Instead, both Volume-MCTS and Open-Loop AlphaZero keep track of the maximum
actual earned reward of each branch, and always select the branch with the
highest maximum value at the end of the episode.

###  D.6 Implementation

Our implementation draws on several existing codebases: an implementation of
AlphaZero-Continuous by Moerland et al. (2018), the pyOptimalMotionPlanning
package developed by Kris Hauser
(https://github.com/krishauser/pyOptimalMotionPlanning). For HER, we draw on
Tianhong Dai’s implementation of HER
(https://github.com/TianhongDai/hindsight-experience-replay) and the
implementation from the authors of USHER
(https://github.com/schrammlb2/USHER_Implementation) (Schramm et al., 2022).
Our POLY-HOOT implementation uses the author’s implementation
(https://github.com/xizeroplus/POLY-HOOT) (Mao et al., 2020).

###  D.7 Environment Details

####  D.7.1 Maze

In this environment, the agent must navigate a maze to reach a goal. Episodes
are 50 steps long. The reward function is 1 in the goal region, and 0 at all
other states. If an agent reaches the goal before the end of the episode, the
episode ends and the agent receives a reward of 1 for each remaining time step
left in the episode.

We tested two sets of dynamics on the maze environment. Geometric dynamics are
simple; the state space and action spaces are both 2-dimensional, and
$s_{t+1}=s_{t}+v_{max}a_{t}$, where $s_{t+1}$ is the next state, $s_{t}$ is
the current state, $a_{t}$ is the action, and $v_{\max}$ is the maximum speed
allowed by the environment. If this movement would cause the agent to collide
with a wall, instead the agent does not move ($s_{t+1}=s_{t}$).

Dubins car dynamics are slightly more complicated. The state space has three
dimensions: two position coordinates and one rotation coordinate. The action
space is two-dimensional. The agent selects a forward/backward speed and a
turning angle, which is bounded to give the agent a minimum turning radius.
The dynamics are as follows: Let $x,y$ be the car’s $x$ and $y$ coordinates.
Let $\theta$ be the car’s rotation coordinate. Let $v_{max}$ be the car’s
maximum speed and $\phi_{max}$ be the car’s maximum steering angle. Let
$a_{0}$ be the first dimension of the action, controlling the car’s speed. Let
$a_{1}$ be the second dimension of the action, controlling the car’s steering.

Then the next state described by the variables $x,y,\theta$ is found by
numerically integrating the differential equation

$\displaystyle\frac{dx}{dt}(t)=a_{0}\cos{\theta(t)}\frac{dy}{dt}(t)=a_{0}\sin{\theta(t)}\frac{d\theta}{dt}(t)=a_{1}$

from time $t$ to time $t+1$. The $x(t+1),y(t+1),\theta(t+1)$ found at the end
of this numerican integration is the next state.

####  D.7.2 Quadcopter

The Quadcopter environment is taken from Sivaramakrishnan et al. (2023). In
this environment, the agent must navigate a quadcopter around a series of
pillars to reach a goal. Episodes are 30 steps long. The reward function is 1
in the goal region, and 0 at all other states. If an agent reaches the goal
before the end of the episode, the episode ends and the agent receives a
reward of 1 for each remaining time step left in the episode. The dynamics of
this environment are given by a Mujoco simulation.

###  D.8 Additional Experiments

In the experiments section, we reported that Volume-MCTS outperformed SST on
reward, but not success rate. Here we provide addition details on this
finding.

(a) Success rate as a function of total environmental interactions

(b) Reward as a function of total environmental interactions

Figure 3: Reward and success rate on Quadcopter environment

Here, success is defined as reaching the goal within the 30-step episode. The
goal state is treated as a s SST and Volume-MCTS reach the goal roughly the
same fraction of the time. However,


# The evolution of the mechanisms of SARS-CoV-2 evolution revealing vaccine-
resistant mutations in Europe and America

Rui Wang1, Jiahui Chen1 and Guo-Wei Wei1,2,3111 Corresponding author. Email:
<EMAIL_ADDRESS>
1 Department of Mathematics,
Michigan State University, MI 48824, USA.
2 Department of Electrical and Computer Engineering,
Michigan State University, MI 48824, USA.
3 Department of Biochemistry and Molecular Biology,
Michigan State University, MI 48824, USA.

###### Abstract

The importance of understanding SARS-CoV-2 evolution cannot be overemphasized.
Recent studies confirm that natural selection is the dominating mechanism of
SARS-CoV-2 evolution, which favors mutations that strengthen viral
infectivity. We demonstrate that vaccine-breakthrough or antibody-resistant
mutations provide a new mechanism of viral evolution. Specifically, vaccine-
resistant mutation Y449S in the spike (S) protein receptor-bonding domain
(RBD), which occurred in co-mutation [Y449S, N501Y], has reduced infectivity
compared to the original SARS-CoV-2 but can disrupt existing antibodies that
neutralize the virus. By tracing the evolutionary trajectories of vaccine-
resistant mutations in over 1.9 million SARS-CoV-2 genomes, we reveal that the
occurrence and frequency of vaccine-resistant mutations correlate strongly
with the vaccination rates in Europe and America. We anticipate that as a
complementary transmission pathway, vaccine-resistant mutations will become a
dominating mechanism of SARS-CoV-2 evolution when most of the world’s
population is vaccinated. Our study sheds light on SARS-CoV-2 evolution and
transmission and enables the design of the next-generation mutation-proof
vaccines and antibody drugs.

Keywords: COVID-19, SARS-CoV-2, evolution, vaccine-resistant mutation,
vaccine-breakthrough, infectivity, Y449S

##  1 Introduction

Started in late 2019, the coronavirus disease 2019 (COVID-19) pandemic caused
by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has had
devastating impacts worldwide, which has plunged the world into an economic
recession. Although several authorized vaccines have offered promise to
control the disease in early 2021, the emergence of multiple variants of SARS-
CoV-2 indicates that the combat with SARS-CoV-2 will be protracted. At this
stage, almost all SARS-CoV-2 vaccines and monoclonal antibodies (mAbs) are
targeted at the spike (S) protein [1], while mutations on the S protein have
been verified to link to the efficacy of existing vaccines and viral
infectivity [2, 3]. Therefore, it is imperative to understand the mechanisms
of viral mutations, especially on the S gene of SARS-CoV-2, which will promote
the development of mutation-proof vaccines and mAbs.

The mechanism of mutagenesis is driven by various competitive processes [4, 5,
6, 7, 8], which can be categorized into 3 different scales with many factors
as illustrated in Figure 1 a: 1) the molecular scale, 2) the organism scale,
and 3) the population scale. From the molecular-scale perspective, the random
shifts, replication errors, transcription errors, translation errors, viral
proofreading, and viral recombination are the main driven sources. Moreover,
the host gene editing induced by the adaptive immune response [8] and the
recombination between the host and virus are the key-driven factors at the
organism level. Furthermore, the natural selection popularized by Charles
Darwin is a critical process, which favors mutations that have reproductive
advantages for the virus to have adaptive traits in evolution. Such
complicated mechanisms of viral mutagenesis make the comprehension of viral
transmission and evolution a grand challenge.

Although there are 28,780 unique single mutations distributed evenly on the
whole SARS-CoV-2 genome, the mutations on the S gene stand out among all 29
genes on SARS-CoV-2 due to the mechanism of viral infection. Under assistant
with host transmembrane protease, serine 2 (TMPRSS2), SARS-CoV-2 enters the
host cell by interacting with its S protein and the host angiotensin-
converting enzyme 2 (ACE2) [9] (See Figure 1 b). Later on, antibodies will be
generated by the host immune system, aiming to eliminate the invading virus
through direct neutralization or non-neutralizing binding [10, 11], which
makes the S protein the main target for the current vaccines. Specifically,
there is a short immunogenic fragment located on the S protein of SARS-CoV-2
that can facilitate the SARS-CoV-2 S protein binding with ACE2, which is
called the receptor-binding domain (RBD) [12]. Studies have shown that the
binding free energy (BFE) between the S RBD and the ACE2 is proportional to
the infectivity [13, 14, 15, 9, 16]. Therefore, tracking and monitoring the
RBD mutations and their corresponding BFE changes will expedite understanding
the infectivity, transmission, and evolution of SARS-CoV-2, especially for the
new SARS-CoV-2 variants, such as Alpha, Beta, Gamma, Delta, and Lambda, etc.
[17]

The current prevailing variants Alpha, Beta, Gamma, Delta, Kappa, Theta,
Lambda, and Mu carry at least one vital mutation at residues 452 and 501 on
the S RBD. Notably, in July 2020, we successfully predicted that residues 452
and 501 ”have high changes to mutate into significantly more infectious
COVID-19 strains” [18]. In the same work, we hypothesized that “natural
selection favors those mutations that enhance the viral transmission” and
provided the first evidence for infectivity-based natural selection. In other
words, we revealed the mechanism of SARS-CoV-2 evolution and transmission
based on very limited genome data in July 2020 [18]. Additionally, we
predicted three categories of RBD mutations: 1) most likely (1149 mutations),
2) likely (1912 mutations), and 3) unlikely (625 mutations) [18]. Up to now,
all of the RBD mutations we detected fall into our first category [3, 19].
Until now, all of the top 100 most observed RBD mutations have BFE change
greater than the average BFE changes of -0.28kcal/mol (the average BFE changes
for all RBD mutations[20]). It is an extremely low odd (i.e.,
$\frac{1}{1.27\times 10^{30}}$) for 100 RBD mutations to accidentally have BFE
changes simultaneously above the average value, which confirms our hypothesis
that the transmission and evolution of new SARS-CoV-2 variants are governed by
infectivity-based natural selection, despite all other competing mechanisms
[18]. Our predictions rely on algebraic topology [21, 22, 23]-assisted deep
learning [24, 18], but have been extensively validated [3, 25]. However,
infectivity is not the only transmission pathway that governs viral evolution.
Vaccine-resistant mutations or more precisely, antibody-resistant mutations,
that can disrupt the protection of antibodies has become a viable mechanism
for new variants to transmit among the vaccinated population since the vaccine
was put on the market. In early January 2021, we have predicted that RBD
mutations W353R, I401N, Y449D, Y449S, P491R, P491L, Q493P, etc., will weaken
most antibody bindings to the S protein [3]. Later on, we have provided a list
of most likely vaccine escape RBD mutations with high frequency, including
S494P, Q493L, K417N, F490S, F486L, R403K, E484K, L452R, K417T, F490L, E484Q,
and A475S [19]. Moreover, we have pointed out that Y449S and Y449H are two
vaccine-resistant mutations, and “Y449S, S494P, K417N, F490S, L452R, E484K,
K417T, E484Q, L452Q, and N501Y” are the top 10 mutations that will disrupt
most antibodies with high-frequency [20]. As mentioned in Ref. [26], RBD
mutations such as E484K/A, Y489H, Q493K, and N501Y found in late-stage evolved
S variants “confer resistance to a common class of SARS-CoV-2 neutralizing
antibodies”, which suggests the viral evolution is also regulated by vaccine-
resistant mutations.

The objective of this work is to analyze the evolution of the mechanisms of
SARS-CoV-2 evolution, driven by complementary viral transmission pathways. We
demonstrate how the interplay among molecular-scale, organism-scale,
population-scale mechanisms of SARS-CoV-2 mutations have affected the
evolution of SARS-CoV-2. As a primary driven source of mutagenesis, the
molecular-based mechanisms such as random shifts, transcription errors,
proofreading, etc., changing the genetic information initially. Next, gene
editing takes charge of the organism-based mechanism, suggesting the host
immune response to the virus [8]. Then, the population-level mechanism governs
the transmission pathways of viral evolution. Two complementary pathways
(infectivity and vaccine-resistance) regulated by natural selection become the
preponderance of evolution-driven force. The RBD mutations regulated by
infectivity-based pathways exist in the prevailing variants, while the
mutations regulated by the vaccine-resistant pathway start to emerge in
countries with relatively high vaccination rates. In this work, 1,983,328
complete SARS-CoV-2 genomes that isolate from patients are decoded by single
nucleotide polymorphism (SNP) calling, from where a total of 28,780 unique
single mutations are detected. Among them, 737 RBD mutations are discovered up
to September 20, 2021 (The detailed information can be found in the Supporting
Information S5). Based on our comprehensive topology-based artificial
intelligence (AI) model to predict RBD mutation-induced BFE changes of RBD and
ACE2/antibody complexes [18, 3], the transmission trajectory of vaccine-
resistant RBD mutations will be analyzed (The detailed information about
methods and model can be viewed in the Supporting Information S1 and S2).
Moreover, vaccine-resistant RBD mutation Y449S that has been found in more
than 1000 isolates will be discussed. Furthermore, the vaccination rates of 12
countries where Y449S is distributed are also analyzed, which provides a
reliable explanation of the relation between the emergence of vaccine-
resistant mutations and the vaccination rate. Such understanding of two
complementary transmission pathways will shed light on the long-term efficacy
of S-targeted antibodies countermeasures and benefit the development of next-
generation mutation-proof vaccines and mAbs.

Figure 1: a The mechanism of mutagenesis. Nine mechanisms are grouped into
three scales: 1) molecular-based mechanism (green color); 2) organism-based
mechanism (red color); 3) population-based mechanism (blue color). The random
shifts (Random), replication error (Rep), Transcription error (Transcr), viral
proofreading (Proof), and recombination (Recomb) are the six molecular-based
mechanisms. The gene editing and the host-virus recombination are the
organism-based mechanism. In addition, the natural selection (Natural) is the
population-based mechanism, which is the mainly driven source in the
transmission of SARS-CoV-2. b A sketch of SARS-CoV-2 and its interaction with
host cell. c Illustration of 25 single-site RBD mutations with top
frequencies. The height of each bar shows the BFE change of each mutation, the
color of each bar represents the natural log of frequency of each mutation,
and the number at the top of each bar means the AI-predicted number of
antibody and RBD complexes that may be significantly disrupted by a single
site mutation. d Illustration of SARS-CoV-2 S protein with human ACE2. The
blue chain represents the human ACE2, the pink chain represents the S protein,
and the purple fragment on the S protein points out the two vaccine-resistant
mutations Y449S/H.

##  2 Results

###  2.1 Evolutionary trajectories of viral RBD single mutations

Studying the mechanisms of SARS-CoV-2 mutagenesis is beneficial to the
understanding of viral transmission and evolution. The mainly driven force of
viral evolution is regulated by natural selection, which is employed by two
complementary transmission pathways: 1) infectivity-based pathway and 2)
vaccine-resistant pathway. We have discussed the infectivity-based pathways in
Ref.[20] and [27]. This section focuses on the vaccine-resistant pathway and
its impact on the transmission and evolution of SARS-CoV-2. To understand the
mechanisms of vaccine-resistant mutations, we first analyze 1,983,328 complete
SARS-CoV-2 genomes, and a total of 28,780 unique single mutations are decoded.
Among them, there are 737 non-degenerate RBD mutations. The infectivity of
SARS-CoV-2 is proportional to the BFE between the S RBD and ACE2 [13, 14, 15,
9, 16]. Therefore, the BFE change induced by a specific RBD mutation reveals
whether the RBD mutation is an infectivity-strengthen or an infectivity-weaken
mutation. Similarly, the BFE change between S RBD and antibody induced by a
given mutation reveals whether this mutation will strengthen the binding
between S and antibody or not. Up to now, we have collected 130 antibody
structures (see the Supporting Information S4), which includes Food and Drug
Administration (FDA)-approved mAbs from Eli Lilly and Regeneron. For a
specific RBD mutation, its antibody disruption count shows the number of
antibodies that have antibody-S BFE changes smaller than -0.3 kcal/mol. The
ACE2-S and antibody-S BFE changes induced by RBD mutations are predicted from
our TopNetTree model [18], which is available at TopNetmAb. All of the
predicted BFE changes induced by RBD mutations can be found at Mutation
Analyzer. Figure 1 c illustrates the top 25 most observed RBD mutations. The
height and color of each bar represent the ACE2-S BFE changes and frequency of
each RBD mutation. The number at the top of each bar shows the antibody
disruption count of each mutation. The detailed information can be viewed in
Supplementary Information S4. It can be seen that 23 mutations have positive
ACE2-S BFE changes, suggesting they are regulated by the infectivity-based
transmission pathway. Howbeit, 2 RBD mutations D427N and Y449S, have negative
BFE changes. Notably, mutation Y449S has a significantly negative BFE change
(-0.8112 kcal/mol) and a pretty large antibody disruption count (89),
revealing a non-typical mechanism of mutagenesis. Such a mutation with
significantly negative ACE2-S BFE change together with a high antibody
disruption count is called a vaccine-resistant or antibody-resistant mutation.
Figure 1 d is the illustration of SARS-CoV-2 S protein (blue color) with human
ACE2 (pink color), and the Y449 residue (purple color) is located on the
random coil of the S protein. Among all of the vaccine-resistant mutations,
Y449S has the highest frequency (1189). In addition, at residue 449, mutations
Y449H, Y449N, Y449D are all vaccine-resistant mutations that have been
observed in more than 20 SARS-CoV-2 genome isolates.

To track the evolution trajectory of vaccine-resistant mutations, the BFE
changes, log2 enrichment ratios 222Log2 enrichment ratio is collected from the
experimental deep mutation enrichment data in Ref. [28], and log10 frequencies
of RBD mutations are analyzed from April 30, 2020, to August 23, 2021, in
every 60 days, as illustrated in Figure 2. Here, the top 100 most observed RBD
mutations are displayed. In Figure 2 a, red stars mark the vaccine-resistant
mutations that have negative BFE changes. Although a few vaccine-resistant
mutations S438F, I434K, Y505C, and Q506K were detected before November 2020,
they had relatively low frequencies. However, since December 2020, such
vaccine-resistant mutations were no longer in the top 100 most observed RBD
mutation list, suggesting that in this period, the evolution of SARS-CoV-2 is
mainly regulated by natural selection through the infectivity-based
transmission pathway. Notably, in May 2021, two vaccine-resistant mutations
Y449S and Y449H, came back to the top 100 most observed RBD mutation list. In
addition, Y449S has a relatively high frequency. Such finding indicates that
natural selection not only favors those mutations that enhance the
transmission but also those mutations that can disrupt plenty of antibodies
since SARS-CoV-2 vaccines started to provide protection among populations in
early May. Similarly, patterns can be found in Figure 2 b, suggesting our AI-
predicted BFE changes are highly consistent with the deep mutational
enrichment ratio from experiments [28].

Figure 2: Most significant RBD mutations. a Time evolution of RBD mutations
with its mutation-induced BFE changes per 60-day from April 30, 2020, to
August 31, 2021. Here, only the top 100 most observed RBD mutations are
displayed. The height and color of each bar represent the log frequency and
ACE-S BFE change induced by a given RBD mutation. The red star marks the
vaccine-resistant mutations with significantly negative BFE changes. b Time
evolution of RBD mutations with its experimental mutation-induced log2
enrichment ratio changes per 60-day from April 30, 2020, to August 31, 2021.
The height and color of each bar represent the log frequency and enrichment
ratio change induced by a given RBD mutation. The red star marks vaccine-
resistant mutations with significantly negative BFE changes.

###  2.2 Evolutionary trajectories of viral RBD co-mutations

The vaccine-resistant mutations are usually found along with other RBD
mutations. Therefore, analyzing the time evolution of RBD co-mutations offers
a better understanding of the mechanisms of vaccine-resistant mutations.
Figures 3 a, b, and c illustrate the time evolution of 2, 3, and 4 RBD co-
mutations with their corresponding BFE changes every 30 days. Here, the height
and color of each bar represent the log10 frequency and total BFE change
induced by a given RBD co-mutation. Considering the number of co-mutations is
quite low in the year 2020, the time range of analysis is set to [01/25/2021,
08/23/2021] for the time evolution analysis of 2 co-mutations. For 3 and 4 co-
mutations, their time ranges are set to [02/04/2021, 08/23/2021] and
[04/25/2021, 08/23/2021], respectively. In Figure 3 a, red star marks the 2
co-mutations with significantly negative BFE changes. At the end of March
2021, vaccine-resistant mutation Y449D showed up with mutation N501Y in some
genome isolates, resulting in a negative BFE change (-0.473kcal/mol) and a
high antibody disruption count (98) for RBD 2 co-mutation [Y449D, N501Y].
However, its global frequency is relatively low. Since late April 2021,
vaccine-resistant mutation Y449S showed up with N501Y, making RBD co-mutation
[Y449S, N501Y] one of the most prevailing vaccine-resistant co-mutations.
Figure 3 d shows the top 25 most observed RBD co-mutations, the length and
color of each bar represent the total BFE change and the natural log of
frequency of an RBD co-mutation. The number at the side of each bar is the
count of antibody disruption. Among the 25 most observed RBD co-mutations,
[Y449S, N501Y] is the only co-mutation with a significantly negative BFE
change and extremely high antibody disruption count (94). Observing the
evolution trajectory of [Y449S, N501Y] shows that the infectivity transmission
pathway regulated by natural selection in the population level is the major
evolution-driven force of SARS-CoV-2 mutagenesis before March 2021. Starting
in January 2021, several vaccines were authorized for emergent use. Two months
later, since many people have been protected by the vaccines, the mutations
that disrupt the binding between the S and antibodies are able to transmit
among vaccinated people, especially in countries with high vaccination rates.
Such a vaccine-resistant pathway reduces the efficacy of vaccines and antibody
therapies, indicating the combat with COVID-19 will be a prolonged battle.

Similar time evolution trajectories are drown for RBD 3 and 4 co-mutations
(see Figures 3 b and c). Apparently, there are no vaccine-resistant 3 and 4
co-mutations at present, which indicates vaccine-resistant co-mutation [Y449S,
N501Y] is quite unique.

Figure 3:  RBD co-mutation analysis. a Time evolutionary trajectory of RBD 2
co-mutations with its mutation-induced BFE changes per 30-day from January 25,
2021, to August 23, 2021. The height and color of each bar represent the log
frequency and ACE-S BFE change induced by a given RBD mutation. Red stars mark
the 2 co-mutations with significantly negative BFE changes. b Time
evolutionary trajectory of RBD 3 co-mutations with its mutation-induced BFE
changes per 30-day from February 24, 2021, to August 23, 2021. The height and
color of each bar represent the log frequency and ACE-S BFE change induced by
a given RBD mutation. c Time evolutionary trajectory of RBD 4 co-mutations
with its mutation-induced BFE changes per 30-day from April 25, 2021, to
August 23, 2021. The height and color of each bar represent the log frequency
and ACE-S BFE change induced by a given RBD mutation. d Illustration of top 25
most observed RBD co-mutations. Here, the length of each bar represents the
total ACE2-S BFE changes induced by a specific RBD co-mutation, the color of
each bar represents the natural log frequency of each co-mutation, and the
number at the side of each bar is the AI-predicted antibody disruption count.

###  2.3 Vaccine-resistant mutations and vaccination rates in 12 countries

Analysis of the vaccination trends and vaccine-resistant mutations leads to a
fundamental understanding of the transmission and evolution of vaccine-
resistant mutations. We investigate the distribution and time evolution of
vaccine-resistant RBD mutation Y449 in 12 countries. As the most observed
vaccine-resistant RBD mutation, Y449S has been detected in 12 countries,
including Denmark (DK), the United Kingdom (UK), France (FR), Bulgaria (BG),
the United States (US), Brazil (BR), Sweden(SE), Canada (CA), Germany (DE),
Switzerland (CH), Spain (ES), and Belgium (BE), as illustrated in Figure 4 a.
Here, 12 countries that Y449S was found in are in blue. The darker the blue
is, the higher frequency of Y449S will be. The number on the side of each
country is the total positive cases up to August 31, 2021. Although DK has the
smallest positive cases among 12 countries, the frequency of Y449S is the
highest. More than 800 patients carry vaccine-resistant mutation Y449S in DK.
All of the Y449S-related cases are found in Europe and America, where the
vaccination rates in those areas are relatively high. Figure 4 b shows the
time evolution of vaccination ratio and the frequency of Y449S in the 12
countries as mentioned above in 30-day periods. The $x$-axis records the date,
which ranges from 12/26/2020 to 08/23/2021. The left-hand side $y$-axis shows
the frequency of Y499S (red lines), and the right-hand side $y$-axis shows the
vaccination ratio. In addition, the orange region shows at least one dose
ratio, while the purple region means the fully vaccinated ratio. It can be
seen that Y449S was first found in BG and the US in December 2020. However,
the frequency of Y449S in BG and the US is quite low before April 2021. After
April 2021, Y449S has been quickly spread out to other ten countries. Among
them, the total number of cases related to Y449S has a rapid increment
tendency, especially in DK, the UK, and FR. Notably, all these three countries
have relatively high vaccination ratios (over 70% up to late August 2021). It
is worthy to mention that the frequency of Y449S is low in DE, CH, ES, and BE,
etc., which is mainly due to the first Y449-related case in these countries
was detected after June 2021. Since then, Delta variants dominated in the
prevailing variants, which gave Y449S a limited chance to spread out rapidly.
Moreover, from Figure 4 , it can be seen that the frequency of Y449S has a
similar growing tendency as the fully vaccinated ratio, suggesting that the
vaccine-resistant mutations will gradually become one of the main evolution
driven forces of SARS-CoV-2, especially in those areas with high vaccination
rates.

Figure 4:  a Distribution of vaccine-resistant mutation Y449S. The color bar
represents the log10 frequency of Y449S in 12 countries: Denmark (DK), the
United Kingdom (UK), France (FR), Bulgaria (BG), the United States (US),
Brazil (BR), Sweden(SE), Canada (CA), Germany (DE), Switzerland (CH), Spain
(ES), and Belgium (BE). The number located at the side of the country shows
the total positive SARS-CoV-2 cases up to August 31. b Time evolution of
vaccination rate and the frequency of Y449S in 12 countries from December 26,
2020, to August 23, 2021. The data is collected per 30-day. The red line shows
the frequency of mutation Y449S. The orange and purple areas represent at
least one dose rate and fully vaccinated rate in each country.

##  3 Conclusion

Due to the appearance of multiple mutations known to reduce the efficacy of
antibody neutralization generated by vaccines, it is vital to better
comprehend the mechanisms of SARS-CoV-2 mutagenesis, which will be of
paramount importance to understanding the transmission and evolution of SARS-
CoV-2. The driven forces of mutagenesis can be categorized into three groups:
1) molecular-scale mechanism, 2) organism-scale mechanism, and 3) population-
level mechanism. As an initial driven source of mutagenesis, the genetic
information is changed by random shifts, viral proofreading, translation
errors, etc., which all belong to molecular-scale mechanisms. Also, regulated
by the host immune system, host gene editing, and rarely occurring host-viral
recombination are two organism-scale mechanisms. The molecular- and organism-
scale mechanisms provide a large number of candidate mutations in the SARS-
CoV-2 genome, while it is the population-scale mechanism that determines what
mutations become dominating.

Natural selection is a population-scale mechanism, which promotes the surge of
the emerging SARS-CoV-2 variants by two complementary pathways: infectivity
and vaccine resistance. The early stage of SARS-CoV-2 evolution was entirely
dominated by infectivity-strengthening mutations. However, since late March
2021, once vaccines had provided protection to highly vaccinated populations,
several vaccine-resistant mutations such as Y449S and Y449H have been observed
relatively frequently. Considering there is still a good portion of the
population who are not vaccinated, infectivity-strengthen mutations still
dominate in the prevailing and future variants. However, antibody-resistant
mutations will become a major mechanism of transmission once most of the
populations carrying antibodies either through vaccination and infection. Our
studies are valuable to the development of the next-generation vaccines and
mAbs, which are of great importance in the long-term combat with SARS-CoV-2.

## Data and model availability

The SARS-CoV-2 SNP data in the world is available at Mutation Tracker. The
most observed SARS-CoV-2 RBD mutations are available at Mutaton Analyzer. The
TopNetTree model is available at TopNetmAb. The detailed methods can be found
in the Supporting Information  S1 and  S2. The validation of our predictions
with experimental data can be located in Supporting Information  S3. The
information of 130 antibodies with their corresponding PDB IDs, the SARS-CoV-2
S protein RBD SNP and non-degenerate co-mutations data can be found in Section
S5 of the Supporting Information.

## Supporting information

The supporting information is available for

  1. S1

Supplementary data pre-processing and feature generation methods

  2. S2

Supplementary machine learning methods

  3. S3

Supplementary validation: validations of our machine learning predictions with
experimental data.

  4. S4

Supplementary table: the top 25 most observed S protein RBD mutations up to
September 20, 2021.

  5. S5

Supplementary data: The Supplementary_Data.zip contains four files:  S5.0.1:
antibodies_disruptmutation.csv shows the name of antibodies disrupted by
mutations.  S5.0.2: antibodies.csv lists the PDB IDs for all of the 130 SARS-
CoV-2 antibodies.  S5.0.3: RBD_comutation_residue_09202021.csv lists all of
the SNPs of RBD co-mutations up to September 20, 2021.  S5.0.4:
Track_Comutation_09202021.xlsx preserves all of the non-degenerate RBD co-
mutations with their frequencies, antibody disruption counts, total BFE
changes, and the first detection dates and countries.

## Acknowledgment

This work was supported in part by NIH grant GM126189, NSF grants DMS-2052983,
DMS-1761320, and IIS-1900473, NASA grant 80NSSC21M0023, Michigan Economic
Development Corporation, MSU Foundation, Bristol-Myers Squibb 65109, and
Pfizer.

## References

  * [1] Jonaid Ahmad Malik, Almas Hanif Mulla, Tahmeena Farooqi, Faheem Hyder Pottoo, Sirajudheen Anwar, and Kannan RR Rengasamy.  Targets and strategies for vaccine development against sars-cov-2.  Biomedicine & Pharmacotherapy, page 111254, 2021.
  * [2] Medini K Annavajhala, Hiroshi Mohri, Jason E Zucker, Zizhang Sheng, Pengfei Wang, Angela Gomez-Simmonds, David D Ho, and Anne-Catrin Uhlemann.  A novel SARS-CoV-2 variant of concern, B. 1.526, identified in New York.  medRxiv, 2021.
  * [3] Jiahui Chen, Kaifu Gao, Rui Wang, and Guo-Wei Wei.  Prediction and mitigation of mutation threats to COVID-19 vaccines and antibody therapies.  Chemical Science, 12(20):6929–6948, 2021.
  * [4] Rafael Sanjuán and Pilar Domingo-Calap.  Mechanisms of viral mutation.  Cellular and Molecular Life Sciences, 73(23):4433–4448, 2016.
  * [5] Nathan D Grubaugh, William P Hanage, and Angela L Rasmussen.  Making sense of mutation: what D614G means for the COVID-19 pandemic remains unclear.  Cell, 182(4):794–795, 2020.
  * [6] Tugba G Kucukkal, Marharyta Petukh, Lin Li, and Emil Alexov.  Structural and physico-chemical effects of disease and non-disease nsSNPs on proteins.  Current Opinion in Structural Biology, 32:18–24, 2015.
  * [7] Peng Yue, Zhaolong Li, and John Moult.  Loss of protein structure stability as a major causative factor in monogenic disease.  Journal of molecular biology, 353(2):459–473, 2005.
  * [8] Rui Wang, Yuta Hozumi, Yong-Hui Zheng, Changchuan Yin, and Guo-Wei Wei.  Host immune response driving SARS-CoV-2 evolution.  Viruses, 12(10):1095, 2020.
  * [9] Markus Hoffmann, Hannah Kleine-Weber, Simon Schroeder, Nadine Krüger, Tanja Herrler, Sandra Erichsen, Tobias S Schiergens, Georg Herrler, Nai-Huei Wu, Andreas Nitsche, et al.  SARS-CoV-2 cell entry depends on ACE2 and TMPRSS2 and is blocked by a clinically proven protease inhibitor.  Cell, 181(2):271–280, 2020.
  * [10] Jiahui Chen, Kaifu Gao, Rui Wang, Duc Duy Nguyen, and Guo-Wei Wei.  Review of COVID-19 antibody therapies.  Annual Review of Biophysics, 50:1–30, 2020.
  * [11] Peter Chen, Ajay Nirula, Barry Heller, Robert L Gottlieb, Joseph Boscia, Jason Morris, Gregory Huhn, Jose Cardona, Bharat Mocherla, Valentina Stosor, et al.  SARS-CoV-2 neutralizing antibody LY-CoV555 in outpatients with COVID-19.  New England Journal of Medicine, 384(3):229–237, 2021.
  * [12] Wanbo Tai, Lei He, Xiujuan Zhang, Jing Pu, Denis Voronin, Shibo Jiang, Yusen Zhou, and Lanying Du.  Characterization of the receptor-binding domain (RBD) of 2019 novel coronavirus: implication for development of RBD protein as a viral attachment inhibitor and vaccine.  Cellular & molecular immunology, 17(6):613–620, 2020.
  * [13] Wendong Li, Zhengli Shi, Meng Yu, Wuze Ren, Craig Smith, Jonathan H Epstein, Hanzhong Wang, Gary Crameri, Zhihong Hu, Huajun Zhang, et al.  Bats are natural reservoirs of SARS-like coronaviruses.  Science, 310(5748):676–679, 2005.
  * [14] Xiu-Xia Qu, Pei Hao, Xi-Jun Song, Si-Ming Jiang, Yan-Xia Liu, Pei-Gang Wang, Xi Rao, Huai-Dong Song, Sheng-Yue Wang, Yu Zuo, et al.  Identification of two critical amino acid residues of the severe acute respiratory syndrome coronavirus spike protein for its variation in zoonotic tropism transition via a double substitution strategy.  Journal of Biological Chemistry, 280(33):29588–29595, 2005.
  * [15] Huai-Dong Song, Chang-Chun Tu, Guo-Wei Zhang, Sheng-Yue Wang, Kui Zheng, Lian-Cheng Lei, Qiu-Xia Chen, Yu-Wei Gao, Hui-Qiong Zhou, Hua Xiang, et al.  Cross-host evolution of severe acute respiratory syndrome coronavirus in palm civet and human.  Proceedings of the National Academy of Sciences, 102(7):2430–2435, 2005.
  * [16] Alexandra C Walls, Young-Jun Park, M Alejandra Tortorici, Abigail Wall, Andrew T McGuire, and David Veesler.  Structure, function, and antigenicity of the SARS-CoV-2 spike glycoprotein.  Cell, 2020.
  * [17] Changchuan Yin.  Genotyping coronavirus SARS-CoV-2: methods and implications.  Genomics, 112(5):3588–3596, 2020.
  * [18] Jiahui Chen, Rui Wang, Menglun Wang, and Guo-Wei Wei.  Mutations strengthened SARS-CoV-2 infectivity.  Journal of molecular biology, 432(19):5212–5226, 2020.
  * [19] Rui Wang, Jiahui Chen, Kaifu Gao, and Guo-Wei Wei.  Vaccine-escape and fast-growing mutations in the United Kingdom, the United States, Singapore, Spain, India, and other COVID-19-devastated countries.  Genomics, 113(4):2158–2170, 2021.
  * [20] Rui Wang, Jiahui Chen, Yuta Hozumi, Changchuan Yin, and Guo-Wei Wei.  Emerging vaccine-breakthrough SARS-CoV-2 variants.  arXiv preprint arXiv:2103.08023, 2021.
  * [21] Gunnar Carlsson.  Topology and data.  Bulletin of the American Mathematical Society, 46(2):255–308, 2009\.
  * [22] Herbert Edelsbrunner, David Letscher, and Afra Zomorodian.  Topological persistence and simplification.  In Proceedings 41st annual symposium on foundations of computer science, pages 454–463. IEEE, 2000.
  * [23] Kelin Xia and Guo-Wei Wei.  Persistent homology analysis of protein structure, flexibility, and folding.  International journal for numerical methods in biomedical engineering, 30(8):814–844, 2014.
  * [24] Menglun Wang, Zixuan Cang, and Guo-Wei Wei.  A topology-based network tree for the prediction of protein–protein binding affinity changes following mutation.  Nature Machine Intelligence, 2(2):116–123, 2020.
  * [25] Jiahui Chen, Kaifu Gao, Rui Wang, and Guo-Wei Wei.  Revealing the threat of emerging SARS-CoV-2 mutations to antibody therapies.  Journal of Molecular Biology, 433(7744), 2021.
  * [26] Sarah A Clark, Lars E Clark, Junhua Pan, Adrian Coscia, Lindsay GA McKay, Sundaresh Shankar, Rebecca I Johnson, Vesna Brusic, Manish C Choudhary, James Regan, et al.  SARS-CoV-2 evolution in an immunocompromised host reveals shared neutralization escape mechanisms.  Cell, 184(10):2605–2617, 2021.
  * [27] Jiahui Chen, Rui Wang, and Guo-Wei Wei.  Review of the mechanisms of SARS-CoV-2 evolution and transmission.  arXiv preprint arXiv:2109.08148, 2021.
  * [28] Thomas W Linsky, Renan Vergara, Nuria Codina, Jorgen W Nelson, Matthew J Walker, Wen Su, Christopher O Barnes, Tien-Ying Hsiang, Katharina Esser-Nobis, Kevin Yu, et al.  De novo design of potent and resilient hACE2 decoys to neutralize SARS-CoV-2.  Science, 370(6521):1208–1214, 2020.