pre-processing with
PyRosetta. Models fine-tuned on Rosetta stability $\Delta\Delta G$ see their
performance improved. However, the best structure-based model (HERMES PR 0.50
+ FT with mean Pearson r of 0.40) still performs significantly worse, on
average, compared to the state-of-the-art sequence-based models (DeepSequence
[26] with 0.50, and ESM-1v [27] with 0.47).

##  IV Discussion

Here, we presented HERMES, an efficient deep learning method for inferring the
effects of mutations on protein function, conditioned on the local atomic
environment surrounding the mutated residue. HERMES is pre-trained to model
amino-acid preferences in protein structures, and can be optionally fine-tuned
on arbitrary mutation effects datasets. We provide HERMES models pre-trained
on a large non-reduntant chunk of the protein structure universe, as well as
the same models fine-tuned on stability effects ($\Delta\Delta G$) computed
with Rosetta. We thoroughly benchmark HERMES against other state-of-the-art
models, showing robust performance on a wide variety of proteins and
functions: stability effects, binding affinity, and several deep mutational
scanning assays. We open-source our code and data used for experiments, where
we provide easy-to-use scripts to run HERMES models on desired protein
structures and mutation effects, as well as code to fine-tune our pre-trained
HERMES models on the user’s own mutation effect data.

##  V Contributions

GMV led the project in both implementations and experiments. MNP effectively
started the project by developing HCNN, and provided code snippets and
guidance. WG assisted the implementation of preprocessing pipelines. ED
collected and curated the structures from the DMS experiments. KB and UW
assisted in running experiments with ESM-1v. AN supervised and designed the
project, and acquired funding. GMV and AN wrote the paper.

##  VI Acknowledgement

This work has been supported by the National Institutes of Health MIRA award
(R35 GM142795), the CAREER award from the National Science Foundation (grant
No: 2045054), the Royalty Research Fund from the University of Washington (no.
A153352), and the Allen School Computer Science & Engineering Research
Fellowship from the Paul G. Allen School of Computer Science & Engineering at
the University of Washington. This work is also supported, in part, through
the Departments of Physics and Computer Science and Engineering, and the
College of Arts and Sciences at the University of Washington.

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## Appendix A

|  | ProTherm | Protein G | VAMP-seq |
---|---|---|---|---|---
|  | 1BVC | 1LZ1 | 2CI2 | 2RN2 | 1PGA | 5BON | 2H11 | 1D5R | Average
Zero-Shot | ESM-1v | 0.49 | 0.49 | 0.21 | 0.27 | 0.24 | 0.48 | 0.54 | 0.44 | 0.40
ProteinMPNN 0.02 | 0.50 | 0.56 | 0.43 | 0.19 | 0.48 | 0.47 | 0.40 | 0.52 | 0.44
ProteinMPNN 0.20 | 0.57 | 0.64 | 0.51 | 0.30 | 0.44 | 0.48 | 0.41 | 0.51 | 0.48
HERMES BP 0.00 | 0.29 | 0.40 | 0.47 | 0.36 | 0.44 | 0.42 | 0.33 | 0.42 | 0.39
HERMES BP 0.50 | 0.52 | 0.42 | 0.52 | 0.58 | 0.57 | 0.50 | 0.41 | 0.49 | 0.50
HERMES PR 0.00 | 0.50 | 0.50 | 0.44 | 0.42 | 0.50 | 0.48 | 0.39 | 0.47 | 0.46
HERMES PR 0.50 | 0.63 | 0.51 | 0.58 | 0.65 | 0.59 | 0.54 | 0.45 | 0.52 | 0.56
Fine-Tuned | RaSP + FT | 0.71 | 0.57 | 0.65 | 0.79 | 0.72 | 0.50 | 0.48 | 0.52 | 0.62
HERMES BP 0.00 + FT | 0.64 | 0.60 | 0.55 | 0.70 | 0.74 | 0.60 | 0.52 | 0.57 | 0.62
HERMES BP 0.50 + FT | 0.71 | 0.57 | 0.63 | 0.78 | 0.76 | 0.61 | 0.52 | 0.56 | 0.64
HERMES PR 0.00 + FT | 0.67 | 0.62 | 0.54 | 0.73 | 0.73 | 0.61 | 0.52 | 0.57 | 0.62
HERMES PR 0.50 + FT | 0.71 | 0.58 | 0.56 | 0.80 | 0.74 | 0.60 | 0.52 | 0.58 | 0.64

Table S1: Pearson Correlation as prediction performance on experimental
$\Delta\Delta G^{\text{stability}}$ for eight test proteins. Same data as
Figure 3.

|  | S669 | Ssym dir. | Ssym rev.
---|---|---|---|---
|  | Per Structure | Overall | Per Structure | Overall | Overall
Zero-Shot | ProteinMPNN 0.02 | 0.42 $\pm$ 0.08 | 0.30 | 0.30 $\pm$ 0.10 | 0.52 | 0.30
ProteinMPNN 0.20 | 0.47 $\pm$ 0.07 | 0.34 | 0.40 $\pm$ 0.09 | 0.56 | 0.39
HERMES BP 0.00 | 0.31 $\pm$ 0.08 | 0.15 | 0.31 $\pm$ 0.09 | 0.47 | 0.21
HERMES BP 0.50 | 0.40 $\pm$ 0.07 | 0.29 | 0.45 $\pm$ 0.08 | 0.56 | 0.30
HERMES PR 0.00 | 0.37 $\pm$ 0.06 | 0.21 | 0.38 $\pm$ 0.09 | 0.53 | 0.17
HERMES PR 0.50 | 0.44 $\pm$ 0.06 | 0.31 | 0.53 $\pm$ 0.06 | 0.61 | 0.30
Fine-Tuned | RaSP + FT | - | 0.39 | - | 0.58 | 0.18
HERMES BP 0.00 + FT | 0.44 $\pm$ 0.07 | 0.32 | 0.48 $\pm$ 0.06 | 0.59 | 0.37
HERMES BP 0.50 + FT | 0.47 $\pm$ 0.07 | 0.38 | 0.54 $\pm$ 0.05 | 0.62 | 0.44
HERMES PR 0.00 + FT | 0.47 $\pm$ 0.06 | 0.34 | 0.53 $\pm$ 0.05 | 0.62 | 0.36
HERMES PR 0.50 + FT |  0.50 $\pm$ 0.07 | 0.39 |  0.59 $\pm$ 0.04 | 0.64 | 0.46

Table S2: Pearson correlation with stability $\Delta\Delta G$ from S669 and
Ssym datasets. “Per Structure” correlations are computed by averaging
correlations across proteins with at least 10 mutations; “Overall”
correlations are instead computed across all mutations among all proteins. For
RaSP we currently report only values that were reported in their paper. Ssym
reverse has no Per Structure correlation since every mutant has its own
structure. Standard error is shown for per-structure correlations.

|  | All Mutations | Single-Point Mutations
---|---|---|---
|  | Per Structure | Overall | Per Structure | Overall
Zero-Shot | ESM-1v | 0.01 | 0.19 | - | -
ProteinMPNN 0.02 | 0.25 $\pm$ 0.03 | 0.29 | 0.28 $\pm$ 0.03 | 0.33
ProteinMPNN 0.20 | 0.24 $\pm$ 0.03 | 0.27 | 0.30 $\pm$ 0.03 | 0.33
HERMES BP 0.00 | 0.21 $\pm$ 0.03 | 0.20 | 0.23 $\pm$ 0.03 | 0.24
HERMES BP 0.50 | 0.24 $\pm$ 0.02 | 0.21 | 0.27 $\pm$ 0.03 | 0.24
HERMES PR 0.00 | 0.26 $\pm$ 0.03 | 0.24 | 0.29 $\pm$ 0.03 | 0.28
HERMES PR 0.50 | 0.29 $\pm$ 0.03 | 0.25 | 0.31 $\pm$ 0.03 | 0.29
Fine-Tuned | HERMES BP 0.00 + FT | 0.31 $\pm$ 0.02 | 0.33 | 0.34 $\pm$ 0.03 | 0.39
HERMES BP 0.50 + FT | 0.30 $\pm$ 0.02 | 0.30 | 0.32 $\pm$ 0.03 | 0.35
HERMES PR 0.00 + FT |  0.32 $\pm$ 0.02 | 0.34 |  0.36 $\pm$ 0.03 | 0.40
HERMES PR 0.50 + FT | 0.31 $\pm$ 0.02 | 0.31 | 0.34 $\pm$ 0.03 | 0.36

Table S3: Pearson correlation of models with Binding $\Delta\Delta G$ from the
SKEMPI 2.0 dataset. Standard error is shown for per-structure correlations. We
do not show it for ESM-1v as we show previously-reported scores.

|  | All Mutations | Single-Point Mutations
---|---|---|---
|  | Per Structure | Overall | Per Structure | Overall
Zero-Shot | ProteinMPNN 0.02 | 0.31 $\pm$ 0.11 | 0.08 |  0.37 $\pm$ 0.11 | 0.24
ProteinMPNN 0.20 | 0.17 $\pm$ 0.14 | 0.07 | 0.26 $\pm$ 0.14 | 0.19
HERMES BP 0.00 | 0.15 $\pm$ 0.04 | 0.14 | 0.16 $\pm$ 0.04 | 0.11
HERMES BP 0.50 | 0.16 $\pm$ 0.04 | 0.21 | 0.16 $\pm$ 0.05 | 0.15
HERMES PR 0.00 | 0.06 $\pm$ 0.08 | 0.10 | 0.10 $\pm$ 0.07 | 0.08
HERMES PR 0.50 | 0.07 $\pm$ 0.07 | 0.20 | 0.09 $\pm$ 0.06 | 0.12
Fine-Tuned | HERMES BP 0.00 + FT |  0.32 $\pm$ 0.07 | 0.14 |  0.37 $\pm$ 0.06 | 0.21
HERMES BP 0.50 + FT | 0.27 $\pm$ 0.06 | 0.18 | 0.30 $\pm$ 0.07 | 0.21
HERMES PR 0.00 + FT | 0.27 $\pm$ 0.08 | 0.14 | 0.33 $\pm$ 0.07 | 0.17
HERMES PR 0.50 + FT | 0.21 $\pm$ 0.06 | 0.18 | 0.25 $\pm$ 0.06 | 0.16

Table S4: Pearson correlation of models with Binding $\Delta\Delta G$ from the
ATLAS dataset, using only wildtype structures. Standard error is shown for
per-structure correlations.

|  | All Mutations | Single-Point Mutations
---|---|---|---
|  | Per Structure | Overall | Per Structure | Overall
Zero-Shot | ProteinMPNN 0.02 |  0.33 $\pm$ 0.10 | 0.13 | 0.37 $\pm$ 0.11 | 0.23
ProteinMPNN 0.20 | 0.18 $\pm$ 0.13 | 0.10 | 0.26 $\pm$ 0.14 | 0.17
HERMES BP 0.00 | 0.29 $\pm$ 0.07 | 0.06 | 0.35 $\pm$ 0.05 | 0.16
HERMES BP 0.50 | 0.28 $\pm$ 0.07 | 0.12 | 0.34 $\pm$ 0.08 | 0.17
HERMES PR 0.00 | 0.29 $\pm$ 0.07 | 0.09 | 0.36 $\pm$ 0.06 | 0.17
HERMES PR 0.50 | 0.28 $\pm$ 0.07 | 0.13 | 0.34 $\pm$ 0.08 | 0.16
Fine-Tuned | HERMES BP 0.00 + FT | 0.32 $\pm$ 0.09 | 0.07 |  0.42 $\pm$ 0.08 | 0.22
HERMES BP 0.50 + FT | 0.31 $\pm$ 0.10 | 0.15 | 0.40 $\pm$ 0.11 | 0.25
HERMES PR 0.00 + FT | 0.30 $\pm$ 0.10 | 0.10 | 0.40 $\pm$ 0.09 | 0.21
HERMES PR 0.50 + FT | 0.29 $\pm$ 0.08 | 0.14 | 0.37 $\pm$ 0.09 | 0.20

Table S5: Pearson correlation of models with Binding $\Delta\Delta G$ from the
ATLAS dataset, using both wildtype and mutant structures. Error bars show the
standard error.

|
---|---
Figure S1: Predictions on Ssym-direct vs. Ssym-reverse. Each point is a
mutation. Prediction values are made according to Equation 2. Remarkably,
predictions on Ssym-reverse are consistently under-shooting.


# Agegraphic Chaplygin gas model of dark energy

A. Sheykhi 111<EMAIL_ADDRESS>Department of Physics, Shahid Bahonar
University, P.O. Box 76175, Kerman, Iran
Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha,
Iran

###### Abstract

We establish a connection between the agegraphic models of dark energy and
Chaplygin gas energy density in non-flat universe. We reconstruct the
potential of the agegraphic scalar field as well as the dynamics of the scalar
field according to the evolution of the agegraphic dark energy. We also extend
our study to the interacting agegraphic generalized Chaplygin gas dark energy
model.

##  I Introduction

Over the course of the past decade, evidence for the most striking result in
modern cosmology has been steadily growing, namely the existence of an exotic
dark energy component which has negative pressure and pushes the universe to
accelerated expansion Rie . Of course, a natural explanation to the
accelerated expansion is due to a positive tiny cosmological constant. Though,
it suffers the so-called fine-tuning and cosmic coincidence problems. A great
variety of scenarios have been proposed to explain this acceleration while
most of them cannot explain all the features of universe or they have so many
parameters that makes them difficult to fit. For a recent review on dark
energy proposals see Pad . Many theoretical attempts toward understanding the
dark energy problem are focused to shed light on it in the framework of a
fundamental theory such as string theory or quantum gravity. Although a
complete theory of quantum gravity has not established yet today, we still can
make some attempts to investigate the nature of dark energy according to some
principles of quantum gravity. The holographic dark energy model and the
agegraphic dark energy model are just such examples, which are originated from
some considerations of the features of the quantum theory of gravity. That is
to say, the holographic and agegraphic dark energy models possess some
significant features of quantum gravity. The former, that arose a lot of
enthusiasm recently Coh ; Li ; Huang ; Hsu ; HDE ; Setare ; Seta1 , is
motivated from the holographic hypothesis Suss1  and has been tested and
constrained by various astronomical observations Xin . However there are some
difficulties in holographic dark energy model. Choosing the event horizon of
the universe as the length scale, the holographic dark energy gives the
observation value of dark energy in the universe and can drive the universe to
an accelerated expansion phase. But an obvious drawback concerning causality
appears in this proposal. Event horizon is a global concept of spacetime;
existence of event horizon of the universe depends on future evolution of the
universe; and event horizon exists only for universe with forever accelerated
expansion. In addition, more recently, it has been argued that this proposal
might be in contradiction to the age of some old high redshift objects, unless
a lower Hubble parameter is considered Wei0 . The later (agegraphic dark
energy) is based on the uncertainty relation of quantum mechanics together
with the gravitational effect in general relativity. The agegraphic dark
energy model assumes that the observed dark energy comes from the spacetime
and matter field fluctuations in the universe Cai1 ; Wei2 ; Wei1 . Since in
agegraphic dark energy model the age of the universe is chosen as the length
measure, instead of the horizon distance, the causality problem in the
holographic dark energy is avoided. The agegraphic models of dark energy have
been examined and constrained by various astronomical observations age ; shey1
; setare .

Among the various candidates to explain the accelerated expansion, the
Chaplygin gas dark energy model has emerged as a possible unification of dark
matter and dark energy, since its cosmological evolution is similar to an
initial dust like matter and a cosmological constant for late times. Inspired
by the fact that the Chaplygin gas possesses a negative pressure, the authors
of Gorini  have undertaken the simple task of studying a FRW cosmology of a
universe filled with this type of fluid. The equation of state of the
Chaplygin gas dark energy obeys Kam

$\displaystyle p_{D}=\frac{-A}{\rho_{D}},$ (1)

where $\rho_{D}>0$ and $p_{D}$ are, respectively, the energy density and
pressure of Chaplygin gas dark energy, and $A$ is a positive constant. This
equation of state has raised a certain interest Baz  because of its many
interesting and, in some sense, intriguingly unique features. Some possible
motivations for this model from the field theory points of view are
investigated in Bil . The Chaplygin gas emerges as an effective fluid
associated with D-branes Bor  and can also be obtained from the Born-Infeld
action Ben . The connection between the holographic models of dark energy and
the Chaplygin gas energy density has been established in seta1 ; seta2 .

Our aim in this Letter is to establish a correspondence between the agegraphic
dark energy scenarios and the Chaplygin gas model. We suggest the agegraphic
description of the Chaplygin gas dark energy in FRW universe and reconstruct
the potential and the dynamics of the scalar field which describe the
Chaplygin cosmology. In the next section we study the original agegraphic
Chaplygin gas model. In section III, we establish the correspondence between
the new model of agegraphic dark energy and the Chaplygin gas dark energy. In
section IV, we extend our study to the interacting agegraphic generalized
Chaplygin gas dark energy model. The last section is devoted to conclusions.

##  II THE ORIGINAL ADE as a Chaplygin gas

We assume the agegraphic Chaplygin gas dark energy is accommodated in the
Friedmann-Robertson-Walker (FRW) universe which is described by the line
element

$\displaystyle
ds^{2}=dt^{2}-a^{2}(t)\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}\right),$
(2)

where $a(t)$ is the scale factor, and $k$ is the curvature parameter with
$k=-1,0,1$ corresponding to open, flat, and closed universes, respectively. A
closed universe with a small positive curvature ($\Omega_{k}\simeq 0.01$) is
compatible with observations spe . The corresponding Friedmann equation takes
the form

$\displaystyle
H^{2}+\frac{k}{a^{2}}=\frac{1}{3m_{p}^{2}}\left(\rho_{m}+\rho_{D}\right).$ (3)

We introduce, as usual, the fractional energy densities such as

$\displaystyle\Omega_{m}=\frac{\rho_{m}}{3m_{p}^{2}H^{2}},\hskip
14.22636pt\Omega_{D}=\frac{\rho_{D}}{3m_{p}^{2}H^{2}},\hskip
14.22636pt\Omega_{k}=\frac{k}{H^{2}a^{2}},$ (4)

thus, the Friedmann equation can be written

$\displaystyle\Omega_{m}+\Omega_{D}=1+\Omega_{k}.$ (5)

Inserting the equation of state (1) into the relativistic energy conservation
equation, leads to a density evolving as

$\displaystyle\rho_{D}=\sqrt{A+\frac{B}{a^{6}}}.$ (6)

where $B$ is an integration constant. We adopt the viewpoint that the scalar
field models of dark energy are effective theories of an underlying theory of
dark energy. The energy density and pressure of the scalar field can be
written as

$\displaystyle\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+V(\phi)=\sqrt{A+\frac{B}{a^{6}}},$
(7) $\displaystyle
p_{\phi}=\frac{1}{2}\dot{\phi}^{2}-V(\phi)=\frac{-A}{\sqrt{A+\frac{B}{a^{6}}}},$
(8)

Then, we can easily obtain the scalar potential and the kinetic energy term as

$\displaystyle V(\phi)=\frac{2Aa^{6}+B}{2a^{6}\sqrt{A+\frac{B}{a^{6}}}},$ (9)
$\displaystyle\dot{\phi}^{2}=\frac{B}{a^{6}\sqrt{A+\frac{B}{a^{6}}}}.$ (10)

Now we are focussing on the reconstruction of the original agegraphic
Chaplygin gas model of dark energy. Let us first review the origin of the
agegraphic dark energy. Following the line of quantum fluctuations of
spacetime, Karolyhazy et al. Kar1  argued that the distance $t$ in Minkowski
spacetime cannot be known to a better accuracy than $\delta{t}=\beta
t_{p}^{2/3}t^{1/3}$ where $\beta$ is a dimensionless constant of order unity.
Based on Karolyhazy relation, Maziashvili discussed that the energy density of
metric fluctuations of the Minkowski spacetime is given by Maz

$\rho_{D}\sim\frac{1}{t_{p}^{2}t^{2}}\sim\frac{m^{2}_{p}}{t^{2}},$ (11)

where $t_{p}$ is the reduced Planck time. We use the units $c=\hbar=k_{b}=1$
throughout this Letter. Therefore one has $l_{p}=t_{p}=1/m_{p}$ with $l_{p}$
and $m_{p}$ are the reduced Planck length and mass, respectively. The original
agegraphic dark energy density has the form (11) where $t$ is chosen to be the
age of the universe

$T=\int_{0}^{a}{\frac{da}{Ha}},$ (12)

Thus, the energy density of the original agegraphic dark energy is given by
Cai1

$\rho_{D}=\frac{3n^{2}m_{p}^{2}}{T^{2}},$ (13)

where the numerical factor $3n^{2}$ is introduced to parameterize some
uncertainties, such as the species of quantum fields in the universe, the
effect of curved space-time (since the energy density is derived for Minkowski
space-time), and so on. The dark energy density (13) has the same form as the
holographic dark energy, but the length measure is chosen to be the age of the
universe instead of the horizon radius of the universe. Thus the causality
problem in the holographic dark energy is avoided. Using Eqs. (4) and (13), we
have

$\displaystyle\Omega_{D}=\frac{n^{2}}{H^{2}T^{2}}.$ (14)

We assume the agegraphic dark energy and dark matter evolve according to their
conservation laws

$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})=0,$ (15)
$\displaystyle\dot{\rho}_{m}+3H\rho_{m}=0,$ (16)

where $w_{D}$ is the equation of state parameter of agegraphic dark energy.
Taking the derivative with respect to the cosmic time of Eq. (13) and using
Eq. (14) we get

$\displaystyle\dot{\rho}_{D}=-2H\frac{\sqrt{\Omega_{D}}}{n}\rho_{D}.$ (17)

Inserting this relation into Eq. (15), we obtain the equation of state
parameter of the original agegraphic dark energy

$\displaystyle w_{D}=-1+\frac{2}{3n}\sqrt{\Omega_{D}}.$ (18)

Differentiating Eq. (14) and using relation
${\dot{\Omega}_{D}}={\Omega^{\prime}_{D}}H$, we reach

$\displaystyle{\Omega^{\prime}_{D}}=\Omega_{D}\left(-2\frac{\dot{H}}{H^{2}}-\frac{2}{n}\sqrt{\Omega_{D}}\right),$
(19)

where the dot is the derivative with respect to the cosmic time and the prime
denotes the derivative with respect to $x=\ln{a}$. Taking the derivative of
both side of the Friedman equation (3) with respect to the cosmic time, and
using Eqs. (5), (13), (14) and (16), it is easy to find

$\displaystyle\frac{\dot{H}}{H^{2}}=-\frac{3}{2}(1-\Omega_{D})-\frac{\Omega^{3/2}_{D}}{n}-\frac{\Omega_{k}}{2}.$
(20)

Substituting this relation into Eq. (19), we obtain the equation of motion of
agegraphic dark energy

$\displaystyle{\Omega^{\prime}_{D}}$ $\displaystyle=$
$\displaystyle\Omega_{D}\left[(1-\Omega_{D})\left(3-\frac{2}{n}\sqrt{\Omega_{D}}\right)+\Omega_{k}\right].$
(21)

Next, we establish the connection between the original agegraphic dark energy
and Chaplygin gas energy density. Combining Eqs. (6) and (13), we obtain

$\displaystyle B=a^{6}\left(9n^{4}m^{4}_{p}T^{-4}-A\right).$ (22)

Using Eqs. (1), (6) and (18) one can write

$\displaystyle
w_{D}=\frac{-A}{\rho_{D}^{2}}=\frac{-A}{A+\frac{B}{a^{6}}}=-1+\frac{2}{3n}\sqrt{\Omega_{D}}.$
(23)

Substituting $B$ in the above equation, we obtain following relation for $A$:

$\displaystyle
A=9n^{4}m^{4}_{p}T^{-4}\left(1-\frac{2}{3n}\sqrt{\Omega_{D}}\right).$ (24)

Therefore the constant $B$ is given by

$\displaystyle B=6a^{6}n^{4}m^{4}_{p}T^{-4}\frac{\sqrt{\Omega_{D}}}{n}.$ (25)

Finally, we rewrite the scalar potential and kinetic energy term as

$\displaystyle V(\phi)$ $\displaystyle=$ $\displaystyle
n^{2}m^{2}_{p}T^{-2}\left(3-\frac{\sqrt{\Omega_{D}}}{n}\right)=m^{2}_{p}H^{2}\Omega_{D}\left(3-\frac{\sqrt{\Omega_{D}}}{n}\right),$
(26) $\displaystyle\dot{\phi}$ $\displaystyle=$ $\displaystyle
nm_{p}T^{-1}\sqrt{\frac{2}{n}\Omega^{1/2}_{D}}=m_{p}H\sqrt{\frac{2}{n}{\Omega^{3/2}_{D}}}.$
(27)

Using relation $\dot{\phi}=H{\phi^{\prime}}$, we get

$\displaystyle{\phi^{\prime}}$ $\displaystyle=$ $\displaystyle
m_{p}\sqrt{\frac{2}{n}{\Omega^{3/2}_{D}}}.$ (28)

Consequently, we can easily obtain the evolution behavior of the scalar field

$\displaystyle\phi(a)-\phi(a_{0})=\int_{a_{0}}^{a}{\frac{m_{p}}{a}\sqrt{\frac{2}{n}{\Omega^{3/2}_{D}}}da},$
(29)

where $a_{0}$ is the present value of the scale factor, and $\Omega_{D}$ can
be obtained through Eq. (21).

##  III THE NEW ADE as a Chaplygin gas

Soon after the original agegraphic dark energy model was introduced by Cai
Cai1 , a new model of agegraphic dark energy was proposed in Wei2 , while the
time scale is chosen to be the conformal time $\eta$ instead of the age of the
universe. This new agegraphic dark energy contains some new features different
from the original agegraphic dark energy and overcome some unsatisfactory
points. For instance, the original agegraphic dark energy suffers from the
difficulty to describe the matter-dominated epoch while the new agegraphic
dark energy resolved this issue Wei2 . The energy density of the new
agegraphic dark energy can be written

$\rho_{D}=\frac{3n^{2}m_{p}^{2}}{\eta^{2}},$ (30)

where the conformal time is given by

$\eta=\int{\frac{dt}{a}}=\int_{0}^{a}{\frac{da}{Ha^{2}}}.$ (31)

The fractional energy density of the new agegraphic dark energy is now given
by

$\displaystyle\Omega_{D}=\frac{n^{2}}{H^{2}\eta^{2}}.$ (32)

Taking the derivative with respect to the cosmic time of Eq. (30) and using
Eq. (32) we get

$\displaystyle\dot{\rho}_{D}=-2H\frac{\sqrt{\Omega_{D}}}{na}\rho_{D}.$ (33)

Inserting this relation into Eq. (15) we obtain the equation of state
parameter of the new agegraphic dark energy

$\displaystyle w_{D}=-1+\frac{2}{3na}\sqrt{\Omega_{D}}.$ (34)

Then we obtain, following the approach of the previous section, the evolution
behavior of the new agegraphic dark energy,

$\displaystyle{\Omega^{\prime}_{D}}$ $\displaystyle=$
$\displaystyle\Omega_{D}\left[(1-\Omega_{D})\left(3-\frac{2}{na}\sqrt{\Omega_{D}}\right)+\Omega_{k}\right].$
(35)

Next, we construct the new agegraphic Chaplygin gas model, connecting the
Chaplygin gas model with new agegraphic dark energy. Identifying Eq. (6) with
(30) we have

$\displaystyle B=a^{6}\left(9n^{4}m^{4}_{p}\eta^{-4}-A\right).$ (36)

Using Eqs. (1), (6) and (34), we reach

$\displaystyle
w_{D}=\frac{-A}{A+\frac{B}{a^{6}}}=-1+\frac{2}{3na}\sqrt{\Omega_{D}}.$ (37)

Combining this equation with Eq. (36), we obtain

$\displaystyle A$ $\displaystyle=$ $\displaystyle
9n^{4}m^{4}_{p}\eta^{-4}\left(1-\frac{2}{3na}\sqrt{\Omega_{D}}\right),$ (38)
$\displaystyle B$ $\displaystyle=$ $\displaystyle
6a^{6}n^{4}m^{4}_{p}\eta^{-4}\frac{\sqrt{\Omega_{D}}}{na}.$ (39)

Finally, we reconstruct the scalar potential and kinetic energy term as

$\displaystyle V(\phi)$ $\displaystyle=$ $\displaystyle
n^{2}m^{2}_{p}\eta^{-2}\left(3-\frac{\sqrt{\Omega_{D}}}{na}\right)=m^{2}_{p}H^{2}\Omega_{D}\left(3-\frac{\sqrt{\Omega_{D}}}{na}\right),$
(40) $\displaystyle\dot{\phi}$ $\displaystyle=$ $\displaystyle
nm_{p}\eta^{-1}\sqrt{\frac{2}{na}{\Omega^{1/2}_{D}}}=m_{p}H\sqrt{\frac{2}{na}{\Omega^{3/2}_{D}}}.$
(41)

Eq. (41) can also be reexpressed as

$\displaystyle{\phi^{\prime}}$ $\displaystyle=$ $\displaystyle
m_{p}\sqrt{\frac{2}{na}{\Omega^{3/2}_{D}}}.$ (42)

Therefore, we can obtain the evolutionary form of the scalar field

$\displaystyle\phi(a)-\phi(a_{0})=\int_{a_{0}}^{a}{\frac{m_{p}}{a}\sqrt{\frac{2}{na}{\Omega^{3/2}_{D}}}da},$
(43)

where $\Omega_{D}$ can be derived from Eq. (35).

##  IV Interacting new agegraphic generalized Chaplygin gas

In this section we extend our study to the generalized Chaplygin gas when
there is an interaction between generalized Chaplygin gas energy density and
dark matter. The total energy density is $\rho=\rho_{m}+\rho_{D}$, where
$\rho_{m}$ and $\rho_{D}$ are the energy density of dark matter and dark
energy, respectively. The total energy density satisfies a conservation law

$\dot{\rho}+3H(\rho+p)=0.$ (44)

However, since we consider the interaction between dark matter and dark
energy, $\rho_{m}$ and $\rho_{D}$ do not conserve separately; they must rather
enter the energy balances

$\displaystyle\dot{\rho}_{m}+3H\rho_{m}=Q,$ (45)
$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})=-Q,$ (46)

where $Q$ denotes the interaction term and can be taken as $Q=3b^{2}H\rho$
with $b^{2}$ being a coupling constant. In the generalized Chaplygin gas
approach Ben , the equation of state (1) is generalized to

$\displaystyle p_{D}=\frac{-A}{{\rho_{D}^{\alpha}}}.$ (47)

The above equation of state leads to a density evolution as

$\displaystyle\rho_{D}=\left({A+{B}a^{-3\beta}}\right)^{1/\beta},$ (48)

where $\beta=\alpha+1$. Thus we have

$\displaystyle w_{D}=\frac{p_{D}}{\rho_{D}}=\frac{-A}{{A+{B}a^{-3\beta}}}.$
(49)

Inserting relation (33) into Eq. (46) and using Eqs. (4) and (5) , we obtain
the equation of state parameter of the interacting new agegraphic dark energy

$\displaystyle
w_{D}=-1+\frac{2}{3na}\sqrt{\Omega_{D}}-\frac{b^{2}}{\Omega_{D}}(1+\Omega_{k}).$
(50)

The evolution behavior of the new agegraphic dark energy is given by

$\displaystyle{\Omega^{\prime}_{D}}$ $\displaystyle=$
$\displaystyle\Omega_{D}\left[(1-\Omega_{D})\left(3-\frac{2}{na}\sqrt{\Omega_{D}}\right)-3b^{2}(1+\Omega_{k})+\Omega_{k}\right].$
(51)

We now establish the correspondence between the new agegraphic dark energy and
generalized Chaplygin gas energy density. Identifying Eq. (30) with Eq. (48)
and using Eq. (32) we get

$\displaystyle\left(3m^{2}_{p}H^{2}\Omega_{D}\right)^{\beta}={A+{B}a^{-3\beta}}.$
(52)

Combining Eqs. (49) and (50) we find

$\displaystyle
A=-\left({A+{B}a^{-3\beta}}\right)\left[-1+\frac{2}{3na}\sqrt{\Omega_{D}}-\frac{b^{2}}{\Omega_{D}}(1+\Omega_{k})\right].$
(53)

Solving Eqs. (52) and (53) we obtain

$\displaystyle A$ $\displaystyle=$
$\displaystyle\left(3m^{2}_{p}H^{2}\Omega_{D}\right)^{\beta}\left[1-\frac{2}{3na}\sqrt{\Omega_{D}}+\frac{b^{2}}{\Omega_{D}}(1+\Omega_{k})\right],$
(54) $\displaystyle B$ $\displaystyle=$
$\displaystyle\left(3m^{2}_{p}H^{2}\Omega_{D}a^{3}\right)^{\beta}\left[\frac{2}{3na}\sqrt{\Omega_{D}}-\frac{b^{2}}{\Omega_{D}}(1+\Omega_{k})\right].$
(55)

Next we regard the scalar field model as an effective description of an
underlying theory of dark energy with energy density and pressure

$\displaystyle\rho_{\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\dot{\phi}^{2}+V(\phi)=\left({A+{B}a^{-3\beta}}\right)^{1/\beta},$
(56) $\displaystyle p_{\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\dot{\phi}^{2}-V(\phi)=-A\left({A+{B}a^{-3\beta}}\right)^{-\alpha/\beta},$
(57)

where we have identified $\rho_{\phi}$ with $\rho_{D}$. Substituting $A$ and
$B$ into Eqs. (56) and (57) one can easily find the scalar potential and the
kinetic energy term as

$\displaystyle V(\phi)$ $\displaystyle=$ $\displaystyle
m^{2}_{p}H^{2}\Omega_{D}\left(3-\frac{\sqrt{\Omega_{D}}}{na}+\frac{3b^{2}}{2}\frac{(1+\Omega_{k})}{\Omega_{D}}\right),$
(58) $\displaystyle\dot{\phi}$ $\displaystyle=$ $\displaystyle
m_{p}H\left(\frac{2}{na}{\Omega^{3/2}_{D}}-3b^{2}(1+\Omega_{k})\right)^{1/2}.$
(59)

Using Eq. (59), Eq. (58) can be reexpressed as

$\displaystyle V(\phi)$ $\displaystyle=$ $\displaystyle
3m^{2}_{p}H^{2}\Omega_{D}\left(1-\frac{\dot{\phi}^{2}}{6m^{2}_{p}H^{2}\Omega_{D}}\right).$
(60)

We can also rewrite Eq. (59) as

$\displaystyle{\phi^{\prime}}$ $\displaystyle=$ $\displaystyle
m_{p}\left(\frac{2}{na}{\Omega^{3/2}_{D}}-3b^{2}(1+\Omega_{k})\right)^{1/2}.$
(61)

Consequently, we can easily obtain the evolutionary form of the field by
integrating the above equation. The result is

$\displaystyle\phi(a)-\phi(a_{0})=\int_{a_{0}}^{a}{\frac{m_{p}}{a}\sqrt{\frac{2}{na}{\Omega^{3/2}_{D}}-3b^{2}(1+\Omega_{k})}da},$
(62)

where $\Omega_{D}$ is given by Eq. (51). In this way we connect the
interacting new agegraphic dark energy with a generalized Chaplygin gas energy
density and reconstruct the potential of the agegraphic Chaplygin gas.

##  V Conclusions

In this Letter, we have established a correspondence between the agegraphic
dark energy scenarios and the Chaplygin gas model of dark energy in non-flat
FRW cosmology. A so-called agegraphic dark energy model has been proposed
recently by Cai Cai1 , based on the uncertainty relation of quantum mechanics
together with the gravitational effect in general relativity. Since the
original agegraphic dark energy model suffers from the difficulty to describe
the matter-dominated epoch, a new model of agegraphic dark energy was proposed
in Wei2  while the time scale is chosen to be the conformal time $\eta$
instead of the age of the universe. We have adopted the viewpoint that the
scalar field models of dark energy are effective theories of an underlying
theory of dark energy. If we regard the scalar field model as an effective
description of such a theory, we should be capable of using the scalar field
model to mimic the evolving behavior of the agegraphic dark energy and
reconstructing this scalar field model according to the evolutionary behavior
of agegraphic dark energy. We have reconstructed the potential of the
agegraphic scalar field as well as the dynamics of the scalar field which
describe the Chaplygin cosmology. Finally, we have extended our study to the
agegraphic generalized Chaplygin gas dark energy model when there is an
interaction between generalized Chaplygin gas energy density and dark matter.

###### Acknowledgements.

This work has been supported by Research Institute for Astronomy and
Astrophysics of Maragha.

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